Ali Daouadji

Guilhem Mollon

Bilel, Lucas, Julien Alejandro Fan

Jacques M Et

Chloé, PJ, Valentin, Valentine, Danaé Jean Samuel

Alexandra Asma

Nathalie Casas

The fault gouge, generated by the wear of previous slips, plays a key role in the slip stability and weakening mechanisms of a fault. For a sheared gouge, physicochemical transformations such as rock dissolution or high-temperature fusion can occur. The latter can give rise to infill materials that lodge within the pores of the gouge and strongly impact the mechanical and rheological behavior of the fault. Through this thesis work, three 2D models of sheared gouges were proposed using the Discrete Element Method for a thorough understanding of : (i) how infill materials (matrix or cement) participate in the weakening of the fault during a slip reactivation, (ii) the rheological behavior observed through the formation of shear bands and its link with the physical and mechanical characteristics of the gouges, (iii) the contribution of each of these properties in the breakdown energy and friction laws observed. A first model highlights three types of cemented materials with different rheological behaviors and shearing resistance, depending on the initial surface percentage of cementation. This study also leads to a redrawing of the breakdown energy dissipated by the fault considering the three mechanisms involved in gouge shearing: fault dilation, Coulomb friction, and the cemented bond failure. Then, a material only consisting of the "matrix" element is modeled to highlight the relationship between the intrinsic properties of the matrix and its rheological behavior. The importance of the percentage of matrix present within the gouge is evaluated in a third model. Finally, we detail an energetic method to link the evolution of each deformation band (Riedel bands) wi th the behavior of the entire gouge. The friction laws resulting from these models can be used in larger-scale dynamic models to study fault seismic and aseismic behavior.

Résumé étendu

Introduction générale

Cette thèse académique a pour objectif d' or er e o r o à de e o o e d le cadre des grands enjeux sociétaux actuels. Alors que les modèles urbains sont de plus en plus repensés pour répondre à la croissance démographique et à la transition énergétique, il est important de disposer d'une production d'énergie durable et fiable ayant un faible impact sur 'e ro e e e ' o e. La géothermie profonde (EGS) permet une continuité temporelle de la production par rapport à d'autres sources d'énergie renouvelables intermittentes. Elle pourrait contribuer à répondre à la demande d'énergie actuelle, mais semble faire l'objet de controverses en raison de la sismicité induite observée dans certains projets [START_REF] Grigoli | Current challenges in monitoring, discrimination, and management of induced seismicity related to underground industrial activities: A European perspective[END_REF]. Lors des séismes naturels, le glissement par frottement libère ' er e accumulée dans le milieu environnant d'une faille. En fonction des conditions de pression et de température dues à la profondeur de la faille, les processus de rupture peuvent se dérouler : de manière lente et continue, réduisant la probabilité qu'une instabilité sismique se produise ou de manière rapide et brutale pouvant générer un fort glissement sismique. Les séismes induits sont régis par les mêmes lois physiques, mais sont en partie générés par les technologies et infrastructures de e o e de e e identifiée comme le matériau d'usure issu des précédents glissements. Elle contribue à la stabilité du glissement et joue un rôle prépondérant dans la libération d'énergie lors du relâchement des contraintes accumulées dans la faille. Au sein d'une gouge de faille mature, la cimentation minérale provenant de la dissolution de la roche, de la fusion, ou des particules de matrice issues de la fragmentation de glissements antérieurs, peuvent remplir les espaces poreux restants entre les particules et modifier l'état global de cohésion du matériau. Ces phénomènes donnent naissance à un nouveau matériau granulaire plus résistant combinant son histoire, l'état de la densité initiale, et le matériau de remplissage présent au sein de la gouge de faille. L'objectif principal de cette thèse est d'étudier le comportement des matériaux de remplissage (ciment et matrice) présents dans les failles à des profondeurs proches de celles observées dans les systèmes de géothermie profonde. Les gouges de faille sont ensuite modélisées grâce à un ' ensuite de pouvoir relier directement le comportement de chaque bande de cisaillement observée dans les gouges de faille au comportement global de la gouge (Chapitre 5, Partie C). Une comparaison des différents résultats obtenus dans les différents modèles numériques avec le modèle classique de slip weakening linéaire est ensuite réalisée (Chapitre 5, Partie A). Le travail de e à de de ro e e e 'eà-dire sans considérer de fluide.

Ce travail de thèse résumé dans ce qui suit est schématisé par la Figure 1 suivante : parois de la faille. La gouge est aussi importante car elle fait le lien entre la tribologie et la mécanique des faille grâce à la théorie du troisième corps et des modèles d'usure : même sans la présence de fluide, la gouge joue un rôle de lubrification dans le contact, Figure 2. Dans le cadre de cette thèse, il a été décidé de rester dans le régime de lubrification limite, qui est le contact solide-solide classique (pas de lubrifiant liquide, contact sec). La gouge de faille est définie comme des particules d'usure, similaire au processus de production du 3ème cor ps. Un autre objectif est de reproduire des grains réalistes dans la gouge avec des formes angulaires et à e e e e 'o e ro e d e oe r de e [ [START_REF] Olgaard | The microstructure of gouge from a mining-induced seismic shear zone[END_REF], [START_REF] Lin | S-C cataclasite in granitic rock[END_REF]]. Image tirée de [START_REF] Reches | Fault weakening and earthquake instability by powder lubrication[END_REF], (b) Schéma de la théorie du troisième corps appliquée au système de la faille.

Les mécanismes de déformation des gouges de faille sont des éléments clé dans la compréhension du comportement des failles. Les zones de déformation sont des structures complexes qui présentent des cinématiques et géométries variables en fonction du mécanisme de déformation adopté et des caractéristiques de la roche. Pour un système de failles suffisamment mature (couche de gouge suffisamment épaisse), la localisation du cisaillement a été observée selon des angles spécifiques, et appelée « bandes de cisaillement de Riedel », ou bandes de déformation cataclastiques, [START_REF] Tchalenko | Similarities between shear zones of different magnitudes[END_REF]. Elles reflètent un champ de contraintes hétérogènes se développant en réponse à la déformation [ [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF], [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF], [START_REF] Kaminskaite | Microstructure and petrophysical properties of deformation bands in high porosity carbonates[END_REF]]. La présence de ces bandes de Riedel a été identifiée comme pouvant être responsable d'un affaiblissement mécanique [ [START_REF] Gu | Development of shear localization in simulated quartz gouge: Effect of cumulative slip and gouge particle size[END_REF], [START_REF] Beeler | Frictional behavior of large displacement experimental faults[END_REF]] et comme étant un indicateur d'un changement de stabilité dans le glissement [START_REF] Byerlee | Structures developed sliding and stick-slip in fault gouge during stable[END_REF]. Il est donc essentiel d'étudier le moment de leur formation au cours du glissement et leur orientation. Le lien entre l'orientation des bandes de Riedel et les schémas de glissement n'est pas encore totalement compris. Une partie de la thèse sera donc consacrée à l'étude de la temporalité des bandes de Riedel en fonction des caractéristiques des gouges. Le mécanisme de glissement étudié a lieu dans une faille mature, c'est-à-dire après de nombreux glissements successifs au cours des temps géologiques précédents. Cette faille a pu avoir été cimentée, ou remplie de particules de matrice, au cours du temps sans glissement (par exemple, dans une période inter-sismique). On peut donc affirmer que l'on étudie la réactivation d'une zone de faille remplie de gouge, sans étudier le processus de fragmentation qui est notre point de départ. Deux types de gouges sont considérées, contenant des matériaux de remplissage (matrice ou ciment) qui peuvent se former dans la zone de faille. La « cimentation minérale », désigne un matériau cristallin provenant de la dissolution ou de la précipitation de la roche [START_REF] Walderhaug | Precipitation rates for quartz cement in sandstones determined by fluid-inclusion microthermometry and temperature-history modeling[END_REF], de la fusion partielle (pseudotachylite, matériau non cristallin), [START_REF] Toro | Pseudotachylytes and earthquake source mechanics[END_REF], [START_REF] Fondriest | Pseudotachylyte Alteration and the Rapid Fade of Earthquake Scars From the Geological Record[END_REF]] ou d'autres processus dérivés de glissements antérieurs, 'e r à re o e re e e er e « Matrice » désigne ici les très fines particules sans cohésion, produites par une comminution locale ou par l'introduction d'autres sédiments [ [START_REF] Lee | Comparison of structural features of the fault zone developed at different protoliths: Crystalline rocks and mudrocks[END_REF], [START_REF] Woodcock | Classification of fault breccias and related fault rocks[END_REF]]. Les particules de matrice sont en partie similaires à un matériau granulaire à haute dimension fractale et présentent les mêmes conséquences mécaniques (zone granulaire plus dense avec réduction de la porosité). e o e r re suffisamment grande pour observer les déformations de cisaillement. ' rê est également de pouvoir modéliser les phénomènes se produisant à l'échelle du grain, en représentant le ciment et la matrice remplissant les pores autour des particules de gouge. La modélisation numérique permet d'accéder facilement aux deux échelles sans avoir à construire un système d'observation complexe. La méthode des éléments discrets (DEM) est également util isée pour comprendre la physique granulaire de la gouge de faille en étudiant les interactions entre les grains et leur influence sur les mécanismes de glissement. L'intérêt est également de partir d'une configuration granulaire initiale identique et de pouvoir facilement faire varier la quantité de matrice et de ciment présente dans les échantillons de gouge pour tester une large gamme de o or e e d re ' o d' od e r e er e e e de er et de suivre l'évolution des liens cohésifs créés entre les grains au cours du temps, car de telles données sont difficiles à voir ou à suivre lors d'une expérience en laboratoire ou in situ.

Chapitre 2 : Modèle numérique et méthodes

La Méthode des Eléments Discrets (DEM) est utilisée en mécanique des failles pour o re dre e ro r de ro e e e de o d oe r de e [ [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF], [START_REF] Da Cruz | Rheophysics of dense granular materials: Discrete simulation of plane shear flows[END_REF], [START_REF] Cho | Development of a shear zone in brittle rock subjected to direct shear[END_REF], (Dorostkar et al., 2017a), (Mollon et al., 2021b)]. Elle permet de décrire l'écoulement d'un matériau granulaire où chaque particule a ses propres déplacements et interactions avec les autres particules. Le code utilisé pour les simulations de cette Thèse, MELODY 2D (Multibody ELement-free Open code for DYnamic simulation), est un code C++ permettant de simuler une grande variété de milieux granulaires en 2 dimensions [ [START_REF] Mollon | A multibody meshfree strategy for the simulation of highly deformable granular materials[END_REF], (Mollon, 2018a)]. Un des intérêts de ce code est sa capacité de prise en compte des formes de grains angulaires grâce à un algorithme de contact en 3 étapes e r er e r e r de o e de or e e d' o ro e Dans le cadre du travail de Thèse, des corps rigides sont utilisés pour les grains de la gouge de faille et les parois rocheuses. Ces corps sont caractérisés par leur masse et leur inertie de rotation avec 3 degrés de liberté (deux en déplacements et un en rotation pour la modélisation 2D). Le contour des particules est alors une e o de oe d e de e e de o er e l'application de lois de contact spécifiques. Les travaux de Thèse étant axés plus particulièrement sur l'étude des matériaux de remplissage, l'objectif est de pouvoir modéliser des particules matricielles ou cimentées. Deux lois de contact sont alors considérées : (a) une loi de contact de type Damped-Mohr-Coulomb (utilisée dans le chapitre 4 pour les particules de la matrice), qui est une loi de DEM classique avec un frottement inter-particulaire uniquement entre les particules en contact, et (b) une loi de contact de type Bonded Mohr -Coulomb avec un frottement inter particulaire et une cohésion. Cette loi de contact est similaire au modèle de particules liées (utilisé au chapitre 3 pour les particules cimentées).

