Ph.D. thesis committee, Mr Eric Maire

Mr Dirk Mohr

Mr Pierre-Olivier Bouchard

Mme Véronique Aubin

Mr Ante

Bakić Special

Mr Eric Maire

Mr Dirk Mohr

Keywords: mortem observations

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Acknowledgement "Real generosity toward the future lies in giving all to the present." Albert Camus

As selected quote outlines, giving your best at the present moment is the best stake for the future. Driven by this idea my path brought me to Paris on Ph.D. thesis in a pretty sudden manner. Once out of the comfort zone, beside new tehnical cognitions associated with the work, life offeres a plenty of room for introducing new people, cultures, nationalities, and consequently improved self-knowledge. This would be my PhD experience sublimed into a single line. Now, I would like to thank all the people who contributed in some way to the work in this thesis.

First of all, I would like to thank my supervisors, Mr. François Hild and Mr. Thilo F. Morgeneyer for their constant support during the thesis. You introduced me to the academic world, gave me necessary advices and suggestions but always leaving plenty of space for critical thinking and decisions. From the very first days of the thesis, back in ESRF Grenoble at experimental sessions with Thilo, although very fresh in the area, I was always asked for opinion about different topics and actions. At that point, this was something suprisingly for me, but this modus operandi combined with your careful and repeated explanations resulted in my boosted self-confidence and motivation for the work. Very soon I started to feel you as my true friends and realised in full sense meaning of François' words on our first meeting: "My Ph.D students are on the first place for me"

(EF), qui représenteront la microstructure du matériau étudié, seront conduites avec des conditions aux limites de Dirichlet extraites des mesures CIV. Les résidus de niveaux de gris pour les mesures par CIV et pour les simulations EF seront évalués à des fins de validation.

Le chapitre 8 compare la qualité des résultats obtenus avec différentes simulations numériques à l'échelle microscopique. Les avantages potentiels de l'utilisation des conditions aux limites mesurées par CIV seront à confirmer par rapport à celles provenant de calculs EF macroscopiques (faiblement ou fortement couplés).

Le chapitre 9 introduit la CIV intégrée afin d'identifier les propriétés d'écoulement plastique de la matrice ferritique à l'échelle microscopique. A la connaissance de l'auteur, c'est la première fois qu'une telle analyse inverse est effectuée à l'échelle microscopique en tenant compte des caractéristiques de la microstructure dans le domaine de l'endommagement ductile.

Les chapitres 7, 8 et 9 sont des résultats d'un travail commun avec des collègues du CEMEF. Cependant, une présentation distincte n'a pas été choisie pour des raisons de logique.

Le chapitre 10 contient des conclusions générales extraites de l'analyse effectuée sur les deux matériaux avec une attention particulière dédiée aux perspectives et aux travaux futurs. Les résultats préliminaires d'une première expérience in situ en nanolaminographie avec un alliage AA2139 T3 seront également discutés dans cette partie.

L'annexe A est une publication de recherche combinant laminographie et calculs de CIV pour une éprouvette CT en alliage AA2139 T3. Grâce à des évaluations simultanées de déformations 3D et de l'endommagement à une distance de 1 mm du fond de l'entaille, un ensemble croisé de bandes de déformation est actif dès le début du chargement dans une région où la fissure sera inclinée. Les bandes ont une activité temporelle intermittente mais sont stables en espace.

Conclusions générales

L'objectif principal de ce travail était l'observation et la quantification des mécanismes d'endommagement ductile à l'échelle microscopique et leur interaction avec les champs de déformation. La compréhension de l'influence des changements de microstructure sur la réponse mécanique macroscopique est d'un grand intérêt pour la communauté de l'encommagement ductile. Comme détaillé dans l'introduction, les efforts récents de modélisation multiéchelle basés sur le travail de Gurson (1977) prédisent la réponse macroscopique à partir de solutions analytiques ou numériques. Bien que prometteurs, les résultats obtenus à l'aide de ces méthodes sont principalement limités aux études de microstructures idéalisées et aux conditions de chargement homogènes. Leur identification (sans parler de leur validation) est encore une tâche difficile. Par conséquent, les méthodes numériques qui utiliseraient le même cadre multiéchelle avec des microstructures réalistes et des conditions aux limites mesurées sont souhaitables. Dans ce contexte, le travail présenté apporte une nouvelle série de données expérimentales concernant les mécan-Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Résumé étendu en français ismes microscopiques conduisant à une rupture ductile pour deux classes de matériaux, à savoir des alliages d'aluminium structurels et la fonte à graphite sphéroïdal. L'aspect particulièrement important de ce travail est l'utilisation de la laminographie, ce qui facilite l'utilisation de plaques minces avec différents états de contrainte dans les régions scannées. Dans tous les cas étudiés, les champs cinématiques mesurés par CIV ont été obtenus.

Pour les alliages d'aluminium, on observe la microstructure et les interactions entre endommagement et déformations à une résolution micrométrique pour différentes géométries d'échantillons et niveaux associés de triaxialité des contraintes. Pour la fonte à graphite sphéroïdal, un cadre combiné numérique-expérimental est introduit pour valider les simulations à l'échelle microscopique. Les simulations EF, qui représentent la microstructure du matériau étudié, sont conduites avec des conditions aux limites de Dirichlet extraites des mesures par CIV. Les résidus de niveaux de gris pour les mesures par CIV et les simulations EF sont évalués à des fins de validation. Ce cadre a également été utilisé pour identifier les paramètres de plasticité de la matrice ferritique.

Alliages d'aluminium

Les zones plastiques proches du fond de l'entaille des éprouvettes CT ont été étudiées en combinant la laminographie in situ et la CIV. Les études de l'activité de la déformation plastique et de la microstructure ont conduit aux résultats suivants :

Champs cinématiques

• plusieurs bandes obliques traversantes forment des motifs de déformation à une distance supérieure à 250µm du fond de l'entaille. Elles sont actives dès les premiers stades de chargement. La formation de de déformation précède la croissance de l'endommagement et la phase d'adoucissement.

• Dans les alliages AA2198 T8, AA2139 T3 et AA2139 T8, les bandes de déformation ont une activité alternée dans des régions de fissuration inclinée (à 1000µm du fond de l'entaille) pendant les étapes de chargement incrémental. l'alliage AA2198 T3 montre une activité de déformation plus stable durant le chargement.

• La localisation spatiale des bandes par rapport à la microstructure est stable, c'està-dire que l'activité plastique ne gomme pas les hétérogénéités des champs de déformation.

• L'espacement typique des bandes est de 150 à 250µm.

• Le rapport de la déformation équivalente à l'intérieur et à l'extérieur des bandes est d'environ 1,5 à 2, c'est-à-dire que les zones à l'extérieur des bandes ne sont pas en élasticité, mais se déforment avec des niveaux plastiques inférieurs. C'est la raison pour laquelle on a évité le terme de localisation des déformations en utilisant la désignation de bandes de déformation.

• Les déformations normales au sens de propagation des fissures restent proches de zéro. Il a été démontré que l'apparition de bandes déformées coïncide spatialement avec l'état de déformation plane pour l'alliage AA2198 T8. Ces résultats ont été confirmés pour l'alliage AA2198 T3.

• La comparaison des résultats pour les nuances AA2198 T3 et T8 révèle que le durcissement (induit par le traitement T3) ne modifie pas significativement le comportement observé pour l'alliage AA2198 T8 (développement précoce de bandes de déformation) bien que le durcissement plastique entrave la localisation.

Processus d'encommagement

• Dans la région de rupture inclinée, les bandes de déformation apparaissent très tôt, avant tout signe d'apparition de l'endommagement. Une germination et croissance nulles ont été quantifiées pour ces échantillons.

• En outre, même au voisinage immédiat du fond de l'entaille, des endommagements importants ne se produisent pas immédiatement après l'apparition de bandes de déformation. Le chemin de rupture final suit le lieu de certaines des bandes de déformation mesurées. On peut donc affirmer que les premières bandes de déformation précèdent le chemin de fissure loin du fond de l'entaille et également en se rapprochant de celle-ci pour les alliages d'aluminium étudiés. L'endommagement se développe de manière très tardive.

• La croissance de l'endommagement plus élevée au voisinage immédiat du fond de l'entaille est une conséquence d'une plus grande triaxialité des contraintes. Deux micromécanismes d'endommagement distincts sont identifiés dans le cas de l'alliage AA2198 T3, à savoir i) un endommagement de cisaillement et ii) des dommages plus classiques dans les zones à forte triaxialité des contraintes. Même dans la zone de rupture plate, le micromécanisme de cisaillement prévaut, ce qui provoque un motif de fissure finale en zigzag très proche du fond de l'entaille. En général, les éprouvettes rompent sous un mode de transition en "toit d'usine" puis incliné (c.-à-d. qu'il n'y a pas de région réellement plane), qui est déterminée par le motif de bandes de déformation détecté au début de l'histoire de chargement. Les analyses EBSD et MEB ont confirmé les résultats obtenus pour l'alliage AA2198 T3 et sont compatibles avec les hypothèses faites.

Fonte

Quatre éprouvettes plates en fonte à graphite sphéroïdal, à savoir, des échantillons à 2 trous (configurations 90 • et 45 • ), un trou central et un échantillon de traction présentés dans l'introduction ont été étudiés. L'endommagement est visualisé par laminographie et tomographie 3D in situ. La CIV est utilisée pour les mesures cinématiques.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Résumé étendu en français

Les principaux résultats concernent l'étude des mécanismes d'endommagement ductile (champs cinématiques et processus d'endommagement, sur la base des échantillons à 2 trous) et un cadre numérique / expérimental (échantillon avec trou central, échantillon à 2 trous, échantillon de traction):

Champs cinématiques

• Dès les toutes premières étapes de chargement, des zones déformées intensément sont observées entre certains nodules de graphite.

• On conclut que la présence de bandes de déformation s'accompagne toujours d'une forte densité de nodules, alors que tous les nodules ne sont pas affectés par le champ de déformation hétérogène.

• Les étapes de développement de ce type de bandes entre certains des nodules diffèrent de celles rapportées pour les alliages d'aluminium.

• La fonte à graphite sphéroïdal représente une autre classe de matériaux par rapport aux alliages d'aluminium et la déformation entre les nodules est principalement un effet structurel. Pourtant, il est confirmé ici que l'état de déformation plane n'est pas une condition nécessaire pour une déformation très précoce et stable.

• Dans un cadre multiéchelle (échantillons à 2 trous) lors du passage de la configuration à 90 • (c.-à-d., striction autour des cavités) à 45 • (c.-à-d. colaescence en cisaillement), le rôle des nodules de graphite en tant qu'initiateurs de localisation devient moins influent, c'est-à-dire que le processus de ruine est régi par la position du trou primaire (usiné). Dans la configuration à 90 • , une zone plastique plus diffuse entre les trous usinés favorise l'influence des vides secondaires (nodules) sur le chemin de fissure. La ruine finale a suivi les premiers chemins entre les nodules qui, parfois, ne représentaient pas la distance la plus courte entre les vides primaires (trous).

• À l'échelle mésoscopique (c.-à-d. la majeure partie du ligament entre les trous usinés), la déformation équivalente cumulée moyenne présente un niveau supérieur pour la configuration à 45 • . La trace moyenne du tenseur de déformation, qui caractérise le changement de volume, ne diffère pas significativement pour les deux cas. En d'autres termes, la plasticité cumulée globale est plus élevée dans la configuration à 45 • principalement en raison de la composante de cisaillement.

• À l'échelle microscopique (zone internodule), les niveaux locaux de déformation de von Mises avant coalescence sont semblables pour les deux configurations, c'està-dire 40% pour les ligaments internodules et 45% pour les ligaments en striction, indépendamment des conditions mésoscopiques.

• La capacité des déformations équivalentes de von Mises à indiquer localement la ruine du matériau est démontrée. Cependant, elles restent insensibles au type de

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging mécanismes d'endommagement et de la rupture finale du ligament. D'autre part, les valeurs locales de traces de déformation présentent une certaine sensibilité à différents mécanismes de ruine du ligament (c.-à-d. que les courbes correspondantes possèdent des pentes différentes en fonction du mécanisme de coalescence sous-jacent). La trace locale ne peut pas être utilisée comme quantité seuil pour différents états mésoscopiques de contrainte, mais limitée à l'état de contrainte défini (configurations à 90 • ou 45 • ), c'est une quantité physique utile pour la distinction des deux mécanismes de rupture du ligament.

Processus d'endommagement

• La décohésion précoce des nodules a été observée dans tous les échantillons étudiés. Une fois décollés, les nouveaux vides ont tendance à croître initialement en mode à triaxabilité élevée et à interagir avec les voisins les plus proches dans un mode de striction interne.

• À des stades très tardifs de déformation, juste avant la rupture finale, l'endommagement se produit qu'à certains endroits. Dans la configuration à 45 • , c'est principalement en raison de l'effet de la bande de déformation déterminé par les positions de vides primaires alors que dans la configuration à 90 • , l'endommagement est le résultat d'amas de vides qui coalescent pour former la fissure principale.

• À l'échelle mésoscopique, le taux de croissance des vides est plus élevé dans la configuration à 90 • . Cependant, les fractions volumiques de vide avant la coalescence sont similaires pour les deux cas, soit 16%. La croissance de vide s'accélère avant la coalescence (à une déformation équivalente de 13%) dans la configuration à 90 • alors que dans la configuration à 45 • , cette tendance n'est pas retrouvée.

• À l'échelle microscopique, la croissance de vide individuel montre que même si les ligaments inter-nodule subissent le même type de rupture, la croissance des vides voisins est plus élevée dans la configuration à 90 • pour les deux types de mécanismes de coalescence.

• Les données cinématiques (traces de déformations) sont un outil utile pour l'estimation initiale des valeurs de niveau de gris appropriées pour les méthodes de seuillage lors du calcul de la croissance des vides à partir des données de tomographie / laminographie.

Cadre numérique / expérimental

• Une procédure de validation innovante combinant la laminographie au synchrotron, la CIV et les simulations EF au niveau microscopique a été implémentée (méthode CVI-EF).

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Résumé étendu en français

• Le maillage EF est adapté à la microstructure réelle sous l'hypothèse que les nodules de graphite soient considérés comme des vides avec une procédure spécialement conçue de (re)maillage par [START_REF] Shakoor | Three-dimensional numerical modeling of ductile fracture mechanisms at the microscale[END_REF].

• Une fois validées, les mesures CIV sont utilisées pour piloter des simulations EF. Ajouté au fait que la microstructure réelle soit discrétisée avec le maillage EF, les déplacements hétérogènes imposés par CIV nécessitent une méthodologie d'adaptation de maille robuste. Cette dernière est utilisée ici et permet de mener des simulations avec une croissance de vide significative jusqu'à coalescence.

• Les résultats de CIV et EF peuvent ensuite être analysés par comparaison directe des solutions de champs cinématiques et du calcul des résidus de niveaux de gris. La première méthode, qui est un comparateur relatif, fournit l'écart entre champs cinématiques obtenus par CIV et EF. Les résidus de niveaux de gris peuvent être calculés indépendamment pour les deux approches donnant des mesures d'erreur précises et locales (c.-à-d. directement par rapport aux voxels des images expérimentales).

• On a étudié différents choix de conditions aux limites pour des simulations micromécaniques par EF basées sur des données de laminographie. La plupart des travaux précédents reposent sur des méthodes multiéchelles pour déduire les conditions aux limites pour les calculs micromécaniques. Indépendamment du choix du couplage entre la macroéchelle et la microéchelle (c.-à-d. couplage fort ou faible), il est montré que les schémas multiéchelles ne peuvent pas capter aussi précisément que la CIV-EF les phénomènes complexes de déformation plastique et de décohésion de vide se produisant dans la fonte. Il est important de noter que les mesures d'erreur locales basées sur les résidus de niveaux de gris peuvent également être appliquées à des simulations micromécaniques initialement non pilotées par CIV. Les erreurs produites par les schémas classiques sont donc mesurées plus précisément, révélant que la morphologie des vides n'est pas bien décrite localement avec ces schémas.

• Enfin, le cadre séquentiel CIV-EF est étendu pour devenir une procédure d'identification intégrée. La CIV mécanique 4D consiste à minimiser les résidus d'appariement combinés avec l'écart d'équilibre global pour identifier les paramètres recherchés. La loi de comportement élasto-plastique de la matrice ferritique de la fonte est non seulement sondée en termes de champs cinématiques induits par la microstructure aléatoire mais aussi par des niveaux de charge globaux.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

List of Figures (a) Strain to fracture for monotonic loading as a function of the stress triaxiality and Lode angle parameter reproduced by Wilkins' model [71].

All stress states representing plane stress are connected with the blue line. for different loading steps (noted in colorbar title) . . . . . . . . . . . . . 5.6 Von Mises incremental strain maps from DVC calculations plotted for the ROI mid-thickness section . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Von Mises incremental strain profiles from DVC calculations. The position of the profile axis ξ is denoted on the strain map illustration . . . . . 5.8 Von Mises strain maps from cumulated incremental DVC calculations plotted for the ROI mid-thickness section . . . . . . . . . . . . . . . . . 5.9 Von Mises strain profiles from cumulated incremental DVC calculations.

The position of the profile axis ζis denoted on the strain map illustration . 5. 10 Lode strain parameter µ ∆ε maps from incremental DVC calculations plotted for the ROI mid-thickness section . . . . . . . . . . . . . . . . . . . 5.11 Von Mises incremental strain ∆ε eq and Lode parameter µ ∆ε for mechanical (loading step (3)-( 4)) and noisy data ("bis" case from the uncertainty analysis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Lode parameter µ ∆ε profile (from Figure 5. 13 (3)-( 4) and ( 6)- (7) with and without assumption of incompressibility . . . 5.15 Projections of the stack of slices into single gray level map by keeping only the minimum level from each of the columns in the stack. The stack incorporates most of the sample thickness, i.e., ≈ 810 µm without the part close to the surface. The damage is shown in black . . . . . . . . . . . . 5.16 3D side views of the thresholded damage features in the scanned region for different loading steps . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Damage features found in the shear zone: (a) growth of big elongated damage feature along shear plane (blue arrow) and normal to the loading direction (green arrow), (b) elongated feature at State (7), (c) intermetallic particle breakage at State (7), (d-e) void sheeting as likely nucleation mechanism for elongated damage features (State (7)) . . . . . . . . . . . 8. 15 Cumulative distribution of gray level residuals at the last incremental loading step considering to (a) all supervoxels, (c) only supervoxels with gray level residuals higher than 40. In (b), the absolute gray level differences for the sFE method with residuals higher than 40 are shown for the same Z midsection as Figure 8.11(d) . . . . . . . . . . . . . . . . . . . . 9.1 Schematic representation of the methods used in the present paper for calibrating material parameters via numerical simulations at the microscale 9.2 Tomography setup with the representation of the tested sample. The reconstructed region of interest is also shown . . . . . . . . . . . . . . . . 9.3 Loading and acquisition histories for the in-situ tensile test . . . . . . . . 9.4 Standard displacement uncertainty as a function of the element size for the three displacement components . . . . . . . . . . . . . . . . . . . . . 9.5 DVC (blue) and FE (red) meshes plotted over the corresponding cast iron microstructure in isometric view and sections normal to X, Y and Z axes. Two regions from the section normal to Z axis are showed enlarged on the right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Characterization of three different mesh sizes used in the work with signed distance fields plotted over the corresponding microstructure sections and with the nodule/matrix interface depiction . . . . . . . . . . . . 9.7 Distribution of mesh quality at the initial state for the three tested meshes 9.8 Sensitivity fields in the longitudinal direction for the last loading step.

Online version: corresponding movie for the 6 analyzed steps. The color bar has a fixed range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Stress-strain for the cases analyzed in Table 9.2. In the legend {p 0 } and perturbed {p 0 } stand for different sets of initial parameters detailed in Table 9.

Introduction

This introduction to the work contains several key aspects that are relevant to its narrower understanding. First, the research objectives and the position of the work The economic sector of industries using ductile metals and alloys is one of the most important in France. The mechanical industry is the sector that employs the largest number of people in France. According to FIM (Fédération des Industries de la Mécanique), more than 11,000 companies are working in the mechanical and metallurgical industry and 45% of these companies are working on metal processing. Despite the increasing application of composite-based components, the development and use of metallic alloys exhibiting high mechanical properties is still very active. However, in nowadays global market, innovation in this sector is essential to remain competitive. It is thus particularly important to produce high added value products and to optimize the time to market of industrial components. To reach this goal, the understanding and modeling of ductile failure of metallic alloys are essential to produce components with higher mechanical strength, lower weight and in a minimum number of forming operations. A better prediction of ductile failure would thus have many benefits in terms of:

• reduction of the number of forming stages required to get a final complex component shape without ductile failure;

• prediction of final mechanical properties for better in-use lifetime;

• reduction of the safety margins used in industry to manufacture metallic components;

• reduction of component weight through the optimization of manufacturing processes and component shapes (without failure).

It is important to underline that these benefits are also significant in terms of environmental issues: lighter components and reduction of the number of forming stages lead to energy savings.

COMINSIDE project

This thesis is a part of the COMINSIDE project (Agence Nationale de la Recherche: ANR-14-CE07-0034-02 grant). The project aims at improving the understanding and prediction of ductile failure for various loading conditions. The study of the project is based on local approaches to fracture, i.e., the study of the mechanisms operating at the microscopic scale. The study of the role of the material microstructure and the strain and damage interactions leading to localization and failure will contribute to a better understanding and modeling of the observed failure mechanisms. Subsequently, through homogenization theories, the identifed properties could be transfered to the macroscale where the material characteristics are mainly defined.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Classical ductile damage models [5,8] are usually not well suited to predict failure for low stress triaxiality values. This is partly due to missing experimental data and validation of models for such loading conditions. In these situations the stress triaxiality is missing as void growth driving force and void shape change effect may become important. Recent advances in terms of X-ray tomography enable for the 3D observation of ductile damage micromechanisms in situ and in a non-destructive way [9]. In particular, laminography makes possible the use of flat samples. The utilization of large specimens allows stress states and boundary conditions to be generated, which could not be assessed 3D and in situ at micrometer scales before. The present project is path-breaking due to the use and combination of 3 innovative techniques:

(i) Laminography: this new non-destructive X-ray technique has been set in-place at the European Synchrotron Radiation Facility (ESRF) and allows for the observation of laterally extended objects [6]. In situ laminography tests now enable for the use of nearly full scale standard flat laboratory specimens and have successfully been carried out [START_REF] Morgeneyer | Ductile crack initiation and propagation assessed via in situ synchrotron radiation computed laminography[END_REF]25]. The observation of microstructure and strain-damage interactions at micrometer resolution for various specimen geometries and associated levels of stress triaxiality (from shear to highly triaxial stress states) become possible and allow us to assess failure mechanisms in situ. This represents a major progress over in situ tomography tests on small stick-like specimens for which only axisymmetric loading conditions with less representative boundary conditions and a more limited range in levels of stress triaxiality are used.

(ii) Digital Volume Correlation (DVC) allows displacement fields to be measured in the bulk of various materials by registering 3D images (e.g., obtained via computed tomography or laminography). In order to obtain successful registration the material needs to possess sufficient amount of natural markers, i.e., gray levels contrast. From this information it is possible to monitor in situ tests and to analyze them quantitatively [START_REF] Bay | Digital volume correlation: threedimensional strain mapping using X-ray tomography[END_REF][START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures[END_REF][START_REF] Verhulp | A three-dimensional digital image correlation technique for strain measurements in microstructures[END_REF]. The global approach to DVC allows the measured displacement fields to be seamlessly coupled with finite element simulations [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF].

In the present project, the analysis of large strain levels and the localization due to plasticity and damage will be carried out.

(iii) 3D modeling of strain and damage interaction at the microscale: it is planned here to mesh 3D real microstructures and to apply representative (i.e., measured) boundary conditions in order to model and validate the mechanisms leading to localization. The interaction between plastic strain and ductile damage (particle fracture/debonding, void growth and coalescence) will be studied at the microstructure scale. Modeling of these mechanisms for various loading conditions is essential to understand the role of the microstructure and loading conditions on particular failure mechanisms such as straight or slant failure modes, shear-based failure, localization bands. These modeling efforts will use the in situ 3D damage and strain data obtained by laminography and DVC as input and comparison/validation data. In terms of modeling, the present project aims at developing advanced nu-Understanding, observation and quantification of ductile failure mechanisms via 3D imaging merical techniques to model the stages of nucleation, growth and coalescence of microvoids for 3D configurations. A 3D mesh of the real microstructure will be obtained based on 3D laminography observations. This 3D mesh will be particularly accurate thanks to the use of anisotropic mesh adaption [4,15]. These meshing and remeshing capabilities will be used to reach large plastic strains and model the nucleation (which has never been done yet for complex 3D microstructures), growth and coalescence stages.

To the author's best of knowledge, this is the first time that laminography, DVC and 3D numerical modeling at the microscale will be coupled.

Positioning of the thesis in the project

Due to the complementarity of the techniques used in the project, the scientific program is split into three main tasks.

Task 1 is devoted to the choice and characterization of materials, to experimental tests and their development, to laminography observations and to damage evolution visualization and quantification. This task, which is coordinated by Centre des Matériaux (CdM) -MINES ParisTech, provides the important input data and the raw images necessary for the project. The choice of two materials is essential: a first material that enables a clear observation of ductile damage mechanisms and accurate DVC analyses thanks to its microstructure, and a second material coming from a real industrial aeronautics application.

Task 2 deals with DVC analyses to measure 3D displacement (and strain) fields. This task, coordinated by Laboratoire de mécanique et technologie (LMT) -ENS Paris -Saclay, provides an efficient analysis of the raw data coming from Task 1. DVC supplies data (e.g., boundary conditions) that are very useful when comparing the measurements to the simulations performed in Task 3. There are very few results concerning the use of DVC to analyze plasticity and damage in ductile materials. The first challenge is to have trustworthy measurements with material microstructures that exhibit very low contrast fields such as aluminum alloys. The second one is to drive a DVC code in an area where the gray level conservation is not satisfied. Damage initiation and growth are two typical mechanisms that may make DVC analyses less secure. The third one is to properly capture large strains and localized bands. For the first one, a coupling between DVC and FE simulations is possible to provide the code with an initial solution. The second one requires the use of meshes and/or kinematic bases that are adapted to localized areas [14].

Task 3, coordinated by Centre de Mise En Forme de Matériaux (CEMEF) -MINES ParisTech, consists in the numerical modeling of strain and damage interaction at micro and macro scales. At the microscale, 3D criteria allowing for the description of nucleation, growth and coalescence of void within a complex real microstructure will be developed and validated thanks to the 3D images coming from Task 1. Relevant boundary conditions will be applied to the RVE thanks to Task 2. This microscale analysis will also be used to enhance the modeling of ductile damage at the macroscale through improved nucleation, growth and coalescence laws. Plasticity and strain fields obtained by simulations will be compared to the measured fields.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging As an overlap of Tasks 1, 2 and 3, a combined computational-experimental framework is introduced to validate numerical simulations at the microscopic scale followed by identification procedures via integrated DVC.

The scope of this thesis is mainly focused on Task 1 and Task 2 (Figure 1), i.e., the work conducted in LMT and CdM by the author. However, strong cooperation between the three laboratories resulted in parts of the work where the contribution from CEMEF is equally present (e.g., for integrated DVC). Splitting of those particular results in their presentation would be against the main idea of the project and they would lose on their quality. Hence, some of the results are reported in coupled experimental-numerical version. In this context one should be aware that all the FE numerical developments are conducted in CEMEF and detailed in the Ph.D. thesis of M. Shakoor [11], currently being extended in the ongoing Ph.D. thesis of V. M. Trejo-Navas.

State of the art

In the following section a state of the art is outlined for the three areas covered in the work, namely ductile failure phenomena, Digital Volume Correlation and laminography. The current open scientific questions and principal motivations for performing this work are detailed along this overview. mechanisms over the last decades. A comprehensive summary of the description and modeling of ductile damage mechanisms can be found in Refs. [3,2,13], while some key features associated with the proposed work will be highlighted in the sequel.

As they will be used extensively in the following some initial definitions concerning stress and strain state will be given immediately. The decomposition of the Cauchy stress tensor σ into its spherical and deviatoric part s reads

σ = σ m 1 + s (1)
The stress tensor invariants are defined as

I 1 = trσ; I 2 = 1 2 s : s; I 3 = det[s] (2)
The hydrostatic stress σ m is related to the first invariant by

σ m = trσ 3 (3)
while the von Mises equivalent stress σ eq is proportional to the square root of the second invariant

σ eq = 3J 2 (4)
An important descriptor for ductile damage is the stress triaxiality ratio [START_REF] Bridgman | Studies in Large Plastic Flow and Fracture: With Special Emphasis on the Effects of Hydrostatic Pressure[END_REF] defined by

T = σ m σ eq = I 1 3 √ 3I 2 (5)
To complete the stress state description, the Lode parameter L [7] is introduced as

L = √ 3tanθ; cos(3θ) = 1 2 det 3 s σ eq (6)
where θ represents the Lode angle. In terms of principal stresses it reads

L = 2σ II -σ I -σ III σ I -σ III (7)
The Lode parameter for strain rate λ was also used [START_REF] Morgeneyer | Flat to slant ductile fracture transition: Tomography examination and simulations using shear-controlled void nucleation[END_REF]

λ = 2ε II -2ε I -2ε III εI -εIII . ( 8
)
The Lode parameter enables for the distinction among deviatoric loadings with same effective stress (J 2 ) by introducing the third invariant J 3 of the stress (or strain) tensor. Under axisymmetric loading with a tensile axial stress, the triaxiality can be varied between -2/3 (biaxial compression) and infinity (hydrostatic tension). In (plastic) plane strain, it can be varied between 0 (pure shear) and infinity (hydrostatic tension). In plane stress, however, the triaxiality can only be varied between -1/3 (compression) and 2/3 (biaxial tension);

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging but in practice it is limited to the narrower range 1/3 to 2/3, except when one stress component becomes negative, as in deep drawing for instance. In notched round bars, T ≈ 0.6 -2 [73] whereas higher values are reached ahead of a crack tip in thick components (i.e., T ≈ 1 -4.8). Lode parameter L is limited to the interval [-1,1]. For axisymmetric and plane strain loadings, L is constant irrespective of the principal stress ratios. On the other hand, in plane stress L depends on the eigen stress ratio. Further, some specific stress states can be introduced for convenience:

• generalized axisymmetric tension

σ II = σ III → L = -1, T ∈ R (9)
• generalized axisymmetric compression

σ II = σ I → L = 1, T ∈ R (10)
• generalized shear

σ II = 1 2 (σ I + σ III ) → L = 0, T ∈ R (11)
• pure shear is the special case of generalized shear at zero stress triaxiality

σ m = 0; σ III = -σ I ; σ II = 0 (12)

Mechanisms

Two generic modes of ductile failure will be studied herein:

(i) failure by void growth and coalescence and (ii) failure by instability.

In the former, voids nucleate and grow driven by plasticity and hydrostatic stress, then eventually coalesce and form the final crack. On the other hand, failure by instability is characterized by localized plastic flow prior to damage onset, i.e., mechanical instability (mainly in form of shear bands) is present before the softening induced by damage can set it.

Failure by void growth and coalescence

Ductile fracture is observed after significant plastic deformation and occurs after microvoids or shear bands develop in the metallic matrix, around inclusions or other discontinuities such as grain boundaries. Classical ductile damage approach distinguishes three principal damage mechanisms, namely, void nucleation, growth and coalescence.

In the following sections they are detailed with special attention dedicated to the aspects Understanding, observation and quantification of ductile failure mechanisms via 3D imaging where the current work can be benefitial for the overall understanding of the underlying phenomena.

In contrast to materials that possess initially large porosities, most of structural materials contain several populations of second phase particles acting as potential void nucleation sites under loading. In a soft matrix, nucleation predominately occurs by particle debonding (e.g., pure Al, nodular cast iron in feritic matrix). Conversely, in hard matrices, particle cracking is usually the nucleation mechanism [1,47,[START_REF] Buffière | Three dimensional imaging of damage in structural materials using high resolution microtomography[END_REF] (see Figure 2). Hence, lower matrix yield strength and hardening exponent will promote nucleation by debonding. On the side of particles, bigger inclusions will experience cracking more likely than debonding due to classical volume effect associated with brittle materials [START_REF] Weibull | A Statistical Distribution Function of Wide Applicability[END_REF]. Higher particle stiffness promotes both cracking and debonding. Finally, stress states with higher triaxiality will induce debonding instead of cracking. Voids can also be initially present in structural materials because of processing steps, e.g. hydrogen related voids in Al alloys [17]. There is usually a clear distinction of scales in nucleation phenomena whereas for the usual dislocation densities in worked metals the cutoff particle size for nucleation is ∼ 0.1 µm [12]. Nowadays, synchrotron light enables one to preform X-ray nanotomography experiments under these resolutions [20]. Pre-existing or nucleated voids act as stress concentrators and they start to grow under plastic flow, for instance, in simple tension at first with growth rate roughly twice that of the specimen but later they extend more slowly [21,[START_REF] Maire | Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography[END_REF]. In this case, the voids also grow laterally to an extent proportional with the increase in stress triaxiality induced by necking. The void enlargement is usually accompanied with void shape changes and this is especially prominent at low stress triaxialities. In this regime, even with slow or inexistent void growth, void distortion can be crucial for damage. In a remote shear field, an initially spherical void without (hard) particles would elongate, then rotate [16,[START_REF] Ueda | In situ laminography study of threedimensional individual void shape evolution at crack initiation and comparison with Gurson-Tvergaard-Needleman-type simulations[END_REF] and eventually close into a penny-shaped crack [10] if localization in the intervoid ligament does not take place before complete closure [19]. However, even under a remotely uniaxial stress field, due to void arrangement, void shearing by a microshear band between the voids may occur [22]. In more realistic situations, particles are present inside the voids and contact issues arise under low levels of stresss triaxiality. This so-called void locking by particle phenomenon leads to a net increase in porosity, even under pure uniaxial Understanding, observation and quantification of ductile failure mechanisms via 3D imaging loading, in contrast with the case without inclusions. If the triaxiality is lower than 2/3, contact regions about the equator of the particle experience compressive stresses [29,31]. Even though not closed, void distortions blocked by hard particles can be damaging in the sense that they lead to plastic flow localization, hence failure [19].

After subsequent growth, the voids eventually start to interact and finally coalesce. The physical background of void coalescence has long been ambiguous and some details are still unclear [26]. There exists three principal modes of void coalescence:

(i) coalescence by internal necking of the intervoid ligament -commonly observed and discussed in detail (e.g., Refs. [64,65], see Figure 3);

(ii) void sheeting consists in local shearing in the intervoid ligament that results in coalescence of distant cavities [28] (Figure 3);

(iii) 'necklace' coalescence, i.e., link up of column of voids. It is found in steels with elongated inclusions [23] or alloys with aligned particle clusters [24]. From the experimental side, capturing coalescence events is a challenging task. Tomography, as prevailing real-time in situ observation technique, provided many useful details about coalescence processes, but it is limited to model materials with large porosity levels (≈ 6.5 % in [21]) and rather simple loading conditions (i.e., uniaxial tension) [21,9]. In this context, synchrotron laminography technique due to its high resolution and possibility of using complex laterally extended sample geometries allows for additional valuable experimental informations [25].

Coalescence of the particle-locked voids should be partly exempt from the previous coalescence mechanism description. Internal necking in this conditions is often not possible despite the close packing of voids due to locking effect [27]. Eventually, coalescence by void sheeting may occur in the case of soft matrices and if the voids are close enough. In hard matrices particles crack and form penny-shaped cracks. They experience limited blunting before linking up together. The closer packed particles coalesce by void sheeting while the linkup between the distant particles is neither shearing nor internal necking [27]. This is an example of ductile failure with limited void growth that occurs even for shear-free remote loading. It is likely that technologically important materials (e.g., high strength steels and aluminum alloys) fail by the abrupt coalescence of crack-like voids through the hard matrix. This ductile failure mechanism is poorly studied in computational micromechanics and represents numerous challenges for future models and simulations. Ultra-high resolution tomography captured some details of these processes [30]. Employing high resolution laminography with subsequent measurement of bulk kinematic fields as proposed in this work can shed some additional light into understanding of the reported ductile failure mechanisms with limited void growth.

Failure by instability

Failure by instability denotes a mode whereby plastic instabilities set in before the onset of damage softening. This phenomenon is still not well understood and details remain unclear, yet some of the important structural Al alloys are reported to fail under such mode [32].

For rate-independent materials the onset of shear bands can be understood in terms of Rice's localization criterion [33] while for rate-dependent plastic flow analyzing the onset of shear localization is based on a linear perturbation analysis [35]. Analytical or computational works conducted on this topic produced the basis of necessary and sufficient conditions (e.g., vertex in the yield surface along with thermal softening, strength differential, plastic anisotropy) for early strain band formations [34,38,36,39,37]. If one or more of the conditions are satisfied there is an enhanced possiblity that failure by instability will occur. Failure by instability due to shear bands reduces the ductility of the material [START_REF] Kamat | Mixed mode fracture toughness of engineering materials[END_REF] and thus the understanding of their origin is of great interest in engineering practice, the final aim being the ability to perform microstructural design that would supress them. In this context, once this failure mechanism is observed, the material characteristics that might have caused it are searched for, i.e., an inverse problem is studied. The difficulties lie in the fact that often two or more mentioned necessary conditions for instabilites are fulfilled in such cases.

One of the manifestation of failure by instability is the flat to slant transition during mode I opening of Compact Tension (CT) specimens made of high strength Al, steel or Ti [44,43,42,41]. This slant mode develops soon after crack initiation with the occurrence of shear bands near the crack tip. Recent high-resolution in situ synchrotron X-ray laminography observations combined with digital volume correlation revealed, for an Al-Cu-Li alloy AA2198, that the shear localization process develops before any damage can Understanding, observation and quantification of ductile failure mechanisms via 3D imaging be observed [32].

The following conditions have been identifed (taken from Ref. [13]) to favor slant fracture: (1) a low strain hardening capacity [41], (2) plastic anisotropy [45]; (3) the possibility to nucleate a second population of voids [40] and plane strain condition [34]. Further, increasing the specimen thickness (i.e., increase of stress triaxiality), flat surfaces and side grooves generally force the crack to remain flat. The physical background of this relatively well-established problem is quite obscure. Quantitative assessment via experiments are still lacking. Some of the aspects that promote/supress failure by instability will be tested in this work on the example of CT specimens made of structural Al alloys. In general, plastic instabilities include strained bands and strain localization at a length scale greater than the mean spacing between material inhomogeneities (voids) and material natural texture (grains, grain boundaries). However, in the other limit, plastic instabilities such as necks and macroshear bands are strongly dependent upon the states of stress and strain as well as boundary conditions. As such, they should be viewed as structural, hence non-intrinsic effects. In this work intrinsic type of material instabilities in form of strained bands [32,46] with more or less intermittent activity will be discussed in detail.

Finite element modeling

Nowadays, accounting for ductile damage in finite element softwares essentially relies on one of the three following types of models:

(i) failure criteria in which a damage variable is computed based on stress and strain fields without any coupling with the material behavior [50,49];

(ii) macroscopic models in which a damage variable, coupled with the mechanical behavior, is computed as a function of stress and strain states [48];

(iii) micromechanics-based models that account for the stages of voids nucleation, growth and coalescence [5,8].

Isolated void models

Back in 1960s McClintock [51] and Rice and Tracey [54] developed models for the growth of isolated cylindrical and spherical voids, respectively, embedded in an infinite medium made of a rigid-perfectly plastic material. Their solution for the growth of a spherical void with radius R in a remote uniaxial tension strain rate field reveals that the spherically symmetric volume changing part of void growth prevails over the shape changing part at high stress triaxialities. In the model the net void growth ratio ln R R 0 (R 0 being initial radius) is an equivalent of damage variable D

Ḋ ≡ Ṙ R = α(λ)exp 3 2 σ m σ εeq ( 13
)
Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

where λ denotes the strain rate Lode parameter, σ m the hydrostatic stress, σ the effective stress (σ = σ eq in case without damage) and εeq the equivalent strain rate. α(λ) = α + βλ where α and β are constants [54]. Hence, already at this point the Lode parameter was introduced as a relevant factor in ductile damage studies, but with very small weight put on its importance. The basic equation was later modified by Huang [52] (updated α). The version for more general loading conditions [54] reads

Ṙ R = 2α sinh 3 2 σ m σ εeq + 2βλcosh 3 2 σ m σ εeq ( 14
)
giving good results for different levels of triaxialty and Lode parameter but only for spherical voids. The model assumes spherical voids to remain spherical upon loading. Under low stress triaxialities, the model cannot reproduce shape change of the voids, thus becomes increasingly inaccurate upon straining and predicts imprecise fracture strains.

Uncoupled models

In industrial applications involving large deformations (e.g., sheet metal forming) porous plasticity models are not used to model response of metals. Instead, non-porous phenomenological plasticity models in uncoupled approaches are widely employed. Uncoupled approaches (i) neglect the coupling between the mechanical fields and damage growth and in general have a much simpler structure than coupled models [START_REF] Benallal | Continuum Damage Mechanics and Local Approach to Fracture: Numerical Procedures[END_REF]. They predict the onset of ductile fracture based only on the macroscopic stress and strain history, hence not coupled with specific macroscopic plasticity model. These models are mainly derived from isolated-void growth models. The well known Beremin [50] and Johnson-Cook [49] models are based on integrated forms of the Rice-Tracey void growth equation. In general, fracture initiation for uncoupled models is described by a criterion

ε f 0 g(T, L)dε eq = D c (15)
where D c is a constant understood as critical void volume fraction for different stress states, ε eq and ε f the current effective strain and its value at failure; g a function of stress triaxialty T (ε eq ) and Lode parameter L(ε eq ) that describes the effect of the damage process. Different variants for g are proposed having damage growth more (semi-coupled) or less (truly uncoupled) derived based on plasticity. In case of radial loadings, T and L are history independent and two-dimensional fracture surface (fracture loci) can be expressed as [START_REF] Cao | A comparative study of three ductile damage approaches for fracture prediction in cold forming processes[END_REF]

ε f = D c /g(T, L). ( 16
)
For identification purposes uncoupled models are convenient since often only one fracture parameter D c is being sought. For more realistic situations, i.e., non-proportional loading paths and non-constant D c (depends on stress state) the previous closed form solution is lost. However, this type of equations found its place in bulk metal forming and many other applications [START_REF] Cao | A comparative study of three ductile damage approaches for fracture prediction in cold forming processes[END_REF].

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

For isothermal conditions, constant strain rates and radial loading paths, the Johnson-Cook model [49] delivers an exponential relationship between the strain to fracture and the stress triaxiality

ε f = D 1 + D 2 exp[D 3 T ] (17)
where D 1,2,3 are model parameters. Among the first, Wilkins et al. [71] proposed an uncoupled model that makes a distinction between asymmetric loading and generalized shear. The criterion for the model reads

ε f 0 1 1 -aσ m α 2 -max s II s III , s II s I β dε eq = D c ( 18
)
where s I ≥ s II ≥ s III are the principal stresses of deviators, and (a, α, β) material-dependent constants. The first term in parenthesis describes hydrostatic-pressure weighting while the second one is asymmetric-strain weighting, which represents the effect of the Lode parameter. Wilkins et al. [71] performed a set of experiments with different specimen geometries to calibrate their model including tensile experiments on notched cylinders and plates as well as tension-torsion experiments. Figure 4(a) (taken from Ref. [START_REF] Mohr | Effect of stress state on ductile fracture[END_REF]) shows the strain to fracture for radial loading as a function of the stress triaxiality and Lode angle using the parameters identified in Ref. [71] for aluminum alloy 6061-T651.

The projection of the calibrated Wilkins model for plane stress conditions (Figure 4(b)) results in non-monotonic relationship between the strain to fracture and the stress triaxiality. Later, Bao and Wierzbicki [72] and Barsoum and Faleskog [73] reported similar relationships that motivated further detailed studies of the effect of the Lode parameter and stress triaxiality on the onset of ductile fracture in metals [START_REF] Xue | Ductile Fracture Initiation and Propagation Modeling Using Damage Plasticity Theory[END_REF]75,76,[START_REF] Lou | Modeling of shear ductile fracture considering a changeable cut-off value for stress triaxiality[END_REF][START_REF] Mohr | Micromechanically-motivated phenomenological Hosford-Coulomb model for predicting ductile fracture initiation at low stress triaxialities[END_REF]. Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Some authors, e.g., Haltom et al. [START_REF] Haltom | Ductile failure under combined shear and tension[END_REF], reported the monotonic relationship between failure strains and stress triaxiality. The "dip" for shear-dominant stress states was not found in these results. Thus, determing the exact ε f -T relationship is per se a challenging task. For instance, measuring strains to failure from the experiments remains difficult [72] whereas hybrid experimental-numerical approach opens new possiblites [START_REF] Papasidero | Ductile fracture of aluminum 2024-T351 under proportional and non-proportional multi-axial loading: Bao-Wierzbicki results revisited[END_REF]. Additionally, no information on the origin of reported ε f -T relationships is available, i.e., a complete physical explanation still needs to be clarified.

To sum up, uncoupled models can be well tailored for narrow stress state conditions and in this context be very useful for industrial applications. Along this assumption, they are quite accurate for predicting strains to failure ε f (T, L) even for complex sample geometries and loading paths. On the other hand, it is shown that the fracture strain in such phenomenological approaches may vary for non-proportional loading paths [START_REF] Benzerga | On the path-dependence of the fracture locus in ductile materials analysis[END_REF]. It is hard to expect the creation of robust (global) ductile damage models within this framework since the physics of the phenomena is not completely taken into account, i.e., general applicability is lacking.

Macroscopic models

In micromechanics-based models, damage corresponds to voids (porosity). In Continuum Damage Mechanics (CDM) models (ii), damage is introduced as a scalar or tensorial internal variable D into macroscopic constitutive equations. There is no explicit physical interpretation of damage at the microscale and CDM models usually assume plastic incompressibility of the matrix. In the well-known Lemaitre model [START_REF] Lemaitre | A continuum damage mechanics model for ductile fracture[END_REF]48] the damage variable D can be physically understood as the ratio of damaged area S D of a unit surface divided by the total surface S. The strain energy release rate Y associated with the damage variable D is given by

-Y = σ 2 eq 2E(1 -D) 2 2 3 (1 + ν) + 3(1 -2ν)T 2 ( 19
)
where E is the Young modulus and ν the Poisson's ratio. Plasticity/damage dissipation potential F is defined as the sum of the von Mises plasticity criterion f and the Lemaitre potential of dissipation

F Y F = f + F Y with        f ( σ, R) = σeq -R -σ Y F Y = S 0 b + 1 1 1 -D -Y S 0 b+1 ( 20
)
where S 0 and b are material parameters, R the scalar variable associated with isotropic hardening and σ Y the yield stress. In the above equation the effective stress tensor ( σ) is defined as the tensor that actually resists to the load

σ = σ 1 -D (21)
Understanding, observation and quantification of ductile failure mechanisms via 3D imaging while the equivalent effective stress ( σeq ) reads

σeq = 3 2 s 1 -D : s 1 -D (22)
A kinematic hardening parameter can be added to the von Mises plasticity criterion if necessary. The growth of internal variables can be obtained via normality rule where the final equation for damage growth reads

Ḋ =    0, if ε eq < ε D - Y S 0 b εeq , otherwise (23)
i.e., damage increases only once the equivalent plastic strain has exceeded a strain threshold ε D . When the damage variable reaches its defined critical value D C fracture initiation occurs.

Coupled models

Void nucleation and growth The coupled porous metal plasticity models (iii) deliver constitutive relations that seamlessly couple damage and plasticity. The first set of micromechanics based yield functions have been derived by Gurson [5] using the upper bound theorem of limit analysis. In the original Gurson model [5] the damage variable is the porosity f and void shape changes are not taken into account. The growth law of the damage variable is given by ḟ

f (1 -f ) = 3 2 q σ σ eq sinh 3 2 σ m σ εeq , (24)
σ eq , σ m and σ being related through Gurson flow potential

Φ GT (σ, f ) ≡ σ 2 eq σ 2 + 2q f cosh 3 2 σ m σ -1 -q 2 f 2 = 0. (25)
In its original form q = 1, while later on heuristic factors q 1,2 were introduced by Tvergaard [53]. The most widely used porous plasticity model is the so-called Gurson-Tvergaard-Needleman (GTN) model. It makes use of a phenomenological extension of Gurson's upper bound yield function for a thick-walled sphere

Φ GT (σ, f ) ≡ σ 2 eq σ 2 + 2q 1 f cosh 3 2 q 2 σ m σ -1 -q 2 1 f 2 = 0. ( 26
)
As initialy postulated, this set of equations gives a good approximation for the hightriaxiality behavior of plastically isotropic materials with equiaxed inclusions and voids. The choice of q was debated over the last decades [53,55]. Physically, it seems that Introduction multiple parameters influence factor q, void interaction being one of them, and there is no universal value for it. The basic form of Gurson's law was built up during the years accounting for plastic anisotropy [56] and including the void shape and rotation effects into the model [57,58,59]. Keralavarma and Benzerga developed a coupled model for general loadings [60] where damage growth is specified not only through the rate of porosity but also change of void shapes. However, general growth equations for the damage state variables, other than porosity, are yet to be developed.

In the original formulation of Gurson's law assuming normality of the flow rule at the macroscopic level and plastic incompressibility of the matrix phase, the change of porosity reads

d f = (1 -f )(ε eq : 1) (27)
The previous equation can be extended by including the effects of nucleation in addition to void growth

d f = (1 -f )(ε eq : 1) + N(dε M ) (28)
where dε M denotes equivalent plastic strain in the matrix. If only plastic strain controlled void nucleation takes place, the nucleation function can be described by a Gaussian probability density function [61]

N =    f N s N √ 2π - 1 2 ε M -ε N s N 2 , for σ m ≥ 0 0, for σ m < 0 (29)
where ε N is average nucleation strain, s N standard deviation and f N the volume fraction of void nucleating particles. Additionally, accelerated decrease in stress carrying capacity after the onset of coalescence can be introduced through critical value f C [START_REF] Tvergaard | Material failure by void coalescence in localized shear bands[END_REF]

f =    f , for f ≤ f C f C + f -f C f f -f C ( 1 q 1 -f C ), for f > f C ( 30
)
where f f denotes the void volume fraction at which the entire load carrying capacity is lost. The GTN model and its extensions for non-spherical voids do not predict any softening under shear loading. Hence, these models cannot predict the experimentally observed formation of primary localization bands under shear loading. To tackle this problem, Nahshon and Hutchinson [62] changed the meaning of f from void volume fraction to damage, and added a contribution of damage growth under shear stresses

d f = (1 -f )(ε eq : 1) + N(dε M ) + k ω f ω 0 (λ) s : dε p σ eq ( 31
)
where s represents deviatoric part of stress tensor, ω 0 (λ) a Lode parameter dependent function and k ω a material parameter. A similar effect is achieved by modeling void nucleation based on the strain rate tensor in Ref. [START_REF] Morgeneyer | Flat to slant ductile fracture transition: Tomography examination and simulations using shear-controlled void nucleation[END_REF]. This phenomenological modification

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging was later altered by Nielsen and Tvergaard [63] reducing the effect of shear damage under generalized shear loading at high stress triaxialities.

Void coalescence One particulary important aspect in ductile damage modeling is coalescence. In general, void growth theories cannot independently and quantitatively predict the limits of ductility. Void growth models assume that plastic flow takes place in the whole elementary volume while in reality certain modes of localized plastic deformation set in and determine lower values of plastic dissipation, hence are more likely to prevail after sufficient microstructural changes [3]. Coalescence models are concerned with the transition from diffuse plastic flow to localized modes within the inter-void ligament of neighboring voids in the form of primary localization band. Internal necking or internal shearing are two possible mechanisms of strain localization. Hence, void coalescence induces directional void growth processes. When it occurs through internal necking in relatively ordered void disributions, this process begins with the onset of necking of the intervoid ligament, while the voids are quite distant from each other, and ends with the voids actually connecting to each other. The issue of determining the direction of internal necking or shearing is not addressed in detail. In practice, possible coalescence directions are affected by the nature of void distributions.

Inital efforts of Thomason [64] and Brown and Embury [65] resulted in coalescence criteria under predominately tensile loads. Both models used a condition of critical distance between the growing voids for the onset of coalescence. The Brown-Embury criterion is met when 2h = 2S -2R where 2h represents the void height, 2S the void distance and R the in-plane void radius. Thus, these conditions are purely geometric and do not include any influence of the perturbed stress state. Later, Thomason [66] proposed a coalescence criterion established in the study of three-dimensional periodic arrays of square-prismatic voids. Plastic flow was described by associated von Mises plasticity and restricted to intervoid ligaments, the regions above and below the voids being modeled as rigid. With internal necking assumed to take place in the x 1x 2 plane, which is normal to the largest tensile stress, Thomason's coalescence condition reads

Φ T (σ, χ, ω) = σ 33 -σ T (χ, ω) = 0 ( 32
)
where χ = R S is the ligament parameter and ω the void aspect ratio, while σ T reads

σ T (χ, ω) = σ(1 -χ 2 ) α T χ -1 -1 ω conditions.
It is worth recalling that the above models are insensitivite to shear effects on coalescence. In this context, Tekoglu et al. [70] recently proposed an extension of Thomason's coalescence criterion to non-axisymetric loadings where both extensional and shear components of the microscopic strain field are considered. Coupling of the Gologanu-Leblond-Devaux (GLD) porous plasticity model with a Thomason-type of coalescence model provides accurate results both in terms of the microstructural evolution and macroscopic response compared to results from axisymmetric unit cell simulations [68]. In particular, the acceleration of void growth during coalescence comes in naturally through the transition from diffuse flow (GLD model) to localized flow at the microscale (coalescence model). More research is needed in this area for void distributions that are neither periodic nor uniformly distributed. Studying nodular cast iron samples within the scope of this work will document new experimental evidence of void coalescence coming from real microstrucures under different stress states.

Let us outline another important approach proposed by Rousselier [START_REF] Rousselier | Ductile Fracture Models and Their Potential in Local Approach of Fracture[END_REF][START_REF] Rousselier | Dissipation in Porous Metal Plasticity and Ductile Fracture[END_REF]. It is based on the thermodynamical framework and introduces a damage variable as a state variable. The associated thermodynamical force and the yield surface must be such that the damage growth law corresponds to mass conservation. Under pure shear damage is still generated inside Rousselier model whereas the Gurson model (in absence of nucleation) does not lead to damage growth. Under pure hydrostatic stress states the Rousselier yield surface has a vertex, which implies that at high stress triaxiality ratios the plastic deformation tensor always keeps a nonzero shear component. The hybrid approach, e.g., coupling of Lemaitre formalism for nucleation and the Gurson model for void growth also represents interesting perspectives.

Summary

As a summary of this state of the art overview outlined are two general issues where the current work can contribute:

• plastic instabilities ahead of notch tip An example that assembles the main difficulties in the field of modeling and simulating ductile failure is testing notched or precracked thin metal sheets. For these samples, during Mode I opening the final crack starts to propagate in a flat mode, then tilts and continues to propagate in a slant manner [START_REF] Mahgoub | Three dimensional stress and deformation fields around flat and slant cracks under remote mode I loading conditions[END_REF][START_REF] Lan | Study of slant fracture in ductile materials[END_REF]. The flat region is normally triangular in shape with an area of large primary dimples (Figure 5). In the slant region, primary dimples are accompanied with secondary dimples. The problem is studied in Refs. [40,[START_REF] Bron | Simulation of the ductile tearing for two grades of 2024 aluminum alloy thin sheets[END_REF] using a simplified FE simulation of the test. These simulations show low hydrostatic stresses in slant regions whereas the strain state corresponds to plane strain conditions, but in the crack propagation direction. The failure of the specimens is essentially controlled by plastic instability (Figure 5), which is due to shear strained bands [32]. In the vicinity of the notch root the hydrostatic stress is high, which promotes void growth. The strain state is not planar, which delays localization [34,[START_REF] Besson | Modeling of Crack Growth in Round Bars and Plane Strain Specimens[END_REF] and thus leads to internal necking between primary cavities. The flat to slant transition was obtained [START_REF] Xue | Damage Accumulation and Fracture Initiation in Uncracked Ductile Solids Subject to Triaxial Loading[END_REF][START_REF] Xue | Ductile Fracture Initiation and Propagation Modeling Using Damage Plasticity Theory[END_REF] Furthermore, having two void populations involved, the material posses (at least) two characteristic length scales. If a nonlocal model is to be used, this would imply to use two different nonlocal variables, each being associated with a particular length. In this work, both flat and slant regions, as well as the transition area, will be inspected by employing laminography and DVC bulk kinematic measurements on CT specimens. The observation of microstructure and strain-damage interactions at micrometer resolution should bring some additional insights into the flat to slant crack transition both in qualitative and quantitative terms.

Material failure in slant regions with low hydrostatic stresses exemplifies the problematic of failure in shear, i.e., is it failure possible under pure shear stress state, what are the underyling micromechanisms and driving forces? These questions will be detailed in Chapter 5 dedicated to the shear sample geometry.

• influence of real microstructure features on ductile damage micromechanisms

In general, micromechanical simulations enabled for a better understanding and quantification of essential failure processes. Up to date, two principal problems are encountered. First, it is extremly complex to check experimentally the identified parameters from these simulations. Second, the proposed models lead to accurate predictions of ductile failure only for idealized materials that correspond to void cell calculations. Engineering materials rarely contain initial porosity, and if they do the voids and particles are randomly distributed. The open question is to what extent nonuniform particle distributions and heterogeneous void nucleation can affect crack initiation and growth. The term heterogenity associated with real microstructure involves [13]:

1. distribution of chemical composition of the second phases or distribution of smaller size particles or defects in the second phases or at the interfaces, 2. distribution of second phase size, shape, orientation and position, 3. distribution of strength mismatch between hard and soft phases, 4. crystallographic orientation of the grain, 5. grain boundary and near grain boundary metallurgical heterogeneities, such as precipitate-free zones.

For the nucleation process the loading history experienced by each feature depends on type (2) and type (5) aspects of heterogenitiy. Type (1) heterogeneities lead to a distribution of particle nucleation strengths but they are also affected by particle size (type (2)). Overall weighted contributions from each listed factor define the onset of void nucleation.

For void growth, although random distributions may not be of significant influence [START_REF] Melander | The effect of void size and distribution on ductile fracture[END_REF], void clusters involving voids of various dimensions showed to be an important factor [START_REF] Achon | Comportement et ténacité d'alliages d'aluminium à haute résistance[END_REF]. Voids nucleated from particle fracture (initially flat) will grow

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging slowly compared with voids nucleated by complete debonding (initially round). The higher stress triaxiality in the softer phase induces higher void growth (multiphase materials) with a concentration factor determined by the soft and hard phase distributions [START_REF] Li | The influence of plasticity mismatch on the growth and coalescence of spheroidal voids on the bimaterial interface[END_REF][START_REF] Yerra | Ductile fracture initiated by interface nucleation in two-phase elastoplastic systems[END_REF]. Significant differences in void growth rates (factor 5 or more) are reported by tracking in situ the growth of individual voids in 3D tomography [START_REF] Lecarme | Heterogenous void growth revealed by in situ 3-D X-ray microtomography using automatic cavity tracking[END_REF] for Ti and [START_REF] Hannard | Characterization and micromechanical modelling of microstructural heterogeneity effects on ductile fracture of 6xxx aluminium alloys[END_REF]

Laminography

For the observation of microstructural changes during ductile tearing, different techniques may be employed. They have different advantages and drawbacks. Scanning Electron Microscopy (SEM) for in situ surface observations may be considered. It clearly benefits from high spatial resolutions [101]. However, ductile failure strongly depends on the levels of stress triaxiality that are mostly unrepresentative, since biaxial, on the specimen surface [54]. Additionally, surface roughness often makes the observations difficult. In situ tomography experiments provide the opportunity to gain valuable insight into the damage development in the material bulk in a non-destructive manner. The difference in damage growth between material bulk and specimen surface was clearly shown [START_REF] Maire | Damage quantification in aluminium alloys using in situ tensile tests in X-ray tomography[END_REF][START_REF] Toda | In situ observation of ductile fracture using X-ray tomography technique[END_REF]. The achieved resolution is in the micrometer range. Different levels of stress triaxiality can be obtained by using axisymmetric smooth and notched specimens. However, the boundary conditions used are far from most standard mechanical tests due to the small (typically 1 mm cross section diameter) size of the specimens that do not allow for realistic plastic zone sizes that are typically of the order of several millimeters for thin ductile materials. Another drawback of this technique is that non-axisymmetric loading conditions that have recently been established to be a key point to full ductile fracture understanding [START_REF] Toda | Mechanics and Simulations of Ductile Fracture[END_REF] cannot be reproduced. Crack propagation with its associated typical flat-to-slant transitions cannot be studied either.

With the progress in synchrotron radiation laminography [6,[START_REF] Maurel | Three-dimensional investigation of thermal barrier coatings by synchrotron-radiation computed laminography[END_REF], these limitations of tomography may be partially overcome as it is now possible to observe objects that are extended in two directions and thin in the third one. For laminography, the rotation axis of the specimen is inclined at an angle θ < 90 • with respect to the beam direction (where θ = 90 • corresponds to the case of standard tomography, see Figure 6). The specimens are typically turned about the normal vector of the sheet plane.

