Laura

Carla

Léa

Martin Morgan

Marie-Anne Hayfa

Maria Thereza

Jennifer Layde

papa, Pablo Merci Maman

Je tiens dans un premier temps à remercier mes directeurs de thèse Driss Oulahna et Abder Michrafy pour ces trois années de travail. Merci de m'avoir accordé votre confiance pour ce projet et de m'avoir guidé tout du long. Je remercie particulièrement Abder pour sa présence, son accompagnement sans faille et son soutien dans les bons et mauvais moments. Merci aussi d'avoir fait le maximum pour me transmettre à la fois des connaissances scientifiques mais également des conseils de vie ! Un grand merci

Introduction

Le procédé de mise en forme par compression de poudre est présent dans de nombreux secteurs industriels. Ce procédé de fabrication s'applique à la mise en forme de poudres de nature métallique, céramique, à visée pharmaceutique ou cosmétique, agroalimentaire ou encore énergétique.

Durant ce procédé, les particules de poudre en vrac sont comprimées dans la cavité d'une matrice par l'application d'une pression par un poinçon pour former un compact solide de densité relativement élevée qui épouse la forme de la matrice et des poinçons utilisés.

Le procédé de compression étant largement répandu, la bonne qualité des compacts de poudre fabriqués est nécessaire afin de répondre aux exigences imposées par les besoins industriels.

En effet, dans certains cas, l'échantillon comprimé produit peut présenter des défauts tels que des fissures qui peuvent à leur tour entraîner la rupture. La présence de ces défauts conduit alors au rejet des pièces produites. Ceci est dû au fait que les compacts avec ce type de défauts peuvent présenter des propriétés mécaniques altérées, et ainsi conduire à la perte des propriétés d'usage.

Afin de palier à cette potentielle non-conformité des pièces il est important d'investiguer sur les différents phénomènes qui interviennent lors de la compression. L'état final des échantillons peut être affecté par un ou plusieurs des facteurs suivants [START_REF] Lachman | The theory and practice of industrial pharmacy[END_REF], Ahlneck et al., 1989[START_REF] Jonsén | Fracture and stress in powder compacts[END_REF], Klinzing et al., 2019):

• La cinétique de compression (charge appliquée, vitesse de compression et temps de maintient en pression)

• Les propriétés de la poudre ou du mélange de poudres utilisé (compressibilité, frottement, cohésion...)

• La géométrie de l'outillage (matrice et poinçons)

• L'interaction entre la poudre et l'outillage (friction et adhésion)

• L'environnement (température et humidité)

Le procédé se déroule tel que présenté dans la Figure 1.1 où les différentes étapes sont schématisées. Ces étapes successives sont décrites de la manière suivante:

• Le remplissage est l'étape initiale. Au cours de cette phase, la cavité de la matrice est remplie par la poudre, puis nivelée à la surface de la matrice.

• La compression consiste en l'application d'une force mécanique sur un lit de poudre [START_REF] Lachman | The theory and practice of industrial pharmacy[END_REF]) conduisant à la densification de celui-ci. Cette opération s'effectue en abaissant le poinçon supérieur dans la matrice.

• La décharge a lieu pendant le retrait du poinçon supérieur (perte de contact entre le lit de poudre et le poinçon supérieur) lorsque la pression axiale est relâchée.

• Éjection est l'étape pendant laquelle le compact est retiré de la matrice. Pour ce faire, le poinçon inférieur remonte dans la matrice.

• Enfin, il y a la phase de relaxation, pendant laquelle, une fois que le compact a été éjecté, il peut se dilater à nouveau jusqu'à atteindre un état d'équilibre. Cette étape est d'intérêt pour le stockage et/ou le transport des compacts. En effet, certains matériaux sensibles à l'humidité peuvent alors s'endommager [START_REF] Hersen-Delesalle | The effects of relative humidity and super-disintegrant concentrations on the mechanical properties of pharmaceutical compacts[END_REF].

À la décharge ainsi qu'à l'éjection les pressions axiale et radiale sont relâchées. C'est à ce stade que l'énergie emmagasinée par le compact, lors de la compression, est libérée sous forme de retour élastique ( [START_REF] Van Der | Stress relaxation of compacts produced from viscoelastic materials[END_REF]. Selon la nature de la poudre, l'expansion ou le retour élastique est plus ou moins important. Cela peut, dans certains cas, conduire à la rupture du comprimé en raison de la décohésion du contact entre certaines particules [START_REF] Mazel | On the links between elastic constants and effective elastic behavior of pharmaceutical compacts: importance of Poisson's ratio and use of bulk modulus[END_REF].

Tel qu'il a été mentionné précédemment, divers facteurs tel que la géométrie ou les propriétés matérielles peuvent entrainer l'apparition de défauts (Akseli et al., 2013). Ceux ci peuvent être répertoriés en trois catégories: décalottage, laminage et fissuration (schématisées dans la Figure 1.3). Diverses études visent à approfondir les connaissances liées aux phénomènes menant à leur apparition (Train, 1956, Balakrishna et al., 1996, Santos et al., 2010et Garner et al., 2014...). L'origine résiderait, entre autres, en des variations de densité, généralement générée par la géométrie et le frottement aux parois (Lannutti, 1997), et/ou des générations de contraintes de cisaillement (Wu et al., 2008). Si ces contraintes sont supérieures à la contrainte de rupture, celles-ci peuvent mener à la rupture du compact.

La géométrie de l'outillage ayant un fort impact sur la distribution de la densité, les contraintes et l'écoulement pendant la compression, il est important de se focaliser sur les études expérimentales et numériques visant à évaluer son impact sur l'état final du compact.

Dans le cadre des applications pharmaceutiques, une large gamme de géométries complexes existent, notamment pour les compacts sécables, convexes ou présentant une empreinte de dosage. Les compacts convexes révèlent par exemple des régions de plus forte densité qui se génèrent aux extrémités, au voisinage de la matrice, dû au déplacement radial induit par la géométrie des poinçons (I. [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF], Djemai et al., 2006[START_REF] Mallender | The variation of coefficient of friction with temperature and compaction variables for iron powder stearate-lubricated systems[END_REF].

D'autres types de géométries présentant des formes de type rainures ou sillons ont été analysées expérimentalement par I. [START_REF] Sinka | Measurement of density variations in tablets using X-ray computed tomography[END_REF]et Laity, 2014 et numériquement par [START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF]. Ces travaux révèlent de l'impact de ces empreintes sur la distribution de densité. Celles-ci pouvant révéler des gradients de densité ainsi que des zones de fragilités.

Objectif de la thèse

L'objectif de ces travaux est d'étudier l'effet d'une géométrie de poinçons spécifiques présentant des singularités géométriques sur le comportement en compression. Plus particulièrement, sur l'écoulement de la poudre au voisinage des singularités, les distributions de contraintes et de densité au sein du compact dans l'objectif d'évaluer la tendance à la rupture causée par la géométrie. La géométrie de compact étudiée sera de forme parallélépipédique avec des rainures apportées par les poinçons inférieur et supérieur de manière symétrique.

De manière plus spécifique les travaux de cette thèse comprennent les éléments suivants :

• Essais expérimentaux visant à évaluer le comportement de deux types de poudres différentes en compression. Ces poudres diffèrent par leur nature et leurs réponses différentes lorsque celles-ci sont comprimées à l'aide d'outillages de géométries variées.

• Implémentation d'un modèle numérique pour la simulation de la compression afin d'analyser la réponse de la poudre au compactage en fonction du motif rainuré des poinçons à travers l'étude de l'écoulement, des distributions de densité et de contraintes générées au sein du compact.

Prise en compte des dimensions du compact étudié pour une modélisation plus précise et prédictive du comportement mécanique de la poudre.

• Analyse Tomographique à Rayons-X afin d'obtenir expérimentalement la distribution de densité d'un compact rainuré et ainsi, par comparaison avec les résultats de la prédiction numérique, de valider le modèle mis en place.

Contenu des chapitres

Le Chapitre 1 présente l'état de l'art du procédé de compression de poudre en matrice. Les aspects principaux du procédé y sont présentés parmi lesquels figurent les étapes successives du procédé (introduites précédemment), les différents mécanismes intervenant pendant le processus ainsi que les potentiels problèmes rencontrés.

De plus ce Chapitre introduit les méthodes de caractérisation de la poudre en compression dont; l'analyse des cycles de compression, les modèles analytiques de Heckel (Heckel, 1961b) et Kawakita (Kawakita et al., 1971), l'analyse du rebond élastique à la décharge ainsi que l'évaluation de la tenue mécanique.

La détermination de la distribution de la densité dans un compact de poudre est importante pour l'évaluation de la qualité du compact. La considération de la densité macroscopique de l'ensemble du compact suppose une distribution homogène. Ainsi, pour les distributions hétérogènes, la quantité de densité au niveau microstructurel doit être évaluée. Une variété de méthodes pour accéder à la grandeur de la densité au niveau microstructurel ont été énumérées par Lannutti, 1997. Une section de ce Chapitre se focalise alors sur les méthodes expérimentales permettant d'acquérir les distributions de densité dans un compact parmi lesquelles on retrouve la micro-indentation (Kandeil et al., 1977[START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF] et la Tomographie à Rayons-X (Busignies et al., 2006[START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF].

Le procédé de compression permet de produire des compacts de différentes géométries, plus ou moins complexes, en fonction de la finalité de l'échantillon. Les géométries peuvent varier d'un engrenage (Cedergren et al., 2003) à un simple compact cylindrique en fonction de la nature de la poudre et donc, de son domaine d'application industriel. La complexité de la géométrie de l'échantillon de poudre comprimée conduit à des distributions de densité non homogènes qui peuvent influencer les performances attendues du compact. En effet, une grande variation de densité se traduit généralement par une plus grande quantité de contraintes et de déformations stockées dans le compact, qui sont libérées au moment de la décharge et de l'éjection, entraînant des dommages potentiels. Étant donné que la géométrie de l'outillage a un fort impact sur la distribution de la densité, les contraintes et l'écoulement pendant la compression, une section de ce Chapitre vise à se focaliser sur les différents travaux traitant de ce sujet spécifique.

L'étude du comportement de la poudre pendant la compression en matrice peut être examinée expérimentalement par le biais d'une presse instrumentée et en utilisant des méthodes analytiques. Cependant, ces modèles analytiques restent limités. En effet, ces modèles ne prennent pas en compte certains paramètres tels que le frottement à la paroi et la vitesse de compression. Pour fournir une compréhension plus approfondies des propriétés locales de la microstructure, ce Chapitre introduit deux approches de modélisation numérique : la Méthode des Éléments Finis (MEF) (Zienkiewicz et al., 1971, Zienkiewicz et al., 2005) ainsi que la Méthode des Éléments Discrets (MED) (Cundall et al., 1979, Cundall et al., 1992).

De par les études présentes dans la littérature, il a été constaté que dans les compacts de poudre, la distribution de densité est importante pour évaluer l'intégrité mécanique d'un compact. Plusieurs aspects tendent à influencer fortement la distribution de densité, notamment le frottement entre la poudre et la matrice, mais aussi la géométrie de l'outillage.

Cette thèse vise à se focaliser sur l'impact de la géométrie de l'outillage.

Le Chapitre 2 présente brièvement les différentes poudres utilisées dans le cadre de cette étude ainsi que les principaux équipement et outillages employés. Les essais de rupture diamétrale révèlent de nombreuses incertitudes pour les deux poudres d'Arbocel, car leur aspect fibreux ne permet pas d'obtenir des profils de rupture clairs, tels que des fractures en couches (laminage) dans le cas de l 'Arbocel BC 1000 (Figure 3.6).

Des essais de micro-indentation permettant d'évaluer la distribution de la densité d'un compact sont présentés dans ce Chapitre pour le cas de compacts parallélépipédiques à deux rainures de MCC 102. En effet, les aspects fragiles des compacts parallélépipédiques réalisés à partir des deux poudres d'Arbocel n'ont pas permis d'obtenir des résultats exploitables pour ce type d'approche expérimentale.

Les essais d'indentation Brinell réalisés sur les compacts de MCC 102 ne permettent pas, avec notre dispositif expérimental, d'obtenir des résultats précis et concluants mais révèlent tout de même des grandes variations de densité relative dans la région localisée entre deux rainures qui se font face. De plus, la micro indentation permet une mesure de surface sur les côtés des compacts. Cependant, la densité surfacique est fortement influencée par le frottement entre la poudre et l'outillage (matrice et poinçons), les densités évaluées ici sont donc certainement plus élevés qu'au centre du compact, étant donné la localisation des points d'indentation. C'est pourquoi une seconde méthode de caractérisation de la densité par Tomographie à Rayons-X sera proposée dans le Chapitre 6. Cette méthode permet de caractériser la distribution de densité au sein d'un compact de manière non destructive et précise.

Les essais de compression réalisée à l'aide de la matrice parallélépipédique et des poinçons rainurés ou plat présentés précédemment (Figure 2.4) révèlent, de manière qualitative, l'effet de la présence des rainures sur l'intégrité finale des compacts. Des exemples de compacts obtenus suite à la compression dans la matrice parallélépipédique et les poinçons rainurés ou non sont présentés dans les Figures 3.9,3.11 et 3.18.

La présence des fissures s'observe de manière systématique dans la région entre deux rainures qui se font face (cas du poinçon à deux rainures). De ce fait cette zone des compacts semble être dans un état de tension durant la décharge entrainant une perte de cohésion entre les particules menant à l'apparition de fissures. L'apparition de ces défauts dépend également de la nature de la poudre tel que le révèlent les essais de compression réalisés avec la MCC 102, où aucun défaut n'a été constaté quelle que soit la géométrie des poinçons. Les observations faites mènent alors à la conclusion que l'apparition de fissure est une conséquence à la fois de la géométrie des poinçons (présence de rainures) et de la nature de la poudre.

Ces essais justifient le besoin de modéliser ce problème afin d'avoir plus de détails sur le développement des champs de contraintes, déplacement et densité durant la compression et la décharge et ainsi d'évaluer de l'impact de la géométrie des rainures en menant une étude paramétrique (Chapitre 4). Concernant la modélisation, le choix a été fait d'employer la poudre MCC 102 en tant que matériau modèle pour la suite de l'étude. Ce choix a été fait car la poudre MCC 102 peut être caractérisée afin d'obtenir des données d'entrée pour la modélisation.

Malgré le fait que les compacts faits à partir de MCC 102 ne présentent aucun défaut, l'étude du comportement de ce matériau, largement étudié dans le domaine de la compression de poudres pharmaceutiques (I. [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF], Choi et al., 2010, Kadiri et al., 2013...), permet d'analyser la création des variables de champs qui rendent possible une compréhension plus approfondie de l'impact des rainures.

Le Chapitre 4 présente le modèle choisi pour représenter le comportement de la poudre en compression : le modèle de Drucker-Prager Cap, largement utilisé lorsque l'on souhaite simuler le processus de compression de poudres pharmaceutiques par la Méthode des Éléments Finis [START_REF] Michrafy | Modelling the compaction behaviour of powders: application to pharmaceutical powders[END_REF][START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF], L. Han et al., 2008, Sinha et al., 2010a, Kadiri et al., 2013). Ce modèle élastoplastique tend à être représentatif d'un milieu poreux en compression et les principaux paramètres qui le constituent sont les suivant : 𝑑, 𝛽, 𝑅 et 𝑃 𝑏 (cohésion, angle de frottement interne, excentricité du Cap et la fonction de durcissement respectivement). S'ajoutent à cela les paramètres élastiques : 𝐸 et 𝜈, le module d'Young et le coefficient de Poisson. La méthodologie employée pour déterminer les paramètres principaux du modèle ainsi que les résultats obtenus pour la poudre MCC 102 sont détaillés.

La méthodologie d'identification des paramètres proposée dans ces travaux se veut hybride étant donné qu'elle se base à la fois sur des essais de compression en presse instrumentée (matrice cylindrique) et sur des essais de compression effectué à l'aide des outillages parallélépipédiques non instrumentés introduits précédemment au Chapitre 2.

La méthode de calibration standard (telle que présentée par Brewin et al., 2008) est généralement utilisée pour mener des simulations de compression sur des compacts de taille similaire à ceux de la calibration. Cette méthode standard a révélé dans notre cas des effets d'échelle nous menant à la proposition de cette méthode dite hybride. Les paramètres ainsi obtenus pour la poudre de MCC 102 ont été validés par simulation numérique au Chapitre 5.

Le Chapitre 5 se focalise sur la modélisation du problème de compression ainsi que sur les prédictions obtenues par la simulation MEF afin d'évaluer l'impact de la géométrie des poinçons (rainures) sur l'état final du compact. La géométrie des rainures induisant des fortes déformations pouvant mener à la divergence du calcul, l'approche par la méthode ALE (Arbitraire Lagrangien Eulérien) a été utilisée comme outil de remaillage de manière à contrôler les distorsions.

Les premiers résultats présentés dans ce Chapitre concernent la validation des propriétés matérielles. En effet, dans le Chapitre précédent, nous avons proposé une méthode de calibration hybride des paramètres DPC combinant des compressions en matrice instrumentée (matrice cylindrique de 1 𝑐𝑚 3 ) et des essais de compression en matrice parallélépipédique.

La validation a été effectuée en comparant les densités relatives finales ainsi que les cycles de compression (charge et décharge) avec l'utilisation d'une matrice parallélépipédique et de poinçons plats (géométrie des poinçons présentée dans la Figure 2.5 et système initial dans la Figure 5.4a) sans frottement dans les conditions de l'expérience. Les données ainsi validées permettront de simuler les essais de compression utilisant les poinçons rainurés (2 et 5 rainures).

Les simulations MEF menées à la fois avec la géométrie des poinçons à deux et cinq rainures (Figure 5.30) révèlent des fortes variations de densité au voisinage des rainures. Dans les deux cas des régions de plus faible densité se localisent dans les coins inférieurs et les zones de plus forte densité sous les pointes des rainures. Globalement le centre des compacts présente une distribution de densité homogène.

La distribution de densité au voisinage d'une rainure (forte sous la pointe et qui diminue en direction des épaules des rainures) est en accord avec des résultats de la littérature (expérimentalement et numériquement) où le poinçon supérieur utilisé présentait également des non linéarités géométriques de ce type là (I. [START_REF] Sinka | Measurement of density variations in tablets using X-ray computed tomography[END_REF], Laity, 2014[START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF].

La région entre deux rainures qui se font face avait été identifiée, dans le Chapitre 3, comme une zone sous tension lors de la décharge pouvant mener à la dégradation de la pièce tel qu'observé avec les compacts d'Arbocel BC 1000. Si l'on observe la distribution de densité dans cette région spécifique, celle-ci se présente sous forme de gradient (densité élevée au niveau des rainures et plus faible au centre du compact). L'impact de la géométrie ainsi que de la position des rainures a alors été évalué sur ce gradient de densité. Les trois paramètres géométriques de la rainure désignés pour cette étude sont la profondeur (𝐷), la largeur (𝑊 ) et l'angle d'ouverture (𝛼) (Figure 5.15).

Les Figures 5.19,5.21 et 5.22 présentent les gradients de densité obtenus avec différentes profondeurs, largeurs et angles d'ouverture de rainure, chaque paramètre étant évalué indépendamment. Tel qu'il est possible de le constater parmi ces trois paramètres sélectionnés c'est bien la profondeur (Figure 5.22) qui influe le plus sur l'évolution du gradient de densité.

La distance de la rainure par rapport à la matrice influe également sur ce gradient, lorsque la rainure est plus proche du centre du compact, la variation de densité est moins impactée par les conditions limites (frottement). La densité dans la région centrale du gradient n'est alors plus croissante mais tend à rester constante. Ces mêmes tendances ont été observées avec les gradients de densité obtenus suite à la compression avec le compact à cinq rainures. Des suggestions de modifications de la géométrie de la rainure dans le but de diminuer le gradient

Conclusion générale

Cette étude se focalise sur le compactage de poudres en matrice avec la prise en compte d'une géométrie complexe. Plus précisément, cette thèse vise à évaluer de l'influence de la géométrie des poinçons sur les propriétés finales des compacts, dépendant à la fois de la nature de la poudre et de la géométrie. Powder die compaction is the dominating forming process for granular materials that develop strength of compacted materials. The process leeds the densification of a powder in the form of a compact of various sizes and shapes. Compaction is achieved by the application of pressure, by one or more punches, on a given quantity of powder contained in a die. This process is widely used in many industrial sectors including, for example; pharmaceutical industry [START_REF] Lachman | The theory and practice of industrial pharmacy[END_REF], bio-based industry [START_REF] Sokhansanj | Binderless pelletization of biomass[END_REF] and automotive industry (Sheng et al., 2004). Indeed, this process offers some advantages (Güner et al., 2015) :

• Ability to produce compacts made of wide spectrum of materials (metallic, ceramic or organic)

• Control over final shape, size and porosity

• Suitable for mass production

The success of the compaction results in the production of conform compacts, meaning defect-free and with the desired mechanical properties in order to meet the desired end use applications of the compacts. For a pharmaceutical compact, for illustration, compacts must have a certain strength for storage, coating and transport operations, but also have to have a constant drug content and good bioavailability properties. In the case of ceramic and metallic powders, for the production of mechanical parts, a good hardness, temperature resistance and wear properties among others has to be met. Consequently, a sintering step is carried out following the compaction process for these types of powders.

Numerous studies have been undertaken on the compaction process and more specifically on the impact of the different mechanisms and parameters involved during compaction that influence the properties of the compact such as density distribution, presence of defects and mechanical strength. Several potential sources of perturbations in the production of compacts can be cited [START_REF] Lachman | The theory and practice of industrial pharmacy[END_REF], Ahlneck et al., 1989[START_REF] Jonsén | Fracture and stress in powder compacts[END_REF], Klinzing et al., 2019):

• The nature of powders and their behaviour in compression

• Compression kinetics (speed, applied pressure, dwell time)

• Tooling (die and punch geometry, surface roughness and wear of tools)

• Interaction between the powder and the tooling (friction, sticking)

• Environment (temperature and humidity)

The geometry plays an important role in the homogeneous or heterogeneous character of the final density distribution. The solid fraction or density distribution plays a relevant role in the quality control of a compact, as it directly affects its mechanical properties (Djemai et al., 2006, Al-Qureshi et al., 2008).

The production of compacts with a complex geometry can be done either through punches and dies of specific shape or through a succession of compressions with various punches. For instance, in the pharmaceutical field, compacts can have scoring lines to facilitate splitting or dosage imprints (I. [START_REF] Sinka | Measurement of density variations in tablets using X-ray computed tomography[END_REF], Santos et al., 2010[START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF]. For the metalic powders sector, where the fields of application are wide, we can find complex die geometries allowing the production of gear-type compacts (Cedergren et al., 2003) or a compression system that usses successive sequencing of punches (Coube et al., 2000[START_REF] Wang | Stresses State and Mechanical Behaviors of the Green Body During Die Compaction and Ejection Process[END_REF], thus allowing to define different compraction and unloading schedules.

Objective

The objective of this work is to investigate the impact of a complex punch geometry with parallelepiped die of large dimensions on the behaviour in compression in order to evaluate the cracking tendency of a compact. The main axes of this work concern:

• Evaluation of the effect of different punch geometries on the compaction of two types of powders that reveal different compaction behaviours and therfore differ in their cracking tendency. Aiming to highlight the competition between the geometry and the fracture potential of a material.

• Implementation of a numerical model for the simulation of the compaction process in order to analyse the response of the powder to compaction in regard to the grooved pattern of the punches through the study of the flow, density and stress distributions generated within the compact.

Consideration of the dimensions of the compact studied for a more accurate modelling of the mechanical behaviour of the powder.

• X-ray tomography analysis of the internal microstructure of the compacts obtained following compaction, allowing the validation of the numerical model used.

Outline

The following items are intended to briefly describe the contents of the seven chapters that constitute this manuscript.

Chapter 1 reviews the state of the art of the powder compaction process. Emphasis is placed on the defects that can result in a compact following the process. In addition, particular attention is paid to the literature dealing with the impact of the complexity of the compact geometry on its density distribution.

Chapter 2 presents the different tools/experimental devices used within the framework of this work allowing the characterisation of the powder properties. The two different powders types are presented along with their main properties.

Chapter 3 is dedicated to the investigation of the behaviour of the powder in compression.

Analyses of the compaction cycles, evaluation of the elastic recovery coefficient and mechanical strength are presented. The compressibility of powders is evaluated using the Heckel and Kawakita models applied to the powders studied. Optical microscopy is used to observe the direct impact of punch size and geometry on the integrity of the compacts. Finally, the micro indentation method is used to analyse the density distribution of a compact.

Chapter 4 introduces the model of the mechanical behaviour of the powder during compaction. The Drucker Prager Cap elasto-plastic model is chosen, and its parameters depend on the local relative density. The parameters describing the model are defined and the methodology for their determination is described. More precisely, a hybrid calibration method is presented, allowing the model to be adapted to larger compact sizes. The results of the calibration of the material properties are presented and discussed in this chapter.

Chapter 5 describes the boundary problem of die compaction under quasi-static conditions using the weak formulation in the context of finite element method. The numerical simulation were performed using the ALE explicit method. The results of the numerical validation of the material property characterisation using the hybrid method are presented. The main numerical results of the compaction are also presented and discussed for different punch geometries. Attention will be paid to the impact of the punch geometry on the estimated local quantities by means of a parametric study.

Chapter 6 focuses on the X-ray tomographic analyses of powder compacts. In this study, the experimental method of X-ray tomography is used as a tool to determine the global density distribution within a compact. The focus is put on two image analysis procedures allowing the extraction of the desired quantity from the tomographic raw data: the relative density.

Chapter 7 aims to validate the numerically predicted results. The results of the relative density distribution obtained by X-ray Tomography and those predicted numerically are therefore compared. The purpose of the validation being to justify the robustness of the model used.

Finally, this manuscript ends with a general conclusion summarising the main points and results of this study as well as the hey perspectives arising from this work.

The chart presented in Figure 1 illustrates the work undertaken during the thesis, highlighting the dual approach, experimental and numerical, carried out in parallel.

Introduction

Compaction process is based on the principle of densification of a set of particles (loose powder bed) by the application of a pressure to obtain a solid form (compact) with a more or less high density. The main objective of the process (whatever the field of application) in order to meet their conditions of use is to form compacts that meet the mechanical strength requirements, that respect the dimentional tolerances with regards to the geometry of the tools used and, that do not present any defects. For example in the case of pharmaceutical applications, compacts have to present a good mechanical strength and bioavailability.

The understanding of the mechanisms involved during compaction allows to prevent the potential appearance of defects or to predict the density generated during loading, unloading and ejection.

In fact, during the forming process the final quality of the powder compact is affected by the tooling geometry, the compression kinetics (applied load, compression speed and dwell time), the properties of the powder or powder mixture used (compressibility, friction, cohesion...) and finally the interaction between the powder and the tooling (friction and adhesion). This chapter aims to review the state of the art of the powder die compaction process. The process steps as well as the mechanisms involves during compaction will be presented. The principal compaction characterisation methods, including analytical models, elastic recovery and mechanical strength, will be intoduced. A particular focus will be placed on the impact of tooling geometries on the integrity of the final compact. Finally, modelling approaches will also be introduced.

Process steps

The main steps of the powder die compaction process can be summarised as shown in Figure 1.1. These steps are filling, loading, unloading, ejection and finally relaxation. Filling is the initial step in the compaction process. During this phase the cavity of the die is filled by the powder and then levelled on the surface of the die. In the case of an automated system, the powder is poured and levelled using a shoe. The filling of the die cavity by the powder is done by gravity or suction (I. [START_REF] Sinka | The effect of processing parameters on pharmaceutical tablet properties[END_REF].

Loading consists on the application of a mechanical force on a powder bed [START_REF] Lachman | The theory and practice of industrial pharmacy[END_REF] leading to densification. This is done by lowering the upper punch into the die. At the beginning of the compaction, the air is expelled from the die as the particles are rearranged. After a certain pressure where rearrangement is no longer possible, the particles undergo deformations that depend on their mechanical behaviour (K. Kondoh et al., 2000). Fragmentary powders will notably follow a successive process of deformation based on rearrangement, reversible and irreversible deformation and fragmentation. Powders with a more ductile character will mainly deform irreversibly.

The following step is unloading. It takes place during the withdrawal of the upper punch (loss of contact between the powder bed and the upper punch) when the axial pressure is released. It is at this stage that part of the energy stored by the compact during compression is released in the form of springback ( [START_REF] Van Der | Stress relaxation of compacts produced from viscoelastic materials[END_REF]. Depending on the nature of the powder, the expansion or springback is more or less significant. This can also, for example, lead to the rupture of the compact due to decohesion between some particles [START_REF] Mazel | On the links between elastic constants and effective elastic behavior of pharmaceutical compacts: importance of Poisson's ratio and use of bulk modulus[END_REF]. Ejection is the stage during which the compact is removed from the die. This is done by the lower punch moving up into the die. During this stage, the formed compact may continue to expand axially and is subjected to shear stresses due to friction with the die wall that can lead to the compact's degradation (Briscoe et al., 1998, Anuar et al., 2010[START_REF] Uzondu | A mechanistic study on tablet ejection force and its sensitivity to lubrication for pharmaceutical powders[END_REF].

Finally, there is the relaxation stage, during which, once the compact has been ejected, it can expand again in time until an equilibrium state is reached. This stage is of interest for the storage and/or transport of compacts. Indeed, some materials sensitive to humidity can crack [START_REF] Hersen-Delesalle | The effects of relative humidity and super-disintegrant concentrations on the mechanical properties of pharmaceutical compacts[END_REF].