Le package Matlab « Packing2D » permet de créer des milieux granulaires avec des formes de particules angulaires et à facettes, qui restent complexes à générer directement à partir du code DEM. Le logiciel est basé sur une méthode Fourier-Voronoï qui génère un ensemble de grains avec une distribution de taille définie par l'utilisateur. Il est aussi possible de contrôler les descripteurs morphologiques des grains tels que l'élongation, la circularité et la rondeur. Ce contrôle est effectué en choisissant un spectre de Fourier qui quantifie les fréquences et les amplitudes des aspérités de la surface des grains. Une transformation de Fourier inverse permet or de er d e re re r e e o o r d' e r e à r e or e de la particule. De plus amples informations sont disponibles dans [START_REF] Mollon | Fourier-Voronoi-based generation of realistic samples for discrete modelling of References granular materials[END_REF]. Trois différents échantillons granulaires sont conçus pour la thèse grâce à ce code, Figure 3: (a) un échantillon de gouge avec des grains angulaires seuls, (b) un échantillon de gouge avec des grains de matrices seuls et (c) un échantillon de gouge composé de grains angulaires et de grains de matrices. Après avoir été soumis à une compaction uni-axiale pour avoir un échantillon de gouge stabilisé, ces trois échantillons sont ensuite insérés dans le modèle de faille DEM créé sous MELODY2D (Figure 4). Tous les modèles utilisés dans la Thèse présentent la même structure et les mêmes paramètres externes : la paroi inférieure est fixe, tandis qu'une contrainte normale 𝜎 𝑁 (direction y) et une vitesse de cisaillement 𝑉 (direction x) sont appliquées sur la paroi rocheuse supérieure. La gravité est ignorée dans le modèle, en supposant que la faille peut être orientée dans une large gamme de directions, et que les forces de gravité sont négligeables par rapport à celles liées aux contraintes normales et déviatoriques appliquées sur la faille. Des conditions aux limites périodiques sont présentes sur les côtés droit et gauche de l'échantillon pour maintenir la continuité du mouvement lors de grands glissements. Le mouvement de la paroi rocheuse supérieure dans la direction y reste libre pour tenir compte de la dilatation ou de la compaction de l'échantillon. Grâce à une étude préliminaire sur l'influence du frottement inter-particulaire pendant les étapes de compaction et de cisaillement, deux états de porosités initiales ont été retenus : un échantillon de gouge « moyennement dense » avec une porosité initiale de 16 % et un échantillon de gouge « dense » avec une porosité initiale de 11 %. Ces deux échantillons sont ensuite insérés dans le modèle 2D de cisaillement direct et une loi de cohésion est appliquée entre les particules pour venir simuler une cimentation entre les grains (de 0 à 95 % de cimentation). Chaque échantillon est ensuite cisaillé à une vitesse de 1 𝑚/𝑠 et sous une contrainte normale de 0 𝑀𝑃𝑎. Pour caractériser les gouges cimentées synthétiques (au sens de Coulomb-Mohr), des simulations biaxiales indépendantes ont aussi été effectuées avec des échantillons de gouge présentant les mêmes caractéristiques que celles utilisées pour les gouges de faille cisaillées. Les gouges simulées présentent des valeurs cohérentes d'angle de frottement interne (entre 30° et 5°) et de cohésion (entre 0 et 70 𝑀𝑃𝑎). Les différentes simulations ont permis de mettre en évidence un pic de résistance au frottement très élevé et rapide pour les matériaux denses et fortement cimentés, et un pic plutôt lisse, retardé et d'amplitude modérée pour les matériaux moyennement denses et moyenne me n t cimentés. Le caractère fragile des matériaux est renforcé par la présence de cimentation, en particulier dans le cas des matériaux denses. Il évolue de manière similaire à la présence de cohésion interne (Coulomb-Mohr) au sein de la gouge et conduit à trois schémas de rupture différents, Figure 5. Les bandes de cisaillement observées nécessitent un taux de dilatation critique de la gouge pour se former. un autre matériau de remplissage, cette fois non cohésif, appelé « matrice ». Il se réfère à des particules très fines produites par fragmentation des roches dû aux précédents glissements dans la faille ou à l'introduction de sédiments dans la gouge de la faille [START_REF] Woodcock | Classification of fault breccias and related fault rocks[END_REF]. Deux autres modèles de gouge de faille granulaire 2D ont été implémentés en DEM afin d'étudier l'influence des caractéristiques des particules de la matrice sur le comportement rhéologique de la faille, ' o o de de de ede e ' e e d o r e e de r e/ r d ' o de o e La Partie A présente des échantillons de gouge contenant uniquement des particules de matrice hexagonales dans le but de caractériser ce matériau. Ce milieu granulaire très dense révèle la formation de bandes de Riedel lors du cisaillement. Cette première partie permet de comprendre les schémas de localisation de ces bandes et leur évolution lorsque différentes caractéristiques physiques et géométriques des particules de matrice sont modifiées. La variation des propriétés géométriques (forme, taille et distribution des grains) ou physiques (raideur et frottement interparticulaire) de la matrice peuvent représenter différentes lithologies rocheuses, avec une grande variété de comportements rhéologiques. Il est apparu que toutes ces variations lithologiques pouvaient être résumées en seulement trois paramètres ayant une influence sur le comportement de glissement et les bandes de Riedel au sein du modèle: (a) le module de cisaillement global de la gouge (modification de la raideur inter-particulaire ou la taille des particules), (b) le rapport entre l'épaisseur de la gouge et la taille des grains (c'est-à-dire le nombre de particules dans l'épaisseur de la gouge), et (c) le frottement inter-particulaire. Une gouge de faille avec un faible coefficient de frottement inter-particulaire, un faible nombre de grains dans l'épaisseur de la gouge ou encore un module de cisaillement élevé présentera, dans les trois cas, une diminution de l'angle d'orientation des bandes de Riedel par rapport à la d re o de e e e e r d o d' e e e r e dû à r e e d' e r re de Riedel avec un grand nombre de bandes de Riedels conjuguées (𝑅') limitant la formation de bande R et retardant l'arrivée d'un état stabilisé. Comme l'angle d'orientation des bandes de Riedels peut être considéré comme un indicateur de stabilité pour la gouge de faille granulaire [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF] od o d' de e r re o e e e de faire passer le comportement de la gouge de faille d'un glissement soudain et sismique à un glissement modéré et lent, ou inversement. Cependant, les conséquences sur la résistance au cisaillement de la gouge, la longueur de nucléation et l'énergie de rupture ne sont pas similaires dans tous les cas et conduisent à différents types de comportement de glissement. Les effets combinés de ces trois paramètres principaux conduisent à différents matériaux de gouge plus ou moins susceptibles de générer des glissements stables ou instables.

La Partie B présente quelques résultats sur des mélanges composites entre des grains anguleux entourés de cellules hexagonales représentant la matrice. Elle met en évidence les similitudes et les différences avec le modèle précédent avec uniquement des particules de matrice (Partie A) et avec de la cimentation (Chapitre 3). également la forme et l'épaisseur des bandes de cisaillement dans les échantillons. L'insertion de particules angulaires dans une gouge initialement composée de cellules matricielles ou, ' jo de e e de r e d lon granulaire renforce la gouge (pic de frottement plus important), et donne des comportements d'affaiblissement différents selon le type de distribution des particules dans l'échantillon, Figure 6. Dans tous les cas, le radoucissement du frottement par glissement sera plus soudain pour un matériau mixte (matrice + grains) que pour r o o e e e d' e e de r e e r o d e e de radoucissement implique également la possibilité de faire évoluer le comportement de la faille vers un glissement sismique. Lorsque que le nombre de particules angulaires est assez élevé, une distribution non homogène des particules dans l'échantillon augmente encore la résistance au cisaillement et l'énergie nécessaire pour affaiblir la faille est beaucoup plus élevée : les bandes de Riedel ne peuvent pas rejoindre les parois rocheuses conduisant à une déformation de toute l'épaisseur de la gouge, Figure 6 D'autres résultats sont nécessaires pour analyser les différents comportements observés tels qu'une modification de la déformabilité intrinsèque du matériau de la matrice, pour représenter la teneur en argile par exemple.

Chapitre 5 : Du bilan énergétique aux modèles simplifiés, vers les simulations dynamiques

Ce dernier chapitre est à la fois une synthèse des chapitres précédents et une ouverture à donner à la thèse. A partir des résultats des chapitres 3 et 4, et en se focalisant sur les lois de frottement et le bilan énergétique, l'objectif est de définir des lois de frottement caractéristiques de chaque type de matériau de remplissage pouvant être injectées dans un modèle de faille dynamique à grande échelle.

La Partie A permet d'étudier plus en profondeur les résultats obtenus dans les chapitres précédents (ciment et matrice), en les interprétant selon un modèle très simplifié tel que le modèle linéaire de radoucisseent par glissement (linear slip weakening). Grâce à cette représentation, il est possible d'extraire la pente de radoucissement associée aux modèles, ce qui donne une information sur la stabilité du système. L'énergie de rupture et la longueur critique de nucléation associées à chaque modèle sont calculées pour être mises en relation avec l'intensité des événements sismiques qui pourraient se produire. Après avoir décrit la méthode utilisée pour récupérer et calculer les différents paramètres, les résultats obtenus pour les différents modèles sont comparés et discutés, en mettant l'accent sur la longueur de nucléation critique associée aux simulations et sur l'influence de chacun des matériaux testés, Figure 7 (a). Les matériaux de remplissage ont un rôle majeur dans la longueur critique de nucléation et l'énergie de rupture libérée. La cimentation à l'intérieur de la gouge semble être le paramètre le plus influent testé. Un matériau cimenté initialement dense aura tendance à avoir une distance de glissement très courte avant que les instabilités ne se produisent (petit 𝐿 𝑐 ) ainsi qu'une énergie de rupture élevée (grand 𝐸 𝐺 ). Même s'ils ont lieu dans une gamme spécifique d'expériences (pression et vitesse de glissement imposées et constantes), les résultats soulèvent la question de or e de e e d' e e d e e r î e e e échantillons cimentés moyennement denses et certains échantillons de matrice peuvent être facilement modélisés avec la forme linéaire classique. Les échantillons cimentés denses présentent une forme bilinéaire décroissante qui peut être comparée à un modèle proposé par [START_REF] Abercrombie | o er o o e r ke o r we ke ?[END_REF]. La fracture réelle est bien connue pour être beaucoup plus complexe, mais les modèles complexes sont difficiles à mettre e oe re. Un équilibre doit être trouvé entre un modèle trop simple et un modèle trop complexe. Pour cette raison, d'autres lois de frottement et d'énergie reliant la rhéologie locale et la mécanique de la gouge sont proposées dans les deux parties suivantes.

La Partie B reprend les résultats du Chapitre 3 sur les modèles cimentés. Afin d'isoler les principaux mécanismes agissant sur le processus de fracture, un découpage du frottement macroscopique total généré lors du cisaillement d'un échantillon de gouge granulaire est proposé. Sur la base du contenu physique du modèle numérique utilisé, trois facteurs sont identifiés comme jouant un rôle dans le frottement effectif total et le bilan énergétique, Figure 7 (b) : la dilatance de la gouge, ou déformation dans la direction perpendiculaire au glissement, les dommages provenant de la rupture des liens cohésifs dus à la cimentation et le frottement généré par les interactions de contact glissant. Cette représentation permet la construction d'un modèle de frottement macroscopique simplifié qui a pour objectif final d'être injecté dans des modèles de rupture dynamique à une échelle supérieure. L'énergie de rupture et le comportement de glissement de la gouge de faille semblent être contrôlés par l'intensité et e or de e ro e ' er e de r re e r e e o o e de l'énergie liée à la dilatance de l'échantillon, qui a toujours un rôle prépondérant dans les mécanismes de rupture et de propagation des fissures. Selon la porosité initiale de l'échantillon ou le pourcentage de cimentation, l'influence de la dilatation n'est cependant pas similaire et n'induit pas le même comportement mécanique. En effet, le matériau le plus susceptible de générer des instabilités de glissement semble être un matériau dense et cimenté (entre 25 % et 75 % de cimentation) : les agglomérats de particules cohésives générés dans la gouge rendent la faille plutôt hétérogène et augmentent la rugosité dans la faille, ce qui augmente l'énergie de dilatation de manière plus significative et implique une réponse de vitesse de glissement beaucoup plus élevée. Le premier affaiblissement observé est un mélange des trois mécanismes (dilatance, frottements et rupture des liaisons cimentées), et présente une augmentation de l'énergie de frottement avec l'augmentation de la cimentation dû à la réduction du nombre de particules en contact. Les résultats montrent un second affaiblissement, principalement dû à la d o de ' er e de d e e od e ro o o ez o quant à la forme finale des courbes obtenues et pourraient être adaptés à d' re contraintes et vitesses imposées. Cependant, la caractérisation des modèles cimentés moyennement denses pourrait être améliorée, car l'énergie totale de rupture ne s'accorde pas très bien avec les données observées.