Tomography setup

Laminography setup However, it is worth noting that the sampling of the 3D Fourier domain of the region of interest is incomplete in laminography [137], which leads to imaging artifacts [START_REF] Xu | Comparison of image quality in computed laminography and tomography[END_REF]. These artifacts can be assumed to degrade the displacement and strain resolutions when digital volume correlation (DVC) techniques are applied [25].

With this technique, damage growth in a carbon fiber reinforced (CFR) composite has been observed in situ [START_REF] Moffat | In situ synchrotron computed laminography of damage in carbon fibreepoxy [90/0]s laminates[END_REF]. Ductile crack initiation and propagation in AA2139 Al alloy has been studied for the first time in 3D and in situ at micrometer resolution [START_REF] Morgeneyer | Ductile crack initiation and propagation assessed via in situ synchrotron radiation computed laminography[END_REF] within thin sheet material. Boundary conditions close to those of structures in service and close to standard mechanical tests to assess the tearing resistance can be applied. The specimen geometry was 60 mm × 70 mm × 1 mm with a 36 mm long notch. The observation zone sampled during one scan remains of the same typical size compared to tomography (≈ 1 mm 3 ), but the reconstruction restrictions leading to the cutting of small samples vanishes and one can work using large scale sheets. It opens the way for the use of flat full scale sheet samples for in situ observations where different stress states can be generated. In-situ laminography mesaurements with micrometer resolution have been successfully performed [25] while nanometer resolution experiment will be probed in the scope of this work.

Digital Volume Correlation

The first 3D bulk measurements using digital volume correlation (DVC) were performed on trabecular bones. The measured kinematics was based upon piecewise constant translations [START_REF] Bay | Digital volume correlation: threedimensional strain mapping using X-ray tomography[END_REF]. From these data, the strain fields of bone tested in compression were obtained. Local rotations were considered later on within the same framework [START_REF] Smith | Digital volume correlation including rotational degrees of freedom during minimization[END_REF] to analyze the deformation of aluminum foam. This type of kinematics was also used to study localization bands in sand. Each individual grain is analyzed independently [START_REF] Hall | Discrete and continuum analysis of localised deformation in sand using X-ray micro CT and volumetric digital image correlation[END_REF].

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Warping of sub-volumes was added to the sought kinematics [START_REF] Verhulp | A three-dimensional digital image correlation technique for strain measurements in microstructures[END_REF]. However, the only information that was kept is the average displacement per sub-volume. With this kind of analyses, measurement uncertainties were assessed when testing various biomaterials [112]. Many of the earlier studies were performed within the field of biomechanics. They were extended to other materials in the middle of the last decade. For instance, aluminum matrices were studied in which particles were added to allow DVC analyses to be conducted [START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures[END_REF].

As in 2D digital image correlation, global approaches were also introduced to register 3D images. They consist in solving the minimization of the sum of squared differences of gray levels over the whole region of interest, and not independently over each sub-volume. First, finite element kinematics associated with 8-noded cubes was considered [START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF]. Second, enriched kinematics were considered [START_REF] Réthoré | Extended threedimensional digital image correlation (X3D-DIC)[END_REF] to account for the presence of cracks in the observed volume made of nodular graphite cast iron. In particular, it was possible to link X-FEM and X-DVC to validate a numerical model of the experiment by comparing measured and predicted stress intensity factor profiles [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF].

A feasibility study of DVC to measure displacement fields in an aluminum alloy with a very low content of intermetallics was performed when the experiment was monitored via laminography [25]. Even though the measurement uncertainties are higher than those reported with tomographic data, they are low enough to follow the kinematics and detect strained bands [32]. Keeping the same spatial resolution, the strain uncertainty can be further decreased by employing additional mechanical regularization procedures acting like low-pass displacement filters [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF][START_REF] Taillandier-Thomas | Localized strain field measurement on laminography data with mechanical regularization[END_REF]. Integrated-DVC (I-DVC) was introduced on the example of nodular cast iron [START_REF] Hild | Toward 4D mechanical correlation[END_REF]. In this work, full integration of sensor and imaging data into numerical procedures for the purpose of identification of constitutive laws and their validation was demonstrated.

Recently, the ability of tracking microstructure kinematics from no more than two radiographs in situations where a complete 3D view would require several hundreds of radiographs was reported [START_REF] Taillandier-Thomas | Soft Route to 4D Tomography[END_REF]. These steps are significant developments toward "computed" 4D tomography, the final aim being continuous filming of the sample via X-rays during the experiment.

Materials

Two material classes that are representative of two generic modes of ductile failure will be examined herein, namely, 21xx aluminium alloys (failure by instability) and nodular graphite cast iron (failure by void growth and coalescence).

• 21xx aluminum alloys aluminum and reduced density. At the time being, fuel efficiency of aircraft has become a deciding factor for the aerospace industry. Addition of Li to Al substantially reduced the weight of aeronautic alloys, since each 1 wt. % Li added to Al reduces its density by 3 % and increases the elastic modulus by about 6 % [START_REF] Ehrström | Metallurgical Design of Alloys for Aerospace Structures[END_REF][START_REF] Warner | Recently-developed aluminium solutions for aerospace applications[END_REF]. Second generation of Al-Li alloys (2090, 2091 and 8090) contain Li in more than 2 % concentrations. Although these alloys have desired lower density, higher modulus and improved fatigue resistance, they exhibit hole cracking and delamination during drilling, lower ductility and fracture toughness in the short-transverse (S-T) direction [START_REF] Kalyanam | Delamination cracking in advanced aluminum-lithium alloys-Experimental and computational studies[END_REF]. The latest, third generation of Al-Li alloys (2xxx series) contains less than 2 % Li and has a higher Cu/Li ratio than the previous generation that resulted in optimized toughness and strength [START_REF] Chen | Ductile tearing of AA2198 aluminium-lithium sheets for aeronautic application[END_REF]. For this reason, they have been considered for fuselage application in the new generation of commercial airplanes.

Aluminum-Copper-Magnesium alloys (AA2139) possess a good combination of strength, fracture toughness (mainly for AA2139 T8) and fatigue resistance. Addition of small amounts of Ag to Al-Cu-Mg alloy stimulates strengthening and can promote precipitation along transgranular slip planes that decrease the amount of grain boundary precipitation. Al-Cu-Mg-Ag is a new high strength alloy used in aircraft industries and is a potential combination to develop new alloys for aerospace industries.

For each of the two alloys (AA2198 and AA2139), two different heat treatment conditions, namely, T351 and T851 will be studied in the work. The sheets are first solution heat treated, stretched between 2-4 % and naturally aged to obtain T351 condition materials, then an artificial ageing step was applied to achieve T851 tempered materials. For brevity, heat treatment T351 will be identified as T3 and T851 identified as T8 in the following, T-51 corresponds to a mechanical treatment (stress relieved by stretching).

• Nodular graphite cast iron (ductile cast iron, spheroidal graphite cast iron or SG cast iron) represents a family of materials with a wide variety of properties for different engineering requirements. The major constituent is iron, with important amounts of carbon and silicon. Cast irons are natural composite materials whose properties are determined by their microstructures, namely, the stable and metastable phases formed during solidification and subsequent heat treatment(s).

The major microstructural constituents of cast irons are the chemical and morphological forms taken by carbon, and the continuous metal matrix in which the carbon and/or carbide are dispersed.

Cast iron has become a popular cast metallic material, which is widely utilised in modern industrial production because of its low cost and desirable properties such as good castability, convenient machining property and wear resistance. Big parts of the annual production of ductile cast iron is in the form of pipes, which are used for water and sewer lines.

Ductile cast iron is specifically useful in many automotive components, where

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging strength needs to surpass that of aluminum but not necessarily that of steel. Other major industrial applications include highway diesel trucks, agricultural tractors and oil well pumps. In this work ductile nodular cast iron EN-GJS-400 has been investigated. It contains graphite spheroids embedded in a ferritic matrix (Figure 3). The latter provides cast iron with good ductility and impact resistance [124]. The tensile and yield stresses are equivalent to low carbon steel. The material was provided by Technische Universität Bergakademie Freiberg (Germany).

More details about the materials (e.g., chemical composition, grain size, size of the nodules, mechanical properties) will be introduced as part of the corresponding chapters in the sequel.

Aluminum alloys with their very low volume fraction of secondary particles will be challenging for DVC registrations. Although an initial feasibility was demonstrated [25] it still needs to be confirmed for higher strain levels, in particular in the presence of damage. In the case of nodular graphite cast iron, the relatively coarse microstructure allows the damage nucleation process to be followed in detail at the achieved micrometer resolution via laminography. In addition the strong gray level contrast of the microstructure is beneficial for DVC measurements [START_REF] Réthoré | Extended threedimensional digital image correlation (X3D-DIC)[END_REF][START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using 3D correlation of laboratory X-ray microtomography images[END_REF].

Structure of the work

The samples geometries imaged via laminography that will be studied in this work are illustrated in Figure 7. The samples are lined as a function of the average stress triaxiality obtained during the experiments in the scanned regions (denoted with red squares). The work contains 9 thematic chapters along with the general conclusions and perspectives (Chapter 10). All thematic chapters present published or (to be) submitted research publications. They are divided into two general groups where Chapters 2-5 deal with structural 21xx Al-alloys while Chapters 6-9 present the results obtained with nodular graphite cast iron. Chapters 1 and 10 equally refer to both materials as shown in Figure 8. In Chapter 2 a CT specimen made of AA2198 T8 alloy is studied. This work is a continuation of the work of Morgeneyer et al. [32] but now with the region of interest shifted from the slant zone to the immediate vicinity of the notch root. Bulk strain fields and damage changes are reported and compared with the findings in the slant zone.

In Chapter 3 a CT specimen made of AA2198 T3 alloy, hence with higher work hardening compared to the T8 heat treatment, is studied with detailed correlations between kinematic measurements and corresponding damage micromechanisms.

In Chapter 4 the kinematic fields are measured in CT specimen made of AA2139 T8

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging in the slant zone. The novel correlation technique projection DIC is proposed and found to be a useful tool in a given situation of plane strain fields. The strain fields are studied in synchronization with the underlying microstructure changes. The findings are discussed in the context of other studied Al-alloys and possible generalizations of the determined deformation mechanisms. Chapter 5 deals with the shear specimen geometry inspired by work of Roth and Mohr [START_REF] Roth | Ductile fracture experiments with locally proportional loading histories[END_REF] made of AA2198 T8 alloy. Performing DVC measurements is achieved on highly localized strain fields with 'smart' initialization fields. Next, the ability to measure Lode parameter fields for incremental strains is shown. Laminography observations in the bulk of the shear ligament reveal damage growth from the middle of the loading process away from free surfaces.

Chapter 6 presents a comprehensive analysis of the two sample configurations inspired by work of Weck et al. [22] made of nodular cast iron (i.e., plates with 2 holes oriented at 90 • and 45 • with respect to the loading direction). The results are presented on global and local scales both in terms of kinematic variables from DVC and porosity changes from laminography scans. The damage mechanisms are analyzed in detail and compared for both configurations.

Chapter 7 introduces a combined computational-experimental framework to validate numerical simulations at the microscopic scale. The studied sample is a central hole geometry and it is made of nodular cast iron. Finite Element (FE) simulations, which account for the studied material microstructure, are driven by Dirichlet boundary conditions extracted from DVC measurements. Gray level residuals for DVC measurements and FE simulations are assessed for validation purposes.

Chapter 8 compares the accuracy of the results obtained with different numerical simulations at the microscopic scale. The validation procedure highlights the benefits of using DVC boundary conditions compared to the ones coming from macroscopic (weakly or strongly coupled) FE calculations.

Chapter 9 introduces Integrated-DVC (I-DVC) in order to identify the ferritic matrix flow properties at the microscale. To the author's best knowledge, this is the first time such inverse analysis is performed at the microlevel accounting for microstructure features in the scope of ductile damage.

Chapters 7, 8 and 9 are the results of a joint project work with colleagues from CE-MEF. However, a separate presentation was not chosen for the sake of logic.

Chapter 10 contains general conclusions extracted from the analysis performed on both materials with special attention dedicated to the perspectives and future work. The preliminary results from a first nano-laminography in-situ experiment with AA2139 T3 CT specimen are also discussed in this part.

Appendix A is the research publication with laminography and DVC results from the CT specimen made of AA2139 T3 alloy. Via simultaneous assessments of 3D strain and damage at a distance of 1-mm ahead of a notch root parallel crossing slant strained bands are active from the beginning of loading in a region where the crack will be slanted. The bands have an intermittent activity but are stable in space.

Evaluation of Measurement Uncertainties of DVC applied to Laminography Data

The effect of different data acquisition parameters on the displacement and strain uncertainties of the kinematic fields are assessed via Digital Volume Correlation (DVC) applied to three-dimensional (3D) data sets imaged by synchrotron laminography. The measurement uncertainty is assessed for i) different materials and associated varying image contrast textures ii) repeated scanning, i.e., uncertainty due to the imaging system and 3D reconstruction iii) a rigid body motion between two scans iv) rescaling of the gray value histogram during 32 to 8-bit data conversion v) changes in the beam properties, i.e., use of a monochromator or undulator vi) changes in camera/detector characteristics and vii) influence of the scanned object geometry. It is found that the amount of image contrast is not the prevalent parameter that controls uncertainty for different materials. Further, the rigid body motion procedure should be preferred over repeated scanning as it provides more conservative measurement uncertainty values. The applied 32 to 8-bit rescaling procedure, beam tuning and detector characteristics hardly affect the measurement uncertainty.

Reproduced from: A. Buljac, L. Helfen, F. Hild and T. F. Morgeneyer. Evaluation of Measurement Uncertainties of DVC applied to Laminography Data. Submitted, 2017.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Introduction

Laminography [10,15] can be considered as a 3D imaging technique complementary to computed tomography and particularly adapted to two-dimensionally extended specimens. Employing synchrotron radiation [11,12] the main advantages offered by laminography are that regions of interest in large plate-like samples can be imaged with high spatial resolutions [12], high energies [20,28] and using different phase-contrast imaging modalities [14,2]. In materials science applications panel-and sheet-like samples can be investigated in situ contrary to tomography, where samples close to matchstick or rod shape [3,27,7] are required. The liberty to employ plate-and sheet-like specimens via synchrotron laminography allows more engineering-relevant loading conditions to be applied during the study of the damage initiation and progression inside materials. For instance, the development of damage has been observed for low stress triaxialities ahead of a notch root during the ductile tearing [31,29,32]. It is however worth noting that the sampling of the 3D Fourier domain of the region of interest is incomplete in laminography [15], which leads to imaging artifacts [46]. These artifacts can be assumed to degrade the displacement and strain resolutions when digital volume correlation (DVC) techniques are applied [30]. Laminography can still be considered as a relatively new technique, especially when considering its use for measuring kinematic fields and strain changes inside material microstructures. It should also be noted that the attainable spatial resolution in synchrotron laminography is higher in directions perpendicular to the rotation axis than in the direction parallel to it. One should therefore also expect higher strain measurement uncertainties along the rotation axis direction.

The aim of the present paper is to assess in a systematic manner the displacement and strain uncertainties measured via DVC when considering synchrotron laminography data. There are various ways of estimating the uncertainties of correlation techniques. When only one acquisition is available, artificial motions i.e., generally uniform translations are applied to create a new volume that is subsequently registered with the reference volume [25,37,4,24,17,26]. This first type of analysis mainly probes the interpolation scheme of the gray levels (GL) to achieve sub-voxel resolutions. The second type of approach also uses only one reference volume but consists of adding noise to create a new volume, which is correlated with its noise-free reference [22,21]. The sensitivity to noise is addressed in this type of procedure. However, to be fully representative the optimal model should incorporate the full covariances associated with radiograph acquisition and 3D reconstruction. Last, the previous two procedures can be applied experimentally [25,23]. This method consists of analyzing two consecutive scans of the sample in a given loading state (it may also be unloaded) without rigid body motion (called bis in the following) between the two acquisitions or with a deliberately introduced one (called rbm). The main advantage is that both sources of error are investigated at the same time and the material of interest. This last method will be adopted in this work.

The standard uncertainties are reported to have levels higher than those observed in standard 2D-DIC techniques when using local or global DVC approaches [25,24,23,26]. For laminography, an additional degradation has been observed [30] for an aluminum al-loy with a very low volume fraction of markers. Since laminography is a very recent technique, there are very few results dealing with DVC calculations [38,30,44,9,32]. In particular, measurement uncertainties have been evaluated [30] for one given microstructure (AA 2139) that was deemed very difficult for DVC calculations since the volume fraction of secondary particles was less than 1 %. This microstructure is also considered hereafter in addition to two other ones of aluminum alloys. A second class of materials presented in this work is nodular graphite cast iron. The microstructure scales and topology significantly differ from aluminum alloys. Hence, X-ray projections, reconstructed volume contrast and the final conditions for DVC registrations are changed.

In the reported analyses a global approach to DVC is chosen in which the displacement field is based on discretizations made of 8-noded elements (i.e., C8-DVC [37]). The correlation technique details are outlined first. Second, the analyzed materials and the way the laminography data are obtained are presented. Next, the measurement uncertainties are evaluated for the chosen microstructures for different isolated cases, namely:

• measurement resolution applied to the repeated scanning (bis) of the object;

• measurement resolution applied to the scanning of the object moved (rigid body motion -rbm) prior the second acquisition;

• influence of 32 bit to 8 bit data conversion (rescaling of the gray value histogramrecad);

• influence of the beam properties (i.e., use of a monochromator or undulator);

• influence of the employed optical camera/detector;

• influence of the microstructre modifications of the scanned material;

• influence of the geometrical properties of the scanned object.

Listed items will be detailed in Section 3.3. Finally, possible relations between the underlying microstructure, sample geometry, experimental setup and uncertainty levels are discussed.

Volumetric Correlation

The DVC technique used herein is an extension of 2D global Digital Image Correlation [5,19]. The reconstructed volume is represented by a discrete scalar matrix of spatial coordinate x of (in this case 8-bit deep) gray levels determined by the microstructure absorption of X-rays. The principle of DVC consists of registering the gray levels f in the reference configuration x and those of the deformed configuration g, which satisfy gray level conservation

f (x) = g(x + u(x)) (1.1)
Understanding, observation and quantification of ductile failure mechanisms via 3D imaging where u is the sought displacement field. However, due to acquisition noise, reconstruction artifacts [46] and the correlation procedure itself [22], ideal matching is not achieved in real examples. Consequently, the solution consists of minimizing the gray level residual

ρ(x) = f (x) -g(x + u(x)) (1.2)
by considering its L2-norm with respect to kinematic unknowns. Since a global approach is used in this work [37], a C 0 continuity condition over the Region Of Interest (ROI) is prescribed to the solution and the global residual to be minimized reads

Φ 2 c = ∑ ROI ρ 2 (x) (1.3)
The displacement field is parameterized by using a kinematic basis consisting of shape functions Ψ p (x) and nodal displacements u p

u(x) = ∑ p u p Ψ p (x) (1.4)
Among a whole range of available fields, finite element shape functions are particularly attractive because of the link they provide between the measurement of the displacement field and numerical models [35,16].

Thus, a weak formulation based on C8 finite elements with trilinear shape functions is chosen [37]. After successive linearizations and corrections the current system to be solved reads

[M]{δu} = {b} (1.5)
where

M i j = ∑ ROI (∇ f • Ψ i )(x)(∇ f • Ψ j )(x) (1.6)
represents the DVC matrix, while vector {b}

b i = ∑ ROI ( f (x) -g(x))(∇ f • Ψ i )(x) (1.7)
contains the current difference between the reference volume f and the corrected deformed volume g = g(x + ũ(x)), ũ being the current estimate of the displacement field. The global residual needs to decrease so that convergence is achieved, i.e., the displacement corrections {δu} become vanishingly small. In this form, the displacement field is regularized in the sense of a continuity requirement that is a priori assumed for the kinematic solution. Moreover, additional mechanical knowledge may be added to help convergence. This type of procedure is referred to as mechanical regularization [44]. It will not be used hereafter.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Analyzed Materials and Configurations

Four different materials will be studied herein:

• AA2139 grade (i.e., Al-Cu-Mg alloy), whose intermetallic volume fraction has been determined to be 0.45 % with an initial void volume fraction of 0.34 % (see Figure 1.1(a)). A feasibility study and analysis has been performed in terms of measurement uncertainties [30]. Rather high levels of uncertainties have been reported and one question is related to the representativeness of these results with other materials such as those studied hereafter. The samples with two different heat treatments are examined, namely, T3 denotes the recrystallized state, solution heat-treated, stretched by 2-4 % and naturally aged to obtain T351 (T3) condition. Conversely, T8 stands for the recrystallized state after an artificial aging treatment. The flat to slant transition has been studied for this alloy in its T3 state [32].

• AA2198 grade (i.e., Al-Cu-Li alloy), which has no initial porosity and 0.3 % volume fraction of metallic inclusions (see Figure 1.1(b)). For this grade, the flat to slant transition associated with ductile fracture has been analyzed thanks to DVC measurements [29,9]. This material has the lowest volume fraction of secondary phases (i.e., particles and voids). Again, two different heat treatments (T3 and T8) have been studied.

• AA6061 grade (i.e., Al-Mg-Si alloy) is a precipitation hardened aluminum alloy, with 0.25 % volume fraction of coarse Mg 2 Si particles, 0.57 % of iron-rich intermetallics and 0.05 % of porosity (see Figure 1.1(c)). Compared to Figure 1.1(a) and (b), less ring artifacts (concentric with the specimen rotation centre) are apparent. Crack initiation and propagation have been investigated for such grade [43] but no DVC has been carried out so far. In this work the T6 heat treatment is inspected.

• Commercial nodular graphite cast iron (EN-GJS-400) represents a second class of materials with different microstructural scales (Figure 1.1(d)). Central hole(s) have been created in the examined plates. In these analyses, an experimental/numerical framework has been developed to validate simulations at the microscopic scale [8,42].

Aluminum alloys

The alloy forming directions will define the measurement frame used herein (even for cast iron). The rolling direction is denoted as L, the transverse direction as T, and the short-transverse direction in the through-thickness as S. The listed microstructures and heat treatment variations are characterized through gray level histograms in Figure 1.2.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging The gray level distributions confirm the poor texture quality where most of the dynamic range (256 gray levels) is just partially used. Conversely, cast iron has a broader gray level distribution due to the presence of two highly contrasted materials (i.e., ferrite and graphite).

Sample geometries and region of interest

The 3D images of aluminum alloys were obtained at beamline ID19 of the European Synchrotron Radiation Facility (Grenoble, France). An inclination angle of the specimen rotation axis of approximately 25 • with respect to the beam normal (θ ≈ 65 • ) was chosen as well as a monochromatic beam of 25 keV X-ray energy. A series of 1500 radiographs is then used to reconstruct 3D volumes by using a filtered-back-projection algorithm [33]. The reconstructed volumes represent the scanned region with a size of 2040 × 2040 × 2040 voxels. The side length of 1 cubic voxel is equal to 0.7 µm. Conducting DVC analyses with the full size reconstructed volumes is computationally too demanding and should not be done anyway since the most outward voxels have only information from a limited range of viewing angles (see Chapter 1: Additional Material). Therefore, only a part of the reconstructed volume called Region of Interest (ROI) is considered herein. For aluminum alloys all examined reconstructed volumes belong to compact tension (CT)-like specimen regions ahead of the notch root (AA2139 and AA2198) or crack tip (AA6061). The experimental setup is illustrated in Figure 1.3. The CT-like specimen has dimensions 60×70×1 mm. It contains either a machined notch of radius 0.17 mm produced by electrical discharge machining or fatigue pre-cracking has been utilized for AA6061.

The first case studied in detail is a CT-like sample made of AA2139 T3. Figure 1.4(a) shows the location of the Region of Interest (ROI) in the reference and rigid body motion (rbm) configuration moved by u T = 22 voxels, u L = 1113 voxels. Its size is 1184 × 512 × 992 voxels. All examined ROIs for aluminum alloys have the same size eliminating the potential ROI size effects on the calculated uncertainties. Also, ROIs are positioned roughly at the same position ≈ 800 µm from the notch root/tip, whenever available original and rbm scans allowed for it. The second case is a CT-like sample made of AA2139 T8. Figure 1.4(b) shows the location of the Region of Interest (ROI) in the reference and rbm configuration moved by u T = 15 voxels, u L = 595 voxels and u S = 3 voxels so that the rotation axis and reconstruction artifacts (seen as rings) are positioned differently in the inspected region.

The third case is a CT-like sample made of AA2198 T3. The fourth case is a CT-like sample AA 2198 T8. Understanding, observation and quantification of ductile failure mechanisms via 3D imaging 0 voxel, u L = 0 voxel and u S = 45 voxels. Since there is a small rigid body motion in the S direction, the rotation axis and reconstruction artifacts (seen as rings) are positioned almost identically in the inspected ROI for both scans.

The fifth case is a CT-like sample made of AA6061 T6. A mechanical shift such that u T = 0, u L = 1330 voxels and u S = 170 voxels has been performed between the two scans as shown in Figure 1.6 where the ROI position is depicted. The rotation axis is positioned differently in the inspected ROI while reconstruction artifacts (seen as rings) are not detected. In AA2139 and AA6061 alloys the voids induce edge-enhancing phase contrast during the scanning procedure resulting in bright void boundaries (Figures 1.4(a), 1.4(b), 1.6 and 1.7(a-b,e)). These so-called cupping artifacts complicate particle separation based on intensity difference. The procedure shown in Figure 1.8 is followed to avoid particles overcounting induced by phase contrast. The voids are first isolated in a binary image. The particles and edge-enhancement artifacts are then separated from the original volume. The two binary images are summed followed by merging the voids and corresponding edge-enhancement artifacts into single objects by Fiji functions (Process/Binary/Open). Hence, after these operations each void + edge-enhancement artifact carries a single contrast having the same status as intermetallic particles. The counting of the objects is preformed by Fiji plugin 3D object counter [6]. The results, in terms of number of contrasted objects per element (isotropic element size = 32 voxels), are listed in Table 1.1. Taking into account the low precision of the proposed method the results show there are no significant deviations (i.e., order of magnitude) for any of the studied alloys. AA6061 T6 alloy has the lowest number of features available per element volume, but this needs to be taken with caution since AA6061 contains larger voids and particles (Figure 1.7(e)) where multiple elements can benefit from a single object, i.e., its boundary where gray level gradient is higher. This is outlined in the second row of Table 1.1 where the number of surface voxels (from all objects) per element volume is displayed. For the commercial nodular graphite cast iron (serial code EN-GJS-400) considered herein two tensile samples with different geometries will be examined. First, a plate with a central hole that localizes stresses in the surrounding ligaments is analyzed (Figure 1.9). One of the ligaments has been scanned during the experiment as shown in Figure 1.9.

The inspected region has a size of 512 × 512 × 512 voxels. The results from this ROI have been used to drive micromechanical simulations [8]. The position of the ROI in the global coordinate system is shown in Figure 1.9 while in the through thickness direction (S) the ROI and sample mid-thickness planes coincide. The 3D images of cast iron were obtained at beamline ID15A of the European Synchrotron Radiation Facility (Grenoble, France) with a 60 keV white beam, using 4000 projections per scan. The minimum specimen to detector distance is 70 mm. The resulting strongly contrasted edges are also expected to contribute to laminography artifacts typical for incomplete sampling. The reconstructed volume has a size of 1600×1600×1600 voxels. The side length of 1 voxel is equal to 1.1 µm.

The second geometry, which is inspired by the work of Weck et al. [45], is shown in Figure 1.10. The scanned region incorporates two holes while the ROI employed in DVC calculations is mainly concentrated in the ligament between the holes. The results from this ROI have been used to outline all the advantages of using measured DVC boundary conditions when running micromechanical numerical simulations [42]. These values are in the range of levels for aluminum alloys (Table 1.1).

Procedures for Uncertainty Assessment

Seven different procedures will be applied to analyze the displacement and strain uncertainties of DVC applied to laminography data:

• repeated scan -bis: two scans are acquired without any motion between the two acquisitions. This procedure is standard when aiming for the resolution of the measurement technique [1].

• rigid body motion -rbm: two scans are acquired when a rigid body motion is applied prior to the second acquisition. This case is deemed difficult [30] because the reconstruction artifacts (e.g., rings) do not follow the motion and may severely bias the displacement measurements with low contrasted images as studied herein (Figure 1.2). To assess the contribution induced by rigid body motions bis and rbm cases are compared.

• 32 to 8 bit conversion -recad: the reconstruction of 3D volumes from radiographs involves at one stage to digitize (i.e., recad) the 32-bit floating point gray levels into 8-bit digitizations given the high level of noise. This transformation is performed manually and may depart from the gray level conservation, which is the underlying principle of DVC (see Equations (1.2) and (1.3)). The influence of different recad details is studied in the case of the AA2139 T3 sample.

• changes in the beam properties: since the data originate from different experimental sessions, the influence of the experimental setup, particulary the monochromator, is also evaluated. Experimental data available for this analysis are from AA2139 T3 and AA2198 T8 alloys observed with a monochromator opposed to AA2139 T8 and AA2198 T3 cases scanned without monochromator.

• detector characteristics: the influence of optical camera/detector is also probed. The scans available for AA2198 T8 alloy are sampled with two different cameras.

• microstructure modification: in the case of AA6061 alloy, the microstructure has been slightly changed (AA6061 modified) so that fewer particles (Mg 2 S) are present. The uncertainty levels for modified and original microstructures will be compared.

• geometry of the scanned object: it has been noticed that the geometry of the specimen (e.g., notch root, holes) in the vicinity of the scanned region may influence measurement uncertainties, especially in rbm cases. This effect will be detailed in the sequel on the AA2198 T3 sample.

The overview of listed procedures and cases in which they will be studied are summarized in Table 1 (i.e., translations and rotations). The difference between a linear interpolation and measured fields is an estimation of the measurement errors [23]. The root mean square error associated with each displacement component is obtained. From that quantity, the displacement uncertainty σ u corresponds to the mean of these three values. Eight-noded cubes are chosen in the finite-element based DVC code (i.e., C8-DVC [37]). The size of each C8 element is the length expressed in voxels of any edge. Different sizes are considered, namely, 16, 24, 32, 48 and 64-voxel elements. However for one case (AA2139 T8) calculations could not be performed with 16-voxel elements because of divergence.

The logarithmic (or Hencky) strains are evaluated by using the mean value of the displacement gradient tensor in each C8 element [18]. The reason for that choice is to account for possible large rotations, even though very small stretches are expected in the following. In the present case, von Mises' effective strain p eq (to assess plastic strains [29,9,32]) is reported. The reason for this choice will become clearer in the sequel since the level of uncertainties will generally be higher than elastic strains in the studied aluminum alloys. Such small uncertainties can only be achieved by resorting to regularized DVC [22,21,44], which is not used herein. Further, it has been checked that the mean strain error was at least one order of magnitude less than the corresponding standard deviation σ ε . Only the latter will be reported hereafter. For bis cases no rigid body motions are carried out between the two acquired scans so that the effect of noise associated with laminography, reconstruction and DVC will be probed. For this analysis experimental data are from AA2139 T8, AA2198 T3 and AA6061 samples. Figure 1.12 shows the standard displacement uncertainty and the corresponding standard effective strain uncertainty as functions of the element size. These results illustrate the classical compromise between spatial resolution and measurement uncertainties [37,22,21], which can be expressed by power law relationships [37,23] where A is a constant expressed in voxels, and B a dimensionless quantity that depends on the way the strains are computed from the measured displacements [23,30]. However, in the case of AA6061 T6, it can be noted that B is not constant, but starts to decrease after reaching certain element sizes resulting in lower strain uncertainty levels compared with AA2198 T3 and AA2139 T8 grades for the element size = 64 voxels. This can be partly explained by fact that DVC benefits from reaching certain element sizes that overcome characteristic voids and intermetallic particles spacings (see Figure 1.6).

σ u = A α+1 α and σ ε = B σ u (1.8)

Effect of rigid body motionrbm case

For rbm cases rigid body motions are applied between the two acquired scans. Experimental data available for this analysis are from AA2139 T3, AA2139 T8, AA2198 T3, AA2198 T8 and AA6061 samples (Figure 1 reconstruction axis has moved with respect to microstructure features has a clear effect on the measurement uncertainties (when compared with bis cases), especially for the cases investigated herein, which are deemed very difficult since the volume fraction of "markers" is very low (Table 1.1). However, the overall trend for rbm cases remains the same as in bis cases in terms of dependence with the element size. Along with previous statement, AA2198 T8 showed the lowest uncertainties since the position of the reconstruction axis and consequently artifacts remained almost the same. Hence, this rbm case is close to bis cases reported before.

Comparison between repeated scanning (bis) and rigid body motion (rbm) cases

In Figure 1.14 displacement and strain resolutions are explicitly compared for bis and rbm cases for two grades of aluminum alloys. After rigid body motions the reconstruction artifacts have changed their relative position with respect to the microstructure features what is reflected into higher uncertainty levels for the rbm cases. This effect can be clearly seen on the mean displacement uncertainties in Figure 1.14(a). It can be noted that the microstructure with no voids and lower contrast (i.e., unfavorable gray level gradient for DVC registration) is more sensitive to rigid body motions. This effect is less pronounced for strain uncertainties (Figure 1.14(b)).

The last results show that to fully evaluate measurement uncertainties it is better to move the sample before the second acquisition so that the rotation axis no longer coincides with that of the first reconstructed volume. Hence, in rbm cases all the detrimental effects are accounted for and the reported uncertainty levels represent the stable limit above which mechanical signal is distinguished from the noise even for large displacement amplitudes.

Effect of 32 to 8 bit conversionrecad case

When dealing with reconstructed volumes in tomography or laminography, postprocessing issues arise because of the data size. That is why the reconstruction of 3D volumes from radiographs involves shift from the 32-bit floating point gray levels into 8-bit digitization. This so-called recad procedure is manually operated. In the following two different recad parameters are utilized, thereby resulting in two different 8-bit reconstructed volumes obtained from the same initial radiographs. They will be distinguished by the suffix A or B in their names. Three combinations are analyzed to measure displacement fields, namely, reference scan A wrt. rbm scan A (denoted as "ref. scan A vs. rbm scan A" in Figure 1 Despite the difference of dynamic range the level of strain uncertainty remains virtually identical for the three combinations for any element size (Figure 1.15(b)). However, for the displacement uncertainty there is gap between "ref. scan B vs. rbm scan B" and the others revealing higher noise levels. This is confirmed by fact that in the "ref. scan B vs. rbm scan B" case, the calculation diverged for an element size = 16 voxels.

Effect of changes in beam properties

The monochromator's main function is to extract from the incoming multichromatic Xray beam a monochromatic beam of adjustable wavelength. The experimental data for AA2139 T3 and AA2198 T8 were obtained with a multilayer monochromator opposed to AA2139 T8 and AA2198 T3 scanned without monochromator. The multilayer monochromator was needed in order to limit the dose rate and high X-ray energies with the undulator source employed, which has rather low fundamental energies and multiple higher harmonics. The cross section of the beam diffracted by the multilayer shows a characteristic modulated intensity pattern [34] stemming from long-range substrate and coating imperfections. These intensity modulations are prone to lead to ring artifacts after laminographic reconstruction. The AA2139 T8 and AA2198 T3 data were scanned without monochromator with radiation from a short-period undulator where higher harmonics are suppressed and thus no monochromator is needed anymore.

In Figure 1.16 both mean displacement and effective strain resolutions for the available data are shown. The data obtained with the monochromator are depicted with dashed lines and suffix "M" in the legend. There is no clear evidence of an influence of the monochromator for AA2198 alloy. As mentioned, AA2198 T8 rbm case is the result of small amounts of rigid body motion, hence these data can be understood as a bis case. This is illustrated in Figure 1.16(a) when compared to 2198 T3 bis data where a good agreement is found except for larger element sizes. Hence, the monochromator presence may induce some additional uncertainty with longer characteristic wavelengths (i.e., rings), which results in a saturation of the uncertainty levels as shown in Figure 1.16(a) (see gap between "AA2198 T3 bis" and "AA2198 T8 rbm M" for large element sizes).

The fact that the "AA2139 T3 rbm M" case significantly deviates from "AA 2139 T8 rbm" should not be directly assigned to the monochromator influence. It is rather a consequence of different levels of applied rigid body motions and more importantly, the volume of the notch present in the reconstructed volume and ROI position with respect to it, which is to be detailed in the sequel. Hence, the solution to probe this effect would be the existence of "AA2139 T8 bis M" data to compare it with its non-monochromator correspondent "AA2139 T8 bis" in conditions where there is no influence of rigid body motions.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging The experimental data for AA2198 T8 alloy are obtained with two different types of detectors, namely the ESRF in house-designed FReLoN camera with 20x (i.e., 10x objective plus 2x eye-piece) optical magnification and the PCO.edge camera (PCO GmbH, Kelheim Germany) with 10x optical magnification (but both with similar scintillators converting the X-rays to visible light).

The FReLoN camera uses a sensor based on the charge-coupled device (CCD) principle while the PCO.edge camera uses a scientific CMOS sensor. Both convertors have a dynamic range of about 14 bits but the full-well capacity of the PCO.edge is ≈ 5x lower so that less X-ray photons are needed to saturate the detector. Consequently, also less photon statistics may be expected, resulting in noisier reconstructed images. To compensate for this effect to some extent, the number of X-ray projection images is increased from 2000 to 3600 images. Even with more images acquired the scan time still takes approximately only a third of the scan time with the CCD camera.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

The resolution of FReLoN camera is 0.7 µm while PCO.edge camera yields 0.65 µm. However, for the PCO.edge camera binning of 2 × 2 × 2 voxels is performed in data postprocessing resulting in a final isometric voxel size of 1.3 µm.

The PCO.edge camera shows slightly better results for the displacement resolution (see Figure 1.17(a)) but this is also partly due to binning and final larger physical size of the voxel. The effective strain resolution reveals similar values for both cases (Figure 1.17(b)), which underlines the same quality for both cameras in terms of final DVC measurement resolutions.

Element size (voxels) (c) gray level histograms of the two grades.

Figure 1.18(c) shows the gray level histograms for the two microstructures. The extinction of white particles results in sharper peak in the histogram, thereby an even lower dynamic range. The standard deviations for gray level distributions are 10.5 for AA6061 and 7.9 for AA6061 modif. In Figure 1.19 the standard displacement and strain resolutions are shown for the two states of AA6061. For the reference AA6061 grade bis and rbm results are shown while in AA6061 modified the bis case is not available. Calculations Understanding, observation and quantification of ductile failure mechanisms via 3D imaging with the element size equal to 24 and 16 voxels are not reported with AA6061 modified since they diverged. This is probably due to the fact the element size is reaching the correlation length of AA6061 modified volumes, which is higher than in the case of AA6061 due to reduction of white particles. For the larger element sizes the results are slightly Element size (voxel) better in the case of the modified microstructure. This indicates that, at least for the rbm case, after reaching certain element size the microstructure with less white intermetallic particles and narrower peak in gray level histogram yields lower resolution levels. However, one should be careful in interpreting these results since ROIs could not be positioned identically with respect to the notch tip and different amounts of RBMs were applied in the two cases. For the basic AA6061 grade a mechanical shift corresponding to u T = 0, u L = 1330 and u S = 170 voxels has been performed between the two scans as shown in Figure 1.6 where the ROI position is depicted while in the AA6061 modified a shift of u T = 5, u L = 441 and u S = 2 voxels is performed and the ROI is positioned closer to the crack tip (not shown herein).

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Let us note that uncertainties for ROIs a and c differ regardless of their similar relative position with respect to the center of rotation. This result just confirms that the impact of the used reconstruction method per se is not a dominant factor in terms of measurement uncertainty.

ROIs 1, 2, 3 are distributed along the L-axis and have their common midplane in the S-direction which is located approximately at the center of the specimen thickness (Figure 1.20(b)). The bis cases show good coincidence for all observed ROIs confirming the previously reported insensitivity. Moving from ROI 1 to 3, i.e., in the L-direction, gradually increases the measurement uncertainties for the rbm cases (Figure 1.22). Here, the change of the standard displacement uncertainty is more pronounced than by moving the ROIs along the T-axis. Despite the fact that the motions are such that the ROIs Understanding, observation and quantification of ductile failure mechanisms via 3D imaging get closer to the rotation axis and then depart again, the uncertainty constantly increases. Hence, there is minor influence of the information density gradient coming from the reconstructed volume, where accessible amount of information decreases along the radial direction by nature of the reconstruction process [15]. It can be concluded that the data acquisition is dense enough and the number of radiographs is sufficient.

Element size (voxel) One detail, which at first glance appears to be common for L-direction array of ROIs is the constantly approaching notch root in the scan before performing rigid body motions. The notch (or hole) vicinity degrades the quality of the reconstructed volume. The notch presence or absence in the monitored cases changes the spatial distribution of the attenuation coefficient µ. The 3D spread function h, whose inverse convolved with a compound image provides a spatial distribution µ, only depends on the experimental setup geometry [15]. The presence of a notch at different distances from the sample center of rotation before and after rigid body motions may cause anomalies, which manifest themselves in the form of smeared inclusions that can be observed in the orthogonal sections of the reconstructed volume displayed in Figure 1.24. The LS sections are obtained through his-togram equalization. In general, the left part of the LS sections (i.e., closer to the notch) displays more smeared inclusions both in the scans before and after rigid body motion. The characteristic void shadows are present for which it can be assumed to affect the measurement results because they are less present in the scans after rigid body motions. In a standard experimental setup and procedure the level of rigid body motion is generally lower, so that the void shadow is always present, which eliminates its influence on measured results. Therefore, it can be expected that the measurement uncertainty of the considered ROI for the actual loading steps will be placed between the bis and rbm results from this analysis.

The change of the shadow positions before and after rigid body motions may affect correlation measurements. It is expected that this effect will increase the standard uncertainties. This phenomenon is illustrated in Figure 1.25 for the displacement field along the L-axis and the corresponding effective strains (Figure 1.26). The spatial distribution of the measurand shows that the zone with highest differences for the two considered scans is not symmetric, but mainly expressed as an inclined band crossing the material. Its corresponding angle with the sample surface may coincide to that between the X-ray beam and the sample surface, which supports the aforementioned idea of a harmful effect caused by the void shadow.

Contrary to the previous rbm case, the spatial distribution for the bis case does not present any structured shape and the overall uncertainties are lower (Figure 1.27). This is expected due to the fact that the notch is present in the first and second scans, so the particle shadows do not change.

The same procedure is also followed for AA2139 T8. These scans were obtained from the same experimental setup as in the AA2198 T3 case (i.e., without monochro-Understanding, observation and quantification of ductile failure mechanisms via 3D imaging ). The difference is probably due to the presence of voids in AA2139 (i.e., less attenuation) and it ensures that the contrast is equally distributed in both scans, which yields lower uncertainty levels. This effect of low sensitivity to rigid body motion was already observed in AA6061 modified case. As expected, displacement field u L and effective strain fields for ROI 2 reveal that there is no concentrated zone with higher uncertainty levels as in the AA2198 T3 case.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging For the other two aluminum alloy grades with rbm cases the effective strain fields are shown in Figure 1.30. For AA2139 T3, bis reference is not known but rbm case produces rather high strain levels. Consequently, it is compared with AA6061 T6, where a clear gap between bis and rbm cases was found (Figure 1.19). In Figure 1.30 there is no clear pattern as that found in Figure 1.26 even though local zones with higher measured strains are distinguished. Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Cast iron

For the central hole experiment (Figure 1.9), three independent repeated scanning (i.e., bis) cases are available, namely, "0 mm" in the undeformed state and two more in deformed states, namely, "0.579 mm" and "0.744 mm" for the applied displacement. Figure 1.31 shows the standard displacement and strain uncertainty levels for the analyzed ROI. An increase of the spatial resolution (i.e., proportional to the element size [21]) is followed by a decrease of the displacement/strain uncertainty since more voxels per nodal displacement are used. The observed differences are due to different initial quality of the underlying scans that originates from focal point shift that occurred during the experiment due to thermal drift (analyzing raw data reveals the best quality scans are indeed for the "0.744 mm" case). Further, once damage sets in, higher gray level gradient is more favorable for DVC registration.

Element size (voxel) The results for the "two-hole sample" (Figure 1.10) consist of bis and rbm scans for the undeformed state. Additionally, to be able to keep large ROI sizes, the original reconstructed volumes are a priori coarsened, each 8 neighboring voxels are averaged to form one supervoxel (i.e., binning of 2 × 2 × 2). Figure 1.32 shows the standard resolution levels for different element sizes expressed in supervoxels. The strain uncertainty levels are slightly lower than those reported in Figure 1.31 for bis cases. The gap between rbm and bis uncertainty levels is here more pronounced for displacement than strain levels. This is probably due to the fact that measured rotations are removed differently: for the displacements they are extracted as mean values calculated on the ROI scale, while in the strain resolution procedure the Hencky strains in its definition exclude rotations more locally at the element level. Gathered together most of the previously reported results are shown in Figure 1.33. Repeated scans (bis) cases are shown with solid lines while rigid body motion (rbm) cases are depicted with dashed lines. AA2198 T8 rbm has overall the best result. Since the amount of rigid body in this case is small and mainly in the S-direction (i.e., artificial rings do not move), this case can be considered as equivalent to a bis case. This is why AA2198 T3 bis and AA2198 T8 rbm results almost coincide. When larger rigid body motions are applied to the AA2198 microstructure, significant differences occur as can be seen for AA2198 T3 rbm. Then, the AA2139 T8 bis curve is an important one since it confirms that the AA2198 microstructure per se yields the lowest measurement uncertainties. Both AA2139 T8 bis and AA2198 T3 bis data come from the same experimental set-up and no rigid body motions occurred between the acquisitions, hence isolated effect of microstructure gray level gradient goodness for registration purposes should be observed. However, AA2139 T8 rbm data overlap with its bis case, which shows an additional criterion when talking about DVC uncertainty, namely, its sensitivity to rigid body motions associated with the sample geometrical features (e.g., notch, drilled holes, crack).

The AA2139 microstructure with initial voids is less affected by rigid body motions since all the other conditions between AA2139 T8 rbm and AA2198 T3 rbm including ROI size, its position and amount of rigid body motion are almost identical (see Figures 1.4(b) and 1.5(a)). According to the presented results it is possible that a microstructure with just intermetallic particles (i.e., without voids, less contrast) is more sensitive to the detrimental effect of rigid body motions as detailed in Section 4.1.8.

The uncertainty levels for cast iron represented by the worst (0 mm) and best (0.744 mm) bis cases from Figure 1.31 are denoted with yellow markers in Figure 1.33. Cast iron strain resolution virtually coincides with bis resolution levels for aluminum alloys. When compared to tomography resolutions [23] of the same type of material but not the same physical voxel size (3.5 µm), it is observed that laminography is more challenging than tomography since there is at least a factor 3 difference between the two uncertainty levels.

Let us note that in some bis/rbm cases presented above the trends between displacement and strain resolutions differ significantly, e.g., for AA2198 T3 there is a significant gap between bis and rbm curves for displacement uncertainties while in the strain resolution graph the values almost coincide. This is probably due to the fact that measured rotations in the displacement uncertainty procedure are extracted globally on the ROI scale, while in the strain resolution procedure the Hencky strains exclude rotations more locally, i.e., at the C8 element level.

The first output of a correlation code to be checked is the residual field ρ(x) and its RMS ρ over the whole ROI. In Ref. [19] it has been shown that when noise is artificially added to images there is an almost 1 to 1 correspondence between the normalized RMS residual (by dynamic range ∆ f ) and the dimensionless noise level √ 2σ. In other words, in such a case the correlation residual is virtually identical to the noise level meaning the registration was totally successful. In all bis and rbm cases reported herein the overall noise level (acquistion+reconstruction) is unknown but it still can be estimated a priori by the RMS value of the residual at convergence. In Figure 1.34 the best and worst cases are extracted and normalized by the overall noise level calculated from the corresponding residual fields. From Ref. [21] it is expected that nodal displacement uncertainties divided by the noise level should follow a power law of the element size with exponent -3/2, and of exponent -5/2 for standard strain uncertainties for white Gaussian noise. The results in Figure 1.34 show that this dependence is not captured, which is due to the fact that the reconstruction step induces spatial correlations (the ring artifacts being one ubiquitous manifestation).

Element size (voxel) It has been shown [5,36] that the resolution of image correlation technique is proportional to the inverse of the DVC matrix [M] (see Equation (1.6)) multiplied by the variance of noise and the physical size of the pixel/voxel. This means that depending on the gradient of the underlying texture the resolution of the correlation technique is not necessarily uniform over the ROI. Figure 1.35 shows the 100 largest eigenvalues of the [M] -1 matrix for the studied microstructures and element size =32 voxels. This graph shows more isolated influence of microstructure quality on the uncertainty levels. Hence, the lines are rather grouped together according to corresponding microstructure with the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging exception of AA2139 T3 recad B where the different recad parameters determined higher values (see section 4.1.4).

However, as reported in Figure 1.15, initially inferior recad parameters do not reflect in significant gap in the final nodal uncertainty values since the noise contribution (acquisition + reconstruction) overcomes the single texture quality. AA2198 T3/T8 have higher eigenvalues than AA 2139 T3/T8 although in the final resolution reports this relationship is inverse with smaller gap meaning the microstructure with more voids could be more prone to capturing additional artifacts. It is also interesting to notice that there is no particular difference between AA2139 T3 and T8 eigenvalues as it could be concluded from final resolution values (see Figure 1.33). This again outlines crucial additional noise influence and confirms the studied materials per se are all equally suitable for laminography and DVC studies.

Conclusion

Laminography is still a recent imaging technique [11,13]. The use of reconstructed volumes for kinematic measurements has only started a few years ago [38,30,44,9,32]. Consequently, there is a need for metrological assessment of the measured displacement fields and the corresponding strain fields. The present paper aims at discussing ways of evaluating measurement uncertainties in this context. The procedures introduced herein are applied to aluminum alloys leading to low contrasted volumes when imaged by X-ray sources and cast iron with low contrast ferritic zones when imaged at high resolutions.

In the studied Al alloys, for any location of the region of interest, the uncertainty level is uniform for repeated scanning (bis) cases. However, when rigid body motions are applied between two acquisitions the ROI position close to the notch degrades the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging uncertainty level for AA2198 and AA6061 aluminum grades. The amount of initial voids or secondary particles is not necessarily sufficient to reach the lowest uncertainty level. In the present case, the AA2198 grade has the lowest volume fraction of the three aluminum alloys but also the lowest uncertainty. This result calls for caution and it is advisable to acquire at least two scans in the reference configuration, if possible by slightly moving the setup between the two acquisitions, to evaluate the practical displacement and strain resolutions of the studied material.

By comparing the uncertainty levels measured with and without rigid body motions, it appears that the uncertainty mainly comes from the laminography technique and its associated artifacts [15,46]. The uncertainty levels are equal or higher with rigid body motions since some of the artifacts are not attached to the microstructure (like, e.g., phasecontrast edge enhancement) but intrinsic of the acquisition set-up (e.g., ring artifacts).

Last, let us note that the procedures discussed herein and applied to reconstructed volumes obtained by synchrotron laminography can also be applied to synchrotron or lab tomography. All these techniques require reconstruction steps from the raw data (i.e., radiographs) to the 3D volumes that induce additional artifacts depending on the equipment itself but also on the reconstruction algorithms.

Chapter 1: Additional Material

The sampling of the 3D Fourier domain of the region of interest is incomplete in laminography. Consequently the most outward voxels have only information from a limited range of viewing angles. This is illustrated in Figure 1.36 for three different sections along the sample thickness. The yellow line indicates the boundary of the region where the sampling is incomplete, which results in blurred reconstructed zones. Section B-B has artifacts due to "local" laminography, i.e., the projections are truncated with respect to the projection of the reconstruction volume, and this induces a directionality in the artifacts. In Section C-C let one note what is commonly called "reconstruction circle" in computed tomography. Outside these areas, the artifacts have a strong radial direction due to the missing information from tangential projection angles. For Section A-A, only the outer most voxels are concerned with such effects.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging In this chapter 3D strain and damage analyses are performed in the immediate vicinity of the notch root of a flat CT-like specimen made of aluminum alloy. Experimental data, partially exploited by [28], were obtained by using synchrotron laminography and the 3D reconstructed volumes are subsequently analyzed via Digital Volume Correlation. These data enable for in situ assessments of strain fields and ductile damage in the zone where the stress triaxiality evolves from elevated to lower levels, which is accompanied by flat-to-slant crack transition. The measured strain field patterns in this area are analyzed herein in a systematic manner by studying the incremental strain activity during several loading steps. It is shown that from the very beginning of the loading history multiple slant strained bands appear in front of the notch root while the corresponding damage growth sets in at later loading stages and higher strains. The activity of the different strained bands at the notch root is alternating between different locations over the loading history. However, the band leading to final rupture is always active. The region where slant fracture occurs is identified to be in plane strain condition with respect to the crack propagation direction.

Introduction

One of the objectives of engineering design processes is to enhance the prediction of failure and to improve the fracture resistance of engineering materials. This is achieved through extensive studies on the material behavior and the implementation of new production technologies. As a consequence, the majority of structural alloys nowadays fails by ductile fracture characterized by three principal stages, namely, nucleation, growth and coalescence of voids. Unlike classical global fracture mechanics, micromechanics-based approaches describe these three stages with more physical insight. Still, a general model

does not yet exist to predict failure. While at higher stress triaxiality levels available material and numerical models keep the pace at increasing accuracy in predictions of ductile failure mode and critical strains [11,42,23], lower stress triaxiality levels still present serious challenges [10,36]. Ductile damage micromechanisms responsible for crack initiation and propagation in such conditions are still not fully understood. In particular, the link between stress triaxiality, Lode parameter and strain to fracture remains an open question. Even a clear definition on how to measure strain to fracture is not established at present [36].

The example that summarizes the main difficulties experienced while modeling and simulating ductile fracture is the flat-to-slant transition. The ductile crack propagation path in sheet-like samples usually consists of a flat part (i.e., normal to the loading direction) with a characteristic triangular shape due to tunneling, which is followed by a slant surface [25,6,2,18,30,4]. Such sudden change of crack growth from mode I to a combination of modes I and III may reduce the material toughness compared with the same samples broken under pure mode I loading. During ductile tearing of alloy sheets the crack can also undergo so-called flip-flopping [9]. A slant crack has numerically been shown to dissipate less energy during the tearing process than a flat crack [4]. The propagating crack is in plane strain conditions with respect to the crack propagation direction [5]. Although considerable efforts have been made, numerical simulations of the flat-to-slant transition by using continuum mechanics without prescribing the crack path remain very challenging. The underlying physics of this failure process remain unclear.

Such strain localization phenomena in metals have been studied theoretically by [35], who found that localization is favored by material behaviors that form a sharp vertex on the yield surface, by material damage and by heating due to plastic deformation, leading to thermal softening at high strain rates. Recently, the influence of strain rate on localization has also been identified [33].

To experimentally assess damage processes in the material bulk, tomography techniques have successfully been used in the past to study failure in structural engineering materials such as Al-alloys [26], steels [24,19] and polymers even achieving nanometer resolutions [31]. However, geometrical conditions for sheet material tests cannot readily be met via tomography as only axisymmetric or stick-shape-like specimens can be tested in situ. Only stopped cracks could be cut-out and assessed in 3D by tomography at micrometer resolution. This limitation is overcome by resorting to synchrotron radiation computed laminography [13,12,1] that provides the opportunity to assess in situ damage Understanding, observation and quantification of ductile failure mechanisms via 3D imaging processes in laterally extended sheet-like specimens using physical boundary conditions of engineering relevance and associated naturally developing plastic zone sizes of the order of several millimeters [30,14,39]. As such, it complements tomographic analyses. For example, it has been shown that pre-existing pores elongate and rotate toward the slant crack during flat to slant crack transition [43].

Experimentally, [28] reported early slant strained bands 1 mm ahead of the notch root of a CT-like specimen without any significant damage development in a recrystallized Al-Cu-Li alloy by using synchrotron laminography [13,15,14] and Digital Volume Correlation with a global formulation [38]. Despite the strong competition with other classes of structural materials such as polymer based composites, new generations of metallic alloys (e.g., Al-Cu-Li) are used for future transport applications as they display great potential for application of new joining techniques [21,20].

The present paper considers the same experimental acquisitions [28] for a different Region of Interest (ROI), which is very close to the notch root, and also encompassing part of the ROI analyzed in the previous study. Such an approach should enable for the observation of strain and damage interactions in the flat-to-slant transition region at micrometer resolution. The underlying question arises: does damage onset or does strain localization occur first when getting closer to the notch root? What is the difference in strain fields between the flat and the slant crack? The paper, which aims to answer these questions, is structured as follows. The imaging and mechanical experimental setup, and material properties are first introduced. The basic principles of DVC together with strain uncertainty assessments and definition of the ROI are outlined next. The results are finally discussed. The focus is on incremental strain activity in the zone ahead of the notch root.

Material and experimental setup

The material of the CT-like specimen studied herein (i.e., AA 2198) is in its recrystallized state and after an artificial aging treatment (T8). It is produced by Constellium C-Tech and represents the latest generation of Al-Cu-Li alloys. Due to its low density, enhanced mechanical properties and corrosion resistance AA 2198 is widely used in high demanding branches such as the commercial aircraft industry. The composition (in wt% ) of the aluminum alloy is 2.9-3.5 Cu, 0.8-1.1 Li, 0.25-0.8 Mg, 0.1-0.5 Ag, 0.04-0.18 Zr, balance Al. The yield strength is ∼ 440 MPa and the ultimate tensile strength is ∼ 500 MPa. Therefore the heat treatment endows the material a low work hardening. The intermetallic particle volume fraction is found to be ∼ 0.3 -0.4 % while the initial porosity is very low (i.e., < 0.03 vol%). The very low volume fraction of secondary phases makes this material very challenging for Digital Volume Correlation (DVC) since it relies on natural markers.

The material processing directions are the rolling direction (L), the transverse direction (T) and the short-transverse direction in the through thickness (S). In the present experiment the loading is applied in the T-direction and the L-direction corresponds to that of crack propagation. More details on the material properties can be found in [8,28].

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

The studied sample is imaged by synchrotron radiation computed laminography, which is a 3D non-destructive experimental technique used for scanning laterally extended sheet-like objects [16,27,7,37]. Unlike synchrotron radiation computed tomography limited to stick-like samples, laminography enables for conducting a whole range of tests using more realistic (i.e., engineering-relevant) boundary conditions. The CT-like specimen, with dimensions 1 ×60 ×70 mm, is prepared by using EDM to produce a notch root of radius r = 0.17 mm. The remaining ligament is 24 mm wide. The testing device (Figure 2.1(a)) applies the load by controlling the notch opening displacement (NOD) while the specimen is placed in a specially prepared frame (not shown here) including two anti-buckling plates that avoid out-of-plane motions [30,29]. sponds to the analysis reported in [28] and ROI 2 is analyzed in this work

After applying each loading step (or NOD step), a scan is acquired while the sample is rotated about the laminographic axis (i.e., parallel to the specimen thickness direction S). This axis is inclined with respect to the X-ray beam direction by an angle θ ≈ 65 • (Figure 2.1(a)). A series of 1500 radiographs are then used to reconstruct 3D volumes by using a filtered-backprojection algorithm [32]. The reconstructed volumes represent the scanned region with a size of 2040 × 2040 × 2040 voxels. The physical size (length) of 1

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging cubic voxel is equal to 0.7 µm. Conducting DVC analyses with the full size reconstructed volumes is computationally too demanding, therefore only a part of the reconstructed volume called Region of Interest is considered herein (i.e., ROI 2 shown in Figure 2.1(b)).

Digital Volume Correlation

Principle

DVC used herein is an extension of 2D global Digital Image Correlation [3,17]. Each voxel inside the reconstructed volume contains the gray level information determined by the material ability to absorb X-rays. The reconstructed volume is represented by a discrete scalar field of the spatial coordinate x. In the present case, the image contrast is mainly due to heavy intermetallic particles of micrometer size. The principle of DVC consists of matching the gray levels f in the reference configuration x and those of the deformed volume g

f (x) = g[x + u(x)] (2.1)
where u is the displacement field with respect to the reference volume. However, in real problems the strict conservation of gray level is not satisfied, especially in CL where deviations appear not just due to acquisition noise but also due to reconstruction artifacts because of missing information (angles) [44]. Consequently, the solution consists of minimizing the gray level residual ρ(x) = f (x)g[x + u(x)] by considering its L2-norm with respect to kinematic unknowns associated with the parameterization of the displacement field. Since a global approach is used in this work, the whole region of interest (ROI) is considered, the global residual Φ c

Φ 2 c = ∑ ROI ρ 2 (x) (2.2)
is minimized with respect to the unknown degrees of freedom u p when the displacement is written as

u(x) = ∑ p u p Ψ p (x) (2.3)
where Ψ p (x) are the chosen displacement fields for the parameterization of u(x). Among a whole range of available fields, finite element shape functions are particularly attractive because of the link they provide between the measurement of the displacement field and numerical models. Thus, a weak formulation based on C8 finite elements with trilinear shape functions is chosen [38].

Strain resolution

The DVC resolution is evaluated by correlating two scans of the unloaded sample with a small rigid motion applied between the acquisitions. Due to the noise contribution and reconstruction artifacts, these two volumes are not identical. Therefore, the measured

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Digital Volume Correlation

displacement field accounts for the cumulated effects of laminography and DVC on the measurement uncertainty. From the mean gradient of the displacement field over each C8 element the Hencky strain tensor and subsequently its second invariant will be used in the sequel as Von Mises' effective strain

ε eq = 2 3 d : d (2.4)
where d denotes the deviator of the strain tensor .