Figure 1.2 -Illustration of a typical compression cycle plot

To illustrate the above, Figure 1.2 shows a typical curve obtained from a die compaction test. This cycle is generally presented in the force-displacement or stress-strain plane. There are two main phases: loading and unloading.

Mechanisms involved during compaction

Powder die compaction consists in partially or totally eliminating the voids existing in the initial powder volume by applying a mechanical stress. To achieve this, various mechanisms operate at the particle level. The particles rearrange, deform elastically, plastically, fracture and consolidate with each other through the formation of bonds between them. These events can occur simultaneously or successively. These events have been studied by Seelig et al., 1946, Hardman et al., 1973and K. Kondoh et al., 2000 among others.

The particle rearrangement takes place at the beginning of the compaction when the air present in the powder bed is expelled from the die (J. [START_REF] Liu | Particle rearrangement during powder compaction[END_REF]. The morphology and the size of the particles [START_REF] Nordström | A particle rearrangement index based on the Kawakita powder compression equation[END_REF] as well as the friction between particles influence this step (He et al., 2017). When the compressive force increases, the particles deform elastically and plastically, thus increasing the points of contact between them. Beyond a certain level of compaction pressure, the particles can fragment, creating finer particles that can in turn undergo deformation (Bateman, 1988).

In addition to the deformation mechanisms occurring at the particle level, other process parameters play a crucial role.

The filling of the die is an important step. The mass and uniformity of the content depend on it (Zakhvatayeva et al., 2018). Indeed, the filling can generate a heterogeneous initial density distribution (Cante Terán et al., 2005) which can influence the final density distribution of the compact (Korachkin et al., 2008). The main parameters that influence the good filling quality are the filling speed, the powder flow and the die geometry (Xie et al., 2006).

The flowability depends on the characteristics of the powder particles such as size and shape as well as on the environmental conditions such as temperature and humidity (I. [START_REF] Sinka | The effect of processing parameters on pharmaceutical tablet properties[END_REF]. For instance, spherical particles with a size of above 20 𝜇𝑚 and a smooth surface exhibit better flowability (Abdullah et al., 1999). [START_REF]The effect of suction during die fill on a rotary tablet press[END_REF] shows that using a suction method during filling increases the flowability in comparison to gravity filling.

Various studies describe the impact of the filling speed and in particular the speed of the filling shoe and the amount of time it remains on the die for its filling. M. Kondoh et al., 1996 evaluates the speed of the filling shoe over a range of 10 to 60 𝑚𝑚/𝑠, revealing that the filling density increases with speed. [START_REF] Larsson | Efficient bonding for dimensional stability[END_REF]Wu et al., 2004 note that for higher filling speeds of up to 200 𝑚𝑚/𝑠 the die does not fill up completely. A lower filling speed is then preferable for a better filling of the die cavity (Rice et al., 1986). Some solutions to filling problems have been brought by the proposal of feed shoes with different features (Zahrah et al., 2001) notably allowing the better filling of dies with complex geometries.

The compaction pressure is decisive to reach the final density of the compact. It is defined in order to ensure that the compact achieves the required specifications in terms of strength and geometrical tolerances (C. [START_REF]The effect of suction during die fill on a rotary tablet press[END_REF]. Both the density and the applied load are closely related, and the reaching of high density values requires high forces to be applied. For certain strain rate sensitive materials, a pre-compression can be applied in order to reinforce the inter-particle bonds and the final strength of the compact [START_REF]Powder compaction: process design and understanding[END_REF]. The use of a dwell time (defined by Çelik, 2016) during compaction also tends to increase the final strength of the compacts (Akande et al., 1998, Anbalagan et al., 2017).

Numerous studies have examined the impact of compaction speed on compressibility and compactibility as well as on the strength of the compact (Roberts et al., 1985, Bateman, 1988[START_REF] Holman | Effect of compression speed on the relationship between normalised solid fraction and mechanical properties of compacts[END_REF], Anthony Armstrong et al., 1989[START_REF] Ishino | Influence of tabletting speed on compactibility and compressibility of two direct compressible powders under high speed compression[END_REF], Drissi-Alami et al., 1993[START_REF] Tye | Evaluation of the effects of tableting speed on the relationships between compaction pressure, tablet tensile strength, and tablet solid fraction[END_REF]. A too high speed of compaction results in the entrapment of gas, the increasing pressure of which can lead to cracking [START_REF] Tanino | Occurrence of capping due to insufficient air escape during tablet compression and a method to prevent it[END_REF], Klinzing et al., 2019). Indeed, a reduction in volume in a very short time prevents the complete evacuation of the gas present in the powder, thus increasing the gas pressure. Pre-compression can be applied to eliminate defects like cracks or laminations that occur during high-speed compaction (Brewin et al., 2008). The impact of velocity also depends on the nature of the powder. Materials which consolidate principally by fragmentation show relatively little velocity dependence (Anthony Armstrong et al., 1989).

Two frictional forces can be distinguished during compaction (Train, 1956[START_REF] Lachman | The theory and practice of industrial pharmacy[END_REF] : interparticle friction which is dominant at low pressures when the particles rearrange themselves and friction between the powder and the die walls. The second friction is the result of the material being pushed against the die and becomes dominant at high pressures.

The friction between the powder and the die walls plays an important role on the density and stress distribution in a compact (Zhou et al., 2002[START_REF] Michrafy | Wall friction in the compaction of pharmaceutical powders: measurement and effect on the density distribution[END_REF], Borowski, 2011). This friction opposes the sliding of the powder with respect to the wall of the die. It is calculated from the axial stress and radial stress transmitted to the die walls (C. [START_REF]The effect of suction during die fill on a rotary tablet press[END_REF].

Experimental studies have shown that friction depends on contact pressure, material, sliding speed and distance, temperature, roughness of the die surfaces and lubrication [START_REF] Mallender | The variation of coefficient of friction with temperature and compaction variables for iron powder stearate-lubricated systems[END_REF][START_REF] Terpstra | Ceramic processing[END_REF][START_REF] Solimanjad | Die wall friction and influence of some process parameters on friction in iron powder compaction[END_REF].

During ejection, the friction between the powder and the die also affects the force required to eject the compact [START_REF] Lachman | The theory and practice of industrial pharmacy[END_REF]. If this force is too high, it can indicate the potential appearance of defects [START_REF] Uzondu | A mechanistic study on tablet ejection force and its sensitivity to lubrication for pharmaceutical powders[END_REF]. This force varies from one powder to another. [START_REF]The physics of tablet compression: IV. Relationship of ejection, and upper and lower punch forces during compressional process: Application of measurements to comparison of tablet lubricants[END_REF] proposes in this context a comparison between different pharmaceutical powders. Sun, 2015 shows that this force tends to increase with high compression speeds for pharmaceutical powders.

Potential problems encountered

During the compaction process, defects can be observed in the compacts as mentioned above.

These defects are still poorly understood and not easily predictable. Three types of defects are generally distinguished; cracking, capping and lamination as shown in Figure 1.3. They can be caused by various sources, such as the compaction parameters (compaction speed (Klinzing et al., 2019), excessive lubrication [START_REF] Lordi | Compaction of pharmaceuticals[END_REF]...), complex tooling geometry that generate density gradients [START_REF] Lachman | The theory and practice of industrial pharmacy[END_REF] or the nature of the powder used due to their mechanical properties (Akseli et al., 2013). can be caused by variations in density within it (Train, 1956, Glass et al., 1997) or process parameters and friction [START_REF] Jonsén | Fracture and stress in powder compacts[END_REF].

These cracks appear during the unloading or ejection stages of the compaction process [START_REF] Inman | Topographic characterization of cellulose bilayered tablets interfaces[END_REF]. When they are located on the surface of the compact, they are sensitive to the stress concentrations that develop during ejection (Garner et al., 2014). In some cases these stresses may lead to micro-crack growth, lamination or even capping.

Background and state of the art

The lamination defects ) appear perpendicular to the direction of compaction (Santos et al., 2010), the compact then separates into several parts horizontally. These defects can be caused by shear stresses or tensile stresses that the compacts can undergo during the unloading and ejection stages (L. H. Han et al., 2008) and excessive springback (Anuar et al., 2010). Indeed, tensile stresses in the centre of the compacts caused by the residual pressure applied by the die on the compacts lead to failure by lamination. In particular, this hypothesis has been validated numerically and experimentally by [START_REF] Mazel | Lamination of biconvex tablets: Numerical and experimental study[END_REF] for compacts of biconvex geometry.

If during compression, air remains trapped in the powder, this can also lead to a lamination failure [START_REF] Mallender | The variation of coefficient of friction with temperature and compaction variables for iron powder stearate-lubricated systems[END_REF]. In practice, there is a play between the punches and the die allowing the air present in the powder bed to escape during the compaction phase, however in certain cases the speed of compaction or the filling of the powder in the die can lead to this type of phenomenon [START_REF] Tanino | Occurrence of capping due to insufficient air escape during tablet compression and a method to prevent it[END_REF], Klinzing et al., 2019).

Capping ) refers to the top of the tablet that detaches from the compact after the compact has left the die (L. H. Han et al., 2008). Balakrishna et al., 1996 evaluate the phenomenon of capping on ceramic powder compacts and describe this mode of failure as a consequence of the radial and axial expansion of the compact during its unloading and ejection stage (Paul et al., 2017). Indeed, when the upper punch is withdrawn during unloading, the compact is subjected to radial stresses and expands axially as a result of Poisson's ratio (Anuar et al., 2010). If this axial stress on the sample is greater than the breaking stress of the powder monolith, a rupture may occur when unloading while the powder compact is still in the die. During ejection, the compact tends towards radial expansion (when the compact starts to exit the die) because the die no longer exerts any counterforce to this expansion [START_REF] Inman | The non-uniformity of microcrystalline cellulose bilayer tablets[END_REF]. Once again, if the radial stress is greater than the breaking limit stress of the compact, we find ourselves in another breaking configuration. In this case, reference is made to circumferential lamination. These failures are generally observed in non-elastic powders (Balakrishna et al., 1996). The combination of the two types of fracture described above constitutes end-capping. The capping phenomenon was studied numerically by Wu et al., 2008 on lactose compacts. A parallel study of both experimental and numerical methods has shown that the shear bands observed during the unloading are responsible for the observed rupture. The numerical tool serves, in this case, as a predictive tool for potential failure. Some powders also cause sticking to the die and punch, known as picking (when a small amount of powder remains on the upper punch) which can lead to loss of integrity of the powder compact [START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF].

Other problems can concern the tools, such as excessive wear of the tools due to the use of numerous compaction cycles (Sandberg et al., 2007, Kramer, 2010). Coatings can be applied to enhance tool life through wear reduction (Brewin et al., 2008).

Powder compaction characterisation

The success, or proper execution, of powder compaction requires a good understanding of the properties of powders and how they behave during compaction. In order to observe the behaviour of a compact, it is possible to examine the evolution of two variables as a function of the compaction load: porosity and tensile strength. The different relationships between these parameters relate to the notions of compressibility and compactibility (Desbois, 2021).

The data generated by the compaction, force and displacement evolution, are used to study the relationship between the applied pressure and the volume of the compact (Çelik, 1992).

Numerous mathematical relationships have been defined between pressure and density, a non-exhaustive list of which has been enumerated by Stanley-Wood, 2013 including those of Heckel and Kawakita widely used for pharmaceutical powders (Çelik, 1992, Denny, 2002, Choi et al., 2010). These highlight the compressibility that is described as the ability of a powder to reduce in volume under pressure (Leuenberger, 1982).

Compactibility is defined as the ability of a powder to densify in such a way as to achieve a certain strength depending on the final use of the tablet. In the case of pharmaceutical compacts, compactibility can be characterised by tensile strength and by hardness tests (Santos et al., 2010).

The in-die springback (from the unloading of the compaction cycle) is used to obtain information on the elasticity of the powder and the expansion measured after ejection also provides information on the elasticity of the compact formed (Çelik, 1992).

Compaction cycle

A common method of assessing the compaction behaviour of materials is the use of compaction force profile as a function of punch displacement. An example of cycle is shown in Figure 1.2.

It allows to evaluate the energy necessary for elastic and plastic deformation (Çelik, 1992). This is dependent on the powder studied (Krycer et al., 1982a) and corresponds to the area under the curve.

The work involved in compaction includes that required to overcome intra-particle friction, friction between the powder and the die and that required to deform the powder (Bateman, 1988).

Springback -relaxation

When the compaction pressure is at its maximum, the compact formed has a certain overall strength. This strength must be high enough to support the stresses produced during unloading and those experienced during ejection. When the upper punch is withdrawn, the compact may expand axially as springback (Krycer et al., 1982b, Bateman, 1988).

The elastic recovery (ER) index is defined as the percentage of axial expansion following ejection in relation to the height of the compact when the load applied is maximal (Jain, Background andstate of the art 1999, Akseli et al., 2013). A detailed visual of the unloading phase is presented in Figure 1.4. It is then possible to visualise in detail the elastic return presented by the compact corresponding to the Δ𝑢 displacement gap. At the end of the unloading, the compact can still expand, we can then distinguish the elastic return in die and after ejection. In both cases the ER index is expressed in the same way.

The following Equation is used to evaluate this springback coefficient (Armstrong et al., 1974, Krycer et al., 1982a) :

𝐸𝑅(%) = ℎ 𝑖 -ℎ 𝑐 ℎ 𝑐 × 100 (1.1)
Where ℎ 𝑐 is the height of the compact in compression (obtained from the compression cycle)

and ℎ 𝑖 the compact height after unloading or measured after ejection.

The springback of compacts depends largely on the elastic properties of the powder. A high springback can lead to capping and lamination (phenomenon detailed in Section 1.4)

of the compact and thus deteriorate the integrity of the sample. The value of springback determined increases with higher compressive forces (Babaei, 2020).

Analytical models

Compressibility is put forward in the analytical models of Heckel (Heckel, 1961b) and Kawakita (Kawakita et al., 1971). Indeed, these describe the evolution of pressure by including quantities such as volume or porosity (relative density) in their models.

However, these equations have their limitations (Rue et al., 1978, Denny, 2002). They do not take into account the effects of the compression speed, the contact time, the dimensions of the die, the shape of the compact and the friction between the powder and the die. Nevertheless, these equations represent a privileged tool for comparing the behaviour of different powders in compression.

Heckel's model

Heckel (Heckel, 1961b andHeckel, 1961a) suggests that the evolution of the density of a powder bed as a function of the applied pressure can be defined by two aspects. The first concerns low pressures where the densification process is linked to the movement of the particles between themselves (namely particle rearrangement). The second relates to the ability of the powder to densify by plastic deformation.

Heckel's equation is the following :

𝑙𝑛 ( 1 1 -𝜌 ) = 𝐾𝑃 + 𝐴 (1.2)
Where, 𝜌 is the density, 𝑃 the pressure applied on the powder bed, 𝐾 a proportionality constant and 𝐴 a constant. This equation allows a qualitative representation of the compaction process in the linear region of the evolution of porosity as a function of pressure.

A typical Heckel plot is shown in Figure 1.5. Background and state of the art Bringing a physical meaning to the parameters of the Heckel Equation (Equation 1.2) allows to describe the compaction behaviour of a specific powder.

As can be seen in Figure 1.5, the constant 𝐴 is the intersection between the extension of the linear part of the curve and the y-axis and the constant 𝐾 is the slope of the line. The 𝐴 constant describes densification by particle movement and rearrangement (Nyström et al., 1993). The inverse of the parameter 𝐾 is recognised as being inversely proportional to the yiel pressure noted 𝑃 𝑦 . This pressure is representative of the ability of the powder to deform plastically under the effect of pressure.

Kawakita's model Kawakita's Equation (Equation 1.3) therefore, introduces a relationship between the applied pressure and the volume variation of the powder bed. Where the coefficient 𝐶 represents the volume variation, 𝑉 0 represents the initial volume of the powder bed and 𝑉 the volume of the powder under pressure. 𝑎 and 𝑏 are characteristic constants of the powder studied.

𝐶 = 𝑉 0 -𝑉 𝑉 0 = 𝑎𝑏𝑃 1 + 𝑏𝑃 (1.3)
This Equation can also be formulated in the following way, allowing to observe a linear variation between the ratio 𝑃 𝐶 and the applied pressure 𝑃 :

𝑃 𝐶 = 1 𝑎𝑏 + 𝑃 𝑎 (1.4)
A standard graphical representation of the result of applying the Kawakita equation to the compression data is shown in Figure 1.6. The parameters 𝑎 and 𝑏 can then be determined from the linear fitting of the obtained curve.

The constant 𝑎 corresponds to the limit value of the relative volume reduction in compression.

This constant can also be associated with the initial porosity of the powder bed. (1 -𝑎) is then equal to the initial relative density. 𝑏 is related to the yield strength (noted 𝑃 𝑘 ) of the particles in the following way 𝑏 = 1 𝑃 𝑘 . This constant can be referred to as the compression coefficient of the analysed powder (Choi et al., 2010, Peciar et al., 2016).

The yield pressure corresponds, in the case of the Kawakita model, to the pressure required to reduce the volume of the powder by 50%.

Mechanical strength

Mechanical strength can be described by crushing strength : hardness (Ridgway et al., 1970, Leuenberger, 1982), tensile strength (Fell et al., 1970, Pitt et al., 1991) and work necessary to cause failure [START_REF] Lachman | The theory and practice of industrial pharmacy[END_REF], Rees et al., 1978). The most commonly used method is to determine the strength of a compact by measuring the force required to break a compact in a diametrical breakage test in order to obtain the radial tensile strength (Çelik, 1992).

The tensile strength is formulated as follows :

𝜎 = 2𝐹 𝜋𝐷𝐻 (1.5)
Where 𝐹 is the rupture force measured, 𝐷 the diameter of a compact and 𝐻 its height.

The characterisation of the compactibility of a powder is done by relating the tensile strength to the density of the compact (Wu et al., 2005b). Several elements can influence the strength of a compact: the bonding between particles (Fell et al., 1971), the size of the particles (Adolfsson et al., 1996), the speed of loading and unloading (Ruegger et al., 2000) as well as the ambient humidity (Ahlneck et al., 1989). For example, smaller particles will tend to form tablets with greater strength [START_REF] Mckenna | Effect of particle size on the compaction mechanism and tensile strength of tablets[END_REF], Kachrimanis et al., 2005).

Experimental density characterisation

The determination of the density distribution within a powder compact is of importance for the evaluation of the quality of the compact. The consideration of the macroscopic density of the whole compact assumes a homogeneous distribution. Thus, for heterogeneous distributions the density quantity at the microstructural level has to be evaluated. A variety of methods for accessing the density magnitude at the microstructural level were listed by Lannutti, 1997. Among the experimental methods used we can cite micro-indentation (Kandeil et al., 1977[START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF], X-ray Tomography (Busignies et al., 2006[START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF], Near Infrared Chemical Imaging (NIR-CI) (Ellison et al., 2008, Ropero et al., 2011) and The Near Infrared Chemical Imaging (NIR-CI) method has been used and presented by Hamad et al., 2007, Ellison et al., 2008and Ropero et al., 2011 in order to propose a fast and non-intrusive method for monitoring density profiles in a compact. This technique allows to generate a chemical image, presenting a spatial distribution of the different chemical elements of the sample on the surface of a compact. By combining chemical and spatial spectroscopic information, this method allows each spatial element (pixel) to be associated with an NIR spectrum. Each NIR spectrum has an absorption rate correlated to a density value.

This section will focus on the presentation of two experimental methods for the determination of the local density of a powder compact: micro-indentation and X-ray Tomography. The general principle of these two methods will be presented as well as examples of their application in the context powder compaction.

Micro indentation

Micro-indentation is a material mechanical characterisation technique used to measure the hardness and / or elasticity properties. The basic principle of this method is based on the penetration of a tip in a material's surface. The tip can have different geometries; pyramidal -Vicker, shperical -Brinell and conical -Rockwell among others.

The general and theoretical aspects of the micro-indentation characterisation methods are presented in more detail in Tabor, 1970. In the context of powder compression, the micro indentation method allows to obtain the density distribution in a compact. This is possible by establishing a relationship between a hardness index (calculated from the indentation present on the sample) and the relative density for a material under study. This method then allows access to a local surface density value.

Micro-indentation has been used in this context for ceramic powder compacts (Kandeil et al., 1977 andKim et al., 2000) but also for pharmaceutical powder compacts (I. [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF]. More specifically, I. [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF] The use of micro-indentation to access the density of a powder compact requires an instrumented apparatus allowing a rigorous positioning of the indentation zone in order to obtain the finest possible distribution such as that presented by I. [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF]. The use of the hardness index, formulated from the dimensions of the indentation left by the indenter on the sample, requires the possibility of making an accurate measurement of the dimensions of the indented mark. [START_REF] Leuenberger | Fundamentals of powder compression. I. The compactibility and compressibility of pharmaceutical powders[END_REF][START_REF] Miguélez-Morán | Characterisation of density distributions in roller-compacted ribbons using microindentation and X-ray micro-computed tomography[END_REF] recommend a measurement of the indented mark area and dimensions with a Scanning Electron Microscope (SEM).

The studies mentioned above also present a calibration method that makes it possible to define a law of evolution between the density and the hardness index. The calibration consists of indentation tests on compacts (generally cylindrical) of known density.

X-ray Tomography

The experimental method of analysis by X-ray Tomography is considered to be non-destructive (does not alter the integrity of the sample). This method allows access to the internal Background and state of the art microstructure of a material by reconstituting the 3D structure [START_REF] Withers | X-ray computed tomography[END_REF] of the object and is based on the ability of X-rays to pass through the material.

Indeed, the principle of tomographic analysis, the experimental device of which is schematized in Figure 1.8, is based on the emission of a beam of x-rays on a sample. The different rays are then more or less absorbed by the sample and these different levels of absorption are collected by the detector.

Figure 1.8 -Schematic representation of an X-ray Tomograph

The tomographic results then make it possible to detect an internal defect in a sample [START_REF] Stock | Recent advances in X-ray microtomography applied to materials[END_REF], Brewin et al., 2008), and also, to access the internal density distribution. Notably in the context of powder compacts this technique is used in the studies of Farber et al., 2003[START_REF] Sinka | Measurement of density variations in tablets using X-ray computed tomography[END_REF], Busignies et al., 2006[START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF]. The projections (radiographs) are the result of the radiation transmitted by the emitter through the material.

Some nuances can be observed as the material, depending on its composition/nature, absorbs the emitted x-rays differently. The grey levels thus obtained on each projection characterise the absorption capacity of each element of matter the volume of which is defined by the size of a voxel. A digital reconstruction algorithm is used to recontruct the 3D representation of the object from the projections. Therefore, the final result of the tomographic analysis is a 3D reconstruction of the internal structure, which can be sliced in a direction of interest and which is defined by a multitude of grey levels (commonly varying between 0 and 255).

The tomographic images (slices) are composed of voxels. Their size depends on the resolution chosen during the analysis. This resolution can go down to the nanometer scale depending on the capacity of the tomograph, in this case it is referred to as nano-tomography. The choice of the resolution must be made according to what is to be analysed or seen. The choice of a small resolution generally means that the field of view (area analysed) is smaller.

An adequate analysis of the grey levels that constitute the image, must then be set up in order to extract the desired information from the tomographic images (density of the material for example). The grey level image can also be evaluated qualitatively to detect, for example, the presence of internal defects (cracks, cavities). Some examples of tomographic results are shown in Figure 1.9 and 1.10 for crack detection and density distribution obtained in the context of powder die compression. In some cases, as a result of the 3D reconstruction of the structure of the analysed object, artefacts can be observed. These can alter the quality of the result and potentially lead to misinterpretation. The observable artefacts can be of different types. For example, Ring artefact, motion artefact or beam hardening which are the most common (Phillips et al., 1997). These artefacts are presented and illustrated in detail by Davis et al., 2006, Schulze et al., 2011[START_REF] Withers | X-ray computed tomography[END_REF] Ring artefacts appear during image reconstruction as an alternation of concentric circles with different contrast. These can be caused by fluctuations in the beam or by defects or miscalibration of the detector elements. However, they are commonly corrected during the reconstruction of the tomographic image or by recalibrating the detector.

Regarding motion artefacts, these are most commonly caused by the movement of the sample during the analysis. They can also be caused by the movement of the source. This artefact blurs the image (presence of duplicate edges for example). The artefact can be corrected by realigning the projections during reconstruction.

The beam hardening artefact can lead to a false interpretation of the material density due to a strong grey level contrast between the centre and the edges. This artefact occurs when beams of different energy are not absorbed in the same way. Indeed, in some cases, the lower energy beams may be absorbed more strongly than the higher energy ones. As a result, the attenuation of a homogeneous sample is not always strictly proportional to the thickness in consequence Beer Lambert's law is no longer valid (Busignies et al., 2006). This artefact can be corrected mathematically during the reconstruction of the tomographic image or by using filters to pre-filter the transmitted beam.

Focus on the impact of tooling geometry

The compaction process allows the production of compacts of various geometries, more or less complex, depending on the end purpose of the sample. The geometries can vary from a gear (Cedergren et al., 2003) to a simple cylindrical compact depending on the nature of the powder and therefore, its industrial field of application. The complexity of the geometry of the compressed powder sample leads to non-homogeneous density distributions which can influence the expected performance of the compact. Indeed, a large variation in density generally results in a greater amount of stored stress and strains in the compact, which are released at the time of unloading and ejection leading to potential damage. As tooling geometry has a strong impact on density distribution, stresses and flow during compression, this section aims to focus on the different works addressing this specific topic.

In the context of the use of powders for pharmaceutical purposes a wide range of different geometries can be found, especially for scored and convex tablets or tablets with a dosage imprint on them. Santos et al., 2010 presents a large panel of the different shapes of compacts that can be found in the pharmaceutical sector. Numerous studies have been carried out, both experimentally and numerically, on how the density and stress distributions develop during the different stages of compression using concave punches to produce convex compacts.

It has been generally observed that the transition from a flat surface compact to a convex surface compact disturbs the density distribution because the geometry influences radial displacement. Indeed, in the case of convex compacts, zones of higher density tend to be created at their extremities, thus in the vicinity of the die (I. [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF], Djemai et al., 2006[START_REF] Mallender | The variation of coefficient of friction with temperature and compaction variables for iron powder stearate-lubricated systems[END_REF]. The works of [START_REF] Sixmith | The effect of punch tip geometry on powder movement during the tableting process[END_REF]Eiliazadeh et al., 2003 deal with the convex aspect of the compact and the direct impact on the axial and radial displacements highlighted using colored layers.

Numerical studies done on the same type of shapes have allowed to provide elements of understanding regarding the generation of density mapping, stresses and displacements induced by the concavity of the punches used (Wu et al., 2008, L. Han et al., 2008and Kadiri et al., 2013). Krok et al., 2014 also proposed a parametric study around the geometry of the upper and lower punches in order to evaluate the impact on stress generation. The presence of high density regions in the vicinity of the die can lead to the generation of shear stresses in these regions (upper corners in contact with the die) which, by propagating towards the centre of the compact, can cause compact rupture (Wu et al., 2008[START_REF] Mazel | Lamination of biconvex tablets: Numerical and experimental study[END_REF].

Concerning the compacts with secability furrows, I. [START_REF] Sinka | Measurement of density variations in tablets using X-ray computed tomography[END_REF] Concerning the compression of metal powder, parts with various geometries can be produced for industrial purposes by means of a sintering step after compaction (Brewin et al., 2008).

Many works concern geometries with shouldered features (Hernández et al., 2012 and[START_REF] Wang | Stresses State and Mechanical Behaviors of the Green Body During Die Compaction and Ejection Process[END_REF]. These samples are thus often made by means of several punches which intervene successively in the stage of compaction (Coube et al., 2000). Many examples of this type of geometry are presented by Brewin et al., 2008, where density disparities in the shoulder regions can result in the appearance of defects such as cracks.

Modelling approaches

As presented earlier in this chapter, studying powder behaviour during die compaction can be examined experimentally through instrumented press and using analytical methods. However, such analytical models are limited to better understand the complex behaviour of powder during compaction. Indeed, these models don't take into account some parameters such as wall friction and compaction speed. To provide a deeper understanding of the local properties of the microstructure, Finite Element Method (Zienkiewicz et al., 1971, Zienkiewicz et al., 2005) and Discrete Element Method (Cundall et al., 1979, Cundall et al., 1992) modelling are more versatiles. These two modelling approaches are used in the context of compaction.

Prediction by simulation provides access to the following knowledge:

• Understanding the behaviour of powder under compression as a consequence of axial loading at different stages of the process

• Determination of stress distributions in a compact, including residual stresses

• Evaluation of the density distribution which can potentially influence the mechanical properties of the compact

• Obtaining information on the origin of the emergence of defects

• Optimisation of the tooling geometry and process

Discrete approach

Initially introduced by Cundall et al., 1979 for soil mechanics, the DEM can be used to simulate powder compaction (Harthong et al., 2009[START_REF] Jain | Mechanical properties of powders for compaction and tableting: an overview[END_REF]. The DEM allows the evaluation of inter-particle (powder grain) interactions in order to predict a solution of a given problem at the macroscopic level. The DEM effectively considers each particle individually and analyses the inter-particle contacts and deformation between particles. In this approach each grain can be considered as a rigid or deformable body.

Furthermore, with the DEM it is possible to take into account the initial geometrical configuration and density distribution of the powder bed before the compression phase (Coube et al., 2005, Wu et al., 2006, Garner et al., 2018). This method specifies the intergranular behaviour based on contact laws (for example Hertz type). This method is of great interest for modelling die filling [START_REF] Tsunazawa | Numerical simulation of industrial die filling using the discrete element method[END_REF], in order to evaluate the flow conditions in different die geometries and for different particle morphologies. Due to the large number of particles to be taken into consideration in order to have a representative system the discrete element method is an expensive method in terms of computational cost.

