Gérard Vignoles

Mohamed Guessasma

Patrick Pizette

Keywords: Méthode des éléments discrets, composite réfractaire, compactage de la poudre, propriétés mécaniques des réfractaires, méthode des éléments nis Discrete element method, refractory powder composite, powder compaction, refractory mechanical properties, nite element

Carbon-bonded alumina refractories are powder composites made of coarse alumina, graphite akes, and a bonding matrix (mixture of ne alumina grains and binder). These composites are used for steel casting applications. The nal refractory properties are governed both by the properties of the composition and by the processing stages (mixing, compaction, and ring). The compaction may lead to density gradients along the geometry resulting in a thermo-mechanical properties mismatch. Eorts have been devoted to modeling the constitutive compaction behavior of refractory industrial parts by the Finite Element Method (FEM) using elastoplastic models. However, the material parameters of such models need to be identied by complex experiments. The Discrete Element Method (DEM) may be an alternative to provide such parameters. The DEM can model the compaction stage and the compacted composite nal mechanical properties by explicitly considering the discrete nature of the powder. Also, DEM may be applied to generate constitutive equations to simulate the compaction at the macro scale using the FEM. In this work, we develop a numerical model based on the DEM to simulate the compaction behavior of powder composites and their mechanical properties after compaction and ring. It is based on X-ray tomography observations and experimental data originating from compaction and triaxial tests. The mechanical input parameters of the DEM are calibrated on experimental compaction data. Four compositions in DEM are represented as a mixture of single particles, and particle clusters with various volume fractions. The binder is represented by a soft elastoplastic shell covering hard alumina particles. The densication behavior during compaction and the green properties depend mostly on the contact between those coated particles. The compacted numerical samples are submitted to various complex stress state conditions to generate yield and fracture surfaces tted with the elastoplastic Drucker-Prager Cap model (DPC). Following the compaction, the compacted numerical samples are virtually red in DEM, and their mechanical and fracture behavior is presented. The numerical results are in good agreement with experiments both qualitatively and quantitatively. In particular, the simulations capture the softening eect of an increasing amount of binder. Finally, an attempt to model the composite compaction behavior in a rigid die using the FEM is presented. It uses the DPC model with the material parameters identied by DEM.

Résumé

Les réfractaires en alumine-graphite sont des composites composés d'alumine grossière, de paillettes de graphite et d'une matrice (mélange de grains ns d'alumine et de liant). Ces composites sont utilisés pour les applications de coulée de l'acier. Les propriétésnales des réfractaires sont dictées à la fois par les propriétés de la composition et par les étapes du du process industriel (mélange, compactage et frittage). Le compactage peut conduire à des gradients de densité le long de la pièce, ce qui entraîne une discontinuité des propriétés thermomécaniques. Des eorts ont été déployés pour la modélisation du comportement constitutif de la compaction des pièces industrielles réfractaires par la méthode des éléments nis (FEM) en utilisant des modèles élastoplastiques. Cependant, les paramètres matériau de ces modèles doivent être identiés par des expériences complexes. La méthode des éléments discrets (DEM) peut être une alternative pour fournir ces paramètres. La DEM peut modéliser l'étape de compactage et les propriétés mécaniques nales du composite compacté en prenant explicitement en compte la nature discrète de la poudre. De plus, la DEM peut être appliquée pour générer des équations constitutives an de simuler la compaction à l'échelle macro en utilisant la FEM. Dans ce travail, nous développons un modèle numérique basé sur le DEM pour simuler le comportement à la compaction de poudres composites et leurs propriétés mécaniques après compactage et cuisson. Il est basé sur des observations de tomographie aux rayons X, et des données expérimentales provenant de tests de compaction et d'essais triaxiaux. Les paramètres d'entrée de la DEM sont calibrés sur les données expérimentales de compactage. Quatre compositions sont représentés comme un mélange de particules discrètes, et de clusters de particules avec diérentes fractions volumiques. Le liant est représenté par une coque élastique souple recouvrant des particules d'alumine. Le comportement pendant le compactage et les propriétés à vert dépendent principalement du contact entre ces particules enrobées. Les échantillons numériques compactés sont soumis à divers états de contraintes complexes pour générer des surfaces de plasticité et de rupture ajustées par le modèle élasto-plastique Drucker-Prager Cap (DPC). Après le compactage, les échantillons numériques sont virtuellement cuits en DEM, et leur comportement à la fracture est étudié. Les résultats numériques sont en bon accord avec les expérimentations à la fois qualitativement et quantitativement. En particulier, les simulations saisissent bien l'eet adoucissant d'une quantité croissante de liant. Enn, une tentative de modélisation du comportement de la compaction du composite dans une matrice rigide à l`aide de la FEM est présentée. Elle utilise le modèle DPC avec les paramètres du matériau identiés par DEM.

have a great retirement and be able to enjoy yourself and your family. Achraf Kallel, who was the pioneer in performing DEM simulations at Vesuvius, thank you for having taken the time to explain to me the simulations you performed and for having given the context of your previous work. Even if we followed very dierent paths in the modeling, it helped me in organizing my ideas and thinking about which direction I should follow when at the very beginning. Thy-Thuy-My, thank you for having spent a lot of your time at the end of my Ph.D. performing and improving the nal compaction FEM simulation. Because of you, I could have a nice closure. Your help was highly appreciated. Thanks also to the VISO team for all the nice dialogues around the coee table, laughs, and scientic discussions.

Between 2018 and 2019, for approximately 9 months, I had the opportunity to work at the SIMaP GPM2 group at the University of Grenoble and spend very nice moments with the Ph.D. students, post-docs, and sta. The times at the bars after a hard working day and hiking in the amazing grenobloise mountains will not be forgotten. I would like to especially thank Kaoutar Radi and Shrey Joshi for all the coee breaks to release all the stress during the Ph.D. and for the many funny moments. Kaoutar, thank you very much for have helped me with the dp3D. Your help was extremely important, mostly at the beginning, when I was a bit lost in using the code. Thank you also for all the encouragement. My thanks go also to Pauline Gravier for have given me good instructions on how post-treating the X-ray tomography images on ImageJ. For Brayan Paredes, who helped in organizing my `cadeaux de thèse' by distance. Unfortunately, I did not have much chance to spend time at Grenoble during your Ph.D., but I wish you the best of luck in your nal path. I am sure you will have a great defense! Coming to Grenoble also allowed me to strengthen a very valuable friendship with my dearest Brazilian friend, queen of magnetism, and Dr., Isabelle de Moraes. Thank you for having listened to me in very tough moments, for all the counseling, and for sharing such nice moments.

Way before this long Ph.D. journey starts in the land of croissants and amazing fromage, I studied for a couple of years at an university in Rio de Janeiro with so many talented and smart students who shared the same dreams as me. This group includes my (almost Italian) friend Rosemere Lima, with whom shared many ambitious plans for after my master's degree. Now, here we are! I am sure you'll do great at your Ph.D. defense! I also had great mentors during my bachelor's and master's studies who may not be forgotten. Mariana Banea and Hector Costa, my passion for materials science and desire of working with research denitely started when working with you both. Thank you for have been this great motivation in my scientic career.

Being here, writing my acknowledgments, after a successful Ph.D. defense would not be possible without having my best friend (who also happens to be my future husband) beside me. François Paquay, thank you so much for being with me in all the stressful moments and for having given me all the support for me to succeed. For never letting me fall, holding my hands in the most dicult periods, bringing me coee and encouragement during the thesis writing breaks, and celebrating all amazing moments. Also, because of you, I could have met your incredible family (Janine, Guillaume, and Christian) that gave me so much support all along this journey. Thanks to all I have greatly improved my French (with a bit of the unique Belgian accent) and amplied my passion for Belgian waes. Coming to France and having met you and your family was one of the most wonderful surprises of my life.

x Thanks to my very big, noisy, crazy, and lovely Brazilian family who have supported and given me so much love all my life. To my beloved and caring friend and brother, Gustavo Bonaldo, with my endless list of nicknames (fonho, Gugu, Gu, Guguzinho,. . . ), who patiently always listened to me (even when I was speaking too much, as always). Finally, but not less important, I want to thank the most amazing and caring woman of my life, my mother, Márcia Bonaldo. Thank you for having shown me and Gu that with hard work and study we could reach anything we want in our lives. For always have supported and believed in myself even when I did not. I don't remember of any moment that you were not present (even now with more than 9 thousand kilometers far away). You are and will always be a great inspiration in my life as a mother, woman, and person. VESUVIUS est un leader mondial dans le domaine de l'ingénierie et de la technologie de l'écoulement du métal en fusion, servant principalement les industries de l'acier et de la fonderie dans le monde entier. Le groupe est organisé en trois divisions (Advanced Refractories, Foundry Technologies, et Flow Control). La dernière division (Flow Control) est principalement associée au processus de coulée continue (Fig. 1.1 a), y compris la fabrication de produits réfractaires. Les pièces réfractaires produites sont surtout élaborées par métallurgie des poudres. Selon l'application, les poudres réfractaires peuvent être compactées de manière uniaxiale ou isostatique, conduisant ainsi à des produits réfractaires pour diérentes applications (Fig. 1.1 b). Le comprimé à vert obtenu (après compactage) est durci à 200°C et cuit (à 900°C). Pendant les cycles de coulée, une face de la pièce réfractaire est en contact avec l'acier fondu à 2000°C et l'autre face est en contact avec l'air (à température ambiante). Ces conditions conduisent le matériau à un choc thermique, ce qui entraîne des ssures et des dommages. De plus, les pièces réfractaires fonctionnent dans un environnement très agressif (oxydation, corrosion). La modélisation du comportement des matériaux est l'un des outils permettant d'étudier les phénomènes de défaillance et d'améliorer la conception des produits. L'équipe de modélisation de VESUVIUS, supervisée par Severine-Romero Baivier, utilise la méthode des éléments nis (FEM) pour modéliser le processus de coulée an d'étudier le comportement thermo-mécanique des pièces réfractaires industrielles cuites, pendant la coulée, complémentée par un protocole expérimental. L'une des données d'entrée les plus critiques pour ces simulations FEM est l'équation constitutive de la poudre. Le programme expérimental nécessaire pour identier les équations constitutives d'une poudre donnée est complexe et prend du temps. Il doit être répété pour chaque composite de poudre.

Deux doctorants de VESUVIUS ont déjà étudié expérimentalement le comportement de ces composites. Diane Dupuy (2013[START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] et Andrzej Warchal (2015[START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF] [START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF] ont étudié les propriétés thermomécaniques de composites réfractaires modèles cuits par VESUVIUS. Ces composites modèles ont une composition simpliée par rapport aux autres réfractaires industriels, ce qui facilite la compréhension de la relation microstructure et les propriétés mécaniques. Dupuy a étudié l'impact de la composition de composites liés à l'alumine et au carbone (avec des variations sur les ratios alumine/carbone et les dimensions des gros grains d'alumine) sur les propriétés thermomécaniques résultantes. Warchal a poursuivi les recherches de Diane mais s'est concentré sur l'eet des antioxydants et des diérents paramètres de cuisson sur les propriétés thermomécaniques des composites cuits. Cependant, les propriétés mécaniques du produit nal cuit ne dépendent pas uniquement du processus de cuisson et de la composition. Pendant le compactage, les pièces réfractaires peuvent entraîner un gradient de densité (certaines régions sont sur-compactées, et d'autres sont sous-compactées). Cette distribution de densité peut conduire à des fortes variations locales des propriétés thermo-mécaniques et, par conséquent, à des concentrations de contraintes et à la rupture potentielle pendant les opérations de coulée. Ainsi, des informations telles que les distributions de densité et de contrainte dans le compact vert peuvent être utiles pour détecter les défauts pendant le compactage et après la cuisson.

Dans ce but, VESUVIUS et le laboratoire SIMaP ont proposé un doctorat en collaboration pour étudier le lien entre la microstructure, la compaction et les propriétés mécaniques des composites à vert et post-cuits. Le travail de ce doctorat comprend des analyses microstructurales, des expériences et des simulations numériques utilisant deux approches : la méthode des éléments discrets (DEM) et la méthode des éléments nis (FEM), la majeure partie étant liée à la DEM.

Achraf Kallel a introduit l'utilisation de la DEM, à VESUVIUS (2016[START_REF] Kallel | New ceramic technologies and novel multifunctional ceramic devices and structures (CERMAT2): internal report[END_REF], pour 1.1. Contexte et objectifs : modélisation des composites à base de poudre réfractaire modéliser la compaction et le comportement mécanique des réfractaires en alumine liée au carbone. Kallel a réalisé des simulations par DEM et FEM, ainsi que des essais expérimentaux. L'intérêt pour VESUVIUS de commencer à utiliser la DEM était principalement due à sa possibilité de représenter la nature particulaire du composite en poudre, contrairement à la FEM qui considère la poudre comme un milieu continu. Le modèle utilisé par Kallel est assez diérent de celui utilisé dans ce travail de doctorat (chapitre 4) et a beaucoup changé depuis le travail de Kallel. Cependant, la philosophie du modèle reste la même, c'est-à-dire qu'il considère un ensemble de particules sphériques qui interagissent entre elles par des forces de contact et sont régies par la conservation de la quantité de mouvement (chapitre 2). Plus important encore, Kallel a démontré la pertinence et la faisabilité de cette approche pour modéliser les composites de poudre VESUVIUS.

Sur la base de ce contextes général, trois axes sont développés dans ce travail de doctorat :

Une caractérisation expérimentale est développée sur les composites à base de poudre réfractaire (analyse par tomographie aux rayons X, compactage, et essais mécaniques sur des échantillons verts et après cuisson). Leur comportement de densication pendant la compaction, leurs propriétés mécaniques/thermomécaniques et leur microstructure sont identiés.

Une étude approfondie des interactions au contact des particules entre les diérents composants (alumine grossière/nes, graphite et liant) pendant le compactage à l'aide de simulations DEM est réalisée. Le comportement mécanique macroscopique résultant est comparé et calibré avec des données expérimentales (propriétés à vert et après cuisson).

Une méthodologie est proposée pour générer des surfaces de plasticité et de rupture en eectuant des simulations numériques DEM complexes sur des échantillons numériques verts. Ces surfaces sont utilisées pour ajuster le modèle élastoplastique Drucker-Prager Cap (DPC), et les paramètres du modèle sont calculés pour alimenter un code d'éléments nis an de simuler le compactage de pièces industrielles.

L'un des avantages des simulations numériques est la possibilité d'étudier l'amélioration de la forme proche de la forme nette (procédé de fabrication en continu de formes très proches de la forme nale et le développement de nouvelles géométries pour améliorer les procédés industriels. Un autre avantage est la réduction du nombre d'expériences nécessaires pour chaque caractérisation de poudre. Ces deux actions peuvent avoir un impact direct sur la réduction de la consommation de matières premières et par conséquent sur CHAPTER 1. INTRODUCTION une baisse de la surface des usines et des émissions de gaz à eet de serre (CO 2 ). En moyenne, pour 2018, 1,85 tonne de CO 2 a été émise pour chaque tonne d'acier produite. L'industrie sidérurgique génère entre 7 et 9% des émissions directes liées à l'utilisation mondiale de combustibles fossiles [START_REF] Association | Steels's contribution to a low carbon future and climate resilient societies[END_REF]. VESUVIUS s'est xé comme objectif global en 2020 d'atteindre une empreinte carbone nette nulle au plus tard en 2050. L'une des actions du groupe consiste à réduire les déchets en récupérant davantage de produits après leur utilisation et en augmentant l'utilisation de matériaux recyclés.

Organisation du projet industriel

Les composites réfractaires liés au carbone ont une microstructure très hétérogène. Le protocole expérimental nécessaire pour caractériser leurs propriétés mécaniques est com-1.2. Organisation du projet industriel plexe, principalement au stade de la compaction. Ainsi, la modélisation par la méthode des éléments discrets (DEM) peut être un outil utile et complémentaire pour aider à comprendre leur comportement et leurs propriétés de compaction puisqu'elle prend en compte la nature particulaire de la poudre. De plus, les simulations DEM créent un lien avec la méthode des éléments nis (FEM) en générant des équations constitutives pour modéliser la pièce industrielle réfractaire.

En tant que partie principale du projet, les simulations DEM sont destinées à répondre à certaines questions fondamentales sur la modélisation d'un composite composé de diérents éléments (alumine, graphite et liant) en termes de compactage et de propriétés mécaniques des échantillons verts/après cuisson. La première question à laquelle il faut répondre est comment modéliser l'interaction entre ces diérents composants à l'échelle du contact entre particules et comment ces interactions vont aecter le comportement macroscopique du composite ? De même, comment modéliser les interactions entre des particules dures (alumine et graphite) avec un composant très mou (liant) ? Comment représenter en DEM diérentes compositions liant la fraction volumique de chaque composant ? Comment ces compositions vont-elles représenter le comportement réel (expériences) de la compaction ? L'expérience de VESUVIUS montre que la teneur en liant a un impact important sur le comportement macroscopique. Ceci est-il conrmé et reproduit par les simulations DEM ? Si oui, les simulations DEM peuvent-elles nous aider à comprendre les mécanismes sous-jacents ? Comment distinguer le comportement mécanique pendant le compactage d'un échantillon cuit numériquement ? Quelle méthodologie sera appliquée pour générer des surfaces de rendement et de fracture à partir d'échantillons compactés numériquement soumis à des tests mécaniques ? Comment faire le lien entre ces surfaces générées à partir de DEM et alimenter un code FEM pour simuler le comportement de compactage d'une pièce industrielle ? Plus généralement, quel degré de complexité faut-il apporter au modèle de DEM pour représenter au mieux la microstructure réelle des composites industrielles tout en gardant un modèle susamment simple pour être utilisé industriellement ? Pour répondre à ces questions, le travail eectué au cours de ce doctorat sera présenté sous la forme de six chapitres.

Le chapitre 2 présente une revue de la littérature. Elle résume les principaux travaux sur la compaction et les propriétés mécaniques et thermomécaniques des matériaux céramiques et réfractaires en termes de caractérisation expérimentale et surtout de modélisation utilisant les approches DEM et FEM. Dans le chapitre 4, une approche originale basée sur la méthode des éléments discrets (DEM) est présentée pour représenter les composites réfractaires d'alumine liés au carbone pendant la compaction et leurs propriétés mécaniques après compaction et après cuisson. Les microstructures des composites sont représentées par un élément de volume représentatif (RVE). Les lois au contact des particules sont décrites pour modéliser les interactions entre les composants élastiques et plastiques (alumine, graphite et liant).

Dans le chapitre 5, des simulations de compactage et d'essais mécaniques sur des composites numériques compactés ont été réalisées, en utilisant les lois de contact décrites dans le chapitre 3, et comparées aux données expérimentales. L'eet des propriétés du liant sur le comportement de compactage de la composition et l'hystérésis pendant la décharge sont présentés. Des surfaces de iso-densité et de fracture sont générées à partir de simulations triaxiales et de traction, fournissant des informations précieuses sur le comportement des composites à vert pour des états de contrainte qui sont diciles à atteindre par des expériences.

Dans le chapitre 6, le comportement mécanique du composite virtuel cuit est étudié en eectuant des simulations triaxiales et de traction en DEM. Les simulations sont comparées aux données expérimentales. La propagation des ssures pendant une simulation de traction est étudiée et corrélée avec les défauts qui ont été insérés numériquement pour décrire les phénomènes liés au chauage/refroidissement du liant. Les surfaces de plasticité et de rupture sont générées à partir de simulations triaxiales et de traction et sont comparées aux expériences.

Enn, dans le chapitre 7, le modèle Drucker-Prager Cap (DPC) est ajusté avec les surfaces de rendement et de rupture DEM. Les paramètres du modèle DPC sont calculés. Une première tentative de modélisation du composite dans une simulation en matrice à l'aide du modèle DPC via un code d'éléments nis est présentée. Les résultats sont comparés aux données DEM et expérimentales.

Introduction

Refractory composite materials for steel casting applications are subjected to harsh working environments (thermal shock, oxidation, corrosion). These conditions promote cracking and material damage. In addition, as detailed in this chapter, refractory parts are processed by powder compaction that may lead to density gradient along the part geometry resulting in thermo-mechanical properties mismatch. These mechanical property dierences may induce stress concentration and failure of the industrial part. The quality of the refractory part and its ability to sustain steel casting conditions depend mainly on its nal microstructure. Understanding the link between the microstructure, the processing stages (compaction, ring), and their nal thermo-mechanical properties are important for developing new products and improving the industrial process.

The compaction of metallic, ceramic, or composite powders has been investigated in the literature both experimentally and numerically. From a numerical point of view, the compaction stage has been studied mainly by the Finite Element Method (FEM) that treats the powder as a continuum and uses appropriate constitutive equations to describe the compaction behavior of the material. The material parameters in these equations need to be identied by complex experiments.

Another method to model the compaction stage is the Discrete Element Method (DEM), which explicitly considers the powder as a collection of discrete particles. The DEM may be used eectively to clarify the link between the particulate microstructure and the behavior of the powder during compaction or to model the nal mechanical properties of the compacted parts. DEM may also be applied to generate constitutive equations to feed a FEM code. These constitutive equations may be generated by conducting numerical DEM simulations under various complex stress state conditions.

In summary, this chapter rst describes the steel-making process (steel casting, refractories used in the casting process). Second, a brief overview of refractory ceramic materials (microstructure, manufacturing stages) is presented. Afterward, the main aspects of the compaction behavior of ceramic powders are discussed. The mechanical properties of refractory powders as a function of temperature are detailed. Also, the use of FEM for these materials is described; in particular, two constitutive equations (Drucker-Prager cap and Biggoni-Piccolroaz models) are presented. The main aspects of DEM are introduced, including the successive steps that a DEM simulation undertakes. Finally, some examples from the literature are using two dierent numerical approaches: the nite element and discrete element methods. CHAPTER 2. LITERATURE REVIEW

Steel-making process

Steel is used for numerous applications: automobile, civil engineering, refrigeration systems, home appliances, cargo shipping, aeronautic industry [START_REF] Association | Overview of the steelmaking process[END_REF]. The world steel production follows a characteristic steady linear growth in the last 70 years (Fig. 2.1). In twenty years (from 2000 to 2020), steel production increased approximately by 120%. Steel may essentially be produced following two routes: blast furnace-basic oxygen furnace (BF-BOF) or electric arc furnace (EAF), see Fig. 2

.2.

A total of 70% of steel is produced using the BF-BOF route. The main dierence between these two routes (EAF and BF-BOF) is that iron ore, coal, and recycled steel are used in the BF-BOF route, while in the EAF, the steel is produced using mostly recycled steel and electricity [Association, 2013a]. In the BF-BOF route, iron ore pellets are introduced into a blast furnace (BF) with coke and limestone pellets to produce pig iron [Ghosh, 2008].

The pig iron is converted to steel in the blast oxygen furnace (BOF) and transferred into the ladle (secondary rening) for further deoxidation, desulfurization, vacuum degassing, and chemistry adjustment [START_REF] Fruehan | Overview of Steelmaking Processes and Their Development[END_REF]]. In the electric arc furnace (EAF) route, electricity is used to melt recycled steel. Additives, such as alloys, are introduced to adjust to the desired chemical composition [Association, 2013a]. In both routes, the molten metal is solidied in the continuous casting process.

Figure 2.1: World crude steel production from 1950 to 2020 (million tonnes) [Association, 2013a] At the end of the blast furnace or electric arc furnace (EAF or BOF) process, the liquid steel from the ladle is transferred to the continuous casting machine. Continuous casting is the link between steel-making and rolling. This method was rst suggested by Henry Bessemer in 1856. Still, it is only during the 1930s and 1940s that continuous casting has become a common production method for nonferrous metals and later from 2.2. Steel-making process Figure 2.2: Overview of the steelmaking process [Association, 2013a] the 1960s for steels [START_REF] Louhenkilpi | Continuous Casting of Steel[END_REF].

During the casting operation, the steel ows at a controlled rate from the ladle to the tundish inside a shroud (Fig. 2.2 b) located at the bottom of the ladle. The tundish is a metallurgical reactor vessel that improves control of steel cleanness, temperature, and composition. Ideally, the steel in the tundish should be free of any inclusions, and it should reach the best conditions for casting: optimal temperature, composition, and dynamic behavior.

The steel ows from the tundish through a submerged entry nozzle (SEN) into a water-cooled copper mold [START_REF] Louhenkilpi | Continuous Casting of Steel[END_REF]]. In the mold, heat transfers cause immediate solidication (cooling-down) of the liquid steel in contact with copper plates, resulting in a solid shell. When this solid shell is sucient to avoid any liquid steel spillage, the strand is cut o and transferred to a rolling mill (Fig. 2.3 a).

Refractory materials, such as stopper-controlled nozzles and slide gates, are used to control the steel ow from the ladle to the tundish and from the tundish to the mold (Fig. 2.3 b). The ladle shroud (1) situated between the ladle and tundish also protects the liquid steel from oxidation. The monobloc stopper (2) ensures perfect steel ow control from the tundish through a submerged entry nozzle (SEN). The SEN (3) distributes liquid steel into the mold to achieve the best solidication pattern and direct eventual inclusions towards the mold [START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF]. In VESUVIUS, these refractory pieces are manufactured out of carbon bonded alumina refractory composites (Al 2 O 3 -C), sometimes CHAPTER 2. LITERATURE REVIEW containing metallic additives, such as MgO-C and ZrO 2 -C (Fig. 2.3 c). An overview of carbon-bonded refractory microstructure and processing stages are presented in the next section 2.3. [START_REF] Routschka | Handbook of Refractory Materials: Design-Properties-Testing[END_REF]].

Refractory ceramic materials

Refractory materials are materials which resist high temperatures. They are mostly used in the steel industry, which by itself represents two-thirds of the applications of these materials [START_REF] Katsavou | Eect of production conditions on structural properties of refractory materials[END_REF]. According to [START_REF] Routschka | Handbook of Refractory Materials: Design-Properties-Testing[END_REF], the refractories are based on 6 dierent oxides: SiO 2 , Al 2 O 3 , MgO, CaO, Cr 2 O 3 , and ZrO 2 (Fig. 2.4), as well as carbon, or a combination of these [START_REF] Ewais | Carbon based refractories[END_REF].

A list of common refractory materials and their applications, with the carbon-containing refractory materials, are presented in [START_REF] Thethwayo | A review of carbon-based refractory materials and their applications[END_REF]. The carbon bonded refractories, for example, are applied to steel casting applications and are present in submerged nozzles, ladle shrouds, monobloc stoppers, and slide gate valves at VESU-VIUS. Because of that, this type of refractories is our main interest, and when possible, we will present some of the related studies described in the literature. More specically, the refractory material studied is a carbon-bonded alumina (Al 2 O 3 -C) without any addi- [START_REF] Routschka | Handbook of Refractory Materials: Design-Properties-Testing[END_REF].

The major components of carbon-bonded refractories are the aggregate, the matrix, and the pores. They are mainly composed of coarse refractory oxide grains and graphite akes. These particles are embedded in a matrix of pyrolytic carbon (binder), ne refractory oxide grains, and sometimes additives. Fig. 2.5 a) shows a scanning electron microscope (SEM) image of a carbon bonded alumina refractory containing coarse alumina oxide grains, a matrix (mixture of ne alumina oxide grains and binder), and graphite akes. Fig. 2.5 b) represents the same microstructure, but with the addition of Al-Si additives as antioxidants. The white and grey spots in the coarse alumina grain correspond to impurities (TiO 2 and SiO 2 ) [START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF].

Carbon-bonded composite refractory parts for continuous steel casting are manufactured from pulverized raw materials (Fig. 2.6). The rst step is mix-manufacturing, where raw materials (refractory oxides, graphite, phenolic resin, and additives) are weighted according to the selected material composition. They are inserted in a mixer where a furfural solvent is added, forming aggregates that are a few tens of millimeters. In the VESU-VIUS concept, the binder is formed by the mixture between phenolic resin, catalyzer, and furfural solvent. Mixing temperature (usually between 50 and 65°C), duration, and nal moisture content are adjusted to obtain the desired mixture consistency [START_REF] Gardziella | Phenolic Resins: chemistry, applications, standardization, safety and ecology[END_REF]. The obtained aggregates are afterward cooled down to room temperature.

These powder pellets are then shaped via two routes: isostatic or closed-die compaction, depending on the application (targeted nal product shape). The component after compaction is denominated as the green body. The green bodies are heated up to [START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF]. 200-250°C in air, which leads to the elimination of solvents and reticulation of the phenolic resin structure (curing), see Fig. 2.6 [START_REF] Fitzer | The eect of crosslinking on the formation of glasslike carbons from thermosetting resins[END_REF]. The curing provides the material with sucient strength for handling and prepares the binder for ring. Afterwards, the cured sample is red (ring process). In the ring, the cured sample is heated up to between 800 and 1000°C. During the ring, the powder sample is submitted to microstructural changes (section 2.1).

To ensure excellence and consistency in the nal product, process and quality control procedures are needed throughout the various processing stages (from compaction to ring). The properties of the nal products are aected by all the processes involved in the ring process. For example, some factors such as the ring temperature or dwell time must be carefully controlled during the heat treatment because they inuence the physical and mechanical properties of the red parts (after curing, in contrast to green parts) [START_REF] Braulio | Magnesia grain size eect on in situ spinel refractory castables[END_REF]. Indeed, some of the problems in the red parts (including incorrect density, size or shape, density gradient, and craking) originate from the compaction stage [START_REF] Ramírez-Aragón | Numerical modeling for simulation of compaction of refractory materials for secondary steelmaking[END_REF]. Thus, understanding densication behavior (during compaction) and thermo-mechanical properties during ring are essential for developing new ceramic/refractory materials and improving near-net-shape ability.

Compaction behavior of ceramic powders

Compaction of ceramic powders is a forming technique in which a ceramic powder is made cohesive through mechanical densication. The compaction process allows parts to be produced industrially to close tolerances. A closed-die pressing operation has typically three steps (Fig. 2.7): a) the powder is inserted in a rigid die, b) the upper punch moves inwards, applying pressure, and the powder is uniaxially compacted up to a specic relative density, c) unloading the compact and ejecting the compact from the die.

During compaction, the stress needed to densify the powder increases further. It can be described generally by a simple equation relating the compaction pressure P to the relative density D:

ln(P ) = AD + B (2.1)
where A and B are empirical constants [START_REF] Niesz | A review of ceramic powder compaction[END_REF]. The powder compaction response can be described using more elaborated models (pressure density-dependent equations). The most applied models are the ones developed by [START_REF] Shapiro | Studies on aging of precipitates and coprecipitation: the compressibility of silver bromide powders[END_REF]Konopicky (1947-1948) [START_REF] Shapiro | Studies on aging of precipitates and coprecipitation: the compressibility of silver bromide powders[END_REF][START_REF] Konopicky | Radex Rundsch[END_REF], Balshin (1949) [Balshin, 1949], [START_REF] Heckel | Density-Pressure Relationships in Powder Compaction[END_REF] (1961), [START_REF] Cooper | Compaction Behavior of Several Ceramic Powders[END_REF] [START_REF] Cooper | Compaction Behavior of Several Ceramic Powders[END_REF], and Kawakita/Ludde [START_REF] Kawakita | Some considerations on powder compression equations[END_REF], and Panelli-Ambrosio (2001) [START_REF] Panelli | A study of a new phenomenological compacting equation[END_REF].

These equations were proposed to investigate the eect of strain hardening on the compaction of porous metals or metal powders. P. J Denny [START_REF] Denny | Compaction equations: A comparison of the Heckel and Kawakita equations[END_REF] described the comparison between two powder compaction equations (Heckel and Kawakita). The author pointed out that in these models, it is assumed that the compact is isotropic. This is a strong assumption since, in closed-die compaction, the dierence in stresses between the axial and radial directions should produce anisotropy. Some of these equations have been applied or modied for ceramic powders, for example, in [START_REF] Ramakrishnan | A compaction study on ceramic powders[END_REF]. Ramakrishnan showed that the Heckel equation tted very nearly to the experimental data (closed-die compaction) when applied to magnesia, zirconia, and titania ceramics.

In [START_REF] Moreno | Densication of Al powder and Al-Cu matrix composite (reinforced with 15% Sal short bres) during axial cold compaction[END_REF], closed-die compaction tests were carried out on pure Al and Al-Cu matrix composites reinforced with short polycrystalline bres. Moreno et al. compared experimental data with the Kawakita, Panelli-Ambrosio, and Konopicky equations. The authors observed a reasonably good tting between experimental data and the models. However, each model highlighted a dierent limitation. For example, the Kawawita equation has shown the best linear t with all experimental data. However, the parameter b (compressibility) was not in agreement with the hardening behavior of the material (aluminum-based metal matrix composites). Panelli and Ambrosio's equations tted well qualitatively the data for all compaction tests, still only over a limited pressure range (>50MPa). Finally, the Konopick model allowed the best characterization of the hardening behavior of the aluminum powder material. More specically, the yield stresses obtained by Konopicky ttings resulted in very good agreement with the yield stresses obtained by experiments.

General powder ceramic materials (e.g., alumina powders) follow a characteristic behavior during compaction. The compaction curve of these ceramic materials is controlled by three dierent stages that show dierent mechanisms. The rst one (stage I) where the compaction pressure increases and the RD remain almost the same (Fig. 2.8). This stage is controlled by intergranular particle friction and grains rearrangement, which does not result in any signicant powder compaction. Next, at stage II, the compaction begins to be controlled by plastic deformation. The transition from stage I to stage II is marked by a clear increase of the RD. Stage III is represented by fracture of powder aggregates [START_REF] Niesz | A review of ceramic powder compaction[END_REF]] [Haussonne et al., 2005] or at very high stresses, fracture of the ceramic particles themselves. In some cases, it might be dicult to identify this transition between stages II and III properly. The plastic deformation in stage II may come from the extrusion or deformation of a viscous binder or the aggregate deformation during compaction. Fig. 2.9 shows typical closed-die compaction curves (axial stress versus relative den-CHAPTER 2. LITERATURE REVIEW sity) of spray-dried alumina powder (Al6SG) [START_REF] Carneim | Response of granular powders to uniaxial loading and unloading[END_REF]] and a carbonbonded alumina composite (Al 2 O 3 -C) containing a mixture of coarse alumina aggregates and graphite akes embedded into a matrix composed of a mixture between ne alumina particles and binder. Fig. 2.9 a) shows a clear quantitative dierence between the two curves. In particular, the pure alumina powder is compacted only to relatively low relative density (less than 60%) and relatively large stresses (> 150 MPa). The powder composite experiences much larger densication compared with the pure alumina powder. However, when plotting the same curves in logarithmic scale (Fig. 2.9 b), the curves of both powder materials show qualitatively similar behavior. The main dierence is a shift in relative density (alumina powder starts with a much smaller density). The two rst stages, I and II (Fig. 2.8), might be identied. However, the transition between stages II and III might be dicult to be recognized in both curves. Still, the phenomena that explain the compaction behavior in the alumina powder in Carneim et al. and the refractory composite are dierent. Carbon-bonded composites have a much more complex behavior than alumina powder ceramics because of the viscous component (binder) added to the mixture. Usually, those composites have a much larger binder content (approximately 22 wt.% in the example in Fig. 2.9), against 2.5-5.4 wt.% in Carneim et al. One of the justications for the density dierence between both materials clearly originates from the binder content.

The compaction behavior in carbon-bonded composites may be divided into two main stages. At the beginning, the grains are rearranged with the increase of pressure, and similar to alumina powder behavior in stage I, the composite relative density (RD) remains almost the same. At this stage, the composite is very porous. However, while the pressure continues to increase, the binder extrudes between aggregates and lls the inter-grains pores increasing the composite RD. The binder, which is viscous-plastic, is the component that most dictates the densication behavior of the composite and is responsible for its plastic behavior. Some authors also studied the spring-back behavior of ceramic materials during the unloading (ejection of the sample from the rigid die). The spring back is the release of energy stored in both the organic and inorganic components [START_REF] Niesz | A review of ceramic powder compaction[END_REF]. So, this energy is accumulated during the compaction (loading) and released during the unloading. The contributing factors to the storage of this energy will depend highly on the material composition. For example, compaction comes mostly from interlocking particles on 'dry powders' (no organic component). However, some ceramic materials contain a binder as a bonding component. Because the binder has a visco-plastic behavior, it is a dominant factor for the energy stored during the compaction and consequently spring-back.

In addition, the eect of the strain rate on the compaction behavior of carbon bonded alumina refractories was studied by [START_REF] Kallel | New ceramic technologies and novel multifunctional ceramic devices and structures (CERMAT2): internal report[END_REF] (Fig. 2.10). The compaction tests were carried out using dierent but constant displacement rates (0.5, 5, and 25 mm/min). Fig. 2.10 a) shows the evolution of the strain rate (s -1 ) with the sample relative density (%). The strain rate can be considered approximately constant along the compaction tests (Fig. 2.10 a). The compaction pressure (or axial stress) against the relative density (RD) curves for each punch velocity is shown in Fig. 2.10 b). The compaction with the highest velocity (25mm/min) exhibits the stiest behavior. More generally, these curves show that the behavior is rate sensitive. This strain rate sensitivity is mainly due to the presence of the visco-plastic binder. The rate sensitivity is more pronounced for high punch displacement rates (that correspond to approximately 10 -2 sec -1 strain-rate).

The eect of the binder/moisture content and the strain rate dependency on the compaction behavior of agglomerated alumina powders was studied by Briscoe et al. [START_REF] Briscoe | Compaction behaviour of agglomerated alumina powders[END_REF]. The authors observed that samples with an excess of binder (4.7 wt.%), 'wet' agglomerates reached a higher relative density at a given compaction pressure when compared to the other samples containing less binder content (2.4 wt.%). Similar to [START_REF] Kallel | New ceramic technologies and novel multifunctional ceramic devices and structures (CERMAT2): internal report[END_REF], the compaction in Briscoe et al. showed a strain rate dependency. The higher the speed rate (mm/s), the lower the resulting relative density. The strain-rate dependence of the compaction of samples with more binder content ('wet' agglomerates) is more pronounced than the samples with less binder ('dry' agglomerates). Indeed, in [START_REF] Tanaka | Eect of Moisture on the Structure and Fracture Strength of Ceramic Green Bodies[END_REF], increasing moisture (binder content) during closed-die compaction led to an increase in the relative density of alumina ceramic compacts. The opposite was found in [START_REF] Carneim | Response of granular powders to uniaxial loading and unloading[END_REF]. Carneim et al. studied the eect of organic binder content and binder plasticity on the stress-strain response of alumina ceramic powders during closed-die compaction tests. The authors observed that the mixture with more binder reached a smaller relative density to a given compaction pressure when comparing the other mixtures with less binder.

The eect of the strain rate on the compaction behavior of agglomerated alumina powders was also studied by Briscoe et al. for alumina powders with variation in the binder content (between 2.4 and 4.7 wt.%). In accordance with the study proposed by Kallel, the compaction behavior showed a strain-rate dependency. The higher the speed rate (mm/s), the lower the resulting relative density. Briscoe et al. showed that the strain-rate dependence of the compaction of samples with more binder content (' wet' agglomerates) is more pronounced than for samples with less binder (' dry' agglomerates). The authors observed that samples with an excess of binder (wt.%) reached a higher relative density at a given compaction pressure as compared to samples containing less binder. Similarly, Tanaka et al. [START_REF] Tanaka | Eect of Moisture on the Structure and Fracture Strength of Ceramic Green Bodies[END_REF] showed that increasing moisture (binder content) in alumina agglomerates during closed-die compaction led to an increase in the relative density of alumina ceramic compacts. In Tanaka et al., the moisture content was rather small (0.3-2.7%).

The opposite was found in [START_REF] Carneim | Response of granular powders to uniaxial loading and unloading[END_REF]. Carneim et al. studied the eect of organic binder content and binder plasticity on the stress-density response of alumina ceramic powders during closed-die compaction tests. In their case, the binder content ranged from 2.5 to 5.4 wt.%. Carneim et al. have shown that it was possible to achieve slightly higher density at a given pressure with less binder. Still, the dierences between the powder with less and more binder were very small (≈ 56-58% at the maximum compaction pressure). Also, the powder sample with less binder started from a larger density (29%) compared with the one with more binder (25%).

Most importantly, the general comparison of the compaction behavior and green properties of general ceramics, metallic-ceramic materials, and pharmaceutical powders with carbon-bonded refractories should be taken cautiously. The alumina powder materials studied by [START_REF] Tanaka | Eect of Moisture on the Structure and Fracture Strength of Ceramic Green Bodies[END_REF], [START_REF] Briscoe | Compaction behaviour of agglomerated alumina powders[END_REF], and [START_REF] Carneim | Response of granular powders to uniaxial loading and unloading[END_REF] have a very small binder content (0.3-5.4 wt.%) if compared with the carbon-bonded alumina composite refractory studied by [START_REF] Kallel | New ceramic technologies and novel multifunctional ceramic devices and structures (CERMAT2): internal report[END_REF] and on materials similar to those that will be studied in this Ph.D. (between 17 and 25 wt.%). Also, for example, in Carneim et al., the binder T g is between -32°C and +35°C, and in Kallel, approximately +60 and +70°C. Contrary to these alumina powder materials, the carbon-bonded refractory has a specic behavior at the compaction stage. These composite materials are generally composed of a carbonaceous matrix (mixture of ne oxide grains and binder). Their plastic behavior derives from grains rearrangement and binder visco-plasticity due to the binder, as seen previously.

The interest in carbon bonded alumina ceramic refractories (Al 2 O 3 -C) is motivated by their use as functional refractory products for ow control continuous casting since they exhibit excellent thermal shock and slag resistance after ring (heating process) [START_REF] Chen | Microstructure and mechanical properties of lightweight Al2O3-C refractories using dierent carbon sources[END_REF]. Although more than 4,000 scientic papers were published in 2021 under the topic' carbon bonded refractory', the main focus has been on investigating their mechanical properties as a function of temperature, not on their compaction behavior and green properties. However, many defects in the ring can be traced to the green microstructure. A key problem is density variation within parts and among parts, since it aects both industrial part quality and the ability to keep tight dimensional tolerances [START_REF] Niesz | A review of ceramic powder compaction[END_REF]. Also, density gradient leads to dierent thermo-mechanical properties of the red part. The compaction behavior of ceramic materials and their green properties after compaction using a numerical approach will be discussed in sections 2.6 and 2.7.

2.5 Thermo-mechanical behavior of carbon-bonded refractories

Young's modulus

The thermo-mechanical properties of carbon-bonded refractories have been thoroughly studied in the literature in the last decades. In particular, the evolution of their Young's modulus with temperature is of clear interest as they are used over a large temperature domain. To relate the microstructural changes of refractory composite materials as a CHAPTER 2. LITERATURE REVIEW function of temperature, the nonlinear evolution of Young's modulus of carbon-bonded alumina refractories (Al 2 O 3 -C) was analyzed by [START_REF] Gault | Ultrasonic Measurements of Young's Modulus of Al2O3-based refractories at high temperatures[END_REF], [START_REF] Werner | Young's modulus of elasticity of carbon-bonded alumina materials up to 1450°C[END_REF], [START_REF] Warchal | Microstructure Evolution During Firing and Resulting Mechanical Properties of Steel Flow Control Refractories Containing Various Additives[END_REF], [START_REF] Luchini | On the nonlinear behavior of Young's modulus of carbon-bonded alumina at high temperatures[END_REF], and in the Ph.D. thesis of [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] and [START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF].

In [START_REF] Werner | Young's modulus of elasticity of carbon-bonded alumina materials up to 1450°C[END_REF], the Young's modulus and the thermal expansion were measured in a temperature range from 25°C (RT) to 1450°C by the impulse excitation technique (IET) in red (between 700-1000°C) carbon-bonded alumina refractories. Based on these two analyses, for a better understanding of the behavior of these properties evolution, the cycle here is divided into 3 stages and summarized (Fig. 2.11): Figure 2.11: Eective Young's modulus (normalized) evolution versus temperature, during heating and cooling, for a carbon-bonded alumina composition [START_REF] Werner | Young's modulus of elasticity of carbon-bonded alumina materials up to 1450°C[END_REF].

Stage I (25-450°C): The Young's modulus is almost constant (8 GPa).