The uncertainty values are evaluated by the standard deviation of the measured displacement and calculated strain fields. Figure 8.7 shows the standard resolution levels for different C8 element sizes expressed in voxels. The results are virtually identical for both studied ROIs. Decreasing the element size is followed by an increase of the strain uncertainty. Keeping the same spatial resolution, the strain uncertainty can be further decreased by employing additional mechanical regularization procedures acting like low-pass displacement filters [22,40]. This type of regularization is not used herein. The element size used in this work is = 16 voxels (length) for all three directions, which yields a standard effective strain resolution of 0.3 %. This value represents the limit below which the estimated strain levels are not trustworthy.

ROI positions

The results from the previously studied ROI 1 (∼ 800 µm from the notch root, see Fig- ure 2.1(b)) display from the first loading step on the formation of slant bands with strain levels at least one order of magnitude greater than the strain resolution [28]. Furthermore,

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging it is shown that at the same loading steps damage did not nucleate. Only at the very ultimate loading steps damage develops. The final crack path follows the strained band. The slant band and the corresponding crack path lead to the following question: what happens closer to the notch root? For this purpose, a second ROI denoted as ROI 2 positioned in the vicinity of the notch root is extracted from the reconstructed volume (Figure 2.1(b)). The two ROIs overlap thereby providing a common ground in both ROIs comparing the results. ROI 2 with dimensions 960 × 1024 × 512 voxels is now extended in the L-direction reaching the notch root. Consequently the size in the S-dimension had to be decreased to be able to run the DVC measurements. Since close to the notch root the crack propagated earlier, all the calculations obtained for ROI 1 could not be performed for ROI 2. Therefore, only the converged solutions available for both ROIs will be discussed in the following. After conducting calculations and analyzing them, reliable results (i.e., before the crack has propagated through ROI 2) for both ROIs are obtained for incremental analyses of loading steps between states (0)-( 1), ( 1)-( 2) and ( 2)-(3). 2). The color bar is limited to a maximum value of 0.08 to emphasize more the levels in the overlapping zone. The strain field is evaluated voxel-wise by using the shape functions associated with the C8 discretization. The effective strain difference is evaluated inside the overlapping region and the corresponding standard deviations are respectively equal to 0.0046 for load step (0)-(1), 0.0044 for load step (1)-(2) and 0.0040 for load step(2)-( 3)). Since the standard strain resolution is equal to 0.003, and the two strain evaluations are obtained for two different ROIs, it is expected that the two quantities are independent. Consequently, the standard deviation of the strain difference is expected to be equal to √ 2 × 0.003 = 0.0042, which is very close to the observed levels (mean value: 0.0043). This result fully validates the fact that the two results obtained on ROI 1 and ROI 2 can be merged into a single field.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Results

Incremental loading steps are especially interesting in terms of plastic activity. There are two different ways of displaying these results. First the fields are shown on the reference configuration of each analysis. Second, all results can be expressed in a Lagrangian way by reprojecting back all the measured incremental fields. However, some data are not available since the ROI 2 in the deformed configuration is significantly larger than ROI 2 in the reference configuration. In the sequel, the first approach will be used since it does not induce any loss of information, but special care has to be exercised to identify common features to make the comparison objective.

All the results reported hereafter have required the DVC analyses to converge, namely, the root mean square (RMS) displacement difference between two iterations is less than 10 -4 voxel. An additional check is related to the residuals ρ.

Figure 2.5 shows one L-section of ROI 2 in the reference configuration f of the three analyses. The incremental effective strain fields display very complex patterns that will be commented in the following. This explains why special care was exercised to make sure that these results are trustworthy. Consequently, sections of the deformed volumes corrected by the measured displacement g[x + u(x)] and the corresponding residual maps ρ are also shown. These last maps do not display areas with high residuals. This is an additional confirmation that, even though the registration is very difficult, the strain fields are deemed trustworthy. Sections normal to L and S-axes (referred to as L and S-sections in the sequel) of incremental effective strain fields are respectively shown in Figures 2.6 and 2.7. L-sections will be displayed with shifts of 200 voxels starting from the location close to the notch root.

For S-sections, the second column of displayed results corresponds to the sample midplane while the remaining two sections are located 50 voxels from the ROI 2 faces. The magnitude of the incremental strain fields close to the notch root are significantly higher than in the remainder of the ROI. To capture interesting features, the color bar scale is limited to a maximum level of 0.1 for all displayed sections. By moving away from the notch root, the incremental strain field that is initially concentrated in a horizontal zone evolves to a slant position by forming several inclined bands. In all cases the difference between strain levels inside and outside the bands are more than one order of magnitude higher than the reported strain uncertainties. Consequently, the reported results are Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Interestingly, ∼250 µm away from the notch root slant bands are observed (Figure 2.6) while the initial notch radius is ∼170 µm. One should note that this is the case already from the first loading step (i.e., NOD = 50 µm) where incremental strain field heterogeneities are observed in front of the notch root (Figure 2.6(a)).

The slant strained pattern is composed of multiple bands, which is different from that reported farther away from the notch [28]. In the region closer to the notch, series of "X" type bands form in the plane normal to the crack propagation direction L. From the loading step (0)-(1) on, one of the slant bands is more active and pronounced than the other ones. For the load steps (1)-(2) (Figure 2.6(b)) and ( 2)-(3) (Figure 2.6(c)) the same observations are reported, namely, close to the notch the incremental strain field morphology is more complex, while farther away from the notch one X-shaped band remains with decreased strain levels. The strain fields corresponding to the outer part of the ROI confirm that the strained bands are inclined in the L-section as they are not in the same location through the thickness. This pattern differs from the one that would be obtained via numerical simulations by using von Mises plasticity and isotropic hardening where typical plastic zone shapes around the notch can be seen [28]. First, one can hardly find such regular shapes with closed contours. Second, unlike numerical solutions, strong incremental strain field dependence with respect to the S axis is observed, therefore plane strain conditions in the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging 1), ( 1)-( 2) and ( 2)-(

Figure 2.8 shows effective strain profiles along the line defined by the intersection of the L-section located ≈ 285 µm from the notch root and the middle thickness plane of the specimen. After the onset of strain localization the other zones of the specimen that are not affected still remain active (i.e., the effective strain level still increases). Each additional loading step induces an accumulation of plastic strain but with significantly lower magnitudes.

In a previous work [34] it has been shown that strain localization is easier under plane strain conditions. It is necessary to point out that plane strain conditions here refer to the T-S plane since strains in the crack propagation direction are close to zero. By means of laminography and DVC this can be checked. Figure 2.9 shows the incremental strain maps for the different strain components in the L-T plane for loading step (0)- (1). The reported normal strain components are calculated as nominal strains.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Despite the fact that the effective strain is high (0.07 to 0.1 or higher) over a large horizontal band, the strain in the crack propagation direction (L) is close to zero in most of the ROI. Only the area close to the notch undergoes necking (negative strains) in this direction. This is consistent with numerical simulations of crack initiation and propagation [5].

For more quantitative assessments of this phenomenon, Figure 2.10(b) shows longitudinal nominal strain profiles in the crack propagation direction plotted for the three different axes ξ i located in the mid-thickness plane of ROI 2 for loading step (2)-( 3). This is representative, as the same trend has been found for all three loading steps (Figure 2.10 (a)).

The LL strain profiles show negative levels ∼ -0.04 close to the notch and up to about 300 µm from the notch. LL-strain mean values close to zero are found farther away from the notch. Subsequent L-sections are reported for the effective strains for loading step ( 2 1), ( 1)-( 2) and ( 2)-(3)

Figure 2.11 shows 3D renderings of the segmented porosity/damage volumes and effective incremental strain fields. As expected from the previous analyses, strained bands are observed from the very beginning of the loading process. Conversely, void nucleation and growth is very limited, except for the loading step (2)-( 3) where the damage activity is more developed mainly in the zone close to the notch root due to higher strains and stresses. Therefore, the observed slant bands are not induced by damage mechanisms. In comparison to the previously reported results [28], it can be stated that even in the area closer to the notch root the same mechanisms prevail, namely, the onset and development of slant bands precedes damage, which points out that plasticity is the driving mechanism for the flat to slant transition. At later stages, very high strains develop within the bands and outside. These high levels of plasticity cause damage nucleation and growth leading to the final failure. This type of ductile failure corresponds to Mechanism 2 identified in Ref. [41]. However, compared with ROI 1, the strained bands in ROI 2 are more numerous and decrease in number when moving away from the notch root.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging It is interesting to overlay the final crack path and effective incremental strain maps for three different positions along the L-direction for each loading step (Figure 2.12). Comparing both fields shows how far one can go in predicting the failure mode by observing early (incremental) strained bands. For loading step (2)-( 3), the L-section ∼ 250 µm ahead of the initial notch root clearly reveals how the above reported complex incremental strain fields resulted in "zigzagged" failure patterns following the strained band orientations. Close to the notch root there is a strong competition between different bands determining the future crack path. Although not simple, the strained bands are regularly distributed, which determines the crack path to be either fully slanted or interrupted by ∼ 90 • changes in the propagation direction.

In the case of longer crack paths, such "zigzagged" patterns are also observed during flipping action where an already slant crack shifts to the opposite side [9]. Regarding the previous observations, this zigzagged pattern can be considered as the result of small perturbations that favor deformation in another band while the previously active band is Understanding, observation and quantification of ductile failure mechanisms via 3D imaging deactivated. It has been argued that this is most likely caused by a mechanical response as resistance to bending of the sample induced by slant crack propagation [9]. What defines the preliminary 'choice' of the crack propagation direction is the asymmetry of the system as well as the loading conditions that favor one of already available strained bands, which ultimately determine the crack path. However, it is difficult to conclude on what causes those initial instabilities.

Conclusions

The plastic zone close to the notch of a CT-like specimen made of 2198-T8 aluminum alloy was studied in the material bulk by resorting to in situ laminography and Digital Volume Correlation. 3D incremental strain fields just in front of the notch root are measured to study the flat-to-slant transition in ductile failure of thin sheets. The understanding of such phenomena may help evaluate the material toughness.

The consistency of the DVC results has been demonstrated by checking the correlation residual fields after convergence. No significant deviation has been observed from noiserelated and reconstruction artifacts. Further, by analyzing overlapping zones between two studied Region of Interests (ROIs), namely, the first one from a previous work [28] and the second one analyzed herein the strain differences are very close to the measurement resolution.

It was shown that several slant strained bands form complex localized patterns when moving at a distance greater than ∼ 250 µm from the notch root. Strained bands show alternating activity in this region by switching on and off during incremental loading steps. These incremental strain fields do not follow the conventional plastic zone in front of notched samples. The number of strained bands decreases with the distance to the notch. Normal strains in the crack propagation direction are confirmed to be close to zero. It was demonstrated that the onset of strained bands spatially coincide with reaching the plane strain state. Furthermore, even in the close vicinity of the notch root, significant damage does not occur immediately after the onset of strained bands, which is believed to be the prevalent mechanism explaining the flat-to-slant transition in the studied aluminum alloy. Consequently, the final fracture path proceeds along the measured incremental strained bands. It can therefore be stated that early strained bands precede the crack path away from the notch root [28] and also getting very close to it for the reported experiment.

Additional effort will be put on similar examinations of other materials of the same aluminum 2xxx series (e.g., materials with high work hardening, different microstructure, different heat treatments) to evaluate if the results obtained herein can be generalized to a greater variety of alloys. The deformation and damage micromechanisms ahead of a notch in a large flat specimen are assessed via in situ synchrotron laminography combined with digital volume correlation for the strain measurement in the material bulk. Despite the strong work hardening of the naturally aged Al-Cu-Li alloy a slant strain localization band is found from the very early loading onward in the slant fracture region. The final slant crack follows the strain band without significant ductile damage evolution. In the high stress triaxiality region close to the notch that undergoes substantial local necking, two damage micromechanisms are observed, namely, i) limited void nucleation and growth from intermetallic particles, and ii) slant shear cracks even starting from the specimen surface and also located in single grains as shown by correlated post mortem EBSD analyses. In the intermediate region a new deformation mechanism of flip-flopping strain bands and a resulting flip-flopping crack are revealed via "projection DIC" measurements.

Reproduced from: A. Buljac, L. Helfen, F. Hild and T. F. Morgeneyer. On deformation and damage micromechanisms in strong work hardening 2198 T3 aluminium alloy. Submitted, 2017.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Introduction

Plastic flow localization is in general an undesirable material behavior that causes loss of ductility and toughness [37,39]. From a macroscopic point of view, geometrical features of an engineering part act an important rule in strain flow localization, which is well studied and elaborated. However, the numerical treatment of localized phenomena as a pure material response [39] independent of local geometrical features is more complex task. This is especially critical when modeling mechanical behavior that is still not well understood. Plastic flow heterogeneity and localization as a driving force for ductile failure may play an important role [53]. Hence, deeper understanding of the underlying mechanisms of ductile failure needs for analyses at the microscale. Classical ductile damage models [14,38] are not tailored to predict failure for low stress triaxialities. More recent studies were performed via phenomenological approaches defining the strain to failure ε f as a function of stress triaxiality η and Lode parameter µ [41,2,12,56]. Experimentally, tension-torsion test results obtained by pure experimental [15] and hybrid experimental-numerical [43] procedures reveal the monotonic dependence of the strain to failure on the stress triaxiality and the characteristic asymmetric dependence on the Lode parameter. It has also been shown that both damage growth [48] and strains to failure [3] are loading path dependent, i.e., nonproportional loading conditions yield different strains to failure for same stress averaged triaxialities. Hence, a general model for proper ductile damage description in the space of fracture loci ε f (η, µ) that would account for any loading history still does not exist. There are various mechanisms that are not described in a good manner (e.g., shear ductile failure, slant cracks and nucleation/coalescence in the context of secondary void populations). In situ observations of the underlying phenomena of damage and plastic flow interactions can be used as reference for the development and validation of new models built at the (sub)micrometer-scale [50,7].

An example that summarizes all the afore-mentioned challenges is ductile tearing. It has been reported [28,6,1,23,33,4] that during mode I opening of CT specimens, the initial crack starts to propagate normal to the loading direction but then tilts and continues in a slant manner as a combination of modes I and III. The investigation of the origin for such behavior has its engineering relevance since it has been shown that mixed-mode I/III leads to reduced toughness when compared to pure mode I fracture [22]. Matching both stress-strain curves and crack propagation path is difficult to achieve numerically. Consequently, CT-like specimens were chosen to carry out in situ tests [32,31,35].

The combined use of laminography [16,18,17] and global Digital Volume Correlation (DVC) [49] enables in situ displacement fields to be measured at the microscale. Laminography as a non-destructive imaging technique allows flat specimens to be analyzed and wider stress states to be achieved when compared with tomography [27,24,29,34,40]. Thanks to DVC, bulk displacement fields can be measured and strain fields calculated [32]. Up to now, each new portion of the results obtained by laminography and DVC reveals interesting phenomena at micrometer resolutions. This is per se a strong motivation for further analyses on other aluminium alloys. However, to be Understanding, observation and quantification of ductile failure mechanisms via 3D imaging consistent it was decided to proceed in a systematic manner by selecting a specific family of alloys using different heat treatments. In particular, 2139 and 2198 aluminium alloys have been studied [32,31,52,8,35]. Representing the latest generation of aeronautical alloys, the understanding of the underlying failure mechanisms is of great interest.

Extensive analyses on CT-like specimen made of AA2198 T8 [31,8] have shown that in a zone close to the notch root (i.e., ≈ 800 µm [31]), and even in its immediate vicinity [8], strained bands appeared early in the loading history. Plane strain conditions in the crack propagation direction predicted in numerical simulations [5] have been confirmed by kinematic measurements [8] while through thickness effective strain distributions varied significantly. Slant strained bands appeared well before damage, i.e., plastic flow is very heterogeneous before any significant signs of void nucleation and growth is detected at micrometer resolutions. The origin and observed behavior (i.e., activation and deactivation) of strained bands at the microscale still remains unclear. The T8 heat treatment induces relatively high yield stress followed by low work hardening. Low work hardening materials are believed to be prone to flow localization [47]. Following the same logic, strong work hardening materials should be more resistant to the flow localization. However, this was not the case for another aluminum alloy (i.e., AA2139 T3). Numerous strain bands were observed. Their intermittent activity and pattern were two key findings [35]. Further, damage nucleation and growth set in very late during the loading history. Since these two alloys were different, the question remains open for AA2139 with T8 treatment and AA2198 under T3 condition. The latter will be studied herein.

In this work, results obtained for a CT-like specimen made of 2198 T3 aluminium alloy are presented. The T3 heat-treatment results in a microstructure responsible for a high work hardening behavior. This is the key feature for conducting such analysis. High work hardening and high strains to failure indicate low level damage development and subsequent late softening phases. Logically, this should be promoted by homogeneous strain flow with less strain heterogeneities. Hence, the strain fields in this high work hardening material will be analyzed in the sequel. Moreover, since similar analyses exist for the T8 heat-treatment [31], the influence of the latter can also be studied.

The paper is structured as follows. The material properties of AA2198 T3 are first introduced. The experimental setup and the laminography imaging technique are outlined next. The basic principles of DVC incorporating strain uncertainty assessments are presented. The results are then discussed. The strain fields as well as damage micromechanisms in the slant region ≈ 1 mm from the notch root are first presented. A similar analysis follows for the regions closer to the notch root. Qualitative 3D observations, but also damage quantifications lead to the identification of two different damage micromechanisms. They are further assessed via post mortem SEM and EBSD microstructure analyses that are correlated with the phenomena observed from the laminography and DVC results. Microscopic observations show that there are no significant differences in the grain structure between T3 and T8 conditions [10]. The grain size measured by using a mean linear intercept method is equal to 200-300 µm along the rolling direction (L) and approximately also in transverse direction (T). It is equal to 25-30 µm in the short transverse direction (S) for the recrystallized sheet [10]. Hence, typical pancake grain shapes in L-T sections are found. In the experiment reported herein the loading is applied in the T-direction and the L-direction corresponds to that of crack propagation.

Neither T 1 nor Θ hardening precipitates have been found in the T3 condition. Most of the dispersoids are Al 20 Cu 2 Mn 3 or Al 3 Zr components. The average dispersoid diameter size is 45 nm with a volume fraction of 1.2-1.8 %. Transmission Electron Microscopy (TEM) also revealed no presence of precipitate decoration of the observed grain and subgrain boundaries for the T3 condition while they are present in the T8 state [10]. There are no precipitate free zones found in the T3 condition, meaning there should not be any significant detrimental effect on the mechanical properties related to grain boundaries.

Intermetallic particles composed of iron and silicon (white) are elongated and aligned along the rolling direction [10]. However, low intermetallic content (≈ 0.3-0.4 vol. %) combined with hardly any initial porosity (i.e., < 0.03 vol. %) make the material challenging for DVC registration as the laminography image contrast is not very pronounced [32], see Figure 3.1(b). The yield strength is ≈ 275 MPa, and the ultimate tensile strength ≈ 385 MPa [10]. Therefore, the heat treatment endows the material high work hardening properties.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Experimental Procedure 121

Laminography

Synchrotron radiation computed laminography is a non-destructive technique for flat and laterally extended 3D objects [19,30,9,45]. This geometry is of particular interest in the field of mechanics of materials since sheet-like samples allow for a wide range of the engineering relevant boundary conditions. Laminography is related to the specimen inclination with respect to the beam direction with an angle θ < 90 • (Figure 3.1(a)); while tomography, as a special case of laminography, is limited to the 90 • angle during the scanning procedure and requires stick-like samples. The CT-like specimen studied in this work has the following dimensions: 60 × 70 × 1 mm. It contains a machined notch 0.17 mm in radius produced by electrical discharge machining (Figure 3.1(b)). A specially designed loading device is utilized with a stepwise loading procedure (Figure 3.1(a)). The loading device incorporates an aluminium alloy frame (not shown in Figure 3.1(a)) to prevent out-of-plane motions of the specimen [33,32]. After applying each loading step, the object is scanned while rotating about the laminograph axis. With the collected radiographs 3D volumes are reconstructed by using a filtered back-projection algorithm [36]. In the reconstructed slices an isotropic voxel size of 0.7 µm is obtained. The 3D images used in this work were obtained at beamline ID19 of the European Synchrotron Radiation Facility (Grenoble, France) with a 25-keV monochromatic beam, using 1,500 projections and θ ≈ 65 • . After scanning the unde-On deformation and damage micromechanisms in strong work hardening 2198 T3 aluminium alloy formed state (0), 12 additional scans are acquired during the stepwise loading procedure, where the last two scans correspond to the final crack propagation and opening.

Digital Volume Correlation

The reconstructed volume is represented by a discrete scalar function of spatial coordinate x of (in the present case 8-bit deep) gray levels determined by the microstructure absorption of X-rays. The basic principle of DVC consists of the determination of the displacement field u for two consecutive scans in the reference, f , and deformed, g, configurations

f (x) = g[x + u(x)] (3.1)
However, due to acquisition noise, reconstruction artifacts [26,55] and the correlation procedure itself [25], ideal gray level conservation (Equation (3.1)) is not satisfied in real cases. Consequently, the solution consists of minimizing the gray level residual

ρ(x) = f (x) -g[x + u(x)] (3.2)
by considering its L2-norm with respect to the kinematic unknowns. Since a global approach is used herein [49], C 0 continuity over the whole Region of Interest (ROI) is prescribed to the solution, and the global residual to be minimized reads

Φ 2 c = ∑ ROI ρ 2 (x) (3.3)
The displacement field is parameterized by using a kinematic basis consisting of fields Ψ p (x) and amplitudes u p

u(x) = ∑ p u p Ψ p (x) (3.4)
At this level of generality, a variety of fields can be chosen [21]. Finite element shape functions are particularly attractive because of the link they provide between the measured displacement fields and numerical models [44,7]. Thus, a DVC formulation based on 8noded cubes (i.e., C8 finite elements) with trilinear shape functions is chosen [49] in this work. Mean gradients of the displacement field over each C8 element are used to calculate the Hencky strain tensor [20] and subsequently its second invariant will be used in the sequel as Von Mises' effective strain

ε eq = 2 3 d : d (3.5)
where d denotes the deviator of the strain tensor .

The reconstructed volume has a size of 2040×2040×2040 voxels. The correlation procedure using the whole volume size would be computationally too demanding. Therefore an extracted Region of Interest (ROI) is used for DVC analyses. Three different ROIs with dimensions 1184 × 512 × 992 voxels are selected. The first one is located approximately 1 mm from the notch root (as in previous works [31,35]), while the other two are Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Experimental Procedure 123 closer (Figure 3.1(b)). A mutual overlap of 112 voxels (i.e., 78.4 µm) is chosen to check the consistency of the results. As could be expected, most of the DVC registrations obtained for ROI 1 could not be achieved for ROI 2 and ROI 3 since complex displacement fields develop in the immediate vicinity of the notch root. Although few of them converged, the results obtained for these ROIs are valuable since they are located in the zone of the flat-to-slant transition.

Before discussing any result, an uncertainty assessment is presented. It consists of analyzing two consecutive scans of the unloaded sample with ("rbm") and without ("bis") applying rigid body motion between the two acquisitions. In the "rbm" case the sample is moved by u T = -15 voxels, u L = 586 voxels and u S = 3 voxels. For ROI 3, the "rbm" case is missing since applying a rigid body motion moved this part of the sample out of the scanned region. The uncertainty values are evaluated by the standard deviation of the measured displacement and calculated strain fields.

The element size used in this work is = 32 voxels (i.e., C8 element length for all three directions), which yields a standard effective (in von Mises' sense) strain resolution of 0.3 % (Figure 3.2). This strain level represents the limit below which strain measurements are not trustworthy. The normal strain uncertainty plots follow the same trend. No significant effect of the ROI location is observed. Conversely, the application of rigid body motions slightly increases the standard strain resolution. 11) and ( 12)) are missing since the crack already propagated through the ligament area (Figure 3.3).

(0) (1) (2) (3) (4) (5) (6) (7) (8)

Projection Digital Image Correlation

To obtain further strain fields in the region close to the notch root for all load increments and link them to the final crack path, a new "projection DIC" correlation procedure is proposed. The idea and assumption are the following. The region farther away from the notch has been shown to be in plane strain condition in the crack propagation direction [8]. As a consequence, the contrast provided over a certain thickness in the propagation direction can be projected onto one single plane and used to perform in-plane correlation (i.e., classical 2D DIC [21]). The result of the procedure where contrasts (the maximum gray level values) from 50 µm thick slices are projected onto one plane (to become a 2D image) is shown in Figure 3.4(b). It can be seen that the contrast is enhanced compared to a simple 2D section (Figure 3.4(a)). This procedure allows to have faster correlations than for a 3D volume in DVC and also helps convergence. With these "pseudo" 2D images, regularized T3-DIC is performed [54] with unstructured triangular mesh with element size of 32 pixels and regularization length [54] of 256 pixels. This configuration resulted in the standard measurement uncertainty of 0.029 % ("bis" case) and 0.03 % ("rbm" case) for the reported normal strains in the loading direction.

A comparison in terms of normal strain in the loading direction is shown for the DVC (Figure 3.4(c) left) and the "projection DIC" (Figure 3.4(c) right) results. The strain fields are very similar. This result validates the so-called "projection DIC" procedure.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging calculations stand for correlation between any reference volume (State (n)) and subsequent (State (n + 1)) configuration. As mentioned before, ROI 1, which is located ≈ 1 mm from the initial notch root (Figure 3.1(b)), occupies the region that will contain a fully slant crack. For ROI 1, direct calculations up to the loading step (7) converged successfully, while the last converged incremental loading step calculation is between States (7) and (8). The final crack in this (slant) region appears at State (11), see Figure 3.3.

For the meaningful observation of the material behavior section B-B of ROI 1 (Figure 3.1(b)) is followed through the loading history (Figure 3.5). The section is positioned ≈ 1058 µm from the initial notch root normal to the crack propagation direction (L). Figure 3.5 shows von Mises effective strains for section B-B for different loading steps. The color bar scale is adjusted to the range of values for the corresponding section. Figure 3.5(a) corresponds to the results for direct calculations while Figure 3.5(b) consists of the results for incremental loading steps. Since incremental calculations update the calculation frame (i.e., nodal positions), special care is exercised to ensure that the same section is selected through the loading process. This is done by tracking several specific microstructural features. Strain heterogeneities appear very early on in the loading process, both in the case of incremental and direct calculations. The maximum level of the strain in the band for the first direct calculation is 1.4 %. Except the band that is the most active, other bands are also observed as part of the strain pattern. They are inclined at about 45 • and crossing each other at a right angle. Activation and deactivation of these background bands results in plastic strain accumulation in all these zones. The final slant crack path follows the most pronounced band already detected at loading step (0)-(3) as will be shown in later sections. Figure 3.6 shows the effective strain profiles plotted along the axis ξ depicted in Figure 3.5(a) for direct calculations. These profiles enable for a quantification of the strained bands. The curves are characterized by a pronounced peak marking the strained band position. The strain levels in the bands, e.g., for loading step (0)-( 3), are at least one order of magnitude higher than the strain resolution. Hence, the measured strain heterogeneities represent a mechanical signal rather than noise. The band width is seen to be ≈ 150-200 µm, i.e., two (or more) times higher than the reported grain size in the short (S) transverse direction. The remaining part of ROI 1 does not behave as an elastic body. The bulk part is less active than that in the band, but deformation there steadily increases over the loading history, which results in the generation of high strains after several loading steps. However, the strain levels in the bands are significantly higher, keeping the ratio between out-of-band and in-band strain levels within the range 0.5-0.55 for incremental and equal to 0.5 for direct calculations over the whole loading history. Hence, despite the stronger hardening in AA2198 T3, these findings confirm the results for AA2198 T8 [31] in the sense of early onset of slant strained bands keeping a constant strain rate ratio between in-band and out-of-band material portions of the ROI. (5) -( 6)

(5) -( 6) The intermetallic particle breakage and debonding from the matrix together with very few initial pores act as nucleation sites for the voids causing ductile damage. Later happenings involve void growth and coalescence marking the loss of load-carrying capacity. These mechanisms would be the main stages of classical ductile damage processes. In the present series of alloys, however, mechanisms derogating from those above mentioned have been found in the sense that straining and localization occurs before significant damage onset.

The volume fractions of voxels in the ROIs having gray levels below the threshold GL = 110 are plotted in history. For the first six loading steps there is virtually no damage growth. After loading step (0)-( 6), the higher triaxiality stress state around the notch root causes higher void growth inside ROIs 2 and 3. For ROI 1 there is no evidence of void volume fraction increase until the final failure (states ( 11) and ( 12)). Hence, under the selected sub-micrometer spatial resolution (i.e., 0.7 µm) and achieved temporal resolution (number of loading steps, see Figure 3.3), nucleation and void growth in the slant region are not detected until the final crack has propagated (i.e., for loading step (0)-( 11)). It is worth noting that heterogeneous plastic strain fields are detected at this location from loading step (0)-(3) on. Hence, the bands developed very early on in the loading history and it is very unlikely that damage softening triggered localization. Furthermore, after the flow localization has initiated, void growth is still very limited up to strain levels (in the bands) of 10 % or more.

Let us observe the loading step (0)-( 7) where damage growth is still not recorded. The mean value of effective strain inside ROI 1 is equal to 7 %. This phenomenon is depicted in Figure 3.8(b) showing mean values together with the standard deviation of von Mises effective strains inside ROI 1 for direct calculations. For the early loading steps the mean effective strain is relatively low (e.g., for the direct calculation (0)-( 4) it is equal to ≈ 0.02) where the standard deviation already takes significant values (e.g., for (0)-( 4) it is equal to 0.008). This is in line with the section view of Figure 3.5, which shows high strain fluctuations. Later on, the standard deviation continuously increases for the direct calculations in proportion with the mean level. This trend suggests at least a conservation or even magnification of the field heterogeneities. Again, this result confirms the leading role of flow heterogeneities in determining the overall strain to failure levels.

The very small rate of void nucleation and growth is in line with macroscale observation of the material where significant work hardening occurs. Conversely, it is shown that low/high work hardening and strained bands are two independent properties of the material, i.e., the first one does not exclude the second one. This observation also implies that concentration mechanisms develop independently of damage growth. Work hardening, as a measure of the material resistance after plastic flow sets in, can be seen as a consequence of the slow rate of microdamage development, but not as a resistance to flow localization at the microscale. The strained bands neither explicitly occur after damage growth, nor caused direct damage growth that occurs only at the very end of the load history. Further information on damage growth inside and outside the strain bands is given in Appendix A. Damage detection via correlation residuals is also shown in the appendix.

An illustration of the final fracture of the scanned sample is depicted in Figure 3.9 by showing its height map in conjunction with a section of an early strain field. The final crack follows the early strain band. Note that the location of the strain band does not coincide perfectly with the crack as the strain field is shown in the initial (i.e., undeformed) configuration. In Figure 3.9, the complete crack surface is first extracted from laminography data followed by assigning a color to each voxel according to its distance to the L-S plane. The crack starts to propagate in a "roof-top" manner without any fully flat location on the crack and then evolves to a fully slant path at ≈ 500 µm from the notch root. No clear link between the microstructure and the final crack can be found. The main crack propagates in an transgranular way, which is not influenced by the grain orientation and size. The strain band orientations and subsequent absence of (detectable) damage growth suggest crystallographic glide as the most likely micromechanism behind the observed failure behavior. As will be shown in the following, post mortem observations of the fractured surface in this zone reveal mostly flat regions with small secondary voids. For section L = 460 µm slant strained bands are observed. Considering the very small rate of damage growth for ROIs 2 and 3 at state (3), see Figure 3.8, one of the strained bands determined the final slant crack path. Interestingly, by moving farther away from the notch root (L = 740 µm) other secondary bands are (re)activated. Last, on section L = 880 µm one of the secondary bands becomes even more pronounced while at the same time previous primary slant bands remain active to a lesser extend. The spatial distribution is even more complex than the temporal one. In particular, the activation of secondary bands normal to the primary one is an important phenomenon for understanding the strain activity.

L =180 µm L =320 µm L =460 µm L =740 µm L =600 µm L =880 µm
The through thickness (S) sections for all three merged ROIs containing the effective (a) and normal strains (b-d) for loading step (0)-( 3) are shown in Figure 3.11. The reported normal strain components are calculated as nominal strains. The ROI boundaries are marked due to higher measurement uncertainties at edges. However, the overall consistency of the results is satisfactory for different calculations (i.e., different ROIs). The color bar scales are adjusted to the data range of the corresponding field, except for the effective and normal strains in the loading direction ε T T where maximum values have been limited to 0.2. This level is chosen to increase the band visibility since considerable variations occur between the strain levels close and farther away from the notch root. For each case, three through thickness sections are displayed where d denotes the distance between the observed sections and the ROI thickness midplane, which is roughly the specimen thickness midplane. From Figure 3.11 it is confirmed that normal strains ε T T and ε SS are responsible for the observed heterogeneous effective strain fields. Normal strain levels in the crack propagation direction ε LL (Figure 3.11(c)) are close to zero except in the necking zone with negative strains in the immediate vicinity of the notch root. Plane strain conditions are observed, which are consistent with earlier observations [5,8]. For normal strains in the through-thickness direction ε SS (Figure 3 the S-axis. The contraction zones with minimum strain levels correspond to the band positions. Shear strains ε LS (not shown) are insignificant and their distribution does not reveal any heterogeneous pattern that matches the position of the effective strain bands. One possible explanation is related to the scale where the underlying shear cannot be properly captured. The strain activity and the roof-top cracking in the intermediate region (i.e., about 460 µm from the notch root) is assessed up to failure. DVC could only converge for the first loading steps. This is due to the complexity and levels of the strain fields but also to the onset of damage. To be able to follow the strain activity to the very end, the socalled "projection DIC" technique has successfully been applied (see also Figure 3.4).

In the early loading steps shown in Figure 3.12 primary strained bands develop from the lower left to the upper right corners of the investigated region. Some 'hot spots' forming a band normal to the main band are also active. From load step (8) on this normal band becomes the most active and the former primary bands are less active. Some bands parallel the former main band in the upper left corner suddenly become active and, surprisingly, this leads to the final roof-top crack that can be seen in the last overlay image.

The corresponding surface strain activity measurements are reported in Appendix A.

A new flip-flopping deformation mechanism is found, which can explain the flipflopping behavior of the crack. This behavior could be linked to the hardening behavior of this naturally aged material. Hardening within bands may make the deformation in another band easier. The presence of several crossing strained bands is clearly the origin of the zigzagging crack near the notch root. This phenomenon is found in different engineering materials [51,35]. There is one damage feature indicated by an arrow that was always present on the main strained band. The nature of this feature will be further assessed in the following sections via EBSD analyses. and3). This is confirmed in Figure 3.13 where the porosity development is assessed by thresholding the gray levels of the reconstructed volumes. It is worth mentioning, as can be seen from Figure 3.1(b), that the ROIs overlap so that some of the damage features are shared by the two neighboring ROIs. ROI 3, which is closest to the notch root, shows the highest porosity content before final crack propagation. ROI 2 experiences smaller void growth but still with some damage growth before final failure while ROI 1 does not show any microstructural changes before failure, which is consistent with the change in the stress state and the observed strain pattern.

Damage micromechanisms in flat region

In Figure 3.14 microstructure states are shown for different loading steps in the midthickness section of the sample. Void growth occurs at late stages, but only in the triangu-Understanding, observation and quantification of ductile failure mechanisms via 3D imaging ). The same procedure as before has been employed, i.e., the porosity corresponds to the volume fraction of voxels having a gray level less than the chosen threshold. A regular trend is identified meaning that moving away from the notch root induces slower and less damage growth. from the notch root. However, this is not sufficient to explain the observed shape of the final failure. Significant necking occurs close to the notch root (see Figure 3.13 for ROI 3 and State (10)). Although damage grows in this zone, the crack starts to propagate from a low stress triaxiality area close to the surface where less void growth occurred. This points out the competition between the two micromechanisms. The first one, which is plasticity driven (i.e., shear-type) and the other one characterized by void growth and a softening phase prior to coalescence and final failure. The latter will be referred to as classical high stress triaxiality induced micromechanism. The flat fracture zone therefore undergoes a combination of these two mechanisms.

(0) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 10
Void growth observed in the vicinity of the notch root can be divided into two types. The first one is characterized by an equiaxial growth of the voids originating from particle breakage/debonding and mostly determined by the existing stress state. The second one corresponds to elongated dark surfaces (≈ 45 • with respect to the loading direction T) in Figure 3.13 that may represent transgranular cracks originating from shear slip systems as will be shown below. Contrary to the slant zone, these elongated features are detected in the flat zone since they also experienced certain opening due to higher hydrostatic parts of the stress tensor.

As a result of this competition, the final crack even in the flat region behaves in a zigzagged manner [13]. There is no regular damage pattern in Figure 3.13 (i.e., State (10), black region on the left being the specimen surface due to necking), which explains such crack propagation. However, from the early loading steps on, zigzagged strain patterns occur in this zone.

Post mortem observations

The above mentioned damage feature is inspected by Electron BackScattered Diffraction (EBSD) analyses (Figure 3.16). Although it inevitably induces certain void growth and edge rounding electrolytic polishing (EP) needed to be resorted to in order to obtain satisfactory diffraction figures. The damage feature dimensions are measured prior to (Figure 3.16(a)) and after EP for which a ≈ 25 % change in length occurred. It can be seen from Figure 3.16(b) that the inspected feature exhibits transgranular failure. The damage feature dimensions measured (from laminography data) at its initiation stage (i.e., State (10), Figure 3.12) are equal to ≈ 140 µm, ≈ 35 µm and ≈ 40 µm for L, S and T directions, respectively. From Figure 3.16, the S component of the grain size measured at the hole position is equal to ≈ 40 µm, thereby confirming that the damage feature originates from transgranular crystal crack event.

Let us note intensely strained bands forming an "X" shape around the observed damage feature. They extend over multiple grains as previously reported. Although initiated in several of them, the crack opened significantly only in a particular one. The inverse pole figure (Figure 6.6(c)) reveals ≈ 001 orientation of the damaged grain, while the neighboring ones have more favorable ≈ 111 orientations. The damaged grain has unfavorably positioned FCC slip systems with respect to the strain direction ε LL and therefore could not follow the surrounding intense deformation (glide) activity. This difference induced Scanning electron micrographs are shown in Figure 6.6. In Figure 6.6(a) a "rooftop" type of the failure path is recognized. The crack starts to propagate in a "roof-top" manner, but since the slant side close to the depicted subset a is being trapped, it continues to propagate in a fully slant manner. Figure 6.6(a) also contains other subset positions that are shown magnified in Figure 6.6(b).

Dimpled zones corresponding to higher triaxiality levels are found around the midthickness plane (subsets b and e), ≈ 250 µm and ≈ 900 µm from the notch root. The slant areas are detected close to the sample surface where the stress triaxiality levels decrease. Last, two locations, in subset b (region b) and between subsets d and e (region d-e) are depicted and shown enlarged in Figure 6.6(c). Region b reveals primary and secondary void growth features but also incorporating final shear surface alignment (i.e., classical equiaxed void growth and shear-type micromechanism). On the other side, region e is a fully slant spot without any signs of void growth and coalescence (i.e., shear-type micromechanism).

The SEM results support the proposed two types of damage growth mechanisms, especially the micrographs under higher magnification (regions b and d-e). For subset e higher stress triaxialities caused some void growth at very late stages of loading when the final crack was already propagating, although no void growth was detected during the loading according to in situ laminography scans. This result highlights the importance of in situ experiments since post mortem observations integrate all the events during the underlying experiment, without the possibility of positioning the events in time and knowing the weighted contribution of each of them to the observed damage mechanisms.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Discussion

Concerning the damage and deformation process, it has been shown here that the void volume fraction and growth ratio at particular instants do not act as driving force for softening, further ductile damage development and final crack propagation direction. Damage behaves more as a state but not process variable. It has been argued that shear slip is likely to occur even in strong work hardening materials. The question of the proper scale for the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging physical modeling of ductile damage mechanisms (at least in this group of materials) can be outlined since there is a clear need for a shift to modified models accounting for the activation/deactivition of (possibly) multiple strained bands. Classical J2 flow theory or damage softening have been shown to not capture the measured bands [31].

Portevin -Le Chatelier effect could be one of the possible origins of slant band development and in particular their intermittent activity [46]. The flip-flopping of the bands would also be consistent with such a behavior. These findings would then be in contrast with the recent work of Ovri et al. [42] where for naturally aged AA2198 alloy negative strain rate sensitivity and associated instabilities have not been found. However, the loading conditions, sample geometries and stress states (i.e., microtensile and nanoindentation tests) during those experiments significantly differ from the present work. Further λ hardening precipitates are found in the studied material. The hardening precipitates with specific spatial arrangements may suppress strain heterogeneities to some extent [11]. However, the strain activity has not been assessed by surface of bulk measurements.

The present findings are also important in the context of similar results obtained on AA2139 and AA2198 aluminium alloys. The results reported herein confirm the conclusions made in previous works [31,8,35]. The formation of strained bands precedes damage growth and the softening phase. The final fracture is driven by heterogeneous plastic flow. Damage growth is limited until the very late stages of the experiment. Portions of the ROI outside the strained bands do not experience only elastic loading, but deform with lower plastic strain levels (i.e., the ratio of the strains within and outside the bands is ≈ 2). Closer to the notch root, slant bands appear very early in the loading process. Higher damage growth in this zone is a consequence of the higher stress triaxiality state in the characteristic triangular zone ahead of the notch root. However, even significantly developed voids do not play a key role in the determination of the final crack path. The specimen fails under a "roof-top" to slant transition mode (i.e., there is no real flat region), which is determined by the strained band pattern detected already at the beginning of the loading history.

Conclusions

It has been shown herein, using 3D imaging and strain measurement techniques in the material bulk, that strong work hardening does not significantly change the results reported on AA2198 T8 grade (i.e., same material but with low work hardening [31]) concerning the early development of slant strained bands ahead of a notch in a large flat specimen. In the present case, the slant strained bands are revealed even earlier in the loading history since more loading steps were performed during the experiment. In the slant fracture region, strained bands appear very early on, before any signs of damage onset. Void nucleation and growth has been quantified. It is found to be very low in the slant fracture region.

SEM fractography revealed that in this region hardly any dimples, not even on a very small scale, are present.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

In the high stress triaxiality region close to the notch, two separate damage micromechanisms are identified, namely, i) shear-type damage (Subsection 3.2) and ii) more classical high stress triaxiality induced damage (Subsection 3.5). Even in the "flat" fracture zone the shear-type micromechanism prevails, which causes the zigzagged final crack pattern close to the notch root. Both EBSD and SEM analyses correlated the results obtained in this work and are consistent with such assumptions.

In the intermediate region, at about 460 µm from the notch root, a new deformation mechanism of flip-flopping slant strained bands has been revealed. It led to a flip-flopping roof-top crack. This observation has been achieved using a new "bulk-DIC" technique. Under the assumption of plane strain condition the in propagation direction, which is satisfied herein, image contrast is projected onto a single slice of material for different load steps. With these 2D projections classical 2D-DIC can be performed. This analysis allowed extremely complex strain fields to be measured even during the onset of damage.

To complete the present findings and previous studies, the AA2139 grade with T8 treatment should also be investigated. Other classes of aluminum alloys (e.g., 6XXX or 7XXX series alloys) could also be analyzed to confirm the presence of slant strained bands.

Chapter 3: Additional Material Additional damage quantification

A similar analysis as in Section 3.2 is repeated, this time by following void nucleation and growth in two extracted subsets inside ROI 1 (i.e., ES1 and ES2, Figure 3.18(a)). ES1 is located inside the zone where two strained bands cross each other and that is active during the whole loading history. ES2 occupies a zone with significantly lower strain activity. Although the present method intrinsically incorporates several imperfections (e.g., the analysis does not always exactly follow the same portion of the material and does not account for overall gray level variations), the overall trend is identical (Figure 3.18(b)). Only after loading step (0)-( 6) does the porosity increase inside the strained band. The void growth ratio inside and outside the bands does not differ significantly, yet the final crack path follows the band position. An additional damage characterization is carried out by observing the final gray level residual ρ(x) of DVC analyses (Subsection 2.3). Not only does it serve as quality inspector, but it also provides useful information to detect damage occurrence for which the gray level conservation (see Equation (3.1)) will be violated. Let us recall that the corrected volumes are updated at each iteration on a sub-voxel scale by using gray level interpolation (in this work a trilinear interpolation scheme has been employed). This means that by using residual fields one can benefit from sub-voxel resolutions in the gray level registration procedure.

In their original form, the residual fields are evaluated voxel-wise (see Equation (3.2)). The first insight can be obtained by observing mean values of the residuals inside ROI 1. They are virtually the same for all calculations and very close to the levels observed for the resolution analysis. These results are shown in Figure 3.19 where all the values are normalized by the dynamic range of the reference volume (i.e., 256 gray levels). The fact that there is a small drift for direct calculations is probably due to the complexity of the kinematic fields, which cannot be fully captured with the chosen discretization.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging For a deeper understanding, a field characterization is required. Figure 3.20 shows the residual distribution on section B-B (Figure 3.1(b)) and that normal to it (mid-thickness ROI section) for the last direct (i.e., (0)-( 7)) and last incremental (i.e., ( 7)-( 8)) DVC registrations. There are no significant features (zones) with high residuals confirming the values reported in Figure 3.19. This result indicates gray level conservation (i.e., no new voids nucleate, nor void growth) and continuity of the displacement field, hence no significant damage growth occurs. Therefore the cumulation of strains in the ROI is mostly due to the strained band activity (as is shown above) and does not induce significant residual deviations. Image Correlation in the zone ahead of the notch root. Microstructure changes along with measured bulk strain fields have been followed throughout the experiment with the special care dedicated to the correlations between the two quantities. The new deformation mechanism has been found. It consists of multiple slant strained bands present from the beginning of the loading process with intermittent plastic activity. The final slant fracture has its origin in heterogeneous plastic flow.

Reproduced from: A. Buljac, L. Helfen, F. Hild and T. F. Morgeneyer. Ductile tearing of aluminum alloy (AA2139 T8) via in situ synchrotron computed laminography and projection digital image correlation. To be submitted.

Introduction/Motivation

Ductile damage studies came to the point where more initimate relationships between the underlying microstructure and corresponding strain fields can be studied in detail [19].

In situ experimental techniques, e.g., computed tomography [7,9] and laminography [6] brought new sets of experimental evidence in context of microstructure changes during mechanical experiments. Combined with Digital Volume Correlation [22] such observations are enriched with the corresponding bulk kinematic fields. Laminography enables for scanning regions of interest (ROIs) in plate-like samples [13] and thereby giving access to a whole range of stress triaxialities (depending on the initial sample geometry) with significant size of plastic zones. In the context of industrial applications, this enables engineering relevant boundary conditions to be prescribed on thin sheets, which are especially interesting objects as part of weight (thickness) reduction design trends.

In this work laminography together with so-called projection Digital Image Correlation (DIC) technique (detailed in the sequel) will be used to study ductile tearing of Compact Tension (CT)-like sample made of aluminum alloy AA2139 T8. It comes as a final part of a series of studies conducted on CT-like samples made of two different aluminum alloys with two different heat treatments, namely, AA2198 T3 [2], AA2198 T8 [15, 1], AA2139 T3 [16] and AA2139 T8. For the first three listed Al-alloys a new deformation mechanism in the form of multiple slant strained bands has been found from the early stages of the deformation process preceding any damage development. For AA2198 T8 alloy one governing band is identified throughout the experiment [15]. The final failure path is following early set strained bands without influence of damage softening, outlining plasticity as the origin for flat to slant transition of final crack.

The heat treatments T8 and T3 endow low and high work hardening of the material respectively. The high work hardening tends to hamper localization [23] but this effect is not found for AA2198 T8/T3 alloys [15,16], at least not for the hardening rates that were obtained with these two heat treatment conditions. In this study the eventual difference between T3 [16] and T8 behaviour will be detailed for AA2139 alloy. Hence, the work will complete the study about this new deformation mechanism found in particularly important structural materials often used in aeronautical applications [26]. The results obtained on AA2139 T8 alloy alone represent important evidence since it is the first study of in-situ CT-like sample made of an alloy with significant initial porosity in combination with low work hardening induced by the T8 heat treatment.

Experimental details

Imaging was performed on KIT's laminography instrument installed at beamline ID19 of the European Synchrotron Radiation Facility (ESRF) in Grenoble (France). An axis inclination angle of about 25 • and a monochromatic beam of 25 keV X-ray energy were used. Volumes were reconstructed from 1500 angularly equidistant radiographs; the exposure time of each projection was 250 ms. The scanned ROI was ≈ 1 mm 3

in volume with an

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging isotropic voxel size of 0.7 µm. The minimum specimen to detector distance was 70 mm leading to relatively strong edge enhancement [5] due to phase contrast. The material of the specimen is a commercial ductile 2139 Al-Cu alloy in artificially aged T8 condition, which endows low work hardening of the material. The sample has been machined symmetrically from 3.2 mm to 1 mm thickness in the short-transverse (S) direction. Testing has been performed in the T (long transverse) -L (rolling direction) configuration. The grain size measured by using a mean linear intercept method is equal to 200-300 µm along the rolling direction (L) and approximately also in transverse direction (T). It is equal to 25-30 µm in the short transverse direction (S) for the recrystallized sheet [4]. Hence, typical pancake grain shapes in L-T sections are found. The material has an initial void volume fraction of ≈ 0.3 %. Further details on the material properties are given in Refs. [11,12]. The notch has been machined via electron discharge machining (EDM) resulting in a radius close to the EDM wire radius of 0.15 mm.

The sample geometry together with the loading device are shown in Figure 4.1(a). An anti-buckling device (not shown in the figure) was used to prevent the sample from significant buckling and out-of-plane motion. The loading has been carried out via a 2screw displacement-controlled wedging device that controls the specimen crack mouth opening displacement (CMOD) similar to that used in Ref. [10]. Hence, the loading has been applied in a stepwise procedure and after each loading step a scan has been acquired. Overall, 10 scans from the experiment will be analyzed in the following analysis starting with Scan (0), i.e., undeformed state, up to Scan (9), i.e., already propagated final crack.

In front of the notch root (starting at distance ≈ 250µm) of CT-like sample plane strain conditions have been reported both in numerical simulations [3] and experimental DVC measurements [1,2]. The plane strain condition corresponds to the crack propagation direction (L). This allows one to project the stack of microstructure sections (normal to the crack propagation direction, positioned in plane strain region) with a thickness of 50 µm into a single section. This procedure is illustrated in Figure 4.1(b).

For the studied material the projection is performed by keeping only minimum gray level values (i.e., voids), maximum gray level values (i.e., intermetallic particles and some white phase contrast fringes around voids) or combination of the two (noted as mix. version) as shown in Figure 4.1(c). The projection allows stronger contrast to be obtained (compared to original one from laminography section shown in Figure 4.1(c)) and is subsequently used in 2D-DIC measurements [24], which are fully representative (Figure 4 in Ref. [2]) of the corresponding measurements from DVC analysis.

These are main elements of so-called bulk-DIC employed in the work. The main benefits from bulk-DIC are significantly faster correlations compared to standard DVC and in this particular case the ability to measure the kinematics for one difficult loading step (i.e., ( 5)-( 6)) which was hardly achieved in full DVC registration. In this work DIC setup with unstructured triangular mesh with element size of 32 pixels and regularization length [24] of 256 pixels have been used. This configuration resulted in the standard measurement uncertainty of 0.07 % for the reported normal strains in the loading direction.

Results

Figure 4.2 shows reconstructed laminography 2D sections (not projected ones) positioned normal to the crack propagation direction (L) at a distance of 865 µm from the initial notch root (see Figure 4.1(b)). State (0) shows as the received material where the aluminum matrix is shown in gray, initial porosities in black while the intermetallics are seen in white. The surface of the sample is additionally marked with red thin lines for easier reading of the figure. One can note that the surface is smooth without machining marks. Upon loading the sample is deformed in the tensile direction but with limited local necking (State (5)). Up to State (5) no signs of significant growth of existing voids nor nucleation of new ones have been noted. At Stage (7) intermetallic particle breakages and isolated void coalescence events are spotted (blue and red arrows). At State (8) voids coalesce inside an inclined band (yellow arrow) in a sudden manner without prior damage growth at this particular location. Later on, the final failure will propagate through this coalesced feature. Hence, just from laminography observations the set of events leading to final failure remains unclear.

State (0)

State ( 5) For this reason strain fields are added to the analysis as shown in Figure 4.3. The strain fields are plotted over the corresponding projected microstructure sections (mix. projection mode from Figure 4.1(c)). The normal strains in the loading direction (ε T T ) from incremental DIC calculations, i.e., between stage (n) and subsequent stage (n + 1), are followed throughout the loading history. From the first studied loading step the strain field is not uniform but exhibits strained bands. The strain pattern consists of multiple slant strained bands where the band oriented from upper left to lower right part of the section seems to be prevalent (Band A). Two bands oriented from the lower left part of the section to the upper right part are distinguished (Bands B and C). Let us note that the bands themselves posses a certain structure (smaller bands), which outlines the underlying multiscale problematic. At this point, the maximum strain levels in the bands are ≈ 1.8 % that is one order of magnitude higher than the reported strain resolution (0.07 %). From Figure 4.2 (State (0)) no such zones with voids (soft zones) are observed in the undeformed state that coincide with the location of the strained bands. Hence, in this 2D section no clear link between the underlying voids and the strained bands is established.

State (7) State (8)
For the second and third loading steps the strain field mainly possesses the initial shape with cumulation of additional strains within and outside the bands. The ratio between the strain levels inside and outside the bands is ≈ 2. This ratio has been already found for other aluminum alloys either with similar or with more work hardening and without initial porosity [15,16,2]. The part of the material outside of the band also deforms plastically and this is the main reason for not using term strain localization but rather strained bands for this phenomenon.

During the following loading step (3)-( 4) most of the background small bands are less active. The strain is mostly concentrated in two half-bands, namely, upper part of Band A and lower part of Band B. Next, during the step (4)-( 5) the upper part of Band A becomes less active, and most of the deformation activity occurs in Band B. During the loading step ( 5)-( 6) the half-band parallel to the original lower part of Band A appears. This zone also showed some plastic activity during first three loading steps. One of the intermetallic particles from this zone is highlighted with a green circle throughout the loading history. In the next loading step ( 6)- (7) this half-band is less active and a single slant Band B dominates the field. In the last correlated loading step ( 7)-( 8) (just prior to the crack propagation) the lower part of Band A is again activated. The final failure in its left part does follow early active lower part of Band B while on the right side it follows the shortly re-activated half band from the loading step ( 5)-( 6) (note green circle position).

Significant damage coalescence (microstructure ( 8)) and subsequent final failure (microstructure ( 9)) occur in the bulk along the precursor band path. The final failure path has zig-zag (or roof top) shape, which could be guessed already from the initial strain fields. In other words, the initial strain maps contain possible paths for the final failure while later strain activity will determine which one is more favorable. For instance, at loading step (2)-(3) the pattern precursor for the future crack can be drawn by following the strained bands (blue dotted lines shown on strain map (2)-(3) and microstructure ( 9)). Later on, the activity and sudden (de)activation of the bands could also be the result of the propagating crack closer to the notch root.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging laminography and projection digital image correlation

(0)-(1) (1)-(2) (2)- (3)
(3)-( 4) ( 4)-( 5) ( 5)-( 6) ( 6)-( 7) microstructure (9) In a previous work [17] it has been shown that strain localization is easier under plane strain conditions and it is confirmed herein. An interesting approach is to compare the reported strain fields with those obtained for late stages of deformation in AA2139 T3

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Concluding remarks 159 grade, in particular State (e) from Figure 6 in Ref. [16] with loading step ( 5)-( 6) from Figure 4.3 in this work. In general, strain field changes prior to failure and failure itself are similar in both cases. In the studied AA2139 T8 alloy the intermittent activity of some bands (half-bands) is observed but the main half-band (lower part of Band B) is active from 1.5 % strain onward highlighting more temporal stability due to material low work hardening compared to the behavior of the same microstructure with higher work hardening [16]. However, the major differences are not found in behaviour of AA2139 alloy grades because both T3 and T8 have half of the bands with intermittent activity regardless hardening exponent. The similar behavior of the material with T3 and T8 heat treatments is confirmed in case of AA2198 alloy in Refs. [15,16].

Concluding remarks

To conclude, in this work 3D in-situ laminography experiment with relatively large aluminum sheet was reported. Based on the previous knowledge of plane strain condition [3,1] bulk-DIC was applied to measure kinematic fields in the section normal to the crack propagation direction (L). Compared to DVC, it represents a much faster way of estimating bulk displacement fields. For AA2139 T8 alloy, with important initial porosity and low work hardening, these results provide unprecedented insights into damage vs. strain field mechanisms in an engineering-relevant sample geometry. In a broader context, these results confirm findings from previous studies on AA2139 T3, AA2198 T3 and AA2198 T8 grades. The new deformation mechanism with "waffle"-like patterns has been found for all four studied aluminum alloys. It consists of multiple slant strained bands present from the beginning of the loading process with more (AA2139 T3, AA2139 T8) or less (AA2198 T3, AA2198 T8) intermittent plastic activity. The final fracture is driven by heterogeneous plastic flow. The observed strain fields cannot be reproduced with classical J2 flow theory [15] or Gurson-Tvergaard-Needleman-type model with Lode parameter-based shear void nucleation [14]. Portevin -Le Chatelier (PLC) effect [20,21], heterogeneous microstructures (randomly distributed soft grains) [18,16] or material processing marks (ex-super grains) [25] could be possible origins of such early heterogeneous field developments. However, complete physical explanations for the origin of slant strained bands, their structure (stable in space) and behavior (intermittent activity) still remain to be clarified.

Introduction/Motivation

An increased interest and invested resources resulted in substantial progress made in understanding ductile damage mechanisms over the last decades [6,33]. However, the predictive capabilities of these models are also known to be limited for arbitrary loading conditions [9,10]. In general, one of the leading reasons for such difficulties is the lack of comprehensive experimental data. The strain to failure ε f is often defined as a function of stress triaxiality η and Lode parameter µ. There exists constant demand for specifically tailored experiments to describe effects of stress state (η, µ) on ductile failure. In the definition of fracture loci ε f (η, µ), Lode parameter and stress triaxiality reported from different experiments are mainly defined as average values in the loading process [2]

η av = 1 ε f ε f 0 η(ε p )dε p , µ av = 1 ε f ε f 0 µ(ε p )dε p (5.1)
where ε p denotes an equivalent plastic strain. Some authors define the stress state with the final values of η and µ just before the onset of failure [3]. Even the measuring of strains to failure ε f from the experiments remains difficult [2] whereas hybrid experimentalnumerical approach opens new possiblites [30]. However, it has been shown that following different loading paths, but having the same average triaxialities, results in different simulated damage growth and strains to failure [4,37,30]. Hence, being able to preform experiments with local triaxialities within controlled and quantified margins is of great interest.

For instance, micromechanical simulations with monotonic and proportional loading histories are conducted successfully by prescribing appropriate boundary conditions [41,39,12]. On the other hand, the validation of the identified properties from this simulations is not an easy procedure. It is difficult to achieve constant stress states in points of interest during the experiment due to the deformation of the sample. In tension of smooth specimens local triaxiality changes significantly (η = 0.33 → 0.8) when the strains localize and necking occurs [13,21].

Two principal ways to avoid the mentioned issues and perform experiments with locally proportional stress states are a) by using specifically designed sample geometries [7] and b) by employing corresponding boundary conditions [20,30]. The first approach is more robust since the reported experiments can be performed on classical tension/compression loading machines. However, when using the specimen geometry for controlling stress states, due to large deformations, the issue of preserving constant stress states during the experiment arises. The second approach is less sensitive to triaxiality variations due to geometry changes but it is more expensive to implement in experimental facilities. Hence, the reproducibility of experiments is less likely.

Structural simulations fail to predict failure under specific loading paths for which they are not tailored. Very often, uncoupled phenomenological approaches are employed where the set of parameters {p} is introduced in the analysis [19,43]. The appropriate experiments need to be designed to be able to identify the material parameters [19,43,2,3]. These models have probable industrial application [10], thereby the corresponding Understanding, observation and quantification of ductile failure mechanisms via 3D imaging experiments for identification of the material properties should be robust and simple to execute. In this context, following the approach with smartly designed sample geometries loaded by standard tension-compression machines for controlling local stress states is a more favorable option.

In Ref. [35] the authors optimized the sample geometries for important stress states by conducting extensive inverse analyses with the objective function being the minimization of deviations from the targeted stress triaxialities during the experiment. For the zero triaxiality level they considered the so-called 'smiley' geometry as a basis for optimization. The original geometry of the sample [23,40] had some drawbacks. The final crack tends to initiate from the free surface where the stress state is not dominated by shear. This observation limits the analysis and the measured values need to be noted only as a lower limit for strains to failure in pure shear. The optimized geometry should ensure the final crack starting within the gage sections.

The results were successfully confirmed by performing a set of experiments with the optimized sample geometries [35]. In the case of the shear sample certain doubts were noted. Even though the measured strains show the progress made with the new set of geometrical features (i.e., more ductility), it is still hard to conclude from surface images whether the final failure initiates on the free surface or within the ligament bulk.