Continuum approach

The FEM approach has often been used to model die compaction of metallic and ceramic powders (İ. Aydin et al., 1996, I. Aydin et al., 1997, Coube et al., 2000, Toussaint, 2001[START_REF] Wang | Stresses State and Mechanical Behaviors of the Green Body During Die Compaction and Ejection Process[END_REF] and pharmaceutical ones [START_REF] Michrafy | Modelling the compaction behaviour of powders: application to pharmaceutical powders[END_REF], Wu et al., 2005a, C. Sinka, 2007, Wu et al., 2008, L. Han et al., 2008, Kadiri et al., 2018).

In contrast to the discrete approach, this one considers the powder bed as a continuous medium. Indeed, in this case, the focus is placed on the macroscopic behaviour of the powder under compression. The continuum medium is described by macroscopic parameters that are Background and state of the art generally depending on the porosity or relative density characterising local state variables.

No reference is made to the mechanisms occurring at the particle scale.

A modelling based on a continuous approach assumes the fact that powder compression can be considered as a process during which large irreversible deformations take place as the state of the powder reaches high densities. This approach takes into account the geometry of the system, complex loading and contact interactions between the powder and the tooling.

Conclusion

The aim of this first chapter was to present the overall context of the powder die compaction process and the various mechanisms involved in the different stages. Approaches aiming to characterise powder properties such as compressibility, elastic recovery or mechanical strength have been detailed. The focus was placed on the general presentation of two experimental methods for obtaining the density distribution within a powder compact. The impact of the tooling geometry on the stress and density distributions and thus on the final integrity of the compact have also been presented. Finally, the different approaches for modelling the behaviour of powders under compression were introduced (macroscopic and microscopic).

In this work, it was chosen to use a macroscopic approach to simulate the compression of powders.

It has been stated that in powder compacts the density distribution is important in order to assess the mechanical integrity of a compact. Indeed, a non homogeneous density distribution can lead to the generation of stress concentrations that tend to relax during unloading or ejection increasing the risk of failure (Coube et al., 2000). Several aspects tend to strongly influence the density distribution, notably the friction between the powder and the die (Zhou et al., 2002[START_REF] Michrafy | Wall friction in the compaction of pharmaceutical powders: measurement and effect on the density distribution[END_REF], Borowski, 2011), but also the geometry of the tooling (I. [START_REF] Sinka | Measurement of density variations in tablets using X-ray computed tomography[END_REF], L. Han et al., 2011, Laity, 2014).

CHAPTER

Materials and characterisations

Introduction

In order to understand the powders behaviour, several devices or experimental apparatus have been used, in particular to obtain compression cycles, to measure the density of a powder or compact, the mechanical strength of a compact or the shape and size of particles.

The different tools (die and compression punch) used for the purpuse of this study will also be described in terms of shape and size as well as the powders used.

Materials

In this work, two powders were used: the microcrystalline cellulose PH102 (MCC 102) and two grades of Arbocel, BC 1000 and BWW 40. These powders differ in their properties and behaviour. They were chosen in order to seek to highlight the importance of powder properties to resist cracking. Indeed, MCC 102 is known as a powder that develops good cohesion between particles and therefore has a better chance to resist cracking during springback. The Arbocel powder with two different particles sizes tends to present the opposite behaviour.

For compaction tests, in particular for the characterisation of the behaviour of the MCC 102 powder (presented in Chapter 4), a Magnesium Stearate (MgSt) powder was used as a solid lubricant. As a matter of fact, magnesium stearate is regularly used for its lubricating properties due to its ability to reduce friction between particles and between the powder and the tooling [START_REF] Strickland | The Physics of Tablet Compression X: Mechanism of Action and Evaluation of Tablet Lubricants[END_REF]. The main characteristics of this powder (average particle size, true density and bulk density of the powder) are shown in Table 2.1. MCC 102 is widely used in the pharmaceutical field as an excipient. However, the powder is also used in other fields such as food and cosmetic industries.

Microcrystalline cellulose (MCC 102)

Mean particle size (𝜇𝑚) True density (𝑔/𝑐𝑚 3 ) Bulk density (𝑔/𝑐𝑚 The true densities of Arbocel were obtained by helium pycnometry, the bulk densities were provided from the supplier.

Equipement

Tools for powder characterisation, mechanical characterisation and observations are presented in this section. These include particle, surface and internal microstructure visualisation equipment, presses and tooling (punches and dies) as well as equipment for fracture strength measurement.

Powder characterisation

Density

The true density or solid fraction is an intrinsic property of the powder and is equal to the mass divided by the true volume of the powder. This density is usually measured by helium pycnometry [START_REF] Viana | About pycnometric density measurements[END_REF]. Here, true densities were measured using the ACCUPYC II 1340 apparatus from Micrometrics. The relative density (𝜌 𝑟 ) is equal to the ratio between the bulk density (𝜌 𝑏 ) and the true density (𝜌 𝑡 ) and is directly related to the porosity of the powder (𝜖).

𝜌 𝑟 = 𝜌 𝑏 𝜌 𝑡 = 1 -𝜖 (2.1)
The main properties of the compacted powders are depending on the relative density. This property is considered as local characteristic of the microstructure which evolves with the applied pressure.

Size and shape

The morphologies of the particles as well as their size can be visualised and obtained using a Scanning Electron Microscope (SEM) (Hitachi TM3030+ Tabletop microscope).

Particle size distribution (when the powder is suitable) is determined by laser diffraction, using the Masterizer 2000 device (Malvern Instruments, UK).

Mechanical characterisation

Compaction presses

Two different compaction presses were used in this work. The Instron® 5567 and the eccentric Frogerais OA press, France.

Instron uniaxial press :

Upper punch holder is equipped with a force and displacement transducer (LVDT). The compression speed is adjustable and can vary from 0 to 500 mm/min. Different compression systems (dies and punches) allowing to produce compacts of different sizes and shapes can be used, they will be detailed later (in Section 2. 3.3). The press can also be used for diametral and unconfined axial rupture tests.

Frogerais eccentric press :

Fully instrumented press which constitues an advantage for detailled powder characterisation, allowing among others models calibration and die-wall friction characterisation. It is equipped with five sensors allowing access to the relative displacement of the upper and lower punches during loading and unloading steps, the axial pressures messured on the upper punch and transmitted to the lower punch and the radial pressure on the die wall.

The press is equipped with a cylindrical die with a height of 10 mm and a diameter of 11.28 mm, corresponding to a volume of 1 𝑐𝑚 3 .

Diametral rupture strength

Rupture tests are used to characterise diametrical strength of compacted powder using Erweka GmbH's TBH 220 device.

The maximum force measurable by the device is 500 N with an accuracy of ± 1 N. The compacts analysed must have a thickness between 0.1 and 10 mm and a diameter between 2 and 28 mm.

Die and punches

Four main punch and die geometries sets were used in the context of this work. Cylindrical die with flat punches and large parallelepipedic die with three different punch topologies.

Cylindrical die

This tool consists of a cylindrical die with an internal diameter of 11.28 mm and flat punch sets of the same diameter, allowing the manufacture of flat cylindrical compacts.

Parallelepiped die

Within the framework of this work, parallelepipedic die and associated punches were designed and manufactured. These were dimentioned at a reduced scale of a real tool in order to be able to carry out laboratory-scale tests on the Instron press (presented in paragraph 2.3.2).

The die is parallelepipedic in shape with dimensions of 1.56 𝑐𝑚 wide, 5.4 𝑐𝑚 deep and 5 𝑐𝑚 high. The total volume of the die is therefore, equal to 42.12 𝑐𝑚 3 . Several sets of punches were also dimensioned and manufactured. The three different punch geometries are shown in

Micro indentation

Micro-indentation (Brinel test) is commonly used to measure relative density of compacts (Kandeil et al., 1977, Kim et al., 2000[START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF][START_REF] Miguélez-Morán | Characterisation of density distributions in roller-compacted ribbons using microindentation and X-ray micro-computed tomography[END_REF] The experimental set-up for this is shown in Figure 2.8. Two types of Brinell (spherical) indenters of radius 0.5 mm and 1 mm (Figure 2.8b) was mounted on the Instron press (Figure 2.8a)

for the micro-indentation tests using a 10 N load cell.

The relative density is a function of a given hardness index. In the case of a Brinell indenter this index (noted 𝐻 𝑏 ) in calculated taking into account the imprint of the indenter (𝑑), the diameter of the indentor (𝐷) and the force applied (𝐹 ) as in the following Equation : The choice was made to take into account the indentation depth and not the hardness index as formulated in Equation 2.2, as the density related index.

𝐻 𝑏 = 2𝐹 𝜋𝐷(𝐷 - √ 𝐷 2 -𝑑 2 ) (2.2)
The principle of the test consists in applying a constant maximal force through the indentor on the surface of the compact. The maximum displacement obtained is then associated with a density value. Therefore, a great displacement corresponds to a lower density and lower displacement to a higher one for an equal indentation force.

In order to formulate a law of evolution of the maximum displacement (𝑑 𝑚𝑎𝑥 ) measured and 𝜌 𝑟 , a first step of calibration is necessary. Homogeneous cylindrical compacts (mixed with 0.5 % 𝑤/𝑤 MgSt) of known density are indented in order to associate this known relative density of the compact with the maximum displacements measured in a series of points. The results are then mapped to obtain a surface density distribution on the compact surface.

Calibration and indentation results will be presented in Section 3.4.

Surface observations and microstructural characterisations

Regarding the optical observations, a Keyence VH1000E optical microscope was used to examine the state of the lateral surfaces of the powder compacts and to highlight potential surface defects. The optic used allows a magnification of up to ×200.

With the objective to characterise the internal structure of a compact, an X-ray Tomography device was used. The Easytom 130, RX solutions tomograph shown in Figure 2.9 installed in a radiation protection chamber. This equipment makes it possible to obtain a succession of images constituting a 3D visual of the observed sample as detailled in Section 1.5.5.2. These images can have a resolution varying between 254 𝜇𝑚 and 5 𝜇𝑚 (maximum magnification) per voxel.

Conclusion

In this chapter the different experimental devices used to characterise a powder in terms of density and particle morphology were listed. The different equipment used to characterise the behaviour in compression have also been described. These tools, in particular the two presses and the breakage tests mentioned, will be used later, in the context of the calibration of the behaviour model presented in Chapter 4.

Microscopy and tomography observation methods have been introduced, and these will be used to observe the surfaces of the compacts produced in Chapter 3 and to determine the relative density distribution in Chapter 6 respectively.

The micro-indentation method has been presented in this chapter as a tool for the determination of the surface density of a compact. This experimental method will be investigated in more detail in the Chapter 3.

CHAPTER

Powder behaviour during die compaction

Highlights

• Depicting the differences in powder behaviour during compression • Die compaction results using parallelepiped die and grooved punches. Results reveal the presence of localised cracks depending on the geometry of punches and on the type of powder

• An insight into the density distribution of a grooved compact obtained by micro indentation

Introduction

When talking about powder die compaction, the first objective is to understand how a powder behaves as it is formed into a compact. In the context of understanding the behaviour of a specific powder, the experimental results of die compaction will be studied in this Chapter.

Firstly the compaction cycles will be outlined. Then, the models of Heckel and Kawakita will be used as an approach in understanding the behaviour of the studied powders during the compression phase. The evaluation of springback rates and mechanical strength will also be presented for the different powders at different compression rates. The impact of punch and die geometry (grooved parrallelepiped tools) will be studied qualitatively. Finally, the micro indentation method introduced in the previous chapter will be employed to study the density distribution.

Results from cylindrical die compaction tests

This section concerns the study of the behaviour of the three powders during compression MCC 102, Arbocel BC 1000 and Arbocel BWW 40 (introduced previously, Section 2.2).

First the compression cycles will be presented, this will be followed by the application of the Heckel and Kawakita models. Secondly, the dimensional variations will be evaluated over time for the three powders in order to account for the elastic return (expansion) of the compacts following their compression. Finaly, the mechanical strenght of cylindrical compacts of the powders will be evaluated through diametrical fracture tests.

Compression cycles

An example of die compression cycles is shown in Figure 3.1 for the three powders presented above (MCC 102, Arbocel BC 1000 and Arbocel BWW 40). These compression tests were carried out in the 11.28 mm diameter cylindrical die mounted on the Instron press. The initial height of the powder bed was of 20 mm for these tests and the maximum compression force was set at 30 kN. It is noticeable that the three powders present different compressibility rates, with the Arbocel BC 1000 powder presenting a compression rate of 92.18 %. For MCC 102 the compression ratio of 82.65 % is reached at maximum load, while for Arbocel BWW 40 a compression ratio of 85.97 % is reached at 30 kN. Both Arbocel powders show a larger settling phase before densification in contrast to the behaviour of microcrystalline cellulose.

Heckel model analysis

The Heckel model (presented in Section 1.5.3) is useful for analysing the compressibility of a powder, i.e. its ability to reduce its volume when pressure is applied. The Heckel equation From the fittings on their linear phase we obtain the results of the model parameters 𝐾 and 𝐴. The yield pressure 𝑃 𝑦 (inversely proportional to the coefficient 𝐾) was also calculated. These results are detailed in Table 3 The slope 𝐾, informs about the plasticity of the powder, the higher the coefficient the more plastic the powder is in compression. It can, therefore, be said that of the three powders presented, MCC 102 has a more significant plastic behaviour than the Arbocel type powders.

The yield pressure (𝑃 𝑦 ) gives us the pressure at which the powder starts to deform plastically.

In general, the higher the yield pressure, the more fragmentary the powder under study will be considered to be. Thus, it is possible to say that the Arbocel BWW 40 has a more The results should be taken with some caution, as the parameters obtained show a certain sensitivity to the applied compression force and subsequently to the selection of the linear section of the curve for the determination of the parameters 𝐾 and 𝐴 [START_REF] Hooper | A modern approach to the Heckel Equation: The effect of compaction pressure on the yield pressure of ibuprofen and its sodium salt[END_REF].

Kawakita model analysis

The Kawakita model (previously presented in Section 1.5. 3.2. The Kawakita curves for the three powders are shown in Figure 3.3. The constant 𝑎 corresponds to the limit value of the relative volume reduction by compression, and thus informs us on the compressibility of the powder as 𝑎 represents the degree of compression at infinite pressure [START_REF] Nordström | Compression analysis as a tool for technical characterization and classification of pharmaceutical powders[END_REF].

Springback

During unloading and relaxation, a powder compact undergoes elastic recovery. Its value varies according to the powder and the compression pressure applied to the powder bed. This springback index is calculated from the height of the compact in compression ℎ 𝑐 (minimum height) and the height measured at the end of ejection (ℎ 𝑖 ). The height of the compacts can also be examined over time to follow the state of relaxation of the compact until stability is reached. This height, which is compared to the height in compression ℎ 𝑐 , is noted ℎ 𝑖 .

The elastic recovery (ER) index is formulated as ;

𝐸𝑅(%) = ℎ 𝑖 -ℎ 𝑐 ℎ 𝑐 × 100 (3.1)
The results obtained are presented for MCC 102, Arbocel BC 1000 and Arbocel BWW 40 compacts compressed at 5 kN and 10 kN (equivalent to a pressure of 50 and 100 MPa between 8 and 11%. During relaxation, the changes take place mainly during the first 24 hours, after which the variations remain low and the height stabilises (case of MCC 102 and Arbocel BC 1000). However, it can be noted that the Arbocel BWW 40 compacts continue to expand during the relaxation phase before stabilising after 48h.

Diametrical fracture strength

The results of diametrical breakage are presented in this section. These tests were carried out with the Erweka apparatus on flat cylindrical compacts of the three powders (MCC 102, Arbocel BC 1000 and Arbocel BWW 40) with relative densities varying between approximately 0.65 and 0.95. They were obtained with compaction pressures varying from 50 to 300 MPa in order to have consistent compacts of Arbocel powder. Each fracture test was repeated three times.

In the case of flat cylindrical compacts, the fracture stress is calculated with the formula Fell et al., 1970). Where 𝐹 is the detected load that initiates the rupture of the compact, 𝐷 and 𝐻 are the compacts' diameter and height respectively.

𝜎 𝑑 = 2𝐹 𝜋𝐷𝐻 (
The results are presented as a function of the relative density in Figure 3.5. The compacts produced for these tests were compressed at variable loads ranging from 5 to 30 kN (i.e. 50 The fracture stresses obtained (𝜎 𝑑 = 6.96 MPa and 𝜎 𝑑 = 6.24 MPa as shown in Figure 3.7) are consistent with the results presented previously in Figure 3.5 for an equivalent density.

Results and observations from parallelepiped die compaction tests

Parallelepipedic die compression tests were carried out for the three powders presented above, varying the different punch geometries (Figure 2.4). The aim of these tests is to highlight the influence that the geometry of the punches can have on the final state of a compact of a particular powder. The initial height of the powder bed was set at 4 cm and the compression load was set at 20 kN for this example.

Compression cycles

The observations are similar to those for the cylindrical die compression tests presented in Section 3.2.1. Indeed Arbocel BC 1000 is more compressible than the other two powders and a long settling phase is also noticeable in the compression cycles of Arbocel BC 1000 and Arbocel BWW 40 powders.

Optical microscope observations

The compression of MCC 102, Arbocel BC 1000 and Arbocel BWW 40 powders reveal differences in the compression behaviour as previously stated. With the introduction of new tooling geometries other aspects become apparent. Indeed, the geometry of punches with grooves can, depending on the powder used, lead to the appearance of localised cracks in particular regions of the compact. This section aims to qualitatively examine the impact of the punch geometry on the integrity of the compacts through the analysis of optical microscope images.

Flat punches

Two-grooved punches

A first overview of compacts compressed using two-groove punches is presented in Figure 3.11. As a reminder, the detailed geometry of the punches was presented earlier in Chapter 2 in Figure 2.6a. The compacts were then produced at different pressures ranging from 10

Concluding remarks

In the case of compressions in a parallelepiped die, the presence of crack-type defects was observed in the case of compacts made with grooved punches (more particularly the twogrooved ones). This observation is systematic in the case of the Arbocel BC 1000 (Figure 3.13). In this context, the cracks are always located between two grooves that face each other, which makes this area of the compact a region of interest because it is prone to cracking.

The tests carried out with the five-grooved punches revealed weak powder flow between successive grooves for both types of Arbocel powder. The crack observed on one of the Arbocel compacts (Figure 3.22) does not allow us to conclude what the impact of the five grooves or of the loading conditions on the appearance of cracks is.

Micro indentation results

The results of micro indentations for local relative density mapping are presented in this section. The brittle aspects of the parallelepipedic compacts made from the two Arbocel powders did not allow to obtain exploitable results for this type of experimental approach.

The results of this experimental method are, therefore, presented for an example of a MCC 102 powder parallelepipedic compact with two grooves compressed at 20 kN. As already mentioned in the presentation of the experimental device, it allows to estimate the relative surface density of a compact through calibration. We propose here to link the relative density to a displacement quantity. The aim is to obtain a law of evolution of the measured depth of intent as a function of the relative density. A large displacement corresponds to a less densified area, while a smaller displacement means that the area under study is more dense.

The indentation tests presented were carried out with a Brinell indenter (Figure 2.8b) of 0.5 mm radius. The applied compression force was set at 9.5 N (load cell of 10 N) at a speed of 0.005 𝑚𝑚/𝑠.

Calibration -on MCC 102 powder

The The micro-indentation results allow, from the maximum displacements obtained and the calibration curve presented previously (𝜌 𝑟 = 0.3393𝑑 𝑚𝑎𝑥 -0.512 ), to represent an evolution of

Results on a two-grooved compact

Discussion

The experimental method of Brinell micro-indentation was presented and applied to the case of grooved MCC 102 compacts. This method gives a first idea of the density distribution obtained following a compression in a parallelepipedic die with grooved punches. The lower part of the compact is slightly less dense (due to the compaction direction), the top and the centre seem to have similar density levels (relatively homogeneous region). On the contrary, in the regions of the compact that are located between two grooves that face each other, more disparity is observed, as can be seen in Figure 3.29 were high density is estimated under the upper groove tip.

The results obtained reveal some drawbacks of the experimental setup. For more precision in the measurement and therefore, a better repeatability, a positioning system is necessary.

In our case, the sample was placed manually under the indenter, which does not allow for accurate positioning of the tested piece due to lack of visibility. In addition, to obtain a more accurate density distribution, many more measurement points are required. The manual positioning of the sample does not allow to make measurement points too close to each other at the risk that they partly overlap. The micro indentation allows a surface measurement on the sides of the compacts. However, the density distribution is strongly influenced by the friction between the powder and the tooling (die and punches), so the points measured here are certainly higher than in the centre of the compact, given that we are measuring on the boundaries. Indentation measurements would be more relevant at the centre of the compact through a cutting operation, however, this would potentialy deteriorate it.

Conclusion

In this chapter the Heckel and Kawakita approaches for the analysis of the behaviour of powders in compression have been applied to the three powders presented (MCC 102, Arbocel BC 1000 and Arbocel BWW 40). The Heckel analysis showed, among other things, that the Arbocel powders (BC 1000 and BWW 40) have higher yield pressures and therefore, a more fragmentary character.

The springback of the compacts was estimated on flat cylindrical compacts under various pressures. The Arbocel BC 1000 has a higher springback index (10.87%) compared to 5.8%

for an MCC 102 compact compressed under the same experimental conditions. Overall, for the three powders studied, most of the expansion is observed at unloading and in the first 24 hours, after which the variations are null or remain low.

The diametrical fracture tests reveal many uncertainties for the two Arbocel powders, as their fibrous shown in the SEM images, does not allow clear fracture profiles to be obtained, such as layered fractures as was noted in the case of Arbocel BC 1000.

An approach to evaluate the density distribution within a compact is proposed and presented for the case of MCC 102. These micro indentation tests do not allow, with our experimental device, to obtain precise and conclusive results. Moreover, this technique only allows a surface density measurement. Therefore, a second method of density characterisation by X-ray Tomography will be proposed in Chapter 6. This method allows to characterize the density distribution within a compact in a non-destructive and precise way.

Finally, the observations of parallelepipedic compacts allowed us to assess the impact of the presence of grooves on the integrity of the compacts in a qualitative manner. The observations led to a conclusion aiming at justifying the appearance of cracks by the geometry of the punches. The presence of cracks is systematically located between two grooves that face each other (case of the two-grooved punch). This suggests that this region may be in a state of stress upon unloading, causing decohesion of the particles and consequently the appearance of cracks. The emergence of cracks also seems to depend strongly on the behaviour of the powder.

Indeed, the MCC 102 compacts did not show any defects whereas the Arbocel BC 1000 compacts show this cracking phenomenon (as seen in Figure 3.30). At this stage it is possible to say that the appearance of cracks is the consequence of the competition between the behaviour of the powder (stress bonding capacity bteween particles of the powder at the end of loading stage) and the presence of grooves on the surfaces of the punches and the tension resulting from the density gradient in the unloading step.

Based on these findings it is therefore important, to understand, through numerical modelling, the development of stress, displacement, density fields etc. during compression with grooved punches and thus to evaluate the influence of the punch relief (groove) on the compacts. To address this modelling aspect, allowing a better understanding of the compression process, the MCC 102 powder was chosen as the model material for the rest of the study. This choice was made because the MCC 102 powder can be characterised to obtain input data for the modelling.

Despite the fact that this powder does not allow for the identification of defects, the study of its behaviour through the analysis of the creation of value fields will allow an in-depth understanding of the impact of the grooves, main objective of this work.

CHAPTER

Introduction

Modelling of powder behaviour is necessary to understand its behaviour during compression.

In this study the powder is considered as a continuous media (macroscopic description).

This means that the model does not refer to the mechanisms that are likely to occur at the particle scale. The models used to represent the behaviour of powders in compression are, in general, elastic-plastic models composed of a yield criterion, a hardening function and a flow rule. These models are generally presented in the plane of hydrostatic pressures and Von Mises stresses.

The most commonly used model in the literature for the modelling of powder compression is the elasto-plastic Drucker Prager Cap (DPC) model, which is widely used when using the Finite Element Method (FEM). This approach has been largely used in the case of metallic (Coube et al., 2000, Toussaint, 2001), ceramic (İ. Aydin et al., 1996, I. Aydin et al., 1997), cosmetic (Diarra et al., 2012) and pharmaceutical powders [START_REF] Michrafy | Modelling the compaction behaviour of powders: application to pharmaceutical powders[END_REF][START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF], L. Han et al., 2008, Sinha et al., 2010a, Kadiri et al., 2013).

The DPC model is defined by a yield surface and a cap surface which evolve with density as the powder densifies (as a result of compression). When the stress state remains within the yield surface the powder is elastically deformed, whereas when the stress state reaches the yield surface it deforms plastically . When the state of stress is on the Cap surface the density increases, on the contrary if it is on the shear surface it decreases, this is known as dilation. This expansion leads to a softening which can lead, depending on the behaviour of the powder, to strain localisation and the appearance of cracks. This chapter presents a calibration method as well as the calibration results for MCC 102 powder. More precisely, a new hybrid calibration method is proposed as the powder compacts studied were made in a parallelepipedic non insturmented die of large dimentions as presented previously (Section 2.3.3). Usualy die compaction simulations are carried out on similar size speciment to the ones used for model calibration. In our case the standard calibration on a cylindrical compact presents scale effects. Hence the standard calibration method, as detaild by Brewin et al., 2008, was used in combination with non instrumented parallelepiped die compression tests. This constitutes what we will refer to as the hybrid calibration method.

Description of the Drucker Prager Cap

Model

The Drucker Prager Cap model aims to represent an elastic -plastic behaviour of powder under compression. As it can be seen in Figure 4.1 the behaviour of powder material is described in terms of hydrostatic pressure 𝑝 (defined in Equation 4.1) and of equivalent Von Mises stess 𝑞 (defined by Equation 4.2).

𝑝 = 1 3 𝑡𝑟𝑎𝑐𝑒(𝜎) (4.1) 𝑞 = √ 3 2 (𝑆 : 𝑆) (4.2)
where 𝑆 is the deviatoric stress defined as 𝑆 = 𝜎 -𝑝𝐼 with 𝜎 and 𝐼 the stress tensor and the identity matrix respectively.

⎯ ⎸ ⎸ ⎷ (𝑝 -𝑃 𝑎 ) 2 + [ 𝑅𝑞 1 + 𝛼 -𝛼 𝑐𝑜𝑠(𝛽) ] 2 -𝑅(𝑑 + 𝑃 𝑎 𝑡𝑎𝑛(𝛽)) = 0 (4.4)
The evolution parameter 𝑃 𝑎 , representing the hydrostatic pressure of the intersection between the Cap and the shear surfaces is defined by the following equation :

𝑃 𝑎 = 𝑃 𝑏 -𝑅𝑑 1 + 𝑅𝑡𝑎𝑛(𝛽) (4.5)
• A transition surface 𝐹 𝑡 links the Cap surface 𝐹 𝑐 and the shear failure surface 𝐹 𝑠 . This surface aims to ensure a smooth transition facilitating numerical computations. The transition surface 𝐹 𝑡 is defined as :

√ (𝑝 -𝑃 𝑎 ) 2 + [ 𝑞 - ( 1 - 𝛼 𝑐𝑜𝑠(𝛽) ) (𝑑 + 𝑃 𝑎 𝑡𝑎𝑛(𝛽)) ] 2 -𝛼(𝑑 + 𝑃 𝑎 𝑡𝑎𝑛(𝛽)) = 0 (4.6)
Along the loading path, as the stress state meets the Equation's 4.4 condition the material densifies ((𝑝,𝑞) on the Cap surface) and the Cap can extend. During unloading, the stress state reaches the shear failure surface.

Plastic flow

The flow criterion presents the evolution of the yield surface during densification. Flow potential 𝐺 𝑠 for the shear surface and 𝐺 𝑐 for the Cap surface are detailed in Equations 4.7 and 4.8. The plastic strain increment (highlighted in Figure 4.2) is defined by :

𝐺 𝑠 = ⎯ ⎸ ⎸ ⎷ ((𝑝 -𝑃 𝑎 ) tan(𝛽)) 2 + [ 𝑞 1 + 𝛼 -𝛼 𝑐𝑜𝑠(𝛽) ] 2 (4.7) 𝐺 𝑐 = ⎯ ⎸ ⎸ ⎷ (𝑝 -𝑃 𝑎 ) 2 + [ 𝑅𝑞 1 + 𝛼 -𝛼 𝑐𝑜𝑠(𝛽) ] 2 (4.8)
𝛿𝜖 𝑝𝑙 = 𝛿𝜆 ( 𝜕𝐺(𝑝, 𝑞) 𝜕𝑝 , 𝜕𝐺(𝑝, 𝑞) 𝜕𝑞 ) (4.9)
Where the flow potential function 𝐺 is equal to 𝐺 𝑠 or 𝐺 𝑐 and 𝛿𝜆 is a multiplier (positive scalar). In our case, it can be seen that 𝐹 𝑐 = 𝐺 𝑐 .

In this context of powder compression the volume deformation increases with the load and consequently the progression of the Cap. The evolution of the Cap surface is defined by the hardening function 𝑃 𝑏 .