Stage II (400-1200°C): The Young's modulus increases signicantly up to approximately 1000°C. This increase is related to microstructural changes. The binder is transformed into pyrolitic carbon, and the alumina, graphite, and binder components expand dierently due to their thermal expansions mismatch (alumina: ≈ 7-9 10 -6 K -1 , graphite: 25-42 10 -6 K -1 , and binder: 3.2-6.9 10 -6 K -1 ). The alumina and graphite have a higher CTE than the binder. Thus, they expand more than the binder. The matrix is also expanded, creating gaps between alumina aggregates and the matrix. Between 1000-1200°C, because of the matrix expansion, some open pores are closed or healed, leading to an approximately constant Young's modulus (18 GPa).

Stage III (1200-1450°C): There is a slight decrease in Young's modulus (from 18 to 17 GPa). This decrease is a consequence of microstructure cracks due to alumina expansion during the heating above the pyrolysis temperature. The Young's modulus slightly increases at the holding time (at 1450°C) (from 17 to 19 GPa). The increase during the holding time is due to some bonding between alumina particles that sinter together. The author also observed lower sintering eects of the microstructure at 1450°C in microstructure with higher graphite and coarse alumina content (more graphite layers surrounding the alumina, inhibiting the sintering and large alumina particles that do not sinter easily).

The cooling can also be divided into three stages (I-III). In stage III, the Young's modulus remains almost constant. In stage II, there is a signicant decrease of Young's modulus due to the detachment of the aggregates from the surrounding matrix. Thus, assuming that the matrix is cohesive, decohesions occur at the interface alumina/matrix and graphite/matrix [START_REF] Luchini | On the nonlinear behavior of Young's modulus of carbon-bonded alumina at high temperatures[END_REF]. The presence of acoustic activity in [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF][START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF] during cooling conrms this type of defect. In stage I (RT), the gap at the interface aggregates/matrix is larger than before due to the over-stressed matrix.

These observations from Young's modulus measurements in [START_REF] Werner | Young's modulus of elasticity of carbon-bonded alumina materials up to 1450°C[END_REF], [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] and [START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF] may be complemented by recent experimental data and scanning electron microscopic (SEM) observations in [START_REF] Luchini | On the nonlinear behavior of Young's modulus of carbon-bonded alumina at high temperatures[END_REF], in similar composites (Al 2 O 3 -C). Fig. 2.12 a) shows the SEM micrographs of the 3 analyzed samples (1-3). Fig. 2.12 a) (1) represents the refractory microstructure after curing (up to 180°C). Fig. 2.12 a) ( 2) is related to the rst heating cycle (up to 500°C). Fig. 2.12 a) (3), after two heating cycles (up to 1,000°C).

The SEM micrographs of these three thermal cycle conditions were analyzed after cooling. It can be observed a matrix microstructural change along with the thermal treatment. The sample after curing (1) presents a matrix with very small and isolated micro-cracks. In (2), the matrix exhibits a few medium-sized network cracks in the matrix. In (3), there is an overall well-developed crack network. Those cracks are related to the release of volatiles and to the high anisotropy of graphite thermal expansion [START_REF] Luchini | On the nonlinear behavior of Young's modulus of carbon-bonded alumina at high temperatures[END_REF].

Luchini et al. also documented the evolution of the alumina/matrix interface during those thermal treatment cycles by SEM (Fig. 2.12 b). ( 1) represents the sample after curing, (2) after one cycle of heating (up to 500°C), and (3) after two cycles up to 1,000°C. The interface alumina/matrix after curing (1) is well-connected. However, the one (2) and two heating cycles (3) were highly aected by the thermal cycles. In (3), it is possible to see a notable void between the alumina aggregate and the matrix. Typical graphite basal plane cracks are shown in Fig. 2.12 c) for a sample heated up to 1,000°C after two cycles. [START_REF] Werner | Inuence of porosity on Youngs modulus of carbon-bonded alumina from room temperature up to 1450 °c[END_REF] also investigated Young's modulus (E) of carbon-bonded refractories, not only as a function of the temperature but also as a function of porosity. The [START_REF] Luchini | On the nonlinear behavior of Young's modulus of carbon-bonded alumina at high temperatures[END_REF]. evolution of the normalized Young's modulus (%) and Young's modulus (GPa) versus temperature for all tested porosity levels (13.9% to 21.7%) are shown in Fig. 2.13. The samples were uniaxially compacted at dierent pressure (20,40,70,100,and 130 MPa), cured (up to 180°C), and red (up to 1000°C). Considering the normalized Young's modulus, the results are qualitatively similar for all porosity levels (Fig. 2.13 a). In Fig. 2.13 b), the Young's modulus decreases with increasing porosity, mainly at high temperatures (1200-1400°C).

Compressive behavior

As seen above, the acoustic measurement of the Young's modulus (E) of red carbonbonded refractories shows that E depends on the porosity (compaction pressure leading to samples with dierent porosity percentages) and testing temperature. Material properties are variable and depend not only on the original design but also on the microstructural and chemical changes produced by thermal conditions [START_REF] Musante | Mechanical behaviour of MgO-C refractory bricks evaluated by stressstrain curves[END_REF]. Many research [START_REF] Werner | Inuence of porosity on Youngs modulus of carbon-bonded alumina from room temperature up to 1450 °c[END_REF].

articles are focused on studying the mechanical properties of these refractories. It has been observed in the literature a broad range of mechanical properties values that mainly depend on the refractory composition, the applied curing/ring temperature, the testing temperature, and the experimental methodology.

Besides Young's modulus, some of the mechanical properties largely presented in the literature include the cold modulus of rupture (CMOR), measured at room temperature, and the hot modulus of rupture (HMOR). [START_REF] Fan | Microstructures and mechanical properties of Al2O3-C refractories with silicon additive using dierent carbon sources[END_REF], [START_REF] Zhu | Eect of ceramic bonding phases on the thermo-mechanical properties of Al2O3-C refractories[END_REF], and [START_REF] Yin | Thermo-mechanical properties of Al 2 O 3 -C refractories with in situ synthesized non-oxide bonding phases[END_REF] performed 3-point bending tests on red Al 2 O 3 -C refractories with a silicon (Si) additive and measured the CMOR at RT, andHMOR at 1400°C ([Zhu et al., 2013], and [START_REF] Yin | Thermo-mechanical properties of Al 2 O 3 -C refractories with in situ synthesized non-oxide bonding phases[END_REF]).

In [START_REF] Werner | Inuence of porosity on Youngs modulus of carbon-bonded alumina from room temperature up to 1450 °c[END_REF], uniaxial compressive tests were carried out in carbon-bonded alumina refractories (without any additive). For this purpose, three compositions were produced (T20, T29, and T20-3). According to their compositions, T20-3 and T20 have more alumina aggregates. T29 has more graphite, and T20 has more porosity. The composite with more porosity (T20) has shown lower values of Young's modulus (E), and the one with more graphite (T29) has shown higher CMOR. A decrease of HMOR for temperature above 1000°C could be observed in T20 and T29. This decrease is because, above 1000°C, the microstructures (T20 and T29) were damaged by a further expansion of alumina and graphite, resulting in microcracking (E decreases). For T20-3, this theory seemed not suitable as E also decreases, but HMOR increases or stays equal. Therefore, the recently created microcracks did not aect the strength.

[ [START_REF] Musante | Mechanical behaviour of MgO-C refractory bricks evaluated by stressstrain curves[END_REF] studied the thermo-mechanical behavior of a set of red MgO-C CHAPTER 2. LITERATURE REVIEW refractories (A1, A2, and B compositions), containing small Al content (2-4 wt%). These composites were produced containing two dierent carbon binders (A1 and A2) and resin (B), with dierent content of MgO aggregates. Stress-strain curves were determined by applying uniaxial compression tests, carried out at room temperature (RT), 600, 1000, and 1400°C (Fig. 2.14). Regarding the stress-strain curves, composite B presented greater mechanical strength (and larger stiness) than composites A1 and A2. Also, composites A1 and A2 exhibited a more softening response. These behavior are determined by the type of binder (resin vs. pitch binder) and the presence of silicon in composite B. At 600°C, a softer response is observed in all composites with decreased mechanical properties (e.g., E and strength). According to the authors ( [START_REF] Musante | Mechanical behaviour of MgO-C refractory bricks evaluated by stressstrain curves[END_REF]), this was due to organic binder transformation at 600°C (the same was observed in [START_REF] Uchida | High-Temperature Properties of Unburned MgO C Bricks Containing Al and Si powders[END_REF]).

Another study on red carbon-bonded refractories (Al 2 O 3 -C) applying two testing temperature (RT and 1000°C) on uniaxial compression tests was described by [START_REF] Musante | High temperature mechanical behavior of Al2O3-MgO-C refractories for steelmaking use[END_REF]. At RT, the softening in the composite with a lower graphite content was attributed to its high true porosity (pores and cracks), favoring the microcracking. Overall, the stiness and the mechanical strength were higher as the testing temperature increased (1000°C).

In summary, the mechanical behavior of carbon-bonded refractories might be associated with: a) microcracking of pre-existent damage within the matrix and between matrix/aggregates; b) sliding and crumbling of graphite akes; c) plastic deformation of these akes; or d) binder plasticity [START_REF] Musante | High temperature mechanical behavior of Al2O3-MgO-C refractories for steelmaking use[END_REF].

Tensile behavior

Stress versus (engineering) strain curves from uniaxial tensile tests performed on Al 2 O 3 -C are shown in Fig. 2.15 [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF][START_REF] Warchal | Microstructure Evolution During Firing and Resulting Mechanical Properties of Steel Flow Control Refractories Containing Various Additives[END_REF][START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF]. It is possible to observe that their tensile behavior does not represent the typical mechanical behavior of general ceramics since carbon-bonded alumina composites exhibit a strong nonlinearity. The investigated refractory is also able to develop an important strain before its rupture. Thence, the shape of the curve resembles that of metals (with much lower strength). However, such behavior does not result from the movement of dislocations but the damaged microstructure (microcracks and decohesions) created during ring. Moreover, after each loading/unloading cycle, the elastic modulus value decreases, suggesting that new defects are created within the material during mechanical loading. Thus, such behavior may be classied as pseudoplastic, and the damage is the one that dictates the overall refractory mechanical behavior [START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF]. [START_REF] Warchal | Microstructure Evolution During Firing and Resulting Mechanical Properties of Steel Flow Control Refractories Containing Various Additives[END_REF][START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF]. [Harmuth, 2010] identied that the strength of refractories is correlated with the type of crack propagation. For this purpose, Harmuth performed a wedge splitting test on CHAPTER 2. LITERATURE REVIEW ve dierent compositions based on carbon-bonded refractories. Afterward, the author used a microscopic fractographic procedure to analyze the fracture surface after the destructive tests. The results have shown that less brittle refractories have a lower amount of trans-granular crack propagation, and higher percentage of crack propagation within the grain/matrix interface. Both strength and specic fracture energy for crack initiation increased with the percentage of trans-granular crack propagation.

Also, [START_REF] Chen | A novel approach to lightweight alumina-carbon refractories for ow control of molten steel[END_REF] performed 3-point bending tests on Al 2 O 3 -C aggregates refractories reinforced by SiC whiskers (4 wt%). The paths of the propagating cracks in two samples after the tests are shown in Fig. 2.16 a). TRS and MRS have dierent types of Al 2 O 3 -C aggregates. The specic crack lengths, in percentage, of the three crack propagation mechanisms for the two samples are plotted in Fig. 2.16 b). The cracks propagated mostly along with the aggregate/matrix interface in sample TRS (>85%).

On the other hand, sample MRS exhibited a more stable fracturing behavior (≈ 48% within the matrix, ≈ 31% within aggregate/matrix interface, and ≈ 20% within the aggregate) compared with TRS. Force-displacement curves from the 3-point bending tests for TRS and MRS samples are shown in Fig. 2.16 c). MRS exhibited a stronger behavior (higher strength) and lower brittleness if compared with TRS. This stability in the fracture is generally associated with a lower brittleness [START_REF] Dai | Observation and quantication of the fracture process zone for two magnesia refractories with dierent brittleness[END_REF][START_REF] Chen | A novel approach to lightweight alumina-carbon refractories for ow control of molten steel[END_REF].

In summary, Chen et al. proved that raw materials (alumina, matrix, graphite) could inuence the microcracking mechanisms. Chen et al. are in agreement with the theory proposed by Harmuth ([Harmuth, 2010]) which says that a larger amount of trans-granular cracks (within the aggregate) lead to a improvement of the material strength. et al., 2017], ceramic materials [START_REF] Aychn | The internal form of compacted ceramic components: a comparison of a nite element modelling with experiment[END_REF][START_REF] Abdullah | Finite element simulation of alumina ceramic powder compaction[END_REF][START_REF] Buljak | Calibration of Drucker Prager Cap Constitutive Model for Ceramic Powder Compaction through Inverse Analysis[END_REF], and for the pharmaceutical industry [START_REF] Michrafy | Modelling the compaction behaviour of powders: Application to pharmaceutical powders[END_REF][START_REF] Michrafy | Wall friction in the compaction of pharmaceutical powders: Measurement and eect on the density distribution[END_REF][START_REF] Kadiri | Pharmaceutical powders compaction: Experimental and numerical analysis of the density distribution[END_REF][START_REF] Sinha | A study on the sensitivity of Drucker-Prager Cap model parameters during the decompression phase of powder compaction simulations[END_REF][START_REF] Garner | The extrapolation of the Drucker-Prager/Cap material parameters to low and high relative densities[END_REF][START_REF] Perez-Gandarillas | Compaction properties of dry granulated powders based on DruckerPrager Cap model[END_REF]. Another model is the Bigoni and Piccolroaz yield/damage function [START_REF] Bigoni | Yield criteria for quasibrittle and frictional materials[END_REF][START_REF] Piccolroaz | An elastoplastic framework for granular materials becoming cohesive through mechanical densication. Part i -Small strain formulation[END_REF][START_REF] Bosi | Experimental investigation of the elastoplastic response of aluminum silicate spray dried powder during cold compaction[END_REF]] [Argani et al., 2016[START_REF] Penasa | Computational modelling and experimental validation of industrial forming processes by cold pressing of aluminum silicate powder[END_REF]. Other constitutive models include the one developed by [START_REF] Fleck | On the cold compaction of powders[END_REF] and the Cam-Clay model [START_REF] Lade | Single hardening constitutive model for frictional materials[END_REF] (which is often used in rock mechanics).

Drucker-Prager Cap model

The Drucker-Prager Cap (DPC) model is used to model the elastoplastic behavior of powders, assuming the material is porous compressible and isotropic. The model provides an inelastic hardening mechanism that accounts for plastic deformation during compaction and volume dilatancy when the powder yields in shear [START_REF] Garner | The extrapolation of the Drucker-Prager/Cap material parameters to low and high relative densities[END_REF]. The DPC model is described by the powder yielding as a function of hydrostatic pressure (p), the modulus of deviatoric stress (q), or equivalent Von Mises stress, and the relative density of the powder. The model is divided into two segments: a shear failure surface (F s ) which characterizes the limit of allowable stress that can lead to material failure, and a cap surface (F c ) that denes the powder densication (hardening), see Fig. 2.17. A transition surface may also be introduced to smoothen the transition between the two surfaces. Here, we will more specically describe the DPC implementation in Abaqus ([Abaqus, 2006]) as this is the FEM model that will be used in this Ph.D.

The domain dened under these surfaces characterizes any stress state under which the material can behave elastically. Therefore, for any stress that ultra-passes this domain, the deformation of the material is considered as irreversible (plastic behavior) [START_REF] Perez-Gandarillas | Compaction properties of dry granulated powders based on DruckerPrager Cap model[END_REF]. The shear failure surface in the p -q plane is represented as a straight line and is dened as:

F s = q -p tan(β) -d = 0 (2.2)
where β is the internal frictional angle, and d is the powder cohesion.

The hydrostatic pressure stress (p) and the modulus of deviatoric stress are dened as:

p = - 1 3 (σ z + 2σ r ) (2.3) q = |σ z -σ r | (2.4)
where σ z and σ r are the axial and radial stresses, respectively. The cap surface hardens or softens as a function of the volumetric plastic strain: volumetric plastic compaction (when yielding on the cap) causes hardening, while volumetric plastic dilation (when yielding on the shear failure surface) causes softening [Han et al., 2008]. The cap yield surface (F c ), that is represented by an ellipse, intersects the shear failure surface (F s ) at p = P a , and the hydrostatic stress axis at p = P b .

The cap yield surface is written as:

F c = (p -p a ) 2 + Rq 1 + α -α/ cos β 2 -R(d + p a tan β) = 0 (2.5)
where R is a measure of the cap eccentricity (it controls the shape of the cap). This parameter is typically between 0.0001 and 1,000 [Abaqus, 2006]. d is the material cohesion.

P a is an evolution parameter that represents the volumetric plastic strain driven hardening/softening. β is the material angle of friction. P b is the hydrostatic yield stress. α is a small number (typically 0.010.05) used to dene a smooth transition surface between the shear failure surface and the cap.

The distance between P b and P a is:

P b -P a = R(d + P a tan β) (2.6)
When the material densies, the yield surface expands and the evolution of the yield surface is represented by the hardening law P b as a function of the volumetric plastic strain (ε pl vol ):

ε pl vol = ln RD RD 0 (2.7)
where RD is the current relative density, and RD 0 is the initial relative density on lling of die.

In the transition region between the shear failure surface and the cap, a transition surface (F t ) is introduced:

F t = [(p -P a )] 2 + q -1 - α cos β (d + P a tan β) 2 -α (d + P a tan β) = 0 (2.8)

Bigoni-Piccolroaz model

A yield/damage function was proposed by Bigoni and Piccolroaz (BP) [START_REF] Bigoni | Yield criteria for quasibrittle and frictional materials[END_REF] for modeling the inelastic behavior of a broad class of pressure-sensitive, frictional, ductile, and brittle-cohesive materials. The yield function represents a single, convex and smooth surface in stress space approaching as limit situations well-known criteria and the extreme limits of convexity in the deviatoric plane (Fig. 2.18). The Bigoni-Piccolroaz (BP model) is seen as a generalization of the von Mises, DruckerPrager, Tresca, modied Tresca, CoulombMohr, modied Cam-clay, Deshpande-Fleck (2000), Rankine, and Ottosen models. The BP model allows the possibility of describing complex yield surfaces. In the BP model, one of the motivations was to have a single criterion relating to dierent types of materials.

To describe the hardening of the powder material during compaction, a suitably `deformable' surface is needed [START_REF] Piccolroaz | An elastoplastic framework for granular materials becoming cohesive through mechanical densication. Part i -Small strain formulation[END_REF]. The yield function has 7 parameters Figure 2.18: Increase in cohesion as related to hardening [START_REF] Piccolroaz | An elastoplastic framework for granular materials becoming cohesive through mechanical densication. Part i -Small strain formulation[END_REF]. and is given as:

F (σ, p c , c) = f (pc, p c , c) + q g(θ)
(2.9) where p c and c are the parameters that depend on plastic deformation and dene the hardening behavior. q is the deviatoric invariant. f (p) is the function describing the dependence on the hydrostatic pressure p. g(θ) describes the dependence on the Lode's invariant θ and is dened as:

g(θ) = 1 cos [βπ/6 -(1/3) cos -1 (γ cos 3θ)]
(2.10)

The yield function is explained in greater detail in [START_REF] Bigoni | Yield criteria for quasibrittle and frictional materials[END_REF]]. Analyzing Figs. 2.17 and 2.18, it shows that the DPC model and the Bigoni-Piccolroaz (BP) models bear some clear resemblance. However, the BP model allows for a continuous transition between the dierent behaviors but at the cost of a larger number of parameters.

Compaction simulations of ceramic powders using the FEM

The compaction modeling of ceramic powder materials using a nite element method may be a useful tool to control the shape and density distribution in the powder compact. The interest by the industry and researchers on this topic has been growing in the last years. The growing interest by the industry is because, during the pressing, the friction between the powder and the die-wall can lead to non-uniform density distribution in the green part, consequently leading to material damage/crack during the ring. Therefore, the simulations can be useful for the improvement of the near-net shape. The FEM of powder compaction has been largely applied in the last years on metal [START_REF] Almanstötter | A modied Drucker-Prager Cap model for nite element simulation of doped tungsten powder compaction[END_REF][START_REF] Zhou | A density-dependent modied Drucker-Prager Cap model for die compaction of Ag57.6-Cu22.4-Sn10-In10 mixed metal powders[END_REF], and pharmaceutical [START_REF] Krok | Numerical investigation into the inuence of the punch shape on the mechanical behavior of pharmaceutical powders during compaction[END_REF][START_REF] Garner | The extrapolation of the Drucker-Prager/Cap material parameters to low and high relative densities[END_REF] powders.

Nowadays, the Drucker-Prager Cap is the most known model to simulate the compaction of powder materials. FEM compaction simulations on ceramic materials have been been extensively described in the literature [START_REF] Aychn | The internal form of compacted ceramic components: a comparison of a nite element modelling with experiment[END_REF] [START_REF] Park | Consolidation behavior of SiC powder under cold compaction[END_REF][START_REF] Zeuch | Mechanical properties and shear failure surfaces for two alumina powders in triaxial compression[END_REF][START_REF] Abdullah | Finite element simulation of alumina ceramic powder compaction[END_REF]] [He et al., 2013[START_REF] Penasa | Computational modelling and experimental validation of industrial forming processes by cold pressing of aluminum silicate powder[END_REF][START_REF] Melo | A validation procedure for numerical models of ceramic powder pressing[END_REF][START_REF] Buljak | Calibration of Drucker Prager Cap Constitutive Model for Ceramic Powder Compaction through Inverse Analysis[END_REF][START_REF] Ni | Establishment of constitutive models and numerical simulation of dry pressing and solid state sintering processes of MgTiO3 ceramic[END_REF] together with experimental characterization. Some commercial FEM software, for example ABAQUS has the DPC model already implemented for compaction simulations with density-dependent. [START_REF] Melo | A validation procedure for numerical models of ceramic powder pressing[END_REF] described an experimental procedure (of closed-die and isostatic compaction) to validate the numerical models used to simulate the compaction of alumina powders. The authors applied the DPC constitutive model to represent the compaction behavior of the material. The closed-die compaction leads to a non-homogeneous distribution in the pressing direction due to the friction between the powder and the die walls. The pressed regions have a larger volumetric strain in the isostatic compaction, and the density distribution is more homogeneous.

Besides, [START_REF] Park | Consolidation behavior of SiC powder under cold compaction[END_REF] performed triaxial compression experimental tests on compacted SiC powder materials under various conning pressure. These data were plotted in the p -q plane and the yield surfaces tted with the DPC and Cam-Clay models. A FEM closed-die compaction model was also proposed using these two models. Similar to [START_REF] Melo | A validation procedure for numerical models of ceramic powder pressing[END_REF], the relative density in [START_REF] Park | Consolidation behavior of SiC powder under cold compaction[END_REF] model was the highest at the corner of the contact surface between the upper punch and the die wall and the lowest at the corner of the contact surface between the lower punch and the die wall. Finally, the FEM calculations were compared with experimental compaction data, showing a reasonable agreement. [START_REF] Ni | Establishment of constitutive models and numerical simulation of dry pressing and solid state sintering processes of MgTiO3 ceramic[END_REF]] also studied the closed-die compaction behavior of MgTiO 3 ceramic powders using FEM simulations from the DPC model, and experimental data. The DPC model parameters were tted and calibrated from a long experimental protocol (uniaxial compression, Brazilian, and uniaxial die compaction tests) under dierent relative densities. Similar to [START_REF] Park | Consolidation behavior of SiC powder under cold compaction[END_REF] and [START_REF] Melo | A validation procedure for numerical models of ceramic powder pressing[END_REF], Ni et al. also observed a powder density gradient (Fig. 2.19) due to the presence of friction between the die walls and the powder.

Recently [START_REF] Buljak | Calibration of Drucker Prager Cap Constitutive Model for Ceramic Powder Compaction through Inverse Analysis[END_REF] performed compaction simulations on a VESUVIUS [START_REF] Ni | Establishment of constitutive models and numerical simulation of dry pressing and solid state sintering processes of MgTiO3 ceramic[END_REF]. ceramic composite on three dierent geometries of the die and the punch (3 dierent congurations) to study the stress distribution within the samples at the end of the compaction. It has been shown that the geometry signicantly inuences the stresses distribution within the powder and the stress path in the p -q plane. Similar to [START_REF] Ni | Establishment of constitutive models and numerical simulation of dry pressing and solid state sintering processes of MgTiO3 ceramic[END_REF], in [START_REF] Buljak | Calibration of Drucker Prager Cap Constitutive Model for Ceramic Powder Compaction through Inverse Analysis[END_REF] As seen in the previous section, powder compaction simulations using the nite element method (FEM) have been extensively described in the literature in the last few years.

The complex behavior of the powders is usually considered by calibrating constitutive equations (e.g., Drucker-Prager Cap and Bigoni-Piccolroaz models) with experimental data. These model parameters are typically measured using standard soil mechanics testing procedures [Pavier & Doremus, 1999]. However, measuring these parameters by experiments can be a dicult task due to its complexity. For example, several triaxial compression experimental tests are needed to describe the behavior of a powder in a complex stress state (p-q plane). As discussed in the preceding section, the experimental program may include uniaxial compression, Brazilian, closed-die compaction tests and even fully triaxial tests.

Those constitutive equations may also be developed from micro-mechanics-based models ( [START_REF] Fleck | On the cold compaction of powders[END_REF][START_REF] Sridhar | Yield behaviour of cold compacted composite powders[END_REF]) together with experimental data. In Fleck, densication is assumed to occur due to plastic deformation at the contact zones between neighboring particles. However, it uses several restrictive assumptions that may be questionable to model complex ceramic composites.

Granular media are discontinuous and heterogeneous by nature. Thus, the particulate nature of granular materials may be somewhat lost with the FEM as it works on a much larger scale. Therefore, it is complex to realistically model the discrete nature of granular matter during compaction with the FEM. Because of this, the Discrete Element Methods (DEM) have been specially developed to model granular materials at the particle scale (section 2.7.2). In the last few years, simulations using the discrete element method (DEM) have been proposed in the literature to release limitations from micro-mechanicsbased models, and better represent the particular behavior of powder materials. One of the many approaches of DEM includes establishing material behavior by considering particles interaction and then using this constitutive data in a continuum model for an engineering part.

What is DEM?

The discrete element method (DEM) simulation considers the behavior of a representative volume element of a particulate matter. In numerical simulations based on DEM, each particle is a set of individual objects that may interact with each other and be governed by the conservation of momentum, dierently than the FEM simulation in which the material CHAPTER 2. LITERATURE REVIEW is represented as a continuum matter (Fig. 2.20). The advantage of DEM is that it allows considering some important behavior aspects of granular materials, for example, particles rearrangement, the creation, and loss of contacts between particles, the densication or dilatation of the material and the inuence of interparticle friction. This approach of modeling the behavior of granular matter was pioneered by Cundall and[START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF][START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF] in which the interaction between two spherical particles is described by regular functions. In this case, the individual particle was modeled considering its mass and moment of inertia, and the interactions between particles by employing dashpots and springs [START_REF] Ransing | Powder compaction modelling via the discrete and nite element method[END_REF]. The expression of force between two particles into contact is in the function of their indentation. The initial approach by Cundall and Strack was applied in the eld of geomaterials (e.g., rocks). DEM has also been applied for diverse types of materials, including ceramic [START_REF] Balakrishnan | Eect of particle size in aggregated and agglomerated ceramic powders[END_REF] powders. DEM has also been used to model fracture or buckling problems for initially continuous materials [START_REF] Tan | Discrete element method (DEM) modeling of fracture and damage in the machining process of polycrystalline SiC[END_REF][START_REF] Nohut | Prediction of crack-tip toughness of alumina for given residual stresses with parallel-bonded-particle model[END_REF][START_REF] Leclerc | Eect of packing characteristics on the simulation of elasticity and brittle fracture by the Cohesive Discrete Element Method[END_REF][START_REF] Tan | Discrete element method (DEM) modeling of fracture and damage in the machining process of polycrystalline SiC[END_REF][START_REF] Kumar | Eect of packing characteristics on the discrete element simulation of elasticity and buckling[END_REF][START_REF] André | Using the discrete element method to simulate brittle fracture in the indentation of a silica glass with a blunt indenter[END_REF]] [Radi et al., 2020], and sintering [START_REF] Martin | Discrete element modeling of metallic powder sintering[END_REF] of powder materials. Still, the interest in compaction behavior [START_REF] Martin | Isostatic compaction of bimodal powder mixtures and composites[END_REF][START_REF] Skrinjar | Cold compaction of composite powders with size ratio[END_REF] Martin et al. [2006a] has been one of the main interests in the last few years.

But, what is the principle of DEM? The dierent steps of a DEM simulation are shown in Fig. 2.21. Here, we will specialize the description to the case of a quasi-static simulation, i.e., a simulation where the mechanical equilibrium of each particle is sought for. These are simulations in which inertial eects can be neglected. This last assumption is reasonable for powder compaction processes.

At each time step, the overlap between particles is detected and used to compute interaction forces. The dierent detailed steps of a typical DEM simulation calculation are described in sequence: In step 1, each particle is displaced by an increment ∆x i that follows a macroscopic deformation imposed by a macroscopic strain rate tensor εii : (2.11) where x i is the position of the particle along the i axis. ∆t is the time step. This rst step makes the particles move along the macroscopic deformation eld (also known as the ane solution) as if the powder is a continuous material. This approach is not always applied in DEM. However, it allows getting closer to quasi-static conditions. This rst step, for example, is very commonly used in molecular dynamics, and it was rst proposed by [START_REF] Thornton | Numerical simulations of deviatoric shear deformation of granular media[END_REF].

∆x i = εii ∆x i ∆t
In step 2, after this ane solution, a list of all particles in contact with each other is determined. As the particles are represented by spheres, a simple calculation distance between them is enough to detect the contacts. Two particles are considered in contact if the distance between the center of the spheres is less than the sum of their radius. Note that to be ecient from a CPU time point of view, this stage is generally using a list of potential contacts (Verlet list) which is not recalculated at each time step and is using a system of cells to easily detect contacts with neigbouring particles. Using these two methods, the contact detection scheme only uses a few % of CPU time in quasi-static conditions.

In step 3, the particles interact with each other depending on the specied contact laws, for example, elasticity, adhesion, contact breakage, etc. Once the type of contact laws are detected, the normal and tangential force applied on each particle contact is computed as a function of the indentation between the particles (normal force) and their relative motion (tangential forces). The contact failures, if the case, is also calculated at this step. This stage is typically one of the most CPU consuming.

In step 4, the total force (including tangential and normal components of contact forces) applied to each particle is calculated. The total force applied on each particle is given as:

F part = f contact (2.12)
where f contact are the forces applied at the particle contact. This step corresponds typically to 25% of the CPU time.

In step 5, the calculation of particle new positions at time t is determined using Newton's second law. The particle acceleration is rst calculated from their forces and masses, then a Verlet-velocity algorithm ( [START_REF] Levesque | Perturbation theory and equation of state for uids[END_REF]) is used to calculate the velocity and the new position of the particles. This step corresponds typically to 5% of the CPU time.

mẍ(t) = F part (2.13)
The DEM described in this manuscript is based on the DEM code dp3D (Discrete Powder 3D), which is developed at SIMaP laboratory and is dedicated mostly to materials science applications.

As stated above, the numerical solution method applied in dp3D is on quasi-static conditions. Thus, phenomena related to the inertia of the system are negligible and have no inuence on the near quasi-static equilibrium (F part → 0 for all particles). At the end of each time step simulation, particles should be close to a mechanical equilibrium conguration. Dierent criteria can be used to ensure that this quasi-static equilibrium is reached. For each particle, the following ratio (F error ) is given as:

F error = |F part | max(|N |) (2.14)
where N is the normal contact force. The ratio should be less than 0.01 to ensure that the mechanical equilibrium of each particle was reached.

To analyze if the simulation is under quasi-static conditions, the normalized kinetic energy per particle can also be calculated (chapter 4):

Ẽkin = E kin n max(N R) (2.15)
where E kin is the total kinetic energy of the particle system, n the number of particles, N the normal force, and R is the particle radius. max(N R) is the maximum of the product of contact force and particle radius in the packing. [START_REF] Agnolin | Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks[END_REF]) showed that the simulations must meet the condition of Ẽkin < 10 -7 to ensure a quasi-static system. For the purpose that this condition is met, low enough strain rates should be employed (chapter 4).

When the quasi-static conditions are met, the mass of the particles is renormalized according to Eq. (2.5). The increase of the mass of the particle allows the rise in the time step. The time step is given as:

∆t ∝ m K n (2.16)
where K n is the normal stiness of the contact.

Here, an introduction was presented to how typical calculations are performed in dp3D using the discrete element method. In chapter 4, the contact laws applied in this Ph.D. work are described in detail.

Application of DEM to composites

Various industrial processes involve the consolidation of mixtures of powders with dierent mechanical properties. Some experimental [START_REF] Lafer | Inuence of alumina inclusions on the densication of superalloy powder[END_REF][START_REF] Roure | Analysis of die compaction of tungsten carbide and cobalt powder mixtures[END_REF][START_REF] Bouvard | Densication behaviour of mixtures of hard and soft powders under pressure[END_REF]) and numerical [START_REF] Martin | Study of particle rearrangement during powder compaction by the Discrete Element Method[END_REF][START_REF] Martin | Isostatic compaction of bimodal powder mixtures and composites[END_REF][START_REF] Skrinjar | Cold compaction of composite powders with size ratio[END_REF]]) studies, for example, were focused on investigating the densication behavior of a mixture composed of hard and soft particles. The multi-particle nite element method (MPFEM) was also applied to model the compression behavior of a mixture of ductile and brittle particle assemblies ( [START_REF] Ransing | Powder compaction modelling via the discrete and nite element method[END_REF], [START_REF] Ransing | Using a deformable discrete-element technique to model the compaction behaviour of mixed ductile and brittle particulate systems[END_REF]), of dierent Al/SiC compositions ( [START_REF] Huang | Multi-particle FEM simulation of 2D compaction on binary Al/SiC composite powders[END_REF]), Cu-W composites [START_REF] Peng | Compaction behavior and densication mechanisms of Cu[sbnd]W composite powders[END_REF]). [START_REF] Leclerc | DEM-FEM coupling method to simulate thermally induced stresses and local damage in composite materials[END_REF] coupled nite and discrete elements to simulate thermally induced stresses and local damage in ceramic-metal composites. [START_REF] Alhajj Hassan | On the suitability of a 3D discrete element method to model the composite damage induced by thermal expansion mismatch[END_REF] simulated the thermal-induced damage in powder composites (continuous material in a heterogeneous medium) using the cohesive beam model. Generally, exist several studies in the literature applying numerical simulations and experiments related to composites made of metal-ceramic materials for the powder-metallurgy process. As far as we know, no study has been reported yet in the literature regarding compaction simulations using DEM to model refractory composites on a mixture of hard particles and a soft binder.

[ [START_REF] Martin | Study of particle rearrangement during powder compaction by the Discrete Element Method[END_REF] performed DEM isostatic simulations of packing with a mixture of ideally plastic soft and hard elastic particles (Fig. 2.22 a). The eect of the presence of hard elastic particles in dierent content (0, 10, 20, 30 and 40% volume) on the macroscopic stresses (normalized) is shown in Fig. 2.22 b). It can be seen that the increase of hard particles leads to an increase in pressure at a given relative density.

Figure 2.22: a) Numerical sample after isostatic compaction (RD=0.9). Hard particles are white and soft particles are grey. b) Evolution of the macroscopic pressure versus relative density (0, 10, 20, 30, and 40% of hard particles). The macroscopic pressure is normalized (Σ 1 is the yield stress of the soft particles). Some authors also investigated the compaction behavior [START_REF] Martin | Isostatic compaction of bimodal powder mixtures and composites[END_REF][START_REF] Wu | Study of the shear behavior of binary granular materials by DEM simulations and experimental triaxial tests[END_REF][START_REF] Ramírez-Aragón | Comparison of Cohesive models in EDEM and LIGGGHTS for simulating powder compaction[END_REF], and sintering [START_REF] Martin | Study of the sintering kinetics of bimodal powders. A parametric DEM study[END_REF] of model powder materials containing dierent particles size. [START_REF] Martin | Isostatic compaction of bimodal powder mixtures and composites[END_REF], for example, mixed the two concepts. Martin et al. studied: i) homogeneous sample in which all particles have the same material properties, but dierent particles size, and ii) heterogeneous sample made of a mixture between soft and hard particles (but also having dierent particles size). In summary, [START_REF] Martin | Isostatic compaction of bimodal powder mixtures and composites[END_REF]) showed that bimodal samples (made of a mixture between small and large particles) exhibited a stier response when compared to monomodal ones during isostatic compaction simulations. Also, the unloading and tensile strength are shown to be dependent on the bimodality of the compact (the tensile strength decreases with the increase of particle size). In other words, the spring back during unloading and the resulting green strength of a bimodal sample was mainly controlled by the smallest particles when the volume fraction of large particles was not too large.

Application of DEM to generate yield surfaces

The modeling of the compaction behavior of powders can be investigated using pressuredensity-dependent equations [R. W. [START_REF] Heckel | Density-Pressure Relationships in Powder Compaction[END_REF], [START_REF] Kawakita | Some considerations on powder compression equations[END_REF]] (section 2.5); and by micro-mechanical based models [START_REF] Fleck | Yielding of metal powder bonded by isolated contacts[END_REF][START_REF] Fleck | On the cold compaction of powders[END_REF][START_REF] Henderson | Micro-mechanical modelling of powder compaction[END_REF]. These models, or some constitutive equations (e.g., Drucker-Prager Cap and Bigoni-Piccolroaz models, sections 2.6.1 and 2.6.2) which parameters can be identied through experimental data may be implemented into a nite element method (FEM) code to simulate the compaction behavior of a part for industrial applications, for example.

The most critical step is the identication of the parameters that are introduced in those constitutive equations. In other words, the experimental characterization to identify those parameters for modeling the compaction behavior of powder materials is a dicult task. In these circumstances, the discrete element method (DEM) may be applied as a tool for generating constitutive equations by modeling the material behavior at the particle contact scale to feed a FEM code. Some authors performed DEM simulations to develop yield surfaces of metal [START_REF] Redanz | The compaction of a random distribution of metal cylinders by the discrete element method[END_REF], ceramic powder materials [START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF], and soil materials [START_REF] De Bono | On the micro mechanics of yielding and hardening of crushable granular soils[END_REF]. Other similar studies used the multi-particle nite element method [START_REF] Abdelmoula | A study on the uniqueness of the plastic ow direction for granular assemblies of ductile particles using discrete nite-element simulations[END_REF][START_REF] Loidolt | Prediction of the anisotropic mechanical properties of compacted powders[END_REF]. These yield surfaces generated by DEM may be used, for example, to t the yield surfaces from a DPC model.

Usually, triaxial compression experimental tests are applied at various conning pressure to build the yield surface. Pizzete et al. [START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF]) performed various DEM simulations on particle aggregates to explore fracture and yield surfaces. The model parameters used in DEM were calibrated with experimental data of UO 2 ceramic powders. Fig. 2.23 shows the various numerical simulations that were carried out in their work.

In [START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF]), the packing was rst compacted uniaxially and unloaded. From the unloading point, several reloading paths (triaxial tests) were applied under compressive strain to explore the yield surface. The reloading simulations stopped when the packing reached the same relative density (D) retrieved at the end of the compaction. Fig. 2.24 a) shows the isodensity contour, which was obtained by connecting those reloading points with the same density (D=0.59). The fracture points were assumed at the maximum axial stress (z-axis) together with a decrease in coordinate et al., 2010]) for the generation of yield and fracture surfaces [START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF].

number.

The eect of the consolidation path on the shape of the isodensity yield surface was also investigated by [START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF]), see Fig. 2.24 b). The isodensity yield points and fracture points of a compact, which have been obtained by closed-die compaction (from Fig. 2.24 a), are compared with a compact obtained by isostatic compaction. It was observed a clear dierence between both surfaces, which shows the eect of the strain history.

Application of DEM to refractories

Generally, refractory materials are very heterogeneous, exhibiting a multi-phase composition. Indeed, their microstructure presents some discontinuities (such as inclusions, cracks, debonding, porosity). One of the main advantages of DEM is the possibility of investigating the interaction between these dierent phases to represent the macroscopic behavior of the composite (Fig. 2.25 a).

However, the application of the discrete element method to model the mechanical behavior of refractory materials is quite recent [M. H. [START_REF] Moreira | Discrete element modeling A promising method for refractory microstructure design[END_REF][START_REF] Ramírez-Aragón | Numerical modeling for simulation of compaction of refractory materials for secondary steelmaking[END_REF][START_REF] Asadi | Numerical modelling of the quasi-brittle behaviour of refractory ceramics by considering microcracks eect[END_REF] and has not been explored yet considering the microstructure of commercial refractory composite materials (e.g., carbon-bonded based composites). Although the possibilities of using DEM on refractory composites are vast (crack initiation/propagation, compaction, porosity, mechanical properties), it has not been much studied in the literature. CHAPTER 2. LITERATURE REVIEW Recently [M. H. Moreira et al., 2020] carried out destructive mechanical tests (Brazilian, uniaxial compression, and 3-point bending) on alumina castable refractories. These tests were performed experimentally and by DEM simulations using the cohesive beam model [START_REF] André | Discrete element method to simulate continuous material by using the cohesive beam model[END_REF][START_REF] Leclerc | Eect of packing characteristics on the simulation of elasticity and brittle fracture by the Cohesive Discrete Element Method[END_REF] (Fig. 2.25 b). In general, the numerical simulations in [M. H. [START_REF] Moreira | Discrete element modeling A promising method for refractory microstructure design[END_REF] showed the capacity of reproducing the mechanical behavior of the experimental tests both qualitatively (crack path) and quantitatively (Young's modulus and failure load), mainly on 3-point bending tests. Nevertheless, considering the 3-point bending and uniaxial compression tests, the maximum stress (fracture stress) from stress-strain curves in DEM was below the experimental stress (16%-29%). The authors [START_REF] Moreira | Discrete element modeling A promising method for refractory microstructure design[END_REF]) assumed that this dierence in the stress may be correlated with the Rankine calibration criterion applied in DEM for calibration.

Also, [START_REF] Ramírez-Aragón | Numerical modeling for simulation of compaction of refractory materials for secondary steelmaking[END_REF] simulated the powder compaction of refractory materials (MgO and Al 2 O 3 ) using the discrete element method (DEM) with two cohesive contact models (described in [START_REF] Ramírez-Aragón | Comparison of Cohesive models in EDEM and LIGGGHTS for simulating powder compaction[END_REF]), see Fig. 2.26. In those simulations, some mechanisms and properties were analyzed, such as the force applied to the punch, porosity, nal shape of the samples after compaction, the inuence of Young's modulus, cohesion energy, and the particles' diameter. The simulations were calibrated and compared with experimental data. The Young's modulus and cohesion energy density (CED) were the input DEM parameters that most impacted the macroscopic compaction behavior (maximum force, porosity). The CED is the ratio between the normal cohesion force and the contact area between the interacting particles. The maximum force (at fracture) increased signicantly with Young's modulus and the decrease of the CED.

A summary of some of the numerical models using DEM applied to ceramic and refractory materials is described in Table 2.3. It includes the material, type of modeling, contact model1 , some of the principal results, and the respective references.

Conclusions

Continuum mechanical modeling with nite element method is very popular to evaluate the compaction behavior of powder materials. The Drucker-Prager Cap (DPC) model is currently one of the most applied constitutive equations for compaction modeling of industrial parts. Furthermore, it is already implemented in commercial FEM software. However, the experimental characterization for the identication of the DPC model parameters is relatively complex and long. For this purpose, using the discrete element method (DEM) may be benecial as it considers the particulate behavior of heterogeneous powder materials. The DEM may be used to provide information on the behavior of the powder for stress states that are impossible or dicult to attain with triaxial tests due to its complexity. As far as we know, DEM has not been applied yet to generate yield and fracture surfaces of composite powder materials, and especially to refractories for steel casting applications.

In short, the methodology that we will adopt in this Ph.D. is rst to characterize the complex behavior of a composite refractory material (from VESUVIUS) experimentally (chapter 3) to represent the compaction behavior (chapter 5) and its mechanical response after ring (chapter 6) using the DEM. One of the most important stages in DEM is to compute the contact forces between particles (section 2.7.2). These forces need to accurately describe the contact between dierent phases of the refractory composite to represent its macroscopic behavior. In our case, the composite is composed of alumina particles, graphite, and binder. These forces are described in detail in chapter 4 through appropriated contact models. We can anticipate that as the binder is not considered a discrete compound, we will need to represent it dierently in the DEM framework. Once these stages are completed (experimental characterization and DEM modeling of the composite), we can deploy the DEM methodology to explore complex stress states to generate yield surfaces that are dicult to obtain experimentally.