La Partie C est construite sur les résultats du Chapitre 4 Partie A, et présente une analyse spatioe ore e de ' tivation des bandes de Riedel et de leur implication dans le comportement global des gouges de faille. Tout d'abord, il est possible d'observer des variations à la fois globales et locales de l'ouverture et du glissement au sein des bandes de Riedel. L'angle d'orientation de ces bandes peut être différent selon que la zone observée se trouve au centre de la gouge ou près des parois rocheuses. Ensuite, l'évolution temporelle des bandes de Riedel avec la distance de glissement peut être observée en se focalisant sur la variation globale d'ouverture de chaque bande. Les bandes primaires et les bandes secondaires apparaissent ensembles, mais les bandes primaires croissent plus rapidement, laissant place aux bandes secondaires seulement au moment du pic de frottement. Lorsque les bandes secondaires deviennent « actives », elles désactivent alors les bandes primaires. Les bandes Y évoluent progressivement jusqu'à l'état d'équilibre où elles deviennent les seules bandes « actives » dans la o e 'e -à-dire le lieu principal des déformations et glissements. Cette représentation des bandes de Riedel est vraiment prometteuse pour réaliser des microanalyses de l'intérieur de chacune des bandes R et Y. Enfin, le lien entre le comportement mécanique à l'intérieur de chaque bande à la cinématique globale de la gouge a été étudié. Une vue d'ensemble de la méthode est proposée avec une description énergétique et des relations appropriées entre chaque bande et la gouge, Figure 7 (c). La contribution énergétique de chaque bande est en accord avec leur localisation dans le temps et l'espace mais donne une autre façon d'observer les mécanismes de Riedel et de conclure sur leur rôle et leur activité. Bien que certaines simplifications et recherches soient encore nécessaires pour obtenir un meilleur modèle énergétique, ce petit modèle simplifié est très prometteur pour étudier le bilan énergétique et le rôle de chaque bande de Riedel dans le mécanisme d'affaiblissement en fonction des caractéristiques de chaque matériau de remplissage. e de e e e o r ' er e de r re 𝐸 𝐺 (en gris). Avec 𝜏 p la contrainte de cisaillement maximale (Pa), 𝜏 ss la contrainte de cisaillement steady-state (Pa) et 𝐷 𝑐 la distance critique de glissement (m). (b) Modèle proposé pour les gouges cimentées avec la décomposition de l'énergie de rupture 𝐸 𝐺 . 𝐸 𝐺 regroupe une énergie de dilatation 𝐸 𝐷𝑖 𝑎 𝑖𝑜𝑛 , une énergie 𝐸 𝐷𝑎𝑚𝑎𝑔𝑒 provenant de la rupture de la cimentation (énergie de création de surface) et 𝐸 𝑓 e r e de ' er e de ro e e de o o 𝐸 𝐻 . (c) Modèle proposé pour les gouges avec matrices avec la décomposition de l'énergie de rupture 𝐸 𝐺 : 𝐸 𝐺 regroupe une énergie de dilatation 𝐸 𝐷𝑖 𝑎 𝑖𝑜𝑛 et une énergie du aux frottements 𝐸 𝑓 .

Conclusions et perspectives

Le gouge de faille, provenant de l'usure des glissements précédents, agit sur la stabilité du frottement et joue un rôle clé dans la libération soudaine d'énergie. Une grande partie des mécanismes de glissement est influencée, voire contrôlée, par les caractéristiques de la gouge de faille et son environnement. Au cours de ce travail de thèse, une modélisation par éléments discrets en 2D a été réalisée pour mieux comprendre (i) comment le matériau de remplissage (matrice ou ciment) participe au renforcement ou à l'affaiblissement de la faille, (ii) les comportements rhéologiques observés mettant en évidence la formation de bandes de cisaillement, et le lien avec les caractéristiques de la gouge, (iii) la contribution de chacun de ces mécanismes dans l'énergie de rupture et les lois de frottement observées. Trois types principaux d'échantillons de gouge ont été générés pour simuler une gouge de faille mature avec un matériau de remplissage, puis insérés entre deux parois rocheuses pour réaliser des expériences numériques de cisaillement direct à l'aide du logiciel MELODY2D. Un modèle de contact sec a été considéré pour étudier les mécanismes sans fluide (à vitesse de cisaillement et contrainte normale constantes). Les simulations numériques réalisées et les analyses associées ont permis de répondre aux questions soulevées dans l'introduction générale et mettent en évidence que les deux matériaux de remplissage étudiés (matrice et ciment) ont un rôle prépondérant dans l'affaiblissement des gouges de failles matures. Ils jouent un rôle dans la r e de o e e e r r de r o o e d ' o e e granulaire et les déformations de la gouge (agglomérats de particules, bandes de Riedel). Dans un second temps, le frottement macroscopique a été décomposé en plusieurs mécanismes r d e d'o e r od e de ro e e o rr e e ê re injecté dans un modèle dynamique à plus grande échelle.

Suite au travail effectué, plusieurs perspectives sont proposées à cette thèse. La recherche effectuée s'est concentrée sur un ensemble d'expériences numériques sous une contrainte normale, une vitesse de glissement et une épaisseur initiale constantes. Il pourrait être intéressant d'étendre la campagne numérique à une gamme plus large de contraintes normales, de e e e d' e r de o e d' orer e o de ro e e e r es basées sur les propriétés micromécaniques de la gouge. L'analyse détaillée de l'évolution des bandes de Riedel pendant le glissement n'a été réalisée que pour un seul échantillon de gouge, il est envisagé de réaliser la même étude sur les autres simulations avec des cinématiques différentes. Les résultats permettront d'observer comment chaque caractéristique de la gouge va influencer l'ouverture et l'activité des différentes bandes de Riedel. Ils permettront également d'enrichir le modèle énergétique proposé en fonction des caractéristiques de la gouge. Maintenant que des modèles énergétiques ont été proposés pour les différentes gouges de failles étudiées, il serait intéressant de les insérer dans des modèles dynamiques à plus grande échelle pour étudier le comportement sismique ou asismique de ces différents matériaux de gouge. Ces modèles seraient très instructifs pour compléter les informations apprises sur les matériaux de remplissage (ciment et matrice) à l'échelle millimétrique et surtout pour tester les lois de frottement à des échelles plus grandes e e r e er 'o je d' e ro e de ' re perspectives sont aussi envisageables telles que ' de de o e de e e e hétérogènes, la modélisation de grains déformables avec la soft DEM pour représenter des matériaux faibles e e ' r e ' o o de ' re de ro de e e od e e CZM ou encore le contrôle de la température dans la gouge.

General introduction

Motivations

During natural earthquakes, frictional sliding releases the stresses accumulated in the prestressed surrounding medium of a fault. Depending on the pressure and temperature conditions due to the depth of the fault, rupture processes can take place: slowly and continuously, reducing the probability that a seismic instability occurs or in a fast and brutal manner which can generate strong seismic sliding. There also exist induced earthquakes, governed by the same physical laws, but partly generated by geoengineering technologies and infrastructures that can modify the surrounding stress field of a fault.

While urban models are increasingly rethought to respond to population growth and energy transition, it is important to have sustainable and reliable energy production with low impact on the environment and human beings. Enhanced Geothermal Systems (EGS) lie in the temporal continuity of the production compared to other renewable energy sources such as solar or wind power, which depend on climatic conditions. They could help meet energy demand but appear to be a controversial issue due to the induced seismicity observed in some projects [START_REF] Grigoli | Current challenges in monitoring, discrimination, and management of induced seismicity related to underground industrial activities: A European perspective[END_REF]. In order to promote the economic viability of geothermal extraction, EGS are generally located on deep rock reservoirs corresponding to depths of 3 to 5 km [(Gentier, 2013), (S.-M. M. [START_REF] Lu | A global review of enhanced geothermal system (EGS)[END_REF]]. At this depth, fault networks are already present and subjected to high stresses reducing the permeability of host rocks. Hydraulic stimulation is often used to reopen existing fault systems and improve fluid transport capacity. Although techniques are available to reduce the increase in pore pressure generated by the introduction of fluid, the effective stress is still reduced within the fault, which may induce fault rupture and re-activate slips in the existing fracture network. The slips generated can either be aseismic (slow slip [START_REF] Cornet | Seismic and aseismic motions generated by fluid injections[END_REF] or e w " d ed ro-e " [START_REF] Talebi | Analysis of the microseismicity induced by a fluid injection in a granitic rock mass[END_REF]. Even if the magnitude is generally smaller than for natural earthquakes (𝑀 𝑤 < ~3), tremors can be felt at the Earth's surface [START_REF] Bourouis | Evidence for coupled seismic and aseismic fault slip during water injection in the geothermal site of Soultz (France), and implications for seismogenic transients[END_REF]. These tremors remain perceptible most of the time and lead to negative reactions from the inhabitants who stand out against this energy [START_REF] Majer | Induced seismicity associated with Enhanced Geothermal Systems[END_REF]. Recently, the relation between scientists working EGS, and citizens living near the EGS project of Strasbourg in France was questioned and reported in an interesting Master Thesis, directly associated with the present Thesis, as a socio-political understanding of the research on EGS and induced-seismicity, [START_REF] Voisin | Savoirs et politiques de la sismicité induite en France[END_REF].

The fault gouge, identified as the wear material of previous slips, contributes to friction stability [START_REF] Marone | The depth of seismic faulting and the upper transition from stable to unstable slip regimes[END_REF] and plays an important role in the sudden energy release during seismic sliding [START_REF] Sammis | The kinematics of gouge deformation[END_REF]. The number and amplitudes of successive slips occurring within the gouge reduce the size of particles towards a fractal distribution, which also reduces pore spaces [ (Sammis & Biegel, 1989), [START_REF] Blenkinsop | Cataclasis and Processes of Particle Size Reduction[END_REF], [START_REF] Muto | Fractal particle size distribution of pulverized fault rocks as a function of distance from the fault core[END_REF]]. Within a mature fault gouge, mineral cementation coming from rock dissolution, melting, or matrix particles from previous slips fragmentation, can fill remaining pore spaces between particles and change the global state of cohesion [START_REF] Philit | Clusters of cataclastic deformation bands in porous sandstones[END_REF]. These phenomena give birth to a new, stronger granular material combining its history, the state of initial density (i.e. porosity within the sample), and infill material present within the granular fault gouge [START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF]. Through this Thesis work, properties of infill materials, which could have been formed on the d' re e ed we e r e e o e d . This work is based on previous research done on deep geothermal energy systems considering slip reactivation in fault zones where fluid may have flowed and on research on fault gouge mechanics. The role played by the fluid itself will not be considered. The results can be either used within the deep geothermal community, as a post or pre-installation state of deep geothermal power plants or within the seismic fault community since they obey to the same basic principles without fluid. My thesis work was carried out in the LaMCoS Laboratory (Contact and Structural Mechanics Laboratory), team TMI (Tribology and Interface Mechanics) and at GEOMAS Laboratory (Civil Engineering Laboratory) at INSA Lyon. We also exchanged a lot with the BRGM through Arnold Blaisonneau, Julie Maury and Théophile Guillon.

Objectives and outlines

The main objective of this thesis is to study the behavior of infill materials in faults at depths close to those observed in deep geothermal energy systems. However, we have restrained our work to dry friction. We have chosen to focus specifically on numerical granular physics to study the behavior of fault gouges within the fault core. For that purpose, we will address and answer the following questions all along the report:

• What role plays cementation in the weakening of mature fault gouges? Do cemented materials promote slips in faults? How can the energy balance of a cemented gouge be decomposed and what mechanism remains most influential? • What role plays the matrix particles in the weakening behavior of mature fault gouges?

How important are the different physical and geometric characteristics of matrix particles in the slip behavior? To which extent is there an influence on the proportion of matrix/grain and their distribution in the fault gouge samples?

Based on the main questions mentioned, a few more questions need to be addressed:

• How can the characteristics of the fault gouge material be related to its rheological properties? Which type of Riedel patterns are observed and with which orientation? • What friction law better represents the behavior of a cemented granular fault? And what frictional models can be used in modeling dynamic slips? • How the behavior of each shear band observed in a fault gouge can be related to the overall behavior of the gouge?