Calibration method

The calibration of the DPC model for a powder material requires a great number of experi- Numerical simulations are often performed under the same conditions as the compaction experiment, which ignores the effect of the size and shape of the compact, although the accuracy of the input data to a numerical simulation is the guarantee of good predictions.

ments
In this work, we had to deal with a change of size and shape (the instrumented cylindrical die has a cross-section of 1 𝑐𝑚 2 while the uninstrumented parallelepipedic die has a cross-section of 8.4 𝑐𝑚 2 ). This change of scale showed that the material data, determined using the cylindrical die, were not valid for the parallelepiped compact, in particular with regard to the hardening function, which has the strongest impact on the compression results. In order to take this scaling effect into account, a hybrid material parameter calibration procedure is proposed.

The following section presents a hybrid procedure for the model caracterisation combining cylindrical die compaction with intrumented die, failure tests on cylindrical compacts and compaction test in the parallelepiped die with flat punches. The combination of the data obtained from these experiments allow us to characterise the powder behaviour predictively during compression inside a non intrumented parallelepiped die.

The model parameters (plastic, elastic as well as wall friction) and their corresponding experimental procedures are summarized in Table 4.1.

Cap surface -𝑃

𝜎 𝑥 = 𝜎 𝑦 = 𝛾𝜎 𝑧 (4.12)
This coefficient 𝛾, defined as the ratio between horizontal (radial) stress and vertical (axial) stress is considered independent of the geometry of the die wall.

In this work, as the parallelepiped die is not instrumented, the coefficients 𝛾 and 𝑎 (density dependent) were calculated from the axial and radial stress of the loading and unloading curve resulting from compression experiments in a cylidrical instrumented die (as firslty mentionned in Table 4.1).

From the axial pressure (𝜎 𝑧 ) resulting from the compression using the parallelepiped die and the 𝛾 and 𝑎 parameters the radial (horizontal) stress for the parallelepiped die can be expressed as Equation 4.13 during loading and Equation 4.14 during the unloading stage.

𝜎 𝑥 =

Cap surface parameters

The evolution parameter 𝑃 𝑎 , the Cap exentricity 𝑅 and the hydrostatic yield pressure 𝑃 𝑏 are calculated from the folowing equations.

𝑃 𝑎 = -3𝑞 -4𝑑𝑡𝑎𝑛(𝛽) + √ 9𝑞 2 + 24𝑑𝑞𝑡𝑎𝑛(𝛽) + 24𝑝𝑞𝑡𝑎𝑛 2 (𝛽) + 16𝑞 2 𝑡𝑎𝑛 2 (𝛽) 4𝑡𝑎𝑛 2 (𝛽) (4.19) 𝑅 = √ 2(𝑝 -𝑃 𝑎 ) 3𝑞 (4.20) 𝑃 𝑏 = 𝑃 𝑎 + 𝑅(𝑑 + 𝑃 𝑎 𝑡𝑎𝑛(𝛽)) (4.21)
The Cap position, defined by 𝑃 𝑏 as expressed in Equation 4.21, also depends on the plastic volumetric strain 𝜖 𝑝𝑙 𝑣𝑜𝑙 . This quantity can be expressed as a function of material density as in Equation 4.22.

𝜖 𝑝𝑙 𝑣𝑜𝑙 = 𝑙𝑛 ( 𝜌 0 𝜌 𝑟 ) (4.22)
where 𝜌 0 is the initial relative density of the loose powder (with no plastic volumetric strain)

and 𝜌 𝑟 the current relative density of the powder durring compression.

Shear surface -𝑑 and 𝛽

Shear surface parameters 𝑑 and 𝛽 can be detemined by at least two failure tests. Commonly the diametral and non confined uniaxial failure tests.

Diametral rupture test

In the case of the diametral rupture test (as presented in Figure 4.4a) the rupture force 𝐹 𝑑 is measured allowing to compute the resulting rupture stress 𝜎 𝑑 calculated from:

𝜎 𝑑 = 2𝐹 𝑑 𝜋𝐷𝐻 (4.23)
Where 𝐷 is the diameter of the compact and 𝐻 it's height.

The corresponding stress state (𝑝 𝑑 ,𝑞 𝑑 ) is defined as :

(𝑝 𝑑 , 𝑞 𝑑 ) = ( 2 3 𝜎 𝑑 , √ 13𝜎 𝑑 ) (4.24)
This rupture test was done with the Erweka THB 220 equipement.

Axial rupture test

For the axial rupture test (as illustrated in Figure 4.4b) the rupture force 𝐹 𝑐 measured allows to calculate the coresponding axial stress 𝜎 𝑐 using Equation 4.25.

𝜎 𝑐 = 4𝐹 𝑐 𝜋𝐷 2 (4.25)
The corresponding stress state (𝑝 𝑐 ,𝑞 𝑐 ) is given by :

(𝑝 𝑐 , 𝑞 𝑐 ) = ( 1 3 𝜎 𝑐 , 𝜎 𝑐 ) (4.26)
These rupture tests were made on compacts with sizes respectig a ratio between the height and the diamater of approximately 2. The tests were done using the Instron 5567 press.

Shear surface parameters

From the radial stress 𝜎 𝑑 and axial stress 𝜎 𝑐 the cohesion (𝑑) and internal friction angle (𝛽) can be calculated using Equations 4.27 and 4.28 respectively.

𝑑 = 𝜎 𝑐 𝜎 𝑑 𝜎 𝑐 + 2𝜎 𝑑 ( √ 13 -2 ) (4.27) 𝛽 = 𝑎𝑡𝑎𝑛 -1 [ 2(𝜎 𝑐 + 𝑑) 𝜎 𝑐
] (4.28)

Elastic properties -𝐸 and 𝜈

The elastic behaviour of the material is considered linear and is density dependent. This assumption is satisfactory when density and pressure levels are sufficiently high.

Elastic properties of the powder were determined from the compression cycle curves (more specifically during the unloading phase) of compacts compressed in the parallelepiped die for 𝐸 and from the cylindrical intrumented die for 𝜈 . The unloading slope 𝑑𝜎𝑧 𝑑𝜖𝑧 is related to the Bulk modulus (𝐾) and the Shear modulus (𝐺) following Equation 4.29.

𝑑𝜎 𝑧 𝑑𝜖 𝑧 = 𝐾 + 4 3 𝐺 (4.29)
In the case of the Drucker Prager Cap model, Bulk and Shear modulus can be related to elasticity variables as:

𝐾 = 𝐸 3(1 -2𝜈) (4.30) 𝐺 = 𝐸 2(1 + 𝜈) (4.31)
By combining previous equations (4.29, 4.30 and 4.31), the Young modulus can be calculated from:

𝑑𝜎 𝑧 𝑑𝜖 𝑧 = 𝐸(1 -𝜈) (1 + 𝜈)(1 -2𝜈) (4.32)
Poisson's ratio is calculated from the slope (𝑎) of the axial -radial stress unloading curve as previously presented in Figure 4.3.

The axial -radial stress increment is also related to the Bulk and Shear modulus :

𝑑𝜎 𝑥 𝑑𝜎 𝑧 = 3𝐾 -2𝐺 3𝐾 + 4𝐺 (4.33)
Combining Equations 4.30 and 4.31 into Equation 4.33, Poisson's ratio is calculated from :

𝑑𝜎 𝑥 𝑑𝜎 𝑧 = 𝑎 = 𝜈 1 -𝜈 (4.34)

Wall friction -𝜇

Friction between die walls and powder leads to the heterogenity of density distribution in the compact (creation of density gradients, [START_REF]Powder compaction: process design and understanding[END_REF]. This heterogeneity may lead to weaken mechanical properties of the sample. The determination of the friction coefficient between a specific powder and die walls requiers an instrumented die. As the parallelepiped die was not instrumented, these experiments were performed on the cylindrical intrumented die.

Wall friction coefficient was calculated as being the ratio between mean tangencial and normal force of die wall from the equation proposed by Cedergren et al., 2002 :

𝜇 = (𝜎 𝑧-𝑢𝑝𝑝𝑒𝑟 -𝜎 𝑧-𝑙𝑜𝑤𝑒𝑟 )𝜋𝑅 2 2𝜋𝑅𝐻𝜎 𝑟 (4.35)
Friction is, as presented, a function of the radial stress (𝜎 𝑟 ), axial pressures measured on the upper and lower punch (𝜎 𝑧-𝑢𝑝𝑝𝑒𝑟 and 𝜎 𝑧-𝑙𝑜𝑤𝑒𝑟 respectively), 𝑅 the radius of the die and 𝐻 the compact height.

Results for MCC powder

In this work DPC model parameters, friction and elastic properties were callibrated for MCC vivapur 102 powder using the hybrid procedure presented above.

The DPC parameters were dertemined with lubricated compacts in order to limit wall friction effects.

Axial and radial stress for the calculation of 𝛾 and

𝑎 coefficients

Radial and axial stress (𝜎 𝑟 and 𝜎 𝑧 respectively) as expressed in Equations 4.10 and 4.11 were measured in instrumented cylindrical die compaction tests (Frogerais OA press).

Shear surface -cohesion 𝑑 and internal friction angle 𝛽

To obrain the diametral rupture strength 𝐹 𝑑 (used to calculate the rupture stress 𝜎 𝑑 in Equation 4.23), cylindrical compacts were manufactured in a lubricated cylindrical die. Rupture tests were performed using the Erweka TBH 220 equipement.

Results of 𝜎 𝑑 are presented in Figure 4.9a as a function of relative density.

For the axial rupture strength 𝐹 𝑐 , cylindrical compacts were manufactured in a 11.28 mm diameter and 90 mm high die compressed at 30 kN on the Introm press.

In order to respect the 𝐻𝑒𝑖𝑔ℎ𝑡 𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 ratio of 2, compacts were manufactured with the objective of obtaining a final height of 22 mm. The mass was varied to obtain a relative density variation range of 0.4 to 0.9. Results of 𝜎 𝑐 are presented in

Conclusion

In this chapter we proposed a hybrid method for the calibration of elastic, friction and DPC parameters. The results have been presented for MCC 102. This method was proposed to counter the scale effect observed between a cylindrical die compression of 1 𝑐𝑚 3 volume and a parallelepipedic die compression of 33.7 𝑐𝑚 3 volume. The characterisation results

presented in this chapter will be validated numerically in the following chapter, by comparing final relative densities as well as experimental and numerically predicted compression cycles within the framework of parallelepiped die compression with the use of flat punches.

In general, numerical simulations of die compression of pharmaceutical powders are carried out on geometries with a similar volume to the volume of the instrumented die, allowing the characterisation of material properties of the powder studied. In these cases, no scaling problems were therefore, highlighted.

Discussion with results from the literature with regards to the use of the hybrid method :

The parameters impacted by the use of the hybrid method are the elastic properties (particularly Young modulus 𝐸) and the cap parameters (𝑃 𝑎 , 𝑃 𝑏 and 𝑅). In fact, in the case of the parameters of the shear surface (𝑑 and 𝛽) and the friction coefficient, only the instrumented cylindrical die compression tests were taken into account.

In comparison with results obtained (parameters calibrated by standard method in a 1 𝑐𝑚 3 instrumented cylindrical die) for a similar MCC PH102 Avicel powder in I. [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF]'s work show differences in pressure for an equivalent range of volumetric strain. In our case the pressures obtained varied in a range going from 5 to 35 MPa for plastic strains that varied between 0.7 and 1.2. In I. Sinka et al., 2003's reaches 25 GPa for MCC Avicel PH101 powder. In our case, the maximum value reached is estimated at 3 GPa, which is significantly lower than the results found in the literature.

These differences highlight the effect induced by the transition from a surface of 1 𝑐𝑚 2 to a surface more than 8 times higher for equivalent volumetric plastic strain, revealing a scale effect.

CHAPTER 5

Boundary problem of die compaction and approximation

Introduction

Modelling of the compression problem is an important step in understanding the phenomena involved in the process. Indeed, it allows to access local fields such as stress, strain and density quantities within the compact as a response to the imposed loading.

The modelling tools developed using the Finite Element Method (FEM) have shown their effectiveness for the study of the behaviour of granular materials in compression. The advantage of this approach is that the behaviour of a porous medium can be studied taking into account the defined loading and tooling (through the definition of boundary conditionsloadings and contacts).

Modelling combined with the FEM has been used in the context of the evaluation of the impact of the friction coefficient on the density distribution by I. [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF], Krok et al., 2014[START_REF] Michrafy | Wall friction in the compaction of pharmaceutical powders: measurement and effect on the density distribution[END_REF] It is indeed the presence of the friction coefficient that induces the generation of strong non-homogeneity in the density distribution (L. Han et al., 2011). This method also finds its interest in the study of the sensitivity to material parameters [START_REF] Mazel | Lamination of biconvex tablets: Numerical and experimental study[END_REF], Sinha et al., 2010b).

In a first part of the Chapter we will introduce the modelling of the problem. The data that constitute it are the behaviour law, in this case the DPC model (introduced in Chapter 4), the definition of the quasi-static framework of the compaction process and of the geometry and boundary conditions of the problem. Material parameters constitute essential experimentaly obtained input data.

The second part of the Chapter deals with the numerical simulation results by the FEM, which is a powerful numerical tool for solving boundary problems where an analytical solution is no longer possible. In this work, the FEM simulation was performed using explicit integration and the Arbitrary Eulerian Lagrangian (ALE) method more adapted to solve large deformation problems. Abaqus/Explicit v6.18 was used in this context. The results will be presented MCC 102 powder.

The first results will concern the validation of material properties. Indeed, in the previous Chapter we proposed a hybrid calibration method for the DPC parameters combining instrumented die compressions (cylindrical die of 1 𝑐𝑚 3 ) and compression tests in parallelepiped die (using flat punches). It is therefore, of interest to validate these experimental data numerically. The data thus validated will enable the simulation of the compression tests using the grooved punches (2 and 5 grooves). Finally, attention will be paid to the impact of the groove geometry and position on the fields generated during compression.

Quasi-static formulation of compaction process

The behaviour of powder during compression is highly non-linear, due to the large deformation, plastic deformation, the contacts and friction between the tools (die and punches) and the powder.

Modelling die compaction is considered here under quasi static conditions. The dynamic effet (speed or acceleration) of time on the behaviour is not considered.

In these conditions, posing the boundary problem of die compaction involves :

• Equilibrium equations In the following these equations that constitute the partial derivate equations of the system to be solved using the FEM approach will be introduced.

Equations involved in the definition of quasi-static equilibrium :

In the case of a quasi-static problem the equilibrium is generaly formulated by the Equation 5.1 on an Ω domain.

𝜕𝜎(𝑡)

𝜕𝑥 + 𝑓 (𝑡) = 0 (5.1) 𝜎(𝑡) represents the time dependent Cauchy stress tensor and 𝑓 (𝑡) the external volumetric forces (gravity) which are negligible for compression. Nevertheless, in the framework of the quasi-static problem, the hypothesis is made that the evolutions are slow enough for a time-dependent solicitation to be solved at each moment as a static problem (Sidoroff, 1980).

The time variable can then be considered as a loading one.

The maximum deformation of a powder bed can reach 80 to 85% in powder die compaction.

We therefore, place ourselves in the framework of the consideration of large deformations. The strain 𝜖 𝑖𝑗 is then formulated as a function of the displacement 𝑢 𝑖 in the following manner :

𝜖 𝑖𝑗 = 1 2 ( 𝜕𝑢 𝑖 𝜕𝑥 𝑗 + 𝜕𝑢 𝑗 𝜕𝑥 𝑖 + 𝜕𝑢 𝑘 𝜕𝑥 𝑖 𝜕𝑢 𝑘 𝜕𝑥 𝑗 ) (5.2)
In the context of elasto-plastic models, the total deformation is decomposed into two components an elastic and a plastic (irreversible) one as suggested in Equation 5.3.

𝜖 = 𝜖 𝑒𝑙 + 𝜖 𝑝𝑙 (5.3)

Elastic Behaviour

The elastic behaviour is assumed to be linear and dependent on the relative density state known at each instant of the medium considered. The elasticity tensor 𝐷(𝜌 𝑟 ) enables to link the stress tensor 𝜎 to the elastic strain tensor 𝜖 𝑒𝑙 as formulated in Equation 5.4.

𝜎 = 𝐷(𝜌 𝑟 )𝜖 𝑒𝑙 (5.4) The elasticity tensor is written in more detail as follows :

𝐷(𝜌 𝑟 ) = 𝐷 𝑒𝑙 𝑖𝑗𝑘𝑙 = 𝐸(𝜌 𝑟 ) 1 + 𝜈(𝜌 𝑟 ) [ 1 2 (𝛿 𝑖𝑘 𝛿 𝑗𝑙 + 𝛿 𝑖𝑙 𝛿 𝑗𝑘 ) + 𝜈(𝜌 𝑟 ) 1 -2𝜈(𝜌 𝑟 ) 𝛿 𝑖𝑗 𝛿 𝑘𝑙 ] (5.5)
Where 𝛿 𝑖𝑗 is the Kronecker delta. In this work 𝐸 and 𝜈 are defined by Equations 4.43 and 4.44.

Plastic behaviour

Plastic behaviour is formulated from Equations 4.3 to 4.9. The parameters intervening in the definition of the powder behaviour were introduced prviously through Equations 4.38, 4.39, 4.40 and 4.42.

The data generated from the fits obtained by the characterisation for elastic data and the DPC model were therefore, directly entered into the input file.

These experimentally obtained material data will be validated numerically, the results will be presented in Section 5.7.1.

To complete the system, contact law, friction conditions and initial conditions, will be detailled in Section 5.5 as the geometry of the material domain will be described.

Time integration

The set of above equations do not have, in general, an analytical solution and require the search for an approximate numerical solution. To do so, the equations are discretised in time using implicit or explicit techniques. The resulting system is then depending only on space variable which will be solved using the Finite Element Method.

For the simulation of powder compaction it was decided to use explicit integration implemented in Abaqus/Explicit. This procedure is recommended for highly nonlinear problems and allows good management of large deformations, which is the case in our study.

Explicit integration allows the solution at a given time to be determined on the basis of the previous known states. This leads to an explicit system for the computation of the approximation. The resolution procedure implemented on Abaqus uses an integration scheme by central differences were the equilibrium at time 𝑡

+ Δ𝑡 2 is calculated from time 𝑡 -Δ𝑡 2 .
The explicit procedure is conditionally stable, meaning that there is a time increment beyond which the result is no longer guaranteed. The determination of the stability limit therefore ensures the convergence of the method.

The stable time increment (Δ𝑡) must thus satisfy the following condition: 5.6) where 𝐿 𝑒 represents the characteristic dimension of a mesh element (corresponding to the smallest length element), 𝜌 the density and 𝐸 the Young's modulus of the material.

Δ𝑡 ≤ 𝑚𝑖𝑛 ( 𝐿 𝑒 √ 𝐸 𝜌 ) (
This condition on the time step leads to an increase of the calculation time. In order to reduce this time it is possible to increase the time increment limit by increasing artificially the density of the material without altering the results, this is called mass scaling. Indeed, mass scaling is widely used in Abaqus/Explicit to improve the computational efficiency of quasi-static analyses. An excessive value of mass scaling can nevertheless lead to the loss of accuracy of numerical predictions. It is therefore necessary to ensure that the kinetic energy of the deforming material does not exceed 5 % of the internal energy over the majority of the analysis [START_REF] Hibbitt | [END_REF].

As mentioned above, once the time discretisation is achieved, the resulting system is depending only on the space variable at each time, which is solved using the FEM method. This method leads to the computation of stress, strain and relative density fields.

Spatial approximation based on the FEM

FEM method can be applied to the system in an Eulerian, Lagrangian or combined Lagrangian Eulerian representation. In Abaqus/Explicit, there exists two types of combined Eulerian Lagrangian methods : the first one called « Coupled Eulerian Lgrangian -CEL » with the ability to simulate a class of problems where the interaction between structures and fluids is important and the second one called « Arbitrary Lagrangian Eulerian Adaptive meshing -ALE » which combines the features of Lagrangian analysis and Eulerian analysis allowing to maintain a high-quality mesh throughout the analysis when large material deformations occur.

It could be mentioned here that in some publications (Coube et al., 2000), the compaction is achieved using Lagrangian approach which is very accurate. However, due to the distorsion of elements in large material deformation, remeshing is needed and was done manually (mesh to mesh mapping method) to keep the good quality of the meshes. This procedure needs to be carrefuly executed as the material data transfer between two meshes may introduce implementation errors. In our study, we preferred to use the ALE adaptive meshing method in which the remeshing is done internally in the software.

Schematic

Lagrangian description:

This method is based on the fact that each node of the mesh is associated with matter during the movement that the discretised space may undergo. Material points are therefore, always associated with real nodes of the mesh. It is more often used in structural mechanics. It is also widely used in soil mechanics, among which we find the compression of powders. This method allows a good visualisation and definition of free surfaces and contact interfaces. However, this description has a notable drawback. Indeed, it does not allow the management of large distortions of the discretised domain, which can lead to the divergence of the calculation or, in the case of convergence, a mesh of poor quality at the end of the simulation.

Eulerian description:

This description is the most commonly used in fluid mechanics. Unlike the Lagrangian description, this one is based on the fact that the material points are not associated with the physical points/nodes of the mesh. We therefore, find ourselves in a description of motion where the mesh is fixed and the material moves freely (depending on the problem being addressed) in a defined domain. In this situation there is a good management of large distortions at the expense of less accurate information at the interfaces.

The ALE method is based on the combination of the two motion description algorithms presented above. The mesh then has more freedom of movement, has a better management of distortions and is more accurate (especially at the contact interfaces) than in a pure

Eulerian or Lagrangian description. This ALE method was first proposed by [START_REF] Hirt | An arbitrary Lagrangian-Eulerian computing method for all flow speeds[END_REF] in the framework of fluid mechanics. In this remeshing algorithm used at each step of the calculation, several iterations of the ALE algorithm are generally required to optimise the mesh. During these iterations the mesh is rearranged (as a function of the distortion experienced) and the material is redistributed (material flow into the new optimised mesh).

This redistribution of material is done from the state of the material data recovered from the previous step.

This method will be used in this study because the geometry of the system (Figure 5.4)

will lead to the generation of strong distortions during the compression calculation. The ALE method then allows us to preserve a good mesh quality (acceptable topology), while following the domain boundaries, along the calculation. In the Abaqus software the use of this method as a remeshing tool is defined through two main parameters; the remeshing frequency and the sweep.

More details on the ALE formulation can be found in [START_REF] Hirt | An arbitrary Lagrangian-Eulerian computing method for all flow speeds[END_REF][START_REF] Liu | An arbitrary Lagrangian-Eulerian finite element method for path-dependent materials[END_REF]and Sarrate et al., 2001.

Geometry and boundary conditions

The initial geometry of the system (before compression) is shown in Figure 5.2 for the case of the two-grooved punch. Initially, the upper punch (more precisely the tip of the groove) is in contact with the powder bed. As a reminder, the precise geometries of the 2 and 5 grooved punches were previously presented in Chapter 2, the boundary conditions are the same for the different configurations of punch geometry.

The initially very elongated parallelepiped geometry has a powder bed (Ω domain) height of 40 mm (along the 𝑧 axis) and a width of 7.83 mm (𝑥 direction). For clarity, in this representation (Figure 5.2) the initial height of the powder bed has been reduced.

The hypothesis of plane deformations was made. This assumption can be made given that the depth (along the y-axis) is significantly larger than the other dimensions. It is then possible to assume that the deformations remain in the plane, i.e. in the (𝑥, 𝑧) plane in the main central region far from the boundary conditions. This allows the transition from a 3D to a 2D calculation, thus considerably reducing the calculation time.

The geometry of the tools (die and punches) allows to define a plane of symmetry, highlighted in red, on the Γ1 surface. The loading is imposed by the upper punch on the powder bed on the Γ2 surface via a predefined displacement which was estimated based on the maximum

Mesh generation

Generating the initial mesh for compression was a delicate task due to the geometry of the lower punch that the powder fits initialy. The initial state of compression, for the simulation We have therefore, chosen to use a preliminary step to the compression calculation. This preliminary calculation consists (as shown in Figure 5.3) in the indentation of a rectangular powder bed by the lower punch, the use of the ALE method allows the powder to deform in good conditions to fit its geometry. The mesh generated by this preliminary step is then used as an orphan mesh, imported into the compression calculation file (stresses and strains are reset, only the deformed mesh is imported).

This method has a real advantage: the simplicity of the generation of the preliminary calculation mesh (case (a) in Figure 5.3), as we have chosen to use regular quadrangular elements. The initial rectangular geometry of the powder bed makes it very easy to generate a first mesh (especially for the mesh convergence calculations which will be presented later in this Chapter). This method also has the advantage that it can be adapted to any punch geometry, especially when studying the impact of the groove geometry on the density, strain and stress fields proposed in Section 5.8.

The initial states (system geometry before compression) shown in Figure 5.4b and 5.4c present the state of the mesh generated from a preliminary step as suggested here. The initial mesh presented is therefore, the imported orphan mesh.

Results of the simulation of the compression problem

The different steps of the calculation tend to reproduce the experimental compression test (typically compression, unloading and ejection).

• The first step, loading or compression consists of a downward movement of the upper punch in the die to densify the powder bed. The input instruction is given by a fixed displacement value.

• The second step is the unloading. This is done by the withdrawal of the upper punch upwards until contact between the upper surface of the powder and the punch is lost.

• The last step is the ejection, during which the compact is ejected from the die by pushing the lower punch upwards.

The simulation results will only be presented for the first two stages (compression and unloading), as at this stage of the study the explicit algorithm for the simulation of this problem leads to severe distortions of the elements leading to the divergence of the calculation during ejection.

The material parameters were used as defined in the Equations 4.38,4.39,4.40,4.42,4.43 and 4.44. The calculations were carried out as quasi-static problems on Abaqus/Explicit v6.18. The dependence of the material parameters on the relative density was defined with a user subroutine VUSDFLD.

The use of this subroutine in our case study is based on the following principle: at each time step of the simulation the subroutine gets the strain increment data. By the following relation

𝜌 𝑟 = 𝜌 𝑟0 𝑒𝑥𝑝 ( ∑ 𝜖 𝑝𝑙 𝑣𝑜𝑙 )
, obtained by Equation 4.22, the relative density associated with the actual strain rate is calculated and is reinjected in the calculation so that the material data is updated. Indeed, the data was implemented in the calculation input file according to the density. The properties are thus updated according to the density state calculated for each element of the mesh. It is on the basis of this updated material data that the new stress state is calculated.

In our problem, the powder is defined as a deformable medium whereas the tooling (die and punches) are defined as rigid parts. A friction coefficient between the tools and the powder bed is set to 𝜇 = 0.35 according to the experimental results presented in the previous Chapter. The friction coefficient is considered to be constant over the entire contact surface.

The initial relative density of the powder bed was set to 0.195 (bulk density divided by the true density of the MCC 102).The mesh that composes the powder bed is made up of CPE4R elements (plane deformation with reduced integration). To prevent excessive element distortions the ALE remeshing tool for Abaqus/Explicit was employed. ALE remeshing parameters were fixed to 5 for the frequency and 3 for the sweep. Allowing the management of an optimal deformation of the mesh during compression. The simulation of the unloading was a delicate step. Indeed, it is during the unloading (as well as during the ejection) that the compact is likely to be damaged. Indeed, during the unloading, the powder compact releases the energy stored during the compression (Riera et al., 2001), during which a decrease and/or reorganisation of the density field is observed. The Abaqus code [START_REF] Hibbitt | [END_REF] recommends the use of the implicit algorithm for springback simulation (in our case this corresponds to unloading). However, as mentioned before, the compression calculations in the following results were carried out with Abaqus/Explicit.

A decompression calculation using the implicit integration algorithm following an explicit compression is not as trivial as could be expected. Indeed, during the importation, some information is lost (field variable -density, deformations are reset...) and the import from explicit to implicit does not allow to import the discrete rigid parts (punches and die) not allowing us to simulate the unloading such as in the experimental tests but only to unload, in a way less faithful to reality, by releasing the residual stresses at the contacts. This importation method is widely used in forming simulations such as stamping or bending .

Therefore, the unloading was carried out explicitly to follow the compression calculation but with a smaller time step.

Material properties validation

In this part, a numerical validation of the material properties is proposed. Indeed, the material parameters obtained by an experimental characterization referred to as hybrid (in The mean squared error estimated between experimental resutls and prediction is of 0.092 and the standard deviation of 0.303 and as can be seen in Figure 5.5 predictions are in good agreement with the experimental results.

Final mean densities were also compared, results are presented in Table 5.1. The final relative density range obtained from predictions remains low for all three cases (lower than 1.9 %) reflecting the homogeneity of the density distribution. The mean relative densities were compared to the measured density (inside and ouside the die). They were experimentaly calculated from the ratio between mass and volume for all three parallelepipedic compacts without grooves. The deviation between in die relative density values remain bellow 4 %.

When comparing predicted densities to the final compact densities (after ejection) the relative difference remains in all three cases below 10 %. This difference can be explained by the fact that we are comparing the values obtained experimentally with the values obtained after unloading and not after ejection. Therefore, the deviation is tolerable. Table 5.1 -Confrontation of experimentaly and predicted relative density (using the same experimental conditions) obtained from parallelepiped die compression using flat punches

Compact with two grooves

The simulation results depend on the mesh used (Coube et al., 2008). Several simulations were therefore, carried out with the calculation parameters described above, varying the number of elements that constitute the mesh. The different mesh refinements tested were :

12194, 19152, 25283 and 34048 elements. The minimum and maximum values of relative density and hydrostatic pressure at the end of the calculation are presented in Figures 5.6a With this mesh resolution the rectangular elements of the mesh are 3.50.10 -1 mm high and 4.66.10 -2 mm wide in their initial configuration.

Loading and unloading results

The radial and axial displacement results are shown in Figure 5.7 at the end of the unloading.