By the end, this Ph.D. aims to propose a numerical tool to feed FEM with yield and fracture surfaces generated by DEM using the DPC model for compaction modeling of industrial parts. Chapter 7 is the rst attempt to show a typical use of this methodology, modeling the compaction of the composite refractory in the closed-die conguration using the Drucker-Prager Cap model.

Chapter 3

Experimental characterization

The materials to be studied in this Ph.D. work are complex heterogeneous powder composites composed of alumina grains and graphite embedded into a phenolic resin. Good knowledge of the microstructure (morphology, composition) is important for the modeling part. Thus, in this chapter, rst, microscopical observations of a carbon-bonded alumina refractory for steel casting applications are described. For this purpose, SEM and X-ray tomography analysis were carried out on a reference composite (sections 3.4 and 3.5). Four dierent composites are studied with distinct compositions. Their manufacturing process comprises four main steps: mix preparation, compaction, curing, and ring. X-ray tomography images are used to highlight the microstructure evolution: before compaction, after compaction, and ring stages. In section 3.6, the theoretical density, the bulk density, and the open porosity of cured and red microstructures are compared. The densication response of these powder materials is also identied through close-die compaction experiments by evaluating the eects of density and composition on the green mechanical behavior. The green mechanical strength is analyzed through uniaxial and triaxial compression tests (section 3.7). The thermo-mechanical behavior of such materials is carried out by TGA (section 3.9.1) and dilatometry (section 3.9.2) experiments. In TGA, the resin weight loss (%) of cured samples is estimated as a function of the testing temperature. In dilatometry characterization, the coecient of thermal expansion of the four composites is measured on both cured and red samples. Finally, their mechanical characterization after the ring is investigated by destructive tests (uniaxial and compression tests and diametrical tests), followed by an estimation of their yield locus and failure envelope in the mean and deviatoric stress plane (section 3.10).

Graphical abstract

Carbon-bonded alumina refractory is a heterogeneous and complex granular composite where hard alumina grains and elongated graphite akes are reinforcing the bonding matrix;

The evolution on the composite microstructures is analyzed through Xray tomography prior compaction, after compaction (green), and after ring grain samples;

Powder composites manufacturing process consists of four main steps: mix preparation, compaction, curing, and ring;

Temperature-dependent experiments are performed on cured and red composites via ThermoGravimetric analysis and Dilatometry;

Closed-die compaction tests are carried out on the powder composites, followed by uniaxial/triaxial compression tests;

Mechanical tests (uniaxial/triaxial compression and Brazilian tests) are performed on red composites.

Introduction

Carbon-bonded refractories are heterogeneous and complex granular composites made of solid alumina grains and graphite embedded into a polymeric resin (transformed into pyrolytic carbon during the nal heat treatment). These materials are designed to resist harsh steel casting environments (between 1500°C and 1600°C) involving thermal shock and oxidation/corrosion mechanisms. Mechanical, physical, and chemical properties should imperatively be controlled in the whole industrial process 'e.g., casting). Each constituent plays a specic role: grains limit the carbon dissolution in steel, graphite brings the non-wetting and enhanced thermal conductivity, whereas the pyrolytic carbon continuum is responsible for the mechanical strength and thermal shock resistance thanks to its limited thermal expansion. It is a delicate balance between dierent properties.

The key for engineering such composite relies on a good understanding of the relationship between the manufacturing conditions, microstructure, and resulting properties.

For this purpose, a few years ago, Vesuvius started studying model refractory composite materials. They are simplied compositions aimed at better understanding their microstructure and mechanical/thermo-mechanical properties with their composition. Thus, for the present Ph.D. work, a model alumina-carbon model material, free of any other additive, was prepared as the reference material along with this study. The other three composites were as well prepared simply by modifying the volume fraction of coarse and ne alumina grains, graphite, and binder content. Their manufacturing process consists of four main steps. First, the mix-manufacturing (or mixing of raw materials) where the powder grains are agglomerated, forming pellets that are a few tens of millimetres. Second, these powders are shaped via two routes: isostatic or closed-die compaction, depending on the application (targeted nal product shape). The sample after the compaction is the 'green body'. Third, the obtained green body is cured at 200°C, leading to evaporation of the solvent and reticulation of the phenolic resin. Finally, the cured refractory is followed by a ring process at approximately 900°C, and the nal properties are measured (chapter 2).

In this chapter, rst, we introduce some essential information on the overall microstructure by SEM and X-ray tomography analysis. The reference composite is rst submitted to image analysis to investigate its microstructure to understand better the mechanical and thermo-mechanical behavior of such carbon-bonded alumina refractories. The physical properties (theoretical density, bulk density, and open porosity) are measured. Likewise, closed-die compaction tests are carried out, followed by mechanical tests. In sequence, thermal analysis is performed to investigate the refractory composite behavior after the curing and ring steps. Finally, mechanical testing is performed to estimate the 3.3. Design of powder composites composite mechanical properties and elastic/plastic behavior after the ring stage.

Design of powder composites

The microstructure of the model composite is shown in Fig. 3.1. It can be thought of as an overlapping hierarchy, with a mesoscale made of coarse hard grains (white fused alumina with 250µm average size) and graphite (<400µm) reinforcing a matrix; itself constituted at microscale of ne grains (calcined alumina <50µm) embedded into a pyrolytic carbonaceous binder (a mixture of resin and solvent). The mix and the granulating process can be decomposed into three main steps. First, the solid raw materials (oxides, graphite, resin) are weighted according to the composite recipe. Second, these raw materials are loaded into a mixer. The solvent is added during the mixing, generating a temperature increase due to frictional forces. The mixing stops when the temperature reaches 50°C [START_REF] Gardziella | Phenolic Resins: chemistry, applications, standardization, safety and ecology[END_REF]. Finally, the excess solvent previously added to disperse the binder properly is removed during a drying cycle, blowing hot air at 100 °C and continuously shaking the mix.

A good mix engineering at this stage means obtaining the desired pellet size and maximizing their packing density: neither dusty (lack of binder) nor sticky (excess of binder) for the next process step [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF]. This powder composite is conditioned in a controlled atmosphere and closed barrel to limit as much solvent evaporation as possible.

In the present work, four dierent composites are investigated with dierent compositions (Table 3.1) with special emphasis on numerical simulations. Mix 1 can be considered the standard composite; mix 2 matrix exhibits a lack of binder (excess of ne grains).

Whereas mix 3 is in the opposite trend (excess of binder and lack of ne alumina grains). Concerning Mix 4, the matrix composition of mix 1 is used but with an overall depleted matrix (40 % on mix 4 against 45 % volume fraction on mix 1, 2, and 3). In turn, mix 4 has an excess of coarse alumina grains and graphite. As can be understood from these descriptions, the objective in mix 2 and mix 4 was to generate a model composite with an excess of solid grains. In contrast, mix 3 presents an excess of the binder. All compared to reference material (mix 1).

Composite

Coarse

SEM observations

To better understand and visualize the microstructure of these refractory materials, some SEM (scanning electron microscope) analyses are carried out on the nal product (red), displayed in Fig. 3.2. The images highlight the above-mentioned hierarchical microstructure with coarse alumina (a few hundred µm) and elongated graphite akes observed at low magnication. The matrix embedding the reinforcement, appearing in grey at low magnication, is composed of ne calcined alumina grains and binder, visible for magnication >x1000. It is worth noting that the geometry of coarse grains is erratic (dispersed in shape but generally angular), whereas the graphite akes are mostly planar. Fine calcined alumina grains exhibit mostly rounded shapes. As mentioned above, these coarse alumina grains (white fused alumina) are visualized in dierent dimensions, with an average size of 250µm. Some more rounded ne calcined alumina grains can also be noticed in Fig. 3.2 d). The graphite akes can be deformed during the compaction stage, and some micro-cracks can be observed at higher magnication. The alumina grains remain unaected (Figs. 3.2 c) and d).

X-ray tomography

In addition to SEM, X-ray computed tomography analysis (CT, done at SIMAP laboratory, Grenoble, France) is performed on mix aggregate samples to examine the overall microstructure arrangement. This investigation is restricted to the reference powder composite material, mix 1. The experiment consists of taking X-ray images of a sample tilted by a small angle for each image. The whole volume of the imaged specimen is reconstructed by assembling the digital images. The resulting 3D image is a superimposed projection of a volume in a 2D plane [START_REF] Salvo | X-ray micro-tomography an attractive characterisation technique in materials science[END_REF]. The CT analyses are performed at three stages of the process (sample size ranging from 0.7 mm to 1.9 mm): on mix aggregate, a compacted green sample, and a red sample (Fig. 3.3) fragments. The resolution of CT images depends on several parameters, including the voxel size applied during the analysis. Typically, the smaller the voxel size is, the better is the image resolution. All samples were rst analyzed at a voxel size of about 0.59 µm -1.19 µm size. The last test was repeated with a higher resolution to highlight potential microstructure damage associated with the ring process, leading to a voxel size of 0.59 µm (Table 3.2). Note that the gray levels in a CT slice image correspond to the X-ray attenuation, i.e., reecting the proportion of X-rays scattered or absorbed as they pass through each voxel. The X-ray attenuation is highly dependent on the X-ray energy, material density, and composition. Typically, alumina grains are denser and absorb more photons by the beam than porosity and graphite. The tomography image also reveals some large pores in the agglomerate sample before compaction that are mainly located around coarse grains (Fig. 3.3 a). Though, this porosity is not observed after the compaction to the same extent. Likewise, some porosity is noticed mainly around the coarse alumina on the red sample (Fig. 3.3 c), attributed to the so-called "bell eect" and within the matrix phase. The bell eect originates from the plastic deformation of the organic binding phase during the ring, accommodating grain thermal expansion. Above 700-800 °C, the binder becomes stier, leading to some porosity surrounding the grains.

Consequently, during the cooling, the shrinkage of grains will not be accompanied by an identical shrinkage in the pyrolytic phase, and the grains will detach from the shell surrounding them. These decohesions within the material microstructure contribute to the decreasing coecient of thermal expansion and the thermal shock resistance [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] [?]. The debonding occurs rst at the interface aggregate-matrix, and after, microcracks are propagated within the matrix leading to a composite macroscopic rupture [START_REF] Schmitt | Micromechanics applied to the thermal shock behavior of refractory ceramics[END_REF]. A higher resolution was required to highlight this eect on coarse grains, but it remains insucient for ne grains.

X-ray tomography also highlighted porosity surrounding graphite akes suggesting that not only coarse alumina but also graphite akes detach during cooling from the surrounding matrix due to the signicantly lower coecient of thermal expansion of the pyrolyzed binder compared with alumina and graphite. The anisotropic thermal expansion of graphite akes could add up to this eect [START_REF] Werner | Young's modulus of elasticity of carbon-bonded alumina materials up to 1450°C[END_REF].

The coarse alumina grains, matrix, graphite, and porosity can be highlighted due to the contrasting shades of grayscale associated with each phase. Thus, these images were binarized using a thresholding method, as white as the highlighted region of interest, using ImageJ and Analysis 3D plugin [START_REF] Boulos | Investigating performance variations of an optimized GPU-ported granulometry algorithm[END_REF] to distinguish the dierent phases of the aggregate and their respective percentage (in pixel size). The plugin was helpful to discriminate the graphite from the grain's outer region as both have a similar gray level. The plugin can also measure the volume of each phase based on the marching cube algorithm. The 3D region is divided into cubes, such that the vertices of each cube are a voxel [START_REF] Lorensen | Marching cubes: A high resolution 3d surface construction algorithm[END_REF]. Nevertheless, in this work, the percentage of each phase was associated with the calculated pixels of thresholded regions from each image slice. Thus, predicting the % variance based on the shape of the agglomerate from each image slice.

An example of a 3D thresholding analysis of a red sample composed of 1218 x 1343 x 1218 pixels (performed at 0.59 µm voxel size) is exhibited in Fig. 3 3.4 d) andphases a), b), andc). It is worth noting that the matrix was not further subdivided into the binder and ne grains because it was not possible to distinguish each other with the selected tomography resolution. Besides the thresholding, a segmentation method was also applied by assigning a range of grey levels (from color 0 to 255, black to white) to distinguish the dierent phases of interest (Fig. 3.5). In summary, image segmentation is a technique of partitioning the original image into distinct phases. The white color (255) represents large and medium alumina grains. The grey color (215) represents the matrix. The darker grayscale (color 120) represents the graphite phase, and the black color (color 0) inside the grain is the porosity. The threshold method was applied using, for instance, tools like 'remove outliers,' 'll holes,' and 'dilate' in the ImageJ program, and by the lter '3D fast median' from the Analysis 3D plugin. The segmentation was applied after the thresholding since the thresholded phases could be checked if correctly separated by comparing with the original CT 3D images.

As mentioned above, the percentage of each phase could be extracted from the thresh- olding (Fig. 3.4) applied to the four X-ray tomography analyses: agglomerate, green, and two red grain samples (performed at two dierent voxel sizes). The percentage of the total quantied white pixels, from each binary slice image, of four distinct phases: coarse/medium alumina grains, matrix (a mixture of ne alumina grains and binder), graphite, and porosity is listed in Fig. 3.6. In contrast with SEM analysis, where the sample needs to be polished and embedded in epoxy resin, this step is not required for X-ray tomography analysis, allowing easier porosity quantication. ) is also an after-red grain sample, analyzed with a higher resolution (0.59µm voxel size). Note that the % of each phase is aected by the X-ray tomography resolution and on the aggregate geometry itself.

The % of each phase, identied from thresholding, is presented in Table 3.3. The open porosity of the red sample was additionally measured by Archimedes' method (16.9% ± 0.1). This value exceeds the values estimated from X-ray tomography. The value at the highest CT resolution (13.4% porosity) is closer to Archimede's value but still lower. This is due to the ner pores not being thresholded by CT even at this resolution. Thus, from this table, and knowing that the same threshold was applied in both analyses, it can be concluded that the tomography analysis resolution impacts the measurement of porosity by image analysis. The volume percentage of each sample is highly dependent on the shape of the grain, and the resolution applied on CT analysis.

The volume fraction percentage of coarse and medium alumina grains is represented by approximately 23% for the rst three CT analyses. And, 36% for the last test (performed at larger resolution), which is closest to the volume fraction of coarse alumina grains of mix 1 (reference composite) recipe, that is 33%. The % of graphite and matrix for mix 1 recipe is 22%, and 45%, that is closest to the image analysis done at the third CT test (21%, and 44% for the graphite and matrix, respectively). However, it is worth highlighting that the volume fraction of the composite recipes do not reect the porosity content, only for raw materials. Also, the quantication of each phase through image analysis can depend on the image resolution, the grain shape, and it is representative of one unique millimetric agglomerate. Table 3.3: Average porosity, alumina grains, matrix, and graphite content (% of white pixels) calculated from all image slices at each grain sample condition (agglomerate, green, and two grains after-red voxel size resolution analyses). After-ring (1) corresponds to the CT analyses using a smaller voxel size resolution, and the after-ring ( 2) is the analysis performed at a higher resolution (0.59µm voxel size). % of volume fraction of the composite recipe (coarse alumina, graphite and matrix).

Physical properties

This section presents the methodology for measuring the theoretical density of the dierent phases (coarse alumina, ne alumina, graphite, and binder). The theoretical density is an important parameter for the measurement of the relative density during close die compaction. In sequence, the powder composites bulk density and open porosity after curing (to 200°C) and after ring are measured by Archimedes' principle.

Theoretical density

The theoretical density (ρ), measured in g/cm 3 , corresponds to the material density, once pressed, is fully dense (no porosity). In this work, the composite theoretical density was estimated by the "Rule of Mixtures," knowing the volume fraction of each constituent (white fused alumina, ne calcined alumina, graphite, and binder) and their respective densities (Table 3.4). The main objective of knowing the composites' theoretical densities is to describe their densication behavior during compaction experiments. The composite theoretical density can be written as:

ρ = ρ i • ν i (3.1)
with ρ as the loose composite density, and ρ i and ν i are the density and volume fraction of each constituent, respectively.

The theoretical density, in g/cm 3 , of the four composite recipes is reported in Table 3.5, considering the raw materials volume fraction related to each composite (Table 3

Bulk density and open porosity

The bulk density (ρ) and the open porosity (P) are measured by hydrostatic weighting based on Archimedes' principle, a known method of Hydrostatic Weighting. These measurements were performed on cured (after a heating treatment close to 200°C) and red small cylindrical samples. Firstly, the weight of a dry sample is measured (m 1 ). Next, the sample is submerged into water (ρ w is the water density, 1 g/cm 3 ) under vacuum.

Thus the liquid begins lling the pores. After approximately 1 hour, the weight of the fully inltrated sample is measured (m 2 ). In water suspension, the weight (m 3 ) of the wet sample is also measured. The bulk density and open porosity can be calculated using the following equations:

ρ = m 1 m 3 -m 2 ρ w (3.2) P = m 3 -m 1 m 3 -m 2 • 100% (3.3)
The bulk density, and the porosity of each composite, after curing and after ring, are listed in Tables 3.6 and 3.7, respectively. A total of two samples for each composite and each condition (cured or red) were analyzed (globally 16 measurements).

It can be noticed in Table 3.6 that all the cured samples show a very low amount of open porosity, except the composite with a lack of binder wt% (mix 2) that shows a signicantly larger amount of porosity (12.5%) as compared to the reference mix 1 (5.1%). The composite with more binder content, mix 3 (4.5%), exhibits a slight decrease in porosity, and mix 4 (6.4%) has more porosity than mix 1. A low binder content results in grains blocking each other, not allowing the sliding of the grains during compaction and leading to higher pore space (mix 2). Fig. 3.7 sketches this phenomena linking binder volume fraction with porosity.

Powder composite

In Table 3.7, the bulk density and open porosity percentage on red samples are shown. In general, the bulk densities of red microstructures are lower than the cured ones due to the resin pyrolysis that mostly happens during the ring stage. Consequently, due to the elimination (evaporation) of some binder content during pyrolysis, some porosity is introduced into the red microstructure. Concerning the recipe containing an excess of calcined alumina and lack of binder (mix 2), the open porosity evolves in the same way as the cured one. The open porosity increases with an excess of ne alumina grains and a lack of binder (mix 2). Nonetheless, the increase of porosity was more signicant at the cured stage than the pyrolyzed one (red). The bulk density and the porosity % of mix 1-4 are in agreement with [START_REF] Werner | Inuence of porosity on Youngs modulus of carbon-bonded alumina from room temperature up to 1450 °c[END_REF] where carbon-bonded alumina refractories (without additives and red at 1,000°C), compacted at 40 MPa, presented apparent porosity and bulk density of approximately 18% and 2.60 g/cm 3 . In [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] the bulk density and open porosity of carbon-bonded composites after the ring varies between 2.86-2.97 g/cm 3 , and the open porosity ranges between 16.20-18.30%.

Green mechanical testing

This section relates to closed-die compaction (nominal pressure of 35 MPa) and further green testing (referred to uncured samples) of the composites (mix 1 to mix 4). The objective is to identify and understand the composition eect on material densication and its green mechanical behavior. The eect of density on these latter properties was further evaluated for dierent compaction pressures on the reference mix 1: 10 MPa, 35 MPa, and 100 MPa.

Close-die compaction

The close die compaction experimental setup consists of a steel die with an internal diameter, in which the mix pellets are inserted. An upper punch moves inwards, applying stress to the powder and compacting it following the set cycle. Once the nominal end stress is reached, the unloading stage is initiated. The compaction aims to replicate the industrial pressing cycle, which is time-controlled.

The powder with a mass of approximately 140-143 g is inserted in a 40 mm diameter die. The cycle is divided into 3 stages: 1) preloading displacement (Fig. 3.8 a), 2) loading 35 MPa, 3) controlled unloading stage. The nominal stress is achieved in approximately 30 seconds, and the unloading takes a similar time. The preloading displacement aims at avoiding overloading due to the large displacement involved in the early compaction stage. It is worthwhile mentioning pieces containing several materials are commonly processed. Therefore, they need to have a normalized testing duration. The rigidity of the equipment is determined by means of the so-called "white test". It consists of performing a compaction experiment without any powder. Thus, the displacement of the device at a given pressure is obtained. Generally, the behavior is linear. The deformation of the equipment is then subtracted from the compaction curve of the studied powder composite. Thus, any recorded displacement corresponds to the actual mix height (h) in the die. Knowing the mix mass and die diameter makes it possible to obtain the density during the test directly. In Fig. 3.9 the close-die compaction and unloading of Mix 1 is shown. It presents the compaction pressure against the mix deformation (h 0 -h) curve, where h 0 =74.56 mm is the initial mix height, and h is the actual mix height that decreases along with the compaction. CHAPTER 3. EXPERIMENTAL CHARACTERIZATION Starting from h=h0, during the rst 15 mm, the pressure remains close to zero due to grains rearrangement and the high porosity; only a few contacts counteract the piston movement. Above 20 mm, the pressure increases signicantly with the powder deformation as the contact area between grains increases. The unloading stage may be divided in two parts: between 35 MPa and approximately 30 Mpa, the powder continues to densify, below 20 MPa the powder experiences some spring-back with some dilatation (decrease of the density). This phenomenon can be explained by the visco-plasticity behavior of the green composite material due to the binder. In the early stage of the unloading, visco-plasticity remains the dominant phenomenon, and the binder phase is "extruded" in-between the grains (density keeps increasing). Below approximately 20 MPa, this mechanism is balanced by elasticity also called the spring back eect. In other words, upon releasing the compaction pressure, the microstructure expands due to the elastic energy stored in the sample and a viscoelastic relaxation caused by the binder. Figure 3.9: Close die compaction-unloading curve: powder deformation (h 0 -h) against the compaction pressure of mix 1. The targeted nominal pressure is 35 MPa. h 0 is the initial powder height (h 0 = 74.56 mm), and h is the actual powder height.

Eect of the composition on the compaction behavior

To evaluate the eect of the powder composite compositions (Mix 1-4) on the densication behavior, the green density is converted into relative density (ρ r ). The relative density is dened as the ratio between the compacted powder density, obtained by pressing at a given pressure, and the theoretical density (ρ) (when the material is considered as fully dense). The theoretical density of the powder composites is listed in Table 3.5, and it was calculated considering the volume fraction of the raw materials in the mix. The relative density (in %) writes:

3.7. Green mechanical testing ρ r = m V 1 ρ 100 (3.4)
where m is the powder mass, which is constant, V is the powder sample volume. The powder volume can be written as πr 2 h with r the powder cylindrical sample radius, and h the powder sample height (which decreases during the compaction). Pressure-relative density (ρ r ) compaction curves of powder composites (mix 1-4) are plotted. The compaction response during the loading is shown in Fig. 3.10 a) and the unloading in Fig. 3.10 b). The objective of these close-die compaction trials is to extract the powder composites densication curves (mix 1-4) with a compaction pressure target equivalent to 35 MPa (applied by Vesuvius for industrial applications) and to estimate the viscoelastic recovery of the powder composite during the unloading stage. Here, the compaction tests are not repeated for each composite. However, VESUVIUS has conducted multiple tests on similar mixes and the standard deviation, at a compaction pressure of 35 MPa, is of the order of 0.3% on the relative density ( [START_REF] Mazerat | VESUVIUS RD VISO Technical Memorandum (Model Mixes 2)[END_REF]). During the compaction, when mix 1, 3, and 4 reach a density of 95%, it becomes more dicult to densify the microstructure because contacts between particles grow signicantly. At the maximum compaction pressure (35 MPa), a relative density of ρ r =96% for mix 3 (larger % of binder) and ρ r =90% for mix 2 (reduced % of binder) are obtained (Table 3.8). Concerning Mix 2, this composite departs from the other materials for the densication behavior. The higher ne alumina particles content (+5%) and the lower CHAPTER 3. EXPERIMENTAL CHARACTERIZATION binder content (-5%) in mix 2, compared with reference mix 1, suggests a stier behavior (larger number of sti contacts between ne calcined alumina grains and coarse alumina grains). Consequently, it becomes more dicult to press the microstructure, thence the smaller relative density. andc) at the end of the unloading (at 0 MPa).

Regarding mix 4, a slight dierence in the relative density, at 35 MPa, compared with the reference mix 1 (-1.5%) can be observed. Mix 4 is a specic composite compared to Mix 1, 2, and 3 as it contains more reinforcement (a larger amount of coarse alumina and graphite). This larger amount of reinforcement makes it slightly more dicult to compact compared with Mix 1 and Mix 3. Still, it does not modify the compaction behavior of Mix 4 drastically as compared to Mix 1, as shown by Fig. 3.10. This phenomenon demonstrates the predominant eect of the binder. Possibly there is enough matrix to ll the inter-coarse grains voids and enough binder to ll the inter-ne grains voids.

Comparing Mix 1 and Mix 3, these two composites present only a 1% relative density dierence at the end of the compaction (35 MPa). These two composites have a similar binder content (22.5% and 25%, respectively) and the same % of matrix content. The binder lls the inter-grain pores, increasing the relative density. These observations suggest that the binder and ne alumina grains are the materials that most impact the powder composites densication behavior. In other words, the compaction behavior is probably dictated mostly by the matrix phase for the refractory composites studied in this Ph.D. Still, it should not be a general statement for other refractories.

The unloading (and spring back eect), are shown in Fig. 3.10 b). At the beginning of the unloading (or when the pressure is still high), the four powder composites continue to densify slightly due to the binder. From approximately 35 MPa to 30 MPa, there is an increase in ρ r by approximately 0.48%, 0.38%, 0.60%, and 0.48% for Mix 1, Mix 2, Mix 3, and Mix 4, respectively. Mix 3, the recipe with more binder in its composition, presents the highest increase in relative density (0.6%). In the opposite, Mix 2, that is the one with less binder content, shows a lower density increase (0.3%) (Table 3.8). During the unloading, below 30 MPa, the spring-back eect leads to a relative density decrease of approximately 0.2%, 0.5%, and 0.2% for Mix 1, mix 3, and Mix 4, respectively. On Mix 3.7. Green mechanical testing 2, this decrease counter balances the initial gain mentioned above: the relative density globally decreases during the unloading stage.

In summary, usually, during compaction, the composite obeys to the interactions between two intricating scales: the reinforcement/matrix, and the ne alumina grains/binder. However, the eect of matrix could be highlighted in these materials densication, but not the eect of the reinforcement (coarse alumina and graphite). Hence, it could be concluded that during the loading, the ne alumina and binder is the pair of raw materials that most impacts the densication of powder composites.

Closed-die compaction at dierent pressures Additional close-die compaction tests are carried out on Mix 1 reference composite, under pressures of 10, 35, and 100 MPa (Fig. 3.11). The objective here is to generate samples at three dierent relative densities for triaxial uniaxial compression tests. For this purpose, 9 samples are generated from compaction tests at each pressure, totalizing 27 compacted samples. At 10 MPa, and 35 MPa, the relative density reached at the nal of the compaction, on average, is 84.5% ± 0.37 and 95.5% ± 0.30, respectively (Fig. 3.11.

In the test at 100 MPa, the samples reached a nal density of RD>100%. But, taking points at the limit of these curves (RD=100%), the compaction pressure encountered is 75.2 MPa ± 2.7. Thus, a pressure of ≈ 75 MPa should be high enough to have Mix 1 fully compacted. Composite ageing eect and moisture CHAPTER 3. EXPERIMENTAL CHARACTERIZATION Some important aspects may aect the compaction behavior of such carbon-bonded alumina refractory composites. Some examples include the interval of time between the mix composite manufacturing and the green characterization tests, the conditions of the mix storage, and the moisture (the solvent content).

The mixes in VESUVIUS are stored in a tightly closed, temperature-controlled tank to avoid signicant solvent evaporation. In addition, it is imperative to use the mix in compaction and green mechanical tests approximately 0-3 months after the mix manufacturing to avoid solvent evaporation and resin reticulation. In this Ph.D., the mixes followed the VESUVIUS protocols for storage and transportation. They were also submitted to closed-die compaction and after-testing (uniaxial/triaxial compression) within 0-6 weeks (approximately one month and a half) after the mixes were manufactured. These actions are important to avoid any discrepancy in the compaction behavior and mechanical properties. The use of the appropriate solvent adjustment also impacts the mix properties. Fig. 3.12 a) exhibits the evolution of the compaction pressure vs. the relative density of the composite Mix 1 (reference) and the same composite compacted 9 months after the manufacturing (old mix). In this example, at 35 MPa, the old mix has approximately a 6% decrease in relative density. This phenomenon can be explained by progressive solvent evaporation and resin reticulation. Thus, after some months, the powder becomes more dicult to compact. Fig. 3.12 b) shows the evolution of the compaction pressure vs. the relative density of the composite Mix 1 (reference) and the same composite, but with dierent solvent contents. The dry mix is the composite with less solvent. The powder drying is carried out in a cement dryer at room temperature for a couple of minutes. The wet mix is the composite with more addition of solvent. In other words, the wet mix is Mix 1, but with an addition of 0.3% of solvent. The dry mix represents Mix 1, yet with 0.3% less solvent. Compared with mix 1 (reference), Mix 1 (wet) exhibits a softer behavior by reaching a larger relative density at a given compaction pressure. On the contrary, Mix 1 (dry) presents a decrease of roughly 5% at the end of the loading, meaning a stier behavior during the compaction. This indicates that compaction behavior is drastically aected by this moisture content and the related binder viscosity.

In summary, composite powder compositions (Mix 1-4) are compacted up to a nominal pressure of 35 MPa, replicating the industrial protocol. The samples obtained with this procedure will be submitted to uniaxial compression tests (crush tests). The powder composite Mix 1 (reference) is used for closed-die compaction up to dierent maximum compaction pressures (10, 35, and 100 MPa). Those compacted compositions will be used for triaxial compression tests (Fig. 3.13). In the following sections, those mechanical tests

Uniaxial compression test

Cylindrical green composite specimens (Mix 1-4) with a height of approximately 40 mm ± 0.3 and diameter equal to 40 mm and compacted at 35 MPa are submitted to uniaxial compression tests (crush tests). As presented before, the relative densities measured after the unloading are approximately 95.7%, 89.9%, 96.5%, and 94.8% for Mix 1-4, respectively (Table 3. 8).

The samples are compressed uniaxially until a failure force (P ) is reached. The uniaxial compression strength (σ c ) is:

σ c = 4P πD 2 (3.5)
where D is the diameter of the sample in mm. In other words, uniaxial compression strength (also known as green strength) is the maximum axial compressive stress that a cylindrical sample of material can withstand before failing. The compressive green strength calculated for the powder composites (mix 1 to 4) is presented in Table 3.9. As seen previously, only one close die compaction for each composition was carried out. Thus, one compacted sample of each composite composition was conducted to uniaxial compression tests (a total of four samples). The standard deviation may be evaluated from the experience of VESUVIUS on the same type of refractory composites, which is approximately 0.02 MPa ( [START_REF] Mazerat | VESUVIUS RD VISO Technical Memorandum (Model Mixes 2)[END_REF]).

Powder composite Compression strength (MPa)

Mix 1 3.85 Mix 2 3.81 Mix 3 4.52 Mix 4

3.52

Table 3.9: Compression green strength (MPa) of powder composites (mix 1 to 4). The samples were compacted previously at 35 MPa and submitted to uniaxial compression tests. The standard deviation is typically around ± 0.02 MPa ( [START_REF] Mazerat | VESUVIUS RD VISO Technical Memorandum (Model Mixes 2)[END_REF]).

The failure surface of the samples after the tests is shown in Fig. 3.14. In general, the samples exhibit cracks parallel to the load direction and a lateral deformation (sample swelling). The images show that Mix 2 and Mix 4 are the ones that visually present more cracks. The binder is a ductile phase that cements the grains together. A lack of binder (Mix 2) clearly makes the compacted composite more brittle. As for Mix 2, Mix 4 incorporates also less binder compared with Mix 1 and has an excess of coarse alumina particles. This last eect may play a role as a more continuous skeleton of coarse alumina grains may form and degrade the ductility of the composite.

Green mechanical testing

Although Mix 2 is more porous than Mix 4, Mix 4 presents somewhat smaller compression strength (Table 3.9). This could be related to an interplay between a larger amount of coarse alumina particles and graphite, and to the smaller amount of ne alumina particles for Mix 4. Mix 3, the composite with more binder volume fraction, exhibits the highest compression strength and is almost crack-free. In summary, the compression strength is mostly impacted by the binder and coarse alumina contents.

Triaxial compression test

The triaxial compression test is an eective test to characterize the mechanical behavior of powder materials by controlling the shear and compressive stress components on the sample. Triaxial compression tests are performed on compacted Mix 1 cylindrical samples (30 mm diameter and approximately 36.6 mm ± 0.12 height), previously compacted to three pressures (10, 35, and 100 MPa) in a close-die conguration (Fig. 3.11), for a better understanding of the composite yielding and fracture behavior after compaction (green). The experimental setup, using an Instron machine device, is shown in Fig. 3.15. The tests are performed at ambient temperature. The radial pressure, or conning pressure, is applied by the oil pump, and the radial and axial pressure, σ 1 , evolves together until a given set point (radial pressure target). Thus, only the conning pressure (σ 2 = σ 3 ) is maintained constant, and only the axial pressure is increased. The entire cycle lasts CHAPTER 3. EXPERIMENTAL CHARACTERIZATION approximately 9 min. The sample axial strain is measured using an extensometer device (MFX). Dierent conning pressures are employed to investigate stress-strain curves and to be compared and calibrated with numerical simulations. Triaxial compression tests are used to analyze the material when submitted to both shear and compressive stresses. This method allows testing with dierent loading paths to establish a complete material characterization [Pavier & Doremus, 1999].

The axial stress (σ 1 ) evolution as a function of the axial strain of mix 1 is presented in Fig. 3.16. Three conning pressure (or radial stress) were applied in these tests, σ 2 = σ 3 = 10, 20, and 30 MPa. These curves show the same qualitative behavior with an approximately linear rst part followed by a non-linear one. The rst part corresponds to the increase of the conning pressure to the set value (10, 20 or 30 MPa). This rst part is terminated for strains smaller that 0.012 whatever the lateral pressure. This roughly linear part of the curve is represented by increasing the three stresses together (σ 1 = σ 2 = σ 3 ). The non-linearity is at above the conning pressure (σ 2 = σ 3 ). For example, when the applied radial stress is 10 MPa, the transition from the linear to the nonlinear part take place at σ 1 = 10 MPa.

The most porous samples (RD=85%) exhibit a more signicant deformation to attain the radial pressure of the curves. This deformation eect decreases with higher relative density samples. For instance, at RD=85%, the axial strain in stress-strain curves is below approximately 0.002-0.012 (for radial pressures of 10 MPa, and 30 MPa, respectively). At RD=95%, the axial strain varied between 0.003-0.009. And, at RD≈100%, the axial strain is between 0.003 and 0.005. Samples with smaller RD are less sti because more porosity in the microstructure means more voids around the grains, so the grains have more space to rearrange during the compaction, increasing the composite deformation for a given stress. The binder may also aect the sample deformation because it can extrude more easily when it ows through larger spaces between grains. Fig. 3.17 shows the samples damage prole after triaxial compression tests performed at 10 MPa, 20 MPa, and 30 MPa radial conning pressure. The sample shrinks mostly in the center with the increase of the conning pressure, showing a concave shape. Also, no visible cracks could be observed, suggesting that the green samples are still very ductile because of the binder eect. In our composites, the stress at which the material yields is not easily detected. Note in Fig. 3.16 that the green composite does not show a clear yield either ultimate stress at any conning pressure.

The identication of the yield and ultimate (maximum) stresses are determined from axial stress-strain curves of triaxial compression tests. We then calculate the deviatoric (q = σ 1 -σ 3 ) and mean (p = (σ 1 + σ 2 + σ 3 )/3) stresses. The yield stress is assumed as the oset of the rst linear part from stress-strain curves, and the ultimate stress as the

) σ 2 = σ 3 =10 MPa. b) σ 2 = σ 3 =20 MPa. c) σ 2 = σ 3 =30 MPa.
CHAPTER 3. EXPERIMENTAL CHARACTERIZATION point in common between the two secants from the second curve part (Fig. 3.18 a). Fig. 3.18 b) shows the deviatoric (q) and mean stresses (p) (in MPa) from triaxial compression tests based on the methodology applied in Fig. 3.18 a). The p vs. q (calculated from ultimate stresses) of samples at RD=95% and RD≈100% turn into a more convex shape. Still, no conclusive informations on the shape of these yield and ultimate surfaces could be taken from Fig. 3.18 b).

Nevertheless, it is probable that even if more triaxial tests were performed applying more conning pressures, predicting the yield and ultimate stresses of green microstructures would remain a dicult task. This is because plastic yielding may arise very soon in these green granular composites and that the elastic core (the part of the stress space inside the yield surface) may be very small in all strictness.

Sample preparation for thermal analysis and for ring characterization

All samples for the thermal analysis and ring characterization (mechanical tests and Brazilian tests) were produced from Vesuvius industrial test tubes. First, the powders are inserted inside a mold (Fig. 3.19 a), and compacted isostatically up to a nominal pressure of approximately 35 MPa, cured, and red to 900°C, following the standard industrial procedure. After the manufacturing process, the test tubes were cut into several cylindrical samples (Fig. 3.19 b) to be tested in Thermo Gravimetric analysis (TGA), dilatometry, uniaxial and triaxial compression, and Brazilian (diametral) tests. The samples dimensions are dependent on the type of analysis and are described in the next sections.

Thermal analysis

Temperature-dependent experiments are performed on cured powder composite samples with ThermoGravimetry analysis (TGA) (section 3.9.1) for investigating the pyrolysis of the binder phase during the thermal treatment. Thermal expansion analysis (TEX), or dilatometry, are carried out on cured (Mix 3) samples and red samples (mix 1-4).

Thermogravimetric analysis

ThermoGravimetric analysis (TGA) was measured by VESUVIUS technicians on aftercuring small cylindrical alumina-graphite powder composite (mix 1-4) samples. TGA was performed under argon atmosphere up to 1450°C with a ramp-up to 5°C min-1 followed by a dwell at the maximum temperature for 8 h (Fig. 3.20 a). The objective of TGA is to measure the weight loss as a function of the composite composition.

The weight loss curves show 3 temperature ranges (Fig. 3.20 b). First, stability up to 200°C corresponds to the curing temperature, followed by a steep weight loss that slows down at 600-800°C. Finally, above 800°C, smaller weight loss is observed depending on the mix composition. During the overall thermal cycle, the global weight loss for mix 1, 2, 3, and 4 is 4.3%, 3.8%, 4.6%, and 4%, respectively (ascending order: mix 3 > mix 1 > mix 4 > mix 2). The weight loss is increased with the binder content because of the binder pyrolysis. [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] performed TGA analysis on cured alumina-graphite powder refractory samples. The derivative curve (DTA) highlighted three peaks. The rst one at around 280°C, the second at 450°C, and the major weight loss occurred between 300°C and 800°C. Below 400°C, weight loss can come from absorbed water and unreacted phenols [START_REF] Costa | Structure-charring relationship in phenol-formaldehyde type resins[END_REF]. Above 800°C, the curve was relatively stable, indicating that the binder pyrolysis was almost nished. Dupuy calculated a global weight loss equivalent to 4.3%,

Dilatometry

Dilatometry is a technique of materials characterization that enables the measurement of a material's thermal expansion and is able to study the evolution of a refractory during the curing and/or ring steps. Dilatometers are usually classied into two categories: horizontal and vertical. Beyond the geometric aspect, the main dierence between these two types of dilatometers is controlling the load applied to the sample. A vertical dilatometer oers good control of the load throughout the thermal cycle. Since the contact pressure can inuence dilatometric results, a very low and constant load was applied throughout the test [START_REF] Schneider | Thermal length changes of some refractory castables[END_REF][START_REF] Wohlfahrt | The inuence of the contact force in dilatometry: force-controlled measuring cells in dilatometry[END_REF].

The TGA analysis was performed by VESUVIUS lab technicians using a horizontal dilatomer that allow larger samples increasing the precision of the expansion measurement. Likewise, in the case of a heterogeneous microstructure, the larger the sample, the more representative of the material. Additionally, for the same measurement error on the length variation ∆L, a larger sample allows a more precise measurement of the thermal expansion.

The dilatometer used in the present study is a Netzsch DIL 402 C (Fig. 3.22). With this device, it was possible to apply a force on the sample varying from 0.5 KPa-1.6 KPa. The dilatometric measurements are carried out on cylindrical samples of 18.8 ± 0.1 mm diameter and 24-26 ± 0.1 mm length under argon, with a 150 ml/min ow rate. The thermal cycle corresponded to a heating up to 1050°C with a 5°C min -1 ramp, followed by a cooling down at 10°C min -1 . The tests were performed in argon atmosphere carried out for 20 min before the start of the test and throughout the thermal cycle. The coecient of linear thermal expansion α T L (or CTE) is obtained from:

α T L = ∆L L 0 1 T -T 0 (3.6)
where α L T [K -1 ] is the coecient of linear thermal expansion (1/°C) at a temperature T (°C). L 0 is the initial length [meters] at an initial temperature T 0 (°C). ∆L is the change in length of the test specimen during heating or cooling in meters. First, to understand the eect of the binder pyrolysis and the refractory composite behavior after the curing step, the dilatometry was performed on cured (up to 200°C) refractory composite (Mix 3). As shown in Fig. 3.23 a), the thermal strain can be divided into four major steps; three during heating and one during cooling. The rst step (1), ranging from room temperature to 330 °C, can further be divided into 2 sub-steps: i) repeat of the curing cycle <200 °C followed by ii) a stable linear behavior [START_REF] Mottram | Thermal expansion of phenolic resin and phenolic-bre composites[END_REF]. Around 360 °C (step 2), this linearity is lost, and shrinkage is observed (the thermal strain becomes negative as the resin is pyrolyzed accompanied by degassing and volume shrinkage [START_REF] Trick | Mechanisms of the pyrolysis of phenolic resin in a carbon/phenolic composite[END_REF][START_REF] Costa | Structure-charring relationship in phenol-formaldehyde type resins[END_REF]. This same step 2 can also be sub-divided into two parts (Fig. 3.23 b-1): from 330 °C to 500 °C (2 a), where the major shrinkage is observed, and from 500 °C to approximately 700 °C (2 b), the CTE becomes more constant (still negative). In summary, the resin pyrolysis continues and starts to transform into rigid pyrolytic carbon during the second step. Indeed, the pyrolysis shrinkage is balanced by the alumina grains expansion but is still in favor of binder shrinkage. In step 3 (Fig. 3.23 b-1), the grain expansion takes over, and the overall expansion becomes positive.

The expansion inversion between steps 2 and 3, at about 700 °C, causes the bell eect 3.9. Thermal analysis and marks the transition from a viscous shrinking binder phase to a rigid pyrolytic carbon.

Since the pyrolytic carbon is stier, the bonding phase can no longer follow the swelling of the alumina particles. Thus, compressive radial and circumferential tensile stresses (that started at step 2 b), which happen at the interface, can cause radial micro-cracks within the pyrolytic carbon [START_REF] Ko | Microstructural changes of phenolic resin during pyrolysis[END_REF]. This phenomenon occurs at two scales within the composite microstructure, rst, at a lower scale within the bonding matrix (between the pyrolytic carbon and ne calcined alumina grains). In second, at a larger scale, between the bonding matrix and the coarse alumina grains.

Finally, during the cooling (step 4), the CTE stabilizes (Fig. 3.23 b-2), with a continuous shrinkage of the material. However, because the white fused alumina aggregates and ne calcined alumina particles have a larger CTE than the pyrolytic carbon, they shrink more. Thus, assuming that at 900 °C, the interface between the alumina grains and the carbon matrix is cohesive, decohesions occur during this stage because of the CTE mismatch, inducing to radial tensile stresses and circumferential compressive stresses in the matrix phase [START_REF] Nazaret | Inuence of the mechanical behaviour specicities of damaged refractory castables on the Young ' s modulus determination[END_REF]. Also, these decohesions (binder bell eect) occurs at two scales, within the matrix, between ne calcined alumina particles and the pyrolytic carbon, and between the matrix and the coarse alumina grains. It is worth noting that this mechanism has been observed for carbon bonded refractories (chapter 2).