In this work a laminography experiment is performed with the optimized shear sample geometry [35] for the aluminium alloy AA2198 T8. This material was already studied with a CT-like sample [27]. The reconstructed scans are registered with Digital Volume Correlation. The resulting bulk displacement and strain fields together with the microstructure changes in the gage volume should give additional insights into what is the set of events leading to the final failure. At low mean hydrostatic pressure stress triaxiality as driving force for void growth [34,16] is missing. Hence the proposed experiment should bring new experimental evidence to what are damage micromechanisms in pure shear stress state, e.g., is the particle debonding present as discussed in [14] or some other phenomena. The key advantage of the present work is the ability to follow bulk microstructure and kinematic changes opposed to the surface observations of shear samples [15] where pure shear stress state cannot be obtained.

When studying ductile fracture one cannot benefit from analytical formulas to define stress states in material points due to the heterogeneous nature of strain fields. Therefore, hybrid experimental-numerical approaches can be employed [24,1]. The 3D tomography/laminography images and kinematic fields measured via DVC can improve existing ductile damage models and serve as rich source of experimental data for identification and validation of parameters [8,38,18]. In this sense, laminography and DVC can help one to bridge the gap between experimental data and their direct interpretation.

Beside the above-mentioned reasons, the motivation per se for conducting this analysis on shear specimen is to conduct DVC registrations in cases with high plastic strain levels and to test the ability of measuring Lode parameter fields in the shear band when expressed in terms of incremental or cumulated strains.

The chapter is structured as follows. The experimental setup and material properties are first introduced. The basic principles of DVC together with the strain definition and uncertainty assessments are outlined next. The results are finally discussed. The focus is on incremental strain activity and damage changes in the monitored shear zone.

Experimental methods

Laminography

The testing device used herein applies the load by manually controlling the displacement via screw rotation. After each load increment, a scan is acquired while the sample is rotated about the laminographic axis (i.e., parallel to the specimen thickness direction). This axis is inclined with respect to the X-ray beam direction by an angle θ ≈ 65 • . The shematic view of the laminography setup together with the loading device is shown in Figure 5.1(a). The experiment was performed at beamline ID19 of the European Synchrotron Radiation Facility (ESRF, Grenoble, France). The 3D images were obtained with a 25 keV monochromatic beam, using 3000 projections per scan. The series of acquired radiographs is then used to reconstruct 3D volumes by using a filtered-back projection algorithm [29]. The parameter optimization has been performed automatically using a GPU-accelerated implementation of this algorithm [42]. The scanned region is positioned in one of the two ligaments where shear occurs (Figure 5.1(b)). The material Tech and represents the latest generation of Al-Cu-Li alloys. In this work it is used in its recrystallized state after an artificial aging treatment (T8). The alloy has low density, good corrosion resistance and enhanced mechanical properties. It is widely used in demanding branches such as the commercial aircraft industry. The composition (in wt% ) of the alloy is 2.9-3.5 Cu, 0.8-1.1 Li, 0.25-0.8 Mg, 0.1-0.5 Ag, 0.04-0.18 Zr, balance Al. The yield strength is ∼ 440 MPa and the ultimate tensile strength is ∼ 500 MPa. Therefore the heat treatment endows the material low work hardening. The intermetallic particle volume fraction is found to be ∼ 0.3 -0.4 % while the initial porosity is very low (i.e., < 0.03 vol%). More details on the material properties can be found in Refs. [11,27]. The very low volume fraction of secondary phases makes this material very challenging for Digital Volume Correlation (DVC) since it relies on natural markers [26].

The grain size measured by using a mean linear intercept method is equal to 200-300 µm along the rolling direction (L) and long transverse direction (T) and 25-30 µm in the short transverse direction (S) for the recrystallized sheet [11]. In the experiment reported herein the loading is applied in the L-direction and the S-direction corresponds to the thickness direction of the sample. Figure 5.1(b) illustrates the grain orientations and their relative size with respect to the scanned region.

Digital Volume Correlation

The reconstructed volume is represented by a discrete gray level field (8-bit deep) of the spatial (voxel) coordinate x. It has a size of 2560 × 2560 × 2300 voxels. The physical size (length) of one cubic voxel is equal to 0.65 µm. Each voxel inside the reconstructed volume contains the gray level information determined by the material ability to absorb Xrays. In the present case, the image contrast is mainly due to heavy intermetallic particles of micrometer size. The part of reconstructed volume that is used in DVC will be designated as Region of Interest (ROI). The ROI position is depicted in Figure 5.1(b) while in the z direction the ROI and sample mid-thickness planes coincide. The original gray level volumes are coarsened, i.e., blocks of 8 voxels are replaced with the corresponding supervoxel having the mean gray level value of the block, to be able to cover larger region in DVC calculations. Therefore an ROI of size 640 × 800 × 688 supervoxels is used for DVC analyses.

DVC used herein is an extension of 2D global Digital Image Correlation [5,17]. The principle of DVC consists in matching the gray levels f in the reference configuration x and those of the deformed volume g

f (x) = g[x + u(x)] (5.2)
where u is the displacement field with respect to the reference volume. However, in real applications the strict conservation of gray level is not satisfied, especially in laminography where deviations appear not just due to acquisition noise but also due to reconstruction artifacts because of missing information (angles) [44]. Consequently, the solution consists in minimizing the gray level residual ρ(x) = f (x)g[x + u(x)] by considering its L2-norm with respect to kinematic unknowns associated with the parameterization of Understanding, observation and quantification of ductile failure mechanisms via 3D imaging the displacement field. Since a global approach is used in this work, the whole region of interest (ROI) is considered, the global residual Φ c

Φ 2 c = ∑ ROI ρ 2 (x) (5.3)
is minimized with respect to the unknown degrees of freedom u p when the displacement is written as

u(x) = ∑ p u p Ψ p (x) (5.4)
where Ψ p (x) are the chosen displacement fields for the parameterization of u(x). Among a whole range of available fields, finite element shape functions are particularly attractive because of the link they provide between the measurement of the displacement field and numerical models. Thus, a weak formulation based on C8 finite elements with trilinear shape functions is chosen [36].

Strain uncertainty

From the mean gradient of the displacement field over each C8 element the Hencky (logarithmic) strain tensor and subsequently its second invariant will be used in the sequel, the von Mises effective strain being

∆ε eq = 2 3 ∆ d : ∆ d (5.5)
where ∆ d denotes the deviator of the logarithmic strain tensor for incremental fields ∆ . Further, the Lode parameter [22] associated with strain increments (µ ∆ε ) is defined as

µ ∆ε = 2∆ε II -∆ε I -∆ε III ∆ε I -∆ε III (5.6)
where ∆ε i (i = I, II, III) represent eigen strains (∆ε I ∆ε II ∆ε III ). The DVC resolution is evaluated by correlating two scans of the unloaded sample with (denoted "rbm") and without (denoted "bis") a small rigid motion applied between the acquisitions. Due to the noise contribution and reconstruction artifacts, these two volumes are not identical. Therefore, the measured displacement field accounts for the cumulated effects of laminography and DVC on the measurement uncertainty [26]. The uncertainty values are evaluated by the standard deviation of the measured strain fields. Figure 5.2(a) shows the standard effective strain resolution levels for different element sizes expressed in supervoxels. Decreasing the element size is followed by an increase of the strain uncertainty.

The element size used in this work is = 16 supervoxels (length) for all three directions, which yields a standard effective strain resolution of 0.51 %. These values represent the limit below which the estimated strains are not trustworthy. The associated spatial resolution is 2 × = 41.6 µm.

DVC calculations

Direct DVC results refer to the solutions obtained by correlating the undeformed volume (state (0)) and one of the deformed configurations (state (n)), while incremental calculations stand for the correlation between any deformed volume (state (n)) and subsequent (state (n + 1)) configuration. Due to large strains in this work incremental DVC calculations will be reported. However, since the global coordinate system is identical for all incremental calculations, the displacement fields can be superimposed [28]. This is done by interpolating incremental calculation results (n)-(n+1) on the deformed mesh from the loading step (0)-(n). The operation results in a part of the boundary nodes from the initial DVC ROI without kinematic information since ROIs in incremental calculations keep a constant size. By conducting such analysis one can obtain DVC registrations from the undeformed configuration to the very end of the loading history. This operation is the basic principle of a cumulation procedure in postprocessing the DVC results and will be used in the sequel.

Last, there are two different ways of displaying incremental results. First the fields are shown on the reference configuration of each incremental DVC analysis. Second, all results can be expressed in a Lagrangian way by reprojecting back all the measured incremental fields as difference between cumulated calculations.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging 3 Results

DVC fields

Beside the scans in the undeformed configuration 8 additional scans are acquired during the experiment, the last one being the scan of the fractured ligament. The load levels associated with the machine displacement are shown in Figure 5 The DVC calculations were challenging since highly localized deformation occurs in a shear plane. Thus, instead of starting with no initialization, the registrations are initialized with a priori chosen displacement fields (Figure 5.4). The initialization fields are obtained by first conducting two independent DVC analyses on two sub-ROIs that are positioned out of the shear plane and mostly experience rigid body motions. The initial guess for the displacement field in the remainder of the ROI is linearly interpolated between the values obtained for the two sub-ROIs. The resulting field served as a good initial guess (see displacement profiles in Figure 5.4) for the corresponding DVC calculations.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging The gray level DVC residuals are shown in Figure 7.8. The fact that ring artifacts can be noted in the residual maps reveals the successful registration where initial differences between the gray level volumes decreased to noise level, i.e., ring artifacts for this case. Had the artifacts disappeared, it would have meant the registration tracked the rings and not the material motions. In the rest of the ROI spots with slightly higher residuals are located in the shear plane. The fact that there exists non-conservation zones of the brightness has its mechanical interpretation and will be detailed in the sequel. Figure 5.6 shows von Mises incremental strain maps plotted on mid-thickness section of the ROI. During the first load step (0)-(1) (field not shown in Figure 5.6) the specimen is mainly in the elastic regime and the resulting strain levels are close to the estimated noise levels. From the second incremental loading step on, the strain field is localized in the shear band. The strain levels in the band are > 5%, namely, one order of magnitude higher than the reported measurement uncertainty (Figure 5.2(a)), which indicates a clear mechanical signal. In the following loading steps the field localizes more into a thinner band. Interestingly, the highly localized band (loading step (4)-( 5)) is not perfectly straight, but shows certain curvatures. This could be induced by the underlying grains and grain boundary positions.

In Figure 5.7 von Mises incremental strain profiles along line ξ are reported for different loading steps. Since incremental results are reported in Lagrangean way (see Section 2.2.2) the strain peak value (band position) remains fixed in the graph. The strong field localization into a thiner band for the last loading steps can be observed quantitatively in the figure .   The cumulation operation (see Section 2.2.2) yields von Mises strain fields between undeformed configuration (0) and subsequent stages (n) as shown in Figure 5.8. Due to this operation part of the boundary nodes from the initial DVC ROI remains without kinematic information as illustrated in the figure. Unlike incremental plastic activity, in these strain maps the band remains significantly wider due to cumulated plasticity from the early loading steps.

In Figure 5.9 von Mises strain profiles along line ζ from cumulated incremental DVC calculations are reported for different loading steps. The results are expressed in Lagrangean way and the band position remains fixed for all loading steps. The strain level inside the band for the last reported loading step is ε f ≈ 0.65.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging This effect is illustrated in Figure 5.11. The graph contains von Mises strain ∆ε eq and Lode parameter µ ∆ε for mechanical (loading step (3)-( 4)) and noisy situations ("bis" case from the uncertainty analysis). In the region out of the band, the Lode parameter displays similar oscillations. In the shear band the Lode parameter is stabilized, thereby indicating higher mechanical signal, with values close to zero accompanied by very high strain increments. According to Equation (5.6) the uncertainty for the numerator of Lode parameter µ ∆ε can be estimated as

Σ µ = Σ I 2 + 4Σ II 2 + Σ III 2 (5.7)
where Σ j ( j = I, II, III) denotes standard uncertainty of the (incremental) eigen strains.

Error bars associated with the Lode incremental strain measurements are then obtained as

e(x) = Σ µ √ 3∆ε eq (x) (5.8)
In this equation it is assumed that the denominator is significantly higher in level than the numerator of the Lode parameter, i.e., in shear case µ ∆ε → 0. Further, the eigen strain difference ∆ε I -∆ε III is assumed to be close to √ 3∆ε eq (this expression is exact in pure shear). The Lode parameter (µ ∆ε ) profiles with the corresponding error bars are shown in Figure 5.12. The values are taken from the profile depicted in Figure 5.13(a) for loading step (3)-( 4). There is a significant error decrease in the zone of the band with a clear mechanical signal. The value of ∆ε eq = 2 % is chosen as certain offset (i.e., factor 4) with respect to the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging reported strain uncertainty of 0.5 %. One can note that the Lode parameter µ ∆ε values are oscillating about a mean value close to zero. The level of oscillations is within the window determined by the measurement error shown in Figure 5.13(b) for the loading step (3)-( 4). This result confirms the ability of DVC to measure Lode parameter µ ∆ε for incremental strain fields. Let us recall that the stress Lode parameter is proportional to strain rate Lode parameter [22]. Hence, in the case of pure shear, zero is also a value expected for incremental strain Lode µ ∆ε measurements as reported in the work.

In the literature [25] the strain rate Lode parameter is often reported in its form for incompressible solutions

µ = 3∆ε II ∆ε I -∆ε III (5.9)
The purpose of Figure 5.14 is to inspect possible differences between the two definitions of Lode parameter. The Lode parameters µ ∆ε for four steps obtained from Figure 5.13(a) are compared with the assumption of the incompressibility. The values for the loading steps 3-4, 4-5 and 5-6 do not differ significantly since there is no large damage growth at this point. However, for the last reported DVC result (loading step 6-7) the difference is more important which outlines the existence of volume variations, i.e., incompressiblity is no longer satisfied when damage grows even in shear. 4) and ( 6)-( 7) with and without assumption of incompressibility

Microstructure changes

The laminography data obtained with high resolution (1 voxel ∼ 0.65 µm) enable us to follow damage changes in what is supposed to be pure shear stress states. the inspected region, most of them in the shear plane. The analysis serves only as ad hoc study to estimate whether there is (or not) any damage nucleation and growth during the loading process. It is confirmed that there is damage growth during loading in the shear plane.

State (4) State ( 5)

State ( 6) State ( 7) State ( 8) For a more detailed analysis, the voids are isolated with a segmentation procedure via thresholding and 3D views from the side of the sample are shown in Figure 5.16 [31,32]. One can follow the full thickness of the sample. Damage mainly cumulates in the bulk of the specimen, away from any of the free surfaces. Although the final failure comes Understanding, observation and quantification of ductile failure mechanisms via 3D imaging in a sudden manner, these representations of damage serve as justification of the sample design [35].

State (0)

State ( 1) State ( 2 .17 shows in more detail different damage features found in the shear zone. In particular, the elongated feature (blue arrow) with its significant size (along loading and thickness directions) seems to be an important factor that influences the final failure (Figure 5.17(a)). This could be the mark of grain boundaries, i.e., intergranular failure since similar elongated features are also found positioned normal to the loading direction (green arrows). Further, the final failure propagates along these elongated features as shown in the last picture of the sequence. Similar elongated features are found at other positions along the thickness of the specimen, one of those is shown in Figure 5.17(b). In addition to the elongated features, intermetallic particle breakage is also found (Figure 5.17(c)) but does not seem to influence significantly the final failure. On the other hand, the particle debonding as important damage micromechanisms under low stress triaxiality [14] is not found herein.

State (4)

State ( 6) State ( 7) State ( 7) State ( 7) State ( 7)

State ( 8) The nucleation mechanism for the elongated features is presumably due to void sheet-ing. It is depicted in Figure 5.17(d) and (e) for two different locations. In Figure 5.17(d) elongated features with already significant size are captured before the linkage into bigger segments. On the other hand, Figure 5.17(e) illustrates void sheeting of smaller and still spherical voids. Interestingly, this type of sheeting with small voids is found all over the section, the latter being relatively close to the sample surface.

Conclusions

In the reported work a region of interest in a flat specimen where local stress states are close to pure shear was imaged in 3D by synchrotron laminography under in situ mechanical loading. The main conclusions are:

• the ability of DVC in its global formulation to preform measurements in challenging conditions with high level strains localized in shear plane has been demonstrated. The correlation is possible thanks to the material natural contrast features, i.e., due to the white intermetallic particles embedded in gray Al matrix (see Figure 5.1). The low level of natural material contrast made the present situation more challenging. However, by means of initialization fields, successful DVC registrations have been obtained, which is confirmed in the residual maps (Figure 7.8).

• the strain to failure in the bulk of material is measured and it is ε f ≈ 0.65.

• incremental strain Lode parameter µ ∆ε fields as post-processed from incremental DVC displacements proved to be sensitive enough to confirm generalized shear stress states, i.e., values close to zero have been found in the shear band. The stress Lode parameter being proportional to strain rate Lode parameter indicates zero as target value for incremental strain Lode measurements reported in this work. This finding can be used in other cases, e.g., Compact Tension mode I opening.

• damage nucleating from the fourth loading step on has been identified in the scanned ligament, mostly concentrated within the shear plane away from free surfaces. The observation that the damage starts away from the notches is the most important finding and an additional proof of proper design optimization for shear sample used herein [35].

• damage features have mainly elongated shapes and cumulate in the bulk of the ligament. Along the elongated features, intermetallic particle breakages are also detected but with minor influence on the final failure of the ligament. Elongated features are suspected to be grain boundaries while void sheeting is the likely nucleation mechanism for them.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Effect of void arrangement on ductile damage mechanisms in nodular cast iron: in situ 3D measurements

Introduction/Motivation

A whole range of engineering materials has been optimized throughout history following the idea of avoiding brittle fracture. This quest resulted in signficant portion of modern materials failing in ductile manners. The constant quest for more fuel efficient and hence lighter design implies extended knowledge about the characteristic ductile damage mechanisms during forming and in-service conditions. Significant efforts in modelling of ductile failure processes have been made during the past few decades which resulted in substantial progress of knowledge [4]. However, the full understanding of ductile failure mechanisms under specific stress states still remains an open question. Low stress triaxialities (<1) induce phenomena that cannot be handled with classical ductile damage models [16,31] that are tailored for higher triaxialities. Phenomenological approaches where the strain to failure ε f is definend as a function of stress triaxiality η and Lode parameter µ [32, 1, 15, 49] yield satisfactory results in industrial contexts but still without general applicability. There is no robust ductile damage model that is able to correctly predict fracture loci ε f (η, µ) under arbitrary loading paths [12,11]. It has been demonstrated in previous works that both damage growth [36] and strain to failure values [3] are loading path dependent, i.e., nonproportional loading conditions lead to different ε f values for similar stress-averaged triaxialities.

The main reason for the above mentioned shortcomings is an incomplete understanding of ductile damage mechanisms. The lack of quantitative experimental data and their questionable interpretation result in case-sensitive model calibrations where the underying physics is not fully captured. 3D experimental methods, e.g., Synchrotron Radiation Computed Tomography (SRCT) [21,20,22,28], allow for in-situ monitoring of the three principal stages of ductile damage, namely nucleation, growth and coalescence of the voids at the microscale. While tomography is limited to scanning stick-like samples with small cross sectional areas, Synchrotron Radiation Computed Laminography (SRCL) [18,17,24,10,35] enables regions of interests (ROIs) in laterally extended objects to be scanned. In the context of mechanical experiments this feature facilitates performing in-situ experiments with sheet-or panel-like samples [51,25] with more engineering relevant boundary conditions, different levels of stress triaxilities and significant sizes of plastic deformation zones. Digital Volume Correlation (DVC) can be employed to measure 3D bulk kinematic fields. The measured displacement fields can be used for driving FE simulations that account for underlying microstructures [8,41]. The proposed method where the FE model accounts for the real microstructure and is driven by measured boundary conditions can be extended to inverse identification purposes at the microstructure level.

In this work in-situ laminography experiments are preformed on nodular graphite cast iron sheet samples that contain two machined holes. The geometry of the samples is inspired by the work of Weck et al. [48]. Two variants of the sample geometry will be studied, namely, a) two holes are machined at 45 • , b) two holes are machined at 90 • with respect to the loading direction. The general idea is to induce natural localization between the machined holes and observe whether there are any fundamental differences between Understanding, observation and quantification of ductile failure mechanisms via 3D imaging damage mechanisms in the localized band for the two cases. The material itself contains two particle populations, namely, graphite nodules considered as voids because of their weak mechanical properties [14,50,6,19] and the population of small particles present in the ferritic matrix before loading. Hence, after machining holes with a diameter one order of magnitude higher than that of nodules, the proposed configuration presents a rather unique opportunity to observe three void populations in a single experiment.

The laminography technique enables us to focus on void growth in the sheet for the first time under such boundary conditions while DVC will be used for measuring bulk kinematics. The estimated strain fields will be utilized for the determination of strain to failure values for 90 • and 45 • configurations in the ligaments between the nodules and on the scale of the macroband between the machined holes. Void growth in the examined region and individual growth of the chosen voids will be reported by using segmentation technique on laminography images. Additionally, mean void growth measurements will be supported by analyzing the volumetric part of the strain tensor. Finally, measured meso and micro coalescence and fracture criteria in terms of both laminography and DVC kinematic data will be analyzed.

Experimental methods

Sample preparation

The studied material is a commercial nodular graphite cast iron with the serial code EN-GJS-400 [19]. The specimen geometry, shown in Figure 6.1, is inspired by the work of Weck et al. [48]. Understanding, observation and quantification of ductile failure mechanisms via 3D imaging can be considered as distinct void populations with respect to the machined holes. In Ref. [48], the authors used machined holes of micrometer size. Consequently, the observation of the secondary void population was very limited due to the current imaging technique resolution [42]. In opposite, the larger size of the holes and graphite nodules in this work shall allow for such observations. Furthermore, the initial small graphite fragments in the ferritic matrix (<1 µm) can also be detected in laminography scans, therefore they can be considered as a third void population as illustrated in Figure 6.2(b).

In-situ Laminography

The load is manually adjusted via a displacement control in the testing device operated by screw rotation. After each load increment, a scan is acquired while the sample is rotated about the laminographic axis (i.e., approximately parallel to the specimen thickness direction). This axis is inclined with respect to the X-ray beam direction by an angle θ ≈ 60 • . The schematic view of the laminography setup together with the loading device are shown in Figure 6.2(a). The 3D images used in this work were obtained at beamline ID15A of the European Synchrotron Radiation Facility (ESRF, Grenoble, France) with white beam of about 60 keV average energy. Using 3,000 projection angles per scan, the series of radiographs Understanding, observation and quantification of ductile failure mechanisms via 3D imaging is then fed into filtered-back projection algorithm [30] to reconstruct 3D volume images. The parameter optimization has been performed automatically using a GPU-accelerated implementation [47]. For each scan, the reconstructed volume has a size of 1600×1600× 1600 voxels. The physical size (side length) of 1 cubic voxel is equal to 1.1 µm. The scanned regions are concentrated in the ligaments between and partly incorporate the two machined holes (Figure 6.1). Overall 24 scans were performed for the experiment with the 90 • configuration while 12 scans were done for the 45 • configuration.

Digital Volume Correlation

The reconstructed volume is represented by a discrete gray level field (8-bit deep) of the spatial (voxel) coordinate x. Using the whole volume size for the correlation procedure would be computationally too demanding. Additionally, to cover larger regions of interest (ROIs) for the DVC calculations the original volumes were binned, i.e., blocks of 8 voxels are replaced with the corresponding supervoxel having the mean gray level value of the block. Therefore an ROI of size 512 × 544 × 544 supervoxels for the 90 • configuration and 240 × 416 × 544 supervoxels for the 45 • configuration is used for DVC analyses. The positions of the ROIs in the global coordinate system are shown in Figure 6.1 while in the z direction the ROIs and samples mid-thickness planes coincide.

The basic principle of DVC [2,7,46] consists of registering the gray levels f in the reference and g in the deformed configuration such that their conservation is obtained

f (x) = g[x + u(x)] (6.1)
where u is the displacement field with respect to the reference volume. Due to the different noise sources in the experiments [27] the strict conservation of gray levels is never perfectly satisfied meaning the solution consists in minimizing the gray level residual ρ(x) = f (x)g[x + u(x)] over the whole ROI (i.e., global approach to DVC) by considering its L2-norm with respect to parameterized kinematic unknowns. The displacement field is written as

u(x) = ∑ p u p Ψ p (x) (6.2)
where Ψ p (x) are the chosen shape functions for the parameterization of u(x). Hexahedral finite elements with trilinear shape functions are chosen in this work [37]. After successive linearizations and corrections the current system to be solved reads

[M]{δu} = {b} (6.3)
where

M i j = ∑ ROI (∇ f • Ψ i )(x)(∇ f • Ψ j )(x) (6.4)
represents the DVC matrix, while vector {b}

b i = ∑ ROI ( f (x) -g(x))(∇ f • Ψ i )(x) (6.5)
Understanding, observation and quantification of ductile failure mechanisms via 3D imaging contains the current difference between the reference volume f and the corrected deformed volume g = g(x + ũ(x)), ũ being the current estimate of the displacement field. The global residual needs to decrease so that convergence is achieved, i.e., the displacement corrections {δu} become vanishingly small. All the results reported hereafter require the DVC analyses to have converged, namely, the root mean square (RMS) of the displacement increment between two iterations is less than 10 -4 voxel.

Definiton of strains and uncertainty

From the mean gradient of the displacement field over each 8-noded element the Hencky (logarithmic) strain tensor and subsequently its second invariant will be used in the sequel, the von Mises effective strain being

ε eq = 2 3 d : d (6.6)
where d denotes the deviator of the logarithmic strain tensor . The DVC resolution is evaluated by correlating two scans of the unloaded sample with (denoted "rbm" in Figure 8.7) and without (denoted "bis" in Figure 8.7) a small rigid body motion applied between the acquisitions. Due to noise contributions and reconstruction artifacts, these two volumes are not identical. Therefore, the measured displacement field accounts for the cumulated effects of laminographic imaging and DVC on the measurement uncertainty [27]. The uncertainty values are evaluated by the standard deviation of the registered strain fields. Figure 8.7 shows the standard strain resolution levels for different element sizes expressed in supervoxels (with respect to the original voxel size). Decreasing the element size is followed by an increase of the strain uncertainty. The element size used in this work is = 16 supervoxels (length) for all three directions, which yields a standard effective strain resolution of 0.78 % for the 90 • configuration and 0.73 % for the 45 • configuration. Since the trace of the strain tensor will be considered in the following, the corresponding standard resolution for the 90 • configuration is 1.81 % while

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging the 45 • configuration has uncertainty of 1.67 %. These values represent the limit below which the estimated strains are considered not trustworthy.

Direct vs. incremental DVC calculations

Direct DVC results refer to the solutions obtained by correlating the undeformed volume (state (0)) and one of the deformed configurations (state (n)), while incremental calculations stand for the correlation between any deformed volume (state (n)) and subsequent (state (n + 1)) configuration. Due to the large cumulated deformations the convergence of direct DVC calculations is hardly achieved for ultimate loading steps, hence an incremental correlation procedure is also followed. When displaying incremental calculation results the fields are shown in the reference configuration of each incremental analysis. Therefore, special care has to be exercised to identify common features to make the comparison objective. Since the global coordinate system is identical for direct and incremental calculations, the incremental displacement fields (from the end of the loading history) can be interpolated on the deformed mesh from the last converged direct calculation. By conducting such analysis one can extend DVC registrations from the undeformed configuration to the very end of the loading history. This operation is the basic principle of a cumulation procedure in postprocessing of the DVC results and will be used in the sequel [29].

Results

The mesoscale in the context of this work refers to the scale of the ligament between the two machined holes while the microscale stands for the scale of the ligaments between two nodule holes.

Ductile damage mechanisms 3.1.1 Laminography observations

.4 shows the mid-thickness section of reconstructed laminography data for the two configurations during the loading history [33,34]. For both samples the final coalesecence occured in the ligament between the machined holes. On the mesoscale, the 90 • configuration shows internal necking while in the 45 • configuration void sheeting prevails. However, one of the general ideas for this work is to study whether there are any differences in ductile damage mechanisms at the microscale between 45 • and 90 • cases.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging measurements mainly consists of void sheeting coalescences in the ligaments between the voids from graphite nodules that often have grown until this point under a plasticity mode involving high triaxiality.

In the 90 • configuration it seems that the influence of the second population (nodules) on the crack path is more pronounced in the sense that there is no regular macrocoalescence event (which here should occur normal to the loading direction connecting the two machined holes) but the final failure propagation is influenced by the graphite nodule positions. Overall, this interplay results in numerous locations (along the thickness direction z) where the final crack has a roof-top shape at the mesoscale. This type of macro-coalescence also often consists of local void sheeting events between the voids (nodules) but the prevailing mechanism is internal necking between them.

DVC residuals observations: nodules and ferritic matrix

Both samples experience early nodule debonding from the ferritic matrix. This mechanism is illustrated in Figure 6.5 by observing the final gray level residual ρ(x). The residuals are a useful tool for precise detection of damage occurrence in cast iron [45]. Let us recall that the corrected volumes are updated at each iteration on a sub-voxel scale by using gray level interpolation (in this work a trilinear interpolation scheme has been employed). Therefore, by using residual fields one benefits from sub-voxel resolutions in the gray level registration procedure. All deviations from C 0 continuity such as nodule/matrix debonding, void nucleation or growth cannot be properly captured by the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Effect of void arrangement on ductile damage mechanisms in nodular cast iron: in situ 3D measurements underlying kinematic basis. In consequence, the hypothesis of gray level conservation is no longer satisfied, which will be reflected in local residual increases. Figure 6.5(a) shows residual maps for the section normal to the y-direction for load step (0)-( 5) for the 45 • configuration. The residual map is shown in overlay with the corresponding microstructure in the background. Note that areas of higher residuals correspond to the positions of the debond zones around nodules. The irregularly shaped nodule (denoted with a white arrow) acts as spot with concentrated residuals, i.e., specifically expressed local debonding, caused probably by high stress concentrations around this spot. Beside debond zones with higher gray level residuals, the residual map in the rest of the ROI shows sufficiently low values indicating successfully converged DVC calculations. In Figure 6.5(b) residual field maps are shown with microstructure sections (see Figure 6.4) for the 90 • configuration. Note early debonding from load step (0)-( 4) onwards. Upon further loading, the residuals mainly cumulate in the ligament between the two machined holes with their peak values along the micro-localization band as shown in Figure 6.8 that will be commented in the sequel.

A 3D representation of the residual fields is illustrated in Figure 6.5(c) where its changes are observed in isometric view with a slightly inclined z axis of the sample toward the observer. The residual fields are from the 90 • configuration. It is explicitly confirmed that most of the residual source is from debond nodules in the part of the ROI lying between the two machined holes.

SEM fractography

There are three distinct coalescence mechanisms found in post mortem observations on fracture surfaces: a) complete internal necking; b) incomplete necking and c) large void sheeting surfaces. Complete necking designates void growth from debond nodules characterized by high plasticity levels that occur on free void surfaces. It is marked with smooth fractured surface spots without additional small voids in the ligaments. When the distance between the nodules is sufficient they do not coalesce completely by a high stress triaxiality mechanism but void sheeting occurs at the very end, which is marked with an additional small void population (no small particles were found inside these voids) resulting in uncomplete necking. At the end, larger surface spots with small void populations are found between the machined holes and some of the nodules when the distance between the two is significant. These mechanisms can be found in both hole configurations. They are functions of stress state but also of morphology/topology context (i.e., distance between the nodules itself, nodules and machined holes, nodule shapes and orientations). On the other side, Zoom G shows the complete internal necking mechanism, i.e., without any additional nucleation in the ligament between the nodules. The fully sheared spots (i.e., without any voids, with final shear surface alignement) are not found herein.

Interestingly, even under high magnifications, no particles have been found in the pores belonging to the void sheeting zone of the fractured surface. If there are any small particles inside, it can be well recognized on the fracture surface as shown in Figure 6.7 (marked with green arrow). Although the objective particle is very small in size, the corresponding spot is marked with significanlty higher void growth and associated plastic deformation. Hence, if there are any particles, i.e., stress concentrators present, they are easily distinguished from the empty ones among the void sheeting population. In this example most of the void sheeting population remains small in size and no particles can be found inside them meaning they probably do not nucleate on the particles but rather as outcome of large plastic deformations of the matrix and associated accumulated dislocations.

As a result of all previously mentioned observations, it can be stated that in 45 • and 90 • cases the underlying ductile damage micromechanisms are not fundamentally different, i.e., they fail locally in a similar manner. As a result of cumulated plastic flow, the voids develop from nodules and re-position to favorable arrangement as precursor for fast 45 • coalescence between the two machined holes in the last scan. However, it is hard to judge what is the role of the nodules and the corresponding microstrained bands (between some of them) on the final macrocoalescence event since it seems it is mainly a mesoscale effect determined by the machined hole positions and not influenced by the local nodule positions.

In the 90 • configuration (Figure 6.8(b)) the final failure is following more clearly the strained band positions between the nodules determined already at the beginning of the loading process. The strained bands are always containing the voids, but not all of the nodules (in the area of the ligament between the machined holes) are equally affected by the band paths. Moreover some of the nodules remain unaffected.

The fact that the final failure does not follow the path of microbands in the depicted section for the 45 • configuration suggests that a significant amount of plastic deformation has been dissipated without direct contribution to the final coalescence propagation. Hence, the process zone is signficantly wider than the crack plane itself. However, large amount of plastic deformation decreases the material hardening rate inside the ligament areas between the nodules making it easier for macrocoalescence to occur in such a sudden manner for the 45 • configuration.

3D analysis

While explaining the discrepancy between the microbands and the final failure trajectory one should keep in mind that this study deals with a 3D effect and the microstructure overview along the thickness directions z can help one to better understand the previous observations, especially in the 45 • configuration as shown in Figure 6.9. In particular, clusters of nodules favorably positioned are found in the neighborhood (on both sides) of the mid-thickness slice shown in Figure 6.8(a). In other words, these nodules are located roughly at 45 • already from the undeformed state. In Figure 6.10 von Mises strains are plotted over the corresponding microstructure section at the aforementioned favorably positioned nodule cluster. Studying plastic activity before mesocoalescence reveals general agreement between the microband positions and the final crack propagation path. This case exemplifies the importance of taking into account the real 3D microstructure compared to periodic distributions often used in micromechanical studies, see e.g., Refs. [44,43,13]. Correlated to this observation, the use of DVC measured heterogeneous boundary conditions for driving micromechanical calculations shows significant improvement in both global and local errors between the simulations and experimental data [41].

The same analysis is repeated for the 90 • configuration in Figure 6.11. Going through the sample thickness reveals that there are numerous spots where roof-top (or V-like) shaped final cracks occur.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging One of these spots is depicted and followed during the loading history together with the corresponding von Mises field overlays in Figure 6.12. It is confirmed that in the 90 • configuration the mesoscale strain localization effect of machined holes is less imposed, i.e., the final coalescence is rather determined by microbands between the nodules that often do not correspond to the shortest distance between the two machined holes. In this context, the 90 • configuration is more influenced by 3D nodule distribution than the 45 • configuration for which mesoscale plastic flow governs most of the process.

The effect of bulk constraints is probed in Figure 6.13 where the strain maps for specific loading steps are plotted over the corresponding microstructure for three different locations along the sample thickness, two of them being closer to the free-surfaces while the middle one is roughly the sample mid-thickness. In the 90 • configuration microbands spread wider when close to the surface where less constraint is present. Conversely, in the bulk the influence of the two holes is concentrating most of the plastic activity in the narrow ligament between them. In the 45 • configuration this is not the case. The bands are strongly concentrated in the ligament irrespective of the section position.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Effect of void arrangement on ductile damage mechanisms in nodular cast iron: in situ 3D measurements

Quantification of the results

In the following section the mean and local strain values are extracted from the measured kinematic fields. Those values are then correlated with the corresponding microstructure changes, i.e., void growth and coalescence effects from laminography data. Since for the late loading steps direct DVC registrations could not be conducted, incremental calculations are performed. Results from these correlations are cumulated to the latest sucessfully converged direct calculations. This operation results in a situation with a part of the boundary nodes from the initial DVC ROI without kinematic information. Hence, a priori analysis is performed and only a part of the intial ROI domain, referred to as sub-ROI (Figure 6.14), with available nodal displacements up to coalescence is used in the following analysis.

Mean kinematic values on sub-ROI level

In Figure 6.14 the relationship between the nominal machine displacement and the mean von Mises strain (mean over the sub-ROI) are reported for the analyzed sub-ROIs of the two samples. The graph is accompanied with microstructure sections for important experimental instants. The last reported values are those immediately preceding overall coalescence. They will be denoted as coalescence strains in the sequel. Note the substantially higher effective strains in the 45 • configuration, therefore exhibiting higher mesoscale ductility.

The coalescence strain level for the 45 • configuration corresponds well with that reported in the literature [48] on Al alloys, i.e., 27% but this may be coincidental. The 90 • configuration has substantial lower coalescence strain (i.e., 16%) than reported in Ref. [48] (i.e., 25.1%), thereby indicating weaker mesoscale geometry effect and important contribution of the material heterogeneites from the mesoscale (nodules) not present in the tested Al alloys [48]. Also, it is noteworthly that the reported data are mean values of the strain fields while the DVC ROI consists mostly of material points inside the macroband for the 45 • sample, while in the 90 • case a significant part of the ROI material points is out of the macroband between the machined holes.

Standard deviations are added to the analysis. The fact that there is almost a linear relationship between the mean and standard deviation values reveals the stable nature of the heterogenity of the strain fields.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Both samples experience positive and stable strain trace from the beginning of the loading process. In the second part of loading history, for the same levels of mean von Mises strains, the 90 • configuration experiences higher strain traces, which indicates higher void growth (∑ ε ii = 0 for constant volume). This result is depicted in Figure 6.17 The ratios between the normal strains ε xx , ε zz and the normal strains in the stretching direction ε yy are shown in Figure 6.17(b). The 90 • configuration experiences stronger necking along the thickness direction. This can be explained by the wider macroband shown in Figures 6. 13.

Conversely, the 45 • configuration has significant relative necking in the x-direction. In the first part of the loading history ε xx /ε yy is close to 0.5 indicating there is low constraint in the x-direction due to the machined holes. However, combined with the low ε zz /ε yy ratio it results in non-isochoric deformation with positive trace from the begining of loading, i.e., increasing volume. Although the normal strain ratios vary significantly during loading, the sum of the two ratios is relatively stable and amounts to ≈0.74 for both configurations.

However, this result does not imply a plane strain condition along the thickness direction in the 45 • configuration as shown in Figure 6.24 (Appendix) where the normal strains ε zz in the thickness direction are plotted for three different positions along the ligament between the holes for the two samples. Both 90 • and 45 • configurations exhibit (mostly) contracted and some zones around nodules with higher resistance to lateral deformation. Although present from the early loading steps on this type of straining between some of the nodules differs from those reported for Al alloys in slanted regions of Compact Tension samples where the plane strain condition along the crack propagation direction has been identified [26,9]. Figure 6.25 (Appendix) reveals that there are normal strains ε xx in the crack propagation direction for both 90 • and 45 • configurations. Nodular graphite cast iron represents another class of material compared to Al alloys and straining between the nodules is mainly a structural effect. Yet, it is confirmed herein that the plane strain condition is not a necessary condition for very early and stable straining.

If the voids are modeled in FE simulations, the J2 flow theory for the ferritic matrix is capable of capturing local straining between the nodules as a structural effect [8,41]. However, Al alloys (21xx series) in plane strain condition develop slant strained bands as pure material response, which is not predictable with Von Mises plasticity, e.g., in ductile tearing [26]. In Figure 6.18 points C and D (i.e., representatives of the internal necking ligaments) are also analyzed. Again, similar values for the coalescence effective strains for the two samples (dashed lines) are found (≈45 %). Note that these levels are close to those obtained in the void sheeting ligaments and seem to be independent of the ligament failure type. In other words, this result underlines the ability of effective von Mises strains to locally predict material failure. However, it remains insensitive to the type of damage mechanisms and final ligament failure, i.e., these items did not reflect in von Mises strains changes (i.e., the slope of the curves for different mechanisms are similar) nor its final values. It is the local nodule arrangement of the real microstructure that is of prime importance, then the nodule shapes and finally matrix defects that will induce different stress states responsible for different ligament failures.

The local strain trace values exhibit some sensitivity to different ligament failure mechanisms (Figure 6.19). For each of the configurations, the two different failure mechanisms can now be distinguished due to the different strain trace levels and ultimate values.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Results

215

This is especially true for the 45 • configuration. Although they fail in a similar manner as shown in Figure 6.14 the strain trace levels for points A and B differ substantially. The same observation is valid for points C and D meaning that the local trace cannot be used as global threshold quantity for different mesostress states, but when limited to the defined mesostress state (in this case 90 • or 45 • configuration) it is a helpful physical quantity for the distinction of the two ligament failure mechanisms.

This example also illustrates the importance of proper scale modeling since on the ligament level both sample geometries failed in a similar manner as already shown in post mortem analyses but not with an identical kinematic history (e.g., different strain trace values). The real added value of this analysis to the SEM images is the possibility of following the kinematics/microstructure changes during the whole experiment. The common way of quantifying the void volume change/damage growth is by segmenting gray level images and following the number of remaining voxels that normally should belong to voids [23]. The analysis based on this approach is performed in this work but with certain improvements. First, by means of DVC measured kinematic fields the initial set of material points is followed throughout the experiment (i.e., a truly Lagrangian measurement). Second, the good choice for gray level threshold, which is often subjective, is probed by employing the measured mean strain trace. Void growth obtained with different thresholds from laminography data (δ f ) is compared with the volume change calculated by the mean strain trace multiplied by the current sub-ROI volume (δV ). The results are

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

shown in Figure 6.20 for the two analyzed samples. The best agreement is obtained for a threshold of 115 gray levels for both samples. However, if one performs more sophisticated procedures while creating binary images that include several operations from image analysis procedure, namely, placing seeds on nodule positions followed by limited threshold growth and fill hole operations on binary images two additional curves are obtained and marked with thicker red lines in Figure 6.20 (denoted as δ f s80u115 and δ f s80u125). When compared to δV values from the DVC analysis the gap is somewhat higher than that obtained with δ f 115 curve.

The reason for this effect is probably due to the fill hole operation in Fiji software [38,40,39] whose effect is illustrated in Figure 6.21. Again this operation prevents the attempts of better calibration of this threshold method to δV values since the fill hole step compensates reducing of the threshold to lower values. Nonetheless, the latter method for thresholding has physically the best sense as shown in Figure 6.21 and the values for δ f s80u115 from Figure 6.20 will be used in the sequel. Kinematic data (strain trace) are shown to be a useful tool for the initial estimation of the proper gray level values for thresholding methods when calculating void growth from tomography/laminography data.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging microstrucutre section threshold GL=115 seed + growth method + ଏll holes GL=115 The void volume fraction is obtained by division with the current sub-ROI volume during the experiment. Hence, when reduced to the corresponding mean effective strain levels, the void growth rate is higher in the 90 • configuration. A change of slope is observed at ≈13% strain, thereby denoting accelerated void growth prior to coalescence while in the 45 • configuration such trend is not found. This goes along with the mesoscale point of view where the 45 • configuration should experience lower void growth and coalescence without any warning in terms of increased damage growth rate. The void volume fractions prior to coalescence are similar for both cases, namely, ≈16%. On the left side of the figure void sheeting ligament voids for 90 • (A1) and 45 • (B1) configurations are followed through the experiment and accompanied by void representations isolated from the original laminography volumes. The 90 • configuration void (magenta) experiences stable growth up to 30 % of local strain when the first signs of internal necking occur on the left side of the void. In the following loading step the void is linked by internal necking with numerous surrounding voids, while many of them are further connected with their subsequent neighbors creating a signficant network of linked voids. This is the reason why the mesurement is not conducted here. One additional figure is added showing just part of the mentioned linked network containing a wide coridor (shaded) between the two mini-void-clusters where sheet coalescence will occur in the following step. A similar scenario is observed for the 45 • configuration void (yellow). Just before the onset of void sheeting shown on most right image, the followed void also interacted with the closest neighbor in an internal necking linkage mode.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

The right side of the figure shows internal necking ligament voids for 90 • (C1) and 45 • (D1) configurations. In the 90 • case the void (blue) grows individually up to 22 % of the measured local strain when it starts to interact with its environment. In the next loading step the void representation shows the onset of internal necking linkage between the monitored void and its neighbor. Note that point C is placed between these two voids (Figure 6.18). Since complete linkage has not occured and only the two illustrated voids are interacting, the measurements are carried on up to the point when more general coalescence occured as illustrated at the ultimate measured point. The 45 • void (red) experiences more stable growth to the very end of the measurements. Only for the last step the studied void coalesces with the group of voids and the linked network is established. The measurement is still performed but the monitored void is previously isolated from the network. The same isolation procedure is followed in all other cases where coalescence partly occurred but measurements were still reported, i.e., only the volume/surface of the controlled void is reported, not those from the whole linked assembly of voids.

Conclusions

The effect of the arrangement of two artificial machined voids on ductile void growth and coalescence mechanisms is studied in flat specimens made of nodular graphite cast iron. The damage changes are imaged via in situ 3D laminography. Digital volume correlation is employed for bulk strain measurement. Two configurations of machined holes are studied i) 90 • with respect to the loading direction to mimic internal necking and ii) 45 • to trigger void sheeting mechanisms between the artificial voids. The machined holes are supposed to play the role of primary voids, the graphite nodules the role of secondary voids. Fractography revealed at a lower scale a smaller third population of voids.

On the mesoscale, it is shown that the 45 • artificial void configuration reproduced the expected shear strain band. For the coalescence mechanism the secondary voids (nucleating from nodules) played a minor role, i.e., the final crack path was not affected by the nodules. In the 90 • configuration, a more diffuse plastic zone between the machined holes favored the influence of secondary (nodule) voids on the crack path, namely, the final failure followed the early prescribed localization paths between nodules that, on occasion, did not represent the shortest distance between the primary voids.

On the microscale, secondary voids (from nodules) in both configurations tended to initially grow in high triaxility mode and interact with the closest neighbors in an internal necking mode. At the very late stages of deformation just prior to the final failure void sheeting coalescence took place in some locations; in the 45 • configuration mainly because of the mesoscale strain band effect determined by the primary void positions while in the 90 • sample void sheeting is the outcome of the favorably positioned clusters of already linked voids that finally needed to coalesce to form the main failure crack. The SEM figures confirmed these micromechanisms at smaller scales in both configurations (Figure 6.6). Nanometric dimples are seen on the fracture surface but their origin remains undetermined. The main conclusion from this part is the complexity of multiscale and Effect of void arrangement on ductile damage mechanisms in nodular cast iron: in situ 3D measurements strong 3D nodule distribution effects revealed in the two samples. The mean effective strain on the level of the sub-ROI (i.e., most of the ligament between the machined holes) shows higher ductility of the 45 • configuration with values close to 27 %. The 90 • sample shows substantially lower ductility on the mesocale, 15% being the value prior to general coalescence (Figure 6.14). This is attributed to the more important role of the secondary voids in the 90 • configuration. Interestingly, the mean trace of the strain tensor, i.e., volume change, does not differ significantly for the two cases (Figure 6.19), which corresponds to early debonding and void growth on the microscale. The overall cumulated plasticity is higher in the 45 • configuration mainly because of the shear component ε xy (Figure 6.16).

For the microscale internodule coalescence analysis two different ligament types for both cases were depicted, namely, one of them experiences void sheeting while the other classical internal necking coalescence. Here the local von Mises strain levels prior to coalescence were substantially higher than that of the mesoscale, but they are similar for both samples (void sheeting ligaments ≈40 %; internal necking ligaments ≈45 %, see Figure 6.18). The normalized individual void growth values in the immediate vicinity of the mentioned ligaments were determined. The results, shown in Figure 6.23, outline that even though the ligaments will end up with the same type of failure, void growth of boundary voids is higher in the 90 • configuration for both types of coalescence mechanisms.

Last, let us note that the present data will be used in the future as input and validation data for micromechanical modeling to probe plasticity and damage models using real microstructures and measured boundary conditions [41].

Introduction

In many applications, metals and alloys for in-use analyses and metal forming raise important issues for ductile fracture, which have led researchers to investigate more and more advanced damage models. A first category of damage models, known as macroscopic models [31,52,38], is used to predict not only damage occurrence but also the softening and transition to fracture. Due to their macroscopic nature, these models are known to have limited predictive capabilities and are usually calibrated and applied for specific loading conditions. For applications such as material forming, where loading may be complex and non proportional, these limitations are problematic [14,13].

A promising alternative is provided by microscopic models [20,47], where the macroscopic constitutive behavior is derived thanks to the homogenization of microscopic calculations. This derivation may be purely analytical [20] or aided by numerical calculations on ideal microstructures [47]. However, the predictive capabilities of these microscopic models are also known to be limited for arbitrary loading conditions [14,13], due to quite restrictive assumptions that are used for their derivation [20,47]. The calibration of these models is a challenging task that is being addressed in the literature using more advanced experimental and numerical procedures. Some variables such as porosity can now be observed experimentally thanks to X-ray imaging techniques [1,8,69,22] with the possibility of considering inclusions and voids individually based on manual [69] or automatic [22] procedures.

Further, simulations are necessary to experimentally relate observed quantities such as porosity and number of fractured/debonded inclusions to mechanical variables such as plastic strain and stress-based criteria. These calculations at the microscale are usually conducted with idealistic microstructures and constitutive behavior, considering homogeneous kinematic or static boundary conditions, which cannot capture the local strain and stress states that inclusions and voids are subjected to, due to their random shapes and distribution [1,44,66,22]. The effect of the three-dimensional random distribution of voids on softening has been studied [18] using different void volume fractions.

The main objective of the current work is to enable for more realistic microscopic calculations thanks to physical models calibrated directly at the microscale to analyze the three steps of ductile damage, namely, void nucleation, growth and coalescence. To achieve such goal, a first step consists of developing a numerical framework allowing for such validation and identification. The proof of concept of such validation framework is the topic of the present paper. The studied material is nodular cast iron, which at the microscale features a ferritic matrix and graphite nodules, with no significant porosity in the initial state. Then, under tensile loading, ductile fracture is mainly driven by a very early debonding of the nodules from the matrix, and coalescence of the subsequent nucleated voids [17,30,68]. Evidence in the literature [17,74,6,30] suggests to model the nodules as voids, as their load carrying capacity is very low under tensile loading. This assumption is made herein and at the microscale the material is considered as a two-phase microstructure with a ferritic matrix and voids.

The final objective of the present work includes enabling for discussions regarding

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging such assumptions, thanks to accurate and local error measurements directly at the microscale. This would help the development of experimentally probed microscopic models that could then be used to improve the common knowledge and understanding of ductile fracture, and deduce more accurate macroscopic responses. In particular, the ultimate aim is to set up a framework that allows failure and coalescence mechanisms and criteria such as internal necking or coalescence via nucleation of voids on a second population of particles [28,29] to be assessed numerically. The methodology proposed to obtain these local comparisons between experimental analyses and numerical simulations is based on the following steps (Figure 9.1):

• X-ray laminography to get 3D pictures of an in-situ test in a synchrotron facility. By post-processing them, one may get, for instance, a first estimate of the morphology of the two-phase microstructure.

• Global digital volume correlation (DVC) to measure displacement fields whose kinematic basis is made of the shape functions of 8-noded elements. These displacement fields serve two purposes. First, they correspond to the kinematic data of the test. Second and more importantly, they will be used as Dirichlet boundary conditions of Finite Element (FE) simulations at the microscale.

• The FE simulations at the microscale explicitly account for the morphology of the studied two-phase material. Therefore the mesh made of 4-noded tetrahedra is adapted to the microstructure with a Level-Set (LS) procedure.

• FE simulations are run with an elastoplastic constitutive equation to model the nonlinear behavior of the matrix.

• Comparisons between experimental measurements and numerical simulations are carried out for the displacement fields and correlation residuals.

Synchrotron radiation computed laminography is a non-destructive 3D imaging technique for scanning regions in laterally extended 3D objects [25,24,43,12,54]. This geometry is of particular interest in the field of mechanics of materials since sheet-like samples allow for a wide range of engineering-relevant boundary conditions. Laminography is related to the inclination of the specimen rotation axis with respect to the beam direction with an angle θ < 90 • ; while tomography, as a special case of laminography, is performed at the 90 • angle during the scanning procedure and requires stick-like samples. However, due to the rotation axis inclination angle, incomplete sampling of the 3D Fourier domain of the specimen is performed and additional artifacts are present [23] but often they are less disturbing than the ones of tomography [72].

Laminography and tomography are nondestructive imaging techniques that are used in material sciences. They enable the material microstructures and their degradation mechanisms to be assessed qualitatively and quantitatively [41]. In-situ experiments can also be performed to analyze the mechanical behavior of different materials [21,11,32,15]. Initially developed at synchrotron facilities [3], these tests are also performed at lab tomography set-ups [10]. The 3D images are acquired at different steps of the loading history while the tested material is still under load for visualization and measurement purposes. For instance, 3D displacement fields can be measured via DVC. DVC is a registration technique to measure 3D displacement fields in the bulk of samples. In its early developments [2,62,7,70], DVC consisted of independently registering small interrogation volumes in the considered Region of Interest (ROI). This type of approach is nowadays referred to as local, i.e., the only information that is kept is the mean displacement assigned to each analyzed Zone of Interest (ZOI) center. Various kinematic hypotheses, possibly accounting for ZOI warping, have been implemented. However, they are different from those made in FE simulations in which the field is continuous and dense over the whole ROI. Global approaches, which appeared more recently [58], consist of performing registrations over the whole ROI by using kinematic fields based on finite element discretizations. Such DVC approaches can be directly linked with numerical simulations of mechanical problems [53,9].

Examples of techniques enabling real microstructures to be meshed can be found in the literature [75,73]. Solutions are yet to be formulated to reduce the cost of these procedures both in terms of memory requirements and computation time. The LS method [49] is a useful tool to represent interfaces in an FE simulation when large deformations and complex topological events occur [63,57,51], though some regularity [61] and conservation [60] issues have to be handled. Distance maps and LS functions have also been used in the field of image processing to extract interfaces and topological data [34]. If FE simulations are to be performed with these interfaces, image processing can be performed directly on the FE mesh, thereby reducing memory requirements for high resolution images and enabling the parallel capabilities of standard FE codes to be used [61].

In the past, 3D meshes stemming from X-ray tomography data have been created for fatigue loading cases e.g. to assess the stress intensity factor ahead of a crack [53]. In Ref. [50] the 3D grain structure and orientation has been assessed via diffraction contrast tomography and subsequently been meshed in 3D to assess the effect of crystallographic orientation on fatigue crack propagation numerically and experimentally. To the best of our knowledge, there are no studies that explicitly mesh real microstructures for the case of ductile damage.

The paper is structured as follows. The experimental setup and the laminography imaging technique are first introduced. The basic principles of Digital Volume Correlation incorporating uncertainty quantification and DVC results are then presented. FE simulation tools including the microstructure meshing procedure and the strategy for applying DVC measured boundary conditions are described next. Last, the results from both methods are compared relatively via kinematic field subtractions and absolutely by computing gray level residuals.

Experimental analyses 2.1 In situ test

The material used in this study is a commercial nodular graphite cast iron (EN-GJS-400). The geometry of the flat specimen with a central hole is shown in Figure 7.2(a). The specimen geometry yields stress triaxialities within the range 0.4-0.5 in the vicinity of the central hole [56]. A stepwise loading procedure resulted in the force-displacement curve shown in Figure 7.

2(b).

The 3D images used herein were obtained at beamline ID15A [16] of the European Synchrotron Radiation Facility (Grenoble, France) with a filtered white beam (average energy around 60 keV), using 4,000 projections per scan. Due to the detector design and the angle between rotation axis and beam, the minimum specimen to detector distance is 70 mm. The resulting strongly contrasted edges are also expected to contribute to laminography artifacts typical for incomplete sampling. All these features induce increased DVC uncertainties compared with the measurement resolutions for tomography [45].

The testing device applies the load by controlling the displacement via manual screw rotation where grips incorporate a force measuring device. After applying each loading step, a scan is acquired while the sample is rotated about the laminographic axis (i.e., Understanding, observation and quantification of ductile failure mechanisms via 3D imaging 3D Imaging (b) Macroscale force-displacement curve parallel to the specimen thickness direction). This axis is inclined with respect to the X-ray beam direction by an angle θ ≈ 60 • . The series of radiographs acquired is then used to reconstruct 3D volumes via filtered-back projection [46]. The scanned region is positioned next to the central hole whose edge is used as orientation feature during the scanning procedure (

Basic principles

DVC used herein is an extension of 2D global Digital Image Correlation [4,27]. The reconstructed volume is represented by a discrete scalar matrix of spatial coordinate x of (in this case 8-bit deep) gray levels determined by the microstructure absorption of X-rays (Figure 7.3). The image contrast is mainly due to graphite nodules of micrometer size. There are also smaller dark features in the ferritic matrix that can either be ). If all segmented small particles are considered the average number of particles per element is 1.6. However, if particles smaller than 1 or 5 voxels are considered as noise, the values are 1.1 and 0.6 particle per element, respectively. Therefore there is contrast within the ferritic matrix that can be registered and allows displacement fields to be measured between nodules. Let us note that the filter has been applied to the 3D image, which has evened out voxel-scale contrast. Conversely, no filtering is performed for the correlation procedure and it may thus be expected that submicrometer contrast also contributes to the determination of the kinematic fields. The principle of DVC consists of registering the gray levels f in the reference configuration x and those of the deformed configuration g, which satisfy

f (x) = g(x + u(x)) (7.1)
where u is the displacement field. However, due to acquisition noise, reconstruction artifacts [72] and the correlation procedure itself [37], ideal match is not achieved in real examples. Consequently, the solution consists of minimizing the gray level residual

ρ(x) = f (x) -g[x + u(x)] (7.2)
by considering its L2-norm with respect to kinematic unknowns. Since a global approach is used herein [58], C 0 continuity condition over the ROI is prescribed to the solution and

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging the global residual to be minimized reads

Φ 2 c = ∑ ROI ρ 2 (x). (7.3)
The displacement field is parameterized by using a kinematic basis consisting of shape functions Ψ p (x) and nodal displacements u p

u(x) = ∑ p u p Ψ p (x). (7.4)
Among a whole range of available fields, finite element shape functions are particularly attractive because of the link they provide between the measurement of the displacement field and numerical models [53]. Thus, a weak formulation based on C8 finite elements with trilinear shape functions is chosen [58]. After successive linearizations and corrections the current system to be solved reads

[M]{δu} = {b} (7.5)
where

M i j = ∑ ROI (∇ f • Ψ i )(x)(∇ f • Ψ j )(x) (7.6)
represents the DVC matrix, while vector {b}

b i = ∑ ROI ( f (x) -g(x))(∇ f • Ψ i )(x) (7.7)
contains the current difference between the reference volume f and the corrected deformed volume g = g(x + ũ(x)), ũ being the current estimate of the displacement field.

The global residual needs to decrease so that convergence is achieved, i.e., the displacement corrections {δu} become vanishingly small. In this form, the displacement field is regularized in the sense of a continuity requirement that is a priori assumed for the kinematic solution. Moreover, additional mechanical knowledge may be added to help convergence. This type of procedure is referred to as mechanical regularization [65]. In the present analyses, mechanical regularization based on the satisfaction of the elastic solution for the inner nodes in the calculated domain has been introduced. In the absence of body forces it implies

[K]{u} = {0} (7.8)
where [K] is the rectangular stiffness matrix associated with the inner nodes. Displacement fields u that do not satisfy the equilibrium will give rise to a gap that will be minimized in addition to the correlation residuals

Φ 2 m = {u} t [K] t [K]{u}. (7.9)
Minimizing the equilibrium residuals incorporates the mechanical admissibility in the solution of the measured kinematic field. However, the previous strategy cannot be used as

Φ 2 b = {u} t [L] t [L]{u} (7.10)
where [L] behaves as a mathematical operator eliminating non-physical high frequency solutions. Since they do not have the same physical units these functionals cannot be added directly [36]. For this reason, a reference displacement field v is needed to normalize them. Additionally, weights w m and w b are given to the normalized functionals, thereby defining regularization lengths that act as cut-off wavelength of low pass mechanical filters [36]. The regularized correlation procedure consists of iteratively solving linear systems

([M] + [N]){δu} = {b} -[N]{u} (7.11)
where

[N] = w m {v} t [M]{v} {v} t [K] t [K]{v} [K] t [K] + w b {v} t [M]{v} {v} t [L] t [L]{v} [L] t [L]. (7.
12)

The normalization displacement field v is associated with a wave vector k so that the two weights are defined as [65]

w m = (2π k r ) 4 , w b = (2π k b ) 4 (7.13)
where r and b represent regularization lengths. By tuning the regularization length the influence of the mechanical regularization functional can be varied, resulting in more or less smooth fields. In strongly regularized solutions, high spatial frequencies are filtered out when not mechanically admissible as part of an elastic solution. Hence, the regularization procedure makes use of constitutive laws and helps correlation registration to converge even in the close to zero-gradient gray level zones as found in the cast iron matrix. Finally, let us stress this does not imply that the final solution is elastic in nature, which will be observed in the case of cast iron. However, regularization can still be used as an ad hoc solution to the problem, giving a rough, but fairly good estimate of the correct solution. This solution is then used as initial guess in the subsequent calculations with decreased regularization lengths. This so-called relaxation procedure [67] is followed herein until the overall weights put on Φ m and Φ b are equal to zero, i.e., no additional requirements besides gray level conservation are assumed. During the relaxation procedure the r / b ratio was equal to one.