This report thesis is structured into five Chapters reported in the following and schematized by Figure 8:

• Chapter 1 gives an overview of the basics of earthquake, fault mechanics as well as the study of friction in faults and empirical models used. It also focuses on the properties of the fault gouge whose mechanical and rheological characteristics are key elements in the understanding of the sliding mechanisms of a seismic fault. This chapter also presents the research strategy adopted for the work. • Chapter 2 describes the numerical models and methods. It starts by presenting the main concepts of DEM, the code used as well as all parameters and contact laws necessary In the first two parts, this chapter gives an overview of the basics of earthquake and fault mechanics as well as the study of friction in faults and empirical models used. These concepts are quite well known, so the advised reader can start reading directly from part 1.3 which focuses on fault gouge properties whose mechanical and rheological characteristics are key elements in the understanding of the sliding mechanisms of a seismic fault. This third part focuses in particular on the research strategy adopted for the thesis.

Earthquakes, faults and rocks

What is an earthquake?

When the energy accumulated in the fault reaches the critical stress allowed by the material, the fault suddenly slips and releases the energy resulting in the main seismic shock. The stress excess is then reduced via energy liberation enabling to accommodate the large-scale motion of e e r ' e The stress drop only concerns shear stress and was estimated to be about ~1-10Mpa in natural faults for large earthquakes (as opposed to larger values in the lab), [START_REF] Kanamori | The physics of earthquakes[END_REF]. This stress drop mechanism is then repeated during geological time as a succession between shear stress accumulation (interseismic period) and shear str ess drop (coseismic period), forming the basis of the seismic cycle [START_REF] Byerlee | Stick slip, stable sliding, and earthquakes-Effect of rock type, pressure, strain rate, and stiffness[END_REF].

Plates tectonics

An earthquake can also be presented as a plate motion, or sudden displacement of the crust inducing the propagation of elastic waves [START_REF] Kanamori | The physics of earthquakes[END_REF] while important earthquakes use to be observed along subduction zones (i.e. the frontier between two converging plates) as presented in Figure 9. Nowadays, geodesy technics let us measure the size and shape of the earth's surface, using Global Positioni ng Systems or Satellite Interferometry. All of these methods give access to patterns of crustal movements (cm/year for example) and strain accumulation within plates and make appear larger movements at the plate boundaries than in plate interiors [START_REF] Uyeda | The new view of the Earth[END_REF].

The size of earthquakes can be measured posterior to an event thanks to their magnitude 𝑀 𝑤 obtained from the seismic moment 𝑀 0 [START_REF] Wyss | Apparent Stresses of Earthquakes on Ridges compared to Apparent Stresses of Earthquakes in Trenches[END_REF], itself determined from seismic data (wave amplitude and normal modes), geodetic data (GPS, InSAR), or geological data (surface break of the fault), [START_REF] Kanamori | The physics of earthquakes[END_REF].

Induced earthquakes?

In addition to the previous definitions of earthquakes, it is also interesting to mention induced seismicity. When the released strain energy is predominantly from natural origin but was provoked by a human action we talk about triggered seismicity and when most of the energy is anthropo e e word " d ed" ed Now d w e r e o o e d infrastructures for geo-applications can modify the surrounding stress field of a fault or add new variables into the global system. These stress changes can be due for example to a decrease dro r o e r o … or re e e eo er e dr r r … of fluid pressure in the subsurface and can be a source of seismicity. [START_REF] Grigoli | Current challenges in monitoring, discrimination, and management of induced seismicity related to underground industrial activities: A European perspective[END_REF] present the industrial activities responsible for induced seismicity (Figure 10 (b)) and their physical mechanisms. We number lots of cases of induced seismicity all over the world, Figure 10 (c), gathered by HIQuake (Human-Induced earthquake) database [START_REF] Wilson | HiQuake: The human-induced earthquake database[END_REF].

Slow or fast earthquakes?

As the seismic cycle is complex, a wide range of earthquakes exists from slow to fast slip events [(J. R. [START_REF] Leeman | Laboratory observations of slow earthquakes and the spectrum of tectonic fault slip modes[END_REF], [START_REF] Stefanou | Control instabilities and incite slow-slip in generalized Burridge-Knopoff models[END_REF]], seismic or aseismic motion [ [START_REF] Bourouis | Evidence for coupled seismic and aseismic fault slip during water injection in the geothermal site of Soultz (France), and implications for seismogenic transients[END_REF], [START_REF] Chen | Scaling of small repeating earthquakes explained by interaction of seismic and aseismic slip in a rate and state fault model[END_REF], [START_REF] Cornet | Seismic and aseismic motions generated by fluid injections[END_REF], [START_REF] Mcgarr | Injection-Induced Moment Release Can Also Be Aseismic[END_REF], [START_REF] Chen | On behaviour and scaling of small repeating earthquakes in rate and state fault models[END_REF]], or large to small events. A distinction must be made between fast and slow earthquakes:

-Fast earthquakes mostly present seismic and unstable slips, with a large number of seismic waves radiating during a few instants (seconds to minutes). High rupture fronts velocities are recovered around 3 km.s -1 and are superior to 3.5 km.s -1 for supershear earthquakes [START_REF] Passelègue | From sub-Rayleigh to supershear ruptures during stick-slip experiments on crustal rocks[END_REF]). -Slow earthquakes (i.e. Slow Slip Events (SSE)), more recently discovered thanks to better GPS technics, have lower involved rupture fronts velocities [START_REF] Bürgmann | The geophysics, geology and mechanics of slow fault slip[END_REF] and present less or negligible radiated energy. Their time duration also varies from a few days to years. The formation of slow slips can be due to several factors or phenomena within a wide range of depth and temperature but the presence of both highpressure fluids and heterogeneous fault-zone structure/composition/assemblage seems to be needed. Changes in fluid pressure, for example, or in the coefficient of friction can switch sliding regimes from seismic to aseismic [START_REF] Bürgmann | The geophysics, geology and mechanics of slow fault slip[END_REF]. Image from [START_REF] Grigoli | Current challenges in monitoring, discrimination, and management of induced seismicity related to underground industrial activities: A European perspective[END_REF].

There is still a debate on scaling laws, on whether slow and fast earthquakes are controlled or not by similar dynamic properties [START_REF] Michel | Similar scaling laws for earthquakes and Cascadia slow-slip events[END_REF]. The understanding of the different behaviors between fast earthquakes and SSE is relevant for fault mechanics, as it can indicate how and why a slip may be initiated or not at a critical frictional stress [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF]. For more details on seismicity (seismic and aseismic motions), and the size of events, the reader may refer to [ [START_REF] Bürgmann | The geophysics, geology and mechanics of slow fault slip[END_REF], [START_REF] Mcgarr | Injection-Induced Moment Release Can Also Be Aseismic[END_REF], [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF], [START_REF] Stefanou | Control instabilities and incite slow-slip in generalized Burridge-Knopoff models[END_REF]].

Both natural or induced earthquakes need a fundamental understanding of fault and rock mechanics. Mechanisms of failure in intact rocks or slips in pre-existing fault networks need to be well understood to reduce earthquake hazards and develop safer geo-energy activities.

In the Thesis, we will apply the seismic cycle at a smaller scale of fault rupture, and focus on the role of fault zone internal structure within the schizosphere part of the Earth. In the next sections, we will present the fault zone structure observed, and the basics of the fault and rock mechanics needed to study fault slip behavior.

Fault zone

When talking about earthquakes, it is also essential to define what a fault is. One may first think of the well-known natural faults, such as the San Andreas Fault (F. [START_REF] Chester | Intemal Structure and Weakening Mechanisms of the San Andreas Fault[END_REF], but the variety of geological variables (lithology, displacement, depth, pre-e r … present within the Earth leads to a wide range of different types of faults and internal fault zone structures that also depend on the depth of formation, tectonic environment, protolith, and hydrothermal context.

Brittle-plastic transition: faults and deformation

The lithosphere is the only place where earthquakes can happen as generally, no earthquake happens within the asthenosphere or beyond. Stresses applied to the lithosphere induce deformation of crustal rocks that transform and store elastic energy. Depending on the tectonic context and stress state, it exists three main types of faults: (i) normal faults, corresponding to an extension of plates, quite rare on the continental crust (New Madrid fault zone), (ii) Reverse fault, or thrust faults which are derived from a compressional tectonic context (Main Himalayan Trusts) and (iii) the strike-slip fault or transform faults, derived from lateral sliding along a vertical plane (North Anatolian fault). Figure 11 schematizes the evolution of rocks' behavior in the lithosphere as a function of temperature and pressure. As in rock mechanics, any type of deformation will first admit a reversible elastic deformation (linear stress-strain dependence) then once the yield stress (elastic limit) of the material is passed three main regimes are observed:

-The first 10 km from the surface, or the upper layer of the lithosphere (i.e. schizosphere), generally deform in a brittle manner. Of course, it depends on the temperature and pressure conditions. Irreversible deformations such as rupture or localizations are observed after reaching the yield stress point: microfractures [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF], which first form and gather to create a macroscopic fault, gouge particles [START_REF] Scholz | Wear and gouge formation in brittle faulting[END_REF], as a consequence of rock fragmentation, or even asperity ploughing [START_REF] Scholz | The role of asperity indentation and ploughing in rock friction. Part 1: asperity creep and stick-slip[END_REF] can be observed during shearing. This brittle mechanism also implies an increase of the rock volume (i.e. dilatancy). -The bottom part of the lithosphere (i.e. plastosphere) is considered as a ductile zone. It belongs to the so-called "plastic" regime where the rock can deform irreversibly within a deformation band (shear zones) or in a more distributed way. Then, we tend either towards stabilization of the stress characteristic in a stable flow or towards hardening by deformation. This regime is thermodynamically promoted at high pressures and temperatures. The high ductility in the plastosphere also leads to a decrease in rock strength [START_REF] Brace | Limits on lithospheric stress imposed by laboratory experiments[END_REF]). -Between these two regimes, a semi-brittle regime exists as a mixture between brittle and crystal plastic processes. This is the so-ed " r e-" r o [START_REF] Byerlee | Stick slip, stable sliding, and earthquakes-Effect of rock type, pressure, strain rate, and stiffness[END_REF], promoted by pressure and temperature increase.

Figure 11. (a) Stress-strain representation of brittle-plastic behaviors, curves exhibit (1) elastic deformation followed by rupture, (2) elastic deformation followed by plastic deformation and then by rupture or (3) elastic deformation followed by plastic deformation with viscous flow or hardening. (b) Evolution of the behavior of rocks in the lithosphere as a function of temperatures and pressures. The brittle-plastic transition mostly depends on temperature (increase in ductility with temperature), pressure (ductility increases with applied stresses), and velocity rate (rapid tests show more brittle behavior and slow tests show more ductile behavior). It should also be noted that the water content influences the strength and deformability of rocks. Image modified from (Jérôme Aubry, 2019)

Internal structure from brittle zone

Even if many models propose a planar representation of two fault surfaces in contact, the reality is obviously more complex. The internal structure of fault zones subjected to external loads and strain rate is progressively damaged by the earthquake cycle and shearing. The investigations [START_REF] Chester | Intemal Structure and Weakening Mechanisms of the San Andreas Fault[END_REF], [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF], [START_REF] Billi | The damage zone-fault core transition in carbonate rocks: Implications for fault growth, structure and permeability[END_REF], [START_REF] Wibberley | Recent advances in the understanding of fault zone internal structure: a review[END_REF], [START_REF] Faulkner | A review of recent developments concerning the structure, mechanics and fluid flow properties of fault zones[END_REF], [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF]] on this internal structure and data collected on faults that have already been sheared several times, propose a large scale fault zone model displayed in Figure 12. For the sake of simplicity, the model described here in detail is a one-core fault zone structure, but multiple cores models have also been discussed [START_REF] Faulkner | On the internal structure and mechanics of large strike-slip fault zones: Field observations of the Carboneras fault in southeastern Spain[END_REF]. The fault can be divided into different, more or less damaged zones, Figure 12 (a) & (c):

1) The core zone, or fault core, is the contact interface of the fault where slip is localized on one or several principal slip surfaces. It is composed of pulverized rock material described as a layer of granular material. The fault core can be separated into two parts: the first one is the Principal Slip Surface [START_REF] Rice | Heating and weakening of faults during earthquake slip[END_REF], generally, fault gouge [START_REF] Woodcock | Classification of fault breccias and related fault rocks[END_REF], which is the central zone with a thickness inferior to [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF]. It has lost the original cohesion of the load-bearing rocks. This gouge is surrounded by zones of breccia with coarser particles on both sides or one side of the fault gouge (brecciation process), Figure 12 (b). Frictional heating can also melt the interface, resulting in a fault core mostly composed of pseudotachylyte [ [START_REF] Sibson | Fault rocks and fault mechanisms[END_REF], [START_REF] Toro | Pseudotachylytes and earthquake source mechanics[END_REF]]. The fault core contributes to friction stability [START_REF] Marone | The depth of seismic faulting and the upper transition from stable to unstable slip regimes[END_REF] and plays an important role in the sudden energy release at the onset of seismic sliding [START_REF] Sammis | The kinematics of gouge deformation[END_REF], (cf section 1.4.1).