Evolution of red materials thermal expansion

Once the material has been red, its dilatometric behavior is drastically dierent (Fig. 3.24 a). ∆L/L 0 increases as the temperature rise, but this curve is not completely linear and is also not completely reversible during cooling. This can be rst explained since the ring is carried out at 900°C whereas the dilatometry analysis was performed at 1050°C. Therefore, at the 1000-1050°C the binder is pyrolyzed leading to a material non-reversible evolution. Also, the re-opening of microcracks and decohesions between coarse alumina grains and matrix occurs, leading to a hysteresis phenomenon. Thus, in a composite containing larger binder volume fraction (Mix 3), the global shrinkage increases, a sign of plastic deformation which can originate from binder pyrolysis or microstructural damage.

The coecient of thermal expansion (CTE) evolution with temperature is shown in Fig. 3.24 b-1). The curves exhibit a division between 2 zones: rst, the CTE is relatively stable below 750°C whereas it increases above this threshold temperature and tends to 8 x 10 -6 K -1 at 1000°C. In the rst zone, the CTE attributed to the matrix is presented in Table 3.10. The CTE of the aluminum oxide (Al 2 O 3 ) is ≈ 6.5 • 10 -6 and the carbon bonding matrix < 4 • 10 -6 . Mix 3 show the lowest CTE (3.4 • 10 -6 K -1 ) due to the lack of calcined alumina and higher binder content. In the opposite direction is mix 2, which K -1 ). This behavior can be explained since Mix 1 and Mix 4 have the same matrix volume fraction %. Even if mix 4 has more volume fraction of coarse alumina grains, the grains interlocking was not enough to drastically aect the thermal expansion coecient.

Powder composite α T L (or CTE) [10 -6 K -1 ] Mix 1 4.3 Mix 2 5.5 Mix 3 3.4 Mix 4 4.2
Table 3.10: Average of the coecient of thermal expansion (10 -6 K -1 ) from the four powder composites (between 20-750°C).

The CTE of these powder composite samples during cooling is shown in Fig. 3.24 b-2). Globally, the CTE increases as the ne alumina content rises and the content of the binder is reduced (Mix 2). Above 700°C, all the CTE are relatively high and similar to each other. Below 750°C, Mix 1 and Mix 4 still exhibit very intermediate similar behavior. In contrast, the mix 2 material presents a lower decrease in CTE, because of the percolation of the calcined alumina grains, occurring even at low temperature and leading to an increase of the CTE [Guillo et al., 2015].

Mechanical testing on red samples

This section describes the mechanical characterization of red powder composite refractories. For this purpose, destructive tests were carried out, such as uniaxial compression, Brazilian (or diametral compression), and triaxial compression tests. Uniaxial and Brazilian tests are performed using dierent testing temperatures.

Uniaxial compression test

Uniaxial compression tests (or crush tests) are carried out on red Mix 1, isostatically compacted at 35 MPa. The crush test consists of placing a cylindrical specimen (machined with at and parallel ends) between two plates of a compression machine and loading it up to breakage (Fig. 3.25). The samples have 29 ± 0.06 mm diameter and 60.6 ± 0.05 mm height. The mechanical tests are carried out at 0.3 mm/min speed until sample breakage, at room temperature (20°C), and at dierent testing temperatures (300, 600, 900, and 1200°C) using a steel inductor (Fig. 3.26). Each crush test is repeated with 4 dierent samples and the standard deviation is calculated.

The applied force and deection are measured by a METEC extensometer on the samples and converted into stress and strain with the following relations:

σ = 4P πD 2 (3.7) ε = ∆L/L 0 (3.8)
where σ is the stress applied on the specimen (MPa), P is the load applied during the crush test (N), A is the sample cross-section area (mm 2 ). ε is the sample strain (mm/mm). ∆L is the sample height variation (mm). L 0 is the initial gauge length of the extensometer (mm).

The mechanical behavior characteristics of each material are analyzed from the stressstrain curves, and the following mechanical parameters are determined: modulus of elasticity (MOE), yield stress (σ y ), and ultimate stress (σ c ) (compressive uniaxial strength). The secant Young's modulus (or MOE) is determined at the linear part of the curve, and the yield stress is determined at 0.02% strain (0.0002 mm/mm). The ultimate stress is considered as the maximum stress from stress-strain curves. The ratio σ y /σ c is assumed as a measure of the deviation from the linear part to the non-linear part of the curve. The axial stress evolution of Mix 1 vs. axial strain is exhibited in Fig. 3.27 a) as the temperature increases. An important feature of these refractory materials is that the ultimate stress is increasing with temperature. We observe a increase of material strength (ultimate stress) with the testing temperature, with a more signicant increase above 600°C. The strength is reached at 900°C and 1200°C. Overall, the composite ductility is highly dependent on the temperature (ductility increases with the rise of the temperature). This can also be conrmed by Fig. 3.27 c) that highlights a decrease of σ y /σ c with the increase of the testing temperature. The MOE has been calculated from the slope between stress-strain curves, and exhibits quite a large standard deviation at high temperature (>600°C). Thus, the evolution of MOE with temperature should be taken with caution. Still, we can observe a decrease or at least a plateau for MOE at high temperature. Such behavior has already been observed in the literature [START_REF] Werner | Young's modulus of elasticity of carbon-bonded alumina materials up to 1450°C[END_REF][START_REF] Werner | Inuence of porosity on Youngs modulus of carbon-bonded alumina from room temperature up to 1450 °c[END_REF]. Werner et al. suggests that the increase in the eective Young's modulus with the temperature is a consequence of the closure of gaps between the particles and the matrix, which is related to the thermal expansion mismatch of the phases (graphite, alumina, and binder).

Brazilian test (diametral compression)

Brazilian test, or diametral compression, is an indirect tensile test that is performed to determine the quasi-static tensile strength of concrete materials [START_REF] Carneiro | Um novo método para determinacao da resistencia a tracao dos concretos[END_REF][START_REF] Carneiro | Une nouvelle methode pour la determination de la resistance a la traction des betons[END_REF][START_REF] Fairbairn | A Tribute to Fernando L. L. B. Carneiro (1913 -2001) Engineer and Scientist who invented the Brazilian Test[END_REF]. In this test, a compression load is applied vertically to the diameter of a cylindrical sample. The fracture occurs vertically because of tensile stresses induced horizontally by the sample geometry (Fig. 3.28). The tests are performed in samples with 39.69 ± 0.06 mm diameter and 35.39 ± 0.12 mm height that was previously compacted isostatically up to 35 MPa. Each Brazilian test is repeated with 7 dierent samples, and the standard deviation is calculated.

The Brazilian trial is carried out until specimen breakage. It allows determining the ultimate tensile strength (σ ult ) via the following equation:

σ ult = 2P πDh (3.9)
where P is the load at sample breakage (N), D corresponds to the sample diameter (mm), and h is the sample height (mm). The evolution of the tensile strength (MPa) or ultimate stress against the testing temperature (°C) (from 20°C to 1200°C) for mix1-4 composite compositions is presented in Fig. 3.29. In the present work, the sample strain was not measured. Usually is not common to evaluate elastic properties from Brazilian tests because the stress state within the sample is not uniaxial. So, strain is usually not measured in such test [START_REF] Belrhiti | Combination of Brazilian test and digital image correlation for mechanical characterization of refractory materials[END_REF]. Overall, the compositions present an increase in the tensile strength from 20°C to 1200°C, meaning that at higher temperatures, the materials become stronger, which is coherent with what we have observed in Fig. 3.27 b) for uniaxial compression tests.

The brittle failure of ceramic materials is caused by randomly distributed defects in the component. The defects occur in ceramic materials as a result of the manufacturing process. These defects can be described as microcracks in fracture mechanics in which size and orientation are random. Therefore, the material strength is the stress at which the most critical microcracks start propagating and growing [START_REF] Zielke | Inuence of carbon content and coking temperature on the biaxial exural strength of carbon-bonded alumina at elevated temperatures[END_REF]. These microcracks act as trigger loading to total failure. Based on the Weibull theory, the probability of failure at given stress σ is:

P f (σ) = 1 -exp - σ σ 0 m (3.10)
where σ 0 is the strength of the material with a failure probability of 63.2%. m is the Weibull modulus which describes the scatter of the material strength and it corresponds to corresponds to the slope of the plot ln(-ln(1 -P f )) versus ln(σ ult ).

Weibull distribution plots are obtained for all the samples (7 total for each composition) and are presented in Fig. 3.30 a). Mix 4 and Mix 1 samples show the highest Weibull modulus (m), about 25.5 and 23.8 MPa, respectively, indicating a lower dispersion of tensile strength values (σ ult ). A lower value of Weibull modulus (m=17.4 MPa) was obtained for mix 2, that is, the composite with less binder content, and consequently more porosity (about 19.7%) against approximately 17% for mix 1, 3, and 4 (see previous Table 3.7). [START_REF] Liu | Eects of pore structure on thermal conductivity and strength of alumina porous ceramics using carbon black as poreforming agent[END_REF] The strength of Mix 1-4 samples with a failure probability of 63.2% (σ 0 ) (Eq. (3.10)) is shown in Fig. 3.30 b). It represents the evolution of σ 0 against the testing temperature, that varied from 20-1200°C. From 20-600°C, the reference composite (mix 1) is the one that presented the highest σ 0 . Mix 2, shows the lowest strength. At the ring temperature (T=900°C), mix 1 shows a strength (σ 0 ) declining followed by a strength recovery (increase) above 900°C (T=1200°C). A reverse behavior was measured by [START_REF] Zielke | Inuence of carbon content and coking temperature on the biaxial exural strength of carbon-bonded alumina at elevated temperatures[END_REF] on ne-grained carbon-bonded alumina ceramic materials submitted to B3B-tests at dierent testing temperatures. A composite material red at 800°C exhibited an increase of σ 0 up to the ring temperature (800°C) and a subsequent decrease in strength for testing temperatures above the ring temperature (T>800°C). The same composite but red at 1400°C shows a steady increase in strength (σ 0 ) with increasing testing temperature. The maximum strength occurred when the testing temperature was equal to the ring temperature. Mix 4 follows the [START_REF] Zielke | Inuence of carbon content and coking temperature on the biaxial exural strength of carbon-bonded alumina at elevated temperatures[END_REF] behavior where the maximum strength value was achieved at the ring temperature (T=900°C) followed by a decrease at T=1200°C. Mix 2 and mix 3 highlighted a steady increase of σ 0 with increasing testing temperature.

The standard deviation in strength (σ 0 ), changing with the testing temperature, and mix composition (mix 1-4) may be related to the microstructure pore size distributions and pore size [START_REF] Werner | Inuence of porosity on Youngs modulus of carbon-bonded alumina from room temperature up to 1450 °c[END_REF][START_REF] Liu | Eects of pore structure on thermal conductivity and strength of alumina porous ceramics using carbon black as poreforming agent[END_REF]. The pore size distribution may be associated with the binder evaporation and void formation due to the thermal expansion mismatch between alumina, graphite, and the carbonaceous matrix. The cracks and voids formed during the cooling stage [START_REF] Werner | Young's modulus of elasticity of carbon-bonded alumina materials up to 1450°C[END_REF][START_REF] Luchini | On the nonlinear behavior of Young's modulus of carbon-bonded alumina at high temperatures[END_REF]] also may aect the microstructure strength.

To note, Weibull statistics are generally obtained with a larger amount of data points as compared to the plots shown in Fig. 3.30 (7 samples of each composition at each testing temperature). Still, this plot allows the evaluation of the σ 0 and m values even for a smaller quantity of samples. This is an useful parameter in the industrial context because it allows obtaining the strength of industrial parts whose volume is dierent from the lab part.

3.10. Mechanical testing on red samples

Triaxial compression test

Triaxial compression tests were performed on red composites (mix 1 to 4) in 29 mm diameter and 60 mm height samples, iso-pressed to 35 MPa. The trials were performed in the same Instron machine used for the green characterization (Fig. 3.15). Thus, dening σ 1 , and (σ 2 , σ 3 ) as the axial and conning stresses, respectively, triaxial tests are carried out to calculate the yield and ultimate stresses and consequently the deviatoric and mean stresses of red powder composites with the eect of the applied conning pressure (σ 2 = σ 3 ). A series of tests with increasing lateral pressure (σ 2 = σ 3 = 0 MPa, 2.5 MPa, 5 MPa, 10 MPa, 20 MPa, 30 MPa, 60 MPa) were performed. For each test, the sample was loaded in the pressure cell, loaded under hydrostatic pressure (σ 1 = σ 2 = σ 3 ), coming from the oil, until the pre-set conning pressure is reached and maintained constant. The axial stress (σ 1 ) is gradually increased until sample failure.

A schematic representation of the three stresses (σ 1 , σ 2 , and σ 3 ) applied to the sample during triaxial tests is given in Fig. 3.31 together with samples failure surfaces after the tests by varying the conning pressure values (σ 2 = σ 3 ). From 0 to 5 MPa, the samples show some thin microcracks in shear. Increasing the conning pressure, the samples (at higher pressure, >15 MPa) exhibit some clear fracture in shear in the center of the geometry. Plots of axial stress (σ 1 ) as a function of the axial strain for the dierent conning pressures used in the triaxial compression experiments are presented in Fig. 3.32 a) mix 1, b) mix 2, c) mix 3, and d) mix 4. The axial stress of the four composite powders increases steadily with increasing conning pressure (σ 2 = σ 3 ). This is simply a consequence of the increasing interlocking of individual particles with increasing compaction pressure [START_REF] Zeuch | Mechanical properties and shear failure surfaces for two alumina powders in triaxial compression[END_REF]] [Mervyn S. Paterson, 2005]. The stress-strain curves show a typical ductile behavior with a roughly linear segment followed by a smooth transition to a non-linear curve. This behavior agrees with stress-strain plots from triaxial compression tests using aluminium silicate green materials [START_REF] Bosi | Experimental investigation of the elastoplastic response of aluminum silicate spray dried powder during cold compaction[END_REF]. However, to our best knowledge, this type of experiments has never applied on red carbon-bonded refractory composites.

Figure 3.32: Axial stress, σ 1 , evolution vs. axial strain (mm/mm) from triaxial tests performed at several conning pressure (σ 2 = σ 3 ). These tests were carried on red mix 1 to 4 powder composites (previously compacted to 35 MPa). Fig. 3.33 is represented by the ultimate axial stress σ 1 (ultimate) evolution with the conning pressure. The ultimate stress is considered as the point in common between the roughly linear curve with a strain oset equal to 0.02% and the secant of the second non-linear curve. It can be seen that the ultimate axial stress increases with the conning pressure, and it highlights a linear tendency (with a tting correlation coecient equal to R 2 = 0.99 for mix 1 plot). Mix 1, 3, and 4 exhibit a very similar behavior. Mix 2, which is the composite with a lack in binder content, exhibits a relatively smaller ultimate axial stress with values 10% (60 MPa) to 33% (0 MPa) smaller than the corresponding ones for Mix 1.

Deviatoric and mean stresses

VESUVIUS fundamental research applies the Bigoni, and Piccolroaz function to characterize red composites and to dene the materials model parameters for nite element modelling (FEM) simulations. Thus, the main objective here was to estimate the yield locus and the failure envelope (critical line) to be further compared with discrete element numerical simulations (presented in chapter 5) of red carbon-bonded alumina composites (Mix 1-4) and to apply the model function on thermo-mechanical modelling through FEM (chapter 6). For this purpose, yield and ultimate stresses estimated from previous destructive testing (uniaxial and triaxial compression, and Brazilian tests), were reported as functions of mean stress (P = -1 3 (σ 1 + σ 2 + σ 3 )), and deviatoric stress in module (Q = σ 1 -σ 3 ).

The P-Q yield and ultimate surfaces of the four powder composites (mix 1-4) were evaluated from Brazilian, uniaxial, and triaxial compressive tests and are shown in Fig. 3.34 a), for the ultimate, and on b) for the yield surfaces. The rst p-q point in the ultimate surface (Fig. 3.34 a) is represented by Brazilian tests where p=2 σ 1 3 , and q= et al., 2000]). The other points correspond to triaxial compression tests at conning pressure varying from 0 to 60 MPa. Note that the uniaxial compression (crush) tests at 20°C and the triaxial tests at σ 2 = σ 3 =0 presented similar ultimate and yield deviatoric and mean stresses because these two tests have a similar stress state.

√ 13σ 1 ([Martin
The evolution of the deviatoric and mean ultimate surfaces have almost a linear tendency. However, the yield stresses exhibit a convex shape and resemble the p-q shape from the Bigoni and Piccolroaz function [START_REF] Bigoni | Yield criteria for quasibrittle and frictional materials[END_REF], which turns pertinent to CHAPTER 3. EXPERIMENTAL CHARACTERIZATION the use of this model to characterize these red powder composites. In both P-Q surfaces (ultimate and yield), decreasing the binder content (Mix 2) leads to a strength material decrease due to the relative density of the composite at the end of the compaction. Mix 1 and mix 4 presented a very similar behavior for the yield surface. Mix 3 (higher relative density) is the one that exhibits the highest strength.

Conclusions

This chapter investigated the microstructure of a carbon-bonded powder composite refractory through SEM and X-ray tomography analysis. Four composites were studied with distinct compositions (mix 1-4). Several experimental techniques on green (after compaction), after-curing, and after-ring samples were carried out. In general, the compaction behavior, mechanical and thermo-mechanical properties could be analyzed.

The composite microstructure is composed of white fused alumina (coarse grains) and graphite, kept together by a reinforcing matrix consisting of calcined alumina (ne alumina grains) and a pyrolytic carbonaceous binder. The SEM analysis of red microstructures showed that coarse alumina grains have irregular shapes. With CT analysis, graphite akes deformation and micro-cracks are observed in a green sample due to the pressing. Porosity was detected around coarse alumina grains and graphite due to the mismatch coecient thermal expansion between graphite, alumina, and the pyrolyzed binder. The coarse alumina grains and graphite are detached from the surrounding matrix. These observations are in agreement with [START_REF] Schmitt | Micromechanics applied to the thermal shock behavior of refractory ceramics[END_REF] and [START_REF] Werner | Young's modulus of elasticity of carbon-bonded alumina materials up to 1450°C[END_REF].

The overall open porosity % in red samples is larger than in cured samples because of the binder evaporation and pyrolysis. Thus, some porosity is introduced in the mi-3.11. Conclusions crostructure. The bulk density and porosity of the powder refractories are in agreement with [START_REF] Werner | Inuence of porosity on Youngs modulus of carbon-bonded alumina from room temperature up to 1450 °c[END_REF].

The binder is the component that mostly aects the compaction behavior of the compositions. The reinforcement (graphite and coarse grains) eect on the compaction behavior could not be clearly identied, maybe because the grains interlocking did not occur. An increase in the spring-back eect as a function of the binder content could be observed during the unloading.

Regarding uniaxial compression tests on green samples, the composite with more binder content exhibited the highest green strength and was almost crack-free. The composite with more reinforcement phase presented the lowest green strength because most likely of coarse grains interlocking, increasing the micro-stress at the contact.

During triaxial compression tests, the axial stress-strain curves showed similar behavior on green samples previously compacted at three dierent pressures. The curves exhibited a rst roughly linear part followed by a non-linearity (axial stress continues to grow, and only the radial stresses remain constant). The most porous samples presented a larger deformation during the rough linear part of the curve, whereas this eect was reduced for denser samples. This could be explained because more porosity leads to more space for grains rearrangement during the compaction.

The thermal expansion analysis on cured samples could be divided into three stages. Up to 360°C, the coecient of thermal expansion becomes negative (binder pyrolysis), causing material shrinkage. Until 700°C, the binder pyrolysis continues and starts to transform into a rigid pyrolytic carbon. Above 700°C, the overall CTE becomes positive because the pyrolitic carbon becomes stier and the thermal expansion of aluminum oxide becomes predominant. Thus the matrix can no longer follow the increasing expansion of alumina grains, creating micro-cracks within the microstructure. These observations agree with previous Vesuvius Ph.D. works [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF][START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF].

The uniaxial compression on red samples show that the material softening increased with the testing temperature. The Young's modulus signicantly increased up to a testing temperature equivalent to 600°C, because of the closure of gaps between the alumina grains, graphite, and the matrix (which is related to the CTE mismatch between them). At T >600°C, Young's modulus decreased due to the closure of the gaps during the binder pyrolysis.

The Weibull approach was used to interpret the data from Brazilian tests. The Weibull modulus (m) has been determined for our powder composites. This is an useful parameter in the industrial context because it allows obtaining the strength of industrial parts whose volume is dierent from the lab part. Mix 4 and Mix 1 are the composites that exhibited the highest Weibull modulus, indicating a lower dispersion of the ultimate strength. Mix CHAPTER 3. EXPERIMENTAL CHARACTERIZATION 2 (lack of binder content) has shown the lowest Weibull modulus that may be associated with the higher porosity.

The deviatoric and mean stresses surfaces of the four red composites were estimated through uniaxial and triaxial compression tests and Brazilian tests. The yield surfaces lead to a convex shape likely similar to the Bigoni and Piccolroaz plasticity model [START_REF] Bigoni | Yield criteria for quasibrittle and frictional materials[END_REF]. Increasing the binder content leads to a strength material increase.

Finally, given the complex mechanical behavior of such heterogeneous composite refractories, one may ask if these experimental observations can be correctly reproduced by numerical simulations? In the next chapter 4, an original approach by discrete element method (DEM) is described to represent a composite made of coarse and ne alumina grains, graphite, and binder. The aim of using a numerical approach was to improve our knowledge of the behavior of a complex industrial composite material. In this respect, we will need to represent each of these phases in the DEM and investigate the possibility of DEM approaching some of the experimental data presented here.

Chapter 4

DEM: model development and description

An original approach is presented in this chapter, which is based on the Discrete Element Method (DEM) of composite powder ceramic materials for steel casting applications, adapted for the green (compaction) and after-ring stages. In section 4.3, the contact laws (The Hertzian and the microscopic bond models) that were implemented in this work are described. From those contact laws, it is possible to compute a macroscopic stress tensor at the macroscopic scale of the entire numerical composite (section 4.4). The composite is represented by a mixture of particle clusters and single particles (section 4.5). While the ne alumina is considered as single non-bonded spherical particles, the graphite ake and coarse alumina are discretized as a set of spherical particles bonded together (particle clusters). The graphite particles bonds may fracture during the compaction, except for the coarse alumina particles since those bond interactions are not considered in the calculations. The binder is not modeled as a particulate component. Instead, it is modeled as an elastic soft-shell covering alumina material, however uniquely at the green stage. The preparation of the numerical composite follows two initial steps: i) the gas of particles and ii) the jamming. At the nal of the jamming, the composite is at a relative density of 0.6 (section 4.6). The 'jammed' packing is compacted and unloaded. The unloaded samples are submitted to a series of triaxial compression and tensile simulations (section 4.7). Isostatic and closed-die compaction simulations are carried out on packings containing a dierent number of particles to determine the minimum number necessary to tend towards an isotropic response and converge onto a constant pressure. Finally, in 4.9.2, the virtual ring process is modeled by introducing bonds to pairs of particles in contact. The bonds fracture mechanism is described, considering the porosity revealed on X-ray tomography images of an after-red sample. In the green stage, the coarse and ne alumina particles are coated by an elastic soft-binder shell;

Isostatic and close-die compaction simulations were performed to verify the minimum number of particles that is necessary to converge towards constant deviatoric and mean stresses;

The numerical ring step is represented by creating bonds between particles that are in contact at the end of the unloading stage;

The damage observed by X-ray tomography images was represented by selectively breaking bonds.

Introduction

This chapter presents an original approach for modeling a powder composite made of graphite, alumina (ne and coarse particles), and binder. The objective of using a numerical method was to improve knowledge on the behaviour of a very heterogeneous and complex industrial composite material. The simulations were focused on analyzing the contribution of each phase to the composite compaction response. The rst insight was that the increase of the volume fraction of hard phases (alumina and graphite) aected the composite relative density. Therefore, increasing the hard phase volume fraction decreases the attainable relative density for a given applied pressure. However, since coarse alumina grains and graphite akes are coated by a soft carbonaceous matrix (mixture of ne alumina particles and binder), this simplistic assumption needs further investigation. Indeed, the identication of constitutive models of refractory composites for compaction and casting applications, based on numerical simulations, is an alternative way to reduce the cost of the experimental method itself.

Consequently, a model based on the Discrete Element Method (DEM) is used within the in-house code dp3D [START_REF] Martin | Unloading of powder compacts and their resulting tensile strength[END_REF] and is performed to characterize the composites macroscopic behavior at the green and after-ring stages. DEM is a powerful tool to simulate the behaviour of a representative volume element of a particulate medium. In this method, each particle moves according to the fundamental principle of mechanics, and it interacts with neighbors through appropriate contact laws. DEM was used to represent each phase correctly and yet, suciently simple to run simulations in a reasonable amount of CPU time. The four composites have distinct compositions, which aect their densication response during the compaction. For this reason, the rst question needs to be answered. Is it possible to reproduce qualitatively or quantitatively the composites mechanical behaviour dierences? For this purpose, the composite is modeled as a mixture of particle clusters and single particles. Both coarse alumina and graphite are considered assemblies of particles bonded together (clusters). The ne alumina is represented by single non-bonded particles. The graphite particle bonds may fracture during the compaction, and the pellets can deform. On the contrary, the coarse alumina bonded particles cannot break. The binder material is not characterized as a discrete element and presents some peculiarities at the contact. This last is modeled as a soft elastic shell surrounding coarse alumina clusters and ne alumina particles. Specic contact laws (Hertzian and microscopic bond models) are employed to model these material phases separately. When two coated particles (ne alumina particle or/and coarse alumina cluster) or a coated and non-coated particles indent each other, the contact's elastic behavior will depend on two dierent conditions. First, when the 4.3. Modeling particulate and continuous materials with DEM interaction is limited to this soft binder shell. Second, once the critical indentation is reached, hard alumina contacts appear. If the contact at the second condition is failed (unloaded), some hysteresis is introduced at the unloading.

Following the compaction and unloading simulations, the numerical red composite is modeled by ascribing bonds to all particles in contact to mimic the cohesion between particles during the ring. As the binder is pyrolyzed during the rst stages of the ring, the matrix is represented by ne alumina particles linked by elastic bonds. In sequence, to represent the binder bell-eect revealed from X-ray tomography images (chapter 3), pairs of bonds are broken between ne alumina particles and between ne alumina-graphite particles.

In summary, the present chapter focuses on developing and validating an original approach based on DEM simulations to represent composite materials made of four distinct entities (ne alumina, coarse alumina, graphite, and a soft binder at the green stage). The DEM contact laws used in this work are rst described for particulate and then applied for a composite approach. The volume fraction of each constituent is calculated based on the composites recipe. The mechanical representation for a green and a virtual red sample is also carefully optimized to represent a carbon-bonded alumina refractory material for steel casting applications.

Modeling particulate and continuous materials with DEM

This section describes the implementation of the principal contact laws used in this work: 1 -The Hertzian contact law (section 4.3.1), and 2 -The microscopic bond model (section 4.3.2). The non-bonded contacts were considered to behave elastically, and the bonded particles may fracture.

The graphite akes and coarse alumina materials are represented as particle clusters. Indeed, the clusters are considered assemblies of single spherical particles. For each cluster, the single particles are bonded together. However, the ne alumina particles are modeled as a set of single spheres with no bonds to represent a more granular macroscopic behavior, characteristic of green materials. In DEM, dierent types of elastic particle contacts were used. First, contact between non-bonded particles (by the Hertzian model), see Fig. 4.1 a). Second, contact between bonded particles (Fig. 4.1 b). These bonds may not fracture during the numerical simulations (Fig. 4.1 b-1), or (Fig. 4.1 b-2) fracture. A fractured bonded contact may after be restored [START_REF] Balakrishnan | Eect of particle size in aggregated and agglomerated ceramic powders[END_REF].

Hertzian contact law

The Hertzian law illustrates the normal force at the contact between two particles with elastic properties, together with the Derjaguin Muller and Toporov (DMT) adhesion theory, initially pioneered by [START_REF] Derjaguin | Eect of contact deformation on the adhesion of elastic solids[END_REF], which adds a tensile force to the standard repulsive Hertzian force. [START_REF] Martin | Inuence of adhesion and friction on the geometry of packings of spherical particles[END_REF]] also applied the DMT and Johnson-Kendall-Roberts (JKR) [K. L. [START_REF] Johnson | Surface energy and the contact of elastic solids[END_REF]] adhesion model to study the eect of adhesion and friction parameters on the geometry of monosized packings of spheres. [START_REF] Martin | Unloading of powder compacts and their resulting tensile strength[END_REF]] also used the DMT theory, however, to evaluate the powder spring back through unloading simulations of an agglomerate made of perfectly plastic spherical particles. [START_REF] Barthel | Adhesive elastic contacts: JKR and more[END_REF] wrote a paper review with methods developed in the past decades concerning adhesive interactions (and contact models, such as DMT and JKR) for elastic contact problems.

The normal force acting between two non-bonded particles with a radii (r i and r j ), and elastic properties (E i , ν i ) and (E j , ν j ), respectively, (n and t as unit normal and tangential vectors), is given by the Hertzian law1 (Fig. 4.2) [START_REF] Johnson | Contact Mechanics[END_REF]:

N Hertz = 4 3 E * R * 1/2 δ 3/2 n (4.1)
The eective Young's modulus between these two particles in contact is:

E * = 1 -ν 2 i E i + 1 -ν 2 j E j -1 (4.2)
The equivalent radius R * is:

R * = 1 r i + 1 r j -1 (4.3)
δ n is the normal indentation. The Derjaguin, Muller, and Toporov (DMT) adhesion theory [START_REF] Derjaguin | Eect of contact deformation on the adhesion of elastic solids[END_REF] describes the elastic contact with adhesion between two spheres, under the action of a normal force (N e ) [Martin et al., 2006a[START_REF] Barthel | Adhesive elastic contacts: JKR and more[END_REF].

N e = N Hertz + N DM T = 4 3 E * R * 1/2 δ 3/2 n -2πωR * (4.4)
while N DM T and N Hertz are the adhesive and repulsive forces, respectively, and ω the work of adhesion (ω = 2γ, γ is the surface energy). Decohesion occurs in the DMT model for a pull-o force N DM T = 2πωR * .

Contacts may also transmit frictional force (HertzMindlin model) in the sticking mode (Eq. (4.5)) while the tangential force is limited during sliding (Eq. (4.6)) by Coulomb friction (friction coecient µ) [START_REF] Martin | Inuence of adhesion and friction on the geometry of packings of spherical particles[END_REF]:

T = -8G * • aδ t if 8G • aδ t < µN Hertz (4.5) T = - δ t |δ t | µN Hertz if 8G * • aδ t ≥ µN Hertz (4.6)
where G * is the equivalent shear modulus and δ t the accumulated tangential displacement at the contact location. Note that the condition for slip (Eq. (4.6)) applies on the repulsive part of the normal force (N Hertz ) and not on the total force N e .

Microscopic bond model

For bonded particles, the contact law is given by the analysis of [START_REF] Jeerson | The elastic response of a cohesive aggregate -A discrete element model with coupled particle interaction[END_REF] who studied the elastic response of bonded contact by the nite element method. Two spherical particles p and q with radii r p and r q , respectively, connected to each other through bonds, transmit normal and tangential forces, and resisting moments (Fig. 4.3).

The normal contact force (N b ) for a bonded contact is given by:

N b = 2ER * 1 -ν 2 f N (a * , Ψ)a * u N (4.7)
where E and ν are the Young's modulus and Poisson's ratio of the particles, f N (a * , Ψ) is a function that depends on the relative radius of the bond (a * = a b 2R * ), Ψ is a geometric factor which allows for bond interaction to be considered, and u N is the accumulated normal displacement at the contact. The normal force can be either in compression (u N > 0) or in tension (u N < 0). The tangential contact force (T b ) is written as:

T b = 4ER * (2 -ν)(1 + ν) f T (a * )a * u T (4.8)
where the function f T (a * ) depends on the bond size a * , and, u T is the accumulated tangential displacement at the contact. f N and f T values vary typically in between 1 and 2. For more details concerning the form of f N (a * , Ψ) and f T (a * ) used in this work, see [START_REF] Jeerson | The elastic response of a cohesive aggregate -A discrete element model with coupled particle interaction[END_REF]] [Martin et al., 2006a[START_REF] Pizette | Green strength of binder-free ceramics[END_REF]. The maximum stress in tension (σ N ) and in shear (σ T ), transmitted by this cylindrical bond of radius a b , can be evaluated by using the Potyondy's model [START_REF] Potyondy | A bonded-particle model for rock[END_REF]:

σ N = N b πa b 2 -4 |M T | πa b 3 (4.9) 4.4. Macroscopic stress calculation σ T = T b πa b 2 + 2 |M N | πa b 3 (4.10)
where M N and M T are the bonds resisting moments in the normal and tangential directions, respectively:

M N = - 8ER * 3 (2 -ν)(1 + ν) f T (a * )θ N (4.11) M T = - 2ER * 3 (1 -ν 2 ) f N (a * , Ψ)θ T (4.12)
where R * is the equivalent radius at the bond between these two spherical particles, and θ N and θ T are the accumulated relative rotations in the normal and tangential directions, respectively.

Regarding the failure criteria applied in our work at a bonded contact, Eqs. (4.9) and (4.10) give at each time step an evaluation of the stress in tension and shear acting on the bond, σ N and σ T , respectively. If one of these stresses is above the tension strength (Σ N ) and shear strength (Σ T ), input parameters in dp3D, the bond is considered as broken (in short, if -σ N > Σ N , or if σ T > Σ T ). In our case, the only tensile force (if the particles further separate) that remains is the one given by the N DM T term from Eq. (4.4). An adhesive surface energy was used as an input parameter for the compaction simulations. A fractured bond may transmit a shear force according to a Hertz-Mindlin friction law (Eq. (4.8)). Correspondingly, a fractured bond in shear continues to transmit a resisting moment in the tangential direction but none in the normal direction.

Macroscopic stress calculation

Once the above described contact laws are implemented in dp3D, it is possible to compute a macroscopic stress tensor at the scale of the entire packing. This is simply carried out in periodic conditions by considering the total contact force at each contact (N+T) and the branch vector connecting the centers of the two particles p and q in contact (l p,q ). The macroscopic stress tensor is calculated from Love's formulation [START_REF] Weber | Recherches concernant les contraintes intergranulaires dans les milieux pulvérulents[END_REF][START_REF] Christoersen | A Micromechanical Description of Granular Material Behavior[END_REF]:

σ ij = 1 V contacts (N i + T i )l pq,j (4.13)
where the summation is correlated to all contacts transmitting forces in the packing and where V is the sample volume (when in periodic conditions, V is the volume of the periodic cell), N i and T i are the i th components of the normal and tangential contact forces at the contact, respectively. l pq,j is the j th component of the l pq branch vector. Likewise, when modeling a composite material in dp3D made of a mixture of distinct materials (e.g., materials 1, 2, and 3 ), it is also possible to calculate the macroscopic stress contribution of these materials separately. Thus, similarly to Eq. (4.13) above, the total stresses associated to material 1, e.g., are:

σ ij,mat1 = 1 V contacts,mat1 (N i + T i )(r p - 1 2 δ n )n j (4.14)
where δ n is the geometrical indentation between the two particles at the contact, n j is the j th component of the normal vector n. And, r p is the particle radius (in this example, the radius of material 1 particles). Note that in Eq. ( 4.14), the sum is made on all contacts that includes a particle of material 1, i.e., contacts between mat 1-1, 1-2, and 1-3.

Composite materials approach

This chapter describes the use of composite materials by mixing particle clusters with single particles. The dierences in the mechanical behavior between a graphite ake, coarse alumina, and a ne alumina particle and how these approaches impact the composite macro-mechanical behavior are discussed. In this work, graphite akes and coarse alumina are considered assemblies of particles (or particle clusters) linked by bonds. As opposed to the graphite, these coarse aluminas cannot present any internal deformation, and consequently, the bonds cannot fracture (section 4.5.1). The determination of the elastic input parameters used in the contact laws is discussed in section 4.5.2.

In Section 4.5.3, an original approach to represent coarse and ne alumina grains embedded into a soft matrix was proposed. The binder is modeled as a soft core-shell material surrounding coarse alumina (cluster) and ne alumina particles. When two coated or a coated and non-coated particles indent each other, three conditions are possible at the contact (soft-soft contact, soft-hard contact, and hard-hard contact).

This idea reects the concept originally developed in the framework of homogenization by Hashin [Hashin, 1964]. Hashin proposed a composite sphere model by using graded spherical particles that are embedded in a continuous matrix (Fig 4.4), and the geometry of these composite spheres are represented by a concentric shell. The aim was to investigate the macroscopic mechanical behavior of heterogeneous viscoelastic media in dilatation and shear. Also, [Gernandt et al., 2011] modeled core/shell hydrogel particles by assigning dierent water solubilities to the core and shell domains of the particle.

However, unlike Hashins' model, our model focus on the behavior of contacts between particles, and the matrix is not represented by a viscous material but by a soft elastic behavior (no viscosity). Our approach could be easily extended to composite applications and characterize complex microstructures in a simpler way.

Bonded particles representation

Seeking for simplicity, we have chosen to represent coarse alumina particles as spherical in DEM. However, this is considered an oversimplication since these particles have a very irregular shape, and exhibit size distribution (250µm average size). Concerning graphite akes, they tend to have rectangular and an elongated geometry (< 400µm size). In DEM, all alumina particle clusters have an equal diameter of approximately 130µm (a total of 4,645 single particles with 7.8µm diameter each). The graphite akes are represented by a parallelepipedic shape with dimensions of (121 x 116 x 17)µm with a total of 1,070 single particles with 7.8µm diameter each. All particles in DEM were considered to have a unique size (7.8µm).

The graphite akes present an elastoplastic behavior, and so, they deform during the simulations (particles bond may break, thus bringing some plasticity at the macroscopic scale). The particles breakage are at the origin that some compaction microcracks are observed with X-ray tomography (chapter 3). The graphite cluster is an assembly of single spherical particles (or particles cluster) linked through elastic bonds. The coarse alumina is also considered a cluster. However, the coarse alumina bonds cannot fracture, contrasting with graphite. In Fig. 4.5, the bond interactions representation between coarse alumina and graphite clusters are highlighted. Even if a coarse alumina cluster contains particles linked by bonds, these bond interactions are not taken into account during DEM simulations (Fig. 4.5 a).

In Fig. 4.5 b), the bond interactions inside the graphite ake are calculated by the bond contact model (Eqs. (4.7) and (4.8)). The graphite is denominated as a cluster. In Fig. 4.5 c), the interactions between two coarse aluminas are described by the Hertzian model (Eq. (4.1)), for a contact between non-bonded particles. In Fig. 4.5 d), the interactions between coarse alumina and graphite are also given by the Hertzian contact model. In Fig. 4.5 e), the interactions between two graphite akes are given by the Hertzian contact model, and inside the graphite, the bond interactions follow the bond model. The DEM model (if single particle or particle cluster) applied to each coarse alumina, ne alumina, and graphite, and the external and internal interaction contact models are summarized in Table 4.1.

The bond representation of the ceramic composite in DEM is shown in Fig. 4.6. In summary, the ne alumina particles are considered a set of single non-bonded particles, and the clusters are particle assemblies linked by internal elastic bonds. Also, to represent a more realistic raw material microstructure, the clusters (coarse alumina and graphite ake materials) are considered fully dense (density RD = 1). Thus, it was imposed an indentation between internal cluster particles such as that the macroscopic relative density of the cluster is equivalent to 1. This indentation is of the order of 0.29R, where R is the sphere radius. Table 4.1: Summary of the applied DEM model for each coarse alumina, ne alumina, and graphite materials (if single particle or particle cluster), and the external and internal interaction contact models.

Elastic properties

The DEM Young's modulus (E) input values for coarse and ne alumina, graphite, and binder raw materials were taken from a previous Ph.D. work by Diane Dupuy [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] and literature data. Regarding the coarse alumina, [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] performed nanoindentation experiments on brown fused alumina (similar to the coarse alumina used in our work) to measure its Young's modulus, and it corresponded to 383 (± 8) GPa. Moreover, alumina exhibits a high Young's modulus of approximately 390 GPa, according to [START_REF] Ashby | Butterworth-Heinemann[END_REF], which is very similar to the value measured by [Dupuy2015]. Therefore, we considered in the DEM modeling that each single alumina particle exhibit Young's modulus of E=380 GPa, and for simplication, each particle that composes coarse alumina clusters also has E=380 GPa (Table 4.2).

The powder composites studied in this work have a carbonaceous binder in their microstructure, i.e., a mixture of phenolic resin and solvent (chapter 3). [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF] measured Young's modulus of an equivalent reticulated binder (after the curing at 200°C) by nanoindentation (7.4 ± 0.1 GPa). Due to their large use, a wide range of elastic parameters for phenolic resins can be found in the literature. They were determined by: molecular dynamics simulations The Young's modulus of graphite was also estimated from the literature data. However, because the graphite material is very anisotropic, and depending on the crystallographic axes (that denes its orientation), the Young's modulus can vary, for example, from 36 to 1060 GPa [START_REF] Pierson | Handbook of Carbon, Graphite, Diamonds and Fullerenes: Processing, Properties and Applications[END_REF]. According to measurements from [START_REF] Dupuy | Comportement thermomécanique de composites réfractaires oxydecarbone[END_REF], the graphite has a Young's modulus of 35 GPa. Therefore, in DEM, we considered that every particle belonging to graphite clusters has Young's modulus equal to 35 GPa.

In the literature, the Poisson's ratio for alumina can range from 0.34 to 0.37 (according to [START_REF] Monk | Computational and Experimental Study of Phenolic Resins: Thermal-Mechanical Properties and the Role of Hydrogen Bonding[END_REF] and [START_REF] Izumi | Molecular Dynamics Simulations of Cross-Linked Phenolic Resins Using a United-Atom Model[END_REF]). For the graphite akes and the binder, it is much more dicult to ascertain a precise value. However, Eqs. (4.2) (elastic contact model), (4.7) and (4.8) (bond contact model), indicate that the terms involving the Poisson's ratio are on the form of 1 -ν 2 . The inuence of the Poisson's ratio is minimal for contact laws. In DEM, the Poisson's ratio for all materials (alumina, graphite, and binder), almost negligible, were set to 0.2 (Table 4

Binder shell representation

As shown in chapter 3, SEM and X-ray tomography images show that the granular composites are composed of solid alumina grains and graphite akes embedded into a polymeric resin. The experimental ndings suggest division into two scales. First, a mesoscale with a mixture between coarse alumina (250µm) and graphite (<400µm) embedded into the matrix (i.e., a mixture of ne alumina particles and binder). Second, a microscale by mixture of ne alumina grains (<50µm) embedded into the binder (micro-scale). Knowing that coarse alumina grains and graphite are embedded into a matrix composed of ne alumina grains and binder, we implemented an original approach by modeling an elastic soft binder shell (with a determined thickness t, surrounding hard alumina particles). A generalization of the Hertzian contact law, that exhibits two dierent rigidities depending on the indentation between two particles, was proposed. This model applies at the surface of coarse alumina clusters. The bond interactions inside a coarse alumina cluster are not calculated (section 4.5.1). Also, only particles at the their surface will be impacted by the soft binder shell model. When two coated materials (e.g., ne alumina particle or/and coarse alumina cluster) with a radius r i and r j , elastic properties (E i , ν i ) and (E j , ν j ), surrounded by a binder shell with thickness t i = δ c × r i and t j = δ c × r j , and Young's modulus (E binder ), indent each other, the Young's modulus (Eq. ( 4.1)) depends on the mutual indentation and the thickness of the binder. δ c is the critical indentation. Fig. 4.7 graphically summarizes the binder model in the case of two coated particles (ne or coarse alumina), and Fig. 4.8 the contact between a coated particle and a non-coated particle (e.g., graphite), during the compaction.