The radial displacement (Figure 5.7a) gives us information about the flow of the powder in the vicinity of the groove. Due to the indentation of the upper groove into the material, the powder flows on both sides of the groove (following the red arrows shown in Figure 5.7a). This punch geometry, therefore, induces a complex flow of powder along the compression, which can lead to the generation of so-called dead zones in the groove shoulders. The axial displacement (Figure 5.7b) highlights the deformation layers induced by the indentation of Regarding the equivalent Von Mises stress, the distributions of stresses are presented in Figure 5.8 for the loading and unloading stages. At the end of loading (Figure 5.8a) the stresses vary between 1.68 and 51.27 MPa with a location of high stress values under the upper punch groove and under the chamfer (at the contact between the die and the upper punch). These high stresses dissipate rapidly as one moves away from these areas. Lower shear stresses are observed in the centre of the compact and in the shoulders of the grooves.

Overall, it can be said that the geometric non-linearities induced by the punch geometry lead to the creation of complex shear stress distributions. Following the unloading (Figure 5.8b), a global decrease of the shear stress is observed in the compact. For example, there is a 30 % drop in the maximum stresses and even a 65 % drop in stresses in the areas where they were the lowest. This sudden decrease in the stress field can be explained by the previously mentioned assumption of linearity of the elasticity (Coube, 1998). It is actually not very appropriate for modelling the behaviour of granular materials, when their stress state is in the elastic region of the field. This is notably the case during unloading.

With regard to the relative density distribution shown in Figure 5.9 at the end of compression and decompression. The areas of highest density are located under the tips of the grooves as well as under the chamfers' end, these maximum densities vary around 0.73 -0.76. The lowest density zone is located between the lower punch groove and the die, namely in the cavity formed by the chamfer. The minimum density observed in this region is of 0.42. More globally, the centre of the compact has a certain homogeneity of density distribution, with a relative density of around 0.58.

The details of the density distribution in the vicinity of the upper and lower grooves are shown in Figure 5.10. It is in these areas that the distribution fluctuates the most. In addition to the areas of high density under the tips of the grooves, we can also see that these of lower density can be distinguished in the groove shoulders. These areas, where the density is estimated to vary between 0.50 and 0.53, then dissipate in the direction of the compact centre.

Impact of friction coefficient

The following concerns the study of the impact of the friction coefficient (𝜇) uppon compression. 𝜇 has a great influence on the results of the powder die compression. In particular, it is acknowledged that friction has a major influence on the density gradients within compacts.

The choice was made to vary this coefficient between 0.1 and 0.5 which are acceptable values in view of the results present in the literature for microcrystalline celulose powders similar to the one studied here (I. [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF], Wu et al., 2008and Krok et al., 2014). The different friction coefficients were applied to the powder -die contact.

The results are presented in Table 5.2. These findings summarise the impact of 𝜇 on the Von Mises stress distribution and relative density in the vicinity of the upper and lower grooves.

It is indeed in these areas that the impact of the friction coefficient is clearly visible. With respect to the Von Mises stresses, the areas with high stress concentration are mainly located under the grooves for low friction coefficients. As 𝜇 increases, the stress concentration and therefore, the shear stresses spread more into the rest of the compact, especially towards the region of the powder that is in contact with the die in the upper part of the compact. On the contrary, in the lower part of the compact these stresses will tend to propagate towards the centre of the compact.

An increase in the coefficient of friction results in a more limited movement of the particles at the contact surface, on the contrary as can be seen with low coefficients of friction, the particles have more ease to slide along the die wall. This allows a better filling of the cavity formed by the chamfer of the lower punch. It is also in this region of the compact where lower densities are observed in cases where 𝜇 is significantly higher.

Compact with five grooves

The simulations were carried out under the same conditions as presented above on the two-grooved punch geometry. As a reminder, the initial configuration of the system before compression is presented in Figure 5.4c.

Figure 5.11 shows the radial and axial displacement fields at the end of unloading in the case of the compression simulation with the five-grooved punches. These fields allow us to visualise the flow of the powder during compression. In the same way as for the two-grooved punch geometry, the axial displacement (along the z-axis in Figure 5.11b) shows the successive layers of deformation of the powder resulting from the specific geometry of the punch. These layers are highlighted by dotted lines. In contrast to the two-grooved case, the waves formed by the deformation layers are smoother. This is due to the difference in groove depth (as detailed in Figure 2.6) and also to the repetition of the groove along the punches. The upper grooves cause a displacement of the powder on the right-hand flanks of the grooves (movement highlighted by arrows in Figure 5.11a). Compression-induced flow in the grooves of the lower punch is also observed. In general, the displacement observed in this case (five-grooved geometry) is lower than in the case of the two-grooved punch due to the difference in their geometric parameters.

The Von Mises stress distributions are shown in Figure 5.12 for loading and unloading. At the end of compression the observed stresses vary between 1.92 and 56.11 MPa. Stress concentration zones are systematically observed under the tips of the grooves (upper and lower punch). In these localized high stress regions the observed stresses vary between 38 and 52 MPa. As already mentioned in the presentation of the results of the two-grooved compact, at unloading a stress drop and reorganisation is observed in the whole compact, i.e.

the high shear zones are no longer so distinct. Attention is now turned to the relative density distribution within the compact shown in Figure 5.13. The areas of high density are, as mentioned in the previous section, under the groove tips. We also observe that the maximum density is located at the end of the chamfer (contact area between the upper punch and die). The maximum density in this region reaches a value of about 0.8.

Under the groove tips, where the density is also high, the values vary between 0.72 and 0.76. If we look at the regions in between the grooves of the upper punch, a relatively disturbed density distribution can be seen, which is also lower than in the overall compact.

Systematically, the density of the powder located on the right flanks of the grooves is higher (between 0.56 and 0.6) than on the left flanks of these same grooves (about 0.5). The density distribution at the end of compression (Figure 5.13a) accentuates this observation. This could be due to the flow of the powder around these grooves, as mentioned above. Indeed, the radial displacement obtained following this compression reveals a movement of the powder towards the right flanks of the grooves, which suggests a better filling of these regions.

The centre of the compact shows a relatively homogeneous density distribution, approximately equivalent to 0.60 -0.61. The grooves of the lower punch also cause disturbances in the density distribution, however, less disparity is observed between the left and right flanks of the grooves (for the two grooves in the centre of the compact). The lowest densities are observed in the area between the groove closest to the die and the die in the cavity formed by the lower chamfer. In this region the densities vary around 0.45 and 0.5.

Impact of the two grooved punch geometry and their position

As we have observed in the previous sections, the shape of the punches and dies (parallelepipedic with grooves) used in this work lead to a generation of complex local quantities distributions. Indeed, the numerical results show us that in the presence of grooves (whether for two-or five-grooved punches) the distribution of density and stresses are rather complex as well as the flow in the vicinity of the grooves.

The presence of grooves generally leads to the generation of localised areas of high density which dissipate rapidly as one moves towards the centre of the compact. This leads to the observation of a density gradient between two facing grooves.This section therefore aims to evaluate the impact of the geometrical size of the groove on the density gradients and distribution of the local quantities obtained by numerical simulation. The impact of the position of the groove along the punch will also be presented (distance of the groove from the die).

Cases considered

In order to carry out this parametric study numerically we have chosen to focus on the reference geometry of the two-grooved punch as presented in Chapter 2 (Figure 2.6a).

Remarks and observations made on the results for the two-grooved and five-grooved punches are similar in the sense that the conclusion is such that in all cases it is the grooved aspect of the punch (whatever it is) that represents a source of complexity in the context of powder die compression.

For the sake of simplicity we have decided, in this parametric study, not to take the chamfer geometry into account. Indeed the size range of the groove that will be presented below is large and the presence of the chamfer (of constant size) brings new problems with regard to the flow of the powder. The reference case as defined by the two-grooved punch then becomes as shown in Figure 5.14 (without chamfer).

Groove geometry variations

From this definition, several cases were considered to highlight the influence of each of these parameters. First, a proportional variation of the groove size was chosen, ranging from a groove of dimension (𝐷, 𝑊 , 𝛼)=(0.6 mm, 0.459 mm, 20°) to (𝐷, 𝑊 , 𝛼)=(2 mm, 1.53 mm, 20°). The different cases examined are listed in the Table 5.3.

The impact of the three parameters defining the groove geometry was then evaluated separately. Table 5.4 presents the details of the punches for the study of the impact of the depth 𝐷. Three cases were considered with a depth varying from 0.8 mm to 1.4 mm, the other parameters were fixed (𝑊 =0.765 mm and 𝛼=20°). Table 5.5 presents, on the other hand, the punch geometries for the analysis of the impact of the groove opening width (𝑊 ). Finally, it is the impact of the opening angle (𝛼) that we wish to evaluate. For this purpose three opening angles have been chosen in addition to the

Groove position variations

The two-grooved punch, considered as the reference punch, has two grooves close to the edges leading to the generation of complex flows and therefore, disparities in the density distribution. It was thus decided to study the impact of the position of the grooves on the punches. To do this, two other punch proposals were made and are presented in Figures 5.16a

Results

The numerical simulations of the compression for the evaluation of the impact of the geometry of the groove were carried out in the same conditions as the calculations presented in the previous section (5.7). As mentioned earlier, the presence of grooves on both the upper and lower punch creates a density gradient between them. This gradient is characterised by high density areas under the tips of the grooves and by lower densities as one moves towards the centre of the compact. The different geometries will therefore, be discussed in terms of their effect on this density gradient and more generally on the observed density distribution.

Figure 5.17 -Schematic representation of the density gradient observable between two grooves that face each other For clarity when describing density gradient curves the sketch in Figure 5.17 shows an example of the relative density gradient that can be observed between two grooves that face each other along the height (𝑧 axis). Three main zones can be distinguished. Zone 1 shows the decreasing density evolution in the vicinity of the lower punch groove, zone 3 similarly shows the evolution of 𝜌 𝑟 at the upper punch groove. Zone 2 exhibits the central region of the compact where the density is relatively constant. The descriptions of these curves when analysing the results will refer to these zones.

Influence of proportional variation (D -W)

In the first instance it is the proportional variation of the slot size that is evaluated. In all the configurations presented previously in Table 5.3, the ratio of the width and depth sizes is constant. The density gradients for the six cases considered are shown in Figure 5.18. The density perturbation (in the region defined by zones 1 and 3) is more local, thus zone 2 is wider when the groove is smaller ((D, W) = (0.6 mm, 0.459 mm)). The slope also evolves in this direction, that is to say that for a small groove the density variation is faster than when the groove is larger. The density variations, although localised, are globally between similar maximum and minimum relative density values, regardless of the groove geometry. The slope values of the curves in zone 1 as well as the maximum and minimum density values are summarised in Table 5.7. In general, the larger the groove, the lower the density of zone 2.

(𝐷, 𝑊 ) (mm) Slope in zone We also note that in the vicinity of the lower groove (zone 1) the variation of the density at the tip of the groove dissipates to a lower density (centre of the compact) more progressively with a small opening angle (10°). ndeed, if the density distribution in the vicinity of the lower groove is examined in Figure 5.20, it can be seen that the regions which define the density progression when moving away from the groove are wider for the groove with an opening angle of 10°. In the case where the opening angle is of 25°where the groove tip is thinner, the transition from high to low densities is more abrupt (Figure 5.20b), thus the perturbation is more local. If we evaluate the difference in density between the minimum and maximum values between the grooves, these vary from Δ𝜌 𝑟 = 0.14 in the case of 𝛼 = 10°and increase progressively to Δ𝜌 𝑟 = 0.18 for 𝛼 = 25°. Indeed the maximum value at the tip of the groove increases with the opening angle. In the cases presented here (in Figure 5.20), the opening angle only has a slight influence on the filling of the powder between the groove and the die. In the different cases treated, the relative density in this zone varies between 0.4 and 0.47.

Impact of the width 𝑊

The density gradients (shown in Figure 5.21) show an impact of the width 𝑊 in the vicinity of the lower punch (zone 1). It is noticeable that for a larger width 𝑊 = 1.071 mm the evolution of the density is more progressive between the tip of the lower groove and zone 2 (around Z = 2-3 mm). On the contrary, the density drop for 𝑊 = 0.612 mm (small width) is more abrupt. This can be confirmed by evaluating the slope values of the gradient curves over the density decay phase in zone 1. The slopes as well as the different Δ𝜌 𝑟 are presented in the Table 5.8. It is noticeable that for a smaller width (𝑊 = 0.612 𝑚𝑚) the slope of zone 1 is three times higher than when the width is larger (𝑊 = 1.071 𝑚𝑚). In the case where 𝑊 = 0.612 𝑚𝑚 the density gap (or gradient) is larger (0.18) than that obtained for a wider opening.

As a reminder, the groove geometry of the punches mentioned here were previously presented in Table 5

Impact of the depth 𝐷

Of the three parameters which impact is evaluated, this one seems to have the most influence on the density gradient. The overview of the density gradient curves in Figure 5.22 shows a direct amplification with a greater groove depth. The slope values for zones 1 and 3 as well as the density deviation for the three depth cases treated are presented in Table 5.9.

Several aspects can be highlighted. Firstly (observable for all the results of this Chapter) the slopes in zone 3 are higher than in zone 1 and the maximum density of the gradient is systematically located at the level of the upper punch groove. This is due to the fact that compression is unidirectional from top to bottom. The upper regions are therefore, typically denser than the lower zones of the compacts. Secondly, the slopes of zone 1 show a strong increase when the groove depth is increased by 0.6 mm. This same observation is made with respect to the values obtained from Δ𝜌 𝑟 Table 5.9 -Relative density decrease slopes in zone 1 and 3 of density gradient and Δ𝜌 𝑟 for variable groove depths

Impact of the groove's position

As described earlier in Figure 5.16, the position of the grooves has been evaluated by proposing a first configuration where the groove is located in the centre of the compact and a second where it is located between the centre of the compact and the reference position.

The relative density distribution within the three formed compacts is presented in Figure 5.24. Moving the groove away from the die dissipates the generation of a low density region between the lower punch groove and the die. The density tends towards a better overall homogeneity. In the case of the central and intermediate groove positions the lowest localized density at the groove shoulders (of the lower punch) is around 0.5 and 0.52. In the case of the so-called reference punch, these same regions have densities of around 0.43-0.45. Compared to the groove of the upper punch, the densities are more similar from one groove position to another. The offset of the groove towards the centre tends to result in a more even behaviour between the left and right sides of the groove in terms of relative density distribution.

Looking at the density gradients generated, presented in Figure 5.25, it is possible to see a change in trend. In the central and intermediate cases the density gradient shows a symmetry between the top and the bottom parts of the compact. On the contrary, in the reference case and in all the gradient results presented so far, zone 2 shows a little variable density but still increasing from the bottom to the top of the compact. The modification of the groove Table 5.10 -Density interval Δ𝜌 𝑟 for variable groove positions

Discussion

Among the results presented in this section several aspects can be noted. In the case of a proportional variation of the D and W dimensions of the groove, the density variation observed is only slightly impacted (Δ𝜌 𝑟 varying between 0.14 and 0.17). On the contrary, a clear change of the density variation speed is observable in zone 1. It is sudden one for a small groove and slower when the groove is larger. The variation of the opening angle (with fixed 𝐷 and 𝑊 ) has little influence on the general shape of the density gradient, but nevertheless, leads to an increase of the minimal and maximal interval of relative density. The variation of the width of the groove leads to an evolution of both Δ𝜌 𝑟 and the slope so that for a small slope (wide opening) the minimalmaximal interval is also small. Similarly, in the context of the variation of the groove depth, steep slopes (rate of change of 𝜌) correspond to a larger gradient.

The impact of the groove geometry on the density gradient can be evaluated with two criteria, the slope of density variation (here in zone 1) and the maximum density gap Δ𝜌 𝑟 .

In the industrial context of this work, the modification of the groove geometry is subject to a constant volume constraint. The following discussions will therefore, take this into account.

The results presented above show, in general, that the width of the groove tip has a strong influence on the increase of the local relative density in this region. Indeed, a thin groove tip (when the geometry of the groove tends towards a triangular shape) leads to a strong density concentration. On the contrary, we notice that when this region is wider, the region of high density dissipates slightly, thus decreasing Δ𝜌 𝑟 .

With this consideration, starting from the reference case of two grooves (geometry presented in Figure 2.6a in Chapter 2) two proposals for groove geometries can be envisaged. These two propositions A and B are schematised in Figure 5.26.

As a consequence of the observation made on the width of the tip of the groove the proposals A and B suggest, in one case, to increase the width of the small base of the groove and in the other case, to keep it constant. The second parameter that significantly influences the density gradient and the maximum deviation Δ𝜌 𝑟 is the depth of the groove. In both cases, a decrease in depth is proposed.

The volume constraint imposes that the relation A 3 = A 1 + A 2 has to be satisfied. This suggests that the grey hatched area lost by the decrease in depth must be compensated by the hatched areas 1 and 2.

The above proposals lead to the modification of the opening angle in order to respect the relation of volumetric conservation. In the case A the geometrical modifications impose the decrease of the opening angle 𝛼, on the contrary in the case B the opening angle is larger.

The impact of the opening angles would then have to be evaluated with these new conditions (width of the groove tip and depth), in particular their impact on the flow of the powder in the vicinity of the groove.

Comparison with the five-grooved punch

In the last section, the impact of the geometrical parameters of a groove was evaluated on the density gradient between two facing grooves. This paragraph aims to discuss the density gradients generated in the case of compression with five-grooved punches, the results of which were presented in Section 5.7.3.

In order to distinguish the gradient results for the three grooves present in the symmetrical part of the five-grooved compact, a numbered scheme is proposed in Figure 5.27. Figure 5.28 shows the density gradients obtained for the grooves numbered 1 to 3 as indicated in Figure 5.27. The results were also superimposed to the density gradient obtained in the so-called reference case (as a reminder, the reference geometry is that of the two-grooved punch (Figure 5.16c)).

The gradients obtained for the grooves numbered 1 and 2 show the same type of evolution, especially in zone 2. This trend changes for the third groove closer to the die. The observed behaviour is similar to that of the density gradient in the reference case. That is to say that zone 2 of the gradient shows an increasing density evolution. These observations are in line with the trends obtained during the evaluation of the impact of the position of the grooves on the density gradient presented in Figure 5.25. The maximum Δ𝜌 𝑟 values obtained for the gradients for grooves 1, 2 and 3 are listed in Table 5.11. For comparison, the maximum density spread obtained in the case of the two-grooved punch is Δ𝜌 𝑟 = 0.16 (see Table 5.10). The groove geometries between the two and five grooved punches (initially presented in Figure 2.6 of Chapter 2) differ in several respects.

These are shown in more detail in Figure 5.29. In the case of the five-grooved punch the groove geometry is shallower (1.3 mm) and the opening angle is larger (25°) and therefore the groove tip is thinner. Despite this thinner groove tip, which tends to localise the density in this area, the gradient (maximum density deviation) is lower in the case of the groove in Figure 5.29a. A greater depth would seem to play here a predominant role on the value of Δ𝜌 𝑟 .

Conclusion

In this Chapter the numerical method used in this work was presented. First, the material properties were numerically validated by compression tests in a parallelepipedic die with flat punches. The numerical results showed a good agreement with the experimental ones.

These validated parameters were then used to simulate the compression of the powder using grooved punches, the detailed geometry of which were presented above (punches with two or five grooves). Global view of the compacts after unloading are presented in Figure 5.30a and 5.30b. The results of the displacement, stress and density fields distribution are also presented and discussed for the compression and unloading stages. Indeed, at this stage of the work, the simulation of the ejection of the compact from the die was not satisfactory due to the generation of distortions leading to the divergence of the calculation at the beginning of this step.

A global observation for both punch geometries can be made, indeed in both cases the density distribution around the grooves shows similarities which can also be found in the literature.

These results show areas of high density under the groove tips (or other type of indentation shape) while regions of lower density can be observed in the groove shoulders. These trends have been observed experimentally by I. [START_REF] Sinka | Measurement of density variations in tablets using X-ray computed tomography[END_REF], Laity, 2014 and numerically by [START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF] The regions between two opposing grooves are in consequence areas where a density gradient can be observed: high density under the groove tips and lower density in the centre of the compact. The values that constitute this gradient depend entirely on the geometry of the groove and on its position. Therefore, the impact of the geometric parameters of the groove on the density gradient between them were examined.

Highlights

• General presentation of the X-ray Tomographic analysis principle • Presentation of two image processing methods for density calibration using X-ray Tomography raw data, grey level images • Results for parallelepiped compacts with two and five grooves

Introduction

The aim of this chapter is to present the X-ray Tomography method for the experimental characterisation of density within a powder compact. The objectives are twofold. Firstly, to characterise the density distribution within a compact and secondly to compare the predictions obtained by FEM simulation with the experimental results in order to validate the numerical predictions (aspect covered in the next Chapter).

X-ray Tomography, a non-destructive analysis which allows the detection of internal defects (among others) is used in this work as a characterisation tool which allows access to the internal microstructure of a sample. This chapter presents the results of relative density distribution obtained from a tomographic analysis of MCC 102 compacts with two or five grooves (Section 2.3.3). Two image analysis methods for generating a density distribution from a tomographic image will be presented and discussed.

X-ray Tomography principle

X-ray Tomography is an experimental method to visualise cavities (Farber et al., 2003), internal defects (Wu et al., 2005a, Wu et al., 2008, Garner et al., 2014[START_REF] Mazel | Lamination of biconvex tablets: Numerical and experimental study[END_REF], of a specimen or to access the density distribution of a compact (Burch, 2002[START_REF] Sinka | Measurement of density variations in tablets using X-ray computed tomography[END_REF], Busignies et al., 2006[START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF] The source of the device emits a beam of x-rays onto the sample. Some of these rays are transmitted and some are absorbed by it. The transmitted beams are received by the receptor, and the intensity of these beams is measured to generate X-ray radiographies. These X-ray radiographies (or projections) at different viewing angles are then transformed into tomographic images (3D reconstruction) by numerical processing. This tomographic image generation is done with an algorithm based on the application of Beer Lambert's law. The 3D tomographic images generated are in the form of multiple cross-sections in the desired plane. Each of these sclices can then be interpreted in terms of density (depending on the image analysis performed). Beer Lambert's law (Equation 6.1) is used to calculate the intensity of a transmitted beam over a thickness 𝑥.

𝐼 𝑥 = 𝐼 0 𝑒 -𝜇𝑥 (6.1)
Where : 𝜇 is the absorption coefficient and 𝐼 0 the intensity of the incident ray (intensity of the emited ray).

These absorption coefficients are then converted into grey levels ranging from 0 to 255. The variations in absorption levels within the tomographic image are an indication of variations in density.

Regarding the tomographic results presented in the continuation of this Chapter, no artefacts were identified. As a reminder, the most commonly found, when present, artefacts were briefly presented in the Section 1.5.5.2 of Chapter 1.

Density calibration by image processing

Two image processing methods to map relative density are presented in this work.

• The most commonly used tomoraphic image analysis used to compute density in based on a calibration method. The purpuse of the calibration is to link grey levels to a specific density. Calibrations were performed by I. [START_REF] Sinka | Measurement of density variations in tablets using X-ray computed tomography[END_REF], Busignies et al., 2006[START_REF] Swaminathan | Modeling the formation of debossed features on a pharmaceutical tablet[END_REF] for powder die compaction and by [START_REF] Miguélez-Morán | Characterisation of density distributions in roller-compacted ribbons using microindentation and X-ray micro-computed tomography[END_REF] the case of roll compacted ribbons.

• A second method can be found in the literature. It is based on the analysis of a binary image generated from the raw tomographic image (grey level image). This method presented by Bloom et al., 2009 andCottrino et al., 2013 is based on the choice of a threshold value, to separate in two (pores and particles) the grey levels which constitute the image.

In this section the detailled methodologies of these two image processing methods are presented. In both cases, image analysis was done with the ImageJ software (Rasband et al., 1997) and, in addition, for the binarization method we also used Matlab. Results obtained from X-ray Tomography image analysis methods will be compared and discussed for the two grooved punch geometry. The relative density maps obtained will finaly be used for comparison with numerical results in order to validate their predictions.

Tomographic analysis and process parameters

In this study, the tablets density distribution of MCC 102 were analysed using an X-ray Tomograph Easytom 130. The scans were performed at a resolution of 17 𝜇m per voxel. The acceleration voltage and intensity were fixed at 60 kV and 133 𝜇A. The transmitted rays were recorded for 1440 projections.

The above menstioned resolution size was chosen to get a complete slice of the compact, in fact the chosen resolution is dependent on the distance between the emitter and the specimen and the size of the area to be analysed. The voltage value was chosen, by trial and error and opperator's know-how, considering the porous aspect of the studied sample.

First processing method : Binarisation

A first method consisting in binarising a grey level image was investigated to extract solid fraction data from tomographic results. The basic principle of the method is presented in the following scheme (Figure 6.2).

Process steps are detailed as follows, steps 1 to 5 were processed on the open source software ImageJ, step 6 on Matlab.

1

Grey level image

The raw material for this characterisation technique is the grey level tomographic image. This image depicts a grey level distribution. Each grey level representig an absorption level as mentionned previously.

2

Choice of a threshold value

The choice of the threshold value is the crutial step in the image analysis process.

Indeed the threshold value is a value that separates grey levels of an image into two, the material and the pores. The choice of the thresholding algorithm that computes this parameter is of importance and has to be chosen with caution as it highly influences the final relative density distribution. The threshold value can be calculated with various algorithms; Otsu (Otsu, 1979), Default (Ridler et al., 1978), Huang [START_REF] Huang | Image thresholding by minimizing the measures of fuzziness[END_REF] and Li [START_REF] Li | Minimum cross entropy thresholding[END_REF] among others available in the ImageJ software.

A proposition for the choice of the threshold value could be to choose the value that allows to obtain the same mean relative density as in the experiment.

3

Generation of the binary image

Once the threshold value (noted 𝑇 ℎ ) is selected, a binary image is generated. As mentionned previously the application of a threshold separates the image grey levels into white and black pixels. The white pixels representing the powder material and the black ones the porosities. During the binarisation process the grey value of each pixel is compared to 𝑇 ℎ creating the separation between powder particles (white pixel)

and voids (black pixels). In order to calculate the solid fraction distribution of the image, the binary image has to be discretised into cells of a defined size. This cell size has an influence on the heterogeneity (or homogeneity) of the final results. It's influence will be discussed in the results section.

5

Calculation of the solid fraction in each cell

The solid fraction of each cell is calculated with the folowing Equation:

𝜌 𝑟 = 𝑊 ℎ𝑖𝑡𝑒 𝑝𝑖𝑥𝑒𝑙 𝑟𝑎𝑡𝑖𝑜 𝑇 𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 : 𝐵𝑙𝑎𝑐𝑘 + 𝑊 ℎ𝑖𝑡𝑒 𝑝𝑖𝑥𝑒𝑙 (6.2)
This ratio is calculated for each cell through a macro code programmed for this purpose.

6

Final density distribution

As the solid fraction of each cell is calculated a color map can be associated to the solid fraction scale range. This leads to a final density map of the studied image.

Second processing method : Calibration

Image calibration with the tomographic analysis of cylindricaly shaped calibration compacts of known density is the most common method for compact density characterisation. The basic principle of this method is to obtain a rule of evolution between the grey levels and the relative density for a particular material. Thus it is possible to associate the grey levels that make up a tomographic image with a value of relative density within the sample made of the same material.

The different steps of this process of analysis are detailed below (see Figure 6.3) and results

for density calibration for the MCC 102 powder will be presented in the following section.

1

Grey level image

As for the binary image analysis method, the starting point is based on the raw tomographic image (grey level image) of the studied sample. In addition, in this case, we also consider the raw tomographic images of calibration compacts.

Calibration compacts are, here, homogeneous cylindrical compacts manufactured in a die of 1 𝑐𝑚 3 of volume with flat punches. These compacts are made at different loading pressures in order to obtain compacts of relative densities with a large variation range.

More details on this procedure will be given in Section 6.5.

2

Mean grey value

In order to obtain the correlation between grey level values and relative density, the histogram of each calibration compacts tomographic analysis is extracted to obtain the mean grey value. The histogram of a tomographic image represents the grey level values distribution of the image.

These data make it possible to associate an average grey level (obtained from the image histogram) with a specific relative density (of the corresponding calibration compact).

3

Evolution law between mean grey value and relative density

The association of a series of average grey levels and relative densities for each calibration compact allow to plot the tendency curve describing the evolution of the relative density as a function of the grey levels.

4 Application to the studied sample

The ImageJ software allows to associate a calibration law dirrectly to the grey level of the image studied. Converting the grey level value information of each pixel into a relative density information. Two image analysis methods have been presented step by step for the determination of relative density distribution inside a compact using an X-ray Tomographic analysis. Results obtained using both methods will be presented, compared and disscussed for the two-grooved case study in Section 6.6.

Calibration results

Calibration to obtain the evolution relationchip between grey level values and relative density was done for the MCC 102 powder. Eight cylindrical compacts of 11.28 mm of diameter of The X-ray Tomographic analysis was carried out on a compact of MCC 102 compressed at a load of 20 kN in the parallelepipedic matrix with two-grooved punches as shown in Figure 6.7. The choice of this default 𝑇 ℎ value enables the generation of the binary image from the grey level one as presented in Figure 6.8.

Generation of the density distribution

To obtain a density distribution starting from a binary type image it has to be discretized.

Thus, a solid fraction value can be calculated in each cell of the diretization grid to generate a mapping. The solid fraction is calculated following Equation 6.2 for each cell. The size of the cells that discretize the image is of importance as it highly impacts the results resolution and therfore, their accuracy. In fact, as can be seen in Figure 6.9, a small cell size leads to a more heterogeneous result. Contrary to this a bigger cell size leads to a coarser density distribution result.