2) The highly fractured damage zone, of 1-100m thickness (depending on the history of previous slips), envelops the core zone. This damage zone is deformed by seismic waves release. It can be composed of foliated and non-foliated cataclasites (granular material) and is generally assumed to be highly permeable regarding the amount of porosity [ [START_REF] Faulkner | A review of recent developments concerning the structure, mechanics and fluid flow properties of fault zones[END_REF], [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF]]. 3) Finally, the quasi-undamaged host rocks are submitted to the external load. They are protected by the inner fault zones which take all the deformation and release energy.

The host rock type, stress and strain rate, deformation mechanisms, or fault architecture also influence the permeability of the fault which is a key parameter for fluid pressure distribution and volume characterization (for Geo-engineering applications for example). The large-scale model presented here describes highly slipped fault zones named as mature faults by [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF], and we will use this definition in the rest of the Thesis. It is important to specify that most of the variables influencing fault zone are scaledependent and that the sum of all these scale-dependent variables leads to a very complex mechanism. Fault zone needs to be both studied at small and large scales if we want to understand the entire mechanism. The seismic or aseismic character of a fault greatly depends on this fault zone anatomy [ [START_REF] Scholz | The Mechanics of Earthquakes and Faulting[END_REF], [START_REF] Wibberley | Recent advances in the understanding of fault zone internal structure: a review[END_REF]] but is also controlled by fault frictional properties, which are essential to determine the way the fault behaves between stable or unstable slips (cf. section 1.3). Now that we have defined what an earthquake and a fault are, it becomes interesting to detail rock mechanics mechanisms to understand the internal phenomena occurring just before a rupture. [START_REF] Wibberley | Recent advances in the understanding of fault zone internal structure: a review[END_REF]. The notion of fracture mechanics was introduced by [START_REF] Griffith | The theory of rupture[END_REF], in order to explain the phenomenon of fracture and propagation. From this definition, three different failure modes can be observed [ [START_REF] Lawn | Fracture of brittle solids[END_REF], [START_REF] Scholz | The Mechanics of Earthquakes and Faulting[END_REF] [START_REF] Irwin | Analysis of stresses and strains near the end of a crack transversing a plate[END_REF] as the energy to fracture the entire material (as opposed to G, energy for a single crack creation), (cf. section 1.3.3). More details on the associated equations, stresses, and energies can be found in [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF]. The energy released by crack propagation in rock generally increases with the length of the fault, until a state of equilibrium is reached (i.e. for critical stress). This maximum strength, or resistance to cracking, depends on the grain size and stress singularities appearing at the edges of the crack.

Figure 13. Sketch of the three main crack propagation modes: Mode I, opening or tensile mode (the tensile, or the crack displacement is normal to the crack), Mode II in-plane shearing (crack displacement in the shearing plane and normal to the crack edge), Mode III, antiplane shear (crack displacement in the shearing plane and parallel to the crack edge). Image from [START_REF] Scholz | The Mechanics of Earthquakes and Faulting[END_REF].

Principal stresses and failure criteria

Friction and fracture patterns show different characteristics [START_REF] Lockner | Rock failure and Earthquakes[END_REF] and need criteria to quantify the critical stress and strength needed for the different mechanisms involved. The stress field observed on both the upper crust or a piece of rock submitted to an external load is defined by its principal stresses σ 1 and σ 3 (in 2D). These stresses come from the eigenvectors of the stress tensor and represent the normal stresses applied on two properly oriented orthogonal facets. Figure 14 (a) presents a relevant type of rock rupture observed for brittle rocks, with oblique sliding with respect to the maximum compression direction (2D). By convention in rock mechanics, positive stress values are compressional and it is assumed that σ 1 and σ 3 are positive in the Earth. The equilibrium condition of a small element subjected to stresses in 2D gives the following equations, with 𝜏 the shear stress and 𝜎 𝑁 the normal stress on the fault plane, and 𝛽 the angle between the shearing plane and the major principal stress 𝜎 1 :

𝜏 = (𝜎 1 -𝜎 3 ) 2 s (2𝛽) (1.1) 𝜎 𝑁 = (𝜎 1 𝜎 3 ) 2 - (𝜎 1 -𝜎 3 ) 2 os(2𝛽) (1.2)
re r er were r de ed or e re o " ro k" o o ed o o er r r material found in the crust or fault as a granular gouge. Considering intact rock first, the rock strength is generally considered to follow the Coulomb-Mohr failure criterion [ [START_REF] Handin | On the Coulomb-Mohr Failure Criterion[END_REF], [START_REF] Jaeger | Friction of rocks and stability of slopes[END_REF]], which states that the maximum tangential stress 𝜏 𝑚𝑎𝑥 that the material can withstand is the sum of a constant term

𝜏 𝑚𝑎𝑥 = 𝐶 𝜎 𝑁 (𝜑) (1.3) 𝜇 𝑓 = (𝜑) ( 1.
)

𝜑 being the internal friction angle of the material, or angle between the failure surface (i.e. envelope) and the horizontal axis when drawn in a Mohr plane. Figure 14 (b) shows the stresses represented in a Mohr plane. It is possible to construct what we call the failure envelope, or envelope of points defining failure for specific range or shear and normal stresses. Considering a cohesionless material, the C value is null and the failure envelope passes through the zerostress value. For more details on failure criteria, the reader can refer to [ [START_REF] Lockner | Rock failure and Earthquakes[END_REF], [START_REF] Jaeger | Fundamentals of Rock Mechanics[END_REF]].

The cohesion value C can be easily determined with Biaxial or Triaxial experiments for a specific temperature, pressure range, and loading rate [ [START_REF] Sammis | Self-similar cataclasis in the formation of fault gouge[END_REF], [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF], [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF], [START_REF] Noël | Time-Dependent Deformations of Sandstone During Pore Fluid Pressure Oscillations: Implications for Natural and Induced Seismicity[END_REF]]. During these tests, the specimen is subjected to both an axial compressive stress (force exerted on the top of the specimen) and a radial confining pressure (exerted by a fluid on the walls of the specimen), Figure 14 (a). Several tests may be carried out on the same sample, to consider the history of stresses. Hubbert-type shear box can also be used to determine the extrapolated cohesion and friction coefficient [START_REF] References Abdelmalak | Tectonophysics Description of new dry granular materials of variable o e o d r o oe e o or laboratory modeling of the brittle crust[END_REF]. If pore fluid pressure 𝑃 𝑓 is present within the fault, the same Mohrcircle representation is possible but considering effective pressure instead of normal stress (𝜎 𝑒𝑓𝑓 = 𝜎 𝑁 -𝑃 𝑓 ).

From Mohr-Circles representation, it is possible to relate 𝛽 to the internal friction angle 𝜑.

Major and minor principal planes can be observed during shearing and each point of the circle represents a fault orientation. The combination of the previous equation and trigonometrical relations enables to link the fault plane orientation angle to the coefficient of internal friction: All these definitions allow us to describe the large-scale behavior of a fault and its formation (rupture tip), i.e. the state of limit stress that a fault can support as well as the type of rupture (brittle, semi-brittle, plastic) observed. However, fault mechanics is also played out on a smaller scale, and by considering the properties of the contact between the two rock walls of a fault. The way the walls slide against each other primarily depends on the frictional properties of the materials. We explore in more detail the frictional and energetical contact theories in the next subsection.

Rock friction and empirical models

From a physical point of view, friction is described as a resisting force opposed to the relative displacement between two contacting surfaces. Friction is the focal point of tribology (the Science of contacts), but it appears to be present wherever a contact is observed between two different bodies. Rock and fault mechanics are not exceptions, and also assign a great importance to this parameter that acts at different scales [START_REF] Jaeger | Fundamentals of Rock Mechanics[END_REF]. This section presents some methodological elements to study and model faults sliding using tribological concepts.

Theoretical concepts

Leonardo Da Vinci brought the first bases of friction and resistance to motion in the 15 th century. Two centuries later in France, Amonton studied the characteristics of two contacting bodies and proposes that (i) the friction force is independent of the size of the contact surface and the second that (ii) the friction force is proportional to the normal load acting on the surface. This effective friction will be used all along the Thesis. As friction coefficient 𝜇 𝑓 is weakly dependent on normal stress [START_REF] Lockner | Rock failure and Earthquakes[END_REF], effective friction 𝜇 mostly depends on the presence of a third body (i.e. material filling the contact interface, as a granular fault gouge (section 1.2.2), and surface geometry (roughness of ro k o er e … er [START_REF] Byerlee | Friction of rocks[END_REF] proposed that for intact rocks, the maximum shear strength before failure was independent of rock mineralogy. [START_REF] Marone | Fault zone strength and failure criteria[END_REF]. Even though these rules are widely used to predict rock strength and effective friction for many materials, certain layered minerals and phyllosilicates [ [START_REF] Niemeijer | Influence of phyllosilicates on fault strength in the brittle -ductile transition: Insights from rock analogue experiments[END_REF], [START_REF] Collettini | Fault zone fabric and fault weakness[END_REF], [START_REF] Collettini | Beyond Byerlee friction, weak faults and implications for slip behavior[END_REF]], gouge material layers [START_REF] Marone | Fault zone strength and failure criteria[END_REF], or fluid-filled faults [ [START_REF] Schmocker | Granular flow and Riedel band formation in water-rich quartz aggregates experimentally deformed in torsion[END_REF], [START_REF] Collettini | Fault zone fabric and fault weakness[END_REF]] re e we ker e or do ' w e e r e [START_REF] Imber | Frictional -viscous flow, seismicity and the geology of weak faults: a review and future directions[END_REF]. Some effective friction coefficients obtained for weaker rocks are also displayed in Figure 15 [START_REF] Collettini | Beyond Byerlee friction, weak faults and implications for slip behavior[END_REF]. For granular materials, powders, and fault gouge, special attention needs to be paid to frictional properties, which are more complex than for classical intact rock (section 1.4).

These new definitions of the contact allow us to characterize the stability of the fault according to empirical friction models that we will present in the following section. Friction is a key parameter to follow the weakening of a fault, and its dual dependency on time and velocity gave birth to friction laws.

Stick-slip and friction scaling laws

Stick-slip and stability

The seismic cycle, previously presented in subsection 1.2.1 with stress drop, is the analogy of e " k-" de o r ed [START_REF] Byerlee | Stick slip, stable sliding, and earthquakes-Effect of rock type, pressure, strain rate, and stiffness[END_REF] during experimental tests on a cylindrical sample of rock. When a slip is initiated within the fault, it can either be slow and stable or sudden with an associated stress drop enhancing dynamic instabilities. These instabilities are followed by a period without movement during which the stresses are restored (stick), and then again by instabilities (slip).

From an engineering point of view, this phenomenon is often presented as an analogy of a massspring sliding system [ [START_REF] Scholz | The Mechanics of Earthquakes and Faulting[END_REF], [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF]], Figure 16 (a). The spring represents the stiffness of the material surrounding the fault and the bloc system is submitted to both external load (normal force N) and constant velocity (V) applied on the free spring extremity. These stick-slip experiments are highly reproduced in laboratories or numerical experiments, as analogs to natural earthquakes [ [START_REF] Byerlee | Stick slip, stable sliding, and earthquakes-Effect of rock type, pressure, strain rate, and stiffness[END_REF], [START_REF] Byerlee | Structures developed sliding and stick-slip in fault gouge during stable[END_REF], [START_REF] Aharonov | Stick-slip motion in simulated granular layers[END_REF]), (J. [START_REF] Leeman | Stiffness evolution of granular layers and the origin of repetitive, slow, stick-slip frictional sliding[END_REF], [START_REF] Dorostkar | Stick-slip dynamics in dry and fluid saturated granular fault gouge investigated by numerical simulations[END_REF], [START_REF] Passelègue | From fault creep to slow and fast earthquakes in carbonates[END_REF]]. These notions are very important for the following since they will allow characterizing the stable or unstable character of the observed fault slips.