When these two particles are in contact (the distance between the two centers is less than r i + r j , two conditions may apply. First, if:

δ n ≤ δ c = t i + t j (4.15)
thus, the two particles in contact are at the binder soft phase branch 1 (Fig. 4.7 a) during the rst stage of the compaction (loading). The elastic modulus of the hard alumina particles i and j is equivalent to the Young's modulus of the binder (E i = E binder , E j = E binder ). Following the loading, during the unloading, the contacts at branch 1 follow the same last path developed by the loading. The Hertz contact force at branch 1 (soft phase) is:

N Hertz = 4 3 E * R * 1/2 δ 3/2 n (4.16)
Second, if the indentation goes further and reaches the critical indentation:

δ n ≥ δ c = t i + t j (4.17)
thus, condition 2 applies, and the two particles interact directly through the hard alumina phase (branch 2). The elastic modulus of particles i and j are multiplied by a factor (M ). The Hertz contact force at branch 2 (hard phase) is:

N Hertz = 4 3 M E * R * 1/2 δ 3/2 n (4.18)
Once the contacts are at the hard phase branch 2 (Fig. 4.7 b), during the rst stage of compaction (loading), they are at the critical indentation (δ c ). In branch 2, the contacts may face three possibilities. First, the normal force at the contact (N Hertz ) may continue increasing during the compression (Fig. 4.7 b) i). When the contact is at the critical indentation, it may appear constant, but it is not. The critical indentation is much smaller because the particle's Young's modulus is much larger at branch 2 than at branch Second, the contact may unload (N Hertz decreases), (Fig. 4.7 ii). Third, the contact may also fail (Fig. 4.7 iii). For the contact to fail, it needs to follow the condition:

δ n ≤ δ c = (t i + t j ) 1 -1 M 2 3
, and in that case no normal force is transmitted at the contact. Following the compaction (during the unloading), the contacts at branch 2 follow the same last path developed by the compaction. Some contacts may present compressive and tangential forces during the rst stage of the compaction (loading). Some contacts may also unload and fail during the unloading simulation and the compaction. When a contact at branch 2 fails (or is unloaded) during the compaction, hysteresis is introduced at the particles' contact. This hysteresis does not occur when the contact is at branch 1. Because of that, after the compaction (during the unloading), the macroscopic hysteresis of the composite will depend on the total number of indentation types (branch 1 or 2) at the end of the compaction. Even if this is a simple condition applied for the unloading, this model still allows us to model the most important aspect of the binder shell; upon indentation, the rigidity of the contact evolves steeply when a critical indentation is reached.

The application of the binder model when a non-coated material particle (graphite) with a radius r k and a coated material particle (ne alumina or coarse alumina), with a radius r i , elastic properties (E k , ν k ) and (E i , ν i ), respectively, indent each other, is shown in Fig. 4.8. This case is a particular case of Fig. 4.7 with t k = 0 (a null binder thickness). At condition 1 (soft contact), because the eective Young's modulus is

E * = 1-ν 2 i E i + 1-ν 2 k E k -1
(Hertzian equation, Eq. (4.1)), and E binder << E k the eective Young's modulus (E * ) is approximately E * = E binder /(1 -ν 2 binder ).

Binder thickness calculation

The binder shell thickness of each microstructure (Mix 1 to 4) is directly related to their composition, calculated from the volume fraction of each component (coarse and ne alumina, graphite, and binder), i, j, and k represent the coarse alumina, ne alumina particle, and graphite, respectively. All particles on DEM have the same radius equivalent to 3.9µm (including the ne alumina) for two main reasons. First, this size (3.9µm) covers the average size range of ne alumina grains (<50µm) measured experimentally. Second, to reduce the time simulation consuming. Until approximately 2 years ago, dp3D could not handle simulations eciently with dierent particle sizes (this was improved recently). The coarse alumina cluster has around 67 µm radius, and the graphite ake is represented by a parallelepiped with dimensions of 121x116x17 µm (Fig. 4.5). Each coarse alumina and graphite ake cluster has a total of 4,645 and 1,070 particles, respectively.

The total binder volume fraction can be written as: ) when the indentation reached a critical value (that remains constant) i.e., condition δ n ≥ δ c = t i + t j applies, the elastic modulus of particles i and j are multiplied by a factor M and the contact enters branch 2. Once in branch 2, i) due to the compression, the normal force at the contact may increase, or ii) the contact may unload, or iii) the contact may fails (if

δ n ≤ δ c = (t i + t j ) 1 -1 M 2 3
) and transmits no normal force.

V f (binder) = (V binder(i) + V binder(j) )/(V i + V j + V k ) (4.19)
where V binder(i) , and V binder(j) are the binder volume surrounding the coarse alumina clusters and ne alumina particles, respectively. (V i , V j , and V k are the coarse, ne alumina, and, graphite volume, respectively (the sum of these three represent the total volume of the microstructure).

V binder(i) is:

n i 4 3 π(r 3 i -(r i -t) 3 ) (4.20)
Likewise, V binder(j) is:

n j 4 3 π(r 3 j -(r j -t) 3 ) (4.21)
The total packing volume (V total = V i + V j + V k ) is: where r i and r j are the radius of coarse alumina cluster and a ne alumina particle. n i and n j are the number of coarse alumina clusters and ne alumina particles, respectively.

t is the binder shell thickness. a, b, c are the graphite cluster dimensions (V k = abc).

The binder volume fraction surrounding coarse alumina clusters is:

V f (alumina)coarse = n i 4 3 π(r i -t) 3 /(V total ) (4.23)
The binder volume fraction surrounding ne alumina particles is:

V f (alumina) f ine = n j 4 3 π(r j -t) 3 /(V total ) (4.24)
Finally, the volume fraction of graphite is:

V f (graphite) = n k • abc/(V total ) (4.25)
Assuming that the binder thickness of coarse alumina cluster and ne alumina particle 4.5. Composite materials approach is equivalent (t i = t j = t), from Eqs (4.23), (4.24) and (4.25) it is possible to calculate the coarse and ne alumina binder shell thickness, the total number of clusters (coarse alumina and graphite) and single particles (ne alumina), as well as the total number of particles used to generate DEM packing composites (mix 1 to mix 4). They are listed in the Table 4.3 that shows the typical numbers of particles introduced in the simulation once a initial number of coarse alumina clusters is chosen. 8 to 10 coarse alumina particles were chosen to obtain a total of the order of 120,000 spherical particles.

Microst An alternative approach to the assumption described above (t i = t j ) is to assume that the volume fraction of binder surrounding coarse and alumina particles are equal (V f i = V f j ). In that case, small alumina particles exhibit much smaller binder thickness than coarse alumina particles. The same volume fraction assumption leads to:

n i 4 3 π • [r 3 i (r i -t i ) 3 ] = n j 4 3 π[r 3 j (r j -t j ) 3 ] (4.26)
For the assumption where V f i = V f j , the binder thickness, and number of particles are given in Table 4.4.

The main implications between these two assumptions were investigated by closeddie compaction DEM simulations using the same macroscopic input parameters. The objective was to analyze the qualitative impact of each assumption on the macroscopic response of the mixture. Typical densication curves (macroscopic axial stress evolution against the packing relative density) of closed-die experiment, of the two distinct assumptions (t i = t j = t, and V f i = V f j ), are shown in Fig. 4.9. The nal relative density (RD) reached at the end of the DEM close-die compaction simulation for t i = t j , and 4.4: 'same volume fraction' assumption (V f i = V f j ): number of particles on dp3D depending on their composition. In this assumption, we consider that the binder volume fraction in coarse and ne alumina is the same.

V f i = V f j is 93.5% and 95.9%, respectively. Thus, only a small dierence in terms of density. However, both curves follow the same slope up to approximately 87%. Above 87%, the blue curve (V f i = V f j ) follows a stier behavior when comparing with the experiment and the 'same binder thickness' assumption gray curve (t i = t j = t). While the 'same volume fraction' follows the same behavior of the experiment. This stier behavior of (V f i = V f j ) above 87% is because the binder thickness of ne alumina particles is smaller, leading to more hard alumina-alumina contacts. The closest correspondence between DEM macroscopic behavior and experimental data suggests that the 'same binder thickness' assumption (t i = t j ) is better approaching the actual composite.

Microstructure generation

A classical dynamic DEM, as introduced by [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF], is used in a quasistatic approach within the in-house code dp3D [START_REF] Martin | Study of particle rearrangement during powder compaction by the Discrete Element Method[END_REF][START_REF] Martin | Isostatic compaction of bimodal powder mixtures and composites[END_REF] to compute the equilibrium positions of particles at each time-step. Fig. 4.10 summarizes the typical procedure to prepare packings. The initial composite assembly is obtained by rst randomly locating bonded particle clusters (graphite and coarse alumina) and non-bonded single particles (ne alumina) in a parallelepipedic periodic cell with (1x1x2) size ratio (Fig. 4.10 a). At this stage, there is no contact between these constituents (ne alumina, coarse alumina, and graphite akes do not contact each other). Suppose a particle belonging to a coarse alumina happens to contact with an already existing ne alumina particle. In that case, another set of random coordinates is drawn until the particle cluster can be located inside the box without any contact.

In this rst step, the particles are located inside the box until the nal packing relative density is equivalent to 0.4. In all subsequent simulations, periodic conditions are used. The particles whose center is outside of the simulation box are translated to the opposite face as shown in Fig. 4.11. For example, a particle i that protrudes outside of the periodic wall through the left periodic face interacts with other particles in the opposite face as if an image of i was created outside of the periodic box on the right side.

Hereafter, by slowly moving the box walls inwards, the packing (RD=0.4) is subjected to isostatic densication with low macroscopic pressure (0.02 MPa) up to a relative density [START_REF] Martin | Inuence of adhesion and friction on the geometry of packings of spherical particles[END_REF][START_REF] Kumar | Eect of packing characteristics on the discrete element simulation of elasticity and buckling[END_REF]. At this step, also called jamming, contact forces appear. However, only elastic interactions are considered (Eq. (4.1)), friction and adhesion between particles are disregarded. In summary, the jamming stage mimics the gradual rearrangement of particles in the die without any plasticity or bond breakage. The 'jammed' packing at RD=0.6 represents the mix in the die before compaction.

Typical DEM simulation conditions

As stated above, all samples were generated by applying periodic boundary conditions and by imposing a suciently slow strain rate to ensure quasi-static equilibrium. The value of the imposed strain-rate is set at each time step, starting from a value of 10 -5 sec -1 ( εz = -10 -5 ), and considering the normalized kinetic energy per particle: [START_REF] Agnolin | Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks[END_REF] Ẽkin = E kin n max(N R) (4.27) where E kin is the total kinetic energy of the particle system, n the number of particles, N the normal force, and R is the particle radius.

Fig. 4.12 shows the evolution of the normalized kinetic energy against the relative density (RD) during a closed-die compaction simulation (chapter 5). The strain-rate and the re-normalized mass of particles are adapted so that Ẽkin remains below 10 -7 in the major part of the simulation (0.65 ≤ RD ≤ 1) (Fig. 4.12). At RD=0.6, the normalized kinetic energy rises to approximately 3.5 • 10 -6 because of the 'abrupt' introduction of adhesion to the compaction. During the jamming, the adhesion is set to zero (friction and adhesion between particles are disregarded). [START_REF] Agnolin | Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks[END_REF] have shown that to ensure quasi-static conditions, the normalized kinetic energy value should stay below 10 -7 . However, the experience in dp3D has shown that if it stays below 10 -6 , the simulation can still be considered as quasi-static. After the microstructures generation (section 4.6), the "jammed" packing is at RD = 0.6 and represents the mix in the die before compaction. The compaction can be done along two routes: by closed-die or isostatic compaction (Fig. 4.13 c). In closedie compaction, the packing is attained along the z direction by applying a compressive macroscopic strain rate ( εz < 0) in the z axis to the simulation box, and a null strain rate in the x and y axis. In isostatic compaction, an identical compressive strain rate is imposed on all axis x, y and z ( εx = εy = εz < 0). Small time-steps are considered in all simulations to ensure numerical stability. The nal density during compaction simulations varied between 0.6 < RD ≤ 1 by imposing stress or relative density targets. At the end of the compaction, the packing, densied up to a given density, is under compressive stresses. The compacted samples are unloaded to obtain a macroscopic stress state with nearly zero compressive stress (Fig. 4.13 d). During unloading, starting initially from an isostatic strain rate conditions ( εx = εy = εz > 0), the strain rates are modied so that the macroscopic stress state becomes isostatic (σ x = σ y = σ z ), see Fig. 4.14. The unloading is stopped when the pressure (P = -1/3(σ x + σ y + σ z )) is equal to 1 MPa. Fig. 4.15 shows the local density of each particle during the three steps: jamming, close-die compaction, and unloading. The particle clusters (coarse alumina and graphite), due to the indentation between the internal particle clusters in DEM (section 4.5.1), are fully dense (density RD=1). This density tracking change is calculated using a Voronoi tessellation (voro++ cell library) [START_REF] Rycroft | VORO++: A three-dimensional Voronoi cell library in C++[END_REF].

Subsequently, the unloaded samples are submitted to a series of triaxial compression simulations tests (Fig. 4.13 e) to analyze the macroscopic microstructure response. In this step, an initial strain rate is applied in the axial (z direction), εz = -10 -5 s -1 (to ensure quasi-static conditions), with a conning radial pressure (σ x = σ y ). This numerical setup allows a representation of a conventional experimental test where the axial stress increases while the radial conning pressure remains constant. These unloaded samples are also tested in DEM tensile tests by applying a tensile strain rate in the z direction ( εz > 0), and null stresses in x and y directions (σ x = σ y =0) up to sample's failure. This methodology is used to generate the yield surface and fracture line (chapter 5).

Representative volume element (RVE)

Due to the complexity of the microstructure geometry, we do not consider the real shape of the actual raw materials (alumina, graphite grains, and binder) in DEM. Still, we developed a simple way to represent their microstructures, depending on their composition (as presented in the sections above). The macroscopic response of the composite should be isotropic (due to the random orientation of the graphite akes, the graphite behaves as an isotropic media). Therefore, isostatic compaction numerical simulations are performed on three-dimensional Representative Volume Element (RVEs) packings containing dierent total particles number. The objective was to determine the minimum number of particles necessary to tend towards an isotropic response (zero deviatoric stress) and converge onto a constant pressure value and a reasonable CPU time. Also, close-die compaction tests were carried out to verify further the convergence of the macroscopic response with the number of particles for a dierent stress state.

For each RVE, ve dierent packings were rst generated from the 'gas of particles' step using dierent random seeds to locate the particles initially in the simulation box (section 4.6). This allows to have the same volume fractions for each packing but with dierent locations of particles. A total of thirty RVEs samples were generated for each compaction type (e.g., close-die or isostatic) to allow deviation measurement. The mean and deviatoric stresses were calculated considering the nal stresses (end of the simulation) in the three-axis (x, y, and z). They are written as:

Σ m = 1/3(σ x + σ y + σ z ) (4.28) 4.8. Representative volume element (RVE) Σ d = σ z -[1/2(σ x + σ y )] (4.29)
where Σ m and Σ d are the mean and deviatoric stress, respectively.

The simulations stop when the packing is with a density of 0.95. The chosen density target (RD=0.95) makes the convergence criterion of the mean and deviatoric nal stresses quite demanding. For large values of relative densities (Fig. 4.9), the compaction curve becomes vertically asymptotic. All numerical samples exhibited an initial size ratio of x : y : z = 1x1x2 x, y, and z axis with periodic boundary conditions.

The stresses are rst measured with isostatic compaction simulations by applying an equal strain rate in x, y, and z axis ( εx = εy = εz < 0). After, the stresses are analyzed with close-die compaction simulations by applying an initial strain rate in z axis ( εz < 0) and a null strain rate in x and y axis ( εx = εy = 0).

The convergence of mean (Σ m ) and deviatoric (Σ d ) stresses, from isostatic DEM simulations, in absolute value, with the total number of particles for Mix 1 (reference composite) is shown in Fig. 4.16. For an RVE with n ≥ 380, 000 particles, e.g., the stresses converge with a low standard deviation. The simulations show that an RVE with n ≥ 117, 605 particles is large enough to ensure a good convergence of the measured macroscopic Σ m and Σ d stresses.

The RVE exhibits an anisotropic macro mechanical response for the compacted packing by isostatic conguration. The deviatoric stress (Σ d ) is non-zero, i.e., even if applying an equal strain rate in the three-axis. This can be explained because we are limited to a nite number of particles (n). By increasing n, this anisotropy decreases with Σ d closer to zero. But, for n higher than 1 million particles, Σ d is still dierent than zero (approximately 4 MPa on average). The second possible reason is the packing size ratio that is two times larger in the axial direction (1x1x2), which translates into axial stress in z that is always slightly smaller than in x and y directions. The reason for this behaviour is not apparent since we are using periodic conditions. The number of contacts between coarse alumina clusters in the x and z may be underestimated because there are only a few of those contacts. Only very large boxes may smooth out this eect.

The ratio of the radial (1/2(σ x + σ y )) to axial (σ z ) stresses has been reported for several powders compacted in closed-die conditions. Typically, it is of the order of 0.5 to 0.75 at high density for metallic powders that deform plastically [START_REF] Brown | Yield behavior of metal powder assemblages[END_REF][START_REF] Mosbah | Experimental techniques for analysis of die pressing and ejection of metal powder[END_REF][START_REF] Sridhar | Yield behaviour of cold compacted composite powders[END_REF]. This ratio leads to Σ d /Σ m =0.3-0.5. For ceramic powders that compact through particles rearrangement and microstructure fracture, the radial to axial stress ratio is reported to be of the order of 0.6 to 0.7 [START_REF] Shima | Compaction-Induced Anisotropy in Internal Structure of Ceramic Powder[END_REF][START_REF] Sinka | Constitutive data for powder compaction modeling[END_REF] leading to Σ d /Σ m =0.37-0.5. In our closed-die compaction simulations, Σ d /Σ m is of the order of 0.34-0.38 for RVE's with n ≥ 117, 605 (Fig. 4.16), that is slightly smaller than the average range encountered in the literature for ceramic materials. The composites studied in our work, when compacted by DEM simulations, present an elastic deformation and a plastic response (due to the particles and clusters rearrangement; and of the binder behavior).

The convergence of Σ m and Σ d , in absolute value, from closed-die compaction simulations is shown in Fig. 4.17. Similar to the isostatic compaction simulations, an RVE with n ≥ 117, 605 particles leads to the convergence of the measured macroscopic stresses. was made where the limit of convergence is ± 35% of Σ d (3.5<Σ d <7.2 MPa). Thus, it was observed that for the 117,605 particles RVE, three samples are already inside this region, out of ve (Fig. 4.18 a).

In Fig. 4.18 b), for closed-die compaction, the average of Σ m and Σ d stresses for n ≥ 117,605 particles is Σ m = -29.4 MPa, and Σ d = -10.4 MPa. We considered that the convergence area is ± 13% of Σ d (-9.1<Σ d <-11.8 MPa). For an RVE with n = 117,605 particles, only one sample does not converge (out of ve). The larger the number of particles in RVE, the higher the number of samples laying inside the convergence region (e.g., n ≥ 117,605 particles).

As well, a lower divergence of the stress state can be predicted, compared with the isostatic compaction variation (Fig. 4.18 a). In isostatic compaction (a), the Σ d is always slightly positive since the σ z < 1 2 • (σ x + σ y ), and in closed-die compaction

Σ d is negative since σ z > 1 2 • (σ x + σ y ).
The aim of the present work is its application within industrial conditions. Thus, it becomes essential to have a reasonable CPU time and suciently accurate material mechanical properties. Therefore, an RVE with 117,605 particles was chosen because the major points are inside the selected convergence region (highlighted in the graph), see Fig. 4.18 b). This way, this RVE provides a microstructure that ensures a reasonably isotropic response, converged macroscopic behavior, and a suciently low CPU time. The total clock time (from the generation of the samples up to the end of closed-die compaction numerical simulations) against the total number of particles (between 14,567 and 1,029,314) is shown in Fig. 4.19. The curve exhibits an almost linear evolution. Each simulation used a total of 8 CPUs (with the openMP parallel option in dp3D). In summary, the dp3D code has been parallelized with a ne-grained method (at the loop level) with openMP directives. Note that because of the openMP parallelization and the cache miss, using more than 8 CPUs does not necessarily result in shorter clock times. Thus 4 to 8 CPUs have been used in all simulations afterward.

Virtual ring step modeling

The forming process of refractory ceramic powders is attained through compaction, curing, and ring. The ring consists in applying heat treatment to the cured material to obtain the red product. Indeed, this nal step, referred to as sintering in powder metallurgy, involves diusion and grain growth.

However, in our work, the ring process takes place at approximately 900°C, which is too low for alumina oxides sintering temperature (around 1400°C) [START_REF] Cutler | Sintering of Alumina at Temperatures of 1400°C. and Below[END_REF]. Anyhow, ring leads to the cohesion of the composite microstructure: i) at the mesoscale (coarse alumina grains, graphite akes, and the matrix), and ii) at the microscale (between ne alumina grains and the carbonaceous binder). Also, during ring, the binder is subjected to pyrolysis that induces modications of its microstructure, accompanied by eliminations of volatile compounds (weight loss) [START_REF] Schmitt | Micromechanics applied to the thermal shock behavior of refractory ceramics[END_REF][START_REF] Zhang | Firing mechanism of oxide-carbon refractories with phenolic resin binder[END_REF]. In this section, the methodology for the generation of virtual DEM red samples is rst described in section 4.9.1. Afterward, the red samples are submitted to mechanical testings (section 4.9.2).

In addition, during ring/pyrolysis, elimination of the binder occurs, introducing some defects within the microstructure [START_REF] Zhang | Firing mechanism of oxide-carbon refractories with phenolic resin binder[END_REF]. The binder burnout can cause the 4.9. Virtual ring step modeling formation of porosity in the body in two dierent ways. The binder can recede in the interparticle porosity, creating a porous shell that increases in size as pyrolysis proceeds. And, capillary pressure in the porous body causes rearrangement of the binder, creating a distribution of porosity in the artifact [START_REF] Wu | Acrylic binders for dry pressing ceramics[END_REF].

Samples generation

To mimic the cohesion between particles reached during the ring, the DEM numerical red microstructure is represented by creating elastic bonds (in compacted samples) in between pairs of particles that are in contact (Fig. 4.20).

In the green DEM microstructure (after compaction), elastic bonds are considered only between particles inside the coarse alumina and graphite particle clusters (Fig. 4.20 a), and the alumina is coated by a soft binder shell. In the virtual ring, elastic bonds are introduced in all particles that are contacting each other on unloaded samples ((Fig. 4.20 b). As well, during ring, the resin is partially volatized, and the rest is carbonized. Furfural creates cohesion between alumina particles. Therefore, the virtual red composite matrix is represented by those elastic bonded ne alumina particles (no soft binder coating).

When two-bonded particles i and j, and radius r i and r j contact each other, interaction at the bond scale is given by the microscopic bond model (Eqs (4.7) and (4.8)). Six bond interactions are possible at the contact between coarse alumina, ne alumina particles, and graphite. There are elastic bonds between ne aluminas particles (1), between ne alumina and graphite (2), between coarse alumina and ne alumina (3), between coarse aluminas (4), between graphite akes (5), and between graphite and coarse alumina (6) (Fig. 4.20 b).

The Young's modulus after the virtual ring stage for coarse alumina and graphite clusters are the same as of the green stage (380 GPa for the coarse alumina, and 35 GPa for the graphite), previously presented in Table 4 ,c) unloading down to 1 MPa pressure (nearly isostatic stress state), e-1) bonds creation. Thus, in e-1), the unloaded sample is prepared to start following the numerical virtual ring step by creating bonds to all particles in contact.

(closed-die compaction) (Fig. 4.21 c) with a nal axial stress target of 35 MPa to mimic the industrial process. The closed-die compaction simulations were carried out with several relative density targets (0.6 < RD ≤ 1). The sample is successively unloaded until equal stress of 1 MPa in the three-axis (Fig. 4.21 d).

Until the unloading step, the methodology applied is the same as for the green characterization. In sequence, the unloaded microstructure is numerically red by creating elastic bonds to all particles that are in contact, whatever their previous history of contacts (Fig. 4.21 e-1).

During the ring, the powder composite microstructure shows some damage and decohesions, mainly due to the mismatch of the coecient of thermal expansion between the alumina, graphite, and binder. The rst damage takes place at below 700°C, due to the resin pyrolysis, giving some plastic deformation to the resin. At this stage, there are still contacts between the grains and binder. However, above 700°C, the pyrolysis of the resin is almost completed, and it is transformed into a more rigid bonding phase (pyrolytic carbon). And, since the coecient of thermal expansion of alumina is higher than that of the pyrolytic carbon, compressive and tensile stresses are created around alumina grains resulting in microcracks within the matrix (Fig. 4.22 b) [START_REF] Tessier-Doyen | Etude expérimentale et numérique du comportement thermomécanique de matériaux réfractaires modèles[END_REF]]. During the cooling, or when the temperature decreases below 700°C, the grains start to pull the carbon and creating decohesion at their interface (called as 'bell eect') (Fig. [START_REF] Tessier-Doyen | Etude expérimentale et numérique du comportement thermomécanique de matériaux réfractaires modèles[END_REF] In general, on powder composites containing a carbonaceous binder, the weak regions after ring are located either in the matrix, which has a lower mechanical strength when compared to the aggregates, or at the aggregatematrix interface [START_REF] Schmitt | Micromechanics applied to the thermal shock behavior of refractory ceramics[END_REF]. This phenomenon occurs in two scales: i) within the matrix (between ne alumina particles and the pyrolytic carbon), and ii) between the matrix and coarse alumina grains. It was also possible to observe decohesions around graphite akes on X-ray tomography images which may be related to decohesions in the interface graphite/matrix [START_REF] Werner | Young's modulus of elasticity of carbon-bonded alumina materials up to 1450°C[END_REF].

Thus, to reproduce these decohesions/cracks within the microstructure to the DEM virtual ring, broken bonds were imposed between two pairs of materials: i) coarse alumina and ne alumina particles, and ii) ne alumina and graphite particles (Figure 4.23). Broken bonds do not transmit tensile forces but may transmit shear forces (friction) and repulsive forces (section 4.3.2).

Testing the numerical red microstructure

Compacted powder composite samples (mix 1-4), by closed-die conguration up to 35 MPa, were submitted to three dierent mechanical testings (uniaxial and triaxial compression tests and Brazilian tests), see chapter 3. The main objective rst was to evaluate their mechanical response when subjected to complex stress-state. And to estimate yield and fracture surfaces. In DEM, the virtual red microstructures (after the broken bonds step), Fig. 4.24 e-2), follow two dierent paths. First, an initial strain rate in tension was applied in the z axis ( εz = 10 -5 s -1 ) until sample's failure (Fig. 4.24 f). Second, to prepare these bonded broken samples (from Fig. 4.24 e-2) for triaxial tests, the samples were subjected to pre-loading tests. Specically, an initial equal strain rate in x, y, and z axis was applied so that the samples could reach an isostatic stress-state (i.e., σ x = σ y = σ z =5.4 MPa), as shown in (Fig. 4.24 g). In the same experimental protocol, the red samples are subjected to a pre-loading of between 5.2 and 5.7 MPa (chapter 3). This is mandatory since the conning pressure (from the oil) and the axial pressure need to be equivalent at the beginning of the triaxial tests.

In sequence, the pre-loaded samples are submitted to a series of triaxial numerical tests (Fig. 4.24 h) by applying an initial compressive strain rate in the z direction ( εz = -10 -5 s -1 ), so that the simulations could have an equal (and constant) stress in the x and y axis (σ x = σ y ). The matrix Young's modulus (=4 GPa) and other dp3D input parameters (e.g., Figure 4.24: After the bonds breakage step, in DEM, e-2), the sample may be submitted to two types of testing. A direct tensile by applying a tensile strain rate in the z direction (f). A two-stages test with rst a pre-loading stage (target: σ x = σ y = σ z = 5.4M P a), and a traxial compression test by submitting an equal (and constant) stress in the x and y axis (σ x = σ y ). bond strength) were calibrated together with these numerical mechanical simulations and experiments to reproduce the red macroscopic behaviour (chapter 6).

Typical stress-strain curves of uniaxial compressive experiment and DEM simulations are shown in Fig. 4.25. In these simulations an initial axial strain rate εz = -10 -5 s -1 was applied, so that an equal and constant null radial stress (σ x = σ y = 0) is ensured (as in the experiment setup). These simulations were performed with microstructures of dierent bond status and using an equal dp3D macroscopic input parameters. Fig. 4.25 a) represents the microstructure with intact particles bonds (no broken bonds), and in Fig. 4.25,Fig. 4.25 b) with broken bonds between ne alumina and coarse alumina particles, and Fig. 4.25 c) with broken bonds between ne alumina and graphite particles. In general, the stress-strain curves exhibit mostly ductile behaviour. However, the ultimate axial stresses (σ z ) in a) and b) are smaller if compared with microstructure c). This can be explained since the normal and tensile forces (Eqs. (4.7) and (4.8)) at the bond depend on the bond interaction. An increase in the number of bonds also increases the material's stiness.

Conclusions

The commercial composite material that we want to quantitatively model in the next chapter is quite complex. It includes four components, coarse and ne alumina, graphite akes, and a carbon binder. These components have complex behavior, and some are drastically modied along the process route (e.g., the binder and the graphite akes) under severe thermo-mechanical conditions. We needed to simplify the composite material's texture to allow DEM modeling with acceptable CPU simulation time. Notwithstanding, these simulations are quite long for a number of particles n ≥ 117,605 (Fig. 4.19). The made assumptions are:

1. Coarse alumina and ne alumina particles are spherical (coarse alumina is represented by clusters of single sphere particles);

2. Graphite akes have rectangular shapes and are also represented by a particle cluster;

3. The graphite internal particle bonds may break during compaction simulations, bringing some macroscopic plastic appearance. On the contrary, the internal particle bonds of the coarse alumina cannot fracture;

4. The coarse alumina cluster and ne alumina particles are coated by a binder shell, but not the graphite;

5. The binder has a soft elastic response (instead of plastic or viscoplastic) in the green state;

6. The matrix of a virtual red microstructure is represented by ne alumina particles bonded together if in contact but are disconnected from coarse alumina and graphite akes (bonds breakage).

The justications (from 1 to 6) for these assumptions are:

1. Coarse and ne alumina particle are clearly not spherical (Fig. 4.23). However, discrete simulations deal essentially with spherical objects. For coarse alumina, it could be envisioned to extract more realistic shapes from X-ray tomography.

However, this would lead to some complexity in the tting procedure as it would introduce some degree of arbitrariness in particle shape choice. Also, we believe that the most important feature, from a mechanical point of view, is the very sti elasticity of alumina, as compared to the binder and the graphite;

2. The reason for the simple rectangular shape adopted for the graphite is to ease DEM model. By the end of the compaction, graphite akes are subjected to meaningful deformation due to the elastic bonds;

3. X-ray tomography proves that the coarse alumina grains do not fracture after compaction. Conversely, the graphite deforms, and it presents some micro-cracks in its interior may be due to the compaction;

4. An alternative to the assumption of the absence of binder around graphite akes would be to have binder around all components. It has not been tested yet. However, all the aluminas are covered with this soft binder, so all contacts between graphite and alumina particles are binder coated (Fig. 4.8). Indeed, the number of contacts between graphite clusters is negligible compared with other types of contacts (e.g., ne alumina-ne alumina and ne alumina-coarse alumina). Therefore, this assumption should not have a large eect on the macroscopic response of the composite, suggesting the irrelevance of a graphite binder coating;

5. This assumption mainly provides a simple way to model the divergence in mechanical response between alumina and the binder. The plasticity of the binder may be considered secondary. The rearrangement between alumina particles (ne and coarse) provides some plasticity at the macroscopic scale, reinforcing the validity of the assumption;

6. SEM observations clearly show decohesions between particles that can be ascribed to the thermal expansion coecient mismatch between the binder and the other components (alumina and graphite). This is indirectly conrmed because the experimental response is best modeled when this assumption is used;

It can be seen from the above lists that the most critical assumptions are linked to the binder. The binder is challenging to model as it transforms drastically along the process (due to its changing phase). Moreover, the binder is not an easily discernible component in a DEM simulation as it is not a discrete particle. Nonetheless, we believe that our assumptions are reasonable. They should help simplify the modeling, preserving the ability to correct discrimination between the four composites (studied in the next chapter 5).

Chapter 5 DEM simulations on green powder refractory materials

This chapter presents numerical simulations using the Discrete Element Method (DEM) to represent the compaction behavior and mechanical properties of carbon-bonded alumina refractory compositions. In DEM, the numerical composite microstructures are represented by a mixture between hard discrete particles (coarse alumina, ne alumina, and graphite) and a soft-binder shell coating alumina particles (chapter 4). The numerical simulations (closed-die compaction and mechanical tests) are carried out using the contact laws described in the previous chapter 4. First, The global view of the qualitative eect of DEM material properties in the compaction/unloading stages and the green strength resulting from numerical simulations are presented (section 6.3). As seen in chapter 4, depending on the particles' indention, coated particles may interact through the soft binder (if suciently indented) or interact directly through their hard branch 2. Therefore, each discrete component (coarse alumina, ne alumina, and graphite) contributes to the composite macroscopic stresses with a dierent magnitude (section 5.4.2). The binder stiness greatly aects the densication and macroscopic hysteresis during unloading (section 5.4.3). The composite composition (mix 1-4) also aects the composite densication and macroscopic hysteresis mostly because of the binder-shell thickness (section 5.4.4). Closed-die compaction and triaxial/uniaxial compression numerical simulations are compared with experimental data (section 5.5). The isodensity yield surfaces and fracture points are generated triaxial and tensile simulations by controlling the radial stresses through the axial and radial strain rates (section 6.5). These surfaces obtained from DEM modeling provide valuable information on the behavior of green composites for stress states that are dicult to attain by experiments. The eect of the composite density, consolidation path, and composition on the shape of the yield surfaces and the fracture points are investigated.

Introduction

The compaction of refractory composite powders is a basic process for refractory production. Understanding composite densication and the resulting mechanical properties is a key factor for developing new products. The most critical step for modeling powder material compaction with the help of the nite element method (FEM) is to formulate constitutive equations. This task is challenging and time-consuming due to extensive experimental testing on green samples. Unlike the continuum mechanics approach (FEM), the discrete element method (DEM) allows the particulate nature of the powder to be considered since the modeling treats the particle contact. DEM provides a tool for generating constitutive equations on a small representative volume to model a full-scale process on FEM.

For this purpose, it is necessary to rst understand the powder compaction behavior through appropriated discrete contact laws to develop yield and fracture surfaces that will be used in FEM. As the particles are assumed to be spherical for simplicity, DEM describes how the contact areas between particles inuence the overall powder macroscopic stress behavior when modeling a small representative volume element. DEM has been the object of many studies on the compaction of ceramic powders. However, despite the importance of such modeling, the use of DEM in carbon-bonded composite refractories has not yet been studied in the literature (chapter 2).

In this context, in this chapter, the compaction behavior is rst investigated by modeling alumina particles coated by a soft binder shell mixed with bonded graphite particle clusters to represent carbon-bonded alumina refractory compositions. Second, it is shown how the interaction between dierent branches aects the macroscopic densication and spring-back during close-die compaction simulations. These data were compared with experimental data. The compacted numerical samples were also submitted to mechanical tests (triaxial and tensile) to generate yield and fracture surfaces [START_REF] Redanz | The compaction of a random distribution of metal cylinders by the discrete element method[END_REF][START_REF] Heyliger | Cold plastic compaction of powders by a network model[END_REF]] [Pizette et al., 2010]. The triaxial simulations were carried out by imposing a constant axial strain rate and varying the radial stress to represent experimental mechanical test conguration. Thus, the possibility of a material failure or densication was explored, providing valuable information on the behavior of the composites for stress states that are dicult to obtain from experiments. These data will provide crucial information for generating constitutive equations for the compaction modeling using FEM (chapter 7), turning these DEM simulations into results that can be exploited in an industrial context.

Material properties

Material properties

In the present work, the ability to model close-die compaction of refractory composites are tested. The compaction and mechanical simulations are carried out using the DEM input elastic material parameters presented in chapter 3 (E for the material's Young's modulus in GPa, and ν for the Poisson's coecient). As previously mentioned in chapter 4, Young's modulus of the alumina and graphite applied in DEM simulations are taken from the literature. The binder elasticity is calibrated with experimental data. According to [START_REF] Gardziella | Phenolic Resins: chemistry, applications, standardization, safety and ecology[END_REF], there are dierent kinds of phenolic resins applications, and Young's modulus can vary from 1.8 GPa to 13 GPa. In this Ph.D. work, the binder softshell Young's modulus is xed at 1.02 GPa for the green stage (compaction and mechanical numerical simulations). This value is considered the best t with experimental data. The impact of the binder stiness in the compaction and spring-back eect in the unloading is explained in detail in section 5.4.3. The DEM material properties applied for the green characterization are listed in Table 5.1. γ is the work of adhesion in J/m², and µ is the particles friction coecient. γ=28J/m 2 , and a standard friction coecient value of 0.2 for all particle materials are chosen. However, it has been observed that µ has only a negligible eect on the compaction behavior. Table 5.1: Material parameters used on DEM simulations for the green characterization (compaction and mechanical simulations).

How these DEM properties were chosen, and how do they aect the composite green mechanical behavior? The mechanical behavior depends on the balance between all input properties. The qualitative impact of the DEM properties in the composite densication (during compaction), at the end of the unloading (spring-back eect), and the composite green strength from uniaxial compressive (crush) test simulations is exhibited in Table 6.2. Adhesion is dictated by the work of adhesion ω (DMT model, chapter 4). RD comp. is the relative density (RD) reached at the end of closed-die compaction simulations, the spring-back (%) is measured as the dierence between the RD comp. and the RD reached at the end of the unloading. The green composite strength is measured as the ultimate stress (the maximum stress which suggests a microstructure failure), in MPa, during uniaxial compressive test simulations ( εz < 0, and σ x = σ y = 0). MATERIALS The increase of adhesion slightly decreases the RD reached at the end of the compaction. Consequently, the rise of adhesion also reduces the spring-back eect during the unloading. However, the increase of adhesion signicantly increases the green strength during uniaxial compressive test simulations because it increases the pull-o force (N DM T , see Eq. (4.4) in chapter 4).

DEM property

RD comp. spring-back (%) green strength (MPa) Work of adhesion, ω, (J/m²) 5.2: Qualitative impact of DEM material properties' on the green mechanical behavior during compaction, unloading, and uniaxial compressive test simulations.

⇑ ↓ ↓ ↑ E binder (GPa) ⇑ ↓ ↑ ↓ σ N and σ T (MPa) ⇑ ↓ - ↑ Table
The binder Young's modulus (E binder ) is the property that most dictates the densication, spring-back, and composite green strength. Concerning the densication, by increasing E binder , the contacts become stier (in comparison with a microstructure with lower E binder ), resulting in larger contact forces. Consequently, the composite exhibits higher macroscopic stresses to a given relative density (more dicult to compact). About the spring-back, the increase of E binder increases the number of particle contacts in the soft binder branch, leading to a larger spring-back during the unloading. The increase of E binder results in a less dense (porous) microstructure, leading to a decrease in the green strength. Details concerning hard and soft contacts and the eect on the compaction and unloading simulations are described in detail in section 5.4.1.

σ N and σ T are the bond particle strength in tension and shear, respectively. The bond strength is active only for graphite. The coarse alumina cluster also has internal particle bonds, but these bonds cannot deform nor break (chapter 4). The graphite bond strength (σ N and σ T ) plays an important role in the composite compaction behavior. The increase in the graphite bond strength leads to stier and stronger graphite akes (smaller composite relative density to given stress), increasing the composite strength. The graphite bond strength has no signicant eect on the spring-back during the unloading. As a note, the graphite Young's modulus also impacts the composite densication and strength. However, since the Young's modulus of the graphite (and alumina) are known properties, we decided not to modify it. Still, the binder is the component that most impacts the composite compaction behavior, spring-back, and strength. The densication of a heterogeneous refractory composite in DEM compaction simulations depends upon the interaction between non-coated graphite and alumina particles (ne alumina particle and coarse alumina) coated by an elastic soft-binder shell, and between coated alumina particles themselves. When two coated or non-coated and coated particles interact, the contact can remain either in the soft shell (branch 1) or reach the hard (branch 2), see chapter 4. The main goal, in this section, is to improve the understanding of the composite densication, represented by a composite numerical microstructure made of both discrete components (coarse alumina, ne alumina, and graphite) and a soft shell surrounding alumina material. The elastic mechanical behavior at the contact is studied rst. The composite (macroscopic) behavior is then investigated (section 5.4.2).

For this purpose, the interaction between i) coated-coated, ii) and non-coated-coated particles are studied. In dp3D, to track the history of particles at the contact, particles can be tagged before running a simulation. Thus, the contact of tagged particles can be followed during the compaction at each time step. A total of four pairs of particles are tagged: i) coarse alumina-graphite, ii) ne alumina-graphite, iii) coarse alumina-ne alumina, and iv) ne alumina-ne alumina. The contacts between two coarse alumina clusters are not presented because this type of contact is very rare. The graphs in the present section originate from a closed-die compaction simulation of Mix 1 (reference composite) with a target of 35 MPa for the axial stress. We focus here on the behavior uniquely at the contact (particle interactions).

The evolution of the normal contact force (N Hertz ), Hertzian model (Eq. (4.16), chapter 4), against the particle relative indentation, δ n /r, (r is the particle's radius) in closed-die compaction numerical simulation is shown in Fig. 5.1 a1) and a2). The evolution of the normal contact force (N Hertz ) against the composite relative density is shown in Fig. 5.1 b). Note that the Hertzian force starts at N Hertz < 0 (tensile), see Fig. 5.1. At the beginning of the compaction, the contact is under tensile because of the abrupt introduction of adhesion to the compaction simulation (jamming has neither friction nor adhesion) (chapter 4).

An example of contact between coated coarse alumina and non-coated graphite particles is represented by contact type number 1) (Fig. 5.1 a2). The contact moves towards the hard branch and the indentation is at the critical value ( δn = t/r = 0.198, see Fig. 5.2 1). In other words, the particles contact indent within the soft binder shell, increasing the normal force signicantly at the contact. Once the contact reaches the hard branch, MATERIALS Figure 5.1: DEM close-die compaction simulation (target σ z =35 MPa). a1) and a2) Normal Hertzian contact force evolution (N Hertz ) versus normalized particle indentation (δ n /r). Contacts type 1) to 4) represent the particle contact between three components (coarse alumina, ne alumina, and graphite). Once the contact reaches the critical indentation value ( δn ), hard contacts appear (see Fig. the contact is at the critical indentation.

Contact type number 2) (Fig. 5.1 a1) represents the contact between a coated ne alumina particle and a non-coated graphite particle where δn is also equal to 0.198 (Fig. 5.2 2). Similar to contact type number 1), the interaction is also at the hard (branch 2). Contact type number 3) (Fig. 5.1 a2), is the interaction between two coated alumina particles (ne alumina and coarse alumina) that reach the hard alumina branch (Fig. 5.2 3).

Contact type number 4) is represented by the contact between two coated ne alumina 5.4. Compaction and unloading simulations particles. Contrary to the previous contacts, the interaction remains in the soft (branch 1). The relative indentation are lower than the critical value ( δn < 2t/r = 0.39), as shown in Fig. 5.2 3) and 4). In this case, if the particles unload, the unloading path will follow the same path given by the loading (no hysteresis).

It is worth mentioning that these curves are examples of the multiple contacts that arise in the simulation. During compaction, some contacts will hit the hard branch (contacts type number 1, 2, and 3) while others will stay in the soft branch (contacts type number 4). Also, the contact path at the hard branch seems vertical when the particles are unloaded, but it has a nite slope dictated by the alumina Young's modulus (380 GPa). In other words, the stiness at the contact when in the hard branch is dictated by the alumina, and in the soft branch, by the binder (alumina Young's modulus binder Young's modulus).

Figure 5.2: Related to contacts 1) to 4) highlighted in Fig. 5.1 above. 1) is the contact between coarse alumina and graphite when hard contact appears. Thus, δn =0.198. 2) is the hard contact between ne alumina and graphite when the contact also reaches the critical value ( δn =0.198). 3) is the contact between coarse alumina and ne alumina particles when the interaction remains in the soft branch ( δn <0.39). 4) also represents a soft-soft contact between two ne alumina particles ( δn <0.39).

In Fig. 5.1 b), the markers (* ) represent the approximate point where the interactions hit the hard branch (contacts type 1, 2, and 3), and are at the critical indentation value. In contact type 1), there is a peak of the normal force at approximately RD=0.7 (contact reaches the hard branch). Above RD = 0.87, the contact is under a tensile force (N Hertz < 0). Above RD = 0.92, the particles return on transmitting a compressive force (N Hertz > 0). Overall, the contacts may unload, fail, and return to the previous contact several MATERIALS times during the compaction.