Direct vs. incremental DVC calculations

Since stepwise loading has been applied, direct DVC results refer to the solutions obtained by correlating the undeformed volume (state (0)) and the corresponding deformed configuration (state (n)), while incremental calculations stand for the correlation between any reference volume (state (n)) and subsequent (state (n + 1)) configuration. Due to the small number of scans available for the experiment, high levels of displacements occur between two consecutive acquisitions. The convergence of direct DVC calculations

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging is hardly achieved under such conditions, hence an incremental correlation procedure is followed (Figure 7.5).

State (0)

State ( 1 Second, all the results can be expressed in a Lagrangian way by re-projecting back all the measured incremental displacement fields on the undeformed frame (by subtracting neighboring direct calculation results). In the sequel, the first approach will be used but special care has to be exercised to identify common features to make the comparison objective.

All the results reported hereafter have required the DVC analyses to converge, namely, the root mean square (RMS) of the displacement increment between two iterations is less than 10 -4 voxel. An additional check is related to the residuals ρ that will be shown in the sequel.

Link between DVC and numerical simulations

The measured displacements are expressed on a regular undeformed mesh made of 8noded cube elements. Full-field results written node-wise are provided as .txt output files of DVC and will be used as boundary conditions for the FE mechanical model. Hence, for the first incremental loading step (0)-( 1) the results are written on the undeformed frame (0), for the second ( 1)-( 2) on the mesh associated with the microstructure at state (1), and so on. An example is shown in Figure 7.5. Since undeformed meshes (blue DVC ROI) with constant (initial) size are considered on the microstructure state (n -1) for loading step (n), with its positions defined relatively to the followed microstructure features, some of the material points present in the DVC ROI at the beginning are lost during the loading history. These regions are depicted with white dotted rectangles in the figure. Hence, if all the incremental displacement fields are cumulated to the initially defined nodes, it will end up with a partial solution. Since the material points used for FE computations need to possess a complete kinematic information through the loading history, just an inner part of the original ROI has been utilized for communication between Understanding, observation and quantification of ductile failure mechanisms via 3D imaging 3D Imaging DVC and FE simulations (red FE ROI), where the ROI size reduction especially refers to the loading direction (y) for which the displacements are the largest.

Once defined, boundaries of the domain are used to extract the corresponding part of the DVC displacement solutions. These displacements are interpolated on the unstructured tetrahedral mesh (T4 mesh) used in numerical simulations. They are used as Dirichlet boundary conditions in the FE simulations. Additionally, since the underlying experiment and consequently DVC calculations are obtained with a small number of loading steps, linear interpolation in time is necessary to properly prepare boundary conditions for FE computations.

Uncertainty quantification

Before presenting any result, an uncertainty assessment is performed. It consists in analyzing two consecutive scans of the unloaded and loaded sample without applying any additional load (i.e., denoted "bis") or prescribing rigid body motions between the two acquisitions [45]. For the present experiment, three independent "bis" cases are available, namely, "0 mm" in the undeformed state and two more in deformed states: "0.579 mm" and "0.744 mm" of the screw displacement (see Figure 7.2(b)). Due to the noise contribution and reconstruction artifacts, these two volumes are not identical. Therefore, the measured displacement field accounts for the cumulated effects of laminography and DVC on the measurement uncertainty. From the mean gradient of the displacement field over each C8 element the Hencky strain tensor and its second invariant (i.e., von Mises effective strain) will be used in the sequel. The uncertainty values are evaluated by the standard deviation of the measured displacement and calculated strain fields. Figure 7.6 shows the standard displacement and strain uncertainty levels for the analyzed ROI. An increase of the spatial resolution (i.e., proportional to the element size or regularization length [36]) is followed by a decrease of Understanding, observation and quantification of ductile failure mechanisms via 3D imaging the displacement/strain uncertainty since more voxels per nodal displacement are available. This first series of results shows that the convergence of unregularized DVC can be achieved with smaller elements than the 32-voxel limit discussed in Section 2.2.1 thanks to the fact that displacement fields are a priori assumed to be continuous over the whole ROI. However to lower the measurement uncertainties, the element edge size was chosen equal to 32 voxels (or 35 µm) in all three directions, which yields a standard displacement and effective strain resolution of 0.2 voxel and 0.2 %, respectively. These values represent the limit below which the measured levels are not trustworthy. As expected, these levels are greater than uncertainties reported in case of tomography obtained on cast iron scans [39] but lower than laminography results on less contrasted aluminium alloy microstructure [45].

It is worth noting that the chosen element size (i.e., = 32 voxels) does not allow the kinematic details induced by the cast iron microstructure to be fully captured. However, the main purpose of the DVC calculations is to provide realistic boundary conditions to the FE simulations. It has been shown that such Dirichlet boundary conditions do not need to be measured at very fine spatial resolutions for identification and validation purposes. However, it is crucial that they capture the mesoscopic kinematics [26].

Keeping the element size identical (i.e., = 24 voxels), the uncertainties can be further decreased (down to 0.05 voxel for displacements or 0.02 % for strains) by utilizing mechanical regularization [37,65]. The results with mechanical regularization are shown with dashed lines in Figure 7.6. A significant gain in uncertainty is achieved since mechanical regularization successfully penalizes high-frequency noise features. A saturation is observed for regularization lengths greater than r = 512 voxels. The observed differences are due to different initial quality of the underlying scans that originates from focal point shift that occurred during the experiment due to thermal drift (analyzing raw data reveals the best quality scans are indeed in the "0.744 mm" case). Further, once damage sets in, higher gray level gradients are more favorable.

DVC results

Figure 7.7 shows microstructure sections for the unloaded state and after the first five loading steps. From the reconstructed raw images early nodule debonding from the matrix is noted. This phenomenon was also observed in the residual maps of DVC analyses for another cast iron grade imaged via tomography [68]. From the macroscopic loaddisplacement curve it can be seen from the sixth loading step on that softening is followed by the final crack propagating through the scanned region and these results, although available, are of less interest for the present numerical modeling procedure. Conversely, the measured kinematic fields for the initial steps are of special interest.

Correlation residuals

While performing DVC calculations at each iteration the deformed volume g is being corrected by the current estimate of the displacement field. The final difference between undeformed and corrected deformed volumes represents the gray level residual field ρ(x).

It does not only serve as correlation quality inspector, but it also provides useful tools for sensitive (precise) detection of damage occurrence [68]. In the present case it will also provide an objective way of comparing measured and simulated displacement fields, as will be shown hereafter.

The kinematic basis utilized herein prescribes C 0 continuity to the measured displacement field. Any deviation from this requirement (e.g., nodule/matrix debonding, void nucleation and growth) cannot be properly captured by the underlying kinematics and will be reflected in a local residual increase. Let us note that corrected volumes are updated at each iteration on a sub-voxel scale by using gray level interpolation (with a trilinear interpolation scheme). This means that by using residual fields one can benefit from sub-voxel resolution in the gray level registration procedure (Figure 7.6). However, a significant amount of damage growth deteriorates gray level conservation and possibly DVC convergence.

In Figure 7.8 the gray level residual fields for the first loading step are shown in isometric (a) and section (b) views as functions of different regularization lengths. It can be noted that areas of higher residuals correspond with the position of the debond zones.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging The relaxation procedure gives more freedom (since debonding and its consequences are far from the elastic assumptions and those of constant volume being made) in the correction of the deformed image as a part of the DVC procedure that results in slightly lower residual levels for the non-regularized approach.

Beside debond zones with higher gray level residuals, the residual map in the rest of the ROI shows sufficiently low values indicating successfully converged DVC calculations. This is confirmed in Figure 7.9 where the root mean square (RMS) residual field Numerical Validation Framework for Micromechanical Simulations based on Synchrotron 3D Imaging is shown for all loading steps and various regularization lengths. Plotting the values for different regularization lengths shows the relaxation procedure, i.e., decreasing elastic regularization, yields slightly better solutions since the elastic behavior requirements are relaxed. The fact that the residuals remain close to the levels observed in the uncertainty analysis is an additional proof of global convergence of DVC.

Displacement fields

Displacement fields in the loading direction for the first loading step and four different regularization lengths obtained through the relaxation procedure are shown in Figure 7.10. These fields correspond to the mid-thickness section of the sample. Only relaxed solutions allow the heterogeneities induced by the nodules to be captured. The mean values of the displacement fields do not differ much but the standard deviation is somewhat increased by decreasing the regularization lengths. This trend can be understood with the decrease of the registration residuals (see Figure 7.9). The relaxed solution (i.e., r = 0) allows more kinematic heterogeneities that endow more precise corrections of the deformed images, i.e., lowers the difference (especially in debond zones) with the undeformed image.

Effective strains

Effective strain maps are shown in Figure 7.11 for the first loading step during the relaxation procedure. Let us note the strain field heterogeneities can be seen starting from r = 2 = 64 voxels. The strain maps plotted over the corresponding microstructure details for first loading step for an unregularized solution are shown in Figure 7.12. It is interesting to note that even at this scale, from the first loading step on, higher strained areas are observed between some of the nodules. It is concluded that the presence of strained bands is always accompanied by nodules, while not all the nodules are affected by the localized strain pattern.

Mechanical simulations

Performing FE simulations with real microstructures and boundary conditions requires a robust and efficient numerical framework. Robustness is important because a real microstructure includes complex features and interfaces. For example, a wide range of biomechanical problems typically requires meshes of human bones or tissues that are accurate enough to reproduce nature but with a mesh quality that allows FE simulations at large strains [75,73,48]. Other materials such as foams and composites induce similar challenges as they feature complex and sometimes tortuous microstructures [33,73]. In the present case, the microstructure consists of voids (Figure 7.3) that are nearly spherical but of different sizes and with a random distribution where some voids may have initially coalesced. Additionally, realistic boundary conditions may induce a non proportional loading path and heterogeneous distortions of the FE domain. In the following, the whole numerical framework for performing the simulations is presented from the mesh generation to the application of boundary conditions and mechanical modeling (Figure 9.1). Particular attention is paid to the computational cost of each operation as it will play a key role for computations on large ROIs.

Meshing X-ray images

Meshing real materials and structures from images is a challenging task for which multiple approaches have been proposed [75,73]. A direct method would consist of first thresholding the image, then applying a contour detection algorithm to build a surface mesh of all interfaces and finally constructing a volume mesh around them [75]. The method used herein first builds a volume mesh, usually from a cube, which can be the image itself or, more likely, the ROI. Then, the image is imported into this mesh and surface meshes are built inside this volume mesh using mesh adaption techniques.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Image import

In general, the whole X-ray image is not considered because it is likely to contain some artifacts and/or noisy areas. A simple way to extract only the regions that are exploitable is to crop the image based on a 3D box. The first step of the meshing procedure is to generate a 3D mesh of this box. When no mesh refinement is used, the box can already be meshed with the chosen uniform isotropic size. When local mesh refinement is used, especially if it is based on geometric error estimators, it is assumed that the initial mesh of the 3D box is globally finer than the smallest size prescribed by this local mesh refinement. After import and meshing of the image, this mesh can be coarsened in areas where refinement is not necessary. The next step is to have a description of the image in the mesh, for example through node-wise gray levels defined at mesh nodes. The image can also be thresholded before import, in which case each gray level will represent not a color but a phase identifier. In order to transfer this information from the image to the mesh, FE interpolation is used. The image is considered as a hexahedral FE grid where voxels are nodes, and field transfer is operated from this grid to the nodes of the 3D box. For each node of the 3D box, finding the hexahedron of the image that contains it can be performed in constant time, since the image is a structured grid.

Once this gray level is obtained, LS reinitialization is performed. LS functions are a useful tool to localize evolving interfaces in multiphase simulations [49,63,51]. They play a key role in the numerical method that will be used to adapt the mesh throughout the simulations. The LS function φ i for phase i is defined as

∀t, ∀x ∈ Ω(t), φ i (x,t) = +d(x, ∂Ω i (t)), x ∈ Ω i (t), -d(x, ∂Ω i (t)), x / ∈ Ω i (t), (7.14)
where Ω(t) is an evolving domain, Ω i (t) is the subset of Ω(t) occupied by phase i, and d(x, ∂Ω i (t)) is the Euclidean distance from point x to the boundary of Ω i (t). LS functions will also play a key role to compute the porosity and for error measurements as they carry all information regarding phase distributions. The difficulty is how to convert a nodewise gray level or phase identifier into a signed distance function. This problem is known as the LS reinitialization problem [64,61]. It is solved herein using a direct brute-force algorithm enhanced with a parallel space partitioning technique [61]. Once LS functions are obtained for each phase of the domain, one may start a simulation using only this implicit representation of interfaces. The FE framework developed in Refs. [59,60] aims at going a little farther and maintaining an explicit representation of interfaces throughout the simulation.

Mesh generation

Mesh generation is directly based on LS functions. Every edge where an LS function φ i has opposite signs on both ends is an edge crossed by an interface. In Ref. [59], an interface fitting procedure was proposed in order to introduce all these intersection points Understanding, observation and quantification of ductile failure mechanisms via 3D imaging in the mesh of the domain. This procedure is likely to introduce very poor quality elements that will render any FE simulation on this mesh impossible. A mesh adaption step is therefore necessary to restore an appropriate mesh quality. In Ref. [60], the topological mesh adaption technique detailed in Ref. [19] was enhanced by taking into account volume conservation during remeshing in order to preserve (at best) the distribution of phases in the initial mesh. As presented in Section 3.5, it is also used during FE simulations, where large deformations and topological events may induce large distortions of the Lagrangian mesh.

The whole image import and mesh generation procedure is summarized in Figure 7.13. A closer look at interfaces is presented on the right side of this figure. While interface fitting produces a smooth surface mesh of the interface, which is actually the zero-LS function, edge lengths are randomly distributed and element quality is deteriorated. This is corrected in the last mesh adaption step.

Definition of a computational domain

For each experiment, 3D X-ray images are reconstructed at various loading steps. For obvious reasons, it is impossible to systematically capture the same part of the specimen, and some features may go in and out of the images. Regarding DVC, this is dependent on whether computations are conducted with a direct or incremental scheme (see Figure 7.5). The ROI may be different depending on the used method as for example more material points of intermediary images are taken into account by the incremental method. However, this is not possible for mechanical simulations, namely, the chosen initial FE domain has to remain inside all DVC ROIs throughout the simulation. This is independent of the DVC method (direct or incremental). In practice, the DVC ROI is chosen as large as possible (ideally, the full X-ray images minus the regions with missing information), and the FE domain for mechanical computations is manually chosen in order to be contained inside all DVC ROIs.

Application of boundary conditions

Once an appropriate domain has been chosen and meshed according to the initial image, boundary conditions have to be applied to follow experimental images through each loading step. This problem is split into two operations that are time and spatial interpolations. Neither the time step nor the FE mesh used for mechanical computations will generally match the loading steps used in experiments and the mesh used for DVC. In practice, a loading step will be split into several time steps, especially at crack initiation. Regarding the FE mesh, it will be fitted to the features of the microstructure as described in Section 3.1. It is worth noting that such fine meshes are not compatible with standard DVC approaches [36], the reason for that being the fact that the number of voxels per nodal displacement cannot be arbitrarily reduced without additional regularization [37,36] (see Figure 7.6). As discussed in Section 2.3, the very details of the displacement field are not needed for Dirichlet boundary conditions when used for validation purposes.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging 3D Imaging node of the FE mesh the element of the DVC mesh containing it, and apply tetrahedral (resp. hexahedral) interpolation inside this element depending on its type. Operation 1 has linear complexity and negligible cost as the number of loading steps is small. The same remark applies to operations 2 and 3 as the DVC mesh is at least one order of magnitude coarser than the FE mesh (in terms of number of elements). Operation 4, however, requires to localize each node of the FE mesh inside the DVC mesh. Optimized algorithms can be found in the literature to perform such operation in logarithmic time [61] but were not implemented here due to the small size of the DVC mesh compared to the FE mesh. Hence, this operation has a cost proportional to the number of elements in the DVC mesh, and quadratic cost considering it has to be run for all boundary nodes of the FE mesh. This cost would have to be optimized if finer DVC meshes were to be used, especially if those meshes were to have an order comparable to that of the FE mesh.

Last, the displacement field interpolated at the boundary nodes of the FE mesh is prescribed by means of Dirichlet boundary conditions on the whole boundary of the FE domain. Therefore, apart from small differences due to interpolation and remeshing, the boundary of the FE mesh is expected to follow exactly the measured displacement field at each loading steps. The errors that will be measured inside the domain are expected to be mainly due to the mechanical modeling.

Constitutive law

As stated in the introduction, graphite nodules are considered as voids in the present computations [17,74,6,30]. These voids are modeled using an isotropic linear elasticity law with very low Young's modulus. With a sufficiently low value, the behavior of a void is obtained. The influence of such penalization technique on void growth was investigated in Ref. [57]. This investigation is again carried out herein.

Regarding the ferritic matrix, von Mises plasticity with power law hardening is assumed [40] σ 0 (ε) = σ y + Kε n , (

where ε is the equivalent (von Mises) plastic strain, σ y the yield stress, K the plastic consistency and n the hardening exponent. This law was fitted against a stress/strain curve for a pure ferritic matrix [74], which is assumed to be representative of the insitu behavior of the ferritic matrix of the studied cast iron. The obtained values are σ y = 290 MPa, K = 382 MPa, n = 0.35; the Young's modulus E = 210 GPa, and Poisson's ratio ν = 0.3. The chosen constitutive law may appear very simplistic. However, in virtually all unit cell calculations or micromechanical models accounting for ductile damage, such types of plasticity postulates are used. It is noteworthy that more complex models can be incorporated in the present framework (e.g., crystal plasticity).

To avoid numerical locking in the plastic regime, a mixed velocity-pressure formulation is used with a P1+/P1 element [5]. Additionally, a Newton-Raphson algorithm is implemented to account for the nonlinear behavior of the matrix [71]. This mechanical solution yields displacement fields at mesh nodes and stress / strain distribution at Understanding, observation and quantification of ductile failure mechanisms via 3D imaging quadrature points, including the equivalent plastic strain. Large plastic strains are considered herein thanks to an updated Lagrangian formulation, which is commonly used in metal forming simulations [71]. This incremental formulation has proven to give accurate results for small time steps (time step convergence is usually carried out in such analyses as shown below).

Mesh motion / adaption

All FE computations are performed in a Lagrangian framework. Therefore, at each time step, the displacement field is applied to move the nodes of the mesh. A mesh adaption technique has already been introduced in order to restore mesh quality when generating body-fitted meshes from images. The same technique is applied during the FE simulation. Due to the presence of large void growth and coalescence events in the images, the boundary conditions may induce very large deformations, which can be local for specific voids, or global to the whole domain. Though no fracture or coalescence criteria such as those used in Ref. [59] are employed in the present simulations, these deformations may induce topological events such as coalescence of neighboring voids. The mesh motion and adaption procedure described in Ref. [60] was designed to handle such issues. This procedure consists of detecting element flip during mesh motion, going back if it occurred, remeshing, and trying again. Doing so, small deformation steps are performed and the robustness of the Lagrangian framework is improved.

Simulation results

The DVC ROI for which boundary conditions are available is of dimensions up to 532 3 voxels. Finding the largest 3D box included in this ROI and in all ROIs through each loading step was performed manually by launching simulations without meshing any microstructure. Box dimensions were reduced progressively until reaching the above condition. The chosen FE domain is of dimensions [-245, 200] × [-17, 200] × [-245, 245] voxels, due to necking in the x-direction and crack opening in the y-direction.

In the following, FE simulation results are presented using this initial domain and, unless otherwise mentioned, DVC boundary conditions using element size = 32 voxels and no regularization. Qualitative comparisons with experiments are presented based on 2D slices of the microstructure. Particular interest is given to void volume change (nodules being already replaced by voids in the initial mesh) in order to enable for quantitative comparisons. More precise error measurements are presented in Section 4.

Mesh generation

To show how different meshing parameters can be used in order to capture different scales of the microstructure, three different sets are tested. The reference configuration defines an isotropic mesh size h2 of 10 µm for elements close to interfaces and 50 µm for elements farther than 100 µm from any interface, the transition between these two mesh sizes being Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Numerical Validation Framework for Micromechanical Simulations based on Synchrotron 3D Imaging linear. A coarser mesh h4 is then obtained by multiplying all these parameters by 2, while a finer mesh h1 is based on a division of these parameters by 2. The resulting meshes have 69, 010 elements for the coarsest one, 1, 857, 466 for the finest one, and 387, 473 for the reference one.

It is important to check that finer scales are better captured with the finest setting. Therefore, the middle slice of the image is compared to the midsection slice for each obtained mesh in Figure 7.14. The coarsest result in Figure 7.14(c) gives a rough approximation of the microstructure, while an acceptable description is obtained in the reference mesh (Figure 7.14(b)). Last, no significant difference is observed between the initial image and the finest mesh in Figures 7.14 For FE simulations, mesh quality is of major importance and should be investigated. The quality distribution is presented in Figure 7.15 for the three settings. A quality of 1 means a well shaped tetrahedron, while a quality of 0 means a degenerated one. Finer meshes induce more elements far from interfaces and render mesh adaption easier. Therefore, the histogram for the finest mesh is slightly shifted to the right. However, the proportion of elements in the first half of the histogram remains negligible for the three configurations, which proves that accuracy is improved with finer meshes but not at the sacrifice of mesh quality. Though this qualitative comparison suggests to use very fine meshes, it is important to see the influence of these parameters on the quantities of interest such as void volume changes for example.

Choice of numerical parameters

To choose appropriate meshing parameters, time step and void penalization coefficient, a sensitivity analysis is conducted. Meshing parameters are tested using the three meshes defined in the previous section. Regarding the time step, the reference is identical to that of the reconstructed images, i.e., there are 7 scans, so 6 DVC displacement fields are available and 6 time increments are performed in the FE simulation. Note that scans were not acquired at uniformly distributed displacement increments as smaller loading steps were performed at the onset of void coalescence (Figure 7.2(b)). A configuration with a time step 10 times smaller is tested (i.e., 60 time increments) to investigate time step influence. Last, the reference configuration uses a Young's modulus for voids 1, 000 times smaller than that of the matrix. A simulation is performed with a penalization of 10, 000 to check the influence of this coefficient.

The results are presented in Figure 7.16 in terms of relative void volume change, which is expressed as the void volume divided by the volume of the FE ROI. While the penalization parameter and the time step do not seem to have a significant influence on the results, the mesh size plays a key role. The image import and mesh adaption methodology introduced in Section 3.1 enable all scales of the initial image to be captured with a finer mesh, and preserve them more efficiently throughout the simulation. Hence, both the initial void volume and its changes in Figure 7.16 are under-estimated with a coarser mesh. These results suggest that the reference penalization parameter and time step are appropriate for the FE simulations, hence they were used in all following results. However, more accurate error measurements are necessary to locally investigate the influence of mesh size (see Section 4).

Discussion

To show the interest of the present method linking synchrotron experiments and FE simulations through the use of DVC, it is important to compare the void volume change 7.16 to experimental observations. This is qualitatively illustrated in Figure 7.17, where void growth in the FE simulation using mesh h1 is compared with experimental images, while quantitative comparisons are presented in Section 4. Results in Figure 7.17 suggest that void coalescence is not only due to an acceleration of void growth in inter-void ligaments. It seems that there is either important matrix softening between neighboring voids, which would imply the presence of a minor void population [29,76] not captured by X-ray images, or a transition from strained bands to localized fracture of these ligaments. Inter-void plastic strained bands are clearly visible in Figure 7. 18.

Comparison with the final crack propagation path in Figure 7.17 reveals the importance of studying more closely the behavior of the matrix in these bands. Crystal plasticity modeling [55] may be important, as grain size in the matrix material is comparable to nodule size [30]. Such discussions are only possible thanks to simulations with real microstructures and measured boundary conditions. As mentioned above, such models can be included in the numerical framework presented herein.

Quantitative comparisons

The whole methodology presented in the previous sections focuses on expressing experiments and FE simulations in the same kinematic space. On the one hand, experimental images are directly exploited by the FE code in order to mesh the initial domain. On the other hand, they are also exploited by DVC in order to compute displacement fields that are then used to drive the FE simulation. In order to check the validity and accuracy of all the computations, these links need to be reversed, as shown in Figure 9.1. The FE simulation results should be compared both against experimental images obtained during loading and measured displacement fields as well. Regarding the validation of DVC measurements, it is common practice to study correlation residuals (see Section 2.4.1), namely, the deformed experimental volumes are corrected using the measured displace-Understanding, observation and quantification of ductile failure mechanisms via 3D imaging ment field and compared to the actual reference volume. This operation can also be applied to the displacement fields obtained by the FE simulation, i.e., compute gray level residuals (see Equation (7.2)) for the FE kinematic fields by correcting reconstructed volumes. The corresponding DVC and FE gray level residual fields can then be compared. DVC displacement fields are applied to the boundaries of the FE domain, but they are also available inside the domain. Hence, DVC and FE kinematic fields can also be interpolated on the same mesh and directly compared.

In the following, displacement fields obtained by C8-DVC are interpolated on the corresponding FE mesh (i.e., T4 unstructured mesh). The meshes are shown in Figure 7.19. The FE domain, which needed to be reduced compared with the DVC ROI for the reasons outlined in Sections 2.2.3 and 3.1.2, contains the fine mesh h2 adjusted to the microstructure details as opposed to the structured and coarser DVC mesh.

The original FE and DVC interpolated displacement fields are then subtracted yielding the displacement difference field. The displacement fields and their difference is shown in Figure 7.20 for the third loading step on the section normal to the y-direction. The Understanding, observation and quantification of ductile failure mechanisms via 3D imaging The displacement differences deviate more than the measurement uncertainty meaning the current DVC and FE results are not fully consistent. However, it is worth noting that for the first four steps the differences remain rather small (i.e., less than 5 times the standard displacement resolutions). Conversely, for the last loading step, the agreement is less good. This is due to the fact that significant void growth has occurred (Figure 7.17) and is not fully captured either by DVC or FE simulations.

The influence of the mesh size used in the FE computations is reported in Figure 7.22 using the three densities defined in Section 3.6.1. The diagrams are showing (a) mean and (b) standard deviation of the displacement differences between measured (via DVC) and FE simulations. The results show there is no significant influence of the mesh size when compared to DVC results. Interestingly, for the last two loading steps, the difference between the three FE simulations increases. This is a further indication that localized phenomena occur and that the FE discretization becomes sensitive to such events.

Although assessing displacement differences is an interesting piece of information, it is not an objective way of validating FE simulations since the meshes are different in DVC calculations and FE simulations (Figure 7.19). In the following it is proposed to use gray level residuals to compare all results even though the experimental solution is unknown. To validate DVC results the corrected deformed volumes ĝ are computed with measured displacement fields. The same procedure is followed by computing gray level residuals with FE computed displacement fields. The T4 mesh together with the computed Understanding, observation and quantification of ductile failure mechanisms via 3D imaging [35,26]. By using the T4 shape functions the displacement results are interpolated voxel-wise and the corresponding deformed volume g(x) is corrected by the computed displacement field u FE (x). The gray level residuals, namely, differences between the reference volume f (x) and corrected deformed volume g(x + u FE (x)) can then be compared for DVC and FE computations. This procedure is illustrated in Figure 7.23 for the third loading step. The microstructure section in the reference and deformed states is shown in addition to their raw difference. The latter shows that the motion of the nodules is not fully accounted for. are plotted together with the corresponding residuals from the uncertainty analyses of the first and fourth loading steps. First, the DVC and FE residuals have very similar levels for the whole history even though the discretizations are very different. This is mainly due to the fact that measured boundary conditions have been applied to the numerical model. As the screw displacement is increased, the two residuals depart from those observed for the uncertainty analysis. This is an indication that the measured and computed displacements are no longer able to capture all the complex phenomena taking place. Model errors are therefore to be expected. On the DVC side it is related to the coarse and continuous C8 discretization. In terms of overall levels, there are two regimes ob-0.4 0.5 0.6 0.7 0.8 0.9 4 served in the standard deviations of the residual fields. For the first five loading steps, the standard deviations vary between 1.2 and 1.5 times the level observed in the uncertainty quantification. Such low levels prove that even though imperfect, the present simulations are consistent with the experimental data at hand. For the sixth loading step, there is a clear degradation of the gray level residuals (i.e., the standard deviation is 2 times the level observed in the uncertainty quantification). This phenomenon is associated with coalescence (see Figure 7.7(f)), which is confirmed in Figure 7.17. In this last regime, the numerical predictions depart from the experimental observations. The influence of the FE mesh is evaluated in Figure 7.25. There is no significant influence of the mesh size on the gray level residuals. The three meshes can be used in the present analyses since they lead to virtually identical global residuals. This last result shows that on the FE side the origin of higher residual is due to the constitutive model that is no longer able to fully capture all the strained bands preceding coalescence mechanisms.

From the previous results (Figure 7.24) it is concluded that the FE solution driven by the DVC boundary conditions is close to the laminography data used herein in the void growth regime. Conversely, void coalescence is not well described. This result validates the numerical framework. However, it does not necessarily validate the constitutive model since up to now only kinematic fields extracted from laminographic scans were probed. Error measurements based purely on kinematic data may not be sufficient because all Hence, static data should also be involved in the analysis. This procedure requires the knowledge of stress profiles along the sample ligament to be able to compare experimentally measured macro-forces and calculated static responses from FE simulations on microscopic level (FE ROI scale).

Conclusions

In this study an innovative validation procedure combining synchrotron laminography, digital volume correlation and FE simulations on the microscale level has been presented (Figure 9.1). Experimental data, namely, reconstructed volumes are collected via laminography while performing a tensile test on a flat specimen of nodular cast iron with a central hole. Bulk kinematic fields in the region close to the central hole are measured via DVC. Since high strain levels occur between subsequent scans regularized approaches to DVC are to be used to converge. A stepwise relaxation procedure has been followed ending with unregularized solutions. The FE mesh is adapted to the real microstructure under the assumption that graphite nodules are considered as voids with a specially developed (re)meshing procedure. The FE simulations are driven by Dirichlet boundary conditions extracted from DVC measurements on all 6 surfaces of the analyzed domain.

The results obtained from DVC registrations and FE simulations are then mutually and separately compared with respect to the available experimental information. DVC results are validated with respect to experimental data through the correlation residuals, which are probed for all voxels belonging to the region of interest and all the analyzed scans. Since the kinematic basis utilized herein prescribes C 0 continuity to the kinematic fields, the gray level residual fields serve as correlation quality inspector and provide a useful tool for damage detection in the zones where the continuity requirement is not met. The overall residual levels are reasonably low thereby indicating successful registrations. The areas of higher residuals correspond to the position of the debond zones that appear very early on, and increase further due to void coalescence.

Once validated, DVC measurements are used to drive FE simulations. Added to the fact that the real microstructure is discretized in the FE mesh, heterogeneous displacements prescribed by DVC require a robust mesh adaption methodology. The latter is used herein and enables simulations to be performed with significant void growth up to coalescence. While microstructures are often idealized in the literature, these promising results open new possibilities for numerical modeling of heterogeneous 3D microstructures having a complex morphology and undergoing large deformations and complex topological events such as those observed in reality. DVC and FE results are then probed by direct comparison of the full-field solutions and computation of the gray level residuals. The first method, which is a relative comparator, provides the gap between DVC and FE kinematic fields. It is shown that differences between the two methods are mainly concentrated in the bulk of the inspected region around the nodules, while boundary zones have mostly zero differences because the boundary conditions are measured via DVC. However, just observing the displacement difference between the two methods does not yield an absolute error. Conversely, the gray level residuals can be computed independently for both approaches. The kinematic fields are probed by computing the deformed volume corrected by the measured or simulated displacement fields. The fact that the gray level residuals remain reasonably low for the FE simulations in the void growth regime is a partial validation. However, void coalescence is not properly captured. This last result calls for more advanced constitutive models for the matrix, the nodules and their interface with the matrix. During coalescence, void nucleation at finer scales and very large strains might play a role. It is associated with damage softening. The present framework will allow, in future analyses, to test this hypothesis using more sophisticated plasticity/damage models.

One of the next steps of the present work will thus deal with the constitutive law of the matrix whose parameters can be modified to get an even better agreement between the experimental data and numerical results. Similarly, the hypothesis of considering nodules as voids can also be probed and debonding can be addressed. The proposed numerical/experimental framework will still be used with sequential or integrated identification procedures [26,42].

To the authors' best knowledge this is the first time 3D calculations are performed on a real microstructure and by using measured boundary conditions up to large plastic strain. This approach will enable for better understanding and modeling of damage mechanisms at microscale levels. The presented numerical framework can easily be generalized for other types of materials that possess heterogeneous microstructures.

Introduction

In the aim for understanding the physical nature of any phenomenon, comparison between experiments and theoretical models is used extensively [1]. In the case of ductile fracture, difficulties arise due to the competition between localization and softening events occurring at different scales [2,3]. In particular, investigations of ductile fracture mechanisms, namely, void nucleation, growth and coalescence at the microscale require 3D observations at this scale. On the numerical side, being able to simulate these mechanisms with comparable geometric and loading conditions is still an open research topic, which is once again due to the scale at which the geometry and the loading have to be identified.

Although 3D in-situ experimental observations (e.g., X-ray tomography and laminography) have provided valuable information for ductile fracture modeling [4,5,6,7,8,9], numerous questions regarding comparisons between models and experiments are still to be answered. For instance, the well-known work of Gurson [10] has motivated researchers to rely extensively on homogenization theory to model ductile fracture based on analytical [10,11,12,13] or numerical procedures [14,12,13,15,16]. Even though interesting results can be obtained on average values, the applicability of these homogenized models to real 3D microstructures is quite limited due to the assumptions of idealistic microstructures and boundary conditions on which they are based [17,18,6,9].

Numerical procedures at the microscale could be applied given that 3D X-ray images of the microstructure are acquired in-situ, that a microstructure meshing technique is available, and that boundary conditions are applied. This is a critical piece of information that will be analyzed herein. There are different ways of performing such microscale simulations:

• The most straightforward approach is Direct Numerical Simulation (DNS), where the microstructure of the whole specimen has to be meshed, hence taking into account only microscale constitutive models. Since the whole specimen is simulated, boundary conditions are applied directly at pins, as in the experiment. However, depending on specimen size, DNS can have a huge cost regarding both experiments and simulations. Scanning the whole specimen could require multiple scans at each loading step, and this large set of 3D data would then have to be meshed.

• To avoid this huge computational cost, full specimen Finite Element (FE) simulations only partially taking into account the microstructure have been considered [19,20,21,22,23,24]. Microscale constitutive models are used in the ROI where the microstructure is meshed, while macroscopic homogeneity is assumed in the remainder of the specimen. Thus, an appropriate macroscale constitutive model has to be defined and identified for this out-of-ROI homogeneous material. This approach will be referred to as strong FE (sFE) coupling.

• Opposed to the previous approach in which the ROI mesh is directly embedded within the specimen mesh, the two FE calculations can be weakly coupled. In this

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging weak FE (wFE) approach [25,26,27], the specimen scale simulation assumes a homogeneous material in the whole domain. A relevant mechanical field is then transferred from this first simulation to the second simulation at the ROI scale, where the microstructure is meshed. This mechanical field defines the boundary conditions for the micromechanical simulation. These conditions can use displacements, forces, or a combination of both.

• In a recent work [28], an alternative option for performing microscale simulations was proposed. Digital Volume Correlation (DVC) was used in order to measure displacement fields between 3D X-ray images taken at consecutive loading steps [29,30,31]. The DVC technique is based on tracking the natural image contrast, herein originating from the heterogeneous microstructure. FE simulations were conducted by applying measured DVC displacements to the boundaries of the meshed ROI. The difference between FE results and DVC displacement fields inside the FE domain could be assessed. Additionally, a better quantification of the error produced by FE models was obtained by computing gray level residuals based on FE and DVC displacement fields. An advantage of this so-called DVC-FE method is that no specimen scale simulation is required, thus no macroscale constitutive model has to be identified. Yet, it requires a thorough uncertainty assessment regarding DVC measurements.

The present paper aims at showing the interest of the DVC-FE approach and compare its predictive capability with simulations at the microscale based upon either weak or strong FE couplings. One key aspect is related to the boundary conditions that are applied to the simulated ROI where the microstructure is meshed. This investigation is conducted with new experimental results using nodular graphite cast iron specimens with a geometry inspired from Ref. [5]. A first test (A) was performed using small loading steps in order to obtain accurate force measurements and 2D surface images to be exploited thanks to global Digital Image Correlation (DIC). The second test (B) was performed using larger loading steps since it was conducted in a synchrotron facility. For this second test, both 3D X-ray scans of the ROI and 2D surface images were acquired (see Figure 9.1). While the DVC-FE methodology presented above uses directly and solely the 3D X-ray data, the two alternatives to obtain boundary conditions for ROI calculations considered in this paper (i.e., wFE and sFE) rely on force measurements and 2D surface images. Force measurements are used to identify the behavior of the material at the specimen scale, as well as 2D surface images thanks to a recent Integrated-DIC framework [32]. In the wFE method, an FE simulation of the experiment at the specimen scale is conducted, and calculated displacement fields are used to drive a second FE calculation at the ROI scale.

In the sFE method, the ROI is embedded and meshed directly inside the specimen mesh and only one FE simulation is conducted. In this second method, the material behavior is modeled using microscale constitutive models in the ROI, while macroscale constitutive models are used in the rest of the domain.

The aforementioned approaches with weak (wFE) and strong (sFE) couplings between specimen and ROI calculations are applied to a real 3D microstructure observed thanks to 3D in-situ laminography experiments. To the best of the authors' knowledge, this is the first time such simulations are conducted and compared locally to experimental observations, this last point being possible thanks to gray level residual computation. The results are then compared to those obtained using the DVC-FE approach [28], showing the interest of this methodology. Details regarding the material, the experiments and the numerical method used for ROI calculations are presented in Section 2, while the technical implementation of each approach for boundary conditions is described in Section 3.

The results are presented and compared based on error measurements with respect to experimental images in Section 4.

2 Experimental and numerical framework

Experiments

The material used in this study is a commercial nodular graphite cast iron with the serial code EN-GJS-400. It features a ferritic matrix and graphite nodules at the microscale (see Figure 8.2), with no significant porosity in the initial state. Under tensile loading, ductile fracture is known to be mainly driven by early debonding of the nodules from the matrix and coalescence of the subsequent nucleated voids [33,34,35]. Previous works [33,36,37,34] suggest to model the nodules as voids, as their load carrying capacity is very low under tensile loading. This assumption is made herein meaning that at the microscale the material is considered as a two-phase microstructure with a ferritic matrix and voids.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

The specimen geometry, which is inspired by the work of Weck et al. [5], is shown in Figure 8.2. The holes have been machined via Electrical Discharge Machining (EDM). The testing device applies the load by manually controlling the displacement via screw rotation. Test (A) is conducted as a pre-test for test (B) to study in detail the particular F F In test (B) after applying each loading step, a scan is acquired while the sample is rotated about the laminographic axis (i.e., parallel to the specimen thickness direction). This axis is inclined with respect to the X-ray beam direction by an angle θ ≈ 60 • . The 3D images used in this work were obtained at beamline ID15A of the European Synchrotron Radiation Facility (ESRF, Grenoble, France) with a 60-keV white beam, using 3,000 projections per scan. The series of radiographs acquired is then used to reconstruct 3D volumes by using a filtered-back projection algorithm [38]. The parameter optimization has been performed automatically using a GPU-accelerated implementation of this algorithm [39]. The reconstructed volume has a size of 1600 × 1600 × 1600 voxels. The physical size (length) of 1 cubic voxel is equal to 1.1 µm. After scanning the undeformed state (0), 12 additional scans are acquired during the stepwise loading procedure, where the last scan corresponds to the fully opened final crack.

The scanned region incorporates two holes while the ROI employed in DVC and FE calculations is mainly concentrated in the ligament between the holes (see Figures 9.1

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging and 8.2). Since the two machined holes have diameters of 500 µm, the nodule population (treated as voids in the FE calculations) in the ligament area with the characteristic size of 60 µm can be considered as a secondary void population. Hence, subsequent concurrent micro and macro plasticity and damage localization phenomena can be observed. In Ref. [5], the authors used machined holes of micrometer size, which made impossible the observation of the secondary void population (this limitation on the size of observable voids is actually due to current imaging techniques [40]). Here, the larger size of the holes, and the large size of the graphite nodules allow for such observations. This is illustrated in Figure 8.3 where mid-thickness sections of the reconstructed volume for the three different load stages are shown. As a consequence of the mentioned multiscale flow conditions, classical void coalescence mechanisms are accompanied by sheet coalescence between the two machined holes that is observed in the last loading step (deformed state (11)).

FE mesh of the microstructure

The framework used for microscale FE simulations is based on previous developments presented in Refs. [41,42,43,44,45]. As explained in Ref. [28], the ROI used for FE simulations has to be included in all DVC domains throughout each loading step. Otherwise displacement fields would not be available as boundary conditions on the whole boundary of the FE mesh. This requirement is necessary for the DVC-FE approach but also for other approaches in order to allow for comparisons with experimental observations. In practice, in order to be as representative as possible, a 3D box as large as possible is chosen. This 3D box has to be small enough to remain in all scanned regions throughout loading. The image meshing technique used to model the microstructure observed in the experimental 3D images starts with standard image processing operations [46,47]. They consist of smoothing the data, applying a gray value threshold to separate matrix and voids, and then converting these binary data into a signed distance function. The latter is interpolated to a first mesh of uniform size (i.e., 10 µm) of the FE ROI using trilinear FE interpolation (i.e., the image is considered as a hexahedral grid where the voxels are

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging nodes). The resulting signed distance function is then regularized based on a recently proposed parallel reinitialization algorithm [43], and used as an intermediary to locate the interface [45]. This mesh generation step is combined with an adaption step taking into account the local maximum curvature of the interface [44]. These different steps are summarized in Figure 8.4. Parameters of the final mesh are defined to have a size of 10 µm close to matrix/void interfaces, and 50 µm at a distance of 100 µm from any interface, the transition in this layer being linear. It can be observed qualitatively in Figure 8.4 that the FE approximation of the geometry is really close to experimental data. A more thorough analysis of the sensitivity of the results to meshing parameters was assessed (not reported herein) and revealed this influence to be negligible, as already validated in a previous study for a different experiment on the same material [28].

Constitutive laws and parameters

Microscale model

The microscale simulations need material models for the graphite nodules and the ferritic matrix. As stated in the introduction, graphite nodules are modeled as voids [33,36,37,34], while the ferritic matrix is considered as elastoplastic with power law hardening

σ 0 (ε) = σ y + K (ε) n (8.1)
where ε is the equivalent (von Mises) plastic strain, σ y the yield stress, K the plastic consistency and n the hardening exponent. A particularity of the present numerical method is that voids are meshed and defined as a purely elastic material with very low Young's modulus with respect to the matrix. A sensitivity analysis regarding this Young's modulus was conducted [28] showing that a ratio of 1, 000 between this modulus and that of the matrix was sufficient. The properties of the matrix are deduced from stress/strain curves [36] and presented in Table 8.1. These data are based on tensile experiments on a purely ferritic material. Thus, it is possible that the actual behavior of the matrix be more complex. An interesting perspective to the present work would be to use micromechanical calculations and local error measurements to study more appropriate microscale constitutive models and identify more realistic material properties. Mechanical solution of the equilibrium equations is based on a mixed velocitypressure formulation solved using a P1 + /P1 element to avoid locking issues [48]. The nonlinear behavior of the matrix requires a Newton-Raphson scheme both for numerical integration of the plastic law and global equilibrium [49]. In order to handle large deformations, an updated Lagrangian scheme is used and the velocity field resulting from the served in Figure 8.3, large distortions will occur inside the ROI. As a result, mesh quality will deteriorate, thus affecting the accuracy of FE solutions. Flip of elements may also occur. These issues are handled herein thanks to an automatic mesh motion and adaption method developed in previous works [43,44,45]. This method was designed to preserve at best the local distribution of volumes between the matrix and the voids while handling large void growth and complex topological events such as void coalescence.

Macroscale model

Both wFE and sFE methods require the identification of material parameters {p} based on experimental measurements acquired during test (A). In the case of wFE, these material parameters are used in the whole domain for the calculation at the specimen scale of test (B). In the case of sFE, these parameters are only used for the homogenized out-of-ROI material, while microscale material parameters (Table 8.1) are used inside the ROI. The identification of material parameters can be performed only using force measurements, or using both force measurements and 2D surface images. The two approaches are detailed hereafter.

Load data

A first identification is conducted using standard global optimization methods [50,51], where the objective function is defined as [52]

E(F(u)) = U 0 (F(u) -F exp (u)) 2 du U 0 F exp (u) 2 du (8.2)
where the forces obtained in an FE simulation F are compared with the sum of reaction forces measured during the experiment F exp , with an integral on the loading path [0, U]. Therefore, only load data acquired during test (A) are taken into account. Note that the displacement u, and all displacements mentioned in this paper, are measured via 2D DIC [53,54,55]. These values were determined directly at the upper and lower parts of the mesh presented in Figure 8.5(a). The force/displacement curve based on measurements and simulations is shown in Figure 8.5(b). The experimental data were acquired up to final failure, but the last loading steps of test (B) were discarded for this inverse analysis because no micromechanical simulation is carried out up to final failure in Section 4. The force/displacement curves show that the numerical approximation is very good, but here only the macroscopic force is compared. Local measurements are considered in the sequel.

Load data and pictures

In order to obtain more realistic material properties more advanced identification techniques can be considered. In Integrated-DIC or DVC [32,56,57], both pictures (i.e., 2D surface images from test (A)) and load (i.e., force measurements from test (A)) data are taken into account. Integrated DIC requires displacement fields to be fully mechanically admissible, i.e., satisfy equilibrium for the chosen constitutive law. Since material parameters are sought, the corresponding sensitivity fields [58] are also needed. The displace-

where the kinematic degrees of freedom u p are linked to the sought material parameters via sensitivities

u p ({p},t) = u p ({ p},t) + δu p δ{p} ({ p},t) T {δp}. (8.4)
The kinematic sensitivities are collected in matrix [S u (t)] and evaluated for each loading step t via FE simulations in which measured displacements are prescribed on the top and bottom boundaries of the considered ROI. These Dirichlet boundary conditions are measured a priori by conducting global DIC on 2D images from test (A). Simultaneously, the load measurements gathered in vector {F exp } are compared with the resultant forces {F FE } from the corresponding FE simulation. As for the kinematic part, the load sensitivity matrix [S F (t)] to the sought material parameters is computed to update {δp} from the current estimate {F FE ({ p})} of the reaction forces. Since both images and loads are used in a single approach and due to different physical natures of the data a Bayesian framework is considered herein, in which each piece of data is weighted by its variance and covariance with all the other data [32,57]. The Integrated-DIC code used herein is an in-house Matlab implementation with C++ kernels (i.e., Correli 3.0 [59]) while the accompanying FE simulations are performed using the commercial package Abaqus/Standard. More details about mechanical correlation can be found in Ref. [32]. Due to very low sensitivity the Poisson's ratio had to be set Understanding, observation and quantification of ductile failure mechanisms via 3D imaging in Integrated-DIC to its initial value of 0.28 while the other elastoplastic parameters have been identified.

Material parameters obtained using both identification procedures are given in Table 8.2. Though these parameters may seem to differ notably, the influence of this difference is significant only at low (i.e., < 0.2) and very large (i.e., > 1) equivalent plastic strain, the latter not being experienced herein. Hence, the results are not expected to strongly depend on the choice of identification procedure apart from the first loading steps, as illustrated in Figure 8.5(b).

Boundary conditions

As explained in Section 1, three different techniques will be used to determine the boundary conditions of FE simulations at the microscopic scale. Two procedures, namely, wFE and DVC-FE, consist of computing or measuring displacement fields and then applying them as Dirichlet boundary conditions on the whole boundary of the FE mesh of the ROI of the microscale calculations. The third procedure (i.e., sFE) has the ROI mesh embedded within the sample geometry and the whole computation is run in a single step. The three approaches and their implementation are detailed hereafter.

Weak Finite Element (wFE) technique

An application of the wFE technique to ductile fracture with meshed microstructure can be found in Ref. [27]. The objective of this study was to assess the influence of macroscopic loading conditions on the microstructure, especially regarding damage localization around inclusions. Although the real microstructure was meshed from a 3D X-ray image of the material, there was no comparison with experiments regarding micromechanical calculations, as proposed herein.

In the present work, a first macroscopic simulation of test (B) is conducted at the specimen scale. The material is considered as homogeneous and modeled using the same elastoplastic model with the power law hardening defined in Equation (8.1) and the two sets of material parameters given in Table 8.2. Since specimens used in tests (A) and (B) had identical geometries, the same mesh as in Figure 8.5(a) is used. The displacement field between each consecutive loading steps where 3D X-ray scans were acquired is stored in the reference configuration. These displacements are then interpolated at the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging On the choice of boundary conditions for micromechanical simulations based on 3D imaging boundaries of the ROI during a second FE simulation at the microscale by means of linear interpolation (both meshes being exclusively composed of tetrahedra).

Strong Finite Element (sFE) technique

In the sFE method, a single FE simulation is carried out. The microstructure is directly meshed within the specimen mesh, with a progressive mesh coarsening from the very fine mesh close to the microstructure to the coarse mesh out of the ROI. Such mesh can be quite complex to build with conventional meshing tools, especially when a real microstructure is considered. This approach has been used for small compact tension specimens in Refs. [19,21,24]. The microstructure of the studied high strength steel featured two populations of particles and voids, namely, one of micrometer size and one of submicrometer size. In these studies, only the major population was meshed and simulated with the sFE method. Promising results were obtained by using a damage model inside the matrix material in order to account for sub-micrometer size voids. However, in the absence of local error measurements and in-situ experiment, it was not possible to quantify locally the accuracy of this micromechanical model. In order to avoid the difficulties linked to real microstructures and their randomness, experiments on tensile specimens with machined micrometer size holes were proposed [5]. Simulations based on this experimental procedure were performed [60,22]. Because the position of the holes and their size was part of the specimen geometry, generating meshes adapted to the microstructure was simplified. However, these studies revealed an influence of a minor void population, which could not be meshed.

The specimen geometry with two machined holes used in the present work is directly inspired from Ref. [5]. Thanks to the millimeter size of the machined holes, and the micrometer resolution of synchrotron laminography, it is proposed herein to mesh both void populations, namely, the machined holes and the nodules (considered as voids herein). This is illustrated in Figure 8.6. This mesh is similar to that shown in Figure 8.5(a), with the difference that this time it incorporates the ROI with its microstructure. In the ROI, the material properties correspond to those of Table 8.1, while in the rest of the specimen the two sets of material properties given in Table 8.2 are considered. Compared to other approaches, this method adds significant computation time, as illustrated by the used mesh of ≈ 1.5 million elements in Figure 8.6. However, it has the advantage that ductile fracture can be studied simultaneously at two scales.

Digital Volume Correlation -Finite Element (DVC-FE) technique

DVC used herein is an extension of global 2D DIC [53,54]. The reconstructed volume is represented by a discrete gray level field of the spatial (voxel) coordinate x. The principle of DVC lies in matching the gray levels f in the reference configuration with those of the deformed volume g such that their conservation is obtained

f (x) = g[x + u(x)] (8.5)
where u is the displacement field with respect to the reference volume. In real experiments the strict conservation of gray levels is not satisfied, especially in laminography where deviations appear not just due to acquisition noise but also due to reconstruction artifacts because of missing information [61]. Consequently, the solution consists in minimizing the gray level residual ρ(x) = f (x)g[x + u(x)] by considering its L2-norm with respect to kinematic unknowns associated with the parameterization of the displacement field.

Since a global approach is used in this work, the whole ROI is considered, the global residual

Φ 2 c Φ 2 c = ∑ ROI ρ 2 (x) (8.6)
is minimized with respect to the unknown degrees of freedom u p , the displacement field being written as

u(x) = ∑ p u p Ψ p (x) (8.7)
where Ψ p (x) are the chosen displacement fields for the parameterization of u(x). Among a whole range of available fields, finite element shape functions are particularly attractive because of the link they provide between the measurement of the displacement field and numerical models. Thus, a weak formulation based on hexahedral finite elements with trilinear shape functions is chosen [62].

Conducting DVC analyses with the full size reconstructed volumes is computationally too demanding. Therefore only a part of the reconstructed volume called DVC ROI is considered herein, as shown in Figure 8.2. Additionally, to be able to keep large DVC ROI sizes, the original reconstructed volumes are a priori coarsened, i.e., each 8 neighboring voxels are averaged to form one supervoxel. By doing this, file sizes are decreased by a factor of 8.

The DVC resolution is evaluated by correlating two scans of the unloaded sample (0) with (denoted "rbm") and without (denoted "bis") a rigid body motion (RBM) applied Understanding, observation and quantification of ductile failure mechanisms via 3D imaging On the choice of boundary conditions for micromechanical simulations based on 3D imaging between the acquisitions. Due to the noise contribution and reconstruction artifacts, these two volumes are not identical. Therefore, the measured displacement field accounts for the cumulated effects of laminography and DVC on the measurement uncertainty [63].

The uncertainty values are evaluated by the standard deviation of measured displacement fields. Figure 8.7 shows the standard resolution levels for different element sizes expressed in supervoxels. Decreasing the element size is followed by an increase of the displacement resolution [53,1,54,64]. The element size used in this work is = 16 supervoxels (length) for all three directions, which yields a standard displacement resolution of 0.25 supervoxel. This value represents the limit below which the estimated displacement levels are not trustworthy. Successful DVC registrations have been obtained for the first 9 incremental (i.e., between Step n -1 and Step n) loading steps and they have been used for DVC-FE boundary conditions. The DVC displacement fields are interpolated at each loading step onto the FE mesh of the ROI using trilinear interpolation (i.e., identical to shape functions of DVC measurements).

Results

In this section, results using the DVC-FE approach are first presented and discussed, as comparisons with experiments are qualitatively and quantitatively possible. Then, comparisons with other approaches are proposed.

All micromechanical simulations (wFE, sFE, and DVC-FE) were performed on a cluster of two nodes with a 1.2 GHz Intel Xeon 20-core processor and 64 GB RAM each. These simulations included ≈ 100 voids meshed with ≈ 1 million elements (≈ 1.5 million elements for the sFE simulation). The computation time remained close to one hour Understanding, observation and quantification of ductile failure mechanisms via 3D imaging even for the sFE simulation. This quite low computation time is very promising in the perspective of applying inverse analyses at the microscale to identify micromechanical properties, especially regarding coalescence modeling. This is helped by the fact that the three approaches avoid the requirement of meshing the microstructure of the whole specimen, as in DNS. In the present case, considering the ratio between the volume of the out-of-ROI material and the volume of the ROI in Figure 8.6, the number of elements that a DNS calculation would require can be estimated at ≈ 100 million elements. Although such calculation is not conducted herein, the aforementioned computation time shows that it could be in the very near future, at least regarding the numerical part, thanks to the proposed methods and High Performance Computing (HPC) capabilities demonstrated in this paper. However, the wFE, sFE, and DVC-FE approaches are, by far, more efficient, as the microstructure has to be modeled only inside the ROI in the FE simulations.

DVC-FE coupling

The results using the DVC-FE approach are shown in Figure 8.8, where both void growth and plastic strain are observed.

Errors are be assessed first qualitatively, for example by comparing the X midsection of the ROI with experimental images, as shown in Figure 8.9. Since DVC boundary conditions are expected to follow experimental images, the matrix/void interfaces in the simulation (in white in the figure) can be superimposed on these images and compared. This figure reveals that interfaces are overall very accurately meshed and tracked during the simulation, up to the last loading step. However, there is an irregularity in the material, namely a non spherical void in the top left region of Figure 8.9. In the undeformed state, this defect is already poorly captured by the meshing technique due to its very small size, and this error accumulates during loading. The same remark applies to small voids that can be seen in the experimental image in Figure 8.9(a), but not in the numerical approximation. This figure shows that void growth in the simulations compares well with what is observed in X-ray images.

For each pair of consecutive loading steps, the scan acquired for the second step can be deformed back with the displacement field obtained by the ROI calculation and the result can be compared with the scan acquired at the first step. That is, by means of a newly developed tetrahedral-DVC code [59,57] FE solutions with corresponding tetrahedral meshes are directly imported in the reconstructed volumes frame where the displacement results are interpolated voxel-wise and the corresponding deformed volume g(x) is corrected by the computed displacement field u FE (x). The gray level residuals, i.e., differences between the reference volume f (x) and corrected deformed volume g(x + u(x)) can then be compared for DVC and FE computations, enabling quantitative and local error measurements. This is shown in Figure 8.10 as standard deviation of residual fields (normalized by dynamic range of the volume, i.e., 256 gray levels) for DVC and DVC-FE calculations. These residuals remain very close to those observed in the resolution analysis for which no strains occurred. Therefore the DVC results are deemed trustworthy.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging 3D imaging Note that the DVC-FE curve is not expected to lie below the DVC one, since the latter is used to drive the former. The error produced by micromechanical models inside the DVC-FE domain is low and it slightly increases at late loading steps (from ≈ 15% initially to ≈ 20% in late loading steps). A look at the Z midsection in Figure 8. 11(a-b) reveals that these differences between simulations (DVC-FE) and experiments (DVC) are concentrated around interfaces. This is expected due to plastic localization. The growth of a minor void population and damage at a lower scale are certainly also an explanation. These minor voids cannot be observed due to the resolution of the synchrotron imaging technique used herein. In Ref. [34] they have been observed experimentally using scanning electron microscopy of fractured surfaces for a similar material. Once the minor voids are imaged with sufficient resolution, the present DVC-FE framework would allow to model them. Hence, the development and application of higher resolution techniques, such as nanolaminography, would be interesting in order to check these assumptions. Yet, Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Once the absolute errors in terms of gray level residuals are estimated, relative comparisons can be shown. DVC displacement fields are applied to the boundaries of the FE domain, but they are also available inside the domain. Hence, DVC and FE kinematic fields can also be interpolated on the same mesh and directly compared as shown in Figure 8.12(a). Again, it is confirmed that the main differences are concentrated in the debonding zones, while the differences close to the boundaries are mostly zero.

Although the results using the DVC-FE method are promising, several aspects remain to be improved. In particular, Figure 8.9 as qualitative and Locally, the differences are mainly concentrated around debond areas (i.e., matrix-nodule interfaces). It is hence important to consider more carefully the nodules and their impact both on the load carrying capacity of the material, and void growth. Additionally, the increase of the error at late loading steps indicates the inability of the microscale constitutive models used for the matrix material to capture the acceleration of void growth, and the subsequent void coalescence. This observation calls for more advanced plasticity models at the microscale that can capture the complex multiscale plastic flow localization in the matrix. The growth of a minor void population and damage at a lower scale could also be a possible explanation. Thus, additional material parameters should be introduced for the matrix material, and the present procedure should be extended to allow for the identification of these material parameters. These developments will extensively rely on the DVC-FE method and its ability to provide experimentally measured boundary conditions for micromechanical simulations, and then compute local and relevant error estimators to assess the validity of these simulations. The extension of the Integrated-DIC framework to 4D analyses [57] will also be considered to conduct inverse analyses based on these error measurements and identify material parameters at the microscale.

Comparisons with wFE and sFE results

For quantitative comparisons between the DVC-FE method and its two alternatives used in this paper, two approaches are proposed. In the first approach, as commonly carried out by most authors, void growth curves are compared, giving only average quantities and global error indicators [65,17,18,6,66,8]. The second approach aims to study which method is closer to experimental observations based on local differences. These local errors are computed by reducing the kinematic data for each simulation back to gray level The 3D box where this experimental void growth is computed remains fixed to the initial ROI. The wFE and sFE curves correspond to simulations using material properties based only on load data (first line of Table 8.2), while the wFE I-DIC and sFE I-DIC curves correspond to simulations using material properties based on Integrated-DIC calibration (second line of Table 8.2). Many of the numerical results show an important decrease of the porosity f at the first loading step, which is not observed for the experimental curve. This is due to the fact that remeshing is extensively used for all simulations. This remeshing has the consequence that interfaces can be slightly smoothened, and void volume can be diffused. Apart from this numerical issue, the curves reveal significant void growth with increasing load for Understanding, observation and quantification of ductile failure mechanisms via 3D imaging This effect can be explained by the fact that the computation used to obtain displacement boundary conditions for the wFE simulation does not take into account damage and the subsequent volume change in the ROI. Hence, the displacement fields that are transferred to the ROI in this case are incompressible in the plastic regime, and neither the void volume nor the ROI volume can evolve. This effect calls for more advanced models (i.e., including damage) to be used at the macroscopic scale.

Regarding comparisons with the experimental curve, void growth seems to be overestimated with the DVC-FE method. This can be explained by the fact that nodules are considered as voids in the simulations, while in reality only the voids nucleated after debonding of nodule/matrix interfaces grow (see Figure 8.9). Thus, it can be assumed that all curves would have a slightly lower slope if nodules were taken into account. The DVC-FE method is the only one that shows a void growth similar to that observed in the experiment.