However, for all discretizations, the density distribution presents maximal density zones at the tip of the grooves, high density regions in the compact's borders. The centre of the compact presents a lower density. When looking closer to the inter grooves region (between two facing grooves) a density gradient can be observed.

As an example to illustrate the impact of the cell sizes, the plots of the relative density evolution between two facing grooves (highlighting the density gradient mentioned above) are presented in Figure 6.10. The density evolution presents the same tendencies for all cell The point cloud constituting the results obtained with a small cell size (5 x 5) presents a strong dispersion of points revealing a non-homogeneous character obtained with this discretization. Indeed, the smaller the cell size, the closer we get to the pixel scale. This leads to density values close to the extremes 1 and 0 (real value information of a pixel).

On the contrary, a coarser discretization resolution (for instance the 25 x 25 cell size) leads to a loss of information, especially when the density variations are strongly localized, as in our case, in the vicinity of the grooves. It is indeed noticeable that the difference between the centre and the tips of the grooves is no longer as significant.

The most satisfactory result would therefore, seem to be the discretization performed with cells of size 10 x 10 pixels. Indeed, it allows to reduce the obtention of a too strong point dispersion while preserving enough information to evaluate the density variation in the vicinity of the grooves and in between them.

It can also be observed that the chosen discretization can lead to an "edge effect" on the density distribution image. This effect results in areas of low density on the edges that are not physically realistic (for example on the top edge of the 20 x 20 cell grid -case 𝑑) of Figure 6.9 and also for upper/ lower and right edge of the case 𝑒) in Figure 6.9). This is in fact an effect introduced by the processing of the image as such, when defining the cell grid that defines the descritization. It happens that for the cells which are located on the extremities of the material, the density calculation is done half on the material and half on the void (space surrounding the sample).

Calibration analysis results

The density distribution result within the compact obtained from the calibration of the MCC 102 presented in Section 6.5 is shown in Figure 6.11.

This final result was obtained by associating each grey level of the image with a relative density according to Equation 6.3. This same distribution was obtained on other slices of the tomographic image showing a homogeneity of the density distribution along the 𝑌 axis. The density distribution shows areas of high density on the grooves, which decrease rapidly with increasing distance. There are also areas of higher density on the four edges of the compact compared to its centre which show a more homogeneous relative density distribution of about 0.55. Regions of lower density are also noted in the shoulders of the grooves, particularly between the grooves and the chamfers located at the left and right ends of the compact. This observation results from the flow of the powder during compression.

Discussion

The binarisation method of analysis of the binary image has the advantage that it does not require calibration and therefore, requires less tomographic analyses. Nevertheless, the method by calibration does not introduce any bias, such as the bias of the choice of the threshold using the binarisation method, which is a delicate choice given its influence on the final results.

The tendencies of the results obtained with these two methods are similar. Areas of high density are present under the tips of the four grooves of the sample and regions of lower density are located in both distributions at the groove's shoulders, which are between the grooves and the chamfers.

It should also be noted that the distribution obtained by calibration allows us to observe a large zone of homogeneity in the centre of the compact. This homogeneity is not as evident in the case of the binary image analysis. Indeed, the binarisation of the tomographic image leads to under or over estimation of the density values. If we look in more detail, and as indicated by the scale of relative density variation in Figure 6.9, the density values obtained The resolution of the tomographic analysis is likely to have a significant impact on the results obtained by binarisation. Indeed, in the results presented in this work, the resolution was defined at 17 𝜇m, this resolution does not allow to highlight the presence of the pores which would be of sizes lower than 17 𝜇m. The use of a higher resolution (meaning closer to the micron scale according to the technical limits of the tomograph used) would allow the detection of the presence or not of these pores. More information on the microstructure of our compact would therefore, probably allow less information to be lost during the binarisation process. This hypothesis has not been validated in our work, but it would be interesting to consider if one wishes to further exploit this method of image analysis for the characterisation of density within a compact.

The second method, calibration, is the one most widely used in the literature when determining the density distribution within a compact, especially a compact made from pharmaceutical powders. In this regard, we have chosen to keep only this method of density characterisation for the rest of the results presented, especially for the five-grooved case study and for the validation of the numerical simulations.

Five grooved compact : Density distribution results

In this case the X-ray Tomographic analysis was carried out on a compact of MCC 102 compressed at a load of 20 kN in the parallelepipedic die with 5 grooved punches as presented in Figure 6.12. Grey level images were processed with the calibration method, given the disadvantages of the binary method which various biases were discussed above, to obtain a density map. The tomography results presented below were obtained using the same parameters as previously presented in Section 6.4. The density distribution within the compact obtained resulting from the calibration method is presented in Figure 6.14. This observation is in line with the observations made on the Arbocel compacts. As described in Chapter 3, it was in these specific areas that the Arbocel BC 1000 and Arbocel BWW 40 compacts were deteriorated due to lack of densification. This behaviour can be attributed to the low flowability of such powders. Indeed, under each indenter of the punch, the powder is mostly displaced toward the bottom and only a part of the powder is drawn in between the neighbored indenters.

More precisely, these inter-groove regions have relative densities varying between 0.55 and 0.59 for the upper part of the compact and between 0.54 and 0.56 for the lower ones. The region located between the lower, right and left, grooves (those closest to the die) and the die is visibly the region of lowest density of this compact with an average relative density of 0.47. This is maybe due to the fact that the powder is building up on the head of the The vast majority of the centre of the compact exhibits a reasonably homogeneous density of 0.62 in average. The same observation was made earlier in the case of the two-grooved compact.

Conclusion

In this chapter the experimental technique of X-ray Tomography is presented as a method to determine the relative density distribution in a powder compact. One of the key points of tomography applied in this context is the image processing to access the density quantity. For this purpose, two image processing methods were presented and discussed (image binarisation and calibration methods). The final choice was made for the calibration method in view of the biases introduced by the method which involves the binarisation of the grey-scale tomographic image.

Relative density distribution results were therefore, obtained and presented for MCC 102 compacts compressed at 20 kN with the two and five-grooved punches.

In both cases, it was found that the density is relatively homogeneous over the whole compact except for the edges and in the grooves vicinity. It is indeed the grooves that cause the creation of regions where the density distribution is complex, high density at the grooves' tips and lower densities in the grooves' shoulders. Similar conclusions were made in the previous Chapter when numerical simulations of powder compression were carried out with the same punch geometry.

The experimental results presented will be compared in more details with the results predicted by numerical simulation in Chapter 7. This comparison allows the validation of the model by judging its predictive character.

Introduction

The validation of the numerical results tends to prove the robustness of the model used. As presented above (Chapter 5), the numerical results allow a fine analysis of the behaviour of the powder during compression. This analysis is done in terms of stress, density and displacement distributions. The comparison between the numerical results and the experimental results (X-ray Tomography in our case -results presented in Chapter 6) allows us to justify the reliability of the numerical model used.

This chapter therefore focuses on the comparison of the results presented previously in order to validate the numerical model as a predictive tool for the modelling of the powder die compaction problem.

Two-grooved compact

To compare the results obtained by X-ray Tomography and the results predicted by simulations, several zones of interest were defined (presented in Figure 7.1). The differences between the measured and predicted densities were estimated. To do this, the relative density values were calculated for the tomography on 10 x 10 pixel cells and for the numerical results on an area corresponding to 18 finite elements (6 x 3). These two sizes correspond to an area equal to 170 𝜇m x 170 𝜇m, each. The relative densities calculated for each zone are presented in Table 7.1. The relative differences between measured and predicted density are also provided. The results in Table 7.1 show good agreement between experimental measurements and numerical predictions. in a way to present some kind of symmetry between the upper and lower part of the compact. This variation of tendency observed with the variation of µ is similar to the one observed with the impact of the distance of the groove from the edge of the compact.

The evolution of the relative density obtained by X-ray tomography was superimposed on the curves obtained numerically. It appears that in the vicinity of the lower groove (𝑍 < 2 𝑚𝑚 ) and in the upper part of the compact (𝑍 > 6 𝑚𝑚) the results are close to the prediction obtained with lower friction coefficients. In the region between 2 and 4 mm in height the relative densities are more in line with those obtained for higher coefficients of friction.

The assumption of a constant friction coefficient is, in fact, not fully representative of the experimental observations. Therefore, the development of a model of the evolution of the coefficient of friction depending on a state variable such as the relative density would be more predictive. Models of this type were proposed by Wikman, 1999 and[START_REF] Staf | Evaluation of an advanced powder-die frictional model[END_REF] For the comparison between the predicted and experimental data, the width of the band on which the relative density was calculated is the same for both cases. Indeed, experimentally the density was averaged over cells of width 170 𝜇𝑚 (10 pixels) which corresponds to about 6 elements of the mesh. The density gradients predicted and presented here are therefore an average of the gradient calculated over a width of 6 elements.

Five-grooved compact

In the same way as for the two-grooved compact, this section intends to compare the density distribution results obtained by tomography and by numerical computation. As a reminder, the density distribution results obtained by tomography are presented in section 6.7 of Chapter 6 and the predicted results, obtained by numerical calculation, are discussed in Section 5.7.3 of Chapter 5.

Conclusion

In this Chapter we sought to validate the numerically predicted results by comparing them with the experimental ones. The experimental results obtained by X-ray Tomography, allowed us to access the density distribution in a plane of interest of the compact (plane containing the grooves). These results then allowed us to verify the validity of those obtained numerically by modelling and numerical resolution by the finite element method.

The results, when superimposed on each other, reveal a good agreement between them both in the case of the two-grooved or five-grooved compacts. This validation confirms the robust and predictive nature of the powder die compression problem modelling.

General conclusion and perspectives

This study focuses on powder die compaction with the consideration of a complex geometry.

More precisely, to evaluate the influence of the grooved punches on the final properties of the compacts, depending on both the nature of the powder and the geometry.

A double approach, experimental and numerical, was considered to obtain elements to solve this problem. The compression tests carried out using the parallelepipedic die with grooved punches allowed us to observe the presence of localized cracks between the grooves. As these observations are not systematic for all materials, the experimental study permitted us to conclude that the appearance of cracks was both a consequence of the behaviour of the powder and the geometry of the punches (grooved or not). The presence of localized cracks in the inter-groove region makes it possible to target this region as being particularly sensitive to stresses, especially during unloading, which can lead to the decohesion of particles and therefore, potentially to the appearance of cracks as in Arbocel BC 1000 compacts. Since the nature of the Arbocel BC 1000 powder does not allow its mechanical properties to be characterised (using de procedure presented in Chapter 4), the choice of material for the numerical modelling was that of MCC 102, considered in this study as a model material.

Numerical modelling of the compression problem added to the study a more in-depth and detailed understanding of the different mechanisms involved during compression by allowing us to access local quantities such as stress fields, density distributions and displacements.

The mechanical behaviour model of the powder in compression chosen was the Drucker Prager Cap model. This model allows to take into account the mechanical properties of the powder such as cohesion, internal friction, the increase of the mechanical strength of the compact with the applied pressure as well as the evolution of its elastic properties. In order to compensate for the large dimensions of the compact on which the compression simulation was performed, a hybrid calibration proposal was made. The hybrid calibration method used to obtain the parameters that define the Drucker Prager Cap model combines the calibration results obtained on a standard-sized cylindrical compact with the compression tests carried out on a parallelepipedic compact with a volume thirty-three times larger. The results obtained with this hybrid calibration notably impacted the values of the elastic properties and of the Cap.

By modeling the powder behavior in compression, the compression problem was approximated using the finite element method. The consideration of boundary conditions (contact and friction) and loading conditions complete the modelling of the boundary problem.

Finite element simulation results were performed on two and five grooved parallelepipedic compacts under the hypothesis of a 2D plane deformation model. A common finding of all the computational results is that the relative density variation between two grooves facing each other presents a density gradient. Indeed, this region (identified as critical) has density peaks just below the groove tips which decrease towards the centre of the compact. As the geometry of the grooves causes a heterogeneous density distribution in its vicinity, a parametric study was conducted on the geometric parameters of a groove. The results show the impact of the different parameters chosen on the density gradient present in this region of the compact. In order to validate the predictions obtained numerically, tomographic analyses were carried out on the same MCC 102 compacts to obtain the density distribution. Two suggestions for the exploitation of the tomographic data were presented and discussed in order to obtain the density. The predicted results were then validated by comparing the density distributions obtained via X-ray Tomography and the density predicted by the finite element calculation.

The overall work in this thesis has shown the complexity of the powder compression process when the tooling used presents a complex geometry and is large in size. From a numerical point of view, the presence of grooves, leading to a complex flow, presented a real challenge.

The explicit method with the ALE approach then showed its effectiveness to solve compression problems generating strong distortion.

The validation of the numerical model by X-ray tomography allows us to affirm the predictive character of the model developed. It also demonstrates that the hybrid procedure allows the identification of DPC and elastic parameters for the simulation. This methodology could then be extended to other shapes and sizes of tooling without instrumentation.

The results of this thesis work led to a publication and an oral presentation at a national symposium :

• Alonso Aruffo, G., Michrafy, M., Oulahna, D., & Michrafy, A. (2022)

Perspectives

This work opened interesting and challenging perspectives that could be further developed :

• Optimisation of the tooling geometry. The parametric analyses carried out have enabled us to highlight the parameters that have a greater or lesser influence on the density gradient observed between two grooves. These density gradients have been identified as a potential cause of defects in materials subject to particle decohesion during unloading.

It would therefore be interesting to optimise this geometry to potentially reduce the occurrence of defects.

• Improvement of the numerical model. The first step would be to take into account the non-linear elastic behaviour during unloading. Indeed, in our work as well as in the majority of studies on powder compaction, the elasticity modulus is assumed to be constant during unloading, which is not really the case. Taking into account a non-linear evolution of the elasticity would allow a better understanding of the elastic recovery of the compacts during unloading and ejection stages. This is particularly important for the prediction of damage into compacts, as springback can initiate cracks (Coube et al., 2000, L. Han et al., 2008).

In a second stage, it would be worth considering the integration of a local damage model of Chaboche and Lemaitre type [START_REF] Jean | Mécanique des matériaux solides[END_REF] or a non-local model of the second gradient type (Peerlings et al., 1996). These methods require a certain investment and would require the implementation of the material behaviour via a UMAT or VUMAT type subroutine. It would then be interesting to apply this type of model to a material that can present cracks such as Arbocel (Figure 7.9) but whose mechanical behaviour would be more easily characterisable. 40.19 (2003), pp. 4989-5000 (cit. on pp. 6, 16, 35). 40.4 (2010), pp. 237-244 (cit. on pp. 8, 26, 30, 54). pp. 123-131 (cit. on pp. 13, 16, 37, 38, 40, 74, 97, 163). Springer, 2008, pp. 197-222 (cit. on 45.10 (2008), pp. 3088-3106 (cit. on pp. 8, 13, 36, 38, 74, 91, 92, 163). 4 (1961), pp. 671-675 (cit. on pp. 6, 27, 28).

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In: Powder metallurgy 55. 1 (2012), pp. 36-44 (cit. on p. 37).

( 116-131 (cit. on pp. 36, 94, 109).

  Concernant les matériaux, le choix s'est porté sur la Micro Cristalline Cellulose Vivapur 102 (MCC 102) fournie par J.Rettenmaier & Söhne GmbH & Co. KG et sur deux grades d'Arbocel : BC 1000 et BWW 40 du fournisseur Kremer Pigmente GmbH & Co. KG. Ces poudres ont été choisies afin de chercher à mettre en évidence l'importance des propriétés de la poudre pour résister à la fissuration. En effet, la MCC 102 est connue comme une poudre qui développe une bonne cohésion entre les particules et qui a donc plus de chance de résister à la fissuration lors du retour élastique. La poudre d'Arbocel avec deux tailles de particules différentes tend à présenter le comportement opposé. Les outils de caractérisation des poudres, de caractérisation mécanique et d'observation sont présentés dans ce Chapitre. Il s'agit notamment des équipements de visualisation des particules, de la surface et de la microstructure interne, mais également de presses et d'outillage (poinçons et matrices) ainsi que des équipements pour la mesure de la résistance à la rupture. Une section de ce Chapitre se focalise de manière plus détaillée à la présentation des outils de compression dimensionnées et usinés pour ces travaux. Les outillages permettent de former des compacts parallélépipédiques avec des rainures. La morphologie des trois types de poinçons est présentée dans la Figure 2.4, on y retrouve des poinçons à deux et cinq rainures ainsi qu'un poinçon plat (sans non linéarités géométriques). Le dispositif expérimental global est présenté sur la Figure 2.7. Celui-ci est doté d'un système de guidage (4 colonnes) qui permet une pénétration parfaite du poinçon supérieur dans la matrice sans risque d'endommager les particularités géométriques des poinçons. De cette manière, aucune action d'alignement ne doit être effectuée par l'opérateur. Le Chapitre 3 présente la synthèse des résultats expérimentaux de compression obtenus. Les modèles analytiques de Heckel et de Kawakita ont été appliqués pour les trois poudres et révèlent notamment que la poudre d'Arbocel BC 1000 présente une plus grande compressibilité et que les deux grades d'Arbocel (BC 1000 et BWW 40) ont un comportement plus fragmentaire que la MCC 102.

  de densité sont proposées. Des études complémentaires sont à envisager afin d'optimiser cette géométrie. Le Chapitre 6 est consacré à l'analyse tomographique des compacts de MCC 102 à deux et cinq rainures. Cette technique expérimentale permet, entre autres, d'accéder à la grandeur de densité relative dans un compact. L'objectif de cette analyse est double. Premièrement d'évaluer la distribution de densité des compacts étudiés et deuxièmement pour valider les résultats prédits par la simulation. Une comparaison entre les résultats tomographiques et numériques sera présentée dans le Chapitre 7. L'analyse tomographique fournit une représentation en 3D d'un échantillon, permettant d'accéder à sa structure interne. Cette représentation tridimensionnelle est construite en tranches successives (chacune étant une image tomographique) pour reconstruire un volume. Deux méthodes d'analyses des données tomographiques sont détaillées et discutées : la méthode par calibration (Figure 6.3), plus souvent retrouvée dans la littérature ainsi que la méthode par binarisation d'image (Figure 6.2

Figure 1 -

 1 Figure 1 -Diagram showing the main steps of the thesis

Figure 1 . 1 -

 11 Figure 1.1 -Visualisation of the different stages of die compaction (upper and lower punch in grey and die in hatched lines)

Figure 1 . 3 -

 13 Figure 1.3 -Schematic representation of (a) capping, (b) lamination and (c) cracking

Figure 1 . 4 -

 14 Figure 1.4 -Detailed view of the unloading phase -demonstration of springback in the die

Figure 1 . 5 -

 15 Figure 1.5 -Illustration of a typical Heckel plot

Figure 1 . 6 -

 16 Figure 1.6 -Illustration of typical Kawakita plot

  Background and state of the art Nuclear Magnetic Resonance (NMR)(Djemai et al., 2006) among others. The Nuclear Magnetic Resonance method was first used by[START_REF] Nebgen | 1H-NMR microscopy of tablets[END_REF] in order to determine the density distribution in a powder compact. This imaging technique exploits the magnetic properties of certain atomic nuclei on the principle of Nuclear Magnetic Resonance. Thus making it possible to obtain a 3D distribution of density and more particularly of the distribution of porosity. In order to obtain this distribution, the analysed sample is placed in a contrast fluid that returns a good NMR signal. The fluid penetrates the pores of the sample under the effect of the vacuum and then the imaging reveals the distribution of porosities (inverse image). The data processing then allows the image to be reconstructed to obtain the relative density distribution. This procedure has been used byDjemai et al., 2006 to reveal complex density distributions in pharmaceutical compacts. More details on the analytical principle of the NMR method are presented byCallaghan, 1993.

  used a hardness index (conical indentor) to generate the density distribution mapping within a biconvex pharmaceutical compact. To illustrate this, the results presented by I. Sinka et al., 2003 are shown in Figure 1.7. A significant number of measurement points are necessary to obtain a density mapping revealing principal isovalues.

Figure 1 . 7 -

 17 Figure 1.7 -Density distribution results obtained by I. Sinka et al., 2003 for biconvex tablets of microcrystalline cellulose MCC Avicel PH 102

Figure 1 . 9 -Figure 1 . 10 -

 19110 Figure 1.9 -Detection of an internal crack by X-ray Tomography in a compact of anhydrous calcium phosphate powder -Mazel et al., 2018

  have highlighted the fact that the presence of this geometrical feature has an impact on the local variation of density in this specific area of the compact. The geometries studied in this research and the results obtained by X-ray tomography are presented in Figure 1.11. They clearly show the density peak generated at the tip of the groove which dissipates towards the centre of the compact and the regions of lower density present on the flanks (or shoulders of the groove). Similar results were obtained by Djemai et al., 2006 and Laity, 2014. This has also been investigated numerically by Kadiri et al., 2018 in the case of scored flat and curved compacts exhibiting non-uniformity in the density distribution at the end of compression.

Figure 1 .

 1 Figure 1.11 -Visualisation of the density distribution observed by X-ray Tomography on two types of scored tablets -I. Sinka et al., 2004

Figure 1 .

 1 Figure 1.12 -Images obtained by X-ray Tomography showing the single and double furrows compacts -Laity, 2014

  There exist different constitutive equations representing the continuum mechanical behaviour of powder during compaction. The main ones are elasto plastic models differing by their plasticity criterion. The commonly used for die compaction modelling are the Cam-Clay model and Drucker-Prager Cap model(Brewin et al., 2008).

Figure 1 . 13 -

 113 Figure 1.13 -Overview of the geometry of the compacts studied in this study

2 . 3 . 5

 235 Surface observations and microstructural characterisations . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Highlights • Summary of the main properties of the powders used • Presentation of the equipment used in this work • Detailed description of the die and punches dimensioned and manufactured for this work

  Microcrystalline cellulose Vivapur 102 powder comes from the supplier J.Rettenmaier & Söhne GmbH & Co. KG. It is a white, non-odorous and non-toxic powder that flows well. A visual of the particles (SEM image) and that of the bulk powder are shown in Figure 2.1.

Figure 2 . 1 -

 21 Figure 2.1 -MCC 102 powder -SEM image and loose powder bed visual

2. 2 . 2

 22 Arbocel BC -1000 and Arbocel BWW -40 Two different grades of Arbocel (powdered cellulose) were considered in this study : Arbocel BC 1000 (Figure 2.2) and Arbocel BWW 40 (Figure 2.3) from the supplier Kremer Pigmente GmbH & Co. KG. This powder is non-toxic and environmentally friendly. It can be used as a thickening agent, for fibre reinforcement or as an absorbent.(a) SEM image (b) Loose powder

Figure 2 . 2 -

 22 Figure 2.2 -Arbocel BC 1000 powder -SEM image and loose powder bed visual

Figure 2 . 3 -

 23 Figure 2.3 -Arbocel BWW 40 powder -SEM image and loose powder bed visual

Figure 2 . 4 ,

 24 Figure 2.4, one punch with 5 grooves, one with 2 grooves and the third without any grooves (flat punch).Figure 2.5 shows a 2D cross-section of the profiles of the three punch types. Details of the geometric sizes of the two grooved punches are shown in Figures 2.6aand

Figure 2 .

 2 5 shows a 2D cross-section of the profiles of the three punch types. Details of the geometric sizes of the two grooved punches are shown in Figures 2.6a and 2.6b. These sketches show in particular the depth, width and opening angle of the grooves. These geometric non-linearities are in the (x, z) plane and are present over the entire depth of the punches (5.4 cm) as can be seen in Figure2.4. In addition to the grooves, a chamfer is located on both sides of the punches bringing complexities to the flow of the powder during compression as will be mentioned in the following chapters. The two grooved punches have a plane of symmetry highlighted in red in Figure2.6. The two grooved punches as well as the flat punch adapt to the parallelepiped die.

  (a) Flat punch (b) Two grooved punch (c) Five grooved punch

Figure 2 . 4 -

 24 Figure 2.4 -Flat and grooved punches for the parallepipedic die

Figure 2 . 5 -

 25 Figure 2.5 -2D schematic cross-section of the three different punches for parallelepiped die compression

  (a) Two grooved punch (b) Five grooved punch

Figure 2 . 6 -

 26 Figure 2.6 -Geometrical dimensions of the grooved punches (mm)

Figure 2 . 7 -Figure 2 . 8 -

 2728 Figure 2.7 -Assembly of the compaction equipment -presentation of the parralelpiped die and its guiding system integrated in the press

Figure 2 . 9 -

 29 Figure 2.9 -Tomography equipement-emitter and detector

Figure 3 . 1 -

 31 Figure 3.1 -Example of compaction cycles for three powders in a cylindrical die with an initial height of 20 mm and a loading force fixed at 30 kN

Figure 3 . 2 -

 32 Figure 3.2 -Heckel curves

Figure 3 . 3 -

 33 Figure 3.3 -Kawakita curves

Figure 3 . 4 -

 34 Figure 3.4 -Evolution of the springback index for cylindrical compacts of MCC 102, Arbocel BC 1000 and Arbocel BWW 40

Figure 3 . 5 -

 35 Figure 3.5 -Evolution of diametrical rupture stress as a function of relative density for flat faced cylindrical compacts

  Figure 3.6 -Examples of fracture profiles obtained from diametrical fracture tests for Arbocel BC 1000 and Arbocel BWW 40 compacts

Figure 3 . 7 -

 37 Figure 3.7 -Curve of applied force versus displacement, resulting from diametrical fracture tests on two Arbocel BC 1000 cylindrical compacts of similar relative density (0.86 and 0.87)

Figure 3 . 8 -

 38 Figure 3.8 -Example of compaction cycles for three powders in a parallelepiped die and flat pucnhes with an initial filling height of 4 cm and a loading force fixed at 20 kN

Figure 3 . 9 -Figure 3 .

 393 Figure 3.9 -Example of powder compacted in a parallelepiped die using flat punches (left to right : MCC 102, Arbocel BWW 40 and Arbocel BC 1000)

Figure 3 .

 3 Figure 3.11 -Example of powder compacted in a parallelepiped die using two-grooved punches (left to right : MCC 102, Arbocel BWW 40 and Arbocel BC 1000)

Figure 3 .

 3 Figure 3.12 -Lateral view (in the x, z plane) of a two-grooved parallelepipedic compact obtained after a 20kN compression of MCC 102 powder

Figure 3 . 13 -Figure 3 . 14 -Figure 3 . 15 -

 313314315 Figure 3.13 -Side view (in the x, z plane) of examples of Arbocel BC 1000 compacts compressed at different forces using two-grooved punches

Figure 3 . 16 -Figure 3 . 17 -

 316317 Figure 3.16 -Side view (in the x, z plane) of examples of Arbocel BWW 40 compacts compressed at different forces using two-grooved punches

Figure 3 .Figure 3 . 19 -Figure 3 . 21 -

 3319321 Figure 3.18 -Example of powder compacted in parallelepiped die using five-groove punches (left to right : MCC 102, Arbocel BWW 40 and Arbocel BC 1000)

Figure 3 . 22 -

 322 Figure 3.22 -Detailed view of a crack observed on an Arbocel BC 1000 compact at a load of 20 kN (Compact presented in Figure 3.20b)

Figure 3 .

 3 Figure 3.23 -Side view (in the x, z plane) of examples of Arbocel BWW 40 compacts compressed at different forces using five-grooved punches

Figure 3 .

 3 Figure 3.24 -Example of upper surface view (in the y, x plane) of a five-grooved parallelepipedic compact obtained after a 10 kN compression of Arbocel BWW 40 powder (compact presented in Figure 3.23a )

  calibration tests were carried out on flat cylindrical compacts of MCC 102, mixed with 0.5 𝑤/𝑤 of magnesium stearate for better density homogeneity within the compact. Eight calibration compacts were made at constant mass and variable load to cover a wide range of densities. These compacts, made with the Frogerais OA eccetric press, have relative densities varying between 0.42 and 0.87, i.e. a compression load varying between 7 and 171 MPa.

( a )Figure 3 . 25 -Figure 3 .

 a3253 Figure 3.25 -Example of two MCC 102 powder cylindrical compacts for micro indentation calibration

Figure 3 . 27 -

 327 Figure 3.27 -Location of the different zones of the compact for the micro indentation tests The micro indentation tests were repeated on three MCC 102 compacts compressed at 20 kN in the parallelepipedic die and with the two-grooved punches. In each zone (1 to 17) shown in Figure 3.27, an indentation test was performed.To illustrate the results of the micro indentation tests, the force displacement curves obtained from one compact are presented in Figure3.28. The trends observed are similar for the other compacts on which the same tests were performed. For the central region of the compact (zone 4 to 12) the results(Figures 3.28b, 3.28c and 3.28d) show that the lower region of the compact seems to be less dense than the upper part of it (observation of a higher 𝑑 𝑚𝑎𝑥 displacement). In the region of the compact made up of zones 1 to 3, an increasing density from the bottom to the top can be observed. Three measurement points do not allow us to observe a particular trend nor do they provide more information on the evolution of the density in this region. Therefore, more measurement points were made in the region defined by zones 13 to 17 (Figure3.27), the results are presented in Figure3.28e. A high density (𝑑 𝑚𝑎𝑥 low) is visible in zone 17 (under the tip of the upper right groove). This density decreases (measurement points 16 and 15) and re-increases slightly in the vicinity of the lower groove (points 14 and 13).

Figure 3 . 28 -

 328 Figure 3.28 -Force displacement curves for micro indentation tests performed on the lateral surface of an MCC 102 compact

Figure 3 .