As seen above, the contacts load and unload all along the compaction. These examples represent only a few contacts from hundreds of thousands of possible particle interactions. Therefore, to better understand whether the contacts remain in the soft or are at the hard branch, the % correlated to the total number of elastic interactions between coated-coated and non-coated-coated particles (from the same close-die compaction simulation presented above) are recorded (Fig. 5.3).

B 1 is the total number of contacts in the soft binder branch. B 2 is the total number of contacts in the hard branch. B 1 or B 2 [mat. i -j] are the number of contacts between two components i and j in soft or hard branches, respectively. These components i, and j include ne alumina, coarse alumina, and graphite. Five dierent combinations between coated-coated and non-coated-coated particles are possible: i) coarse alumina-coarse alumina, ii) coarse alumina-graphite, iii) coarse alumina-ne alumina, iv) ne alumina-ne alumina, v) graphite-ne alumina. Fig. 5.3 shows the evolution of the % of contacts in the hard and soft branches (related to the contact between the components) versus composite relative density. The contact between ne alumina particles corresponds to approximately 80% of the interaction in the soft binder (branch 1). Above RD=0.85, it is possible to observe a transition in the curve slope and the reduction in the % of soft contacts (Fig. 5.3 a) because while the particles are compressed during the compaction, some of these soft contacts move towards the hard alumina branch, as shown in Fig. 5.3 b). Overall, the central part of the contacts stayed in the soft elastic (branch 1), mainly related to ne alumina contacts. This shows that most of the contacts stay in the soft branch and that the binder plays a dominant role for a large portion of the compaction.

The type of contact (if soft or hard) will depend mostly on two factors: i) the binder elasticity and ii) the binder thickness (volume fraction). A softer elastic cover will tend to particles reach more easily the hard (branch 2). Indeed, a composite with a smaller binder thickness (mix 2) will highlight more ne alumina contacts at the hard (branch 2) compared with the other compositions.

In this section, the contacts at the soft and hard branches between particles belonging to distinct materials were investigated. How do these interactions between coated-coated and non-coated-coated particles in the soft and hard branches will aect the composite macroscopic behavior? Is the binder elasticity and thickness (related to the composition) an important factor for the composite densication behavior? These questions are investigated in the following sections.

Macroscopic behavior

During close-die compaction, each discrete material (ne alumina, coarse alumina, and graphite) contributes to the composite densication according to the dierent types of contacts and DEM input properties. The Hertzian elastic model governs the interaction between coated alumina particles and non coated-coated particles (graphite and alumina). The internal (and bonded) graphite particles are represented by the bond model. In the previous section, the behavior at the contact was detailed, taking as example mix 1 (reference composite) during a closed-die compaction simulation. In the present section, the same simulation is investigated but at the macroscopic stress level.

The evolution of the macroscopic axial stress associated with each discrete component (ne alumina, coarse alumina, and graphite) and the total macroscopic axial stress (in z direction) against the relative density is shown in Fig. 5.4. The coarse alumina curve represents the macroscopic axial stress contribution of coarse alumina component to the total macroscopic axial stress during DEM close-die compaction simulations (see Eq. (4.14) in chapter 4). This stress includes contact contribution between coarse alumina-ne alumina particles, coarse alumina-graphite particles, and between coarse alumina particles of distinct clusters. To note: the bonds inside the coarse alumina cluster cannot deform either fracture, and these interactions are not considered in the stresses calculation (chapter 4). Similarly, the graphite curve represents the macroscopic axial stress contribution of graphite component to the total axial stress. It includes contacts between graphite-coarse alumina particles, graphite-ne alumina particles, graphite particles from dierent clus-MATERIALS ters, and internal bonded graphite particles. Finally, the ne alumina curve represents the macroscopic axial stress contribution of the ne alumina component to the total axial stress, including the contacts between ne alumina particles, ne alumina-coarse alumina particles, and ne alumina-graphite particles. The ne alumina contribution always represent more than 50% of the total axial stress, while the graphite is approximately 40% and the coarse alumina less than 10% (Fig. 5.4). Therefore, it can be concluded that the ne alumina is the component that impacts most the composite densication. Contacts between ne alumina particles that are in the soft binder shell contributes most to the ne alumina part (Fig. 5.3). The contacts between ne alumina-graphite and ne alumina-coarse alumina also contribute to the ne alumina stresses because the number of ne alumina particles is much higher than the number of graphite and coarse alumina clusters. The macroscopic stress contribution of coarse alumina is mostly due to the interaction between ne alumina and coarse alumina particles. The total contacts between coarse alumina and graphite in branches 1 and 2 correspond to less than 1% (Fig. 5.3).

The macroscopic stress contribution of graphite, which is consistently a bit lower than the ne alumina contribution, comes from contacts between graphite-ne alumina particles, and between internal bonded graphite particles. There are many more contacts between graphite-ne alumina particles than contacts between coarse alumina-graphite particles (the coarse alumina are in a smaller number). In addition, interactions between bonds that were not fractured and between bonds that were fractured during the 5.4. Compaction and unloading simulations compaction are present inside graphite clusters.

To understand the eect of the contact laws (bond and Hertzian models) on the compaction behavior, Fig. 5.5 shows the composite macroscopic axial stress (target: σ z =35 MPa) versus relative density. The contact laws here are divided into three groups. The rst group (Hertzian model) includes interactions between coated components (coarse alumina-ne alumina particles, coarse alumina particle clusters, and ne alumina particles). Also, contacts between a coated and a non-coated component (coarse aluminagraphite particles and ne alumina-graphite particles) and between graphite particles from dierent clusters (Fig. 5.5 (a)).

The second one (Bond model) is represented by interactions between bonded particles (inside the graphite cluster) in which the bonds remain intact until the end of the compaction (Fig. 5.5 (b)). The third group (Bond model, broken bonds) represents contacts between bonded graphite particles (inside the cluster), fractured during the compaction (Fig. 5.5 (c)). As seen before in chapter 4, the tensile force from these broken bonds comes from the DMT adhesion theory. The shear force comes from the Hertz-Mindlin friction law, and the compressive forces between broken bonds.

The Hertzian contact (Fig. 5.5 (a)) contribution, to the total macroscopic axial stress during the close-die compaction numerical simulation, represents nearly 70% of the composite macroscopic stress. The bond contact model (Fig. 5.5 (b)) has more than 10%, and the contact between broken bonded particles (Fig. 5.5 (c)) less than 20%. The MATERIALS percentages are given from the end of the compaction simulation (RD=95.8%, 35 MPa).

The ne alumina is the component that impacts mostly the macroscopic stresses (Fig. 5.4) due to the many interactions belonging to the Hertzian model. First, these contacts are given from interactions between ne alumina particles in the soft branch 1. In second, from contacts between graphite-ne alumina particles, and in third between coarse alumina-ne alumina particles.

Concerning the bond contact model (Fig's 5.5 b) and c), the bonded particle strength in tension and shear (Eq's (4.9) and (4.10) from chapter 4) plays an important role in the stresses of non-fractured-bonded graphite particles and broken bonded graphite particles. The tendency is that by increasing the bond strength, fewer graphite bonds are broken during compaction, thus increasing the stresses coming from non-fractured bonds and decreasing the stresses from broken bonds.

Eect of the binder elastic modulus

During compaction simulations, the binder soft-shell is the factor that most impacts the composite macroscopic behavior, mainly between ne alumina contacts. During the rst stage of the compaction (loading), the binder greatly inuences the composite densication. After unloading, the binder highly impacts the microstructure hysteresis (or spring-back). Thus, close-die compaction simulations on mix 1 (reference composite) with a target of axial stress equivalent to 35 MPa (σ z =35 MPa) were performed to investigate these phenomena. Only the binder Young's modulus is varied, and other dp3D input microscopical mechanical properties remained the same, such as adhesion energy, friction coecient, bond strength. Elastic material properties are kept constant for the alumina and graphite. A total of ve close-die compaction simulations are performed with binder elastic modulus values (E binder ) varying from 0.1 to 3 GPa (Fig. 5.6), including (E binder =1.02 GPa) that is the best-t with experimental data (Table 5.1). Fig. 5.6 a) shows axial stress (target: 35 MPa) vs. relative density curves during the closed-die compaction simulation. In small binder Young's modulus (0.1-0.5 GPa) values, the curve shows a softer behavior with spike proles (due to the indentation between hard particles). On the opposite, increasing the binder Young's modulus (1.02-3 GPa), the curve exhibits a stier behavior and a smoother prole (no spikes). This smoother prole may be explained because the number of hard branch contacts is smaller on compaction simulations with a higher binder Young's modulus (3 GPa). The evolution of the relative density at 35 MPa and the total % of hard contacts in branch 2, versus the binder Young's modulus is shown in Fig. 5.6 b). At 35 MPa, the relative density (RD) of the numerical composite is 99.4%, 98.7%, 95.8%, 90.9%, and 87.7% for an input dp3D binder Young's modulus (E binder ) value of 0.1 GPa, 0.5 GPa, 1.02 GPa, 2 GPa, and 3 GPa, respectively. GPa): a) evolution of the axial macroscopic stress (σ z ) against the composite relative density (loading). b) evolution of the relative density at the end of the loading (σ z =35MPa) and total % of hard contacts, versus the binder Young's modulus (E binder ). Fig. 5.7 a) shows the compaction loading curves (Fig. 5.6), and the unloading. The shape of the unloading curves becomes quite dierent following the modication of the E binder . The area under the loading and unloading slopes (dissipated energy) is reduced by increasing the binder Young's modulus. As seen in chapter 4, in dp3D, the magnitude of the composite macroscopic spring-back will depend on the type of contact (branch 1 or 2). The binder is modeled as pure elastic, but when two particles in branch 2 are unloaded, they unload all along down to separation with the stiness of branch 2 (the stier one), introducing hysteresis at the contact (chapter 4).

The composite spring-back percentage is calculated as the composite relative density (RD) dierence between the end of the loading (at σ z =35 MPa) and the end of the unloading (at σ z =1 MPa). The evolution of the spring-back % and the % of the total of contacts in the hard branch during the loading, as a function of the binder Young's modulus is shown in Fig. 5.7 b). The % of contacts in branch 2 is calculated by the ratio between the total of contacts in branch 2 and the sum of the number of contacts in branch 1 and branch 2 (at the end of the loading, 35 MPa). Note that the graphite bonded particle contacts are not included because it is not a Hertzian contact.

From a binder Young's modulus between 0.1 GPa and 2 GPa, the % of contacts in branch 2 decreases resulting in the increase of spring-back during unloading. At 0.1 GPa, the composite spring-back % is almost close to zero because the major part of the contacts are in branch 2 (≈ 80%). For E binder ≤2 GPa, the spring-back is increasing with the E binder . The spring-back is almost constant between E binder = 2 -3 GPa because the composite is sti, and it becomes more dicult for the composite to densify (see previous In summary, upon indentation during the loading, the rigidity of the contact evolves steeply when a critical indentation is reached (enter into the hard branch 2). If this contact at branch 2 is unloaded, some hysteresis is introduced. During the rst stage of the compaction (loading), some particles might be unloaded. During the second stage of the compaction (unloading), most contacts will unload with a stiness that depends on the branch it has reached upon loading. Therefore, the spring-back % during unloading will depend on the % of contacts in the hard branch during the loading.

Ideally, the binder should be modeled as elasto-plastic to minimize the composite spring-back impact during the unloading. However, this would increase the model complexity since new contact models should be introduced when allowing both elasticity and plasticity for the binder. The binder is the material that dictates most the material den-sication and spring-back behavior. In the next section, the eect of the composition on the material behavior during close-die compaction simulations will be investigated.

Eect of the composition

In this section, the impact of the composite composition on the composite response through DEM close-die compaction simulations is studied. Each composite has a variation on ne alumina, coarse alumina, graphite, and binder in volume fraction percentage. The studied composites composition is listed in Table 3.1 (chapter 3). Table 4.3 (chapter 4) shows the number of ne/coarse alumina particles, graphite particles, and binder thickness calculated from the composition of the composite materials.

In summary, Mix 2 has a smaller binder volume fraction (%), Mix 3 is the opposite of Mix 2 (more binder vol. fraction %), Mix 4 is the composite with more reinforcement (more graphite and coarse alumina volume fraction %), in comparison with the reference composite (mix 1). The DEM input properties, already presented in Table 5.1 for the green characterization, remains the same for all simulations on the four powder composites, and so only the number of particles and the binder thickness is modied.

The evolution of the axial macroscopic stress, of the four composites (Mix 1-4), with their relative density (RD) during numerical close-die compaction simulations, is shown in Fig. 5.8 a). The simulations stopped at 35 MPa axial stress. By decreasing the binder volume fraction (thickness), the curves exhibit a stier behavior. Mix 4, the composite with more hard particles (coarse alumina and graphite) exhibits a loading curve with more spike proles due to the hard particles indentation.

At 35 MPa, the relative density (RD) of the mix 1-4 are 95.8% ± 0.06, 91.3% ± 0.03, 97.5% ± 0.14, and 94.5% ± 0.38, respectively (Fig. 5.8 b), with the following ascending densication ranking: mix 2 < mix 4 < mix 1 < mix 3. The binder volume fraction increase leads to an increase of the relative density at a given stress. This is explained because of the decrease of the % of hard contacts (branch 2). The increase of the binder volume fraction with the spring-back percentage exhibits almost a linear relationship with the binder content due to the decrease of contacts in the hard (branch 2), as shown in Fig. 5.8 c).

In summary, the increase of soft contacts (and consequently, a decrease of contacts in the hard (branch 2), together with the rise of binder volume fraction, mostly between ne alumina particles, leads to a denser microstructure at the end of the loading. It also leads to a more signicant spring-back after unloading. Thus, the binder is the major factor aecting the composite densication and spring-back eect. For example, a thinner binder shell will increase contacts at the hard alumina branch and, consequently, a smaller composite spring-back eect. MATERIALS Figure 5.8: a) Evolution of the axial macroscopic stress against the composite relative density (RD) (mix 1 to 4), target: σ z =35 MPa. b) evolution of the the relative density at σ z =35 MPa (simulations and experiments) against the % of ne alumina particle contacts in branch 2 (hard branch) with the binder volume fraction (%). c) evolution of the spring-back (%) after unloading against the % of ne alumina particle contacts in branch 2 (hard branch) with the binder volume fraction (%).

The contribution of the discrete materials (ne alumina, coarse alumina, and graphite) to the macroscopic stress during such close-die compaction simulations is also investigated (Fig. 5.9), with a target density RD=0.90. The stresses are calculated from the stress partition as described in chapter 4 (Eq. 4.14). Even though it is clear that the signicant part of contacts is related to the contact between ne alumina particles, the objective here is to evaluate the evolution of the stresses associated with each material depending on the composite composition.

The macroscopic axial stress (in average) correlated to ne alumina material ranges between 8.3 MPa to 14 MPa in the ascending order: mix 4 < mix 3 < mix 1 < mix 2. Mix 2 shows a signicant increase in stress compared with the other compositions. This can be explained because mix 2 is the composite with a smaller soft binder thickness and a Figure 5.9: Contribution of the macroscopic axial stress related to each discrete material (coarse alumina, graphite, and ne alumina) to the total macroscopic axial stress (experiments and simulations) of composites (mix 1 to 4), against the binder volume fraction (in %), during close-die compaction simulations (target: density RD=0.90). larger quantity of ne alumina particles. Thus, more contacts in the hard alumina branch during the compaction (branch 2) increase the macroscopic stresses at a given relative density. Concerning graphite, the macroscopic stresses associated is between 6.1 MPa and 10.4 MPa (mix 3 < mix 1 < mix 4 < mix 2).

Still, mix 2 is the composite with larger stress due to mainly the contacts between ne alumina and graphite particles. Mix 4 is the composite with larger graphite particle clusters, and it comes in second. The macroscopic axial stress related to coarse alumina is very small if compared with the other two discrete materials, and it corresponds to 0.7 MPa-1.3 MPa (mix 1 < mix 3 < mix 4 < mix 2). This is explained because the number of contacts between coarse alumina particle clusters is very small. Finally, the total macroscopic axial stress is in ascending order: mix 3 < mix 1 < mix 4 < mix 2, which is correlated mainly to the ne alumina and graphite discrete materials.

Comparison with experiments 5.5.1 Close-die compaction

The "weakly jammed" packing at RD=0.6 can be compacted following two dierent routes: close-die compaction and isostatic compaction (chapter 4). However, isostatic compaction experiments using small samples are challenging to perform due to the testing MATERIALS complexity. It is necessary to control the stresses in the axial and radial directions (they must be constant and equivalent during the compaction). Because of this, we focused mostly on performing DEM close-die compaction modeling to calibrate the numerical densication behavior with closed-die compaction experimental data. The comparison of the axial stress versus the sample relative density using numerical (DEM) and experimental close-die compaction tests of mix 1 (reference composite) are exhibited in Fig. 5.12. The experiments and the simulations are set to a nal target of 35 MPa to mimic the industrial setup. The rst compaction stage (loading) shows a good agreement in the densication behavior between DEM and experiment. Thanks to the introduction of a soft continuous coat (binder), the present model can account for the eect of the dierent components in the composite compaction behavior. However, the spring-back eect in DEM is overestimated in comparison with the experiment. This is because the binder Young's modulus highly aects the composite spring-back behavior when the ne alumina particles are mainly in the hard alumina (branch 2) (Fig. 5.7). One solution to decrease the macroscopic spring-back would be to change the behavior at the contact during unloading by introducing some plasticity. When two particles coated by the soft binder shell are in branch 2, some hysteresis is introduced at the contact. However, the present DEM simulation presents more contacts at branch 1 (soft).

The nal relative density at the end of DEM and experimental close-die compaction tests are plotted with the compaction pressure (Fig. 5.11). The RD at 10 and 35 MPa from experiments exhibit an RD of 84.5% ±0.38 and 95.5% ±0.3, respectively. At RD=100%, the compaction pressure is 77.5 MPa ±2.3. In DEM 85.3% (10 MPa), 95.8% (35 MPa), and 100% (68.5 MPa). The DEM and experimental data show a good quantitative agreement in RD vs. compaction pressure for the rst part of the closed-die compaction tests (loading) for the Mix 1 composite. The main purpose of this section is to demonstrate the ability of DEM to represent the compaction behavior of the four commercial composites. The evolution of axial stress (compaction pressure) against the sample relative density achieved by DEM and experimental closed-die compaction tests of mix 1-4 (with end target 35 MPa) is shown in Fig. 5.12. The experimental and DEM curves exhibit a good qualitative agreement giving some condence in the ability of DEM to represent the compaction behavior of this type of composite considering their respective compositions.

The comparison of the spring-back eect (or hysteresis) during the unloading on experiments and DEM simulations is shown in Figs. 5.13 a) and b). At the beginning of the unloading, or below σ z ≈ 20 MPa, the DEM curve exhibits a linear behavior followed by a large hysteresis that follows almost the same behavior from the loading curve (Fig. 5.13 b). However, as previously seen in chapter 3, in reality, as the system is viscous, compaction is still progressing during the unloading (Fig. 5.13 a). Also, during the DEM unloading simulations, the spring-back eect is overestimated compared with experiments.

On DEM, the spring-back tends to increase with the increase of the binder volume fraction (Fig. compacted microstructures before mechanical test simulations. As it follows, unloaded packings are used to evaluate the green mechanical behavior with actual triaxial/uniaxial compression testing and numerical simulations (section 5.5.2).

Triaxial and uniaxial compression tests

An axial stress-strain plot of a DEM uniaxial compression test with σ x = σ y = 0 for the reference Mix 1 and the corresponding evolution of the numerical microstructure is shown in Fig. 5.14. Both σ z (axial stress) and the strain is in compression. (1) represents the initial microstructure (after the unloading), ( 2) at the maximum axial macroscopic stress, and it is estimated to be the moment where the packing was fractured, and (3) after the material damage. Since there are no bonds between the ne alumina particles, only a few bonds inside the graphite clusters during the uniaxial compression are broken. The sample deforms to the radial direction due to the axial compression stress, increasing porosity (RD decreases). The green strength depends mostly on the binder content and the microstructure composition (chapter 4). Similar to experiments, the composition with a lack of the matrix content (ne alumina and binder) and more coarse alumina and graphite (Mix 4) exhibits a smaller strength because of the decreased macroscopic stresses associated with ne alumina. Mix 2 (more ne alumina particles and less binder) has shown the largest strength due to increased macroscopic stresses correlated to ne alumina contacts. These alumina particles are coated with a thinner binder shell, leading to increased hard contacts (branch 2). Unlike the experiments, Mix 3 has shown a decrease in strength. The overestimation of the spring-back may aect the composite strength measurement since the numerical composite is more porous than in reality. The axial stress-strain from these 5.5. Comparison with experiments DEM uniaxial compression test simulations are shown in Fig. 5.15 c). Analyzing the linear slopes, Mix 2 exhibits stier behavior due to a lack of binder (more hard alumina particle contacts). Mix 3 (more binder content) is less sti due to the larger number of soft contacts.

The compacted samples (Mix 1) are submitted to triaxial compression tests at three dierent conning pressure (or radial stress), 10 to 30 MPa, and compared with DEM triaxial test simulations. A stress-strain curve from a numerical triaxial simulation with radial stress equivalent to 30 MPa is shown in Fig. 5.16. The evolution of the numerical sample microstructure is also highlighted. As seen in chapter 4, the stresses at the radial direction (σ x = σ y ) are increased together with σ z until the radial stress remain constant, and only σ z continues to increase. Some graphite bonds are broken during the compression simulation. The comparison between experimental and DEM stress-strain curves from triaxial tests are shown in Fig. 5.17. Fig. 5.17 a) and b) represent numerical and real samples compacted through close-die to 10 MPa, and 35 MPa. Fig. 5.17 c) represents the triaxial compression behavior of numerical and real samples previously compacted up to RD ≈ 100%. The DEM stress-strain curves for radial stress between 10 MPa and 30 MPa show MATERIALS good accordance with experiments. This is encouraging since material properties have not been tted for these complex stress states.

The rst part of the experimental stress-strain curves exhibits a rough linear shape that may be explained by the viscous-elastic binder eect, grain rearrangement, and increase of the radial stresses. In DEM, the rough linear part is also because of the radial stresses and sample densication (that could not be studied experimentally). The non-linear curve shape departs since only the radial stress (σ x = σ y ) remains constant, and the axial stress continues to increase. Depending on the radial stress value, the microstructure can be compacted or fractured (next section). In general, the deformation of the linear part of the curves is increased when testing more porous samples due to the more facility of particle rearrangement during the triaxial compressive simulation.

Yield and fracture surfaces

Discrete element modeling allows complex loading to be applied to compacted virtual samples to generate the isodensity yield surface and fracture points, even for stress states that are dicult to attain through experimental mechanical tests. First, the methodology for the generation of these points is carefully described (section 5.6.1). In sequence, the eect of the composite density after compaction (section 5.6.2), the eect between close-MATERIALS die and isostatic compaction samples (section 5.6.3), and of the composition (section 5.6.4) on the fracture and isodensity yield points are also presented. 5.6.1 Identication of the isodensity yield surface and fracture points

The generation of isodensity yield surfaces and fracture points were obtained by applying various reloading paths with constant strain rate in the z -axis and by varying the stress in the x, and y directions. For this purpose, the numerical 'jammed' packings at RD=0.6 are compacted up to a given relative density, unloaded and reloaded to obtain these yield and fracture contours in the stress space. These reloading points are plotted in the mean stress vs. deviatoric stress plane (mean stress Σ m = 1/3(σ x + σ y + σ z ) and deviatoric stress

Σ d = σ z -[1/2(σ x + σ y )]
).

An example of the loading (1) and unloading ( 2) stress path in the stress space for the reference mix 1 numerical sample in closed-die condition (up to an axial stress equivalent to 35 MPa) is shown in Fig. 5.18. The * symbol represents the point of a isotropic stress state at the end of the unloading (σ x = σ y = σ z =1MPa). In this case, the mean and deviatoric stresses are Σ m = = -1MPa and Σ d = = 0MPa, respectively. In [START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF], the reloading is restricted to the case where Σ d >0, however in our case, the deviatoric stress may be negative if σ z >1/2(σ x + σ y ) (more common), or sometimes positive if σ z <1/2(σ x + σ y ).

From the unloading point, we apply rst an uniaxial tensile test with an axial strain rate εz > 0 by pulling the sample in the z -axis. For this simulation, the radial stresses in x and y are equal to zero (σ x = σ y =0), and is represented by the number (3), see Fig. 5.19 3). At this point, Σ m ==-0.2MPa, and Σ d =0.8MPa. At point ( 4), an equal and constant tensile strain rate is applied at the three axis (isostatic tensile test), Fig. 5.19 4), and because the numerical composite has an anisotropic macro mechanical response (RVE study in chapter 4), the three stresses are not exactly equivalent, thus Σ m =1MPa, and Σ d =-0.1MPa.

Several reloading tests are performed to explore whether the material is densied or fractured. The macroscopic fracture on DEM was deduced from the maximum stress on the z -axis component associated with a decrease of the sample relative density. The point (5) from the Fig. 5.18 is represented by an uniaxial compression test previously seen in section 5.5.2 where a compressive axial strain rate is imposed in the z and radial stresses equal to zero (σ x = σ y =0). The sample is progressively damaged (decreasing the RD) during the simulation until a maximum σ z is reached (Fig. 5.20 5). At this point (5), Figure 5.18: Isodensity contour points (RD=95.8%) and fracture points of a compact generated in closed-die compaction from DEM simulation (target: σ z =35MPa), and unloaded. The loading (1) unloading ( 2) reloading (3-9) shows a typical stress path followed by the compact to generate isodensity yield points on the surface and fracture points with a linear shape. The black point (*) relates to the point at which fully isostatic unloading is applied (σ x = σ y = σ z =1MPa). later fractured (decrease of RD). In (7), the sample is rst compacted, and then fractured (decrease of RD), and compacted again (Fig. 5.20). In cases ( 6) and ( 7), the fracture point is dened as the maximum axial stress together with the RD decrease. Similar to [START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF], the link between these fracture points leads to a shape that is likely linear.

In sequence, by increasing the radial stresses (σ x = σ y ) the composite undergoes compaction and is densied until obtaining the same relative density reached at the end of the close-die compaction. This methodology, applied to generate the isodensity yield surface points, was previously used in numerical simulations on granulate materials [START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF]] [Harthong et al., 2012[START_REF] Abdelmoula | A study on the uniqueness of the plastic ow direction for granular assemblies of ductile particles using discrete nite-element simulations[END_REF][START_REF] Loidolt | Prediction of the anisotropic mechanical properties of compacted powders[END_REF]. The points (8) and (9) from Fig. 5.18 are represented by triaxial test simulations with σ x = σ y =20 and 26.5 MPa, respectively. The samples are densied up to RD=0.958 (same density reached at the end of the closed-die compaction), see Fig. 5.21. In (9), at RD=0.958, the three stresses are equivalent (Σ d =0). Further increases of the radial stress, the deviatoric stress becomes positive since σ z <1/2(σ x + σ y ). If continuously increasing the radial stresses, the limit of densication is reached since we stipulate a limit (RD=0.958). This `limit' point is highlighted in Fig. 5.18 with a red arrow. In general, the isodensity yield surface presents a concave shape that was also observed in [START_REF] Bibliography Chen | Contribution à la modélisation de la compression des poudres par une méthode d'éléments discrets maillés[END_REF] (Fig. 4.32) during MATERIALS a study of the compaction modeling of metal powders by a discrete nite element method.

Eect of the density

The same approach from the section above was applied, but focusing on generating the fracture and isodensity yield surfaces from numerical Mix 1 (reference) samples compacted up to various relative density values. The evolution of the yield and fracture surfaces as a function of the RD is shown in Fig. 5.22. They represent samples previously compacted by closed-die conguration from RD=0.8 to RD=1 and summarise the hardening evolution during the increase of the composite density. Like in the previous Fig. 5.18, these surfaces have a similar general shape: a fracture line and a cap that can be tted with the Drucker-Prager/Cap model. Inside the yield surface, the powder deforms elastically, and if the stress state reaches the yield surface, the powder deforms plastically [START_REF] Abdullah | Finite element simulation of alumina ceramic powder compaction[END_REF][START_REF] Shin | Physical interpretations for cap parameters of the modied Drucker-Prager cap model in relation to the deviator stress curve of a particulate compact in conventional triaxial testing[END_REF][START_REF] Zhou | A density-dependent modied Drucker-Prager Cap model for die compaction of Ag57.6-Cu22.4-Sn10-In10 mixed metal powders[END_REF]. A constitutive model based on the Drucker-Prager/Cap criterion for modeling close-die compaction through the nite element method will be described in chapter 7.

Eect of the consolidation path

The consolidation path was rst presented by closed-die compaction since the green experimental characterization was performed on samples compacted by closed-die conguration. However, part of VESUVIUS industrial production is isostatic compaction (e.g., ladle shroud). For this reason, the isodensity yield points and fracture points of a com- pacted sample obtained by closed-die compaction and compacted sample obtained by isostatic compaction are compared (Fig. 5.23). Both samples were previously compacted up to RD=0.95. Also, it is interesting to understand the eect of the strain history of the powder on the behavior of the compact. For isostatic compaction, the method to obtain isodensity yield points is identical to the previous sections.

For both compaction congurations, similarly to section 5.6.1, the points which characterize the Σ d <0 region of the stress space were added. The loading points in red (close-die) and blue (isostatic) are represented by the mean and deviatoric stress state at the end of the compaction (before unloading). As seen in chapter 4, the loading point from isostatic compaction does not present deviatoric stress equal to zero (Σ m =-35.6MPa and

Σ d =3.7MPa
). This is because the composite microstructure cannot have a full isotropic behavior due to the limited number of particles. Both fracture surfaces are very similar (linear shape). However, the shapes of the isodensity yield surfaces are dierent. The MATERIALS one from close-die compaction still resembles mostly a concave shape, and the one from isostatic compaction, a more convex as the one found in literature [START_REF] Akisanya | The yield behaviour of metal powders[END_REF]] [Pizette et al., 2010]. However, in Pizette et al., isostatic and close-die compactions resulted in much more dierence between the yield surfaces (see Fig. 2.24 in chapter 2).

Here the two surfaces are quite similar.

Eect of the composition

The approach described in the previous subsections can also be applied to the four dierent compositions: mix 1, mix 2, mix 3, and mix 4. Here, the composites were compacted by closed-die up to a density RD=0.90. All tested compositions show the same behavior observed in the previous sections: a failure line and a cap surface. Increasing the binder ratio, which is a soft material, leads to a strength decrease. Mix 2 highlighted a concave yield surface format, whereas the other composites have a more convex shape. This is probably due to the larger stresses if compared with the other composites. Observe that in the previous Fig. 5.22 the increase in the sample hardening lead also to a more pronounced concave shape (RD=1) compared with the other samples compacted up to a smaller density.

Conclusions

This chapter has explored the potential of DEM for simulating the compaction of ceramic powder composites. For this purpose, rst, a qualitative study concerning the impact of DEM input properties on the macroscopic composite behavior was investigated (section 6.3). The binder coating is the material that most dictates the macroscopic behavior. During the compaction, the particle interactions can remain in the soft binder shell (branch 1) or move towards the hard alumina (branch 2). These two types of interactions will depend on 1) composition and 2) binder stiness. First, a composite with less binder presents a smaller binder thickness leading to more coated-alumina particles moving to branch 2. Second, a softer binder (lower Young's modulus) leads also to contacts that move in larger numbers to branch 2. Thus, stier contacts (branch 2) result in larger elastic forces at the contact, more signicant macroscopic stresses, and a less dense composite (smaller RD). When the particles at branch 2 (hard) are unloaded (not necessary only during unloading simulations), it introduces some hysteresis to the contact. Thus, the spring-back eect will be smaller in a composite with more contacts at the hard (branch 2).

The eect of each discrete material (ne alumina, coarse alumina, and graphite) on the composite compressive macroscopic stress was investigated. Fine alumina and graphite particles contribute to more than 50% and 42% of the macroscopic axial stress, respectively, whereas coarse alumina to 6%. The stresses corresponding to alumina contacts come from the Hertzian model, whereas the interactions between bonded graphite particles from the bond model. These interactions between bonded particles (i.e., internal bonded particles graphite cluster) can be divided into interactions between non-fractured particles and fractured ones. The Hertzian contact generates approximately 69% of the axial macroscopic stress, and the Bond contact model with a) non-fractured bonds (12%), and b) fractured bonds (18%).

During close-die compaction, some factors can impact the powder composite anisotropy. First, the graphite is highly orientation-dependent (anisotropic), and depending on its orientation (if parallel or perpendicular, e.g.), the graphite Young's modulus can signicantly vary. Second, as the radial and axial stresses are quite dierent, graphite akes will be more oriented in the perpendicular direction of the compression (z-axis), increasing the composite anisotropy as was conrmed by our DEM simulations.

Concerning the comparison between DEM and experimental data on the close-die compaction, the rst stage of the compaction (loading) shows a good agreement when considering dierent compositions. However, the spring-back presented in the second stage (unloading) is overestimated compared to experiments because the major part of the alumina contacts is in branch 1 (soft). One possible solution to minimize this discrepancy MATERIALS would be to introduce some plasticity at the contact at branch 1. The stress-strain curves from numerical and experimental triaxial tests show a good agreement for the reference mix 1, giving some condence in the ability of DEM to represent the green mechanical behavior of such powder composite materials.

Finally, we have shown that it is possible to obtain yield and fracture surfaces in the stress state of a representative volume element of a composite powder. The generation of these surfaces was obtained by applying various reloading paths with constant axial strain rates and varying radial stresses. The triaxial tests are carried out to explore whether the composite is densied or fractured. The macroscopic material failure was estimated from the maximum axial stress and decreased relative density. The connection between these points leads to a linear shape (fracture line). We dene the yield surface as the surface linking the points where the microstructure is densied until obtaining the same relative density reached at the end of the compaction simulations (isodensity points). The main advantage of this approach is to allow probing points in the deviatoric-mean stress space that are very dicult or impossible to be measured experimentally. The generation of these surfaces will be exploited to formulate a realistic constitutive model that will enable the Finite Element Method (FEM) compaction simulation in actual production (chapter 7).

Chapter 6 DEM simulations on red powder refractory materials

In this chapter, the mechanical properties of the virtual red composite (Mix 1) are simulated. For this purpose, DEM triaxial and tensile simulations are performed. Contrary to the green numerical composite, the numerical red composite is characterized by elastic bond connections between ne alumina particles, representing the red matrix. In section 6.3, the principal input DEM parameters are presented, and the qualitative eect of the bond strength in shear and tension during triaxial simulations is described. The mechanical behavior at the ring stage is evaluated by comparing stress-strain curves from triaxial compression tests, and the fracture stress from tensile simulations and Brazilian experiments. The cracks (during tensile simulation) develop mostly where the initial damage was located (section 6.4). The yield and ultimate (maximum) stresses are identied from DEM triaxial and tensile simulations using the same approach used in experiments. The yield and ultimate stresses from DEM simulations are plotted in the mean vs. deviatoric stress plane, generating the yield and ultimate surfaces, and compared with experimental data (section 6.5). These surfaces generated from DEM are reported in two ways: i) comparing experimental data and DEM at a given relative density (of samples previously compacted up to 35 MPa), and ii) at a range of densities (RD=0.8-0.95). Finally, we presented an alternative way to estimate the yield isodensity surface using the same methodology applied in green numerical samples (chapter 5).

Introduction

The manufacturing of continuous steel casting refractories comprises the following steps: powder mixing, compaction, curing, and ring (chapter 2). Thus, modeling the processing steps can be helpful for the computer-aided design of new materials. To this aim, in this Ph.D., the discrete element modeling was divided into two parts: 1) the densication/compaction behavior study of four dierent composites (chapter 5), and 2) the analyzes of the mechanical response of the standard composite (mix 1) after the virtual ring (this chapter).

To our knowledge, the discrete element modeling of red refractory composites has not yet been reported. Some authors [START_REF] Olmos | Sintering of mixtures of powders: Experiments and modelling[END_REF][START_REF] Martin | Study of the sintering kinetics of bimodal powders. A parametric DEM study[END_REF] performed DEM simulations for the sintering modeling of powders considering diusion mechanisms. However, in our case, there is no thermally induced microstructure transformation. The virtual ring in our work was represented by the addition of elastic bonds to all particles in contact. Because part of the binder is burn-out during the ring (chapter 3), the matrix in DEM is represented by ne alumina particles linked by bonds. Successively, to reproduce the binder 'bell-eect' (chapter 3), pairs of bonds are broken between ne alumina particles and ne aluminagraphite particles.

Firing introduces defects (microcracks and decohesions) to alumina-carbon composites, thus degrading their mechanical properties, compared with curing, for example, where there is still some plasticity associated with the binder. These refractory composites are quasi-brittle materials submitted to extreme working temperatures, leading to thermal shock stresses and crack propagation. The present model enables a good representation of the mechanical behavior of the standard powder composite (Mix 1), mainly when observing the stress-strain curves from the triaxial compression tests, bestowing some condence to the model.

Material properties

The triaxial and tensile simulations are carried out on red microstructures by adopting the same DEM input material parameters (chapter 5) as for the green stage (E for the material Young's modulus, µ for the particles friction coecient, and ν for the Poisson's ratio), except for Young's modulus of the matrix that was derived from tting experimental triaxial tests. Since the particles are linked by elastic bonds (except the contacts between ne alumina particles and ne alumina-graphite particles because we have imposed the fracture of these bonds), the bond strength input parameter plays an important role in the macroscopic stresses. Thus, the bond strength in tension (σ N ) and shear (σ T ) 6.4. Mechanical simulations were deduced by tting to the experimental triaxial compression data (Table 6 The qualitative impact of those DEM bond strength values is presented in Table 6.2. The objective is to highlight the eect of increasing bond strength in shear (σ T ) and tension (σ N ) on the ultimate axial stress (or maximum stress) when applying dierent radial stresses. For this purpose, several triaxial tests were performed at several radial stresses and modifying the bond strength. First, the σ N remained the same, and the σ T is increased by 100 MPa. Afterward, σ T remained the same, and only σ N was also increased by 100 MPa. Table 6.2 exhibits the increase of the composite strength (in percentage) when increasing the bond strength in shear or in tension. The bond strength in shear has more impact at larger radial stresses (10-60 MPa).

Radial stress (σ

x = σ y ) (MPa) σ N (MPa) ⇑ σ T (MPa) ⇑ 0 ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ (+2.2%) ↑ ↑ ↑ ↑ ↑ ↑ ↑ (+2.0%) 10 ↑ ↑ ↑ ↑ (+1.7%) ↑ ↑ ↑ ↑ ↑ ↑ (+3.4%) 30 ↑ ↑ (1.6%) ↑ ↑ ↑ ↑ ↑ (+4.7%) 60 ↑ (+1.0%) ↑ ↑ ↑ (+3.8%)
Table 6.2: Triaxial test simulations: example of the qualitative impact of the input DEM bond strength in shear (σ T ) and in tension (σ N ) on the maximum macroscopic axial stress (σ z ) depending on the radial stress applied (σ x = σ y =0, 10, 30, and 60 MPa).

σ N was xed at 78 MPa, and σ T =508 MPa (Table 6.1). Although σ T σ N is not standard, this is the best t with triaxial experimental stress-strain curves (section 6.4.1). The shearing and tensile fracture contacts at the bond level highly depend on the radial stresses and the applied bond strength.

Mechanical simulations 6.4.1 Triaxial compression tests

Triaxial compression test simulations on the reference mix 1 are performed after the virtual ring. Similar to the green simulations, those simulations are carried out by controlling the axial strain rate in the z-direction (in compression) with the radial stresses (σ x = σ y ), also MATERIALS in compression). Thus, the axial stress increases with the radial stress at the beginning of the simulation up to the maximum radial stress, remaining constant. Only the axial stress is increased until a maximum value is reached (fracture point).

As a reminder, the compacted and red samples were used on experimental triaxial compression tests at several conning pressure (or radial stress), varying from 0 to 60 MPa (chapter 3). Stress-strain curves were compared with DEM triaxial test simulations to evaluate the capacity of DEM to represent the mechanical response of such powder composites after the ring stage. The numerical samples were previously compacted by closed-die conguration, unloaded, and numerically red (addition of bonds in all particles into contact and further bond breakage between pairs of particles-see chapter 4). To mimic the experimental tests, a pre-loading test was carried out before the triaxial tests. For this purpose, the samples were isostatically compacted up to the condition: σ x = σ y = σ z =5.4 MPa. During this isostatic compaction, the packing is slightly densied (density increase by ≈ 0.3%). Also, a few bonds are broken, but their number is very small (≈1.3% of new broken bonds). The pre-loading does not cause any noticeable change in the composite mechanical behavior.

Stress-strain curves comparing numerical and experimental tests at radial stresses varying from 0 MPa to 60 MPa are shown in Fig. 6.1. The numerical and actual samples exhibit two characteristic behaviora rst stage roughly linear, followed by a slight transition to a non-linear curve. The increase of the radial stress leads to a larger sample deformation due to the rearrangement of particles. Indeed, this increase of the deformation is accompanied by an increased composite densication (RD increase). The stress-strain curves from triaxial DEM simulations and experiments showed a good agreement giving some condence in the ability of DEM to represent the macroscopic mechanical behavior of the reference powder composite (mix 1) after ring. Recall that the numerical microstructures used for triaxial tests in the red state all come from the DEM simulations of closed-die compaction and unloading. Fig. 6.1 shows that at small radial stresses (0-10MPa) the agreement between DEM and experiments is quite satisfactory. Both curves show a bell shape with maximum stress. However, at larger radial stresses (>15MPa), the agreement is not so good. This might be because we had to chose a larger shear strength (σ T =508MPa) so that DEM an experimental stress-strain curves could t. DEM clearly shows a maximum stress, but still too much bonds are breaking in tension and in shear (Fig. 6.2). This might be an indication that our bond breaking model (Eqs. (4.9) and (4.10), chapter 4) might be too simplistic. Other fracture criterion may lead to a better agreement (e.g., Rankine criteria) [Hamelin et al., 2021].

The evolution of the number of broken bonds in shear (N rupt T ) and in tension (N rupt N ) Figure 6.1: Axial stress-strain curves from triaxial compression tests: comparison between experiments and DEM simulations (radial stress from 0 to 60 MPa).

versus the packing relative density (RD) is shown in Fig. 6.2. It also highlights those numbers at several triaxial compression tests with a variation in the radial stresses (σx = σy=0 → 120 MPa). The initial total number of non-broken bonds is ≈ 315,000 and broken bonds ≈ 50,000 (before triaxial tests).

The number of broken bonds during the pre-loading cannot be visualized in this scale because the number of broken bonds is approximately 10 3 smaller than the other simulations. Since the bond strength in tension is much smaller than in shear (σ N =78 MPa, σ T =508 MPa), the samples present more fractured bonds in tension than in shear. In Fig. 6.2, the curves in graphs a) and b) are plotted up to the point where the packing has reached the maximum axial stress.

As shown in Fig. 6.2 a), there is not a signicant dierence in the number of broken bonds in tension for radial stresses varying between 0 and 120 MPa, with minimum values around number=40,000 (at 0 MPa) and maximum of number=88,000 (at 120 MPa). In MATERIALS contrast, the number of broken bonds in shear signicantly increases with the radial stress (Fig. 6.2 b) even if applying a larger bond strength in shear (σ T =508 MPa). The increase of the radial stresses introduces more shearing at the particle contact. At very large radial stress (120 MPa), the number of broken bonds in tension and shear are not so dissimilar from each other (N rupt N ≈ 88,000, and N rupt T ≈ 65,000). Whereas at 0 MPa, N rupt N ≈ 43,000, and N rupt T ≈ 1,900.

In other words, Figs. 6.2 a) and b) indicate that at small radial stress (0MPa), a small number of total broken bonds (shear plus tension) is sucient to trigger fracture at the macroscopic scale. In contrast, when the radial stress is very large (120MPa), a large number of broken bonds is necessary to get the fracture.

In addition, the evolution of the composite relative density (RD) may present three conditions during triaxial compression simulations. The RD may i) decrease, ii) increase and further decrease, iii) continuously increase. The type of RD condition will depend on the magnitude of the applied radial stress. For σ T =0, for example, the composite is dilating (decrease of RD). By increasing the radial stress, the composite tends to densify (RD increase) more easily. andb) triaxial test simulations in samples, with a radial stress between 0 and 90 MPa, at constant macroscopic input parameters: number of broken bonds (in tension and in shear) against the packing relative density (target: maximum axial stress σ z = σ ultimate ). The initial total number of non-broken bonds is ≈ 315,000 and broken bonds ≈ 50,000.