The comparison between DVC-FE and sFE results using the proposed procedure shows that although void growth and the compressibility effects induced by the presence of voids are taken into account in the sFE method, this void growth is not as significant as with the DVC-FE method. In particular, the slopes of the wFE and sFE curves are clearly lower than what is observed experimentally. Both results are not improved when using Integrated-DIC material parameters. The influence of material parameters seems to be negligible in the present case. This observation remains to be confirmed with local error measurements.

Local error indicators

In order to compare wFE and sFE kinematic fields with X-ray images, special care is taken to subtract the rigid body motions from wFE/sFE calculated displacement fields followed by applying the corresponding rigid body motions measured by DVC. First, mean deformation gradients over the ROI for DVC and FE displacement fields are calculated. Surprisingly, although sFE yields slightly better results than wFE, independently of the material parameters, this difference is negligible. This means that although the sFE method predicts a void growth that is globally closer to experimental data, the shape of these voids is inexact. Therefore, errors that do not appear in global measurements are revealed by local measurements, thereby underlining the interest of the present methodology. The small difference between wFE and sFE results could mean that the constitutive model used at the macroscale corresponds well to the homogenized mechanical response obtained in the ROI, where microscale constitutive models are used.

Regarding the comparison between sFE/wFE and DVC-FE, the gap is increasing during the load history, ending with a deviation that is twice higher with the sFE/wFE methods than with DVC-FE. This confirms the tendency that was observed based on void growth curves. To have a more precise idea of this gap, the cumulative gray level residuals distribution is shown in Figure 8.15(a) for the last loading step. This difference does not seem important because it takes into account a large number of supervoxels belonging to the matrix, where contrast is low. This observation is illustrated in Figure 8. 15(b), where only gray level residuals higher than 40 are shown, hence discarding most supervoxels of the matrix. The cumulative distribution taking into account only the remaining supervoxels (i.e., close to voids or nodules) is shown in Cumulative distribution of gray level residuals at the last incremental loading step considering to (a) all supervoxels, (c) only supervoxels with gray level residuals higher than 40. In (b), the absolute gray level differences for the sFE method with residuals higher than 40 are shown for the same Z midsection as Figure 8.11(d) difference between the DVC-FE results and those obtained with the sFE/wFE methods. The residual is doubled with the latter. This underlines the inability of macro simulations to precisely describe all the micro localization phenomena occurring in the ROI both between the machined holes and the nodules, and between the nodules themselves with the chosen constitutive models. There are nearly zero differences in the center of the ROI in Figure 8.12(b), which indicates that the position of the macroshear band (i.e., between the machined holes) is properly captured by wFE simulations, as well as the kinematics within that band. In the same figure, more significant differences are observed in the rest of the ROI compared with DVC-FE simulations.

Overall, the use of Integrated-DIC material parameters for wFE and sFE methods slightly improves the results. As expected in Section 2.3.2, this gain is concentrated at lower strains while at higher strains the influence of these material parameters is negli-Understanding, observation and quantification of ductile failure mechanisms via 3D imaging gible. Paint cracking and subsequent DIC convergence issues from 2D images (test (A)) could be responsible for decreasing Integrated-DIC performance at late loading steps. This observation adds to the interest for the DVC method, which relies directly on the contrast of the material. Thus, the way the material parameters are identified herein improves the results but only slightly.

Using more complex macroscale constitutive models would certainly improve the results obtained using the wFE and sFE methods. This would also increase the need for identification methods such as Integrated-DIC that take into account field measurements. For instance, the inability of both wFE and sFE methods to correctly predict plastic localization could be linked to an anisotropy of the yield surface at the macroscale, due to the presence of the nodules. Damage and the subsequent volume changes of the homogenized material should also be considered, due to debonding of the nodules from the matrix. These remarks stress out the importance of understanding the influence of the microstructure on the mechanical response of the homogenized material. Therefore, a numerical validation procedure for micromechanical simulations that does not rely on any macroscopic constitutive model, such as the DVC-FE method, is of great interest.

Conclusion

The present paper discusses different choices of boundary conditions for micromechanical Finite Element (FE) simulations based on 3D X-ray data. Because it would require meshes of more than 100 million elements, Direct Numerical Simulation (DNS) is discarded and methods allowing the microstructure to be meshed only in a small Region of Interest (ROI), instead of the whole specimen, are considered. The accuracy of these methods is analyzed based on global and local error estimators relative to the 3D X-ray data acquired at consecutive loading steps for the same microstructure. The latter corresponds to new experimental results on a nodular cast iron specimen with two machined holes. Due to the 45 • alignment of the machined holes with respect to the tensile direction, a macroscopic shear band develops between the holes. Within that macroscopic shear band, ductile fracture due to the nucleation (i.e., by debonding of the nodules), growth and coalescence of microscopic voids is observed.

• In an earlier work [28], it was proposed to measure boundary conditions for micromechanical simulations via Digital Volume Correlation (DVC) directly from consecutive 3D X-ray images. Promising results were obtained for a tensile experiment on a specimen with a central hole. The results presented herein show that this DVC-FE method remains very accurate with a shear band going traversing the ROI (Figure 8.8). Global gray level residuals indicate an increase of the error at late loading steps, which is due to void coalescence (Figure 8.13). Local residuals show that these errors are located close to the debond areas around the nodules, hence calling for more relevant micromechanical models.

• A second method, which is referred to as weak FE (wFE), consists of first con- ducting an FE simulation at the specimen scale, considering a totally homogeneous material, and then using the displacement fields from this first simulation to drive a second simulation at the ROI scale. It is proposed to identify material parameters for the specimen scale simulation based only on load data, and then on both load data and 2D surface images. Independently of the identification method, results show that the wFE method leads to a significant underestimation of void growth, as there is no global void volume change in wFE results (Figure 8.13). The slight improvement of the results when using a material parameter identification method that also takes into account 2D surface images indicates that the investigation of more relevant constitutive models for the specimen scale simulation is worth considering.

• Another method, which is coined strong FE (sFE), consists of embedding the ROI mesh in the specimen mesh. Macroscale constitutive models are used for the out-of-ROI material where the microstructure is not meshed, while microscale constitutive models are used inside the ROI where it is meshed. Void volume change curves indicate a significantly increased void growth compared to wFE results. However, local residuals indicate no significant improvement. This proves that while voids in the sFE simulation grow at a rate that is closer to the experimental observations than in wFE results, the shape of these voids is not accurately predicted. Once again, a slight improvement of the results is observed in the first loading steps when using a material parameters identification method that also takes into account 2D surface images.

As a conclusion, the dependence of both sFE and wFE methods on a specimen scale simulation and a corresponding macroscale constitutive model constitute their main limitation. Although the macroscale constitutive model used herein could be improved, such task is not obvious, especially as large plastic strains and complex damage phenomena are observed locally. This limitation of both sFE and wFE methods is also the main advantage of the DVC-FE method since DVC measurements avoid the use of a specimen scale simulation and a corresponding macroscale constitutive model.

It is worth mentioning that since the DVC-FE method requires in-situ 3D X-ray imaging, it is hardly applicable to thick specimens and industrial applications. Future work will hence lean toward the use of this DVC-FE method for small specimens in order to study more relevant micromechanical models and identify material parameters for these models. These enhanced micromechanical models could then be the basis for the investigation of more relevant constitutive models to be used at the macroscale in, say, wFE and sFE methods. Such approach is likely to be applicable to other materials where micromechanical modeling is gaining an increasing interest, such as metallic foams and composite materials for instance.

Introduction

Motivated by constant quest for more fuel efficient and hence lighter design the substantial progress has been made in understanding and modeling of failure of ductile materials over the last decades [11,10,12]. On the other hand, certain open questions still exist, i.e., robust ductile damage model that would be able to correctly predict fracture loci ε f (η, µ) under arbitrary loading paths is still lacking [6,7]. One of the leading reasons for the above mentioned is an incomplete understanding of ductile damage mechanisms at the lower scales. The current work is motivated by idea of having more realistic macroscopic calculations thanks to physical models calibrated directly at the microscale when analyzing the three steps of ductile damage, namely, void nucleation, growth and coalescence. To achieve this aim, computational-experimental framework allowing for validation and identification of the material properties at the microscale is introduced in the following work.

Damage models are useful to optimize the forming processes when dealing with metals and alloys and to predict the in-service life of structures [1,2]. The first models, which are known as macroscopic postulates, were written at the scale of the volume element in continuum mechanics. They allow damage occurrence and the softening toward fracture to be described [3,4,5]. However, these models are known to have limited predictive capabilities for complex loading paths and are not easy to calibrate [7].

An alternative to the previous constitutive postulates is to derive the macroscopic response from microscopic formulations [8,9]. The first propositions were based on closedform derivations [8]. Numerical simulations were also proposed to enhance the initial propositions [9]. Over the years, many improvements have been proposed to enlarge the validity domain of the developed models [10,2,12]. However, it is worth noting that the calibration of such models is still a (very) difficult task [6]. It calls for advanced observation and simulation techniques.

On the experimental side, computed microtomography was used to directly monitor damage in various materials [13,14,15,16]. Physical damage variables such as porosity can now be observed experimentally thanks to 3D imaging techniques [17,18,19,20]. It is possible to study individually inclusions and voids with manual [19] or automatic [20] procedures. The three ductile damage mechanisms, namely, void nucleation [21,22], growth [23,24,25] and coalescence [26,27,15,28] have been analyzed and quantified thanks to computed microtomography.

Simulations are needed to experimentally relate observed quantities such as plastic strain, porosity and the number of fractured/debonded inclusions to internal variables (e.g., stresses) to derive criteria for nucleation and coalescence. Averaged damage models can also be derived by such simulations. The microscale calculations are usually performed with idealized microstructures and constitutive behavior [10,2,12]. Moreover, uniform kinematic or static boundary conditions are prescribed. Such conditions cannot capture the local strain and stress states that inclusions and voids are subjected to due to their random shapes and distributions [17,32,33,20,31]. The effect of the three-dimensional random distribution of voids on softening has been studied [34] us-Understanding, observation and quantification of ductile failure mechanisms via 3D imaging ing different void volume fractions. Consequently, it is desirable to work with realistic microstructures.

From the acquired tomographies, real microstructures were meshed [35,36]. The level set (LS) method [37] is useful to describe interfaces in FE computations when large deformations and complex topological events occur [38,39,40]. When FE simulations are run with such interfaces, image processing can be directly carried out on FE meshes, thereby making such procedures applicable thanks to parallel implementations [41].

Once such microstructures have been generated, the next step is to start validating the numerical simulations. One key aspect is to be able to perform and monitor in situ mechanical tests [42,43]. Combined with digital volume correlation, displacement fields can be measured within the bulk of imaged materials. The early developments of DVC [44,45,46,47] consisted of independently registering small interrogation volumes in the considered Region of Interest (ROI). Global approaches [48] perform registrations over the whole ROI by measuring kinematic fields that are, for instance, based on FE discretizations. Such DVC approaches were directly linked with numerical simulations of mechanical problems [49,50]. They were also used in numerical framework to validate simulations at the microscale [51,31].

The previously introduced computational-experimental framework [51,31] is further extended herein to perform calibration of material parameters at the microscale via 4D mechanical correlation [52] for nodular graphite cast iron. Up to now, such approaches were only performed at the macroscopic level. Once such calibration has been achieved, the next step is to analyze the development of ductile damage. The proposed framework to calibrate material parameters at the microscale is based on the following steps (Figure 9.1):

• X-ray microtomography to get 3D pictures of an in-situ test in a lab tomograph. By post-processing the reconstructed volume in the reference configuration, estimates of the morphology of the microstructure are obtained.

• Global DVC to measure displacement fields whose kinematic basis is made of the shape functions of 8-noded elements. These displacement fields serve as Dirichlet boundary conditions to FE simulations at the microscale.

• The FE simulations at the microscale explicitly account for the morphology of the studied material. Therefore the considered meshes made of 4-noded tetrahedra (mixed velocity-pressure formulation with P1+/P1 elements) are adapted to the microstructure by using signed distance functions obtained from the microtomographies.

• FE simulations are run with an elastoplastic constitutive equation to model the nonlinear behavior of the ferritic matrix. The corresponding parameters will be calibrated herein. The nodules are assumed to have very low elastic properties [53,54,55,56,51]. More advanced models (e.g., debonding of the interfaces between nodules and the matrix) will not be investigated herein.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

• 4D mechanical registration consists of minimizing the registration residuals combined with the equilibrium gap to calibrate the sought material parameters. This integrated approach allows meshes to be as fine as required as opposed to standard DVC techniques [57,58,59].

Identification and validation framework 2.1 In situ mechanical experiment

The material studied herein is commercial nodular graphite cast iron (EN-GJS-400). The volume fraction of nodules is on the order of 11 %. The matrix is mostly ferritic. A dogbone sample is obtained via electro discharge machining (EDM) from a 1.6-mm thick plate. To ensure that the specimen will break in the ligament area and not in the grips, the central part is thinned with a radius of 20 mm. This zone is 9-mm high, while the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging smallest cross-section of the sample is 1.6 × 1 mm 2 . The tensile test reported herein has been carried out in-situ within the X50+ tomograph of LMT (Figure 9.2). The specimen is pulled in six incremental loading and unloading steps (Figure 9.3) in a custom-made testing machine (Figure 9.2), which is placed on the rotation stage of the tomograph. The first two scans of the reference configuration allow displacement uncertainties to be evaluated via DVC analyses.

Displacement uncertainty

The reported test is first analyzed via global DVC with meshes made of 8-noded cubes (i.e., it will be referred to as C8-DVC in the sequel). Such measurements allow the experimental boundary conditions to be prescribed to the numerical simulations (Figure 9.1). It was shown in earlier works on the same type of material that they are crucial when seeking quantitative comparisons with numerical simulations at the macroscopic [52] and microscopic [51,31] scales.

The analyzed ROI is shown in Figure 9.2. Figure 9.4 shows the change of standard displacement uncertainties as functions of the element size for the three displacement components. The observed trend is to be expected from tomographic data [60] and exemplifies the compromise between measurement uncertainty and spatial resolution [58]. In the following analyses, an element size = 64 voxels is selected, which leads to a displacement uncertainty of the order of 0.12 voxel (i.e., ≈ 0.4 µm). ), the image-based mesh construction starts with standard image processing operations [61,62]. They consist of smoothing the data, applying a gray value threshold to separate matrix and voids, and then converting these binary data into a signed distance function. The latter is interpolated to a first mesh of uniform size (i.e., 40 µm or 80 µm depending on the final mesh size) of the FE ROI using linear FE interpolation (i.e., the image is considered as a hexahedral grid where the voxels are nodes). The resulting signed distance function is then regularized based on a recently proposed parallel reinitialization algorithm [41]. The zero contour of the resulting signed distance function gives the location of the interface [63]. The mesh generation and adaption procedure produce bodyfitted isotropic meshes with the local mesh size adapted to the local maximum principal curvature (in absolute value) of the interface [64]. As explained in Ref. [51], the ROI used for FE simulations has to be included in all DVC domains for any considered loading step. Otherwise displacement fields would not be prescribed on the whole top and bottom boundaries of the FE mesh. In practice, in order to be as representative as possible, a 3D box as large as possible is chosen. Further, to make the following simulations tractable, the meshes should not be made too fine since more than one simulation will have to be carried out to calibrate the plastic parameters of the matrix.

Three different mesh densities have thus been studied to analyze the effect of mesh density on the results reported hereafter (Table 9.1). The first mesh (Mesh 1), which is very coarse (Figure 9.6), has less than 13,000 degrees of freedom. The microstructure Microtomography and Digital Volume Correlation is only crudely represented for this initial discretization. The actual size of the meshed volume in FE calculation is 2.87 mm 3 . Meshes 2 and 3 are finer and describe more faithfully the underlying microstructure (Figure 9.6). The number of degrees of freedom has been significantly increased to ≈ 110, 000 and ≈ 360, 000 for the finer meshes. For FE simulations at these small scales, the mesh quality is of major importance and should be investigated. The quality distribution is reported in Figure 9.7 for the three mesh densities. A quality of 1 means a well shaped tetrahedron, while a quality of 0 means a degenerated element. Finer meshes induce more elements far from interfaces and render Understanding, observation and quantification of ductile failure mechanisms via 3D imaging mesh adaption easier. Therefore, the histogram for the finest mesh has more elements with good to very good quality. As the meshes are coarsened, their quality is substantially decreased. This result can be understood by analyzing Figure 9.6, which shows that the 'nodules' correspond to very elongated elements for Mesh 1, which is less the case for Mesh 3.

Investigated constitutive law

The framework used for microscale FE simulations is based on previous developments presented in Refs. [39,65,41,64,63]. To avoid numerical locking in plasticity, a mixed velocity-pressure formulation is used with P1+/P1 elements [66]. Large transformations are considered with an updated Lagrangian formulation, which is commonly used in metal forming simulations [67]. This incremental formulation has proven to give accurate results for small time steps. The constitutive equation investigated herein is Ludwik's law [68], which describes isotropic hardening as follows σ eq = σ y + K p n (9.1)

where σ eq is the second (von Mises) invariant of the deviatoric part of the Cauchy stress tensor, p the accumulated plastic strain associated with the Cauchy strain tensor, σ y the yield stress, K the hardening modulus and n the hardening exponent. Such type of constitutive equation may seem very crude. This is certainly the case up to an extent that will be investigated herein. However, almost all cell calculations [8,9,10,12] or even with more complex microstructures [34,41,51] use such very simple constitutive laws.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Consequently, the present work aims at calibrating material parameters to be as consistent as possible to experimentally observed microstructures. It is worth noting that the validation framework is compatible with more advanced constitutive postulates. However, the computation time would be increased very significantly (provided the simulations can be carried out with such fine discretizations). Ludwik's law was calibrated against a stress/strain curve for a pure ferritic matrix [54], which is assumed to be representative of the in-situ behavior of the ferritic matrix of the studied cast iron. The obtained values are σ y = 290 MPa, K = 382 MPa, n = 0.35; the Young's modulus E = 210 GPa, and Poisson's ratio ν = 0.3. It was also used to simulate damage growth in cast iron when imaged via synchrotron laminography [51,31]. However, up to now it was not probed against an experiment in which displacement and load data were utilized to calibrate the matrix behavior at the microscale in order to analyze ductile damage. The calibration will use an integrated approach to DVC.

Integrated DVC

When performing integrated DVC the displacement fields are mechanically admissible (i.e., they satisfy equilibrium in the finite element sense for a chosen constitutive law in addition to their kinematic admissibility). The first 3D developments were based on elastic simulations for which a non-intrusive framework was implemented [59]. They were then extended to nonlinear analyses at the macroscale [52]. The FE code is then used to generate kinematic fields that are needed for DVC purposes. When material parameters are sought, the corresponding spatiotemporal sensitivities [69,70] are also needed for nonlinear constitutive postulates. The 4D kinematics is thus parameterized with the sought corrections {δp} to the current estimate of material parameters { p}

u(x,t, {p}) = u(x,t, { p}) + ∂u ∂{p} (x,t, { p}) † {δp} (9.2)
where [∂u/∂{p}] is the matrix gathering all the spatiotemporal sensitivity fields to the sought material parameters, x any voxel within the analyzed region of interest, and t the time. Thus the discretized displacement field becomes

u(x,t, {p}) = ∑ t ∑ n υ n ({p},t)N n (x) (9.3)
in which the kinematic degrees of freedom υ n , which are associated with the shape function vector N n , are not independent but all linked via their sensitivities to the sought parameters

υ n ({p},t) = υ n ({ p},t) + ∂υ n ∂{p} ({ p},t) † {δp} (9.4)
The kinematic sensitivities associated with a chosen spatial discretization are gathered in matrix [S υ (t)] (i.e., (S υ (t)) nm = (∂υ n /∂p m )(t)) that is evaluated for each time step t.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging The finite difference method is used to approximate ∂υ n ∂{p} , i.e.,

∂υ n ∂{p} ({ p},t) ≈ υ n ({ p} + ζ{ p},t) -υ n ({ p},t) ζ p (9.5)
where ζ is preturbation factor. To illustrate such fields, Figure 9.8 shows the longitudinal component of the sensitivity fields associated with the three plastic parameters when each parameter is independently perturbed by ζ = 2 % with respect to its nominal value. Consequently, these fields are expressed in voxels divided by the physical unit of the studied paramaters. It is observed that the first and last loading steps have overall shapes that are different from the intermediate steps (i.e., 2 to 5). This trend may signal the development of different damage mechanisms or regimes, as will be discussed in Section 4. Table 9.1 reports the computation time required to have access to sensitivity fields for 3 investigated material parameters (i.e., four FE simulations). There is a two order of magnitude difference in computation time. It is worth noting that the machines used to run these different calculations had to be adapted to the mesh density i.e., for Meshes 1 and 2, a workstation (6 CPUs) with Intel(R) Xeon(R) E5 processor (32 GB, 3.3 GHz) was used, and a cluster node (16 CPUs) with Intel(R) Xeon(R) E7-8837 processor (252 GB, 2.67 GHz) for Mesh 3. For the finest mesh, each set of sensitivity fields (for 3 material parameters) requires about 28 h of simulation.

4D mechanical correlation consists of minimizing the sum of squared gray level differences directly with respect to {p} {p} meas = argmin over the considered ROI for the reconstructed volumes in the reference configuration I 0 (x) and deformed configuration I t (x). The coefficient γ 2 I denotes the variance associated with acquisition and reconstruction noise of the gray level volumes.

In the present case, load measurements gathered in vector {F meas } are also available. Thus they are compared with the FE predictions in which the reaction force vector {F FE } is due to the fact that measured displacements are prescribed on the top and bottom boundaries of the considered ROI. The global equilibrium gap over the whole loading history reads

ρ 2 F = 1 γ 2 F {F meas -F meas ({p})} 2 (9.7)
where γ 2 F denotes the variance of the measurement uncertainty of load cells. The latter is assumed to be white and Gaussian. The minimization of ρ 2 F alone corresponds to updating the FE model by considering only global equilibrium to estimate the sought parameters {p}. It is referred to as load-based FE model updating (or FEMU-F [71,72]).

In the following, both sets of data, namely, reconstructed volumes and measured loads will be considered in a single approach. A normalization framework is considered [52] in which each considered information is weighted by its merit (i.e., variance and covariance Understanding, observation and quantification of ductile failure mechanisms via 3D imaging with all the other data) for DVC (i.e., χ 2 I ) and global equilibrium (i.e., χ 2 F ). An equal weighting is then assumed between the kinematic and load contributions

χ 2 = 1 2 χ 2 I + 1 2 χ 2 F (9.8) with χ 2 I = 1 |ROI|n t ∑ [0,t max ] ∑ ROI (I 0 (x) -I t (x + u(x,t, {p}))) 2 2γ 2 I χ 2 F = 1 n F n t ∑ t {F meas (t) -F EF (t, {p})} 2 γ 2 F (9.9)
where n F denotes the number of load measurements per scan (i.e., n F = 1 in the present case). With the chosen weighting, when the residuals are close to unity, the only contribution to the error is due to noise associated with tomography and load cell(s) measurements.

If χ I and/or χ F are significantly larger than unity then there are numerical model errors. 4D mechanical correlation thus consists of minimizing χ 2 with respect to the sought parameters

{p} meas = argmin {p} χ 2 (9.10)
A Gauss-Newton scheme is implemented for which the corrections {δp} to the sought material parameters are iteratively updated [52]. Each iteration lasts about one day. To speed-up the simulations, the sensitivity fields are no longer updated when all parameter variations are less than 10 % of their current estimate. Convergence is reached when their relative variations are all less than 0.01 %. The integrated DVC code, Correli 3.0 [73], is a Matlab implementation that uses in a non-intrusive way FE codes to compute the spatiotemporal sensitivity fields in addition to the current estimates of the displacement fields and reaction forces. The inputs to such simulations are the measured displacements of the top and bottom boundaries of the ROI. In the present case, the CimLib library is used with its built-in constitutive laws [65].

3 Simulation results

Mesh sensitivity

Table 9.2 gathers the parameters at the end of the integrated DVC analyses for the three different meshes. The first set of simulations is initialized with the parameters proposed by Zhang et al. [54]. The second set of simulations is performed when the previous initial guess is perturbed by 10 %. Each set of calibrated parameters is different. To better understand the difference of all these parameter sets, Figure 9.9 illustrates the corresponding equivalent stress / strain responses. In comparison with the initial guesses, there are two groups of results. First the two finer mesh densities lead to very similar stress / strain responses. The coarser mesh leads to significantly different responses. This difference is attributed to the fact that Mesh 1 only crudely describes the underlying microstructure (Figure 9.6). The fact that Meshes 2 and 3 yield very similar stress / strain responses indicates that convergence is achieved in terms of matrix behavior accounting for void growth. Since both initial guesses lead to virtually identical stress-strain responses for the three mesh densities (Figure 9.9), only the initialization based on literature data is further discussed. In the present case, it can be noted that there is a clear difference between the initial guess and the converged solution. This is to be expected since the in-situ properties of the present ferritic matrix are not necessarily identical to the ferrite studied by Zhang et al. [54]. However, the initial part of the yielding regime (i.e., up to 5 % equivalent plastic strain) is very close. Conversely, for equivalent strains greater than 5 % the difference is more pronounced. It is believed that this is due to the fact that the constitutive law is probed over a larger strain range in the present case thanks to measurements at the microscopic scale.

The local stress-strain differences (Figure 9.9) will also induce different dimensionless load residuals (see Figure 9.10). There is a significant decrease of the latter ones from the initial guess to the converged solution. The coarse mesh leads to a higher global residual (i.e., 14.6) in comparison with the two fine meshes, which lead to global load residuals 9 times the standard uncertainty of the load cell. Let us recall that the gray level residual stands for the difference between undeformed and corrected deformed volume by the measured (DVC) or the simulated (FE) kinematic fields. Figure 9.11 shows the change of dimensionless gray level residual for the six analyzed loading steps. Even though the C8-DVC analyses are only based on a very coarse mesh (Figure 9.5) they still lead to the lowest gray level residuals. The latter ones are at least equal to 1.6 times those corresponding to the uncertainty analysis discussed in Section 2.2. This trend indicates that the measured kinematics does not completely follow the details of the material microstructure. However, the residuals remain sufficiently low Understanding, observation and quantification of ductile failure mechanisms via 3D imaging to deem the measurements trustworthy and apply them as Dirichlet boundary conditions to the FE simulations.

The same degradation of the gray level residuals is observed for the four analyzed numerical cases. Since measured boundary conditions are considered, gray level residuals can also be evaluated for displacement fields that would not be measurable with a standard DVC code since the mesh size would be too small to ensure good convergence. In the present case, all I-DVC results lead to virtually the same residuals. The fact that there is only a very minor influence of the three discretization is due to the fact that the modeled region of interest is small and that the type of loading is rather simple (i.e., induced by uniaxial tension). Further, they are significantly higher than C8-DVC residuals. This clearly indicates a model error, which has to be analyzed more locally by using the voxel scale representation of the gray level residuals (see Section 4). To conclude the mesh sensitivity analysis of the three mesh densities considered herein, the coarser Mesh 1 is disqualified since it is not sufficiently close to the actual microstructure (Figure 9.6) and it does not predict the overall load level as accurately as the other two meshes. The latter ones lead to very similar load residuals (Figure 9.10) and stress / strain responses (Figure 9.9). Consequently, they are equivalent in overall predictive power with the chosen constitutive framework. However, in terms of overall computation time there is a significant increase, which was anticipated from the computation of the sensitivity fields (Table 9.1), namely, 50 h for Mesh 2 and 236 h for Mesh 3.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Note that identification procedure can start with Mesh 1 for the first parameter estimations and finish with Mesh 3 for more detailed analysis.

Discussion

Since the actual microstructure of the material is explicitly meshed, void growth should be reasonably described, provided the constitutive law of the matrix is representative of the actual material. Conversely, damage nucleation and coalescence are not accounted for. Therefore it is believed that some of the deviations of the previous residuals can be linked to these hypotheses.

Figure 9.12 shows sections of the gray level residuals for the six (in printed version only sixth) analyzed loading steps. For the first loading step, the C8-DVC residuals reveal that early debonding occurs in particular in the lower/left region of the analyzed volume. For the second loading step, coalescence and fracture has already started and matrix/nodule debonding has further developed over a large area. For I-DVC, the chosen constitutive law is not able to fully describe the transverse response of the material. Consequently the gray level residuals are higher than in C8-DVC (see Figure 9.11). Similar observations can be made by analyzing the residuals for steps 3 to 5, which have essentially the same features. Last, further coalescence and fracture develop in the last loading step and the residuals degrade for C8-DVC and I-DVC. It is worth remembering that as the analyzed loading step increases, so does the load residual (Figure 9.10). With the present observations it can be stated that this degradation is linked to coalescence and fracture. Let us also note that apart from matrix/nodule debonding and coalescence, no other zone with high residuals is observed. Consequently, it is believed that damage nucleation, if present, occurs at scales lower than that of the voxels (i.e., 3.4 µm).

Figure 9.13 shows microstructural and equivalent strain maps (from FE calculations) for the six analyzed loading steps (in printed version for the sixth loading step). From the first loading step, the strain field is heterogeneous but the most loaded zone is not the one that will lead to coalescence and fracture. For the second loading step, the zone where coalescence occurs is the most strained but coalescence and more importantly fracture are not captured in FE simulations. Further, there is a second zone where strains concentrate (see loading steps 2 to 5). For the last loading step, the strain levels continue to increase.

The FE framework used herein naturally takes into account coalescence by internal necking without any special criterion. In fact, Figure 9.13 (strain maps, between increments 1 and 4) shows that coalescence instances at about one quarter of the length of the crack are captured in FE simulation. Hence, if the coalescence occurs by internal necking, it should be predictable. If coalescence is not completely captured by FE as in the present case, the individual void growth is probably underestimated. The possible reasons for underestimating void growth originating from the matrix behavior are:

• not considering the effects of softening due to damage at smaller scales. The possible acceleration of void growth when coalescence arises calls for an implementation of a coupled damage model that was not considered herein.

• the present hardening law does not saturate (i.e., there is no plateau) and induces unrealistic hardening. This effect will supress levels of plastic strain in the intervoid ligaments for the high stress states, i.e., postpone void growth and coalescence.

Another reason would be having underestimated the initial void volume fraction in the FE simulation. The quality of the geometrical descriptions of the nodules is still imperfect due to the limited mesh density (even Mesh 3). In this scenario, underestimated initial Understanding, observation and quantification of ductile failure mechanisms via 3D imaging void volume fraction in the FE ROI will generally decrease the void growth rate in the matrix. Also, the topological details of the nodules (voids) which may play important role for the stress localization and void interaction in the inter-void ligaments are not properly captured.

Conclusions

In this paper an innovative validation procedure has been presented (Figure 9.1) to study damage growth and coalescence at the microscale. Experimental data (i.e., reconstructed volumes) are collected via tomography while performing an in-situ tensile test on a dogbone specimen made of nodular cast iron. Bulk kinematic fields in the central region are measured via DVC. The FE mesh is adapted to the real microstructure by assuming that graphite nodules behave as voids with a (re)meshing procedure based on level set fields. The subsequent FE simulations are driven by Dirichlet boundary conditions extracted from DVC measurements. To be as representative as possible of the actual test, the plastic parameters of the ferritic matrix are calibrated utilizing integrated DVC.

The fact that FE simulations are driven by measured boundary conditions allows the simulations to be more representative of the actual experiment compared to the other methods used for obtaining the boundary conditions [31]. It also enables calibration procedures to be run at the microscale by accounting for load and imaged microstructure. Consequently, the constitutive law of the matrix is not only probed in terms of overall load level but also by kinematic fields induced by the random microstructure.

The overall residual levels are sufficiently low thereby indicating successful identification. The areas of higher residuals correspond to the position of the debonded zones that appear early on, and increase further due to void coalescence. The fact that the numerical simulations do not account for the accelarated void growth (softening at smaller scales) in the coalescence zone explains why the gray level residuals degrade between DVC and I-DVC. Even though more advanced constitutive models may be considered in the future, the overall dimensionless residual is only of the order of 8, which is low in comparison to analyses at the macroscale [52].

One of the next steps of the present work will deal with void sheeting coalescence that was not accounted for in the reported results. Different criteria may then be probed. Similarly, the hypothesis of considering nodules as voids can also be relaxed and debonding may be addressed. The proposed numerical-experimental framework can still be used to deal with these different degradation mechanisms.

of the Carnot M.I.N.E.S institute ("CORTEX" project). M. Kuna, L. Zybell and M. Horn from IMFD, TU Freiberg, are thanked for material supply and machining as well as for scientific discussions.

The main objective of this work was the observation and quantification of ductile damage mechanisms at the microscale and their interplay with the local strain fields. The understanding of the influence of microstructure changes on the macroscopic mechanical response is of great interest in the ductile damage comunity. As detailed in the introduction recent multiscale modeling efforts based on the work of Gurson [9] derive the macroscopic response from microscopic formulations of ductile fracture based on closed-form derivations [8,7,2,21] or numerical simulations [11,2,21,3,5]. Although promising, the results obtained using these methods are mainly restricted to studies of idealistic microstructures and loading conditions and their calibration (not to mention validation) is still a difficult task. Consequently, numerical methods that would use the same multiscale framework but accounting for realistic microstructures and measured boundary conditions are highly desirable.

In this context, the present work brings a new set of exprimental evidence to microscopic events leading to final ductile failure for two classes of materials, namely, structural aluminum alloys and nodular graphite cast iron. The particularly important aspect of this work opposed to the existing tomography-based data is the utilization of laminography, which facilitates the use of thin sheet samples with different stress states in the scanned regions. All the reported results are accompanied with measured bulk kinematic fields obtained by DVC.

For aluminum alloys, the observation of microstructure and strain-damage interactions at micrometer resolution for various specimen geometries and associated levels of stress triaxiality are analyzed.

On the cast iron side, a combined computational-experimental framework is introduced to validate numerical simulations at the microscopic scale. Finite Element (FE) simulations, which account for the studied material microstructure, are driven by Dirichlet boundary conditions extracted from DVC measurements. Gray level residuals for DVC measurements and FE simulations are assessed for validation purposes. This framework was also used to calibrate plasticity parameters of the ferritic matrix.

1 Aluminum alloys

Major findings

The plastic zones close to the notch root of CT-like specimens were studied in the material bulk by combining in situ laminography and Digital Volume Correlation. The parallel studies of strain and microstructure activity have yielded the following results:

1. Kinematic fields • several crossing slant bands form strain patterns when moving at a distance greater than ∼ 250 µm from the notch root. They are active from the very early stages of loading. The formation of strained bands precedes damage growth and the softening phase.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

• In AA2198 T8, AA2139 T3 and AA2139 T8 alloys the strained bands show alternating activity in slant regions (≈ 1000 µm from the notch root) by switching on and off during incremental loading steps. AA2198 T3 shows a more stable pattern over the whole loading history.

• The spatial location of the bands compared to microstructure features is stable, i.e., the plastic activity is not evening out the strain fields.

• The typical strained band spacing is 150 -250 µm.

• The ratio of the effective strain within and outside the bands is about 1.5 to 2, i.e., portions of the ROI outside the strained bands do not only experience elasticity, but deform with lower plastic strain levels. This is the main reasoning for avoiding the term strain localization but rather using the term strained bands for their description.

• Normal strains in the crack propagation direction are confirmed to be close to zero [4]. It was demonstrated that the onset of strained bands spatially coincide with reaching the plane strain state for AA2198 T8 alloy. These results are confirmed for AA2198 T3 alloy.

• The comparison of results for AA2198 T3 and T8 grades reveals that strong work hardening (T3) does not significantly change the behavior reported on AA2198 T8 grade [14] concerning the early development of slant strained bands ahead of a notch in a large flat specimen even though work hardening is thought to hamper localization.

Damage process

• In the slant fracture region, strained bands appear very early on, before any signs of damage onset. Void nucleation and growth has been quantified for these samples. It is found to be very low in the slant fracture region.

• Furthermore, even in the close vicinity of the notch root, significant damage does not occur immediately after the onset of strained bands. Consequently, the final fracture path proceeds along the measured incremental strained bands. It can therefore be stated that early strained bands precede the crack path away from the notch root and also getting very close to it for the investigated aluminum alloys.

• Higher damage growth in the immediate vicinity of the notch root is a consequence of higher stress triaxialities. Two separate damage micromechanisms are identified in the case of AA2198 T3 alloy, namely, i) shear-type damage and ii) more classical high stress triaxiality induced damage. Even in the "flat" fracture zone the shear-type micromechanism prevails, which causes the zigzagged final crack pattern very close to the notch root. In general, the specimens fail under a "roof-top" to slant transition mode (i.e., there is no real flat region), which is determined by the strained band pattern already detected

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging at the beginning of the loading history. Both EBSD and SEM analyses correlated the results obtained for AA2198 T3 alloy and are consistent with such assumptions.

Discussion

The new deformation mechanism with "waffle"-like patterns has been confirmed for all four studied aluminum alloys as illustrated in Figure 10.1. It consists of multiple slant strained bands present from the beginning of the loading process with more (AA 2139 T3, AA 2139 T8) or less (AA2198 T3, AA 2198 T8) intermittent plastic activity. The final fracture is driven by heterogeneous plastic flow [15]. Damage growth is limited until the very late stages of the experiment. This type of ductile failure (by instability) could be partly placed under the scope of ductile failure with limited void growth presented in the introduction.

Concerning the damage and deformation interactions, it has been shown that the void volume fraction and growth ratio at particular instants do not act as driving force for softening, further ductile damage development and final crack propagation direction. In the AA2139 T3 case even very concentrated (strained) bands with high von Mises effective strain levels (i.e., 0.10) may become inactive and other bands become active at very late stages of loading. The fact that strained bands may switch off at the end of the test indicates that the mechanisms at play for strain concentration may differ from damage softening.

Under the utilized spatial resolution (≈ 0.7 µm) no evidence of primary or secondary void nucleation can be found inside slant strained bands at their onset, even in the immediate vicinity of the notch root (for AA2198 T3). Consequently, it has been argued that shear slip is likely to occur in this class of materials, including strong work hardening alloys. The question of the proper scale for the physical modeling of ductile damage mechanisms (at least in this group of materials) can be outlined since there is a clear need for a shift to modified models accounting for the activation/deactivition of (possibly) multiple strained bands.

The possible physical explanations for the origin of slant strained bands along with the existing numerical simulation results are discussed next:

• Classical J2 flow theory or damage softening have been shown not to capture the measured bands [14]. With a standard Gurson-Tvergaard-Needleman-model the crack remains flat.

• The observed slant strained bands at early loading stages cannot be reproduced using a Gurson-Tvergaard-Needleman-type model with Lode parameter-based shear void nucleation. The flat to slant crack transition is however successfully reproduced [13].

• 2D plane strain simulations in the slant region accounting for a soft zone yield strain ratios within the band and the surrounding material similar to the measurements for • Following this idea, simulations were performed using a mesh with randomly distributed soft zones to mimic soft grain distributions. The slant strained bands can be obtained. These soft zones could thus be a possible origin of the phenomena found here. The ratio of the strain within the bands and outside the band is, however, substantially lower than in the experimental findings and the intermittent plastic

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging activity is not captured (Appendix A).

• In the previous listed models no ingredients to reproduce the intermittent strained band activity are used. Portevin -Le Chatelier (PLC) effect could be one of the possible origins of slant band development and in particular their intermittent activity [17,18]. The flip-flopping of the bands would also be consistent with such a behavior. It is, however, surprising that the experimentally measured strained bands are stable in space with respect to microstructural features. This is generally not found for moving PLC bands on smooth specimens as in these cases the strained bands move in time and even out strain heterogeneities in the different regions.

Here, however, the notch naturally concentrates the strain fields and may prevent the bands from moving.

• The heterogeneous strain fields could be linked to heterogeneous microstructures and resulting plasticity [16]. Local material heterogeneity caused by crystallographic texture, named roping, could also be a possibility [23]. In that case these local textures are the result of ex-super grains that existed at some point during material processing (so called ghost grains). However, the characteristic lengths identified in Ref. [23] are substantially larger than what is found in this work, namely, more than 1 mm [23] compared to about 200 µm herein. In addition, the bands that are found are slanted and evenly spaced. There is no obvious reason to find such slant grain arrangements in soft zones. Additionally the characteristic spacing and ordered nature of the bands makes a purely material heterogeneity based origin less likely.

• Last, it is important to outline that all previously mentioned simulations are mesh dependent. Hence, nonlocal models are needed to get more reliable predictions of localized phenomena but the regularization length is still to be physically understood and identified. The present finding could also add another length to the commonly used particle or void spacing, which is the strained band spacing.

Furthermore, in Chapter 5, a shear specimen geometry has been studied where the final failure initiated far from the notch (free surface). The measured incremental strain Lode parameter fields proved to be sensitive enough to confirm pure shear stress states, i.e., values close to zero have been found in the shear band. This analysis can also be performed by studying bulk kinematic fields in CT samples under mode I opening. Damage nucleates from the middle of the loading process in the scanned shear ligament, mostly concentrated within the shear plane. The damage features have mainly elongated shapes and cumulate in the bulk of the ligament, away from the free surfaces. Along the elongated features, intermetallic particle breakage is also detected but with minor influence on the final failure of the ligament. Elongated features are suspected to be grain boundaries while void sheeting is the likely nucleation mechanism for them. The shape of elongated damage features corresponds well to the shear-type damage features found in the high stress triaxiality region close to the notch root (AA2198 T3) even though their orientation Understanding, observation and quantification of ductile failure mechanisms via 3D imaging with respect to the grain boundary orientation is different (i.e., in shear elongated features are aligned with the grain boundaries while in CT samples they are inclined at ≈ 45 • ).

Perspectives

For alluminum alloys three principal directions for future work can be defined:

1. To complete the present findings and previous studies, other classes of aluminum alloys (e.g., 5XXXX, 6XXX or 7XXX series) or steels could also be analyzed to confirm the presence of slant strained bands.

2. Beside the reported CT and shear samples, other types of geometries with associated stress triaxialities can be studied. In particular, samples with lower geometrical stress concentration factors (e.g., central hole tensile specimen instead of notched sample) should provide answers in terms of possible generalizations of the recorded deformation mechanisms and its characteristic behavior. For example, depending on these results, i.e., whether moving or still strained bands will be found, more or less weight can be put on the PLC effect as a possible explanation for the observed phenomena.

3. For the usual dislocation densities in metals the cutoff particle size for nucleation is ≈ 0.1 µm whereas the scaning resolution for the reported aluminum alloys results was ≈ 0.7 µm. One aspect of future studies of these alloys may consider in-situ nanolaminography.

First but significant steps toward this achievement have been made in this project. The in-situ experiment with an AA2139 T3 CT specimen was performed at beamline ID 16b at ESRF, Grenoble. For the purpose of the experiment a specific loading machine was designed and machined.

Overall, nine scans with intermediate (1 voxel = 240 nm) and nine scans with high (1 voxel = 100 nm) resolution were obtained during the experiment. A holotomography reconstruction principle combining images taken at four different distances from the detector was used [6,12]. This procedure results in quantitative phase mapping along with the complete three-dimensional mapping of the density in the sample. Thanks to this technique, the voids and intermetallic particles of nanometer size are more easily distinguished from their surroundings.

The scanned region is positioned in the immediate vicinity of the notch root as illustrated in Figure 10. The voids and intermetallic particles possess significantly better visual descriptions compared to microlaminography results. This is illustrated in Figures 10.4 and 10.5 where the portion of the reconstructed volume is followed on one of the sections and in 2D views of the segmented volume. Consequently, the microstrucutre features can be introduced as meshing objects with explicit interfaces in CimLib tools [19,20]. Given that the boundary conditions can be measured by DVC, the aluminum alloys can be treated in a similar manner as was demonstrated in case of cast iron. The proposed numerical/experimental framework can then be used to address different degradation mechanisms and associated constitutive laws. Different criteria, e.g., for nucleation and final coalescence may then be probed.

Major findings

The principal achievements consist of the ductile damage mechanisms study (kinematic fields and damage process, based on 2 hole samples) and the numerical/experimental framework introduced in the work (central hole sample, 2 hole sample, tensile sample):

1. Kinematic fields

• From the early loading steps higher strained areas are observed between some of the graphite nodules.

• It is concluded that the presence of strained bands is always accompanied by high density of nodules, while not all the nodules are affected by the heterogeneous strain pattern.

• Although present from the early loading steps on this type of straining between some of the nodules differs from that reported for Al alloys in slant regions of Compact Tension samples where a plane strain condition along the crack propagation direction has been identified.

• Nodular graphite cast iron represents another class of materials compared to Al alloys and straining between the nodules is mainly a structural effect. Yet, it is confirmed herein that the plane strain condition is not a necessary condition for very early and stable straining.

• In a multiscale framework (2 hole samples) when shifting from the 90 • configuration (i.e., void necking as mesoscale mechanism) to 45 • (i.e., shear colaescence as mesoscale mechanism) the role of graphite nodules as localization inceptors becomes less influential, i.e., most of the failure process is governed by primary (machined) hole position. In the 90 • configuration, a more diffuse plastic zone between the machined holes favored the influence of secondary (nodule) voids on the crack path. The final failure followed the early strained paths between nodules that, sometimes, did not represent the shortest distance between the primary voids (holes).

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

• On the mesoscale (i.e., most of the ligament between the machined holes) the mean cumulated effective strain shows a higher level for the 45 • configuration. The mean trace of the strain tensor, i.e., volume change, does not differ significantly for the two cases. In other words, the overall cumulated plasticity is higher in the 45 • configuration mainly because of the shear component.

• On the microscale (internodule area) local von Mises strain levels prior to coalescence are similar for both configurations, i.e., ≈ 40 % for void sheeting internodule ligaments and ≈ 45 % for internal necking ligaments irrespective of the mesocale conditions.

• The ability of effective von Mises strains to locally indicate the material failure is demonstrated. However, it remains insensitive to the type of damage mechanisms (i.e., void necking or void sheeting) and final ligament failure.

On the other hand, the local strain trace values exhibit some sensitivity to different ligament failure mechanisms (i.e., corresponding curves possess different slopes depending on underlying coalescence mechanism). The local trace cannot be used as global threshold quantity for different mesostress states, but when limited to the defined mesostress state (in this case 90 • or 45 • configurations) it is a helpful physical quantity for the distinction of the two ligament failure mechanisms.

Damage process

• Early debonding of nodules has been noted in all studied samples. Once debonded, the new voids tend to initially grow in high triaxility mode and interact with the closest neighbors in an internal necking mode.

• At the very late stages of deformation just prior to final failure void sheeting coalescence takes place in some locations. In the 45 • configuration it is mainly because of the strain band effect determined by the primary void positions while in the 90 • configuration void sheeting is the outcome of the favorably positioned clusters of already linked voids that finally needed to coalesce to form the main crack.

• On the mesoscale the void growth rate is higher in the 90 • configuration. However, the void volume fractions prior to coalescence are similar for both cases, namely, ≈ 16%. Accelerated void growth prior to coalescence (at ≈ 13% effective strain) is found in the 90 • configuration while in the 45 • configuration such trend is not found.

• On the microscale the normalized individual void growth outlines that even though the corresponding inter-nodule ligaments will end up with the same type of failure, void growth of neighboring voids is higher in the 90 • configuration for both types of coalescence mechanisms.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Cast iron

• Kinematic data (strain trace) are shown to be a useful tool for the initial estimation of the proper gray level values for thresholding methods when calculating void growth from tomography/laminography data.

Numerical/experimental framework

• An innovative validation procedure combining synchrotron laminography, digital volume correlation and FE simulations on the microscale level has been implemented (DVC-FE method).

• The FE mesh is adapted to the real microstructure under the assumption that graphite nodules are considered as voids with a specially developed (re)meshing procedure [19,20].

• Once validated, DVC measurements are used to drive FE simulations. Added to the fact that the real microstructure is discretized in the FE mesh, heterogeneous displacements prescribed by DVC require a robust mesh adaption methodology. The latter is used herein and enables simulations to be performed with significant void growth up to coalescence.

• DVC and FE results can then be probed by direct comparison of the full-field solutions and computation of the gray level residuals. The first method, which is a relative comparator, provides the gap between DVC and FE kinematic fields. Conversely, the gray level residuals can be computed independently for both approaches giving accurate and local error measurements (i.e., directly with respect to experimental gray level scans).

• Next, different choices of boundary conditions for micromechanical Finite Element (FE) simulations based on 3D X-ray data are studied. Most previous works have relied on multiscale methods to deduce boundary conditions for micromechanical calculations [22,1,10]. Independently on the choice of coupling between macroscale and microscale mechanics (i.e., strong or weak coupling), it is shown that multiscale schemes cannot capture as accurately as DVC-FE the complex plastic localization and void coalescence phenomena occurring in the ROI. Importantly, the local error measurements based on gray level residuals can also be applied to micromechanical simulations initially not driven by DVC. The errors produced by the mentioned multiscale schemes are thus measured more precisely, revealing that void morphology is not well described locally with these schemes.

• Finally, the sequential DVC-FE framework is upgraded to the integrated identification procedure. 4D mechanical registration consists of minimizing the registration residuals combined with the global equilibrium gap to calibrate the sought material parameters. Hence, the elasto-plastic constitutive law of the cast iron ferritic matrix is not only probed in terms of kinematic fields induced by the random microstructure but also by overall load levels.

3 Discussion and future work

The complexity of multiscale and strong 3D nodule distribution effect is revealed in the two hole samples confirming the importance of employing real microstructures in numerical microsimulations. Let us focus on the 90 • configuration in the following discussion as more representative in the sense of multiscale influences at play. It is the local nodule arrangement of the real microstructure that is of prime importance, then the nodule shapes and matrix defects that will induce different stress states responsible for different ligament failures. The strained areas induced by nodule positions are present from the beginning of loading and the final failure is following one of these paths. Hence, there is no initial primary plastic localized band stretched over several internodular ligaments but it is rather heterogeneous from the begining (Figure 10.6(a)).

For convenience, the failure process is divided into five steps. Damage growth as nodule debonding is present from the early loading steps (Stage 1, 2). The voids grow and coalesce in an internal necking mode both inside and outside of the most active bands (Stage 3, 4). Only at the very end of the loading process most of the plastic flow localizes into a single thin band as precursor to the final failure (Stage 5).

Although the ductile failure of cast iron follows the three principal stages of ductile failure by void coalescence the plastic instability mode is present. One can argue that it is a more structural effect and graphite nodules due to their low mechanical properties can be modeled as voids (i.e., initial softening is present from the begining) but still the very early presence of strained areas between nodules in what will be the final failure path is a remarkable event. It is important to outline that this deformation mode can be fairly well predicted by meshing the microstructure and using von Mises plasticity in FE simulations (Chapters 7 and 8).

For comparison purposes, these phases are also illustrated for the corresponding set of events found in aluminum alloys in Figure 10.6(b). In aluminum alloys the newly found deformation mechanism is present from the begining of the loading process as precursor to softening (Stage 1). The "waffle"-like strain band pattern shows intermittent activity (Stage 2, 3), i.e., bands switch on and off even at late stages of deformation but following the initial pattern (i.e., they are stable in space). At later stages of deformation (Stage 3, 4) damage starts to cumulate in the form of shear-type damage and more classical high stress triaxiality induced damage leading to the final failure (Stage 5). These deformation mechanisms cannot be captured with classical J2 flow theory [14] (as it was the case for cast iron) or Gurson-Tvergaard-Needleman-type model with Lode parameter-based shear void nucleation [13]. Portevin -Le Chatelier (PLC) effect [17,18], heterogeneous microstructures (randomly distributed soft grains) [16,15] or material processing marks (ex-super grains) [23] could be possible origins of such early heterogeneous field developments. The complete physical explanations for the origin of slant strained bands, their structure (stable in space) and behavior (intermittent activity) still remain to be clarified. However, the initial results on nanolaminography scans reveal some signs of similar effects as in cast iron where intermetallic particles act as flow inceptors. These results should be studied in more detail by analyzing the corresponding intermediate resolution Understanding, observation and quantification of ductile failure mechanisms via 3D imaging scans of the same regions, i.e., observing material heterogeneities as part of the multiscale analysis.

One of the principal aspects for the future work on the ductile damage mechanisms in cast iron (or other heterogeneous materials) is putting the present observations in a statistical frame. It would consist in developing more automated procedures for tracking larger numbers of voids with different shapes and sizes, detecting nodule clusters and association of the microstructure morphology changes with local kinematic values measured via DVC.

Additionally, studies of different cast iron sample geometries can be added to the existing analysis. Especially interesting would be testing CT samples in order to compare them with those outlined for the aluminum alloys. Testing shear samples made of cast iron is another interesting perspective in the context of observing shear mechanisms inside void containing material.

On the side of the experimental/numerical framework, to the author's best knowledge, this is the first time 3D calculations are performed on a real microstructure and by using measured boundary conditions up to large plastic strains. While microstructures are often idealized in the literature, these promising results open new possibilities for numerical modeling of heterogeneous 3D microstructures having a complex morphology and undergoing large deformations and complex topological events such as those observed herein. Its main advantage with respect to the existing literature is that it does not require any macroscopic measurements or simulations. This approach will enable for a better understanding and modeling of damage mechanisms at microscale levels. The presented numerical framework can easily be generalized for other types of materials that possess heterogeneous microstructures.

The fact that the gray level residuals remain reasonably low for the FE simulations of void growth in the studied cast iron serves as their partial validation. However, void coalescence is not properly captured. This last result calls for more advanced constitutive models for the matrix, the nodules and their interface with the matrix. During coalescence, void nucleation at finer scales and very large strains may play a role. The present framework will allow, in future analyses, to test this hypothesis using more sophisticated plasticity/damage models. Similarly, the hypothesis of not considering nodules as voids can also be probed and debonding can be addressed.

From an experimental point of view, performing high resolution laminography experiments where full sample sections would be scanned is of particular interest. Given that the full sample section needs to be incorporated in the scan, there exists a trade off between high resolution scan, limited scanned region and employement of large sheet samples with significant plastic zones as used in the present work. Hopefully, by smart sample design the compromise solution may be found. In the context of I-DVC this would enable for a more detailed representation of the microstructure topology (which was the case for tomography scans) inside FE meshing procedures but at the same time ability of using load data from the experiment as fully representative for equilibrium gap methods.

It is worth mentioning that since the DVC-FE method requires in-situ 3D X-ray imaging, it is hardly applicable to thick specimens and industrial applications. Future work Understanding, observation and quantification of ductile failure mechanisms via 3D imaging will hence lean toward the use of this DVC-FE method for small specimens in order to study more relevant micromechanical models and identify material parameters for these models. These enhanced micromechanical models could then be the basis for the investigation of more relevant constitutive models to be used at the macroscale in, say, wFE and sFE methods. Such approach is likely to be applicable to other materials where micromechanical modeling is gaining an increased interest, such as metallic foams and composite materials for instance.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Introduction

The understanding and prediction of strain and damage interactions in ductile alloys remain a challenging topic due to the development of novel alloys and joining techniques, e.g., for transport applications [1,2]. This understanding is important for micromechanics-based development of new materials and the assessment of structural integrity.

The prediction of failure at various stress states still remains a challenge [3]. In particular, strain localization phenomena are important issues as they may reduce the toughness of structures [4]. Their prediction is difficult as the roles of plasticity and damage softening on localization need to be clarified [5]. Such strain localization phenomena in metals have been studied theoretically by Needleman and Tvergaard [6] who found that they are favored by material behaviors that form a sharp vertex on the yield surface, by material damage and by heating due to plastic deformation leading to thermal softening at high strain rates. The effect of softening in bands due to strain rate sensitivity has also been investigated [7]. Huang and Hutchinson assessed numerically the effect of voids and their distribution on strain localization in a band. Fleck et al. [8] analyzed theoretically the effect of nucleation and growth of voids in a shear band and showed that damage softening favors localization.

Another poorly understood phenomenon linked to localization is slant fracture during ductile tearing [9]. Slant fracture is hard to reproduce numerically [10]. This could be linked to numerical problems due to mesh dependence of local models [11]. Regularized models may overcome this issue but the regularization length then needs to be identified in an appropriate way. Slant fracture has successfully been simulated using shear void nucleation based on the Lode parameter [12] that actually resembles the shear modification of the Gurson model [13]. In Ref. [14] it has been shown via a computational cell simulation that the dissipated energy reaches a minimum when the crack is slanted. Experimentally, slant fracture is widely observed in thin walled structures [15] and can also lead to flip-flopping phenomena [16,17] that are neither understood nor reproduced numerically.

Shear banding has also been attributed to Portevin-Le-Chatelier effect that leads to equally spaced rough bands on the surface of a Kahn tear test specimen made of Al-Li alloy [18]. In Ref. [19] the Coulomb fracture model and the Portevin-Le-Chatelier (PLC) model or dynamic strain aging (DSA) are formulated at the slip system scale and slant fracture could be reproduced.

In ductile damage modeling, where nucleation, growth and coalescence of voids is generally taken into account, efforts have been made to account for void shape effects. In particular, the changes of ellipsoidal voids have been accounted for in homogenization models [20]. It has been shown numerically that void reorientation can increase the susceptibility of the material to shear localization for triaxialities ranging between approximately 0.3 and 0.8 [21]. Void coalescence in combined tension and shear fields has been studied numerically in Ref. [22]. However, these models need to be validated via in situ experimental observations.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging With the recent developments in 3D imaging techniques such as synchrotron tomography it has become possible to follow and quantify damage processes in situ at micrometer resolutions [23]. Even nondestructive observations at resolutions of tens of nanometers are now possible [24,25]. In situ tomography is limited to axisymmetric (or stick-like) specimens with diameters of the order of 1 mm, which, in practice, may be relaxed by a factor of 3 to 5 if local tomography can be used [26]. In contrast, regions of interest in flat specimens can be imaged via synchrotron laminography [27,28]. This technique has been applied to assess the underlying physical mechanisms of ductile crack initiation and propagation [29,30].

To evaluate 3D displacement fields in the bulk of imaged samples, two techniques are used, namely, feature tracking and volume correlation. The tracking technique consists of following the motion of particles or other features. It is performed either manually [31] or automatically [32,33]. Digital volume correlation (DVC) provides either a cloud of 3D displacements for local-based registrations [34,35,36] or spatially continuous and dense displacement fields for global registrations [37,38,39]. For both techniques, additional markers have sometimes to be added to enable the tracking or registration procedure to be performed [40,35,41]. However, these additional markers may alter the deformation and degradation mechanisms. Even though the volume fraction of secondary features is very low in the alloy investigated in this study (i.e., less than 1 % volume fraction) it was decided not to add any additional marker since they may become damage nucleation sites. In a feasibility analysis, it has been proven that global DVC could be used to measure 3D displacement fields even under such extreme conditions [42].

For an Al-Cu-Li alloy with low work hardening it has been shown with these techniques that slant strained bands precede the onset of damage and the slant crack [43]. This region of the sample has been shown to be in a plane strain state with respect to the crack propagation direction [44]. It has also been shown that the ratio between the strain in the band and outside of the band was of the order of 2 [43]. In other words the material outside the bands also deforms during every load step, except for the very last ones, but to a lesser extend than the material within the band. For the last load step deformation only occurs within the band. If the term "localization" refers to a state where plastic yielding takes place within the band and the outer region is plastically inactive, localization has only been measured at the very latest step. However, strain concentrations in a slant strained band were observed very early on and the scene for localization and fracture was already set. It is of interest for the understanding of strain and damage interactions if this way to deform is a general mechanism for these types of materials or only found in some particular cases.

In this paper the interactions between strain, porosity and the final crack are assessed via DVC for laminography data already reported in Ref. [30], namely, for a ductile aluminum alloy containing pre-existing voids (AA2139 T3). After introducing the material and the experimental approach, the strain fields and their corresponding microstructure are investigated in a region 1-mm ahead of the notch root. Incremental and cumulated strain fields will be given. The response of one void in such a strain field is then assessed. The findings are compared with a Gurson-Tvergaard-Needleman-type model simulation Understanding, observation and quantification of ductile failure mechanisms via 3D imaging using shear-based void nucleation in terms of strain fields. The strain distribution will be further discussed using 2D plane strain simulations that account for soft bands or randomly distributed soft zones.

Experimental configuration 2.1 Studied Material

The investigated material is an Al-Cu-Mg alloy (AA2139) in a naturally aged (i.e., T3) condition. The material processing directions are the rolling direction (L), the transverse direction (T) and the short-transverse direction (S) in the through thickness direction.

Laminography

Laminography imaging was performed on the instrument installed by the Karlsruhe Institute of Technology at the imaging beamline ID19 [47] of the European Synchrotron Radiation Facility (ESRF, Grenoble, France) in the conditions described by Morgeneyer et al. [30,42]. The main advantage of laminography compared to tomography is that large flat specimens can be used. Synchrotron laminography benefits from various contrast modes including propagation-based phase contrast used herein in addition to the conventional absorption employed in laboratory X-ray imaging. For large source-specimen distances, large sample environments (e.g., mechanical testing devices) can be employed for high-resolution imaging without risk of collision during the scanning motion. Similar to tomography, a series of digital radiographs is acquired while rotating the sample about the laminographic axis [47]. The latter is inclined with respect to the transmitted beam at an angle 0 • < θ < 90 • , see Figure A.3. With the flat specimen oriented approximately perpendicular to the rotation axis, the average beam transmission can be optimized during the entire laminographic scan. Using filtered back-projection algorithms [48] different planes or complete 3D images with down to sub-micrometer resolution zoomed onto the large specimen can be reconstructed by a single scanning motion.