As presented in Figure 16 (b), slip instabilities may occur if the stiffness of the fault K (i.e. stiffness of the loading system for experimental setup) appears to be lower than 𝑐 . In this case, the slope of the force-slip curve passes through a possible friction peak ( 𝑐 > ). Conversely, sliding can occur in a stable manner if > 𝑐 (Figure 16 (b), blue line).

Figure 16. (a) a block-slider model submitted to normal and tangential forces with a stiffness K and a constant velocity V. In the first phase, the spring stores energy, and the tangential resisting force is strong enough to maintain a static contact of the system: this is the stick phase (or interseismic period). Even if the shear strength is increasing, there is no displacement of the mass block. In a second phase, the external force applied to the spring becomes higher than the resisting force, triggering sliding. The block mass is suddenly put into motion and slides until part of the stored energy is released: this is the slip phase (b) Schematic diagram which illustrates the limit of stability as a force-displacement.

Friction scaling laws

Since stick-slip is defined as the analogy of a seismic cycle, it becomes interesting to establish frictional laws to define and model precisely the contact between two fault walls sliding against each other. Frictional forces evolution can be modeled by constitutive laws grouped into two main classes: the slip-dependent laws (considering that friction only depends on fault slip and basic geometry) and the rate-and-state dependent laws (friction is a function of slip velocities and state variables), [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF].

Slip-dependent laws, slip weakening -First observations of the weakening effect were made by [ [START_REF] Dieterich | Time-dependent friction in rocks[END_REF], [START_REF] Scholz | Detailed studies of frictional sliding of granite and implications for the earthquake mechanism[END_REF]]. The slip weakening is related to fracture mechanics and slipping zone and is mainly based on surface geometry and contact properties. It is thus influenced by both the roughness of the fault surface and the thickness of the gouge [START_REF] Marone | Scaling of the critical slip distance for seismic faulting with shear strain in fault zones[END_REF]. Figure 17 shows two schematic views of slip-weakening, (a) is the theoretical " e r-we ke " w e most simplified view of a weakening from the peak of friction to residual friction, and (b) is the same slip weakening but with a more realistic aspect of a nonlinear weakening, [START_REF] Ohnaka | The physics of rock failure and earthquakes[END_REF]. The shape of the slip weakening curve can differ as a function of fault characteristics and other representations are employed [ [START_REF] Kanamori | Microscopic and Macroscopic Physics of Earthquakes[END_REF], [START_REF] Abercrombie | o er o o e r ke o r we ke ?[END_REF]].

In the case of a classic stick-slip phenomenon, [START_REF] Rabinowicz | The nature of the static and kinetic coefficients of friction[END_REF] demonstrated the existence of a critical sliding distance 𝐷 𝑐 , at the end of which the friction has dropped from 𝜇 , the peak of static friction to 𝜇 𝑑 , the residual friction. This well-known critical slip distance 𝐷 𝑐 is the slip distance corresponding to a drastic decrease in frictional strength, and is considered to be the distance necessary for the renewal of the contact surfaces [START_REF] Dieterich | Modeling of Rock Friction Experimental Results and Constitutive Equations[END_REF]. We can thus define the slope of friction drop as follows:

𝑐 = |𝜇 𝑑 -𝜇 p |𝜎 𝑁 𝐷 𝑐 (1.9)
The critical sliding distance is very large for natural earthquakes (cm to meter), and from 10 -5 to 10 -3 for lab experiments at low slip velocity [START_REF] Marone | Scaling of the critical slip distance for seismic faulting with shear strain in fault zones[END_REF]. As first proposed by

T

𝑐 > unstable [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF], these classical slip weakening models may be incomplete and weakening mechanisms do not depend on interface properties only [ [START_REF] Barras | Emergence of Cracklike Behavior of Frictional Rupture: The Origin of Stress Drops[END_REF], [START_REF] Barras | The emergence of crack-like behavior of frictional rupture: Edge singularity and energy balance[END_REF]]. It is interesting to note that the magnitude of the stress drop with the stick-slip phenomenon also varies with the loading rate, fault geometry, elastic properties, etc.

Figure 17. (a) Linear slip weakening, (b) Non-Linear slip weakening, with 𝜇 0 the initial static friction value, 𝜇 the peak of static friction (after healing), 𝜇 𝑑 the dynamic or residual friction. ∆𝜇 is the static friction difference, ∆𝜇 the friction drop from the peak value to its residual value, or breakdown friction.

Rate and state friction laws -Other authors have tried to create a more general model to represent these mechanisms, not only based on "static" and "dynamic" friction. In order to describe the seismic movements of rocks, [START_REF] Dieterich | Modeling of Rock Friction Experimental Results and Constitutive Equations[END_REF] rod ed e " r e d e variab e o e w or ro k r o " w e e re e r w ed o laboratory and numerical observations, showing that the frictional force is logarithmically dependent (i) on the time that surfaces are in contact and (ii) on slip velocity (Sliding friction decreases with increasing velocity rate, also known as "velocity weakening") [ [START_REF] Dieterich | Modeling of Rock Friction Experimental Results and Constitutive Equations[END_REF], [START_REF] Rice | Stability of Steady Frictional Slipping[END_REF], [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF]]. The first Rate and State friction laws have been ed " w" [START_REF] Dieterich | Modeling of Rock Friction Experimental Results and Constitutive Equations[END_REF] and depend on two empirical constants a and b. There are also different versions of the RSF law, such as the slip law of [START_REF] Ruina | 1983_slip_instability_RUINA.pdf[END_REF], or the PRZ law of [START_REF] Perrin | Self-Healing Slip Pulse[END_REF] involving a different static friction drop as a function of time. From these laws, it is also possible to determine the frictional stability of the fault, and whether the fault motion is seismic or aseismic. These laws have the advantage to reproduce both laboratory experiments for low sliding velocities (i.e. from 0.001 to 10 m/s), [ [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF], [START_REF] Lockner | Rock failure and Earthquakes[END_REF], [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF], [START_REF] Im | The transition from steady frictional sliding to inertia-dominated instability with rate and state friction[END_REF]], and observations of frictional sliding [ [START_REF] Chen | Scaling of small repeating earthquakes explained by interaction of seismic and aseismic slip in a rate and state fault model[END_REF], [START_REF] Chen | On behaviour and scaling of small repeating earthquakes in rate and state fault models[END_REF]]. However, RSF laws remain complicated to understand and to interpret, and they are also not representative of some fault behavior for earthquake nucleation (for high stresses, fluid pressures, and high temperatures) [ [START_REF] Stesky | Friction in faulted rock at high temperature and pressure[END_REF], [START_REF] Passelègue | From fault creep to slow and fast earthquakes in carbonates[END_REF]] or earthquake propagation (high slip rate) [ (Brodsky & Kanamori, 2001), [START_REF] Mizoguchi | Reconstruction of seismic faulting by highvelocity friction experiments: An example of the 1995 Kobe earthquake[END_REF], [START_REF] Sone | Frictional resistance of faults during accelerating and decelerating earthquake slip[END_REF]]. We will not enter more in detail in the RSF laws because, as we elaborate later, our numerical models do not include any rate-or state-dependence. The reader may refer to [ [START_REF] Dieterich | Modeling of Rock Friction Experimental Results and Constitutive Equations[END_REF], [START_REF] Rice | Stability of Steady Frictional Slipping[END_REF], [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF]].

In this thesis, we restrict our focus on the most immediate (fracture-related) slip weakening phenomena and dedicate consideration to simpler friction laws. We then still need to look into the energy budget of the sliding fault to complete the knowledge of the mechanical fault behavior from an energetical point of view.

Energy budget and dynamic weakening

Throughout an earthquake, energy is dissipated during fault sliding by the means of different mechanisms. Figure 18 schematizes the total energy budget ∆𝑊 (J.m -2 ) of this rupture propagation based on classical slip weakening models [ [START_REF] Kanamori | Microscopic and Macroscopic Physics of Earthquakes[END_REF], [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF], [START_REF] Abercrombie | o er o o e r ke o r we ke ?[END_REF], (J. Aubry et al., 2018)]. ∆𝑊 is the total deformation energy dissipated in the process and is the sum of a fracture energy 𝐸 𝐺 , a radiated energy 𝐸 𝑅 (propagating by the mean of elastic waves) and a frictional energy 𝐸 𝐻 (dissipated within the slip zone by frictional heating):

∆𝑊 = 𝐸 𝐻 𝐸 𝐸 𝐺 (1.10)
Although 𝐸 𝐻 represents an important part of the total energy budget, it does not strongly influence rupture processes that mainly depend on 𝐸 𝐺 [START_REF] Fulton | Experimental constraints on energy partitioning during stick-slip and stable sliding within analog fault gouge[END_REF]. The radiated part 𝐸 is only a few percent of the total work and can be measured with the stress drop value and total displacement Ds of the fault, Figure 18. This radiated energy released during an earthquake is linked with the magnitude of the earthquake thanks to the seismic moment 𝑀 0 (depending itself on the fault displacement, its Young modulus and its surface pf sliding).

Figure 18. (a) A theoretical model of energy budget with ∆𝑊as the sum of a fracture energy 𝐸 𝐺 , a radiated energy 𝐸 𝑅 (propagating by the mean of elastic waves) and a frictional energy 𝐸 𝐻 (dissipated within the slip zone). 𝜏 0 is the initial static shear stress, 𝜏 1 the dynamic shear stress, 𝐷 𝑐 is the slip weakening distance and 𝐷 the end of slip displacement. This energy budget model is based on linear slip weakening theory. This is a simplification of the total balance of energy which also depends on the size of earthquakes and additional thermal weakening [START_REF] Kanamori | Microscopic and Macroscopic Physics of Earthquakes[END_REF].

Even though many studies focus on the understanding and distribution of these different energies, they remain complex in their relationship with the different sliding mechanisms. Literature reports a large range of phenomena occurring during faulting, after the end of the first slip weakening phase, and mainly observed under high pressure and velocity and within a very thin slipping zone [ [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF], [START_REF] Toro | Fault lubrication during earthquakes[END_REF]] e e "d we ke " e o d ere e w the so-ed " we ke ", need a certain amount of sliding before occurring [ [START_REF] Kanamori | Microscopic and Macroscopic Physics of Earthquakes[END_REF], [START_REF] Scholz | The Mechanics of Earthquakes and Faulting[END_REF], [START_REF] Rice | Heating and weakening of faults during earthquake slip[END_REF], [START_REF] Acosta | Dynamic weakening during earthquakes controlled by fluid thermodynamics[END_REF]], and are generally caused by thermal effects. The temperature rise can lead to several dynamic weakening mechanisms enhanced with water presence such as: (i)

𝐷 0 𝜏 0 𝐸 𝐸 𝐻 𝜏 1 𝜏 ∆ = 𝐸 𝐻 𝐸 𝐺 𝐸 𝑅 𝐷 𝑐 𝐸 𝐺 𝐸 𝑅
Rock melting, large slip and high slip velocities, [ [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF], [START_REF] Otsuki | Fluidization and melting of fault gouge during seismic slip: Identification in the Nojima fault zone and implications for focal earthquake mechanisms[END_REF], [START_REF] Niemeijer | Frictional melting of gabbro under extreme experimental conditions of normal stress , acceleration , and sliding velocity[END_REF]), (Di Toro et al., 2011), [START_REF] Violay | Effect of water on the frictional behavior of cohesive rocks during earthquakes[END_REF], [START_REF] Aharonov | A Physics-Based Rock Friction Constitutive Law: Steady State Friction[END_REF], [START_REF] Fondriest | Pseudotachylyte Alteration and the Rapid Fade of Earthquake Scars From the Geological Record[END_REF], (Mollon et al., 2021b)] ; (ii) Flash Heating, asperity contact [ [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF], [START_REF] Rice | Heating and weakening of faults during earthquake slip[END_REF], [START_REF] Acosta | Dynamic weakening during earthquakes controlled by fluid thermodynamics[END_REF], [START_REF] Cornelio | Effect of Fluid Viscosity on Fault Reactivation and Coseismic Weakening[END_REF]]; or (iii) Thermal pressurization, fluid within porosities [ [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF], [START_REF] Wibberley | Internal structure and permeability of major strike-slip fault zones: the Median Tectonic Line in Mie Prefecture[END_REF], [START_REF] Otsuki | Fluidization and melting of fault gouge during seismic slip: Identification in the Nojima fault zone and implications for focal earthquake mechanisms[END_REF] [START_REF] Rice | Heating and weakening of faults during earthquake slip[END_REF], [START_REF] Violay | Pore fluid in experimental calcite-bearing faults: Abrupt weakening and geochemical signature of co-seismic processes[END_REF], [START_REF] Acosta | Dynamic weakening during earthquakes controlled by fluid thermodynamics[END_REF], [START_REF] Lambert | Rupture-dependent breakdown energy in fault models with thermo-hydromechanical processes[END_REF], [START_REF] Badt | Thermal Pressurization Weakening in Laboratory Experiments[END_REF]].