 3 Figure 3.29 -Results of the relative density evolution obtained on an average of three compacts between two grooves (zones 13 to 17)

Figure 3 .

 3 Figure 3.30 -Side-by-side view of two Arbocel BC 1000 compacts, one without any geometric particuliarities and the second compressed with grooved punches

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Description of the Drucker Prager Cap Model . . . . . . . . . . . . . . . . . . . . . 4.3 Calibration method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Cap surface -𝑃 𝑎 , 𝑅 and 𝑃 𝑏 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Shear surface -𝑑 and 𝛽 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Elastic properties -𝐸 and 𝜈 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Wall friction -𝜇 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Results for MCC powder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Axial and radial stress for the calculation of 𝛾 and 𝑎 coefficients . . . . 4.4.2 Shear surface -cohesion 𝑑 and internal friction angle 𝛽 . . . . . . . . . . 4.4.3 Cap surface -𝑃 𝑎 , 𝑅 and 𝑃 𝑏 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Elastic properties -Young modulus 𝐸 and Poisson's ratio 𝜈 . . . . . . . 4.4.5 Wall friction -𝜇 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Highlights • Presentation of the Drucker Prager Cap (DPC) constitutive model for powder behaviour during compression • Proposal of a hybrid DPC parameters calibration method to overcome scale effects • Calibration results for MCC 102 powder and discussion with literature results

Figure 4 . 1 -

 41 Figure 4.1 -Drucker Prager Cap yield surface in the hydrostatic pressure (𝑝) -Von Mises stress (𝑞) plane

Figure 4 . 2 -

 42 Figure 4.2 -Evolution of irreversible plastic strain increment on shear surface

Figure 4 . 4 -

 44 Figure 4.4 -Failure tests for shear surface parameter calibration

Figure 4 . 5 -

 45 Figure 4.5 -Typical compression cycle in the stress strain plane

Figure 4 . 6 -Figure 4 . 7 -Figure 4 . 8 -

 464748 Figure 4.6 -Compression cycle of MCC 102 at 110 MPa in a cylindrical intrumented die -vizualisation of radial pressure and upper (axial) and lower (transmitted) punch pressures

Figure 4 .

 4 9b as a function of relative density.Finally, from the results obtained for the diametral and axial rupture stresses the cohesion (𝑑) and the internal frcition angle (𝛽) were calculated according toEquations 4.27 and 4.28 as a function of relative density. Final results are presented inFigure 4.10a and 4.10b.

( a )

 a Diametral rupture stress 𝜎 𝑑 (b) Uniaxial rupture stress 𝜎𝑐

Figure 4 . 9 -

 49 Figure 4.9 -Rupture strength results from diametral and axial rupture tests, function of relative density

Figure 4 . 10 -

 410 Figure 4.10 -Shear failure surface parameters : Internal friction angle and cohesion results as a function of relative density (computed from Equations 4.28 and 4.27 respectively)

  , 4.12a and 4.12b.

Figure 4 . 11 -

 411 Figure 4.11 -Hydrostatic yield function 𝑃 𝑏 as a function of plastic volumetric strain

Figure 4 . 12 -Figure 4 . 13 -Figure 4 . 14 -

 412413414 Figure 4.12 -Cap surface parameters, (a) 𝑃 𝑎 , (b) 𝑅 as a function of relative density

  work, for the same values of plastic strain, the measured pressures varied between 20 and 150 MPa.Concerning the Young modulus, if we compare the results obtained with the works presented by I.[START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF] L. Han et al., 2008, the values of 𝐸, reach for a relative density of 1 (full dense material), much higher values than in our study. In fact, the results presented by I.[START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF] for MCC Avicel PH102 powder the maximum Young's modulus obtained reaches 9 GPa and in L. Han et al., 2008's work the maximum Young's modulus

5. 1

 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2 Quasi-static formulation of compaction process . . . . . . . . . . . . . . . . . . . . 94 5.3 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 Spatial approximation based on the FEM . . . . . . . . . . . . . . . . . . . . . . . . 97 5.5 Geometry and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.6 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.7 Results of the simulation of the compression problem . . . . . . . . . . . . . . . . 102 5.7.1 Material properties validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.7.2 Compact with two grooves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.7.3 Compact with five grooves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.8 Impact of the two grooved punch geometry and their position . . . . . . . . . . 113 5.8.1 Cases considered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.8.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.8.4 Comparison with the five-grooved punch . . . . . . . . . . . . . . . . . . . . 125 5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Highlights • Numerical validation of model parameters of MCC 102 (elastic and Drucker Prager Cap parameters) obtained using a hybrid method (presented in Chapter 4) • Die compaction simulation using an explicit and ALE resolution method : Simulation of loading and unloading of die compaction in the cases of two-and five-grooved punches • Results of die compaction, revealed generation of density gradients between facing grooves. Investigation on the impact of the grooves' geometry and position on the density gradient

•

  Strain displacement relation (case of large deformations) • Powder behaviour law • Contact behaviour and frictions laws • Boundary conditions (for imposed external forces and displacements) • Initial conditions

Figure 5 . 1 -

 51 Figure 5.1 -Example of Lagrange, Euler and ALE representations after indentation of studied area by a punch -adapted and modified fromBakroon et al., 2017a and Bakroon et al., 2017b

Figure 5 . 2 -

 52 Figure 5.2 -Boundary conditions -Illustration with the example of the system with two-grooved punches

Figure 5 . 3 -

 53 Figure 5.3 -Explanatory sketch of the use of a preliminary step (a) for the generation of the initial mesh (b) for compression

Figure 5 . 4 -

 54 Figure 5.4 -Initial geometry (powder and tooling) for the flat punch and the two grooved punches -(𝑥, 𝑧) plane

Chapter 4 )

 4 need a first validation through a numerical calculation under the same conditions as the experimental tests. Validation was done by comparing the final relative densities and compression cycles (loading and unloading) with the use of a parallelepiped die and flat punches (punch geometry presented in Figure 2.5 and initial system in Figure 5.4a) without friction as in the experiment. The predicted compression force -displacement curves for three loading values (10 kN, 20 kN and 30 kN) are presented in Figure 5.5. These predicted cycles were superimposed to the experimental ones.

Figure 5 . 5 -

 55 Figure 5.5 -Experimental and predicted compression cycles in parrallelepiped die using flat punches with no friction for three compression forces 10 kN, 20 kN and 30 kN

and 5 .

 5 6b respectively as a function of the number of elements. The results obtained show a certain stability with respect to the mesh. The results presented in the rest of the Chapter were therefore, obtained with meshes consisting of 19152 elements.

( a )

 a Minimal and maximal relative density results (b) Minimal and maximal pressure results

Figure 5 . 6 -Figure 5 . 7 -

 5657 Figure 5.6 -Evolution of maximal and minimal values of relative density and hydrostatic pressure as a function of element numbers of the mesh

( a )Figure 5 . 8 -

 a58 Figure 5.8 -Von Mises stress distribution after compression and unloading (Pa) -Two-grooved punch

( a )Figure 5 . 9 -Figure 5 . 10 -

 a59510 Figure5.9 -Relative density distribution at loading and unloading stages -computed as a field variable in a user subroutine from Equation4.22 -Two-grooved punch

2 -

 2 Impact of the friction coefficient varying between 0.1 and 0.5 on the Von Mises equivalent stress and relative density distributions in the vicinity of the upper and lower grooves

Figure 5 .

 5 Figure 5.11 -Radial (𝑥 axis) and axial (𝑧 axis) displacements results after unloading (m) -Five-grooved punch

( a )Figure 5 .

 a5 Figure 5.12 -Von Mises stress distribution after conpression and unloading (Pa) -Five-grooved punch

( a )Figure 5 . 13 -

 a513 Figure 5.13 -Relative density distribution after compression and unloading -Fivegrooved punch

Figure 5 .

 5 Figure 5.14 -Presentation of the reference punch (without chamfer) for the parametric study of the geometric parameters of the grooves

Figure 5 .

 5 Figure 5.15 -Visualisation of the three main geometric parameters of a groove

D 3 -(Table 5 . 4 - 5 -

 3545 Summary of the magnitudes of depth (𝐷) and width (𝑊 ) with proportional variation, 𝛼 fixed at 20°D Summary of punch geometries to highlight the impact of depth variation (𝐷). Width (𝑊 ), and opening angle (𝛼) fixed at 20°r eference case. The opening angles vary between 10°and 25°, the corresponding punches are in Table5.6. Summary of punch geometries to highlight the impact of width variation (𝑊 ). Depth (𝐷), and opening angle (𝛼) fixed at 20°𝛼 Summary of punch geometries to highlight the impact of opening angle (𝛼). Depth (𝐷) and width (𝑊 ) fixed as in the reference case (𝐷, 𝑊 )=(1.7, 1.301)

  and 5.16b. As a reminder, the geometry of the reference punch is shown in Figure 5.16c . As a reminder details of the geometric sizes of the two-grooved punches are shown in Figures 2.6aand 2.6b.

( a )Figure 5 . 16 -

 a516 Figure 5.16 -Punches geometry for different groove position

Figure 5 . 18 -

 518 Figure 5.18 -Density grandient visualisation for proportional variation of depth and width in mm (constant opening angle of 20°)

Figure 5 . 19 -

 519 Figure 5.19 -Density grandient visualisation for different opening angles

( a )

 a 𝜌𝑟 -𝛼=10°(b) 𝜌𝑟 -𝛼=25°F igure 5.20 -Relative density distribution in the vicinity of the lower punch groove for an opening angle of 10°and 25°I

Figure 5 .

 5 Figure 5.21 -Density grandient visualisation for different widths (mm)

Figure 5 . 22 -

 522 Figure 5.22 -Density grandient visualisation for different depths (mm)

Figure 5 . 23 -

 523 Figure 5.23 -Density distribution for different groove depths

Figure 5 .

 5 Figure 5.24 -Density distribution for different grooves positions allong the upper and lower punches

Figure 5 . 25 -

 525 Figure 5.25 -Density gratient visualisation for different groove positions on upper and lower punches

Figure 5 .

 5 Figure 5.26 -Suggestion of two groove geometries for the decrease of the density gradient between two grooves with a constant volume requirement

Figure 5 . 27 -Figure 5 . 28 -

 527528 Figure 5.27 -Numbering of the grooves present in the five-grooved punch configuration (from 1 to 3)

( a )Figure 5 . 29 -

 a529 Figure 5.29 -Geometrical details of a groove for each of the two punches with two and five grooves (units in mm)

( a )Figure 5 . 30 -

 a530 Figure 5.30 -Global presentation of relative density distribution in the compacts at the end of the unloading process with the two proposed punch geometries

Figure 6 . 1 -

 61 Figure 6.1 -X-ray Tomography experimental setup

Figure 6 . 2 -

 62 Figure 6.2 -Sequence of steps for the analysis of a tomographic image by image binarisation

Figure 6 . 3 -

 63 Figure 6.3 -Sequence of steps for the analysis of a tomographic image by image calibration using cylindrical calibration compacts of known density

Figure 6 . 5 - 4 Figure 6 . 6 -

 65466 Figure 6.5 -Histogram (grey level values distribution) example extracted from the central slice of the calibration compact of 0.66 of relative density predented in Figure 6.4

Figure 6 . 7 -

 67 Figure 6.7 -Two grooved compact manufactured at 20 kN without lubricationestimated relative density of 0.58

Figure 6 . 8 -

 68 Figure 6.8 -Conversion of grey level tomographic image to binary image with the Default arlgorithm computed threshold presented in Table 6.1

Figure 6 . 9 -

 69 Figure 6.9 -Relative density mapping for cell sizes discretization varying from an a) 5 x 5 to e) 25 x 25 pixel area

Figure 6 . 11 -

 611 Figure 6.11 -Relative density distribution at the centre of the two-grooved compact presented in Figure 6.7

Figure 6 . 12 -

 612 Figure 6.12 -Five grooved compact of MCC 102 manufactured at 20 kN without lubrication

Figure 6 . 13 -Figure 6 . 14 -

 613614 Figure 6.13 -Relative density evolution directions in the vicinity of a groove -Five grooved compact

Figure 7 . 1 -

 71 Figure 7.1 -Location of areas of interest for simulations -tomography density comparison -case of the two-grooved compact

Figure 7 . 2 -

 72 Figure 7.2 -Comparison between experimental and predicted relative density evolution between two facing grooves (along the 𝑧 axis) -two-grooved compact

Figure 7 . 3 -Figure 7 . 4 -

 7374 Figure 7.3 -Relative density evolution obtained from X-ray Tomography between two facing grooves -variation of cell size

Figure 7 . 5 -

 75 Figure 7.5 -Location of areas of interest for simulations -tomography density comparison -case of the five-grooved compact

Figure 7. 5

 5 As seen in the simulation results in Section 5.8.4 of Chapter 5 density gradients generated between the grooves that face each other evolve as a gradient. The supperposition of the density gradients predicted and obtained by X-ray Tomography are presented in Figures 7.6, 7.7 and 7.8 for the three grooves present in the symmetrized part of the compact. The numbering of the grooves is provided in Figure 5.28 in Chapter 5. The experimental points are in good agreement with the predictions obtained by X-ray Tomography regarding the density gradients between the different grooves. Indeed, as seen in the Section 5.8.4 of Chapter 5, the density variations show a change in trend with an increasing distance of the groove from the die. The gradient in Figure 7.8 (closer to the edges) shows an increase in density in its central zone. On the contrary, the curves in Figures 7.6 and 7.7 have a more constant density in this same central region of the gradient (namely between 4 and 10 mm high).

Figure 7 . 6 -Figure 7 . 8 -

 7678 Figure 7.6 -Comparison between experimental and predicted relative density evolution between the 1 𝑠𝑡 facing grooves (along the 𝑧 axis) (five-grooved compact)

Figure 7 . 9 -

 79 Figure 7.9 -External and internal representation of an Arbocel BC 1000 compact with two grooves

  main steps of the thesis . . . . . . . . . . . . . . . . . . 1.1 Visualisation of the different stages of die compaction (upper and lower punch in grey and die in hatched lines) . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Illustration of a typical compression cycle plot . . . . . . . . . . . . . . . . . . 1.3 Schematic representation of (a) capping, (b) lamination and (c) cracking . . . 1.4 Detailed view of the unloading phase -demonstration of springback in the die 1.5 Illustration of a typical Heckel plot . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Illustration of typical Kawakita plot . . . . . . . . . . . . . . . . . . . . . . . 1.7 Density distribution results obtained by I. Sinka et al., 2003 for biconvex tablets of microcrystalline cellulose MCC Avicel PH 102 . . . . . . . . . . . . 1.8 Schematic representation of an X-ray Tomograph . . . . . . . . . . . . . . . . 1.9 Detection of an internal crack by X-ray Tomography in a compact of anhydrous calcium phosphate powder -Mazel et al., 2018 . . . . . . . . . . . . . . . . . 1.10 Detection of an internal crack and density distribution by X-ray Tomography of compacted aluminium powder parts -McDonald et al., 2009 . . . . . . . .1.11 Visualisation of the density distribution observed by X-ray Tomography on two types of scored tablets -I. Sinka et al., 2004 . . . . . . . . . . . . . . . . 1.12 Images obtained by X-ray Tomography showing the single and double furrows compacts -Laity, 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Overview of the geometry of the compacts studied in this study . . . . . . . . 2.1 MCC 102 powder -SEM image and loose powder bed visual . . . . . . . . . . 2.2 Arbocel BC 1000 powder -SEM image and loose powder bed visual . . . . . 2.3 Arbocel BWW 40 powder -SEM image and loose powder bed visual . . . . . 2.4 Flat and grooved punches for the parallepipedic die . . . . . . . . . . . . . . 2.5 2D schematic cross-section of the three different punches for parallelepiped die compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Geometrical dimensions of the grooved punches (mm) . . . . . . . . . . . . . 2.7 Assembly of the compaction equipment -presentation of the parralelpiped die and its guiding system integrated in the press . . . . . . . . . . . . . . . . . . 2.8 Tools for micro-indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Tomography equipement-emitter and detector . . . . . . . . . . . . . . . . . 3.1 Example of compaction cycles for three powders in a cylindrical die with an initial height of 20 mm and a loading force fixed at 30 kN . . . . . . . . . . . 3.2 Heckel curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Kawakita curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Evolution of the springback index for cylindrical compacts of MCC 102, Arbocel BC 1000 and Arbocel BWW 40 . . . . . . . . . . . . . . . . . . . . . 3.5 Evolution of diametrical rupture stress as a function of relative density for flat faced cylindrical compacts . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Examples of fracture profiles obtained from diametrical fracture tests for Arbocel BC 1000 and Arbocel BWW 40 compacts . . . . . . . . . . . . . . . 3.7 Curve of applied force versus displacement, resulting from diametrical fracture tests on two Arbocel BC 1000 cylindrical compacts of similar relative density (0.86 and 0.87) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Example of compaction cycles for three powders in a parallelepiped die and flat pucnhes with an initial filling height of 4 cm and a loading force fixed at 20 kN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Example of powder compacted in a parallelepiped die using flat punches (left to right : MCC 102, Arbocel BWW 40 and Arbocel BC 1000) . . . . . . . . . 3.10 Lateral surface view (in the x, z plane) of the flat parallelepipedic compacts obtained after a 20 kN compression . . . . . . . . . . . . . . . . . . . . . . . 3.11 Example of powder compacted in a parallelepiped die using two-grooved punches (left to right : MCC 102, Arbocel BWW 40 and Arbocel BC 1000) . 3.12 Lateral view (in the x, z plane) of a two-grooved parallelepipedic compact obtained after a 20kN compression of MCC 102 powder . . . . . . . . . . . . 3.13 Side view (in the x, z plane) of examples of Arbocel BC 1000 compacts compressed at different forces using two-grooved punches . . . . . . . . . . . 3.14 Detailed view of a crack observed on an Arbocel BC 1000 compact at a load of 10 kN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Example of an Arbocel BC 1000 compact compressed at 20 kN with twogrooved punches. Illustration of the deformation layers between two grooves . 3.16 Side view (in the x, z plane) of examples of Arbocel BWW 40 compacts compressed at different forces using two-grooved punches . . . . . . . . . . . 3.17 Detailed view of cracks observed on Arbocel BWW 40 compacts at a compression load of 10 kN and 20 kN . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Example of powder compacted in parallelepiped die using five-groove punches (left to right : MCC 102, Arbocel BWW 40 and Arbocel BC 1000) . . . . . . 3.19 Lateral view (in the x, z plane) and upper surface view (in the y, x plane) of a five-grooved parallelepipedic compact obtained after a 30 kN compression of MCC 102 powder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Schematic representation of the density gradient observable between two grooves that face each other . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.18 Density grandient visualisation for proportional variation of depth and width in mm (constant opening angle of 20°) . . . . . . . . . . . . . . . . . . . . . . 5.19 Density grandient visualisation for different opening angles . . . . . . . . . . 5.20 Relative density distribution in the vicinity of the lower punch groove for an opening angle of 10°and 25°. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.21 Density grandient visualisation for different widths (mm) . . . . . . . . . . . 5.22 Density grandient visualisation for different depths (mm) . . . . . . . . . . . 5.23 Density distribution for different groove depths . . . . . . . . . . . . . . . . . 5.24 Density distribution for different grooves positions allong the upper and lower punches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.25 Density gratient visualisation for different groove positions on upper and lower punches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.26 Suggestion of two groove geometries for the decrease of the density gradient between two grooves with a constant volume requirement . . . . . . . . . . .5.27 Numbering of the grooves present in the five-grooved punch configuration (from 1 to 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.28 Density gradient visualisation for the three grooves of the symetrical part of the five-grooved compact -superposition with the density gradient obtained in the reference case with two grooves . . . . . . . . . . . . . . . . . . . . . . 5.29 Geometrical details of a groove for each of the two punches with two and five grooves (units in mm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.30 Global presentation of relative density distribution in the compacts at the end of the unloading process with the two proposed punch geometries . . . . . . . 6.1 X-ray Tomography experimental setup . . . . . . . . . . . . . . . . . . . . . . 6.2 Sequence of steps for the analysis of a tomographic image by image binarisation135 6.3 Sequence of steps for the analysis of a tomographic image by image calibration using cylindrical calibration compacts of known density . . . . . . . . . . . . 6.4 Tomographic image slice of a calibration compact (example of the case 𝜌 𝑟 = 0.66) and identification (in red) of the 11 mm circular analysis zone . . . . . 6.5 Histogram (grey level values distribution) example extracted from the central slice of the calibration compact of 0.66 of relative density predented in Figure 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Plot presenting the evolution of X-ray Tomography grey level according to the relative density for MCC 102 powder. Fitting curve : 𝐺𝐿 = 173.57𝜌 𝑟 + 82.428, 𝑅 2 = 0.9888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Two grooved compact manufactured at 20 kN without lubrication -estimated relative density of 0.58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Conversion of grey level tomographic image to binary image with the Default arlgorithm computed threshold presented in Table 6.1 . . . . . . . . . . . . . 6.9 Relative density mapping for cell sizes discretization varying from an a) 5 x 5 to e) 25 x 25 pixel area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Relative density evolution between two facing grooves for 5 x 5, 10 x 10, 15 x 15, 20 x 20 and 25 x 25 pixels cell size . . . . . . . . . . . . . . . . . . . . . . 6.11 Relative density distribution at the centre of the two-grooved compact presented in Figure 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Five grooved compact of MCC 102 manufactured at 20 kN without lubrication146 6.13 Relative density evolution directions in the vicinity of a groove -Five grooved compact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Relative density distribution at the centre of the Five grooved compact presented in Figure 6.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Location of areas of interest for simulations -tomography density comparison -case of the two-grooved compact . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Comparison between experimental and predicted relative density evolution between two facing grooves (along the 𝑧 axis) -two-grooved compact . . . . . 7.3 Relative density evolution obtained from X-ray Tomography between two facing grooves -variation of cell size . . . . . . . . . . . . . . . . . . . . . . . 7.4 Relative density evolution along the 𝑧 axis between facing grooves -comparison with numerical results computed with differents friction coefficients varying in range from 0.1 to 0.5 (dotted lines) . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Location of areas of interest for simulations -tomography density comparison -case of the five-grooved compact . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Comparison between experimental and predicted relative density evolution between the 1 𝑠𝑡 facing grooves (along the 𝑧 axis) (five-grooved compact) . . . 7.7 Comparison between experimental and predicted relative density evolution between the 2 𝑛𝑑 facing grooves (along the 𝑧 axis) (five-grooved compact) . . . 7.8 Comparison between experimental and predicted relative density evolution between the 3 𝑟𝑑 facing grooves (along the 𝑧 axis) (five-grooved compact) . . . 7.9 External and internal representation of an Arbocel BC 1000 compact with two grooves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Summary of 𝑇 ℎ values obtained from four distinct thresholding algorithms, comparison between the mean relative density calculated from binary image with the mean relative density of the 20 kN compacted sample (𝜌 𝑟 = 0.58) . . 7.1 Average relative density 𝜌 𝑟 values calculated from X-ray Tomography and numerical prediction for the regions defined in Figure 7.1 . . . . . . . . . . . 7.2 Average relative density 𝜌 𝑟 values calculated from X-ray Tomography and numerical prediction for the regions defined in Figure 7.5 . . . . . . . . . . . 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  Cottrino et al., 2013) S. Cottrino, Y. Jorand, E. Maire, and J. Adrien. "Characterization by X-ray tomography of granulated alumina powder during in situ die compaction." In: Materials characterization 81 (2013), pp. 111-123 (cit. on p. 133). (Coube et al., 2005) O. Coube, A. Cocks, and C.-Y. Wu. "Experimental and numerical study of die filling, powder transfer and die compaction." In: Powder Metallurgy 48.1 (2005), pp. 68-76 (cit. on p. 38). (Coube et al., 2000) O. Coube and H. Riedel. "Numerical simulation of metal powder die compaction with special consideration of cracking." In: Powder Metallurgy 43.2 (2000),

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  Coube, 1998) O. Coube. "Modelisation et simulation numerique du compactage de poudre en matrice fermee avec etude des formations de fissures." PhD thesis. Paris 6, 1998 (cit. on p. 107). (Coube et al., 2008) O. Coube and P. Jonsén. "Case Studies: Discussion and Guidelines." In: Modelling of Powder Die Compaction.

  (L. Han et al., 2011) L. Han, P. R. Laity, R. E. Cameron, and J. A. Elliott. "Density and plastic strain evaluations using small-angle X-ray scattering and finite element simulations for powder compacts of complex shape." In: Journal of materials science46.18 (2011), pp. 5977-5990 (cit. on pp. 36, 40, 94). (L. H. Han et al., 2008) L. H. Han, J. Elliott, S. Best, R. Cameron, A. Bentham, A. Mills, G. Amidon, and B. Hancock. "Numerical simulation on pharmaceutical powder compaction." In: Materials Science Forum. Vol. 575. Trans Tech Publ. 2008, pp. 560-565 (cit. on p. 25). (Hardman et al., 1973) J. Hardman and B. Lilley. "Mechanisms of compaction of powdered materials." In: Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 333.1593 (1973), pp. 183-199 (cit. on p. 22). (Harthong et al., 2009) B. Harthong, J.-F. Jérier, P. Dorémus, D. Imbault, and F.-V. Donzé. "Modeling of high-density compaction of granular materials by the discrete element method." In: International Journal of Solids and Structures 46.18-19 (2009), pp. 3357-3364 (cit. on p. 38). (He et al., 2017) Y. He, T. Evans, A. Yu, and R. Yang. "DEM investigation of the role of friction in mechanical response of powder compact." In: Powder Technology 319 (2017), pp. 183-190 (cit. on p. 22). (Heckel, 1961a) R. Heckel. "An analysis of powder compaction phenomena." In: Trans. Metall. Soc. AIME 221 (1961), pp. 671-675 (cit. on p. 28). (Heckel, 1961b) R. Heckel. "Density-pressure relationships in powder compaction." In: Trans Metall Soc AIME 221.

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  Klevan et al., 2009) I. Klevan, J. Nordström, A. Bauer-Brandl, and G. Alderborn. "On the physical interpretation of the initial bending of a Shapiro-Konopicky-Heckel compression profile." In: European journal of pharmaceutics and biopharmaceutics 71.2 (2009), pp. 395-401 (cit. on p. 28). (Klinzing et al., 2019) G. R. Klinzing and G. M. Troup. "Modeling the air pressure increase within a powder bed during compression-a step toward understanding tablet defects." In: Journal of Pharmaceutical Sciences 108.6 (2019), pp. 1991-2001 (cit. on pp. 3, 15, 23-25). (K. Kondoh et al., 2000) K. Kondoh, R. Watanabe, and H. Hashimoto. "Analysis of compaction behaviour of wet granulated aluminium alloy powder." In: Powder metallurgy 43.4 (2000), pp. 359-363 (cit. on pp. 21, 22). (M. Kondoh et al., 1996) M. Kondoh and S. Takemoto. "Visualization of powder behavior for gravity filling." In: Journal of the Japan Society of Powder and Powder Metallurgy 43.11 (1996), pp. 1339-1342 (cit. on p. 23). (Korachkin et al., 2008) D. Korachkin, D. T. Gethin, R. W. Lewis, and J. H. Tweed. "EFFECT OF DIE FILLING ON POWDER COMPACTION." In: International Journal of Powder Metallurgy 44.1 (2008) (cit. on p. 23). (Kramer, 2010) D. Kramer. "Compaction Equipment: Powder Compaction Tooling-Technology to Reduce Cost and Increase Quality." In: European Congress and Exhibition on Powder Metallurgy. European PM Conference Proceedings. The European Powder Metallurgy Association. 2010, p. 1 (cit. on p. 25). (Krok et al., 2014) A. Krok, M. Peciar, and R. Fekete. "Numerical investigation into the influence of the punch shape on the mechanical behavior of pharmaceutical powders during compaction." In: Particuology 16 (2014), pp.

  3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Results from cylindrical die compaction tests . . . . . . . . . . . . . . . . . . . . . 3.2.1 Compression cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Heckel model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Kawakita model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Springback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Diametrical fracture strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results and observations from parallelepiped die compaction tests . . . . . . 3.3.1 Compression cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Optical microscope observations . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Micro indentation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Calibration -on MCC 102 powder . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Results on a two-grooved compact . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Les résultats respectifs des Chapitre 5 et 6 sont comparés en termes de distribution de densité relative. Pour ce faire, des zones d'intérêts ont été sélectionnées (sous les rainures, dans les régions de plus faible densité, sur les bords et au centre) afin de comparer les densités relatives prédites et mesurées. Dans le cas du compact à deux rainures un écart maximal de 7.7 % à été noté sur les bords du compact. Dans les régions au voisinage des rainures les écarts ne dépassent pas 3.24 %. Pour le compact à cinq rainures l'écart maximal calculé atteint 6.36 % sur les bords, dans le reste du compact les écarts ne dépassent pas 1.9 %. En plus de ces comparaisons, les gradients de densité ont également été mis en parallèles. Un exemple de la superposition entre le gradient mesuré par tomographie entre deux rainures et le gradient obtenu à partir des calculs numériques dans le cas du compact à deux rainures est présenté dans la Figure7.2. Les résultats numériques et expérimentaux sont en bon accord.Ces observations permettent de vérifier de la validité et du caractère prédictif des résultats numériques obtenus par simulation éléments finis.

	Dans les deux compacts les distributions obtenues présentent des régions de forte densité sous
	les rainures qui se dissipent vers de plus faibles densités dans les épaules des rainures et vers
	le centre du compact. Globalement la région centrale des compacts révèle que la densité y est
	relativement homogène. Ces observations rejoignent celles faites dans le Chapitre précédent
	où les simulations numériques ont été faites dans les mêmes conditions expérimentales.