An inclination angle of the rotation axis of about 25 • with respect to the beam normal was chosen as well as a monochromatic beam of 25 keV X-ray energy. Volumes were reconstructed from 1,500 angularly equidistant radiographs using a microscope-based in-Understanding, observation and quantification of ductile failure mechanisms via 3D imaging direct detector system with ×20 magnification [49] and an ESRF Frelon camera. The exposure time of each projection was 250 ms. The scanned region is ≈ 1 mm 3 in volume with a voxel size of 0.7 µm. This is where the flat-to-slant transition is typically observed [50]. The minimum specimen to detector distance is 70 mm leading to relatively strong edge enhancement due to phase contrast. The resulting strongly contrasted edges facilitate the detection of voids and damage. For easier data handling, all 3D images consisting of 32-bit floating point values were converted into 8-bit gray level 3D images using the same linear dependence. As the same affine transformation is performed after reconstruction, the gray level distributions between the images are comparable. The final reconstructed volumes have a size of 2040 × 2040 × 2040 voxels. However, only the voxels where all projections contribute to the reconstruction exhibit all the directional information available, show similar noise statistics and therefore have been considered herein.

In situ mechanical experiment

To investigate strain and damage interactions in situ, a loading device was adapted to the environment given by the laminography scanning geometry (Figure A.4(a)) [30]. In particular, the device was designed to present a low profile perpendicular to the sample plane, thereby avoiding obstruction of the beam during rotation. The screws (see Figure A.4(b)) allow a stepwise crack mouth opening displacement (CMOD) to be prescribed [30]. Before each "load step," the region ahead of the notch was scanned by laminography. A load step consists of turning the 2 screws of the loading device leading to incremental openings of the notch, i.e., a stepwise monotonic and quasi-static loading was applied.

The specimen (with dimensions 60 × 70 × 1 mm) contains a round notch with a radius of R = 0.17 mm and a 24-mm ligament (see Figure A.4(b)). The loading configuration is T-L. Being close to standard tearing test geometries, this specimen provides relevant engineering conditions and allows for the development of plastic zones sizes as found in standard tests. The region ahead of the notch is loaded in tension and bending. A simple anti-buckling device (i.e., flat plates) is used to limit out-of-plane displacements.

All the mechanical parameters are characterized as functions of the nominal CMOD obtained by the opening of the loading device in this study. The definitions of CMOD, Understanding, observation and quantification of ductile failure mechanisms via 3D imaging that case, the previous option cannot be used and extrapolated values should be assessed. The extrapolated value corresponds to the displacement of the closest node for which a measured estimate exists.

In the present DVC analyses, a volume of interest has been extracted from the scanned volume. Within this volume an initial ROI of 864 × 608 × 992 voxels (i.e., ≈ 600 µm ×425 µm ×700 µm) has been considered for correlation purposes and has been discretized with 32-voxel C8 elements. With such a discretization, ≈ 54, 000 kinematic degrees of freedom are measured. The choice of the element size in any DVC calculation results from a compromise between measurement uncertainty and spatial resolution [56,42] (here twice the element size, i.e., ≈ 45 µm). To evaluate the resolution of the measurement technique, two scans of the undeformed configuration were reconstructed when a small motion was prescribed between the two acquisitions. For 32-voxel (≈ 22 µm) elements, the standard displacement resolution was found to be of the order of 0.2 voxel, and the corresponding standard effective strain resolution equal to 0.5 %. These values are the largest levels below which the measurement cannot be distinguished from noise. From the loading state CMOD = 2.313 mm onward, the ROI was shifted away from the notch by 280 µm to avoid the damaged region and reach correlation convergence. In addition, a bigger ROI has been used (i.e., 1248 × 512 × 992 voxels) to be able to follow two voids in the microstructure in spite of large strain levels.

Based upon these results, the convergence criterion of the correlation calculations, which corresponds to the mean displacement increment between two consecutive iterations, is chosen to be equal to 10 -4 voxel. It was checked that the mean correlation residual has reached a steady state for this criterion. The results are also checked by computing the correlation residuals voxel-wise, and their average value (i.e., Φ c or equivalently ρ c ). The mean and local residual levels are similar to those found by considering two subsequent acquisitions in which no deformation of the sample has occurred (i.e., for the resolution analysis). Consequently, the reported results are deemed trustworthy.

The correlation analyses reported herein consist of estimating the total displacement field for each analyzed step with respect to the reference configuration up to loading step for which the CMOD is equal to 1.625 mm. From these Lagrangian measurements, the incremental displacement fields are directly given by the total displacement difference corresponding to two consecutive steps. The Green-Lagrange strain fields are estimated either with the total displacement fields, or with the incremental displacement fields. In both cases, the mean values per C8 element are reported. For the displacement field between the initial state and any load step for which the CMOD is greater than 1.5 mm the direct correlation could not yield a converged solution as strains are very high and damage nucleation started to set in. For this reason incremental correlations taking load state CMOD = 1.5 mm as a reference have been carried out. This procedure was also performed for the subsequent load steps.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Figure A.6(a) shows the strain field for a load increment starting from the unloaded state to load step CMOD = 1.5 mm. This is the largest direct correlation step that can be performed. Several intermediate steps were performed during the in situ experiment that will be shown in the sequel. When CMOD = 1.5 mm, there is no crack that has initiated from the initial notch. The von Mises effective strain field shows a heterogeneous distribution with an average level ≈ 10 %. Several parallel slant bands are observed at this relatively early stage of loading. Three bands oriented from the lower left part of the section to the upper right part seem to be prevalent. In this 2D section no clear link between the underlying voids and the strained bands is established. The magnitude of the strain in the bands is about twice as high (i.e., 13 %) as in the material between the bands. This ratio has already been found for an aluminum alloy with less work hardening and initial porosity [43]. However, the latter material mainly showed one localized band and not several parallel bands as seen here. In addition, the slant strained band was less diffuse than in the present case. On this 2D section two particular voids of the underlying microstructure are encircled. They are also highlighted for the following load increments. This representation allows the spatial location of the strained bands to be assessed with respect to their underlying microstructure and if these bands "stick" to the microstructure.

In The comparison of the strained bands with respect to the underlying microstructure shows that the band is motionless in the material coordinate system, namely, the same voids and particles are found within the band. The void and particle evolution seems very limited (i.e., void nucleation, growth or shape change are not striking at this scale of observation). It should however be noted that at this loading stage there is already a crack propagating from the notch root (i.e., outside the ROI investigated herein), which may influence the strain fields.

The following load increment, from CMOD = 2.0 mm to CMOD = 2.313 mm, generates an activity of the two upper strained bands ( For the next load increment the extracted volume has been increased symmetrically in the loading direction in order to be able to still follow the two voids encircled in the previous sections. A small incremental step from CMOD = 2.313 mm to CMOD = 2.375 mm is shown. The small remote load increment leads to high local strains as the crack approaches this region. From the lower left part of this region to the upper right part only one strained band is active (

(e)).

There is still very little damage to be seen in the microstructure at this location and, if any, it is rather diffuse and also located in regions that do not belong to any strained band.

In Figure A.6(f) the same strained band from the lower left to the upper right corner of the image is mainly active. Only one out of the two bands normal to this one, i.e., the upper band, is now active. The latter is particularly active in the lower part of the investigated region. It is noteworthy that at these late stages of loading there is still not only a single strained band active but a competition between several bands that have undergone tens of percents of strain. The next load step from CMOD = 2.438 mm to CMOD = 2.5 mm is the last one for which a trustworthy correlation could be achieved (Figure A.6(g)), namely, the residuals were still very close to those observed when two undeformed volumes are registered. It can be seen for the underlying microstructure that a notable damage evolution now sets in. In the region where the strained bands cross, voids grow. The upper one of the two encircled voids shows a reorientation that has already been reported [50] for the same experiment. With the present work the corresponding strain fields for the void shape changes are known. .6(h) shows the microstructure of load step CMOD = 2.5 mm without the corresponding strain field. Voids in the region where the two strained bands cross grow further. This highlights that in this area the main assumption for trustworthy DVC, which is the conservation of gray levels is no longer satisfied as damage sets in. However, the measured strain concentration where the two strained bands cross is plausible, as, in addition, void growth is seen here that is surely a result of highly localized strain activity. The lower half of the strained bands is more active than the upper half. In Figure A.6(i) the final crack path is observed (i.e., CMOD = 2.625 mm). A roof top crack is seen, which has not fractured the full section yet. The crack has followed the second slant strained band that is present from the very early loading onward. The two voids that had been encircled in the previous load increments now contribute to the final fracture surface.

In summary, several slant, crossing and parallel strained bands with intermittent activity were found. Failure occurred along one band that was already active since the very beginning of the loading history.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)

is consistent with the fact that the strain field shown in Figure A.6(b) was "added." The maximum effective strain in this slice is now increased to ≈ 0.24. In Figure A.7(c) a cumulated field is shown corresponding to strains between load step CMOD = 0.0 mm and CMOD = 2.313 mm. The effective strained band pattern is still similar to the previous ones but the maximum magnitude has now increased to ≈ 0.33. The bands do not move in space and, as a consequence, do not even out the strain distribution but keep a concentrated pattern made up of several bands. This is surprising as the regions between the strained bands could be expected to be softer than the bands and easier to deform if a uniform material behavior is assumed. Local variability in stress/strain behavior could be at the origin of such behavior. Damage evolution is not striking until this loading state and thus should not be of prior importance to softening.

The strain profile along a line perpendicular to the strained bands is shown in Figure A.8. This graph confirms the spatial stability of the strained bands. The ratio of the effective strain in the main bands over the strain outside the bands lies between 1.5 and 2. For the band that will lead to the final failure this ratio is somewhat lower. This result is comparable to the strain ratio that was reported for AA 2198 T8 [43]. The main difference is that several bands are active in the present case.

Small load increments from the initial state

In this section strain fields for small steps are reported when DVC is performed from the unloaded step (CMOD = 0.0 mm) to the load step before crack initiation (i.e., CMOD = 1.5 mm). The incremental strain fields in this load interval from the unloaded state are also given. The latter investigation is made in order to interpret the incremental strain activity (∆p eq ) as could also be done in an FE simulation that accounts for instabilities.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging shows the strain field obtained for the first load step carried out during the in situ test (i.e., CMOD = 0.5 mm). In this step a maximum effective strain of 0.03 is obtained. However, the three strained bands from the lower left corner of the image to the upper right corner can already be seen. These bands remain visible for a substantially larger load increment (Figure A.6(a)). Damage nucleation or growth is negligible at this step of loading. A similar result was also found for AA2198 T8 [43]. The strained bands found herein are less localized. In the subsequent load steps the void grows in the loading direction and turns toward the orientation of the strained band. During the next load steps, void growth mainly occurs along the strained band direction. This is consistent with a local mode I and III loading state [9] that lets the void grow in a mixture of tensile and simple shear fields [21].

This void reorientation has been shown numerically to facilitate localization phenomena. In the final stages smaller damage features become visible that are consistent with void nucleation and growth on a second population of particles at high levels of strain. However, very narrow crack regions are also seen in heterogeneous from early loading stages onward no substantial void growth occurs until elevated levels of strain (more than 0.30 in the present case). This is also in agreement with results obtained by unit cell calculations [46]. They predict for a stress triaxiality of 1.25 a void volume (vv) of 3 times the initial volume at about 0.15-0.35 strain. However, void coalescence is predicted in unit cell calculations at higher void volume fraction level than in the experiment (about 12 times the initial vv) and also at higher strain (0.3-0.5). In the experimental data void growth acceleration occurs at 0.3 strain and for a vv that is 3 times higher than the initial one, i.e., substantially earlier than in the unit cell calculations. The complex strain field encountered herein, which may lead to mixed tensile and shear states, may be at the origin of this behavior [21]. It is worth noting that only one void has been studied. An investigation considering more voids would be useful to gain a better statistical representation of these processes.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging The strain profiles normal to the meshed soft band are reported in Figure A.17. The equivalent plastic strain increases in both regions (i.e., within and outside the band). The strain ratio is of the order of 2 for a perturbation x = 10 %. shows the change of the strain ratio for different perturbation levels. The stronger the perturbation the higher the initial strain ratio and the earlier the final increase of the strain ratio. In the experimental case the strain ratio is still of the order of 2 for a strain outside the band of ≈ 0.22. A soft perturbation of the order of 10 % reproduces rather well these experimental findings.

Randomly distributed soft zones

There is no evident reason for soft zones in the material to be distributed in slant bands. Consequently soft zones with a size of 50 µm×50 µm are randomly distributed in an additional simulation. These dimensions are close to the mean grain size of the studied aluminum alloy. The soft zones were randomly distributed in the central part of the meshed region. The soft phase fraction has been chosen to be the same as in the previously reported calculation with a soft slant band. The soft zone distribution is depicted in The strain partitioning is further investigated via plane strain simulations where a slant soft zone is modeled. A plane strain condition had been identified for the crack propagation direction in the region where the crack is slanted [44]. For a perturbation of 10%, strain partitioning between the band and the surrounding material, and a strain level at localization are in agreement with the experimental findings. A softening effect is thus identified rather than shear localization where all strains would localize within the band.

The question that arises is what could be at the origin of these bands. The heterogeneous strain fields could be linked to heterogeneous microstructures and resulting plasticity [58]. Local material heterogeneity caused by crystallographic texture, which is referred to as roping, could also be a possibility [59]. In that case these local textures are the result of ex-super grains that existed at some point during material processing. However, Understanding, observation and quantification of ductile failure mechanisms via 3D imaging the characteristic lengths identified in Ref. [59] are substantially larger than what is found herein, namely, more than 1 mm [59] compared to about 200 µm herein. In addition, the bands that are found are slanted and evenly spaced. There is no obvious reason to find such slant grain arrangements in soft zones. Additionally the characteristic spacing, ordered nature and intermittent activity of the bands makes a purely material heterogeneity based origin less likely.

Using a mesh with randomly distributed soft zones to model soft grain distribution, slant bands are obtained at early loading stages under plane strain conditions. These soft zones may be a possible origin of the phenomena reported herein. However, the strain ratio within and outside the bands is not similar to the experimental findings. Using different surface fractions or softness, results might be obtained that could be closer to the experimental findings. However, the intermittent activity of the bands is not captured through these effects.

In the present models no ingredients to reproduce the intermittent strained band activity are used. Origins of the intermittent nature of the strained bands could be DSA (i.e., Portevin-Le-Chatelier effect [60,19]). This observation is also supported by the fact that even at very late stages of deformation strained bands may become inactive and damage softening does not seem to influence these processes on the first order. It is, however, surprising that the experimentally measured strained bands are stable in space with respect to microstructural features. This is generally not found for moving PLC bands on smooth specimens as in these cases the strained bands move in time and even out the strain heterogeneities in the different regions. Here, however, the notch naturally concentrates the strain fields and may prevent the bands from moving. Similar strained bands have been found on Kahn tear test sample surfaces made of Al-Li alloy [18] after tearing. Deformation patterns in slant crossing bands ahead of notches have also been suggested in early works on these phenomena [61].

Further, it is experimentally shown that inside the strained bands voids change shape and rotate to align along the direction of the band. The question arises whether the shape evolution of pre-existing voids can lead to strain localization and damage softening. It is noteworthy that this evolution takes place only at the latest stages of deformation. It is thus more likely that this void growth is the result of previous, and thus governing, deformation and concentration in strained bands. It is certainly of interest to investigate the effect of void shape changes in such strained bands on the fracture resistance of the material. It is likely that there will be an additional localization effect due to void rotations [21].

The problem in modeling slant fracture remains the mesh dependence of the simulations [11]. The slant fracture found herein was reproduced by a shear void nucleation term based on the Lode parameter [12]. However, the mesh dependence is still an issue. In addition, several slant bands could not be reproduced. To create bands with intermittent activity a PLC-type model may be used [19] but mesh dependence and computation times are also problematic for such types of models.

Experiments on other types of Al alloys and specimen geometries should also be performed to confirm the presence of slant bands and PLC effects. Nonlocal models are needed to get more reliable predictions of localized phenomena but the regularization Understanding, observation and quantification of ductile failure mechanisms via 3D imaging length is still to be physically understood and identified. The present finding could also add another length to the commonly used particle or void spacing, which is the strained band spacing.

Conclusions

Damage and strain interactions have been assessed in situ and within the material bulk during ductile tearing. The region of interest is located 1080 µm ahead of a notched 1mm thick plate made of ductile aluminum alloy containing initial porosities (≈ 0.33 vol%) and particles (≈ 0.45 vol%). This analysis is made possible by the combination of in situ synchrotron laminography and digital volume correlation using the material microstructure contrast. The overlay of strain and microstructure maps has yielded the following results:

• Several parallel, crossing and slant strained bands are active from the very early stages of loading (i.e., max. 0.03 effective von Mises strain).

• The analysis of incremental strain fields shows that the activity of the bands is intermittent over the whole loading history.

• The spatial location of these bands compared to the microstructure is stable, i.e., the strain activity is not evening out the strain fields.

• Even very concentrated (strained) bands with high von Mises effective strain levels (i.e., 0.10) may become inactive and other bands become active at very late stages of loading.

• The typical strained band spacing is 150 -250 µm.

• The ratio of the effective strain within and outside the bands is about 1.5 to 2.

• In the local area of observation substantial void growth could not be detected at micrometer resolution before the occurrence of strained bands.

• The fact that strained bands may switch off at the end of the test indicates that the mechanisms at play for strain concentration may differ from damage softening.

• The observed slant strained bands at early loading stages cannot be reproduced using a Gurson-Tvergaard-Needleman-type model with Lode parameter-based shear void nucleation. The flat to slant crack transition is however successfully reproduced.

• 2D plane strain simulations accounting for a soft zone yield strain ratios within the band and the surrounding material similar to the measurements for a 10-% weakening. A softening effect is thus identified rather than shear localization where all

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging strains would localize within the band. Using randomly distributed soft zones, slant strained bands can be obtained. The ratio of the strain within and outside the bands is, however, substantially lower than in the experimental findings.

Last, investigations on other Al-alloys and specimen geometries would be useful to confirm the present results, which have also be found on a low work hardening alloy (i.e., AA2198 T8 [43]). Titre : Compréhension, observation et quantification des mécanismes de rupture ductile par imagerie 3D

Mots clefs : endommagement ductile, laminographie, corrélation d'images volumiques, alliages d'aluminium, fonte, simulations micromécaniques Résumé : Au cours des dernières décennies, des efforts importants ont été menés dans la modélisation des processus de rupture ductile entraînant des progrès substantiels. Cependant, la compréhension complète des mécanismes de rupture ductile dans des états de contraintes spécifiques demeure une question ouverte. Ceci est dû au manque de bases des données expérimentales et à la non validation des modèles pour ces conditions de chargement. Dans ce travail, les acquisitions de données sont principalement obtenues en utilisant la laminographie, ce qui rend possible l'imagerie de régions d'intérêt d'échantillons plats. L'utilisation d'éprouvettes larges (et minces) permet de générer différents états de contraintes et des conditions aux limites pertinentes pour l'ingénierie, qui ne pouvaient pas être évaluées jusqu'à présent en trois dimensions et en essais in-situ à des échelles micrométriques. La corrélation d'images volumiques (DVC) est utilisée pour mesurer les champs de déplacement à l'intérieur des échantillons en acquérant des images de laminographie 3D. Deux classes de matériaux représentatives de deux modes génériques de rup-ture ductile ont été examinées, à savoir les alliages d'aluminium (rupture par instabilité) et la fonte à graphite sphéroïdal (rupture par croissance de vide et coalescence). L'observation de la microstructure et les interactions déformationsendommagement pour différentes géométries d'échantillons et pour différents niveaux de triaxialité des contraintes associés ont été étudiées pour des alliages d'aluminium à une résolution micrométrique. De plus, un cadre combiné numériqueexpérimental (DVC-FE) est introduit pour valider les simulations numériques à l'échelle microscopique pour la fonte à graphite sphéroïdal. Les simulations par éléments finis (FE), qui représentent la microstructure des matériaux étudiés, sont conduites avec des conditions aux limites de Dirichlet extraites des mesures DVC. Enfin, le cadre DVC-FE a été amélioré et utilisé comme une procédure d'identification intégrée pour l'étude du comportement élasto-plastique de la matrice ferritique de la fonte, non seulement en termes de champs cinématiques induits par la microstructure aléatoire, mais aussi avec les niveaux de charge globaux.

Title : Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Keywords : ductile damage, laminography, digital volume correlation, aluminum alloys, cast iron, micromechanical simulations Abstract : In the last few decades significant efforts have been made in modeling ductile failure processes resulting in substantial progress. However, the full understanding of ductile failure mechanisms under specific stress states still remains an open question. This is partly due to missing experimental data and validation of models for such loading conditions. In this work, data acquisitions are mainly obtained by using laminography, which makes the imaging of regions of interest in flat samples possible. The use of large (and thin) specimens allows various stress states and engineering-relevant boundary conditions to be generated, which could not be assessed in three dimensions and in-situ at micrometer scales before. Digital Volume Correlation (DVC) is used for measuring displacement fields in the bulk of samples by registering 3D laminography images. Two material classes that are representative of two generic modes of ductile failure have been examined, namely, Al-alloys (failure by instability) and cast iron (failure by void growth and coalescence). The observation of microstructure and strain-damage interactions at micrometer resolution for various specimen geometries and associated levels of stress triaxiality are studied for Al-alloys. Additionally, a combined computational-experimental (DVC-FE) framework is introduced to validate numerical simulations at the microscopic scale for nodular graphite cast iron. Finite Element (FE) simulations, which account for the studied material microstructure, are driven by Dirichlet boundary conditions extracted from DVC measurements. Last, the DVC-FE framework is upgraded to an integrated identification procedure to probe elasto-plastic constitutive law of the cast iron ferritic matrix not only in terms of kinematic fields induced by the random microstructure but also by overall load levels.

  (b) Strain to fracture for plane stress as a function of stress triaxiality. Figures are taken from Ref. [128] . . . . . . . . . . . . . . . . . . . . . . 13 5 Flat to slant transition during mode I opening of CT samples. The laminography/DVC section refers to AA2198 T8 alloy studied in this work. Post mortem SEM figure is taken from Ref. [127] . . . . . . . . . 19 6 Tomography and laminography experimental setup . . . . . . . . . . . . 22 7 Sample geometries used on laminography experiments lined as function of average stress triaxiality in the scanned zone . . . . . . . . . . . . . . 25 8 Structure of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.1 Mid-thickness L -T slice in 3D reconstructed volumes of three different aluminum alloys and one cast iron sample. The picture definition is 2040 × 2040 pixels for aluminum alloys and 1600 × 1600 pixels for cast iron. The physical length of one voxel is 0.7 µm for aluminum alloys and 1.1 µm for cast iron. Especially for (a) and (b), ring artifacts from static features on the projection radiographs are clearly visible. . . . . . . . . . 46 1.2 Gray level histograms of the investigated aluminum alloys and cast iron . 47 1.3 Experimental setup schematically shown for the aluminum CT-like specimens investigated in this work [30] . . . . . . . . . . . . . . . . . . . . 48 1.4 Region of interest considered in the uncertainty analysis of the volume made of (a) AA2139 T3 and (b) AA2139 T8. The squares on the right denote 4 DVC elements with isotropic size = 32 voxels . . . . . . . . . 48 Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures1.5 Region of interest considered in the uncertainty analysis of the volume made of (a) AA2198 T3 and (b) AA2198 T8. The squares on the right denote 4 DVC elements with isotropic size = 32 voxels . . . . . . . . . 1.6 Region of interest considered in the uncertainty analysis of the volume made of AA6061 T6. The squares on the right denote 4 DVC elements with isotropic size = 32 voxels . . . . . . . . . . . . . . . . . . . . . . 1.7 Isometric isolated view of voids (blue) and particles (yellow) (+ edgeenhancement artifacts around voids due to so-called "phase contrast", an interference phenomenon due to Fresnel diffraction) in 3D reconstructed volumes (256 × 256 × 256 voxels) of the studied aluminum alloys . . . . 1.8 Procedure followed when isolating voids and particles for AA2139 and AA6061 alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Schematic view of the sample with the scanned region close to the central hole. Section of the reconstructed volume with the position of the region of interest (ROI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 The sample geometry with the scanned region between the two holes. Section of the reconstructed volume with ROI position . . . . . . . . . . 1.11 Isometric isolated view of nodules and secondary void population in 3D reconstructed volume (256 × 256 × 256 voxels) of the studied cast iron . 1.12 Standard displacement (a) and strain (b) resolutions as functions of the element size for bis cases . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Standard displacement (a) and strain (b) resolution as functions of the element size for rbm cases . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Standard displacement (a) and strain (b) resolutions as functions of the element size for bis and rbm cases . . . . . . . . . . . . . . . . . . . . . 1.15 Standard displacement (a) and strain (b) resolutions as functions of the element size for recad cases. Two different recad parameters resulted in 2 different sets of scans, namely: reference scan A and rbm scan A; reference scan B and rbm scan B. Three combinations are analyzed to measure displacement fields: reference scan A vs. rbm scan A, reference scan B vs. rbm scan B and reference scan A vs. rbm scan B (crossed) . . 1.16 Standard displacement (a) and strain (b) resolutions as functions of the element size for different experimental setup . . . . . . . . . . . . . . . 1.17 Standard displacement (a) and strain (b) resolutions as functions of the element size for different cameras used in the experiment . . . . . . . . . 1.18 L-T slice in 3D reconstructed volumes of AA6061 (a) and AA6061 modified (b). The squares on the top denote 4 DVC elements with isotropic size = 32 voxels; (c) gray level histograms of the two grades. . . . . . . 1.19 Standard displacement (a) and strain (b) resolutions as functions of the element size for the two grades of AA6061 alloy . . . . . . . . . . . . . 1.20 Regions of interest considered in the uncertainty analysis of the volume made of AA2198 T3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures 1.21 Standard displacement (a) and strain (b) uncertainties for DVC analyses associated with the ROIs shown in Figure 1.20(a) . . . . . . . . . . . . . 1.22 Standard displacement (a) and strain (b) uncertainties for DVC analyses associated with the ROIs shown in Figure 1.20(b) . . . . . . . . . . . . . 1.23 Standard displacement resolution (a) and measured displacement (b) by components for ROI 1, 2 and 3 from 2198 T3 sample . . . . . . . . . . . 1.24 Scanned volume section LS considered in the uncertainty analysis of the volume made of AA2198 T3 before (a) and after (b) rigid body motion. The vertical black line denotes the axis of the reconstructed volume. The LS projections are obtained through histogram equalization . . . . . . . 1.25 Displacement fields u L for ROI 3 (a); ROI 2 (b) and ROI 1 (c) when measured with 32-voxel elements for the rbm case (AA2198 T3) . . . . . 1.26 Von Mises strain fields for ROI 3 (a); ROI 2 (b) and ROI 1 (c) when measured with 32-voxel elements for the rbm case (AA2198 T3) . . . . . 1.27 Displacement fields u L for ROI 3 (a); ROI 2 (b) and ROI 1 (c) when measured with 32-voxel elements for the bis case (AA2198 T3) . . . . . 1.28 Standard strain uncertainties for DVC analyses associated with ROI 1 and 2 positioned as in Figure 1.20(b) for AA2139 T8 . . . . . . . . . . . . . 1.29 Scanned volume section LS considered in the uncertainty analysis of the volume made of AA 2139 T8 before (a) and after (b) rigid body motion. The LS projections are obtained through histogram equalization . . . . . 1.30 Effective strain fields for AA2139 T3 and AA6061 T6 when measured with 32-voxel elements for the rbm case. The ROIs are positioned as shown in Figure 1.4(a) and 1.6 respectively. . . . . . . . . . . . . . . . . 1.31 Standard displacement (a) and strain (b) resolutions as functions of the element size for cast iron (central hole geometry from Figure 1.9) . . . . 1.32 Standard displacement (a) and strain (b) resolution as functions of the element size expressed in supervoxels for the specimen geometry illustrated in Figure 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33 Standard displacement (a) and strain (b) resolutions as functions of the element size for all analyzed materials. "CI tomo. rbm" denotes the corresponding tomography resolutions [23] of the same type of material but different physical voxel size (3.5 µm . . . . . . . . . . . . . . . . . . . . 1.34 Normalized standard displacement (a) and strain (b) resolutions as functions of the element size. The solid line shows an interpolation by -3/2 power law (a) and -5/2 power law (b) . . . . . . . . . . . . . . . . . . . 1.35 The largest 100 eigenvalues of the inverse DVC matrix [M] for the studied microstructures and element size =32 voxels . . . . . . . . . . . . . . . 1.36 Reconstruction artifacts in laminography. The yellow line indicates the boundary of the region where the sampling is incomplete . . . . . . . . . 2.12 Incremental effective strain fields printed over portion of reconstructed volumes showing the final crack path for loading steps (0)-(1) (a),(1)-(2) (b) and (2)-(3) (c). L-sections are located at a distance d ∼ 250 µm, d ∼ 390 µm and d ∼ 530 µm from the notch root . . . . . . . . . . . . . 2.13 Movies showing L-sections of incremental effective strain field for the three analyzed loading steps. The results of both ROI 1 and ROI 2 are shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 (a) Shematic view of the laminographic setup with the loading device, CT-like sample and the scanned region (red zone). (b) 2D sections of laminography data indicating the positions of the Regions of the Interest (ROIs) examined in this work . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Standard von Mises strain resolution as a function of the element size used in the correlation procedure . . . . . . . . . . . . . . . . . . . . . . 3.3 Notch Opening Displacement (NOD) for the step-wise loading procedure followed in this work; mid-thickness section showing the final crack propagated through the specimen ligament area for loading step (0)-(11) . . . 3.4 Comparison of DVC and "projection DIC" results. (a) 2D sections of reconstructed laminography data for load steps (0) and (3). (b) Projected 2D sections of reconstructed laminography data (the maximum values were taken over a thickness of 50 µm). (c) Comparison of normal strain fields in the loading direction obtained by DVC (left) and "projection DIC" (right) on projected 3D data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Digital volume correlation results. Section normal to the crack propagation direction (i.e., section B-B, Figure 3.1(b)) showing von Mises effective strain fields for direct (a) and incremental (b) calculations . . . . . . 3.6 Von Mises effective strains along the line ξ depicted in Figure 3.5(a) for direct digital volume correlations . . . . . . . . . . . . . . . . . . . . . 3.7 B-B section through 3D laminography data showing the microstructure for different loading steps . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 (a) Void fraction development over the loading history for the three inspected ROIs (see Figure 3.1(b)). The shadow region denotes the final crack propagation through the ligament as shown in Figure 3.3. (b) Mean values and the standard deviation of von Mises effective strain fields inside ROI 1 for direct calculations . . . . . . . . . . . . . . . . . . . . . 3.9 Final crack position in conjunction with the initial von Mises effective strain map plotted along the section normal to the crack propagation direction (L) for loading step (0)-(3). The height of the final roof-top crack is shown with the color map. Left: view at 1 mm from the notch root; Right: view in the crack propagation direction . . . . . . . . . . . . . . 3.10 Sections normal to the crack propagation direction showing von Mises effective strain fields for loading step (0)-(3) for different distances from the notch root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures 3.11 Through-thickness sections of the merged ROIs 1, 2 and 3, where d denotes the distance from the ROI mid-thickness plane. Positions of the L-L and R-R sections are depicted on the section normal to the crack propagation direction (right column). (a) Von Mises effective strains ε eq , (b) normal strains in the loading direction ε T T , (c) normal strains in the crack propagation direction ε LL , (d) normal strains in the through thickness direction ε SS . Normal strain components are defined as nominal strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Incremental normal strain maps (in the loading direction) in ROI 2 (i.e., 460 µm from the notch root) for different load steps and overlay of the last possible incremental strain field with the final crack path . . . . . . . 3.13 Thresholded gray level views of ROIs 1, 2 and 3 for the undeformed configuration and last three loading steps . . . . . . . . . . . . . . . . . . . 3.14 Mid-thickness sample section of the reconstructed volume for different loading steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Porosity changes in the immediate vicinity of the notch root (Figure 3.14) during the loading history . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Electron backscattered diffraction taken at the one side of the broken CTlike specimen on the section normal to the crack propagation direction. (a) View before electro-polishing. (b) Grain texture revealing the transgranular nature of the inspected damage feature. (c) Inverse pole figure showing the grain orientations where the damaged grain with its 001 orientation is sandwiched between 111 oriented grains . . . . . . . . . . . . 3.17 Scanning electron micrographs taken at one side of the broken CT-like specimen. (a) Maginification level ×70, triangular zone at the notch root followed by slant surfaces. (b) Maginification level ×700, magnified subsets a-f depicted in (a). (c) Maginification level ×2000, zoomed details from subset b and from the zone between subsets e and d. . . . . . . . . 3.18 (a) Positions of the subsets depicted on the thresholded von Mises effective strain field. (b) Void growth over the loading history inside two subsets of ROI 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 Standard deviation of the residual fields for incremental and direct calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Gray level residual distribution over section B-B (Figure 3.5(b)) and the section normal to it (S-S) for the last converged direct (left) and incremental (right) calculations . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 (a) 2D images of the CT like specimen surface during loading. (b) Gray level residual field from the last direct DIC calculation (between undeformed State (0) and State (11)). (c) Normal strains in the loading direction ε T T for direct DIC calculations . . . . . . . . . . . . . . . . . . . . Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures 4.1 Laminography experiment: (a) loading device with the specimen; (b) reconstructed volume with positions of the projected sections used in bulk-DIC procedure; (c) single and projected laminography sections for different modes of projection to enhance the contrast in the pictures . . . . . . 4.2 Laminography section normal to crack propagation direction (L) at distance of 865 µm from the initial notch root for different loading steps. The green zoomed region corresponds to the location where significant damage coalescence (State (8)) and subsequent final failure occur. The blue and red arrows point to the intermetallic particle breakages and isolated void coalescence events while the yellow ones denote the final coalescence event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Normal strain fields in loading direction (ε T T ) plotted over corresponding projected microstructure sections (mix. projection mode from Figure 4.1(c))158 5.1 (a) Schematic view of the laminography setup; (b) shear sample geometry and position of the scanned zone. In the section of the scanned region the representation of microstructure grains is shown [11] . . . . . . . . . . . 5.2 Standard strain resolutions as functions of the element size . . . . . . . . 5.3 Load levels plotted against machine displacements. Each symbol depicts a scan acquistion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Displacement profiles from initialization and final DVC field for one loading step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Gray level residual maps plotted for the mid-thickness section of the ROI

  (a)) for loading step (3)-(4)) with the corresponding error bars. The area between dashed lines corresponds to the localization band position, i.e., ∆ε eq > 2 %. The values of Lode parameter inside shaded areas are theoretically non-admissible . . Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures 5.13 (a) Lode incremental strain parameter µ ∆ε profiles from incremental DVC calculations. The position of the profile axis is denoted on the Lode map shown in the corner; (b) Part of the Lode parameter µ ∆ε profile in the shear band for loading step (3)-(4)) with the corresponding error bars . . . . . . 5.14 Part of the Lode parameter µ ∆ε profile in the shear band for loading steps

6. 1

 1 Drawings of the two samples with the scanned regions and the regions of interest (ROIs) for DVC analyses . . . . . . . . . . . . . . . . . . . . . 6.2 (a) Laminography setup schematic view; (b) sample with scanned region: 3 void populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Effective strain resolution (a) and strain trace resolution (b) as functions of the element size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Mid-thickness microstrucutre sections of reconstructed laminography data for the 90 • (a) and 45 • (b) configurations during the loading history . 6.5 DVC gray level residual fields: (a) 45 • configuration section normal to the loading direction y; (b) 90 • configuration section normal to the thickness direction z; (c) isometric view of the residual field for the 90 • configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 SEM observations for the 45 • (a) and 90 • (b) configurations . . . . . . . 6.7 SEM observation of void sheeting coalescence for the 90 • configuration . 6.8 Von Mises strain maps plotted over the corresponding microstructure sections at the sample mid-thickness: (a) 45 • ; (b) 90 • configurations . . . . 6.9 Microstructure sections at different positions along the thickness direction z for two distinct loading steps of the 45 • configuration. l denotes the distance from the reference sample surface whereas the mid-thickness section from Figure 6.8(a) is denoted with dotted lines. One of the sections with premature failure is marked with dashed lines . . . . . . . . . 6.10 Von Mises strains plotted over the corresponding microstructure section with favorably positioned nodule cluster for the 45 • configuration. The distance from mid-thickness plane is l = 220 µm . . . . . . . . . . . . . 6.11 Microstructure sections at different positions along the thickness direction z for three distinct loading steps for the 90 • configuration. l denotes the distance from the reference sample surface . . . . . . . . . . . . . . . . 6.12 Von Mises strains plotted over the corresponding microstructure section for the 90 • configuration. The distance from mid-thickness plane is l = 275 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Von Mises strains plotted over the corresponding microstructure sections along the thickness direction z for 90 • (a) and 45 • (b) configurations. l denotes the distance from the ROI mid-plane in the thickness direction z . 6.14 Mean effective strains from sub-ROI versus nominal machine displacement for the two samples are denoted with solid lines. The right axis shows standard deviations marked with dashed lines. Each marker stands for the scan taken during the experiment. Microstructure sections are also shown for the important experimental moments that are associated with the kinematic measurements . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Mean normal strains ε ii (i = x, y, z) plotted against the mean von Mises strains for the 90 • and 45 • configurations . . . . . . . . . . . . . . . . . 6.16 Shear strains ε i j (i, j = x, y, z) plotted against the mean von Mises strain for both configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Mean strain trace (a) and negative ratio (b) between the normal strains ε xx , ε zz and the normal strains in the stretching direction ε yy plotted as functions of the mean effective von Mises strains for the two configurations213 6.18 Local values (inter-nodule) of von Mises strain for the points of interest denoted on the right (C, D) and in Figure 6.14 (A, B) plotted against the mean von Mises strain values in the corresponding sub-ROIs. The dashed line without markers denotes one-to-one relationship for more convenient reading of the graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.19 Local strain (inter-nodule) trace values for the points of interest depicted in Figures 6.14 and 6.18 plotted against the mean von Mises strain levels in the corresponding sub-ROIs . . . . . . . . . . . . . . . . . . . . . . . 6.20 Void growth obtained with different threshold values from laminography data (δ f ) compared with the volume change calculated by the mean strain trace multiplication with the current sub-ROI volume (δV ) for the 90 • (a) and 45 • (b) configurations . . . . . . . . . . . . . . . . . . . . . . . . . 6.21 Comparison of two different threshold methods applied in this work. The voxels counted as voids are colored black . . . . . . . . . . . . . . . . . 6.22 Void volume fraction versus mean von Mises strain on the level of sub-ROI for the two configurations . . . . . . . . . . . . . . . . . . . . . . . 6.23 Volume (solid) and surface (dashed) growth of voids from nodules A1, B1, C1 and D1 normalized with their initial volumes/surfaces plotted against the corresponding local von Mises strains for points A, B, C and D (see Figure 6.14 and 6.18 for positions of the points). . . . . . . . . . Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures 6.24 Normal strains ε zz plotted over the corresponding microstructure sections for three different positions along the ligament between the holes for 90 • (a) and 45 • (b) configurations. X denotes the distance from ROI mid plane in x direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.25 Normal strains ε xx plotted over the corresponding microstructure sections for three different positions along the ligament between the holes for 90 • (a) and 45 • (b) configurations. l denotes the distance from the ROI mid-plane in the thickness direction z . . . . . . . . . . . . . . . . . . . 7.1 Schematic representation of the methods used in the present paper for validating numerical simulations at the microscale . . . . . . . . . . . . . 7.2 (a) Schematic view of the sample with the scanned region close to the central hole. Section of the reconstructed volume with the position of the region of interest. (b) Macroscale force-displacement curve . . . . . . . . 7.3 Reconstructed volume as a spatial discrete matrix of gray level values . . 7.4 3D rendering of all dark features of the nodular cast iron reconstructed volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Difference between incremental and direct DVC procedures . . . . . . . 7.6 Standard displacement (a) and effective strain (b) resolutions as functions of the element size or regularization length . . . . . . . . . . . . . . . . . 7.7 Through thickness section showing nodules embedded in the matrix for the undeformed state (a) and after the first five loading steps (b)-(f) . . . . 7.8 (a)-(c) Isometric view of the gray level residual field and (d)-(f) through thickness section with the residual map for first loading step and three different regularization lengths . . . . . . . . . . . . . . . . . . . . . . . 7.9 Root mean square gray level residual (normalized by dynamic range of the volume -256 gray levels) for various regularization lengths (expressed in voxels) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Displacement fields of mid-thickness section for the first loading step and four different regularization lengths . . . . . . . . . . . . . . . . . . . . 7.11 Effective strain maps of mid-thickness section for the first loading step and four different regularization lengths . . . . . . . . . . . . . . . . . . 7.12 Effective strain maps plotted together with the corresponding microstructure sections for the first loading step (unregularized DVC) and different positions along the thickness direction . . . . . . . . . . . . . . . . . . . 7.13 Example of mesh generation from a 2D binary image . . . . . . . . . . . 7.14 Matrix and nodules at midsection of: (a) the image, (b) the reference mesh h2, (c) the coarse mesh h4, (d) the fine mesh h1 . . . . . . . . . . . . . . 7.15 Distribution of mesh quality at the initial state for the three tested mesh settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.16 Change of void volume (wrt. total ROI volume) with applied macroscopic displacement for various numerical parameters . . . . . . . . . . . . . . Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures 7.17

  Comparison between the void/matrix interface obtained by the FE simulation (thin white line) and the experimental images at a macroscopic displacement of: (a) 0 µm, (b) 330 µm, (c) 578 µm, (d) 744 µm, (e) 868 µm, (f) 910 µm, (g) 950 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.18 Equivalent plastic strain obtained by the FE simulation superimposed with experimental images at a macroscopic displacement of: (a) 578 µm, (b) 910 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 7.19 (a) DVC (blue) and FE (red) meshes plotted over the corresponding cast iron microstructure in (a) isometric view and (b-d) sections normal to X, Y and Z axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.20 Section normal to y-direction: (a) DVC displacement field magnitude, (b) FE displacement field magnitude, (c) absolute difference between them and (d) DVC and FE meshes for the third loading step. All reported values are expressed in voxels (1 voxel ↔ 1.1 µm) and correspond to the FE mesh h2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.21 (a) Mean values and (b) standard deviations of the difference between the measured simulated displacement fields for the FE mesh h2. Standard deviations are normalized by the corresponding DVC uncertainties . . . . 256 7.22 (a) Mean values and (b) standard deviations of the difference between the measured and simulated displacement fields for different FE mesh sizes. Standard deviations are normalized by the corresponding DVC uncertainties257 7.23 Microstructure section in the (a) reference and (b) deformed states. Absolute gray level differences between the two initial states (c) and after correction with DVC (d) and FE (e) displacements (obtained on h2 mesh) for third loading step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 7.24 Standard deviation for the dimensionless gray level residual fields for all loading steps (FE results obtained with h2 mesh) . . . . . . . . . . . . . 258 7.25 Standard deviation gray level residuals for all loading steps with three different FE discretizations . . . . . . . . . . . . . . . . . . . . . . . . . 259 8.1 Schematic view of the sample with zoomed region between the holes showing on the right: DVC (blue) and FE (cyan) meshes plotted over the corresponding cast iron microstructure in isometric view. On the left: surface image with speckle pattern and DIC mesh from test (B) . . . . . 274 8.2 (a) Sample geometry with the scanned region between the pin holes; (b) section of the reconstructed volume with ROI position . . . . . . . . 275 8.3 Mid-thickness section of the reconstructed volume for three different loading steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures 8.4 Image immersion and meshing. (a) Initial laminography 2D section, (b) signed distance function computed thanks to image processing, (c) signed distance function interpolated and reinitialized on the FE mesh [43], (d) conforming FE mesh generated and adapted to interfaces and local maximum curvature, (e) zoom on the FE mesh, (f) comparison between initial laminography 2D section and interfaces in the final FE mesh (in white) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Identification of macroscopic material parameters based on test (A) data. (a) Mesh used for specimen scale calculations, (b) comparison between the simulated force/displacement curves and experimental data . . . . . . 8.6 Inside view of the mesh used in sFE calculations. (a) Full specimen, (b) zoom on the ROI. The three shades of blue represent, from lighter to darker, the homogenized out-of-ROI material, the matrix, and the voids . 8.7 Standard displacement resolutions as functions of the element size expressed in supervoxels for two different scans . . . . . . . . . . . . . . . 8.8 ROI calculation results using the DVC-FE approach showing the 3D meshed voids and the equivalent plastic strain on sections when: (a) u = 0 (undeformed state), (b) u = 83 µm, (c) u = 192 µm, (d) u = 321 µm . . . 8.9 ROI (blue line) calculation results using the DVC-FE approach comparing the numerical matrix/void interface (white line) with experimental images for the X midsection when: (a) u = 0 (undeformed state), (b) u = 83 µm, (c) u = 192 µm, (d) u = 321 µm . . . . . . . . . . . . . . . . . . . . . . 8.10 Standard deviation for the dimensionless gray level residual fields for all loading steps. For comparison purposes, the dashed line corresponds to the resolution analysis for the so-called "bis" case (see Subsection 3.3) . 8.11 Absolute gray level differences at the Z midsection after correction with DVC (a), DVC-FE (b), wFE (c) and sFE (d) displacements for the ninth loading step. Note that due to different rigid body motions for each simulation, the Z midsections slightly differ . . . . . . . . . . . . . . . . . . 8.12 Mid-section normal to z-direction showing absolute difference between: (a) DVC and DVC-FE displacement fields, (b) DVC and wFE displacement fields for the ninth loading step. The black area at the top of subfigure (b) represents a zone out of the DVC ROI. The color bar range is set according to the data set from sub-figure (a) in order to have more convenient visual comparison. The displacement difference is expressed in supervoxels (1 supervoxel ←→ 2.2 µm . . . . . . . . . . . . . . . . . 8.13 Void volume change curves for all the approaches investigated in this paper 8.14 Standard deviation for the dimensionless gray level residual fields for all loading steps and investigated cases . . . . . . . . . . . . . . . . . . . . Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures

2 .

 2 Initial response stands for σε values obtained with initial set of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Initial and final dimensionless residual for the different mesh sizes used in integrated DVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 Initial and final dimensionless gray level residuals for different mesh sizes used in the I-DVC procedure and for DVC calculations . . . . . . . . . . 9.12 Gray level residuals for I-DVC (left) and C8-DVC (right) analyses for the last loading step. Online version: corresponding movie for the 6 analyzed steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.13 Microstrucutre section and equivalent strain field (from FE calculations) plotted on the deformed configuration for the last loading step. Online version: corresponding movie for the 6 analyzed steps . . . . . . . . . . . 9.14 Technical drawing of the sample . . . . . . . . . . . . . . . . . . . . . . 10.1 Deformation mechanism found for all four studied aluminum alloys . . . Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures 10.2 Nanolaminography results: (a) position of the scanned region; (b) section of the full reconstructed volume for different loading steps. The physical size of 1 voxel is 100 nm for all three directions . . . . . . . . . . . . . . 10.3 DVC results obtained on nanolaminography data: effective strain fields plotted over the corresponding microstrucutre. The results are obtained without (top) and with (bottom) mechanical regularization. The physical size of 1 voxel is 100 nm for all three directions . . . . . . . . . . . . . . 10.4 Portion of the reconstructed volume from nanolaminogrpahy data. The voids and intermetallic particles posses satisfactory visual description important for meshing purposes in CimLib tools. The physical size of 1 voxel is 100 nm for all three directions . . . . . . . . . . . . . . . . . . . 10.5 Nanolaminography data: 2D views of the portion of the segmented volume for different loading steps. The physical size of 1 voxel is 100 nm for all three directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Ductile failure mechanisms elaborated in the work: (a) failure by void growth and coalescence (example: nodular gaphite cast iron, 2 hole sample -90 • configuration); (b) failure by plastic instability (example: aluminum alloy AA2139 T8) . . . . . . . . . . . . . . . . . . . . . . . . . A.1 (a) Pseudo 3D cube showing the grain shapes (optical microscopy after etching). (b) 2D sections of reconstructed 3D synchrotron tomography data (after Ref. [45]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Engineering stress-strain curve for AA2139 T3 when loading in the L and T directions. The thickness variation is also given (after Ref. [46]) . . . . A.3 Schematic drawing of the laminography setup (after Ref. [42]) . . . . . . A.4 (a) Photograph of the specimen, loading device and anti-buckling frame mounted onto the laminography specimen holder; (b) drawing of the flat (1 mm) CT-like specimen and loading device; (c) measured crack length versus CMOD curve (every dot on the curve corresponds to one load increment and corresponding scan); (d) 2D section of reconstructed laminography data at mid thickness for load step CMOD = 2.125 mm . . A.5 (a) 3D rendering of the initial microstructure and the ROI. (b) 3D rendering of the effective strain field in the extracted volume when CMOD = 1.5 mm. (c) 2D overlay of strain field over the initial microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Overlay of incremental effective strain fields (half transparent) over the microstructure 1080 µm ahead of the notch root. (a)-(g) 2D sections of laminography data and effective strain fields. The microstructure is shown in state n for incremental correlations between n and n + i. Two reference voids are encircled in white on the microstructure for every load step; (h),(i) microstructure showing damage and the final crack path; (j) load steps corresponding to the shown sections . . . . . . . . . . . . . . . . . Understanding, observation and quantification of ductile failure mechanisms via 3D imaging List of Figures A.7 2D sections of cumulated effective von Mises strain fields. (a) Direct correlation from CMOD = 0.0 mm to CMOD = 1.5 mm. (b) One cumulation from CMOD = 0.0 mm to CMOD = 2.0 mm. (c) Two cumulations from CMOD = 0.0 mm to CMOD = 2.313 mm (the coordinate system is shown in its initial state) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 (a) 2D section of cumulated fields (effective von Mises strain) with indication of the location for the strain profile. (b) Strain profile for the direct and cumulated strain fields from CMOD = 0.0 mm to CMOD = 2.313 mm A.9 2D section of 3D laminography and DVC data. Overlay of the smallest possible direct effective strain fields (half transparent) over the microstructure in its initial state . . . . . . . . . . . . . . . . . . . . . . . . A.10 2D section of 3D laminography and DVC data. Overlay of the smallest possible incremental effective strain fields (half transparent) over the microstructure in its initial state . . . . . . . . . . . . . . . . . . . . . . A.11 2D section of 3D laminography and DVC data. Overlay of incremental effective strain fields n to n + 2 (half transparent) over microstructure (the microstructure in its initial state) . . . . . . . . . . . . . . . . . . . . . . A.12 2D section of 3D laminography and DVC data. Overlay of incremental effective strain fields n to n + 3 (half transparent) over the microstructure in the initial state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.13 3D void visualization of reconstructed laminography data. (a) Throughthickness section with location of the void 1125 µm from the notch. (b) Overlay of strain field for step CMOD = 2.5 mm to CMOD = 2.563 mm over microstructure at CMOD = 2.5 mm. (c)-(j) Void growth. (k) Indication of the corresponding load step . . . . . . . . . . . . . . . . . . . . . A.14 Void volume fraction for one void (see Figure A.13) versus measured von Mises effective strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.15 Results of a 3D simulation of the CT-like specimen using a Gurson-Tvergaard-Needleman model combined with a shear void nucleation term. (a) Stress field in the loading direction. (b) Equivalent strain field in the plastic zone. (c)-(i) incremental strain fields in the ROI for load steps corresponding to Figure A.6 1050 µm ahead from the notch root. (j) Final crack position. (k) Indication of the corresponding load step . . . . . . . A.16 2D plane strain simulation using a slant soft band. (a) 2D mesh. (b) Equivalent plastic strain field when x = 10 % . . . . . . . . . . . . . A.17 Equivalent strain profiles normal to the band (see Figure A.16(a)) for a 10 % perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.18 Strain ratio versus strain outside the band for different perturbation levels (2 % ≤ x ≤ 14 %) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.19 2D plane strain simulation using randomly distributed soft zones. (a) 2D mesh. (b) Equivalent plastic strain field when the mean level is equal to 0.14 and x = 10 % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Task 1 :Figure 1 :

 11 Figure 1: Complementarity of the techniques used in the COMISNIDE project with denoted scope of the current work

Figure 2 :

 2 Figure 2: Void nucleation as result of (a) particle debonding and (b) particle fracture.SEM figures are extracted from Ref.[47]

Figure 3 :

 3 Figure 3: Ductile damage stages of void nucleation, growth and coalescence shown in the example of nodular cast iron sample studied in this work

Figure 4 :

 4 Figure 4: (a) Strain to fracture for monotonic loading as a function of the stress triaxiality and Lode angle parameter reproduced by Wilkins' model [71]. All stress states representing plane stress are connected with the blue line. (b) Strain to fracture for plane stress as a function of stress triaxiality. Figures are taken from Ref. [128]

  using an explicit FE Understanding, observation and quantification of ductile failure mechanisms via 3D imaging simulation but comparisons with experiments are missing. Very recently [127] an l a m i n o g r a p h y & D V C S E M

Figure 5 :

 5 Figure 5: Flat to slant transition during mode I opening of CT samples. The laminography/DVC section refers to AA2198 T8 alloy studied in this work. Post mortem SEM figure is taken from Ref. [127]

Figure 6 :

 6 Figure 6: Tomography and laminography experimental setup

Figure 7 :

 7 Figure 7: Sample geometries used on laminography experiments lined as function of average stress triaxiality in the scanned zone

Figure 8 :

 8 Figure 8: Structure of the work

Introduction

  Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Bibliography Chapter 1

Figure 1 . 1 :

 11 Figure 1.1: Mid-thickness L -T slice in 3D reconstructed volumes of three different aluminum alloys and one cast iron sample. The picture definition is 2040 × 2040 pixels for aluminum alloys and 1600 × 1600 pixels for cast iron. The physical length of one voxel is 0.7 µm for aluminum alloys and 1.1 µm for cast iron. Especially for (a) and (b), ring artifacts from static features on the projection radiographs are clearly visible.

Figure 1 . 2 :

 12 Figure 1.2: Gray level histograms of the investigated aluminum alloys and cast iron

Figure 1 . 3 :

 13 Figure 1.3: Experimental setup schematically shown for the aluminum CT-like speci-mens investigated in this work[30]

Figure 1 . 4 :

 14 Figure 1.4: Region of interest considered in the uncertainty analysis of the volume made of (a) AA2139 T3 and (b) AA2139 T8. The squares on the right denote 4 DVC elements with isotropic size = 32 voxels

Figure 1 .

 1 5(a) shows the location of the Region of Interest (ROI) in the reference and rbm configuration moved by u T = -15 voxels, u L = 586 voxels and u S = 3 voxels. The rotation axis and reconstruction artifacts (seen as rings) are visible.

Figure 1 . 5 :

 15 Figure 1.5: Region of interest considered in the uncertainty analysis of the volume made of (a) AA2198 T3 and (b) AA2198 T8. The squares on the right denote 4 DVC elements with isotropic size = 32 voxels

Figure 1 .

 1 5(b) shows the location of the Region of Interest (ROI) in the reference and rbm configuration moved by u T =

Figure 1 . 6 :

 16 Figure 1.6: Region of interest considered in the uncertainty analysis of the volume made of AA6061 T6. The squares on the right denote 4 DVC elements with isotropic size = 32 voxels

Figure 1 . 7 :

 17 Figure 1.7: Isometric isolated view of voids (blue) and particles (yellow) (+ edgeenhancement artifacts around voids due to so-called "phase contrast", an interference phenomenon due to Fresnel diffraction) in 3D reconstructed volumes (256 × 256 × 256 voxels) of the studied aluminum alloys Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Figure 1 . 8 :

 18 Figure 1.8: Procedure followed when isolating voids and particles for AA2139 and AA6061 alloys

Figure 1 . 9 :

 19 Figure 1.9: Schematic view of the sample with the scanned region close to the central hole. Section of the reconstructed volume with the position of the region of interest (ROI)

Figure 1 . 10 :

 110 Figure 1.10: The sample geometry with the scanned region between the two holes. Section of the reconstructed volume with ROI position

Figure 1 . 11 :

 111 Figure 1.11: Isometric isolated view of nodules and secondary void population in 3D reconstructed volume (256 × 256 × 256 voxels) of the studied cast iron

1

 1 Repeated scanbis case

Figure 1 . 12 :

 112 Figure 1.12: Standard displacement (a) and strain (b) resolutions as functions of the element size for bis cases

Figure 1 . 13 :

 113 Figure 1.13: Standard displacement (a) and strain (b) resolution as functions of the element size for rbm cases

Figure 1 . 14 :

 114 Figure 1.14: Standard displacement (a) and strain (b) resolutions as functions of the element size for bis and rbm cases

Figure 1 . 16 :

 116 Figure 1.16: Standard displacement (a) and strain (b) resolutions as functions of the element size for different experimental setup

Figure 1 . 17 :

 117 Figure 1.17: Standard displacement (a) and strain (b) resolutions as functions of the element size for different cameras used in the experiment

Figure 1 . 18 :

 118 Figure 1.18: L -T slice in 3D reconstructed volumes of AA6061 (a) and AA6061 modified (b). The squares on the top denote 4 DVC elements with isotropic size = 32 voxels;(c) gray level histograms of the two grades.

Figure 1 . 19 :

 119 Figure 1.19: Standard displacement (a) and strain (b) resolutions as functions of the element size for the two grades of AA6061 alloy

4. 1 . 8 Figure 1 . 20 :Figure 1 . 21 :

 18120121 Figure 1.20: Regions of interest considered in the uncertainty analysis of the volume made of AA2198 T3

Figure 1 . 22 :Figure 1 . 23 :

 122123 Figure 1.22: Standard displacement (a) and strain (b) uncertainties for DVC analyses associated with the ROIs shown in Figure 1.20(b)

Figure 1 . 24 :

 124 Figure 1.24: Scanned volume section LS considered in the uncertainty analysis of the volume made of AA2198 T3 before (a) and after (b) rigid body motion. The vertical black line denotes the axis of the reconstructed volume. The LS projections are obtained through histogram equalization

Figure 1 . 25 :Figure 1 . 26 :

 125126 Figure 1.25: Displacement fields u L for ROI 3 (a); ROI 2 (b) and ROI 1 (c) when measured with 32-voxel elements for the rbm case (AA2198 T3)

Figure 1 . 27 :Figure 1 . 28 :

 127128 Figure 1.27: Displacement fields u L for ROI 3 (a); ROI 2 (b) and ROI 1 (c) when measured with 32-voxel elements for the bis case (AA2198 T3)

Figure 1 . 29 :

 129 Figure 1.29: Scanned volume section LS considered in the uncertainty analysis of the volume made of AA 2139 T8 before (a) and after (b) rigid body motion. The LS projections are obtained through histogram equalization

Figure 1 . 30 :

 130 Figure 1.30: Effective strain fields for AA2139 T3 and AA6061 T6 when measured with 32-voxel elements for the rbm case. The ROIs are positioned as shown in Figure 1.4(a) and 1.6 respectively.