(J. [START_REF] Chester | Fracture surface energy of the Punchbowl fault, San Andreas system[END_REF] found that fracture energy 𝐸 𝐺 is small compared to other energies (~1% for fault gouge according to [START_REF] Fulton | Experimental constraints on energy partitioning during stick-slip and stable sliding within analog fault gouge[END_REF]). However small, this energy is needed to weaken a fault during an earthquake and to allow fracture propagation. Linking friction model and crack model [START_REF] Kanamori | The physics of earthquakes[END_REF], the fracture energy 𝐸 𝐺 corresponds to the work needed to drop the shear strength 𝜏 0 to the residual strength 𝜏 1 for a slip weakening distance Dc:

E G = ∫ (𝜏 (𝑈) -𝜏 1 ) 𝑑 𝑈 𝐷 𝑐 0 (1.11)
Recent researches on the fracture process also try to understand how and to which extent the frictional rupture can be viewed as an ordinary fracture process [ [START_REF] Barras | Emergence of Cracklike Behavior of Frictional Rupture: The Origin of Stress Drops[END_REF], [START_REF] Barras | The emergence of crack-like behavior of frictional rupture: Edge singularity and energy balance[END_REF]] offering some insight into open questions relating to energy budget. However, even though some crucial information can be found in the following papers [ [START_REF] Kanamori | Microscopic and Macroscopic Physics of Earthquakes[END_REF], [START_REF] Abercrombie | o er o o e r ke o r we ke ?[END_REF], (J. [START_REF] Chester | Fracture surface energy of the Punchbowl fault, San Andreas system[END_REF], [START_REF] Rivera | Representations of the radiated energy in earthquakes[END_REF], [START_REF] Fialko | Fracture and Frictional Mechanics -Theory[END_REF], (J. Aubry et al., 2018), [START_REF] Okubo | Dynamics, radiation and overall energy budget of earthquake rupture with coseismic off-fault damage[END_REF], [START_REF] Barras | Emergence of Cracklike Behavior of Frictional Rupture: The Origin of Stress Drops[END_REF], [START_REF] Barras | The emergence of crack-like behavior of frictional rupture: Edge singularity and energy balance[END_REF]], earthquake energy budget still needs further research in the relationship they have with each internal mechanism happening during slipping.

In this Thesis, we restrict our focus on the most immediate (fracture-related) slip weakening phenomena and disregard all thermal effects, which are supposed to happen only after a certain amount of slip. A fault zone remains a very complex system in which many parameters interact at different scales. We would now like to add another degree of complexity by considering the properties of a granular fault gouge that can be found in the main slipping zone defined in (1.2.2). We propose for this purpose a look into granular material science and tribology to detail current knowledge on this material and its interaction with fault slip mechanisms. Section 1.4 will detail the granular fault gouge importance, its rheology, and the link with our research strategy. [START_REF] Lockner | Rock failure and Earthquakes[END_REF] reported that a wide range of parameters can affect the strength of both faults and rocks including a variation on mineralogy, porosity, cementation, packing geometry of gouge, surface roughness and wavelength, angularity and size of gouge particles, temperature, pore-fluid pressure, and composition, deformation rate, deformation and stress history, fabric, foliation, or other anisotropic properties, etc. Figure 19 [START_REF] Byerlee | Stick slip, stable sliding, and earthquakes-Effect of rock type, pressure, strain rate, and stiffness[END_REF], [START_REF] Sammis | The kinematics of gouge deformation[END_REF], [START_REF] Biegel | The frictional properties of a simulated gouge having a fractal particle distribution[END_REF], [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF], [START_REF] Mair | Friction of simulated fault gouge for a wide range of velocities and normal stresses[END_REF], [START_REF] Anthony | Influence of particle characteristics on granular friction[END_REF]], and also numerically by the mean of Discrete Element Modelling (DEM) [ [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF], [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 2. Effects of particle size distribution and interparticle friction on mechanical behavior[END_REF], [START_REF] Guo | Influence of normal stress and grain shape on granular friction: Results of discrete element simulations[END_REF], [START_REF] Da Cruz | Rheophysics of dense granular materials: Discrete simulation of plane shear flows[END_REF], [START_REF] Cho | Development of a shear zone in brittle rock subjected to direct shear[END_REF], [START_REF] Zhao | Particle mechanics model for the effects of shear on solute retardation coefficient in rock fractures[END_REF], [START_REF] Gao | Modeling of Stick-Slip Behavior in Sheared Granular Fault Gouge Using the Combined Finite-Discrete Element Method[END_REF]].

The role of granular fault gouge

As for intact rock, fault gouge stability mainly depends on normal and deviatoric stresses applied to the fault. High stress will tend towards a fractured regime with unstable deformations while low stress will be more likely to result in an unfractured regime (stable deformation), [ [START_REF] Byerlee | Structures developed sliding and stick-slip in fault gouge during stable[END_REF], [START_REF] Mair | Friction of simulated fault gouge for a wide range of velocities and normal stresses[END_REF]]. Elements (a), (b), and (c) are widely studied in the literature and known as major elements acting in the fault behavior. Although they still need further research they will not be the main subject of this manuscript. The reader may access some interesting information on these topics in the following papers: external constraints [ [START_REF] Byerlee | Stick slip, stable sliding, and earthquakes-Effect of rock type, pressure, strain rate, and stiffness[END_REF], [START_REF] Stesky | Friction in faulted rock at high temperature and pressure[END_REF], [START_REF] Mair | Friction of simulated fault gouge for a wide range of velocities and normal stresses[END_REF], [START_REF] Guo | Influence of normal stress and grain shape on granular friction: Results of discrete element simulations[END_REF], [START_REF] Niemeijer | Frictional melting of gabbro under extreme experimental conditions of normal stress , acceleration , and sliding velocity[END_REF], [START_REF] Violay | An experimental study of the brittle-ductile transition of basalt at oceanic crust pressure and temperature conditions[END_REF], [START_REF] Geng | Time and Temperature Dependent Creep in Tournemire Shale[END_REF], [START_REF] Bedford | The Role of Grain Size and Effective Normal Stress on Localization and the Frictional Stability of Simulated Quartz Gouge[END_REF], etc.], rock properties and lithology [ [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF], [START_REF] Anthony | Influence of particle characteristics on granular friction[END_REF], [START_REF] Fournier | Insights to slip behavior on rough faults using discrete element modeling[END_REF], [START_REF] Insua-Arevalo | Influence of the microstructure and roughness of weakness planes on the strength anisotropy of a foliated clay-rich fault gouge[END_REF], and fluid [ [START_REF] Violay | Pore fluid in experimental calcite-bearing faults: Abrupt weakening and geochemical signature of co-seismic processes[END_REF], [START_REF] Violay | Effect of water on the frictional behavior of cohesive rocks during earthquakes[END_REF], [START_REF] Scuderi | Physicochemical processes of frictional healing: Effects of water on stick-slip stress drop and friction of granular fault gouge[END_REF], [START_REF] Scuderi | The role of fluid pressure in induced vs. triggered seismicity: Insights from rock deformation experiments on carbonates[END_REF], [START_REF] Bayart | Slippery but Tough: The Rapid Fracture of Lubricate d Frictional Interfaces[END_REF], (Dorostkar et al., 2017a), [START_REF] Acosta | Dynamic weakening during earthquakes controlled by fluid thermodynamics[END_REF], (Cornelio, Spagnuolo, et al., 2019).

In this chapter and the rest of the Thesis, we will focus on the fault gouge material, its mechanical and rheological properties as well as the potential infill material present within the gouge and their influence on fault behavior. Section 1.4.1 describes the main fault gouge characteristics from a mechanical and granular point of view. Subsection a. answers why the gouge is so important and makes the link between tribology and fault mechanics by the mean of the Third Body Theory and wear models. It also details the importance of a lubricant presence within the contact, even without fluid. Subsection b. explains how the gouge deforms, showing the different kinds of deformation observed within a granular fault gouge in direct shearing and the main shearing bands formed all along the gouge l ife ede d d … Then section 1.4.2 restrains the Thesis to the study of infill material within the fault core (i.e. cement and matrix material). It proposes a description of these materials and presents some work already done in the research field. Finally, section 1.4.3 presents a small review on numerical modeling and fault gouge modelization and concludes on the modeling choices kept for the rest of the Thesis.

Research Strategy #1: focus on fault gouges mechanics and rheology a. Gouge as a geological third body Wear and grain fragmentation

As previously discussed in section 1.2.2, the fault core mainly contains fragmented rock particles coming from host rocks shearing. These granular particles can also be defined as wear material from damage or erosion of surfaces in contact [START_REF] Scholz | Wear and gouge formation in brittle faulting[END_REF]. As shown in Figure 20 (a), wear rate first presents a period of rapid wear due to surface asperities, followed by a lower stable wear rate, corresponding to the established gouge layer [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF]. The wear rate mainly depends on lithology (ex: sandstone wears much faster than granite) and on the applied normal stress, showing a behavior close to that of Archard's law [START_REF] Archard | Contact and rubbing of flat surfaces[END_REF]. As wear continues, the initially rough surfaces in contact become, at some point, totally separated by gouge particles, meaning that the thickness of the gouge zone increases linearly with total slip for brittle faults [START_REF] Scholz | Wear and gouge formation in brittle faulting[END_REF]. The experimental wear rate was also found to be smaller than the one of natural fault due to surface roughness in contact.

A particle size reduction is observed from the comminution process [ [START_REF] Sammis | The kinematics of gouge deformation[END_REF], (F. [START_REF] Chester | Intemal Structure and Weakening Mechanisms of the San Andreas Fault[END_REF], [START_REF] Sibson | Thickness of the Seismic Slip Zone[END_REF]] also called grain breakage [ [START_REF] Daouadji | An elastoplastic model for granular materials taking into account grain breakage[END_REF], [START_REF] Daouadji | An enhanced constitutive model for crushable granular materials[END_REF]] in granular mechanics. The number and amplitudes of successive slips reduce the size of particles towards a fractal distribution (depending on the type of mineral), which also reduces pore spaces [ (Sammis & Biegel, 1989), [START_REF] Olgaard | The microstructure of gouge from a mining-induced seismic shear zone[END_REF], [START_REF] Blenkinsop | Cataclasis and Processes of Particle Size Reduction[END_REF], [START_REF] An | Particle Size Distribution of Cataclastic Fault Materials from Southern or d Pure and Applied Geophysics[END_REF], [START_REF] Billi | Fractal distribution of particle size in carbonate cataclastic rocks from the core of a regional strike-slip fault zone[END_REF], [START_REF] Billi | Grain size distribution and thickness of breccia and gouge zones from thin (<1 m) strike-slip fault cores in limestone[END_REF], [START_REF] Muto | Fractal particle size distribution of pulverized fault rocks as a function of distance from the fault core[END_REF]]. The evolution of this fractal dimension also relies upon the amount of shearing (time scale) and the distance from the fault core (geometric scale), as in Figure 20 (b). Mineral grains morphologies can be very diverse, but granular gouges generated by comminution are expected to exhibit rather rough and angular shapes [ [START_REF] Olgaard | The microstructure of gouge from a mining-induced seismic shear zone[END_REF], [START_REF] Sammis | The kinematics of gouge deformation[END_REF], [START_REF] An | Particle Size Distribution of Cataclastic Fault Materials from Southern or d Pure and Applied Geophysics[END_REF], [START_REF] Lin | S-C cataclasite in granitic rock[END_REF]], Figure 20 (c). A collection of these angular shapes can be observed in Appendix 1.A.

Figure 20. (a) A complete wear curve for Westerly granite, for an applied normal stress of 3 MPa. The first part is the running-in wear, which can evolve to steady-state wear at a constant rate. Image from [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF]. (b) Fractal dimension of the ATTL and San Andreas Fault quartz grains as a function of the distance from the fault core. (c) Pulverized rock of the Arima-Takatsuki Tectonic Line (ATTL) fault gouge, photomicrograph, Both images are from [START_REF] Muto | Fractal particle size distribution of pulverized fault rocks as a function of distance from the fault core[END_REF].