Examples of triaxial compression DEM simulations with radial stresses at 0 MPa and 30 MPa are shown in Fig. 6.3. The microstructure representation of the total number of broken bonds (N rupt N +N rupt T ) per particle at the maximum axial stress (σ z ) are shown.

Mechanical simulations

The graphs are plotted as stress-strain plots with the evolution of the total number of broken bonds. The number of broken bonds increases with the axial stress, even after the maximum axial stress point has been reached. At the maximum stress, the total number of broken bonds at 0 MPa and 30 MPa is approximately 45,000 and 89,000, which are coming mostly from contacts in tension, as previously seen in Fig. 6.2. The evolution of the ultimate and yield stress against the radial stress (or conning pressure) of experimental and DEM triaxial compression is shown in Figs. 6.4 a) and b). As previously seen in chapter 3, the yield strength from experimental triaxial tests is dened by the 0.02% oset strain from strain-stress curves. Therefore, this same method CHAPTER 6. DEM SIMULATIONS ON FIRED POWDER REFRACTORY MATERIALS was also applied for the strain-stress curves from numerical triaxial simulations to compare numerical and experimental data. In Fig. 6.4 c) two examples of stress-strain curves from a triaxial experimental test and a numerical for radial stresses equivalent to 30 MPa are plotted, and the yield strength points are highlighted. However, this is, of course, an assumption. Except for the ultimate (or maximum) stress, it is impossible to precisely determine the material yielding point due to its mechanical response and testing complexity. Nonetheless, the ultimate and yield points from numerical triaxial simulations exhibit a good agreement with experiments.

Tensile tests

Tensile test simulations were applied on mix 1 virtual red samples previously compacted uniaxially at 10, 35, ≈ 70 MPa (RD=100%) and unloaded. These tensile simulations were carried out using the same material properties from Table 6.1. A uniaxial tensile strain rate is applied in the z -direction ( εz > 0), and null stresses in x and y directions (σ x = σ y =0) until sample's failure (chapter 3). The tensile strength (maximum axial stress) from DEM simulations slightly increases together with the sample density and varies between 1.18-1.72 MPa (Fig. 6.5 a). Comparing the tensile strength from numerical and real samples compacted at 35 MPa, those values are 1.39 ±0.14 (DEM) and 3.32±0.16 (experiment). Thus, the tensile strength from DEM underestimates the strength from experimental Brazilian tests. This dierence is likely linked to the more complex stress state (with compressive stress components) in the Brazilian test as compared to the purely tensile test in DEM. Also, the boundary conditions in the Brazilian tests are quite dierent from the periodic boundary conditions used in DEM.

The objective of Figs.6.5 c) and d) is to study the evolution of the bonds damage during the tensile simulation. Fig. 6.5 c) represents the initial state of broken bonds. Pairs of bonds were broken to mimic the voids introduced in the ring process (chapter 4). The two gures in sequence (Fig. 6.5 d) give the additional bonds damage (not showing the initial one) evolution during the tensile test when: 1) the sample has reached the maximum stress, and 2) after sample fracture. Fig. 6.5 e) are zoomed pictures from the previous image to highlight the crack path. It is possible to observe that the cracks develop mostly where the initial damage was located (around coarse alumina clusters). Some graphite bonded particles that are close to the coarse alumina clusters also break.

The same problem was observed in [START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF] where the tensile strength from DEM tensile simulations and Brazilian experimental tests were compared using aggregate powder ceramic materials (DEM strength was underestimating experimental values). In Pizette et al., a contact fracture stress parameter (σ f ) was used to describe the cohesion between the particles. In our work, the bond strength is the parameter that dictates the tensile strength in DEM. MATERIALS

Generation of the yield and ultimate surfaces

The virtual red samples of the reference composite (mix 1) are submitted to a series of mechanical test simulations, as seen in the previous section 6.4. In this section, meandeviatoric surfaces are generated from the triaxial compression and tensile test simulations. The objective is to evaluate the material fracture-yield surfaces shape and compare it with experimental data.

These surfaces are reported as functions of mean P = -1 3 (σ x +σ y +σ z )) and deviatoric (Q = |σ z -(1/2(σ x + σ y ))|) stresses for the triaxial tests (DEM and experiment) and uniaxial tensile simulations (DEM). Concerning the Brazilian tests, P = (2σ z )/3 and [START_REF] Martin | Cohesion and dilatation of powder compacts containing hard phase particles under highly deviatoric stress states[END_REF]). The P-Q stresses from triaxial tests are plotted in the positive quadrant. Fig. 6.6 shows the ultimate and yield surfaces (P-Q) comparison between DEM simulations and experimental data (Mix 1). In DEM, the ultimate surface reefers to the maximum axial stress (σ z ) from the triaxial stress-strain curves (Fig. 6.1 and from tensile 6.5. Generation of the yield and ultimate surfaces stress-strain curves (Fig. 6.5 b). The deviatoric stress above 60 MPa of radial stress (σ x = σ y ) is almost constant as the dierence between the maximum axial stress and the radial stress is ≈ 72-79 MPa. Thus, the deviatoric stress at the ultimate surface is never null (Fig. 6.6 a). The DEM ultimate surface is plotted up to a triaxial simulation with σ x = σ y =120 MPa. Concerning the yield surface (Fig. 6.6 b), the determination of the yield stress from those stress-strain curves is not trivial.

Q = √ 13σ z (

Additional method for the generation of fracture and isodensity yield surfaces

We have seen in the previous section that estimating the yield stress is dicult. Because it is not possible to predict with certainty the material yield point, an alternative way to estimate the yield material surface is proposed (Fig. 6.8). In this example, the yield contour is represented by the connection between isodensity points (RD=95.8%), i.e., when the sample is at the same density from the end of compaction. With this method, the point with the largest value of mean stress (P) is characterized by a zero deviatoric stress (σ x = σ y = σ z ). The fracture points are considered as the maximum axial stress point together with the sample relative density decrease. Similar to chapter 5, by increasing the radial stress, the sample may experience fracture or densication. In other words, the transition between the fracture and isodensity surface is when the sample densies (increase in RD). This same methodology was applied in chapter 5 for generating fracture and yield surfaces for the green stage. Three conditions may apply during triaxial simulation tests, which depend on the applied radial stresses. First, the packing may fracture (detected from the maximum axial stress together with a sample RD decrease). Second, by increasing the radial stresses (σ x = σ y =30 MPa), the packing may rst densify and further fracture (RD decrease together with the maximum stress), as shown in Fig. 6.8 b). Or, third, only densied (RD increase). This last condition meets mostly on simulations with higher radial stresses (σ x = σ y >80 MPa), as shown in Fig. 6.8 c). These two stresses (fracture and isodensity 6.6. Additional method for the generation of fracture and isodensity yield surfaces points) are represented by the number 1) and 2) in the mean-deviatoric stress plane Fig.

a).

The comparison between two methods used to generate the DEM surfaces in the mean vs. deviatoric stresses is shown in Fig. 6.9. The fracture points coincide with the ultimate surface (Fig. 6.9). Contrary to the yield surface, increasing the radial stress, we observe a transition from the fracture to the isodensity surface (sample densication) until Q=0.

However, it is dicult to fully validate this methodology since no work has been published using this method for red samples yet. The fracture-isodensity contour resembles somewhat the yield/damage curve shape from the model proposed by Bigoni and Piccolroaz model [START_REF] Bigoni | Yield criteria for quasibrittle and frictional materials[END_REF] which presents mostly a convex shape, dierently from the green isodensity contour that has a concave shape (chapter 5). MATERIALS Figure 6.9: Comparison between the two methods applied to generate surfaces from DEM simulations in the mean vs. deviatoric stress plane: a) yield and ultimate surfaces, and b) fracture and isodensity surfaces.

Conclusions

In this chapter, an original approach to investigate the potential of DEM for simulating the mechanical behavior of virtual red reference refractory composite (mix 1) was presented. The numerical microstructures tested in this chapter originated from the compacted samples investigated in the preceding chapter 5. We decided to focus on the DEM ring evaluation only for the reference composite to demonstrate the potentiality of the method. The same approach could, of course, be used for the other mixes.

First, the qualitative impact of the bond strength on the composite macroscopic behavior through triaxial compression simulations was presented in section 6.3. The t between experiments and simulations showed that larger bond strength is necessary for shear compared to the bond strength in tension. The consequence is that the number of broken bonds in tension is larger than in shear. Increasing the radial stresses in x and y directions introduces a larger shearing at the particle contacts. Thus, the number of broken bonds in shear is signicantly increased with the rise of the radial stresses. However, the change in the radial stresses seemed to not signicantly aects the number of broken bonds in shear, which remained approximately the same in all simulations.

It is not clear why σ T needs to be signicantly larger than σ N . Still, with this choice of bond strength, stress-strain curves from triaxial compression numerical simulations were compared with experimental data showing a good agreement. This gives some con-dence in the ability of DEM to model the mechanical behavior of after-red samples. The stress-strain curves exhibit two characteristic behaviors. The rst is a roughly linear shape. The second is a transition to pseudo-plastic behavior due to the rearrangement of particles (the model does not allow plastic deformation at the ring). Larger radial stresses lead to larger deformation as a consequence of the sample densication. Regarding the tensile numerical simulations, the DEM ultimate axial stress (or fracture point) underestimates the experimental data points. This dierence in the stress is due to the complex loading conditions applied in the Brazilian test (experiment) with some compression stress components that are not rendered by a simple uniaxial DEM tensile simulation.

The ultimate and the yield surfaces (P-Q) were generated from triaxial compression and uniaxial tensile DEM simulations. The ultimate surface is referent to the connection of maximum axial stress points and the yield surface from yield axial stress points (i.e., oset 0.02% strain). Similar to experiments, the ultimate and yield surfaces never reach a condition where the deviatoric stress equals zero (Q=0). The yield function represents a single, convex and smooth surface in stress space approaching as limit situations wellknown [START_REF] Bigoni | Yield criteria for quasibrittle and frictional materials[END_REF]. We presented an alternative way to estimate the yield surface contour so that Q=0. This methodology was applied in chapter 5 by connecting isodensity points (RD at the nal of the compaction). No previous work has been shown the ability of DEM to connect isodensity points to represent the yield locus for refractory composite materials after the ring yet. The surfaces may be further compared with the yield/damage model (Appendix E), introduced by Bigoni and Piccolroaz (BP) [START_REF] Bigoni | Yield criteria for quasibrittle and frictional materials[END_REF][START_REF] Piccolroaz | An elastoplastic framework for granular materials becoming cohesive through mechanical densication. Part i -Small strain formulation[END_REF]] [Argani et al., 2016]. The BP model is used by VESUVIUS to simulate the thermo-mechanical behavior of powder composites using a nite element code.

Introduction

Carbon-bonded alumina refractories have been widely used as functional refractories such as nozzles, sliding gate plates, and stoppers in the continuous casting process of steelmaking. These industrial parts are shaped by the powder compaction process uniaxially or isostatically. The compaction may induce a density gradient in the nal part leading to thermo-mechanical properties dierences along with the piece and consequently to cracking phenomena and material damage. Therefore, better homogeneity should be achieved by optimizing the part geometry. Understanding powder compaction behavior is of practical importance to improve product development eciency and manufacturing performance.

As a recall, the green characterization in this Ph.D. was divided into three parts: experimental testing, DEM, and FEM simulations (Fig. 7.1). First, experimental closed-die compaction and triaxial/uniaxial compression tests were performed on refractory composites (chapter 3). A DEM model was used to describe the compaction behavior and mechanical behavior. These DEM simulations were calibrated and compared with experimental data. In other words, the green DEM composite (after compaction) was submitted to a series of mechanical simulations (tensile, uniaxial/triaxial compression tests) to generate the isodensity and the fracture surfaces of green numerical samples (mix 1). These surfaces can be tted with the DruckerPrager Cap model (DPC), which is currently commonly used for modeling compaction of metal powders [START_REF] Almanstötter | A modied Drucker-Prager Cap model for nite element simulation of doped tungsten powder compaction[END_REF][START_REF] Zhou | A density-dependent modied Drucker-Prager Cap model for die compaction of Ag57.6-Cu22.4-Sn10-In10 mixed metal powders[END_REF], ceramic powders [START_REF] Abdullah | Finite element simulation of alumina ceramic powder compaction[END_REF] and for application to pharmaceutical powders [START_REF] Michrafy | Modelling the compaction behaviour of powders: Application to pharmaceutical powders[END_REF][START_REF] Michrafy | Wall friction in the compaction of pharmaceutical powders: Measurement and eect on the density distribution[END_REF][START_REF] Kadiri | Pharmaceutical powders compaction: Experimental and numerical analysis of the density distribution[END_REF]] [Han et al., 2008]. The DPC model parameters are calculated and presented in the section 7.3. The advantage of DEM is that these surfaces are dicult to obtain from experiments due to the complex mechanical behavior of the green composite. Also, if an experimental program is able to provide these surfaces, it needs to be repeated for each new powder. These parameters are implemented in ABAQUS (FEM commercial software) with density dependant evolution to simulate the composite compaction response in closed-die conguration. The FEM results are then compared with experimental data. The objective is to calibrate some properties not identied in the closed-die compaction experiment (mix Poisson's ratio and friction coecient) and to validate the DPC parameters (section 7.4).

Parameters identication (DPC model) from DEM simulations

The Drucker-Prager Cap model (DPC) is commonly used to model the elasto-plastic behavior of powders, assuming the material is porous, compressible, and isotropic. The model describes the yielding of the material as a function of the hydrostatic stress (P) and the Von Mises (Q) equivalent stress. The DPC model is composed of a shear failure surface and a cap. The isodensity and fracture points generated from DEM simulations and plotted in the P vs. Q plane (chapter 5) are tted with a line and a convex contour to identify the DPC model parameters. As a recall, the discrete packings were compacted by closed-die, unloaded, and reloaded (triaxial compressive and tensile test simulations). These reloading points were plotted in the P vs. Q plane, forming the fracture and the isodensity points (chapter 5). The fracture DEM points are tted with the shear failure In addition to the DPC cap model parameters, the FEM simulation needs Young's modulus and a Poisson's ratio. The powder Young's modulus is estimated from DEM uniaxial compression (crush) tests. These tests were carried out on compacted Mix 1 samples at dierent relative densities (0.7-1). Concerning the Poisson's ratio, we will see in the next section how we calibrate this value for the powder.

Closed-die compaction conguration (lab scale)

The material parameters from the Drucker Prager Cap model, identied in the previous section (Fig. 7.4), are used in ABAQUS to simulate the compaction of the reference refractory composite (mix 1) in a closed-die conguration. The compaction is modeled in a 2D axisymmetric model. Linear quad elements (CAX4R) with increasing mesh density are employed in the mix part (Fig. 7.5 b) and a more coarse mesh in the rest of the part (punch and die). The main objectives of this simulation using the nite element method are rst to calibrate complementary parameters not identied by experiments (mix Poisson's ratio and friction coecient). Second, to validate the DPC model parameters previously calculated from DEM mechanical tests.

Punch and die are represented by an alloy steel 42CrMo4 (see properties in Table 7.1). The major part of the powder material properties is referent to the DPC model (section 7.3). The powder Young's modulus was calculated from DEM uniaxial compression tests of compacted numerical samples at dierent densities (Fig. 7.4 f). The powder Poisson's ratio and the friction coecient are calibrated and represented as best t properties (compared with experimental data). The friction coecient at the mix/die interface is equal to 0.1. The Poisson's ratio increases with the powder relative density (0.38-0.39). The lling mix is compacted by the upper punch with a displacement boundary condition starting from a height of 60 mm. Similar to the compaction experiment, only the top punch moves (single action pressing) . The diameter of the punch is 30mm. A userdened eld is used (USDLFD), which is updated each time step using the volumetric plastic strain (ε pl vol ), to include the dependency of DPC parameters with the evolution of the relative density (RD). The volumetric plastic strain is equal to ln( RD RD 0 ) (Eq. (2.8), chapter 2). RD is the nal relative density, and RD 0 is the initial (RD 0 =0.6). The user-dened subroutine USDLFD is shown in the Appendix D.

Density distribution

The resulting distribution of relative density after compaction at a compaction pressure of 35MPa is shown in Fig. 7.7 a). The compaction pressure shown in the curve is the average contact pressure (CPRESS) at the interface mix/upper punch. CPRESS is a variable eld output request for general contact in ABAQUS/Explicit. This variable is calculated as the magnitude of the net contact normal force (the CNORMF vector) per unit area (it is an unsigned value) [Abaqus, 2006]. The contact pressure from ABAQUS is the stress that is the most similar to the compaction pressure measured in the experimental closed-die compaction. The density distribution map exhibits a larger density at the top corner and the lowest at the bottom corner because of the displacement of the upper punch. This density gradient is more signicant at the interface mix/die because of the friction coecient. The maximum and minimum relative density, at 35 MPa, are ≈ 99% (top) and 94% (bottom). The heterogeneous density map obtained from single action die compaction due to the friction with the die has been observed experimentally [START_REF] Yanai | Characterization of ceramic powder compacts[END_REF][START_REF] Fruhstorfer | Inuence of particle size distributions on the density and density gradients in uniaxial compacts[END_REF], and using numerical simulations [START_REF] Bibliography Kim | Densication behavior of ceramic powder under cold compaction[END_REF] [?] [START_REF] Wu | Modelling the mechanical behaviour of pharmaceutical powders during compaction[END_REF]. The density map that we obtain in Fig. 7.7 a) is in good qualitative agreement with these studies. In particular, we reproduce the eect of the friction between the powder and the die at the corners of the sample. The compaction pressure evolution versus relative density curves of closed-die compaction simulations (DEM and FEM) and experiment is presented in Fig. 7.7 b). A good qualitative and quantitative agreement between the experimental and FEM curves is observed. The fact that the DEM curve is very similar to the FEM one is encouraging. However, care should be taken when comparing DEM and FEM compaction simulations as the boundary conditions are quite dissimilar. In DEM, we consider a fully periodic condition, and the macroscopic stresses are calculated from Love's formulation (Eqn. (4.13), chapter 4). In FEM, there is strong interaction between the powder and the die/punch (coecient of friction). Also, similar to the experiments, in FEM, the motion is given only by the top punch (in the y-axis). In DEM, both the top and bottom sides of the in Figs. 7.9 a) and b). Maximum stresses are highlighted in the top corner (node 1) because this region is over-compacted (RD ≈ 99%). Node 1 is the most critical point because it is the one closest to the failure line and during the unloading it might appear some tensile stresses, leading to material failure. The lowest stresses are at the bottom corner in the under-pressed region (RD ≈ 94%). The non-uniform stress distribution is exhibited mostly in the interface mix/die due to the friction coecient. Fig. 7.9 c) shows the DPC surfaces and the loading path performed during closeddie simulations (DEM and FEM) until a compaction pressure of 35 MPa. Also, the unloading path of the DEM simulation is exhibited. The curves (1), (2), and (3) are correlated to the tagged nodes in three dierent regions of the powder. To note, these nodes are the same from the highlighted in the previous Fig. 7.8 b). The Von Mises stress (Q) and hydrostatic pressure (P ) evolution with the relative density, during the compaction simulation, referent to each node is extracted. The maximum (1), middle (3), and minimum (2) nodes correspond to a relative density of ≈ 99%, 97%, and 94%, respectively, at a compaction pressure of 35MPa (Fig. 7.7 a). Indeed, the nodes exhibit a Q of between 6.3MPa and 19MPa, and a P of around 14MPa and 30MPa.

The stresses dierences between the FEM simulations and DEM come from the axial and radial stresses. The dierence between the axial and radial stresses in FEM are larger than in DEM, increasing the Mises stress during the FEM compaction simulations.

The triangles highlighted in Fig. 7.9 c) represent the points (P, Q) at a compaction pressure of 35MPa (CPRESS in FEM and σ z in DEM). It is observed a reasonable good accordance of relative density with the DPC surfaces. For example, the RD at 35MPa (node 1) is equal to ≈ 99% and is between the density surfaces of 98% and 100%. The same was observed for nodes (2) and (3).

Conclusions

In this chapter, we presented a rst attempt of modeling the powder composite (mix 1) closed-die compaction using a nite element code (ABAQUS), applying the Drucker-Prager Cap (DPC) model.

For this purpose, rst, the DEM fracture and isodensity points plotted in the P vs. Q plane (chapter 5) are tted with a shear failure surface and a cap. This tting allowed the identication of the DPC model parameters. The shape of the failure surface and the cap are very sensitive to the parameters. The shape of the cap depends on the yield stress, cohesion, evolution pressure, cap eccentricity, and angle of friction. In contrast, the shear failure surface depends on the cohesion and the angle of friction.

Second, the FEM closed-die compaction simulation uses the DPC model to dene the powder material properties. The compaction pressure evolution against the relative density of the FEM numerical simulation and experiment has shown a good agreement. Also, the DEM curve has shown a very similar behavior compared to FEM and experiment, which is encouraging. However, it is important to mention that care must be taken when comparing DEM and FEM compaction simulations since the boundary conditions are very dierent. In DEM, the model uses a fully periodic condition that does not take into account the real die geometry nor the frictional eects between the powder and the die. In FEM, there is an interaction between the powder and the die/punch (coecient of friction). Also, similar to the experiments, in FEM, the motion is given only by the top punch (in the y-axis). In DEM, both the top and bottom sides of the periodic cell move (in the z-axis). The density distribution map was more signicant in the interface powder/die because of the friction coecient. A maximum density was exhibited in the top right corner and a lower density in the bottom corner.

Finally, the Von Mises (Q) and hydrostatic (P) stresses distribution in the die are exhibited. Also, the loading path during the numerical closed-die compaction simulations (DEM and FEM) in the P vs. Q plane are plotted and compared. The Von Mises stress from DEM is smaller than from FEM simulations because the axial and radial stresses in DEM are larger. Consequently, the dierence between the axial and radial stresses in FEM is larger. For more accurate analyzes, additional experimental closed-die compaction may be carried out measuring the radial stress in the die (e.g., using strain gages). So, the P vs. Q stresses from experiments may also be plotted and compared with numerical simulations. This chapter should be seen as a proof of concept of the methodology that can be used in an industrial setting to couple experiments, DEM and FEM. The methodology was exemplied here on Mix 1 and a simplistic cylindrical geometry. The advantage of FEM is that once the constitutive equation is validated, it can vary the geometry of the tools and their kinematics to come as close as possible to the real industrial conditions. An unloading sequence might be interesting to analyze where the defects might appear.

Conclusions

Carbon-bonded alumina refractories are very heterogeneous composites made of a mixture between a soft binder and hard particles (alumina and graphite). Understanding the relationship between the microstructure, compaction behavior, and mechanical properties is essential to developing new refractory products and optimizing new powder compositions.

The literature review presented in chapter 2 provided an insight into the current understanding of the compaction behavior, and the mechanical and thermo-mechanical properties of ceramic powders and refractory composites, combining both numerical and experimental techniques. The numerical simulations were described using the Discrete Element Method (DEM) and the Finite Element Method (FEM). DEM is a powerful tool that can link the microstructure of powder materials with their compaction behavior and mechanical properties. However, the use of DEM to model refractory materials is quite recent [START_REF] Moreira | Discrete element modeling A promising method for refractory microstructure design[END_REF] and [START_REF] Ramírez-Aragón | Numerical modeling for simulation of compaction of refractory materials for secondary steelmaking[END_REF]). As far as we know, no work has been published yet concerning the modeling of commercial refractory composites, such as carbon-bonded alumina, using DEM simulations.

This thesis aimed at modeling the compaction behavior and the mechanical properties of VESUVIUS carbon-bonded alumina composites using numerical simulations (chapters 4 to 7) coupled with experiments (chapter 3). The numerical modeling was divided into a micro-scale using DEM (considering the particulate matter of the composite powder), presented in chapters 4 to 6. And, on the macro-scale, using data generated by DEM to simulate the closed-die compaction considering the powder composite as continuum matter using a nite element code (chapter 7).

In this Ph.D. work, an original approach was developed using the discrete numerical model based on a Representative Volume Element (RVE) to represent the microstructure of carbon-bonded alumina refractory compositions. Seeking simplicity, we have chosen to represent coarse alumina as spherical in DEM. The graphite in DEM was represented by rectangular geometry but with no shape variation. The real composite microstructure is characterized by irregular coarse alumina grains and graphite pellets embedded into a matrix (ne alumina grains and binder). Four compositions were studied with variations on the volume fraction of each component. The composite in DEM was modeled by a mixture of single ne particles and bonded particle clusters (coarse alumina and graphite). Two contact models were used and described in chapter 4. We have also shown the importance of carefully choosing the number of discrete elements to ensure a good compromise between CPU time and model accuracy.

Coarse and ne alumina hard particles were coated by an elastoplastic soft binder coreshell on DEM (chapter 4) at the compaction and green stage (Fig. 8.1 a). The aim of this model was to represent in a simplied manner the presence of a soft shell covering hard particles. The number of particles was calculated based on the volume fraction of each component (ne/coarse alumina particles, graphite, and binder) on the composition. We have shown that the macroscopic compaction behavior of the composite depends mostly on the binder content.

In other words, the macroscopic stress needed to attain a given relative density increases with a decreasing binder shell thickness (related with its volume fraction) and binder stiness (Fig. 8.1 b). During the unloading (after compaction), the soft binder shell in DEM is responsible for the composite hysteresis eect, which increases with the contacts in the soft phase (chapter 5). The number of contacts in the soft or hard phases depends on these two mechanisms: binder thickness and stiness. In other words, the binder results in the introduction of plasticity to the composite during the unloading. The graphite deforms during the compaction (bonds are broken), also contributing to the composite plastic response. Similar to the graphite, the coarse alumina was modeled as bonded particles cluster; however, they do not deform (bonds are never broken). ), the stages I and II referent to the grains rearrangement and plastic deformation, characteristic of general powder ceramics compaction behavior (chapter 2), may be identied in both experiments and DEM simulations. However, stage III (fracture) could not be identied. In some cases, it might be dicult to determine the transition between stages II and III. In addition, the beginning of the compaction curves from numerical and experimental tests exhibit a dierence in terms of initial relative 8.1. Conclusions density. This dierence is explained since, in DEM, the packing has started to compact at 60%. Instead, the compaction experiment has an initial density of approximately 53%. Still, the overall densication behavior from DEM simulations shows a good agreement with experimental data. After compaction and unloading, the green numerical samples in our work were submitted to tensile, uniaxial/triaxial compression test DEM simulations. Uniaxial and triaxial compression test simulations and experiments stress vs. strain curves of the standard composite (mix 1) were compared and have shown a good qualitative and quantitative agreement. A methodology to generate isodensity yield surfaces and fracture lines was described by performing tensile and triaxial compression simulations (chapter 5). The advantage of this method is the possibility of probing points in the deviatoric and mean stress space. This is very dicult or impossible to be measured by experiments (chapter 3). These surfaces were tted with the elastoplastic Drucker-Prager Cap (DPC) model (chapter 7). In that sense, DEM simulations carried out on a periodic representative volume element may be seen as numerical experiments that replace, or more realistically complement, laboratory experiments.

The virtual ring stage in DEM was represented by ascribing bonds in all particles in contact from compacted numerical samples (chapter 6). The matrix was characterized by bonded ne alumina particles. Pairs of bonded particles were broken (between ne alumina and coarse alumina particles, and ne alumina and graphite particles (chapter 6) to represent the binder 'bell eect' eect related to the binder heating-cooling down (chapter 3). Triaxial experimental compression tests and simulations have shown a good qualitative and quantitative agreement. The tensile strength (maximum axial stress) from tensile simulations was underestimated compared with Brazilian experimental tests.

However, this dierence comes from the fact that complex loading conditions of Brazilian tests are not well rendered by simple uniaxial tensile simulations. The stress state in this test introduces both tensile and compressive components. The axial stresses from the DEM uniaxial tensile simulation are uniquely under tension. Still, observing the fracture (broken bonds propagation) of the numerical sample during the tensile simulation, it was possible to highlight that the cracks propagated mostly within the regions with a pre-existence of defects (fractured bonds to represent the binder 'bell eect').

Finally, we have shown in chapter 7 the potential of DEM to generate constitutive equations to feed a compaction simulation using the Finite Element Method (FEM). The axial stress vs. relative density of closed-die numerical simulations (FEM and DEM) and experiment exhibited a good quantitative and qualitative agreement. The DPC model may be used on more complex shapes to model a VESUVIUS industrial compaction process.

Future work

Several interesting paths for future work can be considered to complement the results on the compaction and mechanical properties of refractory composites, mainly using numerical simulations (DEM and FEM). Some guidance and suggestions are proposed in the following.

Applications using DEM

Combining X-ray tomography images with DEM First, the DEM model developed in our work could be extended to combining X-ray tomography or SEM images with DEM to obtain more realistic structures. As seen in this Ph.D. work, we did not consider the real shape of the actual raw materials (alumina grains and graphite pellets) and the porosity distribution. Still, we developed a simple way to represent in DEM these complex and heterogeneous microstructures, depending on their composition. The advantage of using more realistic microstructures would be to consider the eect of grain size distribution, tablets orientation/aspect ratio, and other structural features on the composite compaction behavior and mechanical properties considering the real shape of the raw materials and porosity distribution. Fig. 8.3 shows an attempt of combining X-ray tomography images (mix 1 composite) with DEM packing. In this example, the X-ray images were cut in cubes of 300x300x300 pixel-size and post-treated using a segmentation method to separate the dierent phases in gray tunes. In sequence, these images were combined with dp3D to generate the DEM packing. It is a very rough 8.2. Future work demonstration, but it exemplies the possibility of combining DEM with image analysis. This approach has been used in previous Ph.D. works on nacre-like materials ( [START_REF] Radi | Bioinspired materials : Optimization of the mechanical behavior using Discrete Element Method[END_REF]) and silica aerogel composites ([Hamelin, 2021]) combining DEM with EBSD or X-ray tomography.

Fracture behavior of virtually red refractory composites

The microscopic elastic material properties for particles and contacts for the virtually red samples were calibrated with experimental and numerical triaxial compression tests. Those elastic properties mainly included Young's modulus of the alumina, graphite, and binder, and the bond strength (in tension and shear), as seen in chapter 6. A supplementary approach would be to model qualitatively and quantitatively the composite fracture properties, including their stiness, strength, and toughness using appropriate contact models. Also, we saw in chapter 6 that the cracks propagate mainly due to the presence of defects. In other words, the cracks propagated in the regions where some bonds were previously broken. An alternative would be to evaluate the crack propagation on more realistic packings by considering the eect of particle size distribution and geometry. Fig. 8.4 shows the discrete numerical samples representing SiC ceramics during mechanical test simulations: (a) uniaxial compressive test, (b) Brazilian test, and (c) 3-point bending tests. The cracks are highlighted in red [START_REF] Tan | Discrete element method (DEM) modeling of fracture and damage in the machining process of polycrystalline SiC[END_REF].

Particle packing methodology

VESUVIUS group is constantly studying alternatives to achieve a composite microstructure that oers good mechanical properties. A possible avenue for optimisation is by considering the particle packing (before the compaction). [START_REF] Recarey | Advances in particle packing algorithms for generating the medium in the Discrete Element Method[END_REF], for example, presented a method for packing particles using the discrete element method combining the Monte Carlo technique. Particles' dimensions can follow any given distribution and are generated with Monte Carlo techniques. The packing algorithm provides that the growth direction can be outwards or inwards (Fig. 8.5), for example. The modeling of the packing of the particles could be extended to more realistic DEM composite microstructures by considering the irregular geometry of the raw materials and particles distribution to help VESUVIUS in the design of new compositions. Level set methods are currently under development at SIMaP to replace simple spherical discrete elements by arbitrary shapes.

Applications using FEM

Compaction simulation of the industrial part An additional path is to perform compaction simulations of industrial parts using the data generated by DEM simulations. The objective would be to help VESUVIUS to identify the regions over-pressed (higher RD) and under-pressed (smaller RD) regions and predict the nal compacted shape from the initial one to improve the near-net-shape and future material development. For this purpose, a rst attempt of simulating isostatic compaction of an industrial part using the industrial process set up was performed. The simulation is actually under development by a VESUVIUS colleague (Thy-Thuy-My), but some preliminary results are presented. As a reminder, the yield and fracture surfaces generated by DEM simulations were tted with the DPC model, and its parameters were calculated (chapter 7). The Drucker-Prager Cap (DPC) model parameters applied on FEA closed-die compaction in chapter 7 are also used for the reference composite (mix 1) in the isostatic compaction simulation. In this simulation, the powder is essentially lled inside a rubber bag, sealed, and subjected to a 35MPa all-round pressure to produce the compacted green ladle shroud component. The isostatic compaction model in ABAQUS is represented by an axisymmetric model composed of the mix, an elastomeric (rubber) bag, the mandrel, and the closure. The mold is modeled using a nite deformation hyperelastic formulation ( [START_REF] Henderson | Bag design in isostatic pressing[END_REF]). The closure and mandrel are represented as a rigid body. During the compaction, a uniform pressure of 35MPa is applied to the bag surface. Fig. 8.6 shows the initial shape (before compaction) of the industrial part at a density of RD=60% followed by the nal shape with a density gradient. More work needs to be performed, including a comparison of the dimensions of the numerical model and the actual industrial part after compaction and unloading. Also, an additional calibration of the DPC parameters might be necessary depending on the geometry and shape of the industrial part. Still, this example shows a possible path for simulating complex industrial powder processes at VESUVIUS: numerical (DEM) and laboratory experiments feed FEM constititive equations that are used to model real industrial parts up to the nal shape of the component.

Thermo-mechanical simulation of the compacted industrial part Concerning the FEM simulations approach, thermo-mechanical simulations could also be performed on compacted industrial parts. The compaction induces a density gradient in the nal part leading to thermo-mechanical property dierences along the part. Consequently, when under operation, this density gradient may lead the piece to cracks and damage. A rst attempt of simulating the casting, considering the operation parameters set-up, was carried out. The casting considers the elastoplastic properties of the composite (from experimental data) with temperature dependence. The VESUVIUS modeling 8.2. Future work team actually performs thermo-mechanical simulations on industrial parts coupled with a experimental characterization and the Bigoni-Piccolroaz elasto plastic model. However, these measurements may lead to an error. Also, in this case, the density gradient of the industrial part is not taken into account. Fig. 8.7 shows a rst attempt of modeling the casting simulation of the compacted industrial part (mix 1) comparing cases 1 (homogeneous density) and case 2 (density gradient). The gure highlights the accumulative plastic strain distribution at the end of the thermo-mechanical simulation (Fig. 8.7 b). In these preliminary results, it is possible to observe a dierence in the distribution of the accumulative plastic strain between both cases. This indicates that not only the part geometry impacts on the nal material properties, but also the density distribution.

  Contexte et objectifs : modélisation des composites à base de poudre réfractaire 1.1 Contexte et objectifs : modélisation des composites à base de poudre réfractaire

Figure 1

 1 Figure 1.1: a) Exemples d'opérations de coulée continue. b) Exemple de produits réfractaires VESUVIUS utilisés pour la coulée continue.

Figure 1 . 2 :

 12 Figure 1.2: Représentation du contexte autour de ce travail de thèse. Utilisation de la modélisation des matériaux en poudre réfractaire et motivations.

  Dans le chapitre 3, la méthodologie expérimentale et les résultats sont décrits. Tout d'abord, la microstructure d'un composite lié au carbone est présentée à partir d'une analyse par MEB et tomographie à rayons X. Le comportement de compactage, les propriétés mécaniques et thermomécaniques des échantillons verts et après cuisson sont également CHAPTER 1. INTRODUCTION présentés.

  et propriétés mécaniques à vert ; Propriétés thermo-mécaniques des échantillons après cuisson.

Chapitre 4 .

 4 DEM : développement et description du modèle Modélisation en DEM des réfractaires composites à base d'alumine liée au cabon ; Lois de contact ; Représentation du liant, de l'alumine et du graphite sur le DEM ; Modèle à vert = cuisson numérique. Chapitre 5. Simulations DEM sur les matériaux réfractaires en poudre Simulations de compactage et d'essais mécaniques ; Eet du liant à l'échelle sur la réponse macroscopique ; Comparaison avec les données expérimentales ; Génération de la surface de plasticité isodensité et de la surface de rupture. de la surface de plasticité et de la surface de rupture. Chapitre 7. Modélisation par éléments nis Calibration des surfaces avec le modèle DPC (identication des paramètres) ; Simulation de la compaction en matrice fermée avec le modèle DPC en utilisant un code d'éléments nis.

Figure 2 .

 2 Figure 2.3: a) A schematic representation of the steel casting process (from ladle to mold). Figure adapted from VESUVIUS website. b) and c) Refractories used in steel continuous casting. Figures are adapted from [Routschka & Muthnow, 2012].

2. 3 .

 3 Refractory ceramic materials tives, prepared specically for this PhD work (chapter 3). Although a model refractory, it represents quite well the typical refractory produced by VESUVIUS.

Figure 2 .

 2 Figure 2.4: A base material pyramid with the location of the refractory materials [Routschka & Muthnow, 2012].

Figure 2 .

 2 Figure 2.5: a) Microstructure of alumina carbon bonded refractory and b) containing Al-Si alloy as antioxidant (additives)[START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF].

Figure 2 . 6 :

 26 Figure 2.6: Schematic representation of refractory manufacturing. Raw materials are compacted following two routes (closed-die or isostatic congurations) depending on the nal application (e.g., slide gate or ladle shroud products). The compacted pieces are then red.

Figure 2 .

 2 Figure 2.7: Schematic representation of the closed-die compaction experiment. a) The composite powder is inserted in a rigid steel die. b) The upper punch moves inwards and the powder is compacted. c) Unloading (the compact is ejected from the die).

Figure 2 . 8 :

 28 Figure 2.8: Densication curve of general ceramic powders and the dierent stages of compaction. The gure is reproduced from [Haussonne et al., 2005].

Figure 2 .

 2 Figure 2.9: Typical curve from a closed-die compaction test (alumina ceramic powder and refractory composite): evolution of the axial stress against the relative density in a) linear and b) logarithmic scales.

Figure 2 .

 2 Figure 2.10: Closed-die compaction experiments carried out at dierent strain rates (s -1 ) and displacement rates of a Al 2 O 3 -C refractory composite with ≈ 20 wt.%: (a) evolution of the strain rate (s -1 ) with the relative density. (b) evolution of the relative density with the compaction pressure.

Figure 2 .

 2 Figure 2.12: SEM micrographs of a carbon bonded alumina refractory. a) (1) cured at 180°C; (2) one cycle heating up to 500°C; (3), 2 cycles up to 1000°C. Crack location and its extension are highlighted in red. b) Alumina matrix interface: (1) cured at 180°C; (2), 1 cycle heating up to 500°C; (3), 2 cycles up to 1000°C. c) Typical graphite crack of a sample heat treated up to 1000°C after 2 cycles [Luchini et al., 2018].

2. 5 .

 5 Figure 2.13: a) Normalized Young's modulus versus temperature (E 0 is the Young's modulus at ambient temperature). b) Young's modulus versus temperature for all tested porosity levels (13.9%→21.7%), measured by impulsive excitation technique (IET). Figures are adapted from [Werner et al., 2014].

Figure 2 .

 2 Figure 2.14: Stressstrain curves of MgOC refractories from uniaxial compressive tests at dierent testing temperatures [Musante et al., 2012].

Figure 2 .

 2 Figure 2.15: Stressstrain (engineering) curves of uniaxial tensile tests carried out on Al 2 O 3 -C refractories by a) [Dupuy, 2015] and b)[START_REF] Warchal | Microstructure Evolution During Firing and Resulting Mechanical Properties of Steel Flow Control Refractories Containing Various Additives[END_REF][START_REF] Warchal | Study of the inuence of additives (antioxidants) on the thermomechanical properties of carbon-bonded refractory composites[END_REF].

Figure 2 .

 2 Figure 2.16: 3-point bending test (Al 2 O 3 -C refractories with Si as additive): a) Crack propagation in the samples after the tests. b) Percentages of the three crack propagation mechanisms in the samples (TRS and MRS) after the tests (P A , P AM , P M ): crack propagation within the aggregates, along the aggregate/matrix interface and in the matrix, respectively). c) Load-displacement curves obtained from the tests [Chen et al., 2020].

Figure 2 .

 2 Figure 2.17: a) Drucker-Prager/Cap model: yield surfaces in the pq plane [Abaqus, 2006]. b) Schematics of a density-dependent Drucker-Prager Cap model [Han et al., 2008].

Figure 2 .

 2 Figure 2.19: The distribution of relative density (SDV1): a) after closed-die compaction. b) after unloading [Ni et al., 2021].

Figure 2 .

 2 Figure 2.20: Representation of discrete and nite element scales.

Figure 2 .

 2 Figure 2.21: Representation of the successive steps present in a DEM simulation.

Figure 2 .

 2 Figure 2.23: Schematic of some DEM simulations carried out in Pizette et al. ([Pizetteet al., 2010]) for the generation of yield and fracture surfaces[START_REF] Pizette | Compaction of aggregated ceramic powders: From contact laws to fracture and yield surfaces[END_REF].

Figure 2 .

 2 Figure 2.24: Aggregated ceramic powders. a) Isodensity contour (D=0.59) and fracture line of a compact generated in closed-die compaction from DEM simulation. b) Isodensity yield points and fracture points for compacts obtained by both closed-die and isostatic compaction [Pizette et al., 2010].

Figure 2 .

 2 Figure 2.25: (a) Schematic representation of features of a refractory material and its connection with DEM. (b) Experimental setup and DEM representation [M. H. Moreira et al., 2020].

Figure 2 .

 2 Figure 2.26: Results obtained in simulations relating to the DOE (design of experiments) used to calibrate the DEM models. Final shape quality of the compacts obtained in (a) EDEM and (b) LIGGGHTS [Ramírez-Aragón et al., 2020].

Contents 3 . 1

 31 Graphical abstract . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Design of powder composites . . . . . . . . . . . . . . . . . . . 3.4 SEM observations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 X-ray tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Theoretical density . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Bulk density and open porosity . . . . . . . . . . . . . . . . . . 3.7 Green mechanical testing . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Close-die compaction . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Uniaxial compression test . . . . . . . . . . . . . . . . . . . . . 3.7.3 Triaxial compression test . . . . . . . . . . . . . . . . . . . . . 3.8 Sample preparation for thermal analysis and for ring characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Thermogravimetric analysis . . . . . . . . . . . . . . . . . . . . 3.9.2 Dilatometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Mechanical testing on red samples . . . . . . . . . . . . . . .3.10.1 Uniaxial compression test . . . . . . . . . . . . . . . . . . . . . 3.10.2 Brazilian test (diametral compression) . . . . . . . . . . . . . . 3.10.3 Triaxial compression test . . . . . . . . . . . . . . . . . . . . . 3.10.4 Deviatoric and mean stresses . . . . . . . . . . . . . . . . . . . 3.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Graphical abstract

Figure 3 . 1 :

 31 Figure 3.1: Schematic representation of a alumina-graphite powder composite aggregate, composed of coarse grains and graphite akes embedded into a continuum matrix.

Figure 3 . 2 :

 32 Figure 3.2: SEM images at increasing magnications: a)x56, b)x200, c)x600, and d)1995x. Typical alumina-graphite refractory composite (mix 1) composed of white fused alumina grains and graphite akes embedded into a carbonaceous bonding matrix.

Figure 3 . 3 :

 33 Figure 3.3: Evolution of mix 1 aggregate microstructure (2D slice from a 3D fragment image accessed through CT analysis). The CT was performed on an aggregate pellet (a) before compaction (after the drying step). (b) after uniaxial compaction to 35 MPa. (c) compacted sample after the ring stage, (d) schematic representation of pyrolysis when the temperature is close to 1000 °C.

  .4. The white colors correspond to the thresholded regions of interest. The images on the left correspond to CT images reduced into 8-bit layers, and the ones on the right are the result of thresholding based on the CT image's grayscale range. Hence, three main phases could be distinguished: graphite (Fig. 3.4 a), matrix (Fig. 3.4 b), and coarse/medium alumina grains (Fig. 3.4 c). The porosity could be identied by the dierence between the thresholded grain contour (Fig.

Figure 3 . 4 :

 34 Figure 3.4: Representation of thresholding (white as the region of interest) applied in a binarized 3D image. The phases were separated into a) graphite, b) matrix, c) large/medium alumina grains. In d), the threshold highlights the whole grain.