). Les deux méthodes d'exploitation des données ont été appliquées au cas du compact à deux rainures. Suite aux conclusions émises, seule la méthode par calibration a été appliqué au compact à cinq rainures. En effet, le choix final s'est porté sur la méthode de calibration en raison des biais introduits par la méthode qui fait intervenir la binarisation de l'image tomographique.

Les résultats finaux , obtenus par calibration, sont présentés dans les Figures 6.11 et 6.14.

Le Chapitre 7 se focalise sur la validation du modèle numérique qui tend à justifier de la robustesse du modèle appliqué.

  sur lequel la simulation de compression a été réalisée, une proposition de calibration hybride a été faite. La méthode de calibration hybride utilisée pour obtenir les paramètres qui définissent le modèle Drucker Prager Cap combine les résultats de calibration obtenus sur un compact cylindrique de taille standard avec les essais de compression réalisés sur un compact parallélépipédique d'un volume trente-trois fois supérieur.Pour la modélisation, le problème de compression a été approché par la méthode des éléments finis. La prise en compte des conditions aux limites (contact et frottement) et des conditions de chargement complètent la modélisation du problème aux limites. Les résultats de la simulation par éléments finis ont été réalisés sur des compacts parallélépipédiques à deux et cinq rainures. Un constat commun à tous les résultats de calcul est que la variation de densité relative entre deux rainures se faisant face présente un gradient de densité. En effet, cette région (identifiée comme critique) présente des pics de densité juste en dessous des extrémités des rainures qui diminuent vers le centre du compact.

	Perspectives
	Ces travaux ont ouvert des perspectives intéressantes et prometteuses, détaillées ci-dessous,
	qui pourraient être exploitées par la suite :
	• Optimisation de la géométrie de l'outillage. Les études paramétriques réalisées ont per-
	mis de mettre en évidence les paramètres qui ont une influence plus ou moins grande
	sur le gradient de densité observé entre deux rainures. Ces gradients de densité ont
	été identifiés comme une cause potentielle de défauts dans les matériaux sujets à la
	décohésion des particules lors de la décharge. Il serait donc intéressant d'optimiser
	cette géométrie pour potentiellement réduire l'apparition de défauts.
	powder compaction with consideration of a deep grooved punch. Powder Technology,
	395, 681-694
	• Impact of punch geometry on compacted powder properties, STPMF : Science et
	Technologie des Poudres et Matériaux Frittés, Juillet 2021

Une double approche, expérimentale et numérique, a été envisagée pour obtenir des éléments de réponse à cette problématique. Les essais de compression réalisés à l'aide de la matrice parallélépipédique à poinçons rainurés ont permis d'observer la présence de fissures localisées entre les rainures. Ces observations n'étant pas systématiques pour tous les matériaux, l'étude expérimentale nous a permis de conclure que l'apparition de fissures était à la fois une conséquence du comportement de la poudre et de la géométrie des poinçons (rainurés ou non). La présence de fissures localisées dans la région inter-rainures permet de cibler cette région comme étant particulièrement sensible aux contraintes, notamment lors de la décharge, qui peuvent conduire à la décohésion des particules et donc, potentiellement à l'apparition de fissures comme dans les compacts Arbocel BC 1000. La nature de la poudre d'Arbocel BC 1000 ne permettant pas de caractériser ses propriétés mécaniques (en utilisant la procédure présentée au Chapitre 4), le choix du matériau pour la modélisation numérique a été celui de la MCC 102, considéré dans cette étude comme un matériau modèle.

La modélisation numérique du problème de compression a ajouté à l'étude une compréhension plus approfondie et détaillée des différents mécanismes impliqués lors de la compression en nous permettant d'accéder à des quantités locales telles que les champs de contraintes, les distributions de densité et les déplacements. Le modèle de comportement mécanique de la poudre en compression choisi est le modèle de Drucker Prager Cap. Ce modèle permet de prendre en compte les propriétés mécaniques de la poudre telles que la cohésion, l'angle de frottement interne, l'augmentation de la résistance mécanique du compact avec la pression appliquée ainsi que l'évolution de ses propriétés élastiques. Afin de compenser les grandes dimensions du compact La géométrie des rainures entraînant une distribution hétérogène de la densité dans son voisinage, une étude paramétrique a été menée sur les paramètres géométriques d'une rainure.

Les résultats montrent l'impact des différents paramètres choisis sur le gradient de densité présent dans cette région du compact. Afin de valider les prédictions obtenues numériquement et ainsi de valider le modèle utilisé, des analyses tomographiques ont été réalisées sur les mêmes compacts de MCC 102 pour obtenir la distribution de densité.

L'ensemble des travaux de cette thèse a montré la complexité du procédé de compression de poudre lorsque l'outillage utilisé présente une géométrie complexe et est de grande taille. D'un point de vue numérique la présence des rainures, entrainant un écoulement complexe, a présenté un réel challenge. La méthode explicite avec l'approche ALE a alors montré son efficacité pour résoudre des problèmes de compression générant des fortes distorsions. La validation du modèle numérique par tomographie à rayons X nous permet d'affirmer du caractère prédictif du modèle mis en place. Elle démontre également que la procédure hybride permet l'identification des paramètres DPC et élastiques pour la simulation. Cette méthodologie pouvant alors être étendue à d'autres formes et tailles d'outillage ne disposant pas d'instrumentation. Les travaux de cette thèse on fait l'objet d'un article scientifique et d'une présentation orale dans un colloque national : • Alonso Aruffo, G., Michrafy, M., Oulahna, D., & Michrafy, A. (2022). Modelling • Amélioration du modèle numérique. La première étape serait de prendre en compte le comportement élastique non-linéaire lors de la décharge. En effet, dans notre travail ainsi que dans la majorité des études sur la compression de poudres, le module d'élasticité est supposé constant lors de la décharge, ce qui n'est pas vraiment le cas. La prise en compte d'une évolution non-linéaire de l'élasticité permettrait de mieux comprendre le retour élastique des compacts lors des étapes de décharge et d'éjection. Ceci est particulièrement important pour la prédiction des défauts dans les compacts, car le retour élastique peut initier des fissures (Coube et al., 2000, L. Han et al., 2008). Dans un second temps, il serait intéressant d'envisager l'intégration d'un modèle d'endommagement local de type Chaboche et Lemaitre (Lemaitre, 1988) ou d'un modèle non-local de type second gradient (Peerlings et al., 1996). Ces approches demandent un certain investissement et nécessiteraient l'implémentation du comportement du matériau via une routine de type UMAT ou VUMAT. Il serait alors intéressant d'appliquer ce type de modèle à un matériau pouvant présenter des fissures comme l'Arbocel mais dont le comportement mécanique serait plus facilement caractérisable.

  Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

	Introduction and objective

  Equipement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Microcrystalline cellulose (MCC 102) . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Arbocel BC -1000 and Arbocel BWW -40 . . . . . . . . . . . . . . . . . . 2.3 2.3.1 Powder characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Mechanical characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Die and punches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Micro indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 2 .

 2

	3 )

1 -MCC vivapur 102 powder properties

Table 2 . 2 -

 22 PowderMean particle size (𝜇𝑚) True density (𝑔/𝑐𝑚 3 ) Bulk density (𝑔/𝑐𝑚 3 ) Arbocel BC 1000 and BWW 40 powder properties

	Arbocel BC-1000	700	1.595 ± 0.0025	0.05 ± 0.02
	Arbocel BWW-40	200	1.682 ± 0.0046	0.13 ± 0.02

  Micro indentation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Results from cylindrical die compaction tests . . . . . . . . . . . . . . . . . . . . . 3.2.1 Compression cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Heckel model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Kawakita model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Springback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Diametrical fracture strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results and observations from parallelepiped die compaction tests . . . . . . . 3.3.1 Compression cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Optical microscope observations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 3.4.1 Calibration -on MCC 102 powder . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Results on a two-grooved compact . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5

Table 3 . 1 -

 31 Heckel constants and mean yield pressures

	Powder	K (1/MPa)	A	𝑟 2	𝑃 𝑦 (MPa)
	MCC 102	0.009	0.766 0.9988	111
	Arbocel BC 1000	0.0071	0.5912 0.9951	140
	Arbocel BWW 40	0.0056	0.8492 0.9999	178

.1.

As suggested by Peciar et al., 2016 the linear regressions were done in such a way as to have an 𝑟 2 greater than 0.99, in order to obtain more accurate results.

Table 3 . 2 -

 32 Kawakita constants and mean yield strengthThe parameters 𝑎, 𝑏 and 𝑃 𝑘 obtained from the application of the Kawakita equation ( 𝑃 𝐶 =

	𝑉0-𝑉𝑝
	𝑉0

3) allows, as well as the Heckel model, to represent the evolution of the pressure with the volume reduction of the powder during compression. The Kawakita equation introduces the parameter 𝐶 defined by 𝐶 = to represent the volume reduction. Where 𝑉 0 is the initial volume of the powder and 𝑉 𝑝 the volume under compression.

  to determine each parameter of the model. As presented previously five parameters are required ; 𝑑 and 𝛽 for the shear failure surface 𝐹 𝑠 and 𝑃 𝑎 , 𝑃 𝑏 and 𝑅 for the Cap surface

	𝐹 𝑐 .
	The standard procedure for calibration was presented by Brewin et al., 2008. This standard
	experimental procedure requires a fully instrumented die (usualy for cylindrical shaped

compacts or as proposed by Zhang et al., 2010 for a cubic die) for the determination of the Cap surface parameters. Concerning the shear failure surface parameters, at least two failure tests are required (usualy diametral compression and uniaxial compression).

1 Hydrostatic pressure and Von Mises stress

	Model parameters	Intermediate parameters	Experimental method
			Radial stress (𝜎 𝑟 ) and axial stress
	-	𝛾 and 𝑎	(𝜎 𝑧 ) : compression in a cylindrical
			intrumented die
			Axial stress (𝜎 𝑧 ) : parallelepiped
	𝑝, 𝑞	𝜎 𝑧 , 𝛾 and 𝜎 𝑥 (𝜎 𝑧 , 𝛾)	die compression,
			𝛾 : instrumented cylindrical die
			Diametral compressive test rupture
	𝑑, 𝛽	𝐹 𝑑 and 𝐹 𝑐	Force (𝐹 𝑑 ) and uniaxial failure test rupture force (𝐹 𝑐 ) on cylindrical
			compacts
	𝑃 𝑎 , 𝑃 𝑏 , 𝑅	𝑝, 𝑞, 𝑑 and 𝛽	As above
			Shear modulus (𝐺) and Bulk mod-
			ulus (𝐾) : Unloading curve from
	𝐸, 𝜈	𝐺, 𝐾 and 𝑎	parallelepiped die compaction
			𝑎 : compression in a cylindrical in-
			strumented die
	𝜇	-	Compression in a cylindrical instru-mented die

𝑎 , 𝑅 and 𝑃 𝑏

4.3.1.

For the standard cylindrical compact case, hydrostatic pressure 𝑝 and Von Mises stress 𝑞 are calculated from radial (𝜎 𝑟 ) and axial (𝜎 𝑧 ) stresses obtained from compression tests in a cylindrical intrumented die.

Table 4 .

 4 Radial stress can be expressed depending on a parameter 𝛾(𝜌 𝑟 ) (density dependent) during the loading stage following Equation4.10.

	𝜎 𝑟 = 𝛾(𝜌 𝑟 )𝜎 𝑧	(4.10)
	During unloading radial stress varies linearly according to the axial stress as sugested in
	Equation 4.11.
	𝜎 𝑟 = 𝑎(𝜌 𝑟 )𝜎 𝑧 + 𝑏(𝜌 𝑟 )	(4.11)
	The work presented by Zhang et al., 2010 presents the DPC model calibration using an
	instrumented cubic shaped die. They considered the following relationship between 𝜎 𝑥 (or
	𝜎

1 -Summary of powder behaviour model parameters and the experimental procedures to obtain then

Figure 4.3 -Radial -axial stress during loading -unloading cycle in a cylindrical instrumented die 𝑦 ) and 𝜎 𝑧 :

Table 4 . 2

 42

	The compression results of parallelepiped die compression tests at different loadings are
	presented in Table 4.2.
	Compression pressure (MPa)	Plastic volumetric 𝑣𝑜𝑙 strain 𝜖 𝑝𝑙	Relative density (under 𝜌 𝑟𝑐 compression)	Final relative density 𝜌 𝑟𝑓
	5.84	0.69	0.420	0.398
	8.17	0.79	0.465	0.440
	11.67	0.88	0.508	0.479
	15.17	0.97	0.553	0.514
	17.51	0.99	0.565	0.530
	21.01	1.05	0.598	0.560
	23.34	1.08	0.620	0.580
	35.02	1.20	0.700	0.650

3 Cap surface -𝑃 𝑎 , 𝑅 and 𝑃 𝑏 Hystrostatic pressure 𝑝 and Von Mises stress 𝑞 were computed as in Equations 4.17 and 4.18 from the 𝛾 factor and axial stress obtained from compression in parallelepiped die using flat punches and a lubricated die. -Compaction results from MCC 102 powder in parallelepiped die and flat punches Following calculation of hydrostratic pressure and Von Mises stress, the evolution parameter 𝑃 𝑎 , the hydrostatic yield pressure 𝑃 𝑏 and the Cap exentricity 𝑅 were calculated as functions of relative density and plastic volumetric strain from Equations 4.19, 4.20 and 4.21. Results are ploted in Figures 4.11

Table 5 .

 5 1 Δ𝜌 𝑟 (𝜌 𝑟𝑚𝑖𝑛 -𝜌 𝑟𝑚𝑎𝑥 )

	(0.6, 0.459)	-0.1552	0.16	(0.59 -0.75)
	(0.8, 0.612)	-0.1168	0.14	(0.59 -0.74)
	(1.0, 0.765)	-0.0792	0.16	(0.58 -0.74)
	(1.4, 1.071)	-0.0672	0.15	(0.58 -0.73)
	(1.7, 1.301)	-0.0720	0.16	(0.57 -0.73)
	(2.0, 1.530)	-0.0608	0.17	(0.56 -0.74)

7 -Relative density decrease slope in zone 1 of density gradient and Δ𝜌 𝑟 for homotetical groove size variation

Impact of the opening angle 𝛼

Table 5 . 8 -

 58 .5. Relative density decrease slope in zone 1 of density gradient and Δ𝜌 𝑟 for variable groove widths

	Width 𝑊 (mm) Slope in zone 1 Δ𝜌 𝑟
	0.612	-0.0987	0.18
	0.765	-0.0792	0.16
	1.071	-0.0338	0.13

6.6.1 Binary image analysis results 6.6.1.1 Choice of the threshold value

	Threshold arlgorithm	Threshold value 𝑇 ℎ	Mean relative density 𝜌 𝑟	Deviation from experimental density (%)
	Default	173	0.577	0.52
	Huang	175	0.525	9.48
	Otsu	174	0.551	5.00
	Li	175	0.525	9.48

Table 6 . 1 -

 61 Summary of 𝑇 ℎ values obtained from four distinct thresholding algorithms, comparison between the mean relative density calculated from binary image with the mean relative density of the 20 kN compacted sample (𝜌 𝑟 = 0.58) As discussed previously, the choice of the threshold value is a delicate task. The decision was made to choose it's value in order to get a mean relative density equivalement to the mean relative density estimated experimentally from the geometric dimentions and mass of the compact presented in Figure6.7.Four threshord algorithms were applied to the studied image. Resulting threshold (𝑇 ℎ ) values are presented in Table6.1. As mentioned above this value separates the grey level values into two categories: material (white pixels) and porosities (black pixels).

Table 7 .

 7

	Zone	X-ray Tomography relative density	FEM simula-tion relative density	Relative devi-ation (%)
		0.725	0.712	1.79
		0.537	0.534	0.55
		0.595	0.596	0.16
		0.684	0.665	2.77
		0.602	0.610	1.32
		0.609	0.576	5.41
		0.622	0.577	7.70
		0.684	0.672	1.75
		0.522	0.526	0.76
		0.501	0.487	2.79
		0.524	0.507	3.24
		0.493	0.487	1.21
		0.609	0.580	4.76
		0.618	0.589	4.69
		0.591	0.598	1.18
		0.556	0.577	3.77

1 -Average relative density 𝜌 𝑟 values calculated from X-ray Tomography and numerical prediction for the regions defined in Figure

7

.1

Table 7 . 2 -

 72 Table 7.2.The vast majority of the different areas evaluated show good agreement with a relative error less than or equal to 4 %. Only the areas numbered 11 and 12 show a larger deviation : 6.13 % and 6.36 % respectively. The trends observed are similar in both cases, namely that the regions of high density are located under the tips of the grooves and under the chamfer of the upper punch, the areas of lower density are well located in the inter-groove regions and the centre of the compact has a more homogeneous density. Average relative density 𝜌 𝑟 values calculated from X-ray Tomography and numerical prediction for the regions defined in

	Zone	X-ray Tomography relative density	FEM simula-tion relative density	Relative devi-ation (%)
		0.728	0.719	1.23
		0.558	0.549	1.61
		0.589	0.584	0.84
		0.730	0.722	1.09
		0.592	0.568	4.05
		0.568	0.571	0.52
		0.751	0.747	0.53
		0.616	0.628	1.94
		0.791	0.776	1.89
		0.688	0.675	1.88
		0.685	0.643	6.13
		0.550	0.515	6.36
		0.468	0.463	1.06
		0.736	0.715	2.85
		0.546	0.547	0.18
		0.540	0.537	0.55
		0.737	0.733	0.54
		0.554	0.554	0.00
		0.533	0.532	0.18
		0.752	0.749	0.39
		0.618	0.611	1.13
		0.610	0.611	0.16
		0.631	0.621	1.58

  . Modelling powder compaction with consideration of a deep grooved punch. Powder Technology, 395, 681-694 • Impact of punch geometry on compacted powder properties, STPMF : Science et Technologie des Poudres et Matériaux Frittés, July 2021

  p. 105). (Cundall et al., 1992) P. A. Cundall and R. D. Hart. "Numerical modelling of discontinua."In: Engineering computations(1992) (cit. on pp. 6, 38).(Cundall et al., 1979) P. A. Cundall and O. D. Strack. "A discrete numerical model for granular assemblies." In: geotechnique 29.1 (1979), pp. 47-65 (cit. on pp. 6, 38). (Davis et al., 2006) G. Davis and J. Elliott. "Artefacts in X-ray microtomography of materials." In: Materials science and technology 22.9 (2006), pp. 1011-1018 (cit. on p. 34). (Denny, 2002) P. Denny. "Compaction equations: a comparison of the Heckel and Kawakita equations." In: Powder Technology 127.2 (2002), pp. 162-172 (cit. on pp. 26, 27). (Desbois, 2021) L. Desbois. "Comportement mécanique des poudres et comprimés pharmaceutiques en compression: dépendance à la vitesse de déformation." PhD thesis. Université de Bordeaux, 2021 (cit. on p. 26). (Garner et al., 2018) S. Garner, J. Strong, and A. Zavaliangos. "Study of the die compaction of powders to high relative densities using the discrete element method." In: Powder Technology 330 (2018), pp. 357-370 (cit. on p. 38). (Glass et al., 1997) S. J. Glass and K. G. Ewsuk. "Ceramic powder compaction." In: Mrs Bulletin 22.12 (1997), pp. 24-28 (cit. on p. 24). (Güner et al., 2015) F. Güner, Ö. N. Cora, and H. Sofuoğlu. "Numerical modeling of cold powder compaction using multi particle and continuum media approaches." In: Powder Technology 271 (2015), pp. 238-247 (cit. on p. 15). (Hamad et al., 2007) M. L. Hamad, C. D. Ellison, M. A. Khan, and R. C. Lyon. "Drug product characterization by macropixel analysis of chemical images." In: Journal of pharmaceutical sciences 96.12 (2007), pp. 3390-3401 (cit. on p. 31). (L. Han et al., 2008) L. Han, J. Elliott, A. Bentham, A. Mills, G. Amidon, and B. Hancock. "A modified Drucker-Prager Cap model for die compaction simulation of pharmaceutical powders." In: International Journal of Solids and Structures

  Kachrimanis et al., 2005) K. Kachrimanis and S. Malamataris. "Compact size and mechanical strength of pharmaceutical diluents." In: European journal of pharmaceutical sciences 24.2-3 (2005), pp. 169-177 (cit. on p. 30). (Kadiri et al., 2013) M. S. Kadiri and A. Michrafy. "The effect of punch's shape on die compaction of pharmaceutical powders." In: Powder technology 239 (2013), pp. 467-477 (cit. on pp. 8, 36, 74, 91). (Kadiri et al., 2018) M. S. Kadiri and H. Moussaoui. "Numerical analysis of the density distribution within scored tablets." In: International Journal of Innovation Engineering and Science Research (2018) (cit. on pp. 36, 38). (Kandeil et al., 1977) A. Kandeil, M. De Malherbe, S. Critchley, and M. Dokainish. "The use of hardness in the study of compaction behaviour and die loading." In: Powder Technology 17.3 (1977), pp. 253-257 (cit. on pp. 6, 30, 31, 47).(Kawakita et al., 1971) K. Kawakita and K.-H. Lüdde. "Some considerations on powder compression equations." In: Powder technology4.2 (1971), pp. 61-68 (cit. on pp. 6, 27, 29).(Kim et al., 2000) K. Kim, S. Choi, and H. Park. "Densification behavior of ceramic powder under cold compaction." In: J. Eng.Mater. Technol. 122.2 (2000), pp. 238-244 (cit. on pp. 31, 47).

Mots-clés : Compression de poudre, Modélisation éléments finis, Tomographie rayons-X, Compact parallélépipédique rainuré, Calibration hybride modèle DPC

Acknowledgements

Christine Espinosa d'avoir présidé ce jury de thèse. Un grand merci à vous tous d'avoir accepté d'évaluer mes travaux ainsi que pour

MCC powder were manufactured. For a greater homogeneity of density distribution inside the compacts, MCC 102 was mixed with 0.5 𝑤/𝑤 of Magnesium Stearate. The mixture was done using a Turbula mixer (Type T2F Willy A. Bachofen, Basel, Switzerland) at 50 rpm during 5 minutes.

The powder mass was kept constant at 0.343 ± 0.004 g and different compression pressures ranging from 7 MPa to 171 MPa were applied to obtain a large range of densities, varying between 0.43 to 0.87. Compacts were produced on the uniaxial press (Frogerais OA, France).

Their mean relative densities were calculated from mass and volume after ejection. Volume was calculated from measured diameter and height using a micrometer (Mitutoyo).

X-ray Tomography analyses of the compacts were done on the Easytom 300 Tomograph with the same process parameters as evoqued in Section 6.4.

The tomographic images of each compact were then processed following the steps described in Figure 6.3. to avoid the friction effects, a circular section was defined as the analysis area for the extraction of the average grey level for each compact. This analysis area is highlighted in red in Figure 6.4. For each compact 50 slices were extracted from the tomographic images at the centre of the compacts. The historgrams of these sclices were extracted indicating the mean grey level value of each slice. An example of hystogram is presented in Figure 6.5 for the central slice of the compact presented in Figure 6.4 of a relative density equal to 0.66. Globaly the average mean grey value was computed on a total volume of 80.76 𝑚𝑚 3 for each compact.

In Figure 6.6 the mean grey values computed for each calibration sample are plotted. The grey level (𝐺𝐿) presents a linear evolution (Figure 6.6) according to the relative densities of the calibration compacts. The linear fitting curve is given by Equation 6

Highlights

• Comparison of experimental and numerical results obtained from previous Chapters

• A good agreement between X-ray Tomgraphy results and predictions was found

The regions located on the upper and lower edges (zones 6, 7 and 13, 14 respectively) show large deviations of up to 7.7 %. For the other regions presented, the calculated relative deviations are less than 3.7 %, which is satisfactory given that the experimental results are compared to the numerical results at unloading and not at ejection.

As already mentioned, the area between two opposing grooves shows strong density variations in the form of a gradient. This gradient is formed by the presence of highly dense areas under the tips of the grooves and by lower densities in the centre of the compact.

The region between the grooves is of interest as it is, due to the presence of this gradient (as described in the previous Chapters), subject to density variations and may be in a state of tension during unloading and potentially lead to localized cracking in this area.

The prediction of relative density in this area is presented in Figure 7.2, and has been superimposed with the results obtained experimentally by tomography.

The data presented in Figure 7.2 show a good agreement between the experimental and numerical results. A density difference of about 23 % can be observed between the centre and the tips of the grooves.

The choice of the cell size (10x10 pixels) for the presentation of the tomographic results was made in order to obtain a result that is as representative as possible of the density variation.

Indeed, a cell size that is too small (close to the pixel scale) leads to very heterogeneous results with scattered experimental points. A more global vision (larger cell size) tends, on the contrary, to homogenise the density evolution and does not allow the observation of localised density variations as first observed in Figure 6.10. To illustrate this, the evolution of relative density between two grooves presented in Figure 7.2 is also presented in Figure 7.3 from measurements made with different cell sizes ranging from 5x5 to 20x20 pixels. Therefore, the cell size of 10 times 10 pixels seems to be a good balance.

In Chapter 5, the impact of the friction coefficient on the density distribution is discussed.

The high density areas present on the right and left edges of the compact raise the question of the correct prediction of the coefficient of friction during its experimental characterisation.

To recall, experimentally from the instrumented cylindrical matrix compression test, 𝜇 was determined to be equivalent to about 0.35 for microcellulose crystalline 102.

The plot (Figure 7.4) shows the evolution of the relative density between facing grooves of the compact. These experimental results were compared with the numerical ones obtained for different coefficients of friction (ranging from 0.1 to 0.5). Let us recall that for the purpose of describing the gradient between two grooves, reference is made to three zones distinguished in Figure 5.17.

The relative density curves (predicted values) of the compact show the significant impact of the friction coefficient on the relative density gradients. In the case where the friction coefficient is the highest (𝜇 = 0.5) the density gradient is more important with a difference between the minimum and maximum density of 0.17. Moreover a stronger variation of density is observed in the zone 2 of the gradient (central part). On the contrary, in the case where the friction coefficient is lower (𝜇 = 0.1) the density tends to be more constant in this zone 2 2 (1989), pp. 143-150 (cit. on pp. 3, 15, 30). 3 (1996), pp. 239-254 (cit. on pp. 38, 74).

(Babaei, 2020) P. Babaei. 4,16,25,26,35). Journal 25 (2007), pp. 4-22 (cit. on pp. 23, 24, 38).

( 2 (2008), pp. 121-129 (cit. on pp. 4, 25, 36, 38, 109, 132).

( Vol. 1977. MCgraw-hill London, 1971 (cit. on pp. 6, 38).

Abstract Flow and compaction behaviour of powders with deep grooved punches

The production of compacts with complex geometries by means of powder die compaction can be a delicate task. The compression process, used in many industrial sectors, is based on the densification of a powder bed by the application of an external load to obtain compacts. In some situations, the presence of defects (for example, cracks) in the compacts can be observed following the compression process. These lead to the degradation of the mechanical properties of the compacts and therefore their properties of use, thus leading to the non-conformity of the sample. Defects in powder compacts can be mainly due to the homogeneity of the compact, the mechanical behaviour of the powder and the complexity of the geometric shape of the sample. The work of this thesis contributes to the analysis of the compaction of pharmaceutical powders on large compacts with complex geometry (presence of grooves).

In particular, to further understand the impact of the geometry of the compression tools on the final state of the compacts, both experimental and numerical studies were carried out. In order to model the mechanical response of the powder in compression to an external load, the Drucker Prager Cap (DPC) model was employed. To overcome a scaling problem, a hybrid calibration method of the DPC model was introduced in this work to provide a more realistic prediction of the powder behaviour in compression. The numerical results obtained by finite element simulation were validated by means of tomographic analysis.

Keywords: Powder die compression, Finite element modelling, X-ray Tomography, Parallelepiped grooved compact, Hybrid DPC calibration procedure

Résumé

Écoulement et comportement en compression de poudres avec des poinçons à rainures profondes

La fabrication de pièces à géométries complexes, par compression de poudres, peut s'avérer délicate. Le procédé de compression, utilisé dans de nombreux secteurs industriels, consiste à densifier un lit de poudre par l'application d'une charge afin de produire des compacts. Dans certaines situations la présence de défauts (fissures par exemple) dans les compacts est observable suite à la compression. Ceux-ci mènent alors la dégradation des propriétés mécaniques des compacts et donc leur propriétés d'usage, leur présence entraine alors la non-conformité de l'échantillon. Les défauts sur des compacts de poudre peuvent trouver leur origine principalement dans l'homogénéité du compact, le comportement mécanique de la poudre ainsi que la complexité de la forme géométrique du compact. Les travaux de cette thèse apportent une contribution dans l'analyse de la compression de poudres pharmaceutiques sur des compacts de grande taille à géométrie complexe (présence de rainures). Plus particulièrement, pour aller plus loin dans la compréhension de l'impact de la géométrie des outillages de compression sur l'état final des compacts, une étude à la fois expérimentale et numérique a été mise en place. Afin de modéliser la réponse mécanique de la poudre en compression soumise à une charge externe, le modèle de Drucker Prager Cap (DPC) a été employé. De manière à pallier à un problème d'échelle une méthode hybride de calibration du modèle de DPC a été mise en place dans le cadre de ces travaux pour obtenir une prédiction du comportement de la poudre en compression plus réaliste. Les résultats numériques obtenus par simulation éléments finis ont été validés par le biais d'analyses tomographiques.