Figure 1 . 31 :

 131 Figure 1.31: Standard displacement (a) and strain (b) resolutions as functions of the element size for cast iron (central hole geometry from Figure 1.9)

Figure 1 . 32 :

 132 Figure 1.32: Standard displacement (a) and strain (b) resolution as functions of the element size expressed in supervoxels for the specimen geometry illustrated in Figure 1.10

Figure 1 . 33 :

 133 Figure 1.33: Standard displacement (a) and strain (b) resolutions as functions of the element size for all analyzed materials. "CI tomo. rbm" denotes the corresponding tomography resolutions [23] of the same type of material but different physical voxel size (3.5 µm

Figure 1 . 34 :

 134 Figure 1.34: Normalized standard displacement (a) and strain (b) resolutions as functions of the element size. The solid line shows an interpolation by -3/2 power law (a) and -5/2 power law (b)

Figure 1 . 35 :

 135 Figure 1.35: The largest 100 eigenvalues of the inverse DVC matrix [M] for the studied microstructures and element size =32 voxels

Figure 1 . 36 :

 136 Figure 1.36: Reconstruction artifacts in laminography. The yellow line indicates the boundary of the region where the sampling is incomplete

Figure 2 . 1 :

 21 Figure 2.1: (a) Shematic view of the laminography experimental setup with scanned region in front of the notch root (red) and reconstructed volume. (b) Reconstructed volume sections with ROI positions and relevant dimensions for DVC analyses. ROI 1 corresponds to the analysis reported in [28] and ROI 2 is analyzed in this work

Figure 2 . 2 :

 22 Figure 2.2: Standard resolution of von Mises' effective strain as a function of the element size expressed in voxels for a rigid-body-motion case. Two ROIs are considered in the DVC analyses (see Figure 2.1(b))

Figure 2 . 3 :

 23 Figure 2.3: Notch opening displacement (NOD) for the studied loading steps

Figure 2 . 4 :

 24 Figure 2.4: T (a) and S (b) middle sections displaying incremental effective strain maps for both ROI 1 and ROI 2 for loading step (1)-(2)

  Slant strained band development during flat to slant crack transition in AA 2198 T8 sheet:in situ 3D measurements

Figure 2 . 5 :

 25 Figure 2.5: L-sections ∼ 390 µm away from the initial notch root. The first column corresponds to reconstructed laminography data in the reference configurations f , the second column shows incremental effective strain maps, the third column displays the deformed volume corrected by the measured displacement, and the fourth column the gray level residuals for loading steps (a): (0)-(1), (b): (1)-(2), (c) (2)-(3)

Figure 2 . 6 :

 26 Figure 2.6: Sections normal to the L-direction of incremental effective strain fields for loading step (a): (0)-(1), (b): (1)-(2) and (c): (2)-(3). d is distance from the initial notch root

Figure 2 . 7 :

 27 Figure 2.7: Sections normal to S-direction of incremental effective strain fields for loading step (a): (0)-(1), (b): (1)-(2) and (c): (2)-(3). l is distance from the center line

Figure 2 . 8 :

 28 Figure 2.8: Effective strain plotted along axis ξ parallel to the T-direction 285 µm from the notch root and on the mid-plane for loading steps (0)-(1), (1)-(2) and (2)-(3)

Figure 2 . 9 :

 29 Figure 2.9: Middle S-sections of normal (nominal) strains in T, L and S-direction together with incremental effective strain field for the corresponding slice for loading step (0)-(1). The color bar has been limited to highlight strain concentration areas

  )-(3) (Figure 2.10(b)). It can be seen that reaching the plane strain condition is accompanied with the existence of multiple slant strained bands. Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Results 105 ξ, Load step (0)-(1)

Figure 2 . 10 :Figure 2 . 11 :

 210211 Figure 2.10: (a) nominal strain profiles in the crack propagation direction (L) plotted along axis ξ for loading steps (0)-(1), (1)-(2) and (2)-(3). (b) nominal strain profiles in the crack propagation direction for loading step (2)-(3) plotted along three different axes ξ i on the mid-plane. The L-sections of incremental effective strains are also shown. d is distance from the initial notch root

Figure 2 . 12 :

 212 Figure 2.12: Incremental effective strain fields printed over portion of reconstructed volumes showing the final crack path for loading steps (0)-(1) (a),(1)-(2) (b) and (2)-(3) (c). L-sections are located at a distance d ∼ 250 µm, d ∼ 390 µm and d ∼ 530 µm from the notch root

Chapter 2 :

 2 Additional MaterialUnderstanding, observation and quantification of ductile failure mechanisms via 3D imaging

  (a) step (0)-(1) (b) step (1)-(2) (c) step (2)-(3)

Figure 2 . 13 :

 213 Figure 2.13: Movies showing L-sections of incremental effective strain field for the three analyzed loading steps. The results of both ROI 1 and ROI 2 are shown

Figure 3 . 1 :

 31 Figure 3.1: (a) Shematic view of the laminographic setup with the loading device, CTlike sample and the scanned region (red zone). (b) 2D sections of laminography data indicating the positions of the Regions of the Interest (ROIs) examined in this work

Figure 3 . 2 :

 32 Figure 3.2: Standard von Mises strain resolution as a function of the element size used in the correlation procedure

Figure 3 .

 3 2 shows the strain resolution levels for the three inspected ROIs. Let us emphasize that these uncertainty levels incorporate cumulated effects of laminography and DVC. A stepwise loading procedure resulted in the Notch Opening Displacement (NOD) history shown in Figure 3.3. Measurements are taken approximately 200 µm from the initial notch root and in the notch as shown in Figure 3.1(b). NOD data for the two ultimate loading steps (State (

Figure 3 . 3 :

 33 Figure 3.3: Notch Opening Displacement (NOD) for the step-wise loading procedure followed in this work; mid-thickness section showing the final crack propagated through the specimen ligament area for loading step (0)-(11)

Figure 3 . 4 :

 34 Figure 3.4: Comparison of DVC and "projection DIC" results. (a) 2D sections of reconstructed laminography data for load steps (0) and (3). (b) Projected 2D sections of reconstructed laminography data (the maximum values were taken over a thickness of 50 µm). (c) Comparison of normal strain fields in the loading direction obtained by DVC (left) and "projection DIC" (right) on projected 3D data

Figure 3 . 5 :Figure 3 . 6 :

 3536 Figure 3.5: Digital volume correlation results. Section normal to the crack propagation direction (i.e., section B-B, Figure 3.1(b)) showing von Mises effective strain fields for direct (a) and incremental (b) calculations

Figure 3 . 7 :

 37 Figure 3.7: B-B section through 3D laminography data showing the microstructure for different loading steps

  Figure 3.8(a). The curves correspond to an estimate of the damage variable (i.e., void fraction) inside ROIs 1, 2 and 3 (Figure 3.1(b)) during the loading Understanding, observation and quantification of ductile failure mechanisms via 3D imagingResults 129

Figure 3 . 8 :

 38 Figure 3.8: (a) Void fraction development over the loading history for the three inspected ROIs (see Figure 3.1(b)). The shadow region denotes the final crack propagation through the ligament as shown in Figure 3.3. (b) Mean values and the standard deviation of von Mises effective strain fields inside ROI 1 for direct calculations

Figure 3 . 9 : 131 3. 3

 391313 Figure 3.9: Final crack position in conjunction with the initial von Mises effective strain map plotted along the section normal to the crack propagation direction (L) for loading step (0)-(3). The height of the final roof-top crack is shown with the color map. Left: view at 1 mm from the notch root; Right: view in the crack propagation direction

Figure 3 . 10 :

 310 Figure 3.10: Sections normal to the crack propagation direction showing von Mises effective strain fields for loading step (0)-(3) for different distances from the notch root

Figure 3 . 11 :

 311 Figure 3.11: Through-thickness sections of the merged ROIs 1, 2 and 3, where d denotes the distance from the ROI mid-thickness plane. Positions of the L-L and R-R sections are depicted on the section normal to the crack propagation direction (right column). (a) Von Mises effective strains ε eq , (b) normal strains in the loading direction ε T T , (c) normal strains in the crack propagation direction ε LL , (d) normal strains in the through thickness direction ε SS . Normal strain components are defined as nominal strains.

  .11(d)) there exists a strong dependence along Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Results 133

3. 4

 4 Incremental flip-flopping strain activity and final failure in the intermediate region (ROI 2)

Figure 3 .

 3 Figure 3.8(a) reveals damage growth in the zones closer to the notch root (e.g., ROIs 2and 3). This is confirmed in Figure3.13 where the porosity development is assessed by thresholding the gray levels of the reconstructed volumes. It is worth mentioning, as can be seen from Figure3.1(b), that the ROIs overlap so that some of the damage features are shared by the two neighboring ROIs. ROI 3, which is closest to the notch root, shows the highest porosity content before final crack propagation. ROI 2 experiences smaller void growth but still with some damage growth before final failure while ROI 1 does not show any microstructural changes before failure, which is consistent with the change in the stress state and the observed strain pattern.In Figure3.14 microstructure states are shown for different loading steps in the midthickness section of the sample. Void growth occurs at late stages, but only in the triangu-

Figure 3 . 12 :

 312 Figure 3.12: Incremental normal strain maps (in the loading direction) in ROI 2 (i.e., 460 µm from the notch root) for different load steps and overlay of the last possible incremental strain field with the final crack path

Figure 3 . 13 :Figure 3 . 14 :

 313314 Figure 3.13: Thresholded gray level views of ROIs 1, 2 and 3 for the undeformed configuration and last three loading steps

Figure 3 . 15 :

 315 Figure 3.15: Porosity changes in the immediate vicinity of the notch root (Figure 3.14) during the loading history

Figure 3 . 16 :

 316 Figure 3.16: Electron backscattered diffraction taken at the one side of the broken CTlike specimen on the section normal to the crack propagation direction. (a) View before electro-polishing. (b) Grain texture revealing the transgranular nature of the inspected damage feature. (c) Inverse pole figure showing the grain orientations where the damaged grain with its 001 orientation is sandwiched between 111 oriented grains

Figure 3 . 17 :

 317 Figure 3.17: Scanning electron micrographs taken at one side of the broken CT-like specimen. (a) Maginification level ×70, triangular zone at the notch root followed by slant surfaces. (b) Maginification level ×700, magnified subsets a-f depicted in (a). (c) Maginification level ×2000, zoomed details from subset b and from the zone between subsets e and d.

Figure 3 . 18 :

 318 Figure 3.18: (a) Positions of the subsets depicted on the thresholded von Mises effective strain field. (b) Void growth over the loading history inside two subsets of ROI 1

Figure 3 . 19 :

 319 Figure 3.19: Standard deviation of the residual fields for incremental and direct calculations

Figure 3 . 20 :Figure 3 . 21 :

 320321 Figure 3.20: Gray level residual distribution over section B-B (Figure 3.5(b)) and the section normal to it (S-S) for the last converged direct (left) and incremental (right) calculations

  section: min. GL values projected section: max. GL values projected section: mix.

Figure 4 . 1 :

 41 Figure 4.1: Laminography experiment: (a) loading device with the specimen; (b) reconstructed volume with positions of the projected sections used in bulk-DIC procedure; (c) single and projected laminography sections for different modes of projection to enhance the contrast in the pictures

Figure 4 . 2 :

 42 Figure 4.2: Laminography section normal to crack propagation direction (L) at distance of 865 µm from the initial notch root for different loading steps. The green zoomed region corresponds to the location where significant damage coalescence (State (8)) and subsequent final failure occur. The blue and red arrows point to the intermetallic particle breakages and isolated void coalescence events while the yellow ones denote the final coalescence event

Figure 4 . 3 :

 43 Figure 4.3: Normal strain fields in loading direction (ε T T ) plotted over corresponding projected microstructure sections (mix. projection mode from Figure 4.1(c))

Figure 5 . 1 :

 51 Figure 5.1: (a) Schematic view of the laminography setup; (b) shear sample geometry and position of the scanned zone. In the section of the scanned region the representationof microstructure grains is shown[11]

Figures 5 .Figure 5 . 2 :

 552 Figure 5.2: Standard strain resolutions as functions of the element size

. 3 .Figure 5 . 3 :

 353 Figure 5.3: Load levels plotted against machine displacements. Each symbol depicts a scan acquistion

Figure 5 . 4 :Figure 5 . 5 :

 5455 Figure 5.4: Displacement profiles from initialization and final DVC field for one loading step

Figure 5 . 6 :

 56 Figure 5.6: Von Mises incremental strain maps from DVC calculations plotted for the ROI mid-thickness section

Figure 5 . 7 :

 57 Figure 5.7: Von Mises incremental strain profiles from DVC calculations. The position of the profile axis ξ is denoted on the strain map illustration

Figure 5 .

 5 Figure 5.10 shows the Lode parameter maps for incremental strains µ ∆ε plotted for the ROI mid-thickness section. From the second loading step on, the position of the band can be distinguished from the rest of the ROI. The values in the band are close to zero. However, in the rest of domain with DVC elements with low strains (close to noise level) the Lode parameter cannot be assessed. It is important to recognize and separate these two types of data.

Figure 5 . 10 :

 510 Figure 5.10: Lode strain parameter µ ∆ε maps from incremental DVC calculations plotted for the ROI mid-thickness section

Figure 5 . 11 :

 511 Figure 5.11: Von Mises incremental strain ∆ε eq and Lode parameter µ ∆ε for mechanical (loading step (3)-(4)) and noisy data ("bis" case from the uncertainty analysis)

Figure 5 . 12 :

 512 Figure 5.12: Lode parameter µ ∆ε profile (from Figure 5.13(a)) for loading step (3)-(4)) with the corresponding error bars. The area between dashed lines corresponds to the localization band position, i.e., ∆ε eq > 2 %. The values of Lode parameter inside shaded areas are theoretically non-admissible

Figure 5 . 13 :

 513 Figure 5.13: (a) Lode incremental strain parameter µ ∆ε profiles from incremental DVC calculations. The position of the profile axis is denoted on the Lode map shown in the corner; (b) Part of the Lode parameter µ ∆ε profile in the shear band for loading step (3)-(4)) with the corresponding error bars

Figure 5 . 14 :

 514 Figure 5.14: Part of the Lode parameter µ ∆ε profile in the shear band for loading steps(3)-(4) and (6)-(7) with and without assumption of incompressibility

Figure 5 .

 5 15 shows stacks of reconstructed volumes projected onto a single slice by keeping only the minimum gray level values from each of the columns in the stack. The minimum gray levels indicate void nucleation and growth. The slices are positioned normal to the thickness direction. The initial stack incorporates most of the sample thickness without the part close to the surface where paint and laminography artifacts would corrupt the analysis. One can note that from the fourth loading step (3)-(4) on damage features cumulate in Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Figure 5 . 15 :

 515 Figure 5.15: Projections of the stack of slices into single gray level map by keeping only the minimum level from each of the columns in the stack. The stack incorporates most of the sample thickness, i.e., ≈ 810 µm without the part close to the surface. The damage is shown in black

Figure 5 . 16 :

 516 Figure 5.16: 3D side views of the thresholded damage features in the scanned region for different loading steps

Figure 5 . 17 :

 517 Figure 5.17: Damage features found in the shear zone: (a) growth of big elongated damage feature along shear plane (blue arrow) and normal to the loading direction (green arrow), (b) elongated feature at State (7), (c) intermetallic particle breakage at State (7), (d-e) void sheeting as likely nucleation mechanism for elongated damage features (State (7))

Figure 6 . 1 :

 61 Figure 6.1: Drawings of the two samples with the scanned regions and the regions of interest (ROIs) for DVC analyses

Figure 6 . 2 :

 62 Figure 6.2: (a) Laminography setup schematic view; (b) sample with scanned region: 3 void populations

Figure 6 . 3 :

 63 Figure 6.3: Effective strain resolution (a) and strain trace resolution (b) as functions of the element size

Figure 6 . 4 :

 64 Figure 6.4: Mid-thickness microstrucutre sections of reconstructed laminography data for the 90 • (a) and 45 • (b) configurations during the loading history

Figure 6 . 5 :

 65 Figure 6.5: DVC gray level residual fields: (a) 45 • configuration section normal to the loading direction y; (b) 90 • configuration section normal to the thickness direction z; (c) isometric view of the residual field for the 90 • configuration

Figure 6 .

 6 6(a) shows the complete ligament surface for the 45 • configuration together with two enlarged regions. In Region A one detail is depicted and additionally enlarged in Zoom C. They are representative of incomplete internal necking (void impingement) mechanisms.

Figure 6 .Figure 6 . 6 :

 666 Figure 6.6: SEM observations for the 45 • (a) and 90 • (b) configurations

Figure 6 . 7 :

 67 Figure 6.7: SEM observation of void sheeting coalescence for the 90 • configuration

Figure 6 .Figure 6 . 8 :

 668 Figure 6.8 shows von Mises strain maps plotted over the corresponding microstructure sections at the sample mid-thickness. The DVC registrations could not be performed until the very end of the procedure due to (very) large deformations and significant cumulated damage (i.e., no contrast conservation). The last recorded band positions are plotted over the microstructure (yellow and red lines) to ensure better visualization up to final coalescence.From Figure6.8(a) of the 45 • configuration early and stable strained bands are observed between the nodules. The macroband is located between the two machined holes, but it is rather thick (indicating non locality) and not homogeneous but determined by the local heterogeneities, i.e., nodule positions. The incremental DVC results are available up to the penultimate loading step.

Figure 6 . 9 :

 69 Figure 6.9: Microstructure sections at different positions along the thickness direction z for two distinct loading steps of the 45 • configuration. l denotes the distance from the reference sample surface whereas the mid-thickness section from Figure 6.8(a) is denoted with dotted lines. One of the sections with premature failure is marked with dashed lines

Figure 6 . 10 :

 610 Figure 6.10: Von Mises strains plotted over the corresponding microstructure section with favorably positioned nodule cluster for the 45 • configuration. The distance from mid-thickness plane is l = 220 µm

Figure 6 . 11 :

 611 Figure 6.11: Microstructure sections at different positions along the thickness direction z for three distinct loading steps for the 90 • configuration. l denotes the distance from the reference sample surface

Figure 6 . 12 :

 612 Figure 6.12: Von Mises strains plotted over the corresponding microstructure section for the 90 • configuration. The distance from mid-thickness plane is l = 275 µm

Figure 6 . 13 :

 613 Figure 6.13: Von Mises strains plotted over the corresponding microstructure sections along the thickness direction z for 90 • (a) and 45 • (b) configurations. l denotes the distance from the ROI mid-plane in the thickness direction z

Figure 6 . 14 :Figure 6 . 15 :Figure 6 . 16 :

 614615616 Figure 6.14: Mean effective strains from sub-ROI versus nominal machine displacement for the two samples are denoted with solid lines. The right axis shows standard deviations marked with dashed lines. Each marker stands for the scan taken during the experiment. Microstructure sections are also shown for the important experimental moments that are associated with the kinematic measurements

Figure 6 . 17 :

 617 Figure 6.17: Mean strain trace (a) and negative ratio (b) between the normal strains ε xx , ε zz and the normal strains in the stretching direction ε yy plotted as functions of the mean effective von Mises strains for the two configurations

Figure 6 . 18 :

 618 Figure 6.18: Local values (inter-nodule) of von Mises strain for the points of interest denoted on the right (C, D) and in Figure 6.14 (A, B) plotted against the mean von Mises strain values in the corresponding sub-ROIs. The dashed line without markers denotes one-to-one relationship for more convenient reading of the graph

Figure 6 . 19 :

 619 Figure 6.19: Local strain (inter-nodule) trace values for the points of interest depicted in Figures 6.14 and 6.18 plotted against the mean von Mises strain levels in the corresponding sub-ROIs

Figure 6 . 20 :

 620 Void growth obtained with different threshold values from laminography data (δ f ) compared with the volume change calculated by the mean strain trace multiplication with the current sub-ROI volume (δV ) for the 90 • (a) and 45 • (b) configurations

Figure 6 . 21 :Figure 6 . 22 :Figure 6 .

 6216226 Figure 6.21: Comparison of two different threshold methods applied in this work. The voxels counted as voids are colored black

  Effect of void arrangement on ductile damage mechanisms in nodular cast iron: in situ 3D measurements3.4.2 Local valuesFor the local scale analysis between void growth and kinematic data the following approach is applied. Void growth associated with the 4 nodules positioned in the immediate vicinity of points A, B (void sheeting ligaments from Figure6.14) and C, D (internal necking ligaments from Figure 6.18) is followed during the experiment. The positions of the nodules are also depicted in Figures 6.14 and 6.18. Their names are associated with the corresponding points of interest, hence A1, B1, C1, D1. The volumes of the nodule+void (due to early debonding and growth) are normalized by the corresponding initial nodule volume. The same procedure is repeated with the surfaces of the voids. The counting is performed by Fiji software [5, 38, 40, 39] and the results are shown in Figure 6.23. Note that even on the local scale where both ligaments experience a similar type of failure void growth is still higher for the 90 • configuration. No significant deviations from this conlusion have been observed for the void surface evolutions.

Figure 6 . 23 :

 623 Figure 6.23: Volume (solid) and surface (dashed) growth of voids from nodules A1, B1, C1 and D1 normalized with their initial volumes/surfaces plotted against the corresponding local von Mises strains for points A, B, C and D (see Figure 6.14 and 6.18 for positions of the points).

Figure 7 . 1 :

 71 Figure 7.1: Schematic representation of the methods used in the present paper for validating numerical simulations at the microscale

Figure 7 . 2 :

 72 Figure 7.2: (a) Schematic view of the sample with the scanned region close to the central hole. Section of the reconstructed volume with the position of the region of interest. (b) Macroscale force-displacement curve

Figure 7 . 2 . 2

 722 2(a)). The reconstructed volume has a size of 1600 × 1600 × 1600 voxels. The physical size (length) of 1 cubic voxel is equal to 1.1 µm. Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Experimental analyses 233 Digital Volume Correlation The correlation procedure using the whole volume size would be computationally too demanding. Therefore an extracted ROI of size 532 × 532 × 532 voxels is used for DVC analyses. The position of the ROI in the global coordinate system is shown in Figure 7.2(a) while in the z direction the ROI and sample mid-thickness planes coincide.

Figure 7 . 3 :

 73 Figure 7.3: Reconstructed volume as a spatial discrete matrix of gray level values

Figure 7 . 4 :

 74 Figure 7.4: 3D rendering of all dark features of the nodular cast iron reconstructed volume

Figure 7 . 5 :

 75 Figure 7.5: Difference between incremental and direct DVC procedures

Figure 7 . 6 :

 76 Figure 7.6: Standard displacement (a) and effective strain (b) resolutions as functions of the element size or regularization length

  Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Numerical Validation Framework for Micromechanical Simulations based on Synchrotron 3D Imaging

Figure 7 . 7 :

 77 Figure 7.7: Through thickness section showing nodules embedded in the matrix for the undeformed state (a) and after the first five loading steps (b)-(f)

Figure 7 . 8 :

 78 Figure 7.8: (a)-(c) Isometric view of the gray level residual field and (d)-(f) through thickness section with the residual map for first loading step and three different regularization lengths

Figure 7 . 9 :

 79 Figure 7.9: Root mean square gray level residual (normalized by dynamic range of the volume -256 gray levels) for various regularization lengths (expressed in voxels)

  r = 0 voxel (relaxed)

Figure 7 . 10 :

 710 Figure 7.10: Displacement fields of mid-thickness section for the first loading step and four different regularization lengths

Figure 7 . 11 :

 711 Figure 7.11: Effective strain maps of mid-thickness section for the first loading step and four different regularization lengths

Figure 7 . 12 :

 712 Figure 7.12: Effective strain maps plotted together with the corresponding microstructure sections for the first loading step (unregularized DVC) and different positions along the thickness direction

  (a) and 7.14(d).

Figure 7 . 14 :

 714 Figure 7.14: Matrix and nodules at midsection of: (a) the image, (b) the reference mesh h2, (c) the coarse mesh h4, (d) the fine mesh h1

Figure 7 . 15 :

 715 Figure 7.15: Distribution of mesh quality at the initial state for the three tested mesh settings

Figure 7 . 17 :

 717 Figure 7.17: Comparison between the void/matrix interface obtained by the FE simulation (thin white line) and the experimental images at a macroscopic displacement of: (a) 0 µm, (b) 330 µm, (c) 578 µm, (d) 744 µm, (e) 868 µm, (f) 910 µm, (g) 950 µm

Figure 7 . 18 :

 718 Figure 7.18: Equivalent plastic strain obtained by the FE simulation superimposed with experimental images at a macroscopic displacement of: (a) 578 µm, (b) 910 µm

Figure 7 . 19 :

 719 Figure 7.19: (a) DVC (blue) and FE (red) meshes plotted over the corresponding cast iron microstructure in (a) isometric view and (b-d) sections normal to X, Y and Z axes

  Figure 7.23(d-e) shows the gray level residual after correcting with the displacement fields measured via DVC and computed with the FE simulation with h2 mesh. When compared with Figure 7.23(c) it is concluded that DVC and FE results are close to the experiment. There still are some very small areas close to the nodule interface that are not properly captured by the measured and simulated fields. On the DVC side, this is an indication that Numerical Validation Framework for Micromechanical Simulations based on Synchrotron 3D Imaging DVC (blue) and FE (red) mesh

Figure 7 . 20 :

 720 Figure 7.20: Section normal to y-direction: (a) DVC displacement field magnitude, (b) FE displacement field magnitude, (c) absolute difference between them and (d) DVC and FE meshes for the third loading step. All reported values are expressed in voxels (1 voxel ↔ 1.1 µm) and correspond to the FE mesh h2

Figure 7 . 21 :Figure 7 . 22 :Figure 7 . 23 :

 721722723 Figure 7.21: (a) Mean values and (b) standard deviations of the difference between the measured simulated displacement fields for the FE mesh h2. Standard deviations are normalized by the corresponding DVC uncertainties

Figure 7 . 24 :

 724 Figure 7.24: Standard deviation for the dimensionless gray level residual fields for all loading steps (FE results obtained with h2 mesh)

Figure 7 . 25 :

 725 Figure 7.25: Standard deviation gray level residuals for all loading steps with three different FE discretizations

Figure 8 . 1 :

 81 Figure 8.1: Schematic view of the sample with zoomed region between the holes showing on the right: DVC (blue) and FE (cyan) meshes plotted over the corresponding cast iron microstructure in isometric view. On the left: surface image with speckle pattern and DIC mesh from test (B)

Figure 8 . 2 :

 82 Figure 8.2: (a) Sample geometry with the scanned region between the pin holes; (b) section of the reconstructed volume with ROI position

Figure 8 . 3 :

 83 Figure 8.3: Mid-thickness section of the reconstructed volume for three different loading steps

Figure 8 . 4 :

 84 Figure 8.4: Image immersion and meshing. (a) Initial laminography 2D section, (b) signed distance function computed thanks to image processing, (c) signed distance function interpolated and reinitialized on the FE mesh [43], (d) conforming FE mesh generated and adapted to interfaces and local maximum curvature, (e) zoom on the FE mesh, (f) comparison between initial laminography 2D section and interfaces in the final FE mesh (in white)

Figure 8 . 5 :

 85 Figure 8.5: Identification of macroscopic material parameters based on test (A) data. (a) Mesh used for specimen scale calculations, (b) comparison between the simulated force/displacement curves and experimental data

Figure 8 . 6 :

 86 Figure 8.6: Inside view of the mesh used in sFE calculations. (a) Full specimen, (b) zoom on the ROI. The three shades of blue represent, from lighter to darker, the homogenized out-of-ROI material, the matrix, and the voids

12 Figure 8 . 7 :

 1287 Figure 8.7: Standard displacement resolutions as functions of the element size expressed in supervoxels for two different scans

Figure 8 . 8 :

 88 Figure 8.8: ROI calculation results using the DVC-FE approach showing the 3D meshed voids and the equivalent plastic strain on sections when: (a) u = 0 (undeformed state), (b) u = 83 µm, (c) u = 192 µm, (d) u = 321 µm

Figure 8 . 9 :

 89 Figure 8.9: ROI (blue line) calculation results using the DVC-FE approach comparing the numerical matrix/void interface (white line) with experimental images for the X midsection when: (a) u = 0 (undeformed state), (b) u = 83 µm, (c) u = 192 µm, (d) u = 321 µm

Figure 8 .Figure 8 . 10 :

 8810 Figure 8.10: Standard deviation for the dimensionless gray level residual fields for all loading steps. For comparison purposes, the dashed line corresponds to the resolution analysis for the so-called "bis" case (seeSubsection 3.3)

  Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Figure 8 . 11 :

 811 Figure 8.11: Absolute gray level differences at the Z midsection after correction with DVC (a), DVC-FE (b), wFE (c) and sFE (d) displacements for the ninth loading step. Note that due to different rigid body motions for each simulation, the Z midsections slightly differ

Figure 8 . 12 :

 812 Figure 8.12: Mid-section normal to z-direction showing absolute difference between: (a) DVC and DVC-FE displacement fields, (b) DVC and wFE displacement fields for the ninth loading step. The black area at the top of sub-figure (b) represents a zone out of the DVC ROI. The color bar range is set according to the data set from sub-figure (a) in order to have more convenient visual comparison. The displacement difference is expressed in supervoxels (1 supervoxel ←→ 2.2 µm

Results 291 Figure 8 . 13 :

 291813 Figure 8.13: Void volume change curves for all the approaches investigated in this paper

  and 8.11(c-d) while the displacement difference between DVC measurements and wFE calculated fields is shown in Figure8.12(b).

Figure 8 . 14 :

 814 Figure 8.14: Standard deviation for the dimensionless gray level residual fields for all loading steps and investigated cases

Figure 8 .Figure 8 .

 88 Figure 8.15: Cumulative distribution of gray level residuals at the last incremental loading step considering to (a) all supervoxels, (c) only supervoxels with gray level residuals higher than 40. In (b), the absolute gray level differences for the sFE method with residuals higher than 40 are shown for the same Z midsection asFigure 8.11(d)

Figure 9 . 1 :

 91 Figure 9.1: Schematic representation of the methods used in the present paper for calibrating material parameters via numerical simulations at the microscale

Figure 9 . 2 :Figure 9 . 3 :

 9293 Figure 9.2: Tomography setup with the representation of the tested sample. The reconstructed region of interest is also shown

Figure 9 . 4 : 2 . 3

 9423 Figure 9.4: Standard displacement uncertainty as a function of the element size for the three displacement components

Figure 9 . 5 :

 95 Figure 9.5: DVC (blue) and FE (red) meshes plotted over the corresponding cast iron microstructure in isometric view and sections normal to X, Y and Z axes. Two regions from the section normal to Z axis are showed enlarged on the right

Figure 9 . 6 :

 96 Figure 9.6: Characterization of three different mesh sizes used in the work with signed distance fields plotted over the corresponding microstructure sections and with the nodule/matrix interface depiction

Figure 9 . 7 :

 97 Figure 9.7: Distribution of mesh quality at the initial state for the three tested meshes

Figure 9 . 8 :

 98 Figure 9.8: Sensitivity fields in the longitudinal direction for the last loading step. Online version: corresponding movie for the 6 analyzed steps. The color bar has a fixed range

  (x) -I t (x + u(x,t, {p})))

  On the Calibration of Elastoplastic Parameters at the Microscale via X-ray Microtomography and Digital Volume Correlation

Figure 9 . 9 :

 99 Figure 9.9: Stress-strain for the cases analyzed in Table9.2. In the legend {p 0 } and perturbed {p 0 } stand for different sets of initial parameters detailed in Table9.2. Initial response stands for σε values obtained with initial set of parameters

Figure 9 . 10 :

 910 Figure 9.10: Initial and final dimensionless residual for the different mesh sizes used in integrated DVC

Figure 9 . 11 :

 911 Figure 9.11: Initial and final dimensionless gray level residuals for different mesh sizes used in the I-DVC procedure and for DVC calculations

Figure 9 . 12 :

 912 Figure 9.12: Gray level residuals for I-DVC (left) and C8-DVC (right) analyses for the last loading step. Online version: corresponding movie for the 6 analyzed steps

Figure 9 . 13 :

 913 Figure 9.13: Microstrucutre section and equivalent strain field (from FE calculations) plotted on the deformed configuration for the last loading step. Online version: corresponding movie for the 6 analyzed steps

Understanding, observation andFigure 10 . 1 :

 101 Figure 10.1: Deformation mechanism found for all four studied aluminum alloys

Figure 10 . 2 :Figure 10 . 3 :

 102103 Figure 10.2: Nanolaminography results: (a) position of the scanned region; (b) section of the full reconstructed volume for different loading steps. The physical size of 1 voxel is 100 nm for all three directions

Figure 10 . 4 :Figure 10 . 5 :

 104105 Figure 10.4: Portion of the reconstructed volume from nanolaminogrpahy data. The voids and intermetallic particles posses satisfactory visual description important for meshing purposes in CimLib tools. The physical size of 1 voxel is 100 nm for all three directions

  Figure A.1(a) shows the grain size (mean value: 60 µm) in the L direction, in the T (mean value: 52 µm ) and S (mean value: 24 µm) directions. These data have been obtained from optical microscopy on etched sections using a mean linear intercept method[45].

  Figure A.1(b) shows typical L-S and T-S sections of the material obtained by tomography. Intermetallic particles are seen in white and the aluminum matrix in gray. The intermetallic content has been determined to be 0.45 vol% and the initial void volume fraction 0.34 % [46]. The spacing of the particles and voids has been estimated using Feret dimensions of Voronoi cells around particles and pores. The spacing is in the order of 25 µm [46], which is an average upper bound.

Figure A. 1 :

 1 Figure A.1: (a) Pseudo 3D cube showing the grain shapes (optical microscopy after etching). (b) 2D sections of reconstructed 3D synchrotron tomography data (after Ref. [45])

Figure A. 2 :

 2 Figure A.2: Engineering stress-strain curve for AA2139 T3 when loading in the L and T directions. The thickness variation is also given (after Ref. [46])

Figure A. 3 :

 3 Figure A.3: Schematic drawing of the laminography setup (after Ref. [42])

Strain and damage interaction 367 3

 367 Strain and damage interaction 3.1 Incremental fieldsThe objective of this work is to investigate the interactions between damage and strain during ductile tearing and, in particular, during the flat-to-slant crack transition. The challenge is to show both values simultaneously. Figure A.5 highlights the way in which this simultaneous assessment of strain and damage is performed. A volume of interest has been extracted (Figure A.5(a)).

Figure A. 5 :

 5 Figure A.5: (a) 3D rendering of the initial microstructure and the ROI. (b) 3D rendering of the effective strain field in the extracted volume when CMOD = 1.5 mm. (c) 2D overlay of strain field over the initial microstructure

  Figure A.5(b) shows the 3D von Mises effective strain field in the ROI for a DVC analysis between the undeformed state (i.e., CMOD = 0.0 mm) and when CMOD = 1.5 mm. In Figure A.5(c) a new way to simultaneously show microstructure and strain fields is given. The strain field is made half transparent and it is laid over the underlying material microstructure as observed by laminography (i.e., particles and porosity). The 2D slices (strain and microstructure) have been obtained at the same location in the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging investigated 3D volume. The underlying microstructure section shows the microstructure before loading as this is the reference state for the performed correlation. This representation has been used for different load steps and incremental correlations in Figure A.6. The underlying reference microstructure (at step n) is shown with an overlay of the incremental von Mises effective strain field for the correlation between steps n and n + 1.

  Figure A.6(b) an overlay of the incremental strain field between CMOD = 1.5 mm and CMOD = 2.0 mm is shown over the underlying microstructure of the load step corresponding to CMOD = 1.5 mm. Out of the three slant strained bands that were active in the first load step the upper one is the most active. The other two bands are less active. Compared to Figure A.6(a) the strained band has shifted downward, but this is only due to a shift of the entire region of interest compared to an arbitrarily chosen reference point.

  Figure A.6(c)). The lower band of the two is again located above the two encircled voids as for the first step (Figure A.6(a)).However, in this step there is another very active band oriented normal to the two previous bands. Even more slant crossing bands are active in the center of the specimen. They form a 45-degree tilted square divided into 4 smaller squares yielding an inclined "waffle-like" pattern that shows characteristic distances between the strained bands of the Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Figure A. 6 :

 6 Figure A.6: Overlay of incremental effective strain fields (half transparent) over the microstructure 1080 µm ahead of the notch root. (a)-(g) 2D sections of laminography data and effective strain fields. The microstructure is shown in state n for incremental correlations between n and n + i. Two reference voids are encircled in white on the microstructure for every load step; (h),(i) microstructure showing damage and the final crack path; (j) load steps corresponding to the shown sections

  Figure A.6(d)). It is still located above the two encircled voids. Normal to this band two other parallel strained bands are active. The strain field is now substantially more localized than in Figure A.6(c). A very similar strain pattern is seen for the next load step from CMOD = 2.375 mm to CMOD = 2.438 mm (Figure A.6

Figure A. 8 :

 8 Figure A.8: (a) 2D section of cumulated fields (effective von Mises strain) with indication of the location for the strain profile. (b) Strain profile for the direct and cumulated strain fields from CMOD = 0.0 mm to CMOD = 2.313 mm

Figure A. 9

 9 Figure A.9(a) shows the strain field obtained for the first load step carried out during the in situ test (i.e., CMOD = 0.5 mm). In this step a maximum effective strain of 0.03 is obtained. However, the three strained bands from the lower left corner of the image to the upper right corner can already be seen. These bands remain visible for a substantially larger load increment (Figure A.6(a)). Damage nucleation or growth is negligible at this step of loading. A similar result was also found for AA2198 T8[43]. The strained bands found herein are less localized.

Figure A. 9 :

 9 Figure A.9: 2D section of 3D laminography and DVC data. Overlay of the smallest possible direct effective strain fields (half transparent) over the microstructure in its initial state

Figure A. 13 :

 13 Figure A.13: 3D void visualization of reconstructed laminography data. (a) Throughthickness section with location of the void 1125 µm from the notch. (b) Overlay of strain field for step CMOD = 2.5 mm to CMOD = 2.563 mm over microstructure at CMOD = 2.5 mm. (c)-(j) Void growth. (k) Indication of the corresponding load step

  Figure A.14: Void volume fraction for one void (see Figure A.13) versus measured von Mises effective strain.

Figure A. 16 :

 16 Figure A.16: 2D plane strain simulation using a slant soft band. (a) 2D mesh. (b) Equivalent plastic strain field when x = 10 %

Figure A. 17 :

 17 Figure A.17: Equivalent strain profiles normal to the band (see Figure A.16(a)) for a 10 % perturbation

Figure A. 18

 18 Figure A.18 shows the change of the strain ratio for different perturbation levels. The stronger the perturbation the higher the initial strain ratio and the earlier the final increase of the strain ratio. In the experimental case the strain ratio is still of the order of 2 for a strain outside the band of ≈ 0.22. A soft perturbation of the order of 10 % reproduces rather well these experimental findings.

Figure A. 18 :

 18 Figure A.18: Strain ratio versus strain outside the band for different perturbation levels (2 % ≤ x ≤ 14 %)

Figure A. 19

 19 Figure A.19: 2D plane strain simulation using randomly distributed soft zones. (a) 2D mesh. (b) Equivalent plastic strain field when the mean level is equal to 0.14 and x = 10 %

1

  Complementarity of the techniques used in the COMISNIDE project with denoted scope of the current work . . . . . . . . . . . . . . . . . . . . . 5 2 Void nucleation as result of (a) particle debonding and (b) particle fracture. SEM figures are extracted from Ref. [47] . . . . . . . . . . . . . . . 8 3 Ductile damage stages of void nucleation, growth and coalescence shown in the example of nodular cast iron sample studied in this work . . . . . . 9 4

  for Al alloys.

	Chronologically, coalescence is influenced by all mentioned factors while the com-
	plexity of the problem in coalescence itself continues to evolve. After the first
	coalescence occurs inside void clusters and/or in zones involving high triaxiality
	and/or large plastic deformation larger flat voids are created. New morphological
	features interact with individual voids or yet other coalesced clusters. The rate of
	these events depends on stress states at given locations, i.e., clustering effects are in
	convolution with the probability of finding a cluster in a region of high-triaxiality.
	Note that the creation of new coalesced features can also perturb existing stress
	fields again adding complexity to the problem.
	Hence, studying ductile failure in real microstructures is still an open subject of
	investigation. In the context of this work, performing in-situ experiments on engi-
	neering materials with sufficient resolutions in the laminography technique should
	give some new answers on numerous questions. Laminography enables one to use
	specimen geometries that induce different stress triaxilities in the scanned region,
	hence different failure mechanisms can be observed and quantified.

Table 1 . 1 :

 11 Number of features per element when the element size is = 32 voxels

	Al alloy	AA2139 T3 AA2139 T8 AA2198 T3 AA2198 T8 AA6061 T6
	features per el.	3.5	2.6	4.7	3.3	1.0
	surface voxels per el.	396	248	223	178	642
	Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

  .2. All the previous cases are analyzed as follows. DVC analyses are run to measure displacement fields. The latter ones are interpolated to account for rigid body motions

	detector microstructure modification geometry of the sample	√	√	√
	Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

  .15); reference scan B wrt. rbm scan B (denoted as "ref. scan B vs. rbm scan B"); reference scan A wrt. rbm scan B (denoted as "crossed").

	0.2 0.25 0.3 0.35 0.4 Mean displacement resolution (voxel)	Element size (voxel) 30 40 ref. scan A vs. rbm scan A 20 ref. scan B vs. rbm scan B crossed	50	60 70
			(a)
	Equivalent strain resolution	10 -3 10 -2 10 -1 10 -4	20 ref. scan A vs. rbm scan A 30 ref. scan B vs. rbm scan B crossed	40	50	60 70
			Element size (voxel)
			(b)
	Figure 1.15: Standard displacement (a) and strain (b) resolutions as functions of the
	element size for recad cases. Two different recad parameters resulted in 2 different sets
	of scans, namely: reference scan A and rbm scan A; reference scan B and rbm scan B.
	Three combinations are analyzed to measure displacement fields: reference scan A vs.
	rbm scan A, reference scan B vs. rbm scan B and reference scan A vs. rbm scan B
			(crossed)

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

  Understanding, observation and quantification of ductile failure mechanisms via 3D imagingThe CT-like specimen used in the present experiment is made of 2198 T3 aluminium alloy in its recrystallized state, solution heat-treated, stretched by between 2-4 %, and naturally aged to obtain the T351 (i.e., so-called T3) condition. It is produced by Constellium C-Tec and represents the latest generation of aluminum-copper-lithium alloys. The composition of the Al-Cu-Li alloy is gathered in Table3.1. Compared with the first two generations of Al-Li alloys, AA2198 has higher copper and lower lithium contents. This property results in an increased strength and toughness of the material that is widely used in the transportation (e.g., aeronautical) industry.

	On deformation and damage micromechanisms in strong work hardening 2198 T3
	aluminium alloy
	2 Experimental Procedure
	2.1 Material

Table 3 .

 3

	Element	Cu	Li	Zn	Mn	Mg	Zr	Si	Ag	Fe
	wt. %	2.9-3.5	0.8-1.1	≤0.35	≤0.5	0.25-0.8	0.04-0.18	≤0.08	0.1-0.5	≤0.01
	at. %	1.23-1.48 3.11-4.26	≤0.14	≤0.25 0.25-0.88 0.01-0.05	≤0.08 0.02-0.06 ≤0.05

1: Chemical composition limits of AA2198 in weight percent (wt. %) and atomic percent (at. %)

[10]

  Numerical Validation Framework for Micromechanical Simulations based on Synchrotron 3D Imaging such for boundary nodes, except those that belong to a free edge. For the other boundary nodes, an edge regularization is considered in the same way as what was recently proposed for 2D situations[67, 65]

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

  Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

	Numerical Validation Framework for Micromechanical Simulations based on Synchrotron
					3D Imaging
	Relative inclusion + void volume [-]	0.1 0.15 0.2 0.25 0.3 0.35	0	0.2 h2:time step/10 Screw displacement [mm] 0.4 0.6 h4 h2 h1 h2:penalization/10	0.8
	Figure 7.16: Change of void volume (wrt. total ROI volume) with applied macroscopic
		displacement for various numerical parameters
	curves in Figure

Table 8 . 1 :

 81 Elasto plastic properties of the ferritic matrix

	E (GPa)	ν	σ y (MPa) K (MPa)	n
	210	0.30	290	382	0.35

Table 8 . 2 :

 82 Elastoplastic properties of nodular cast iron obtained using two inverse analyses on test (A). The first one is only based on load data. Integrated-DIC also uses kine-

			matic data
	Method	E (GPa)	ν	σ y (MPa) K (MPa)	n
	Load data	187	0.28	64	520	0.19
	Integrated-DIC	136	0.28	220	410	0.33

  On the choice of boundary conditions for micromechanical simulations based on 3D imagingBy employing a polar decomposition on the latter, FE and DVC rotations are evaluated while the mean values of kinematic fields represent the corresponding translations. From the initial FE displacement solutions then are subtracted mean FE translations and rotations and added the corresponding mean DVC translations and rotations. Hence, from the wFE/sFE displacements fields, rigid body motions originating from the FE simulations are first eliminated, and then rigid body motions associated with the experiment (i.e., measured with DVC) are applied. This is performed in order to have equal conditions for all presented methods when reducing them to gray level residual images. The resulting errors are shown in Figures 8.14

Table 9 . 1 :

 91 Structural details of the three studied meshes. The computation time refers to sensitivity fields (for 3 investigated material parameters) calculations

	Mesh elements	nodes	time (CPUs used)
	Mesh 1 18, 785	4, 293	10 3 s (3)
	Mesh 2 202, 376 38, 688	2 × 10 4 s (6)
	Mesh 3 675, 752 122, 078	10 5 s (16)

Table 9 . 2 :

 92 Plastic properties of nodular cast iron obtained via I-DVC

	Method	E (GPa) ν σ y (MPa) K (MPa)	n
	Initial parameters [54]	210	0.3	290	382	0.35
	I-DVC, Mesh 1	210	0.3	232	303	0.23
	I-DVC, Mesh 2	210	0.3	235	313	0.19
	I-DVC, Mesh 3	210	0.3	245	330	0.21
	Initial parameters (perturbed)	210	0.3	260	420	0.31
	I-DVC, Mesh 1	210	0.3	206	349	0.22
	I-DVC, Mesh 2	210	0.3	211	364	0.21
	I-DVC, Mesh 3	210	0.3	219	371	0.23

  Cast ironBeside CT samples, other sample geometries can be probed by in-situ nanolaminography experiments. The shear geometry is particulary intriguing option due to the limited knowledge about underlying micromechanisms under low stress triaxiality.Four different flat specimens made of nodular graphite cast iron, namely, 2 hole samples (90 • and 45 • configurations), central hole and tensile sample presented in the introduction have been studied in this work. The damage changes are imaged via in situ 3D laminography and tomography. Digital volume correlation is employed for kinematic measurements.

	2 Cast iron

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Volumetric Correlation

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Uncertainty quantifications

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Conclusions

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Results

Ductile damage in shear sample: 3D in situ measurements

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Experimental analyses

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Mechanical simulations

Acknowledgments

The financial support of the Fédération Francilienne de Mécanique and Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant for COMINSIDE project) is gratefully acknowledged. Constellium C-Tech is acknowledged for aluminum alloys supply. M. Kuna, L. Zybell and M. Horn from IMFD, TU Freiberg, are thanked for cast iron supply and machining. We would like to acknowledge the European Synchrotron Radiation Facility for provision of beamtime at beamline ID15 and ID19 (experiments MA1006, MI1149, ME1366, MA2183).

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Bibliography Acknowledgments

The financial support of the Fédération Francilienne de Mécanique and Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant for COMINSIDE project) is gratefully acknowledged. Constellium C-Tech supplied the material. We acknowledge the European Synchrotron Radiation Facility for provision of beamtime at beamline ID19 (experiment MA1006).

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Acknowledgements

The financial support of the Fédération Francilienne de Mécanique and Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant for COMINSIDE project) is gratefully acknowledged. Constellium C-Tec supplied the studied alloy. The authors also acknowledge the European Synchrotron Radiation Facility for provision of beamtime at beamline ID19 (experiments MI-1149 and MA-1631).

Acknowledgements

The financial support of Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant for COMINSIDE project) is gratefully acknowledged. Constellium C-Tec supplied the studied alloy. The authors also acknowledge the European Synchrotron Radiation Facility for provision of beamtime at beamline ID19 (experiments MI-1149 and MA-1631).

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Acknowledgements

The financial support of Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant for COMINSIDE project) is gratefully acknowledged. We acknowledge the European Synchrotron Radiation Facility for provision of beamtime at beamline ID19 (experiment MA 2183). D. Mohr and C. Roth from ETH, Zürich are thanked for providing sample design geometry as well as for scientific discussions.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Acknowledgements

The financial support of the Fédération Francilienne de Mécanique and Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant for COMINSIDE project) is gratefully acknowledged. We acknowledge the European Synchrotron Radiation Facility for provision of beamtime at beamline ID15 (experiment ME 1366). M. Kuna and L. Zybell from IMFD, TU Freiberg are thanked for materials supply and machining as well as for scientific discussions. F. Gaslain CdM, MINES ParisTech is thanked for SEM analysis.

Acknowledgments

This study was performed within the COMINSIDE project funded by the French Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant). We also acknowledge the European Synchrotron Radiation Facility for provision of beamtime at beamline ID15, experiment MA 1932. It is also a pleasure to acknowledge the support of BPI France ("DICCIT" project), and of the Carnot M.I.N.E.S institute ("CORTEX" project). M.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Acknowledgments

This work was performed within the COMINSIDE project funded by the French Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant). The authors also acknowledge the European Synchrotron Radiation Facility for provision of beamtime at beamline ID15, experiment ME 1366. It is also a pleasure to acknowledge the support of BPI France ("DICCIT" project), and of the Carnot M.I.N.E.S institute ("CORTEX" project). M. Kuna and L. Zybell from IMFD, TU Freiberg are thanked for materials supply and machining as well as for scientific discussions.

Acknowledgements

The financial support of Agence Nationale de la Recherche (ANR-14-CE07-0034-02 COMINSIDE and ANR-10-EQPX-37 MATMECA grants) is gratefully acknowledged. It is also a pleasure to acknowledge the support of BPI France ("DICCIT" project), and Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Acknowledgements

The financial support by Fédération Francilienne de Mécanique and Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant for COMINSIDE project) are gratefully acknowledged. Constellium C-Tec is acknowledged for material supply. The authors would also like to acknowledge ESRF for providing beamtime (experiment MA1006), Dr. M. Mavrogordato and Prof. I. Sinclair for help with the experiment. They particularly thank Prof. J.W. Hutchinson for discussing strain partitioning.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Bibliography

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Chapter 5

Ductile damage in shear sample: 3D in situ measurements

In situ experiment imaged via Synchrotron Laminography is performed on thin sheet sample with an optimized geometry that induces pure and stable shear stress states in the two ligaments. One of the ligaments is scanned and the gray level volumes are registered via Digital Volume Correlation. The strain fields show the expected localization in the shear plane. The Lode parameter fields associated with incremental strain measurements are reported for the analyzed zone. The strains close to failure are measured in the material bulk. In the shear zone damage onset in the form of elongated features is detected from the middle stage of the experiment. Damage is mainly concentrated in the gage section subjected to shear and away from free surfaces, in particular away from the notches.

Reproduced from: A. Buljac, L. Helfen, F. Hild and T. F. Morgeneyer. Ductile damage in shear sample: 3D in situ measurements. To be submitted.

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging Effect of void arrangement on ductile damage mechanisms in nodular cast iron: in situ 3D measurements

The effect of the arrangement of two artificial voids on void growth and coalescence mechanisms is studied in flat specimens via in situ 3D laminography imaging and digital volume correlation for bulk strain measurement. Two configurations of machined holes are studied i) 90 • with respect to the loading direction to mimic internal necking and ii) 45 • to attempt to trigger a void sheeting mechanism between the artificial voids. The machined holes play the role of primary voids, the graphite nodules that of secondary voids. At a lower scale there is a smaller third population of voids. The 45 • void orientation successfully causes a shear band between the primary voids. However, the growth and coalescence mechanisms of voids from graphite nodules are similar for the two configurations. Internal necking is found for nodules that are close to each other and void sheeting for large ferrite ligaments between nodules. Strain measurements in the region between machined holes show necking for both cases as well as volume increase, i.e. positive trace of the strain tensor, confirmed by image analysis. Strain to failure in this inter-void region is measured in the bulk and also locally, between nodules and for different coalescence configurations, where it yields similar values.

Reproduced from: A. Buljac, L. Helfen, F. The application of boundary conditions is presented here for time t, which is not necessarily a time at which a scan was acquired, and for boundary nodes of the FE mesh, which are not necessarily nodes of the DVC mesh. Note that t is an artificial and dimensionless time linked to the loading steps (i.e., the material behavior is rate-independent). The automatic procedure implemented in the FE code reads:

1. find the two loading steps taken at instants t a and t b such that t a ≤ t < t b , with no loading step between t a and t b , 2. read the undeformed DVC mesh and the DVC displacement field at t a , 3. perform linear time-interpolation, On the choice of boundary conditions for micromechanical simulations based on 3D imaging

The present chapter addresses the challenge of conducting Finite Element (FE) micromechanical simulations based on 3D X-ray data, and quantifying errors between simulations and experiments. Standard methods used in the literature to conduct FE simulations at the microscale are often based on multiscale schemes. Relevant mechanical fields computed in an FE simulation at the specimen scale are used as boundary conditions for the micromechanical simulation, where the real microstructure is meshed from 3D X-ray images. These methods hence rely on an identification of material behavior at the macroscale, say, using force measurements and 2D surface images. In this work, both multiscale schemes and DVC-FE approach are applied to new experimental results on a nodular cast iron specimen with machined holes. Ductile fracture due to the nucleation, growth and coalescence of microscopic voids between the machined holes is observed in-situ thanks to synchrotron 3D imaging. The objective of this chapter is to assess the accuracy of boundary conditions for each approach and conclude on the optimal choice. Based on both average and local error measurements, it is shown that void growth is underestimated with multiscale schemes, while predictions are significantly improved with the DVC-FE approach.

Reproduced from: M. Shakoor, A. Buljac, J.

On the Calibration of Elastoplastic Parameters at the Microscale via X-ray Microtomography and Digital Volume Correlation

A combined computational-experimental framework is introduced herein to validate numerical simulations of ductile damage at the microscopic scale. A dogbone sample, which is made of spheroidal graphite cast iron, is loaded in tension and imaged via in situ tomography at micrometer resolution. The region of interest is analyzed via Digital Volume Correlation (DVC) to measure kinematic fields. Finite Element (FE) simulations, which account for the studied material microstructure, are driven by Dirichlet boundary conditions extracted from DVC measurements. The plastic behavior of the ferritic matrix is calibrated via integrated DVC. The three mechanisms of ductile damage are then analyzed in view of the predictions of numerical simulations at the microscopic scale. On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)

Strain and damage interactions during tearing of a ductile Al-alloy with high work hardening are assessed in situ and in 3D combining two recently developed experimental techniques, namely, synchrotron laminography and digital volume correlation. Via simultaneous assessments of 3D strain and damage at a distance of 1-mm ahead of a notch root of a thin Compact Tension-like specimen, it is found that parallel crossing slant strained bands are active from the beginning of loading in a region where the crack will be slanted. These bands have an intermittent activity but are stable in space. Even at late stages of deformation strained bands can stop their activity highlighting the importance of plasticity on the failure process rather than damage softening. One void is followed over the loading history and seen to grow and orient along the slant strained band at very late stages of deformation. Void growth and strain are quantified. Gurson-Tvergaard-Needleman-type simulations using damage nucleation for shear, which is based on the Lode parameter, are performed and capture slant fracture but not the initial strain fields and in particular the experimentally found slant bands. The band formation and strain distribution inside and outside the bands are discussed further using plane strain simulations accounting for plastic material heterogeneity in soft zones.

Reproduced from: T. F. Morgeneyer, T. Taillandier-Thomas, A. Buljac, L. Helfen and F. Hild.

On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3). Journal of the Mechanics and Physics of Solids, 96:550-571, 2016.

On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3) ,d). The crack length is measured in the center of the 3D image/specimen and may, due to crack tunneling, be more accurate (and pessimistic) than traditional surface observations.

3D kinematic measurements

The acquired scans can be used to measure 3D displacements fields by registering one gray level volume in the reference configuration f and a second volume g in the deformed configuration via DVC. Two different types of approaches can be followed. First local approaches consist of subdividing the region of interest (ROI) into a set of independent sub-volumes that are registered [34,51,35,36]. An alternative approach is global so that the warping of the reference volume is analyzed globally and not locally [37]. One of the advantages of the latter approaches is that finite element kinematics can be considered thereby allowing for regularized [52,53] or integrated [54,55] approaches.

In the following C8-DVC [37] will be considered. It consists of discretizing the volume of interest with 8-noded cubes (i.e., C8 elements) whose edge length is denoted by .

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Experimental configuration 365

Trilinear displacement interpolations are used within each considered element. The DVC procedure consists of minimizing the global correlation residuals

with respect to the degrees of freedom associated with the parameterization of the displacement field u(x), where x denotes the location of all the voxels belonging to the ROI. The overall quality of the registration is measured by the global dimensionless residual

where ∆ f = max ROI fmin ROI f denotes the dynamic range of the gray levels of the ROI in the reference configuration. When a series of volumes in the deformed configuration is analyzed, two different strategies can be followed. First, when the strain levels remain lower than 10 %, the multiscale implementation [37] usually allows the whole series to be analyzed with f being the volume of the first scan [38]. Second, when larger strains occur, a first step of the analysis consists of evaluating the incremental displacement fields for which the deformed volume of the previous calculation becomes the reference volume of the next one. Once this first series of calculations is performed, it is possible to cumulate the displacements in a Lagrangian way.

For example, let us denote u 01 (x 0 ) the displacement field of scan 1 with respect to scan 0 expressed in the reference frame of scan 0. An incremental calculation between scan 1 and 2 will yield the displacement field u 12 (x 1 ) expressed in the frame of scan 1. To be cumulated, the incremental displacement field u 12 has to be evaluated for the location of x 0 in the deformed configuration x1 (x 0 ) = x 0 + u 01 (x 0 ). Knowing the measured displacement field u 12 (x 1 ), it can be interpolated to be evaluated at x1 by using the shape functions of the C8 discretization. Let û12 denote this interpolated displacement field. The cumulated displacement field then reads

In general, some of the nodal positions may require an extrapolation, namely, they lie outside the ROI in the reference configuration. In that case, among many options, two are implemented. If the aim is to stop the analysis at this stage, no extrapolation is required, they are considered as NaN (i.e., not a number) and they are not visualized. Direct calculations may also be performed in which this first evaluation of the cumulated displacement field corresponds to the initial guess of a direct calculation from scan 0 to 2 [43]. In

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging

Cumulated fields

In this section some cumulated strain fields are assessed for very large load intervals from the unloaded state. This is only possible via numerically cumulating the incremental displacement fields as explained above. Due to the large strains direct correlations are no longer possible. Accumulation of the fields is only possible up to CMOD = 2.313 mm as after this step the 3D ROI has been shifted away from the notch to avoid damaged areas for correlation purposes. For the cumulated displacement fields the effective strain fields are then derived and shown in -c) the information on the displacement is no longer available. This is due to the fact that the ROI size was kept constant for all correlations and that the material elongated along the loading direction. to the second load step (i.e., CMOD = 0.75 mm). The strain field has a similar pattern as the first one, but the strain magnitude increases up to 0.05. For the subsequent load steps (Figures A.9 The noisy nature of the field is due to a lower signal to noise ratio of the measurements (i.e., the maximum strain is about 0.02). The three strained bands can still be guessed. In the subsequent load step the upper strained bands seem more active than the lower ones. The next three incremental fields appear noisy due to the low maximum strain level (as low as 0.02 for Figure A.10(d)). However, the strain activity is higher in the lower part of the field.

In Figure A.11, incremental strain fields in the same loading interval (CMOD = 0.0 mm to CMOD = 1.5 mm) are also given. However, bigger load increments are assessed in the correlation (n to n + 2) to obtain a higher signal to noise ratio. shows the incremental strain field from load step CMOD = 0.75 mm to CMOD = 1.125 mm. A similar strain pattern as in the initial step is observed. The strain activity is somewhat higher in the upper part of the region. In Figure A.11(c) the largest strain activity is seen in the lower part of the 2D section. This activity may be consistent with Portevin Le Chatelier (PLC)-like behavior spreading the strain in an even manner. However, the characteristic strain pattern always remains, i.e., most of the strain activity takes place in the strained bands and not in the surrounding strain "valleys."

Void shape evolution in strained bands

In previous sections the existence of intermittent strained bands has been shown. In this section the interest lies on the evolution of a pre-existing void (i.e., hydrogen micropore [57]) up to fracture in such a strained band. The void is located 45 µm farther away from the notch than the 2D section shown in the previous figures. It is situated at about mid-thickness of the specimen (see Figures A.13(a-b) [50]). This void was located in the final strained band that led to fracture as illustrated by the overlay of the strain field over the underlying microstructure (

Simulations

To interpret these experimental results, and in particular the slant strained band formation at early loading stages, simulations are carried out. In Ref. [50] Gurson-Tvergaard-Needleman-type simulations consistent with the experimentally observed crack growth have been carried out for the present material and sample geometry. These simulations did not show inclined bands in the strain fields nor slant fracture. A shear-based void nucleation term is used [12], which had yielded slant fracture, to compare the initial measured and predicted strain fields and final fracture.

3D Gurson-Tvergaard-Needleman-type simulations

The modification for void nucleation under shear [12] is briefly recalled. The idea is to use the Lode parameter of the strain rate tensor to detect if the material deforms under a shear strain state and nucleate voids in that case. The Lode parameter for the strain rate tensor assuming incompressibility reads

where ṗi are the eigen values of the plastic strain rate tensor. Using the Lode parameter of the strain rate tensor, the shear-controlled nucleation rate is then expressed as ḟn = A n ṗ (A. It can be seen that the initial simulated strain fields do not resemble the measured ones. The simulated strain fields resemble those of von Mises-type plasticity simulations or classical Gurson-Tvergaard-Needleman-type simulation [43] with a characteristic circular strain pattern. This is expected as the shear nucleation terms only sets in after a critical strain of 0.10. Only at late stages do the strain fields begin to be slanted. In contrast to the experimental findings only one band is active and not several parallel strain bands with intermittent activity. The slant fracture and flat-to-slant crack transition are successfully reproduced in this simulation. The fact that the initial strain fields are not found by the simulation indicates that the plasticity model needs to be modified to enhance the predictions.

Effect of material heterogeneity 4.2.1 Soft slant bands

The measurements of the strain fields ahead of the notch show parallel strained bands that are stable in space but have intermittent activity. The ratio of the strains within the bands and outside is of the order of 1.5-2.0 (see Figure A.8). To interpret further the existence of these bands and in particular the strain distribution 2D simulations are performed to account for material heterogeneity [58]. The simulations are carried out in plane strain conditions as it has been shown that crack propagation occurs under such conditions. Isotropic hardening is described by R(p) = R 0 1 + K 0 p + K 1 1e -k 1 p + K 2 1e -k 2 p (A.9)

where the material parameters R 0 , K 0 , K 1 , k 1 , K 2 , k 2 are given in Table A.1. The weakening is described by multiplying the hardening law with a perturbation factor 0 < x < 1 R(p) = (1x)R 0 1 + K 0 p + K 1 1e -k 1 p + K 2 1e -k 2 p (A.10)

The mesh with the soft zone and the line along which the strain profile is analyzed are given in Figure A.16(a).

Understanding, observation and quantification of ductile failure mechanisms via 3D imaging