Figure 3 . 5 :

 35 Figure 3.5: Example of a segmentation method by assigning a range of grey levels from white to black (see greyscale legend). The segmentation was applied in the image volume (3D) by considering the four distinct phases (coarse/medium alumina grains, matrix, graphite, and porosity) identied previously at the threshold (Fig. 3.4).

  Fig. 3.6 a) shows the % content of each phase of the agglomerate grain sample (before compaction), Fig. 3.6 b) of the green grain sample (after compaction at 35 MPa), Fig. 3.6 c) the red sample (from the X-ray tomography at 1.08µm voxel size), and Fig. 3.6 d

Figure 3 . 6 :

 36 Figure 3.6: Percentage of white pixels (based on the threshold from image analysis), at each image slice (2D), of four phases: coarse/medium alumina, matrix, graphite, and porosity. a) before the compaction, b) after compaction, and c) and d) after ring. c) represents the tomography analyses performed with a voxel size equivalent to 1.08µm, and d) 0.59µm.

Figure 3

 3 Figure 3.7: Schematic representation of the link between the binder content during the mixing and the appearance of porosity after the compaction. a) pellet with a sucient % of binder (the binder covers all grains), and ii) lack of binder content (uncompleted binder covering).

  3.7. Green mechanical testing

Figure 3 . 8 :

 38 Figure 3.8: Representation of close die compaction cycle (representative of the industrial cycle) with three steps. 1) pre-loading displacement controlled test. 2) compaction pressure target equivalent to 35 MPa. 3) controlled unloading stage.

Figure 3 .

 3 Figure 3.10: Comparison of close die compaction experiments for mix 1 to 4 to a nominal pressure of 35 MPa. Compaction pressure (MPa) evolution versus the relative density (%).

Figure 3 .

 3 Figure 3.11: Mix 1 close-die compaction curves at 10 MPa (RD≈84%), 35 MPa (RD≈95%) and at RD=100% (all samples).

Figure 3 .

 3 Figure 3.12: a) Ageing eect: the compaction curve in grey (dashed) represents the compaction of the old Mix 1. The black curve represents Mix 1 reference (data used in our work). b) Eect of the moisture in the composite densication. This gure shows the dierences between the dry Mix 1 (less solvent content), wet Mix 1 (more solvent content) compaction curves, and the Mix 1 reference.

Figure 3 .

 3 Figure 3.13: Summary representation of the green characterization. Close die compaction experiments are carried out and the compacted samples are used to uniaxial and triaxial compression tests.

Figure 3 .

 3 Figure 3.14: Crack orientation of powder composites (mix 1 to 4) after uniaxial compression trials (crush tests). The samples were previously compacted to 35 MPa in a closed-die. The relative density of the samples after the close-die compaction are between approximately 89% and 95%.

Figure 3 .

 3 Figure 3.15: Experimental set-up developed for triaxial compression tests. The radial pressure (or conning pressure) is applied by pumped oil. The samples axial strain are measured using an extensometer device.

Figure 3 .

 3 Figure 3.16: Axial stress (σ 1 ) vs. axial strain curves from triaxial compression tests carried out at 10 MPa, 20 MPa, and 30 MPa conning (radial) pressure (all samples). These tests were done on compacted samples compacted to a) 10 MPa (RD=85%), b) 35 MPa (RD=95%), and c) 100 MPa (RD≈100%) on the close-die conguration. A total of 3 samples at each conning pressure setup were tested.

Figure 3 .

 3 Figure 3.17: Mix 1 samples damage prole after triaxial compression tests performed at: a) σ 2 = σ 3 =10 MPa. b) σ 2 = σ 3 =20 MPa. c) σ 2 = σ 3 =30 MPa.

Figure 3 .

 3 Figure 3.18: a) Methodology adapted to predicting the yield and ultimate stresses from stress-strain curves of triaxial compression tests on Mix 1. b) Deviatoric-mean stresses (calculated from yield and ultimate stresses).

Figure 3 .

 3 Figure 3.19: Test tube after ring (cut into cylindrical samples).

Figure 3

 3 Figure 3.20: Thermogravimetric analysis (TGA) performed on powder composites (mix 1 to 4). a) Temperature (°C) against the cycle time (in minutes) applied for all TGA analysis. b) Measured weight loss (in %) against the temperature (°C) for mix 1 to 4.

Figure 3 .

 3 Figure 3.21: Thermogravimetric analysis (TGA) performed on a cured refractory composite [Dupuy, 2015].

Figure 3

 3 Figure 3.22: a) Netzsch dilatometer device. b) Details of the heating/measuring unit, vacuum, and sample holder unit.

Figure 3 .

 3 Figure 3.23: a) thermal expansion of cured mix 3 (during pyrolysis) versus temperature during the heating (1-3) and cooling (4). b-1) and b-2) coecient of thermal expansion (CTE) evolution against the temperature during the heating, and cooling, respectively.

Figure 3 .

 3 Figure 3.24: a) Thermal expansion of red (pyrolyzed materials) powder composites (mix 1 to 4) against the temperature during the heating and cooling, and b) coecient of thermal expansion (CTE) evolution during the heating segment (b-1), and cooling (b-2) against the temperature.

Figure 3 .

 3 Figure 3.25: a) Uniaxial compression test set-up. b) observed failure mode (shear failure) in mix 1 (reference powder composite) samples after uniaxial compression tests performed at ambient temperature (20°C).

Figure 3 .

 3 Figure 3.26: a) Uniaxial compression test set-up with heating. b) observed failure mode (shear failure) in mix 1 (reference powder composite) samples after uniaxial compression tests performed at 300, 600, and 900°C.

Figure 3 .

 3 Figure 3.27: Mix 1 uniaxial crush test trials performed at 20°C (ambient temperature), 300°C, 600°C, 900°C, and 1200°C. a) axial stress (MPa) evolution with the axial strain (mm/mm). b) Ultimate and yield stresses points against the temperature (°C). c) Modulus of elasticity (Young's modulus) against the temperature (°C).

Figure 3 .

 3 Figure 3.28: a) Brazilian test set-up. b) observed failure modes in mix 1 (reference) samples after uniaxial compression tests performed at 20°C, 300°C, and 900°C.

Figure 3 .

 3 Figure 3.29: Evolution of ultimate strength (MPa) against the temperature (°C) during Brazilian tests of mix 1-4.

  investigated the Weibull modulus to evaluate the stability of carbon-bonded porous alumina ceramics strength through three-point bending tests. It was observed that the increase of sample porosity resulted in a decrease of Weibull modulus (m). Liu et al. have shown m values of 5.51-14.05 MPa.

Figure 3 .

 3 Figure 3.30: a) Weibull plot of the test series regarding mix 1-4 (reference composite) at ambient temperature (20°C). b) strength of the material with a failure probability of 63.2% for mix 1-4 at all testing temperatures (20-1200°C).

Figure 3 .

 3 Figure 3.31: a) representation of the three stresses (σ 1 , σ 2 , andσ 3 ) during triaxial tests. σ 1 is the axial stress, and σ 2 , and σ 3 are the conning pressure. b) Mix 1 specimens after failure (fracture surfaces).

Figure 3 .

 3 Figure 3.33: Ultimate axial stress, σ 1ultimate , evolution with the conning pressure (σ 2 = σ 3 ) of powder composites (mix 1-4).

Figure 3 .

 3 Figure 3.34: Deviatoric (σ d = σ 1 -σ 3 ) stress against mean stress (σ m = (σ 1 + σ 2 + σ 3 )/3) generated for the a) ultimate and b) yield surfaces.

  abstractThe contact laws used in DEM are the Hertzian (elastic) and the microscopic bond model;A composite ceramic material is represented by mixing of single spherical particles and particles clusters;

Figure 4 . 1 :

 41 Figure 4.1: Two particles contact is possible: (a) Hertzian (or elastic) contact between non-bonded particles, and (b) elastic contact between bonded particles. (b-1) These bonds may not break (non-fractured bonds) or (b-2) fracture. These fractured bonds may be restored (new contact may appear between particles initially broken).

Figure 4 . 2 :

 42 Figure 4.2: 2D sketch of a typical contact geometry between two elastic spherical particles with radius r i and r j , overlap δ n , and normal and tangential vectors (n and t) at the contact.

Figure 4 . 3 :

 43 Figure 4.3: Bond representation between two spherical particles p and q transmitting normal and tangential forces, and resisting moments.

Figure 4 . 4 :

 44 Figure 4.4: Composite spheres model proposed by Hashin ([[START_REF] You | Discrete element modeling to predict the modulus of asphalt concrete mixtures[END_REF]).

Figure 4 . 5 :

 45 Figure 4.5: Bonds interaction representation between particles clusters (coarse alumina and graphite ake): a) the bond interactions inside the coarse alumina are not taken into account; b) the bond interactions inside the graphite are calculated and are given by the bond contact model; c) the interactions between two coarse aluminas are given by the Hertzian contact model (no bonds); d) the interactions between coarse alumina and graphite materials are given by the Hertzian contact model; e) the Hertzian contact model describes the interactions between two graphite.

Figure 4 . 6 :

 46 Figure 4.6: Typical discrete microstructure after compaction: from blue (no bonds) to red (with bonds). Coarse alumina and graphite akes are considered as an assembly of particles linked by bonds, and ne aluminas as single non-bonded particles. Note how graphite akes may deform while coarse alumina particles cannot.

CHAPTER 4 .

 4 DEM: MODEL DEVELOPMENT AND DESCRIPTION 1.

Figure 4 .

 4 Figure 4.7: Two elastic particles coated with thickness t i = δ c × r i and t j = δ c × r j . a) the contacts remains within the soft elastic binder phase (branch 1). b) when the indentation reached a critical value (that remains constant) i.e., condition δ n ≥ δ c = t i + t j applies, the elastic modulus of particles i and j are multiplied by a factor M and the contact enters branch 2. Once in branch 2, i) due to the compression, the normal force at the contact may increase, or ii) the contact may unload, or iii) the contact may fails (if

Figure 4 . 8 :

 48 Figure 4.8: Schematic representation of the contact between coated (alumina) and noncoated (graphite) material. The contact while reaching branch 1 or 2 is applied by the same methodology shown in Fig. 4.7 as an example of contact between two coated particles.

Figure 4 . 9 :

 49 Figure 4.9: Typical densication curves from DEM close-die compaction simulations at the same macroscopic input parameters performed in our work. (a) 117,605 particles (t i = t j , 'same binder thickness'), Table 4.3. (b) 112,026 particles (V f i = V f j ), Table 4.4. (c) Closed-die compact experiment. Target: axial stress = 35 MPa. Details concerning DEM compaction simulations are described in the next chapter 5.

Figure 4 .

 4 Figure 4.10: Representation of the microstructures generation in DEM. a) ne particles and particle clusters are randomly located inside a box with periodic conditions up to a density RD=0.4. b) the packing in densied until RD=0.6.

Figure 4 .

 4 Figure 4.11: a) Rigid walls condition. b) Periodic condition. The particles in dotted lines and whose center is found outside the simulation box are translated to the opposite face [Parant, 2016].

Figure 4 .

 4 Figure 4.12: Typical DEM close-die compaction simulation performed in or work (chapter 5): kinetic energy (in m•J) against the relative density (RD) with an axial stress limit (target: σ z =35 MPa).

Figure 4 .

 4 Figure 4.13: DEM simulations steps performed in this work: a) Particles/aggregates are randomly located inside a box with periodic conditions up to a density RD = 0.4. b) the packing is densied until RD = 0.6. c) close-die or isostatic compaction following various nal target densities d) the compacted sample is unloaded to 1 MPa pressure e) Triaxial and tensile test simulations.

Figure 4 .

 4 Figure 4.14: Typical DEM unloading simulation in our work: macroscopic isostatic stress state (σ x = σ y = σ z ).

Figure 4 .

 4 Figure 4.15: Local density map of a sample during jamming, close-die compaction, and unloading steps (calculated by Voronoi tesselation). The clusters (coarse alumina and graphite) are considered fully dense (density RD = 1).

Figure 4 .

 4 Figure 4.16: Evolution of the macroscopic deviatoric and mean stress (in module) against the total number of particles, at the end of isostatic compaction simulations (nal density target RD = 0.95).

Figure 4 .

 4 Figure 4.17: Evolution of the macroscopic deviatoric and mean stress (in module) against the total number of particles, at the end of closed-die compaction simulations (nal density target RD = 0.95).

Figure 4 .

 4 Figure 4.18: Mean and deviatoric stress points from DEM: a) isostatic compaction simulations, and b) closed-die compaction simulations, depending on the total number of particles (nal density target RD = 0.95).

Figure 4 .

 4 Figure 4.19: Simulations clock time from samples generation until the end of closed-die compaction. Using 8 CPU's with the openMP parallel option in dp3D.

Fig. 4 .

 4 Fig. 4.21 summarizes the procedure used for the preparation of red numerical samples for virtual mechanical characterization. The packing at RD = 0.6 is compacted uniaxially

Figure 4 .

 4 Figure 4.20: a) Bond representation in green and virtual ring steps: a) bonds representation of a green microstructure (coarse alumina cluster surrounded by a soft-binder shell), and b) red microstructure representation with bonds between all particles in contact. Six elastic bond interactions are possible at the contact: 1) between ne aluminas particles, 2) ne alumina and graphite, 3) coarse alumina and ne alumina, 4) coarse aluminas, 5) graphite akes, and 6) graphite and coarse alumina.

Figure 4 .

 4 Figure 4.21: Summary of the dierent steps needed to prepare a numerical microstructure for mechanical characterization. a) gas of particles up to RD=0.4. b) closed-die compaction with a target axial load of 35 MPa or density (0.6 < RD ≤ 1), c) unloading down to 1 MPa pressure (nearly isostatic stress state), e-1) bonds creation. Thus, in e-1), the unloaded sample is prepared to start following the numerical virtual ring step by creating bonds to all particles in contact.

4 .

 4 22 c) [S. et al., 2010].

Figure 4 .

 4 Figure 4.22: Representation of stresses appearing during the pyrolysis eect: a) before the ring process b) during the ring c) after the ring (cooling-down eect). ( [Nicolas[START_REF] Tessier-Doyen | Etude expérimentale et numérique du comportement thermomécanique de matériaux réfractaires modèles[END_REF]

Figure 4 .

 4 Figure 4.23: X-ray tomography images showing voids around coarse alumina grains and graphite, and in e-2), to represent these voids, we broke bonds between two pairs of materials: 1) between coarse alumina clusters and ne alumina particles, and 2) between ne alumina particles and graphite.

Figure 4 .

 4 Figure 4.25: Typical uniaxial (σ x = σ y = 0) response of numerical samples under 3 dierent assumptions on bond status. Comparison with experimental curve. a) no bonds broken between particles. b) bonds broken between ne and coarse alumina. c) broken bonds between ne and coarse alumina, and between ne alumina and graphite.

  the discrete components (alumina and graphite), and the behavior at the contact features were investigated through DEM compaction simulations; Experiments were compared with DEM simulations (close-die compaction, and mechanical tests); The of binder volume fraction (%) corresponds to an of the material stiness (larger relative density); The of the DEM binder Young's modulus or the of the binder volume fraction (%) correspond to an of the number of contacts in the hard branch, and consequently to a of the springback eect during the unloading. The generation of isodensity yield surfaces and fracture points are obtained through triaxial and tensile DEM simulations by imposing stresses;

  Materialγ (J/m²) µ σ N = σ T (MPa) Young's modulus (

5. 4 .

 4 Compaction and unloading simulations 5.4 Compaction and unloading simulations 5.4.1 Close-die compaction: behavior at the contact

  Figure 5.1: DEM close-die compaction simulation (target σ z =35 MPa). a1) and a2) Normal Hertzian contact force evolution (N Hertz ) versus normalized particle indentation (δ n /r). Contacts type 1) to 4) represent the particle contact between three components (coarse alumina, ne alumina, and graphite). Once the contact reaches the critical indentation value ( δn ), hard contacts appear (see Fig. 5.2). b) N Hertz versus composite relative density (RD). * represents the point where the interactions move towards the hard branch.

Figure 5 .

 5 Figure 5.3: DEM close-die compaction simulation (target σ z = 35 MPa): % of the number of interactions in the soft branch a), and in the hard branch b). B 1 is the total number of contacts in the soft binder branch. B 2 is the total number of contacts in the hard branch.

Figure 5 . 4 :

 54 Figure 5.4: DEM close-die compaction: contribution of the three dierent discrete materials (ne alumina, coarse alumina, and graphite) to the axial macroscopic stress (nal target: σ z =35 MPa).

Figure 5 . 5 :

 55 Figure 5.5: DEM close-die compaction: contribution of the contact models to the axial macroscopic stress (target: σ z =35 MPa). a) Hertzian model (contact between non-bonded particles), b) bonded model (contact between bonded particles inside the graphite), and c) between fractured bonds.

Figure 5 . 6 :

 56 Figure 5.6: Mix 1 (reference composite) close-die compaction simulations (E binder =0.1-3 GPa): a) evolution of the axial macroscopic stress (σ z ) against the composite relative density (loading). b) evolution of the relative density at the end of the loading (σ z =35MPa) and total % of hard contacts, versus the binder Young's modulus (E binder ).

Fig. 5 .

 5 Figure 5.7: Mix 1 (reference composite) close-die compaction simulations (E binder =0.1-3 GPa): a) evolution of the axial macroscopic stress (σ z ) against the composite relative density (loading and unloading curves). b) the total number of contacts (%) in the hard alumina branch during the loading, and the spring-back (%) after unloading, as a function of the binder Young's modulus.

Figure 5 .

 5 Figure 5.10: Mix 1 close-die compaction experiment (time-controlled test) and DEM close-die compaction simulation: evolution of the axial macroscopic stress, in absolute value, in the z direction (σ z ) versus the composite relative density. The evolution of the numerical microstructure during the compaction is also highlighted. Target: σ z =35 MPa.

Figure 5 .

 5 Figure 5.11: Final relative density reached at the end of the loading during close-die compaction experiments and DEM simulations (Mix 1) against the axial stress (or compaction pressure) in MPa.

5

 5

  Figure 5.12: Close-die compaction (loading stage) of Mix 1-4: comparison between experiments (time-controlled test) and DEM simulations. Axial stress (σ z ) against the sample's relative density (target: σ z =35 MPa).

Figure 5 .

 5 Figure 5.13: Close-die compaction (loading and unloading stages) of Mix 1 to 4: a) experiments (force-piloted test), and b) DEM simulations. Axial stress (σ z ) against the sample's relative density (target: σ z =35 MPa).

Figure 5 .

 5 Figure 5.14: DEM uniaxial compression test simulation with σ x = σ y = 0 using a compacted mix 1 reference composite packing: axial stress-strain plot and the corresponding sequences of deformation with (1) initial microstructure (after unloading), (2) maximum stress, and (3) after material damage.

Figure 5 .

 5 Figure 5.15: a) Comparison between the maximum stress (compression strength) through DEM uniaxial compression test simulations (σ x = σ y = 0) and uniaxial compression test experiments against the binder volume fraction % (mix 1-4). b) Typical axial stress-strain curves from DEM simulations.

Figure 5 .

 5 Figure 5.16: Stress-strain curve from DEM triaxial test simulation at σ x = σ y = 30 MPa.(1) initial microstructure (after unloading), (2) microstructure at an axial strain equivalent to 8 %.

5. 6 .

 6 Figure 5.17: Triaxial test experiments and triaxial DEM simulations. Evolution of the axial (σ z ) against the axial strain in z direction. The triaxial tests were made by applying radial stress in x and y directions equal to 10 MPa, 20 MPa, and 30 MPa. The samples were tested experimentally and numerically. a) triaxial tests at microstructures compacted, in close-die compaction, at 10 MPa, and unloaded. b) the same as before but in samples compacted at 35 MPa from close-die. In c), the triaxial DEM test was made with microstructures compacted at approximately 70 MPa, and experimentally at 100 MPa (samples are at RD ≈ 100% after compaction).

Figure 5 .

 5 Figure 5.19: Evolution of the axial stress as a function of the microstructure's relative density. a) DEM uniaxial tensile test simulation (σ x = σ y =0). The imposed strain rate in x, y, and z are in tension (positive value). b) DEM isotensile test simulation where the strain rate in x, y, and z are equivalent and in tension ( εx = εy = εz ).

Figure 5 .

 5 Figure 5.20: Examples of DEM triaxial test simulations by imposing a negative strain rate (in compression) in the three directions, and a σ x = σ y equivalent to a) 0 MPa, b) 5 MPa, and c) 10 MPa. The fracture points are considered as the maximum stress together with a decrease in RD.

Figure 5 .

 5 Figure 5.21: Examples of DEM triaxial test simulations with σ x = σ y equal to a) 20 MPa, and b) 26.5 MPa. The yield isodensity points are considered as the points where the microstructure achieves the same relative density as for the end of closed-die compaction simulations (RD=95.8%).

5. 6 .

 6 Figure 5.22: Representation of mix 1 composite hardening after closed-die compaction to dierent relative densities (from RD=0.8 to RD=1).

Figure 5 .

 5 Figure 5.23: Fracture points and isodensity (RD=0.95) yield surfaces obtained from various reloading DEM simulations (triaxial and tensile) on the mix 1 reference composite compacted by closed-die and isostatic congurations.

Figure 5 .

 5 Figure 5.24: Fracture points and isodensity (RD=0.9) yield surfaces obtained from various reloading DEM simulations (triaxial and tensile) on the four dierent composites (mix 1 to 4) compacted by closed-die conguration.

  -die compaction (up to 35 MPa) → unloading → virtual ring → pre-loading → triaxial tests (variation in the radial stresses); Triaxial compression / tensile test simulations on red virtual samples (Mix 1) previously compacted at 35 MPa: comparison with experimental data; Triaxial compression: axial strain rate and radial stresses. Ultimate and yield surfaces from simulations and experiments are plotted in the deviatoric and mean stress space (in modulus) and compared; These surfaces were generated from Brazilian tests (experiments), uniaxial tensile simulations, and triaxial compression tests (experiments and simulations).

Figure 6 .

 6 Figure 6.2: a) DEM pre-loading (nal target: σ x = σ y = σ z = 5.4M P a),and b) triaxial test simulations in samples, with a radial stress between 0 and 90 MPa, at constant macroscopic input parameters: number of broken bonds (in tension and in shear) against the packing relative density (target: maximum axial stress σ z = σ ultimate ). The initial total number of non-broken bonds is ≈ 315,000 and broken bonds ≈ 50,000.

Figure 6 . 3 :

 63 Figure 6.3: Two examples of triaxial compression numerical simulations at a) σ x = σ y =0 MPa and b) 30 MPa. Evolution of the total number of broken bonds (N rupt N +N rupt T ) in red versus the axial stress-strain. The total number of broken bonds per particle was highlighted when at the maximum axial stress.

Figure 6 .

 6 Figure 6.4: a) Evolution of the ultimate and the b) yield axial stresses: comparison between DEM and experimental triaxial compression tests. c) Stress-strain curves exemplifying the estimation of the axial yield stress (radial stress equal to 30 MPa).

Figure 6 .

 6 Figure 6.5: a) Comparison of the tensile strength achieved by DEM tensile test simulations and Brazilian tests (experiment). b) DEM tensile test (strain-strain). c) Initial predamage (broken bonds between ne alumina-coarse alumina, and ne alumina-graphite particles). d) additional bonds damage (not showing the initial one) at 1) the maximum stress (or sample failure) and 2) after sample failure. e) image (with zoom) highlighting some of the cracks propagation.

Figure 6 . 6 :

 66 Figure 6.6: Deviatoric-mean stress (in modulus) of mix 1: comparison between experiments and DEM simulations for a) the ultimate surface (maximum stresses) b) yield surface.

Figure 6 .

 6 Figure 6.8: a) Deviatoric vs. mean stresses (fracture and isodensity surface). b) Triaxial compression test (radial stress of 30 MPa). The fracture point is the maximum axial stress together with a sample relative density decrease. c) Triaxial compression test (radial stress of 90 MPa). The isodensity point is when the sample is at the same density from the compaction end (RD=95.8%).

  isodensity and fracture surfaces from DEM simulations with the Drucker-Prager Cap (DPC) model; Identication of the DPC model parameters; Finite element closed-die compaction simulation of the powder composite using the DPC model; Comparison between numerical (DEM and FEM) and experimental data.

Figure 7 . 1 :

 71 Figure 7.1: Schematic of the validation of the Drucker-Prager Cap (DPC) model. Experimental closed-die compaction data → DEM (generation of the isodentity yield surface and fracture line) → DPC model parameters identication → FEA modeling (closed-die compaction conguration).

7. 3 .

 3 Parameters identication (DPC model) from DEM simulations surface and the DEM isodensity points with the cap (Fig.7.2). To cover a large range of densities, the DPC surfaces are plotted for densities between 0.7 and 1.

Figure 7 . 2 :

 72 Figure 7.2: Drucker-Prager Cap model: Von mises equivalent stress against the hydrostatic pressure of mix 1 composite compacted (by closed-die conguration) up to densities varying from 0.7 to 1. The model is represented by a failure line and the yield surface.

Fig. 7 .

 7 Fig. 7.3 exemplies the identication of the DPC model parameters for a shear failure and cap surfaces of the packing at RD=0.95. Using the Abaques vocabulary associated to the DPC parameters, the shape of the cap surface is determined by the yield pressure (p b ), the cohesion (d), the evolution pressure (p a ), the cap eccentricity (R), and the angle of friction (β). The linear shear failure surface function depends on d and β. Concerning the calibration of the DPC parameters, rst, the cohesion is identied from the linear function (y = (tan β) • x + d = 0). The pressure p a , p b and the eccentricity (R) can be tted to the numerical DEM points. The evolution of the DPC parameters (cohesion,

Figure 7 . 3 :

 73 Figure 7.3: Typical representation of the identication of the Drucker-Prager Cap model parameters (isodensity RD=0.95).

Figure 7 . 4 :

 74 Figure 7.4: Drucker-Prager Cap model parameters: a) cohesion (d), b) angle of friction (β), c) cap eccentricity (R), d) pressure evolution (p a ), e) hydrostatic pressure yield surface (p b ), f) mix 1 Young's modulus (estimated from DEM uniaxial compression test simulations).

Figure 7 . 5 :

 75 Figure 7.5: Finite element closed-die compaction simulation (mix 1). Density distribution: a) before the compaction, and b) at the end of compaction. c) mesh representation (initial state, before compaction).

Figure 7 . 6 :

 76 Figure 7.6: Evolution of the Poisson's ratio against the relative density during DEM closed-die compaction simulation until axial stress of 35 MPa.

Figure 7

 7 Figure 7.7: a) Relative density distribution in the die during FEM closed-die compaction. b) Axial stress (or compaction pressure) versus relative density curves: comparison between numerical and experimental data. The RD from FEM was measured as the average density of each element.

Figure 7

 7 Figure 7.8: a-1) Axial stress and a-2) radial stress evolution vs. relative density: comparison between numerical simulations (DEM and FEM). b) Distribution of the axial stress (S22) along the powder inside the die at a compaction pressure of 35MPa.

Figure 7

 7 Figure 7.9: a) Distribution of the Von Mises (Q) and hydrostatic (P) stresses in the die at the end of the compaction (35MPa). b) Hydrostatic (P) versus Mises (Q) loading path during the nite element closed-die compaction at the nodes showing maximum (1), medium (2), and minimum (3) stresses. Comparison with the DEM loading path (closeddie compaction with an axial stress target of 35MPa).

Figure 8 .

 8 Figure 8.1: a) Representation of the discrete numerical sample of heterogeneous carbonbonded refractory composites b) Evolution of the axial stress vs. relative density of closed-die compaction numerical simulations performed in this Ph.D. work.

Figure 8 .

 8 Figure 8.2: a) Comparison between an experimental and numerical densication curve (axial stress vs. relative density) in a) a normal and b) logarithm scales.

Figure 8 .

 8 Figure 8.3: a) X-ray tomography images (mix 1 standard composite). b) images posttreatment (segmentation method), and c) discrete element packing representing images from b).

Figure 8 . 4 :

 84 Figure 8.4: Discrete element packings representing SiC ceramics. Simulations of a) uniaxial compressive test, b) Brazilian test, and c) 3-point bending test [Tan et al., 2009].

Figure 8 .

 8 Figure 8.5: a) Example of outwards packing. b) Example of inwards packing. Particles comprising the advancing front are in dark grey [Recarey et al., 2019].

Figure 8 . 6 :

 86 Figure 8.6: Isostatic compaction simulation of an industrial piece (ladle shroud) on ABAQUS using the DPC parameters (mix 1). a) initial shape (RD=60%), before compaction. b) after compaction (average RD ≈ 90%) showing a density gradient.

Figure 8 .

 8 Figure 8.7: a) Final shape of the industrial part after isostatic compaction. b) casting (thermo-mechanical) simulation with temperature dependence. Case 1 represents a homogeneous density (no density distribution). In case 2, the industrial part presents a density gradient after compaction. This gure shows the accumulative plastic strain distribution after the simulation in both cases 1 and 2.

  ].

	CHAPTER 2. LITERATURE REVIEW 2.6. Numerical modeling using the nite element method (FEM)
	Refractory 2.6 Numerical modeling using the nite element method Experimen-tal method Testing T (°C) Mechanical properties References Magne-(FEM)
	Al 2 O 3 -C (red) Continuum mechanical modeling with the nite element method (FEM) has been largely tostrictive ultrasonic pulse echo 20-1800 E: 99-103 GPa [Gault et al., 1985] applied to evaluate the compaction behavior of powder materials. Using FEM to model
	Al 2 O 3 -C (Si additive) (red) compaction necessitates implementing a constitutive equation that is able to reproduce 3-point bending test RT E: 1.1-3.0 GPa / CMOR: 4.5-15.6 MPa [Fan et al., the complex behavior of a powder. In particular, the phenomenological model needs to 2011] reproduce the powder behavior under both isostatic (pressure sensitive) and shear stress
	Al 2 O 3 -MgO-C (red) components. In other words, the model needs to reproduce the fact that a powder densi-Uniaxial compressive test 20-1000 E: 12-37 GPa / σ R : 38-76 [Musante et al., 2011] es under pressure and may dilate or fracture if shear components are applied. Typically MPa the Drucker-Prager-Cap elastoplastic model can reproduce quite simply such a behavior.
	MgO-C (red) The Drucker-Prager-Cap elastoplastic model has been applied for powder metallurgy [Al-Uniaxial compressive test E: 3-46 GPa / [Musante 20-1400 σ R : 4-68 MPa et al., 2012] manstötter, 2015] [Zhou
	Al 2 O 3 -C (red)	IET / Uniaxial compressive test	25-1450	E: 4.2-9.6 GPa / CMOR: 6.3-8.6 MPa / (RT) G: 2.2-3.8 GPa	[Werner et al., 2013, 2014]
	Al 2 O 3 -C (Si additive) (red)	3-point bending test	RT/1400	CMOR: 3-23 MPa / HMOR: 14-22 MPa	[Zhu et al., 2013]
	Al 2 O 3 -C (red)	Tensile tests /US long bar mode	25 (RT)	E: 4.4-10.3 0.1%) GPa / σ R : 2-4 MPa (ε R :	[Dupuy, 2015]
	Al 2 O 3 -C (red)	Tensile tests	25 (RT)	E: 9.7 GPa / σ R : 4 MPa (ε R : 0.1%)	[Warchal et al., 2017; Warchal, 2018]
	Al 2 O 3 -C (cured)	IET	27-1000	E: 11-22 GPa	[Luchini et al., 2018]
	Al 2 O 3 -C (Si additive) (red)	3-point bending test	RT/1400	CMOR: 10.6-13.4 MPa 25.6-34.6 MPa / HMOR:	[Yin et al., 2019]
	Table 2.1: A summary table of some thermo-mechanical properties cited in this section
	showing the dierent compositions (of based bonded-carbon refractories), experimental
	method, testing temperature and references (IET: impulse excitation technique).

  a long experimental protocol was also applied for determining the DPC model parameters for modeling the closed-die compaction.

				2.7. Discrete Element Method
	2.7 Discrete Element Method
	2.7.1 Introduction
	Material	d (MPa)	β (°)	R	Reference
	99.99% -Al 2 O 3	5.5	16.5	0.5	[Aychn et al., 1996]
	94% Al 2 O 3	2.3	26.7	1.7	[Zeuch et al., 2001]
	SiC	0.4	55	1.3	[Park & Kim, 2001]
	Al 2 O 3	4	44	0.5	[Shin & Kim, 2015]
	Al 2 O 3	7	65	1.77	Melo et al. [2018]
	MgTiO 3	3.2	64.8	0.59	[Ni et al., 2021]
	Al 2 O 3 -C	3.2	56.8	0.7	[Buljak et al., 2021]
	Table 2.2: A summary table containing some measured Drucker Prager Cap model prop-
	erties (applied for FEM compaction simulations) presented in the literature (maximum
	values), including the studied material, and references.

Table 2 .

 2 3: A summary table of some of the numerical models, using the discrete element method, applied to ceramic and refractory materials.

2.8. Conclusions

Table 3

 3

		alumina Fine alumina Graphite binder
	Mix 1 (standard)	33	22.5	22	22.5
	Mix 2	33	27.5	22	17.5
	Mix 3	33	20	22	25
	Mix 4	36	20	24	20

.1: Composition of the dierent recipes, given in volume fraction (%).

Table 3 .

 3

	Experiment	Voxel size (µm) Energy spectrum (keV) Grain size (mm)
	Agglomerate	1.19	29	1.9
	Green	1.08	30	1.2
	After-ring (1)	1.08	29	1.5
	After-ring (2)	0.59	99	0.7

2: X-ray computed tomography (CT) experiments parameters applied in all analysis and the approximate grain size (measured from image analysis), performed on the reference composite (mix 1).

Table 3 .

 3 .1). 4: Theoretical density (g/cm 3 ) of raw materials used in this work (coarse alumina, ne calcined alumina, graphite, and binder).

	CHAPTER 3. EXPERIMENTAL CHARACTERIZATION
	Raw material	Theoretical density (g/cm 3 )
	white fused alumina	4.00
	calcined alumina	3.94
	graphite	2.20
	binder	1.23
	Powder composite Theoretical density (g/cm 3 )
	Mix 1	2.96
	Mix 2	3.10
	Mix 3	2.91
	Mix 4	2.98

Table 3 .

 3

5: Absolute (or theoretical) density, in g/cm 3 , of the four composite powders.

Table 3

 3 Bulk density (g/cm 3 ) Porosity (%)

	Mix 1	2.80 ± 0.00	5.1 ± 0.1
	Mix 2	2.77 ± 0.00	12.5 ± 0.1
	Mix 3	2.76 ± 0.00	4.5 ± 0.0
	Mix 4	2.79 ± 0.00	6.4 ± 0.0
	Table 3.6: Bulk density and porosity percentage of the four composite powders (after
	curing).
	Powder composite Bulk density (g/cm 3 ) Porosity (%)
	Mix 1	2.71 ± 0.00	16.9 ± 0.1
	Mix 2	2.70 ± 0.00	19.7 ± 0.1
	Mix 3	2.67 ± 0.00	17.1 ± 0.0
	Mix 4	2.71 ± 0.00	17.0 ± 0.0

.7: Bulk density and porosity percentage of the four composite powders (after ring).

  Powder composite ρ r at 35 MPa (%) ρ r at 30MPa (%) ρ r end of unloading (%) Table3.8: Measurement of the relative density (ρ r ) from close-die compaction tests: a) at the end of the loading (at ≈ 35 MPa), b) during the unloading (at ≈ 30 MPa),

	Mix 1 (standard)	95.5	95.9	95.7
	Mix 2	90.0	90.5	89.8
	Mix 3	96.0	96.6	96.5
	Mix 4	94.0	95.1	94.7

Table 4 . 2

 42 .2).

	Material	E (GPa) [Eq. (4.1)] ν [Eq. (4.2)]
	Coarse alumina	380	0.2
	Fine alumina	380	0.2
	Graphite	35	0.2
	Soft-binder shell	1.02	0.2

: Elastic material parameters used in DEM simulations for the green characterization.

Table 4 .

 4 . Raw material Binder thick. (µm) Nb clusters Total nb of part.

		Coarse alumina	0.774	8
	Mix 1	Fine alumina	0.774	53,695	117,605
		Graphite	-	25
		Coarse alumina	0.569	8
	Mix 2	Fine alumina	0.569	54,589	118,499
		Graphite	-	25
		Coarse alumina	0.893	8
	Mix 3	Fine alumina	0.893	53,199	117,109
		Graphite	-	25
		Coarse alumina	0.771	10
	Mix 4	Fine alumina	0.771	53,192	132,812
		Graphite	-	31

3: Number of particles on dp3D depending on their composition. In this case ('same binder thickness' assumption), the binder thickness of ne and coarse alumina are considered equal (t i = t j ).

  Microst. Raw material Binder thick. (µm) Part./aggl. nb Total nb of part.

		Coarse alumina	6.306	10
	Mix 1	Fine alumina	0.493	39,896	112,026
		Graphite	-	24
		Coarse alumina	5.104	10
	Mix 2	Fine alumina	0.343	45,418	119,688
		Graphite	-	26
		Coarse alumina	6.873	10
	Mix 3	Fine alumina	0.582	37,363	109,493
		Graphite	-	24
		Coarse alumina	5.311	11
	Mix 4	Fine alumina	0.493	37,355	118,410
		Graphite	-	28
	Table

  .2. The Young's modulus of the matrix is 4 GPa. This value was set after calibration trials with experimental results (chapter 6).

	Material	E (GPa) [Eqs. (4.7), (4.8)] ν [Eqs. (4.7), (4.8)]
	Coarse alumina	380	0.2
	Graphite	35	0.2
	Matrix	4	0.2
	Table 4.5: Elastic material parameters used in DEM simulations for the numerical virtual
	ring step.

  .1).

	Material	µ (4.5)(4.6) E (GPa) (4.2) ν (4.2) σ N (MPa) (4.9) σ T (MPa) (4.10)
	Coarse alumina	0.2	380	0.2	78	508
	Graphite	0.2	35	0.2	78	508
	Matrix	0.2	4	0.2	78	508
	Table 6.1: Material parameters used on DEM simulations for the after-ring characteri-
	zation.

2.4. Compaction behavior of ceramic powders

2.5. Thermo-mechanical behavior of carbon-bonded refractories

2.6. Numerical modeling using the nite element method (FEM)

2.7. Discrete Element Method

The contact models used in our work are described in chapter 4.

3.6. Physical properties

For convention, the tensile and compressive forces are negative, and positive, respectively.

Σ m =-1.3MPa, and Σ d =-4.1MPa. (6) and (7) are examples of triaxial test simulations with radial stress equivalent to 5 and 10 MPa. In(6), the sample is rst compacted and

Acknowledgments

The Ph.D. is a `grand voyage' where as soon as you take the train you do not know exactly

Chapter 7

Finite element modeling

In this chapter, we use the information gathered by DEM simulations to implement a constitutive equation for the refractory composites into a Finite Element code. The aim of this chapter is to provide an example of the possibilities oered by the mutual use of the discrete and nite element methods (DEM and FEM) for the simulation of an industrial Vesuvius compaction process. In more detail, the tting of the isodensity and fracture points, generated from DEM simulations (chapter 5) of green samples (mix 1), with the Drucker-Prager Cap (DPC) model are shown. The DPC model parameters are identied (section 7.3). Finite element compaction using a closed-die conguration is described in section 7.4 using the DPC model. The main objectives are rst to calibrate complementary parameters not identied by closed-die compaction experimental tests (composite Poisson's ratio and friction). Second, to validate the measured DPC parameters. The resulting density distribution within the powder composite inside the die is obtained. The evolution of the compaction pressure vs. the composite relative density during FEM numerical simulation and is compared to experimental closed-die compaction results. Finally, the distribution of the Von Mises (Q) and the hydrostatic stress (P) inside the die is highlighted. The loading path during closed-die compaction is plotted in the P vs. Q plane, comparing numerical (DEM and FEM) data. The equivalent Von Mises stress is calculated as:

where σ x , σ y , and σ z are the principal stresses. τ xy , τ yz , and τ zx are the shear stresses.

The Hydrostatic stress is given by:

Figs. 7.8 a-1) and a-2) show the evolution of the axial and radial stresses, respectively, versus the relative density from FEM and DEM simulations. The curves (1), (2), and (3) are correlated to the tagged nodes in three dierent regions of the powder. These regions are highlighted in Fig. 7.8 b) which also exhibits the axial stress distribution along with the powder at a compaction pressure of 35 MPa. The maximum (1), middle (3), and minimum (2) nodes correspond to a relative density of ≈ 99%, 96.7%, and 94%, respectively, at 35MPa (Fig. 7.7 a).

Node 1 experiences the largest axial stress during compaction. This is a direct eect of the relative motion between the die and the powder, which is at its maximum at the top right corner of the sample. Friction between the die and the powder is thus increasing the local stress necessary to plastically deform the powder at this location. The reverse is true for node 2, which experiences very small relative motion. Node 3 is both further away from the cylindrical die and from the two at pistons.

Although the axial stress is clearly aected by the location in the sample, Fig. 7.8 a-2) indicates that the radial stress is nearly independent of the location.

The DEM simulations use full periodic conditions. The direct comparison with FEM simulations introducing die geometrical eects and frictional eects between the die and the powder is not fully meaningful. Still, we note that the stress evolution of node 3, which is at the center of the sample and further away from the eect of the cylindrical die and from the eect of the punches, is in a fairly good agreement with the DEM simulations.

The Von Mises equivalent stress (Q) and hydrostatic pressure (P ) during the compaction numerical simulations (DEM and FEM) are calculated using Eqn's. (7.1) and (7.2) above. The distribution of the Q and the P of the powder inside the die, at the end of the closed-die compaction at a compaction pressure (CPRESS) of 35 MPa, is given Appendix A

Eective thermal conductivity

This section presents preliminary results on identifying the composite (Mix 1) eective thermal conductivity (TC) using commercial software (GeoDict), specically with the ConductoDict module. This module computes the eective thermal [START_REF] Panerai | Micro-tomography based analysis of thermal conductivity, diusivity and oxidation behavior of rigid and exible brous insulators[END_REF][START_REF] Pabst | Relative Young's modulus and thermal conductivity of isotropic porous ceramics with randomly oriented spheroidal pores Model-based relations, cross-property predictions and numerical calculations[END_REF] and electrical conductivity of porous and composite materials. The measurement of the eective TC is carried out from X-ray tomography images and DEM numerical samples. The results are compared with an experimental measurement carried out in VESUVIUS. The TC of the material phases are set as 12 W/mK (coarse alumina), 7 W/mK (matrix), 200 W/mK (graphite), and porosity (0 W/mK) [START_REF] Kallel | New ceramic technologies and novel multifunctional ceramic devices and structures (CERMAT2): internal report[END_REF]. The TC analysis in Geodict was performed applying the ring temperature. From X-ray tomography images X-ray tomography images from Mix 1 after ring are cut in volumes of 300x300x300 pixels and transferred to GeoDict. A total of 7 cubic samples were used.

From DEM samples Numerical composites compacted up to dierent relative densities (RD=0.7-1) using dp3D. The discrete element samples in 3D are converted into a "raw" format (voxelization algorithm) and transferred to Geodict.

Appendix B

Graphite akes morphology before and after DEM compaction simulations The graphite's Young's modulus (E) and some other properties, such as the thermal expansion, are highly orientation-dependent, with dierent values between the parallel and perpendicular directions [START_REF] Luchini | On the nonlinear behavior of Young's modulus of carbon-bonded alumina at high temperatures[END_REF], and highly aects the anisotropy of composite powder materials. In closed-die compaction, the signicant dierence between the axial and radial pressures must produce some anisotropy to the composite. According to [START_REF] Klemm | Microstructure and mechanical properties of ne grained carbon-bonded Al2O3-C materials[END_REF], more oriented graphite akes perpendicular to the pressing direction increases the anisotropy of green samples compacted by closed-die conguration. A study on the evolution of the graphite akes orientation after closed-die and isostatic compaction simulations on discrete element samples and generated from X-ray tomography images could be interesting. Appendix D

User subroutine USDFLD

The DPC parameters are density-dependent. A USDFLD subroutine [START_REF] Kallel | New ceramic technologies and novel multifunctional ceramic devices and structures (CERMAT2): internal report[END_REF] was used, on the closed-die compaction simulation, to calculate the relative density of each element as a eld variable.