Granular gouge as a lubricant

From an engineering point of view, any type of lubrication between two solids in contact, may it be fluid or solid, is supposed to decrease friction at the interface. The lubricant may have a different role on the mechanical contact depending on the quantity and thickness between the two contacting bodies [START_REF] Hardy | Some Problems of Lubrication[END_REF] and is seen as a good ingredient in tribological applications by decreasing friction and reducing energetic loss. The role of fault gouge is similar at the fault scale, the presence of gouge induces an important weakening in comparison with fault friction without or with negligible wear formation, e e d ere e o d w er ee' r e in section 1.3.1 for gouge material e w o red o " owder r o " [START_REF] Reches | Fault weakening and earthquake instability by powder lubrication[END_REF] who also proposed a model for fault weakening as velocity rate increases. The accumulation of wear material also tends to stabilize slip relatively to shear between bare rock surfaces [ [START_REF] Scholz | Wear and gouge formation in brittle faulting[END_REF], [START_REF] Biegel | The frictional properties of a simulated gouge having a fractal particle distribution[END_REF], [START_REF] Marone | Frictional behavior and constitutive modeling of simulated fault gouge[END_REF], [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF], [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF], [START_REF] Reches | Fault weakening and earthquake instability by powder lubrication[END_REF]].

To illustrate the evolution of the different contact mechanisms, with or without lubricant (i.e. fluid or powder), Stribeck proposed in the early 1900s a model of friction coefficient evolution as a function of a hydrodynamic number proportional to (viscosity*velocity)/load. The Stribeck curve was initially proposed for tribological application but was recently used and compared to fluid-lubricated faults [ [START_REF] Spikes | Mixed lubrication -An overview[END_REF], (Kanamori & Brodsky, 2001), (Cornelio, Spagnuolo, et al., 2019), [START_REF] Cornelio | Parametric analysis of the elastohydrodynamic lubrication efficiency on induced seismicity[END_REF]]. e e o e o " rd od " w o ed out by [START_REF] Godet | The third-body approach: a mechanical view of wear[END_REF], inspired by lubrication theories, to characterize the material or medium at the contact interface between two solids, Figure 21 (c).

rd od e " d" lubricant (as fluid already present within the fault or artificially injected (as with EGS)), but in e e o " r " o rd body is still present and is composed of wear particles, Figure 21 (a) and (b). In this mechanism, the granular layer has several interesting functions as (i) supporting the load applied on the contact, (ii) participating to velocity accommodation, and (iii) separating surfaces in contact to avoid direct interaction [START_REF] Godet | The third-body approach: a mechanical view of wear[END_REF]. (c) schema of the third-body theory applied to the fault system. More information on the third body concept are available in [ [START_REF] Iordanoff | Numerical study of a thin layer of cohesive particles under plane shearing[END_REF], [START_REF] Fillot | Wear modeling and the third body concept[END_REF]].

[ [START_REF] Mair | Influence of grain characteristics on the friction of granular shear zones[END_REF], [START_REF] Nouguier-Lehon | Influence of particle shape and angularity on the e o r o r r er er International[END_REF], [START_REF] Anthony | Influence of particle characteristics on granular friction[END_REF]] have shown that angular and faceted particle shapes (instead of circular grains) led to higher friction coefficients and different mechanical behaviors. These results also need to be related to the roughness of the bare rock which increases the shear ratio observed for any shape of particles.

As particle shape influences frictional properties, it also controls stick-slip dynamics: spherical particles mostly show a production of stick-slip motion, whereas angular particles promote stable sliding [START_REF] Mair | Influence of grain characteristics on the friction of granular shear zones[END_REF]. The influence of the grain shape on friction increases with the decrease in the normal stress [START_REF] Guo | Influence of normal stress and grain shape on granular friction: Results of discrete element simulations[END_REF].

Within this Thesis, we decided to stay in the boundary lubrication regime, which is the classical solid-solid contact (no liquid lubricant, dry contact). We define the fault gouge as wear particles, as the process of 3 rd body production, and friction as the rheology of this 3 rd body (similar to fluid viscosity). Another objective is to reproduce realistic granular shapes of gouge with angular and faceted shapes as found in fault core (Appendix 1.A).

The evolution of gouge properties with slip and its influence on slip mechanism and rheology are described within the next subsection b.

b. Deformation mechanisms and rheology of fault gouges

The understanding of fault gouge deformation mechanisms is a key element in the construction of fault models. Deformation zones are complex structures, they have variable aspects (geometry and kinematics), depending on the mechanism of deformation adopted and rock characteristics, section 1.2.3. A word about dilatancy? [START_REF] Marone | Frictional behavior and constitutive modeling of simulated fault gouge[END_REF] indicate that r r o e ow r er " e ore e " e or d ed dilation of fault gouges (dilation implying a variation in the porosity state of the granular gouge sample). It is clear that shear between bare rock surfaces does not involve dilation as great as observed in the case where a granular gouge is present [ [START_REF] Marone | Frictional behavior and constitutive modeling of simulated fault gouge[END_REF], (Kilgore et al., [START_REF] Beeler | Frictional behavior of large displacement experimental faults[END_REF]] and that denser granular samples show larger dilating behavior than a loose sample. However, shear localization can also form within fault gouge to accommodate the inelastic shear strain. They appear as a common phenomenon in granular mechanics, widely studied both experimentally and theoretically [ [START_REF] Dunn | Porosity Dependence and Mechanism of Brittle Fracture in Sandstones[END_REF], [START_REF] Vardoulakis | Calibration of constitutive models for granular materials using data from biaxial experiments[END_REF], [START_REF] Antonellini | Microstructure of deformation bands in porous sandstones at Arches National Park, Utah[END_REF], [START_REF] Antonellini | Microstructure of deformation bands in porous sandstones at Arches National Park, Utah[END_REF], [START_REF] Haines | Shear zones in clay-rich fault gouge: A laboratory study of fabric development and evolution[END_REF]].

Riedel shear bands

For a sufficiently mature fault system (sufficient gouge layer), shear localization was observed at specific angles and called Riedel Shear bands (or cataclastic deformation bands) [START_REF] Tchalenko | Similarities between shear zones of different magnitudes[END_REF]. They reflect a heterogeneous stress field developing in response to deformation [ [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF], [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF], [START_REF] Lockner | Rock failure and Earthquakes[END_REF], [START_REF] Kaminskaite | Microstructure and petrophysical properties of deformation bands in high porosity carbonates[END_REF]]. Figure 22 schematizes the main deformation bands forming within the shearing gouge. Secondary Riedel bands in the R orientation (sense of shear zone) are commonly observed to be predominant, but in some cases, antithetic shears (i.e. conjugate e r or e ' or e o ro w r r ow [ [START_REF] Davis | Conjugate Riedel deformation band shear zones[END_REF], (Y. [START_REF] Katz | Geometry and kinematic evolution of Riedel shear structures, Capitol Reef National Park, Utah[END_REF], [START_REF] Misra | Formation of Riedel shear r re r r er Findings from analogue shear experiments and theoretical analyses[END_REF]]. Y surfaces are parallel to the direction of shearing and can also form within the boundary shear bands B, also parallel to the shearing direction but located at the boundaries [ [START_REF] Gu | Development of shear localization in simulated quartz gouge: Effect of cumulative slip and gouge particle size[END_REF], [START_REF] Blanpied | Frictional slip of granite at hydrothermal conditions[END_REF]]. In some granular materials, such as Clay-rich fault gouge [START_REF] Rutter | Comparative microstructures of natural and experimentally produced clay-bearing fault gouges[END_REF], a primary foliation P can be observed normal to 𝜎 1 (angle 135° from the direction of shearing in the simple shear and progressively rotating until 180°). Extension fractures T, normal to the least principal stress direction can form for low and medium-grade metamorphic terranes or high fluid pressure. Even though Riedel shears were first used to describe brittle deformation, a foliation can also be observed within ductile deformation in the R orientation [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF]. The shear zone at e o o e r o ore o e r re er e ed " e r zo e r re" or " ede e r r re" (Y. [START_REF] Katz | Geometry and kinematic evolution of Riedel shear structures, Capitol Reef National Park, Utah[END_REF]. The presence of Riedel bands (R) has been identified as being responsible for a mechanical weakening [ [START_REF] Gu | Development of shear localization in simulated quartz gouge: Effect of cumulative slip and gouge particle size[END_REF], [START_REF] Beeler | Frictional behavior of large displacement experimental faults[END_REF]] and to be an indicator of a change in stability in frictional sliding [START_REF] Byerlee | Structures developed sliding and stick-slip in fault gouge during stable[END_REF]. It is therefore essential to study the timing of their formation during the sliding and their orientation.

𝑅 𝑅 thickness 𝜎 𝑁 𝑃 𝜎 1 𝜎 3

The temporality of Riedel bands and their consequences

The onset of Riedel bands is mainly associated with a significant dilatancy rate and a reduction in the comminution rate of the granular layer [ [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF], [START_REF] Borja | Shear band in sand with spatially varying density[END_REF]]. The interplay between porosity [ [START_REF] Dunn | Porosity Dependence and Mechanism of Brittle Fracture in Sandstones[END_REF], [START_REF] Antonellini | Microstructure of deformation bands in porous sandstones at Arches National Park, Utah[END_REF]] and particle size distribution [ [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF], [START_REF] Antonellini | Microstructure of deformation bands in porous sandstones at Arches National Park, Utah[END_REF]], confining pressure [START_REF] Byerlee | Structures developed sliding and stick-slip in fault gouge during stable[END_REF], clay or weak mineral content [ [START_REF] Haines | Shear zones in clay-rich fault gouge: A laboratory study of fabric development and evolution[END_REF], [START_REF] Kenigsberg | The Effects of Shear Strain, Fabric, and Porosity Evolution on Elastic and Mechanical Properties of Clay-Rich Fault Gouge[END_REF]] and amount of strain may develop these Riedel bands at different stages and with different characteristics [ [START_REF] Antonellini | Microstructure of deformation bands in porous sandstones at Arches National Park, Utah[END_REF], [START_REF] Philit | Elaboration d'un modèle structural, pétrophysique et mécanique des failles en milieu gréseux poreux; implication pour la migration et le piégeage des fluides[END_REF]]. Figure 23 schematizes the main stages of Riedel shears formation gathered from different studies [ [START_REF] Tchalenko | Similarities between shear zones of different magnitudes[END_REF], [START_REF] Byerlee | Structures developed sliding and stick-slip in fault gouge during stable[END_REF], [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF], [START_REF] Gu | Development of shear localization in simulated quartz gouge: Effect of cumulative slip and gouge particle size[END_REF], [START_REF] Beeler | Frictional behavior of large displacement experimental faults[END_REF], [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF], [START_REF] Hadizadeh | Shear localization, velocity weakening behavior, and development of cataclastic foliation in experimental granite gouge[END_REF], [START_REF] Kenigsberg | The Effects of Shear Strain, Fabric, and Porosity Evolution on Elastic and Mechanical Properties of Clay-Rich Fault Gouge[END_REF], [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF], [START_REF] Bedford | The Role of Grain Size and Effective Normal Stress on Localization and the Frictional Stability of Simulated Quartz Gouge[END_REF]]:

•

Step 1 -elastic phase. Due to external loading, the granular layer is compacted and reorganized (particle movement or grains fragmentation) during the elastic phase. The global stiffness increases and the porosity is reduced, controlling the mechanical et elastic responses [START_REF] Kenigsberg | The Effects of Shear Strain, Fabric, and Porosity Evolution on Elastic and Mechanical Properties of Clay-Rich Fault Gouge[END_REF]. The stress ratio (i.e. effective friction) increases progressively until it reaches the maximal shear strength where Riedel bands formation is observed.

•

Step 2 -weakening. The R-bands are still prominent, but Y-bands begin to form. They are sufficiently developed to take part in the control of gouge behavior and mechanical properties. Interparticle bridges (or force chains networks) are formed to support the shear loads applied to the assembly. They can sometimes correspond to T-bands. These chains are often inclined at 45 ° from the direction of shearing and evolve throughout the shear [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF]. Conjuga e ede ' re o o er ed w the images, Figure 23 (b), but if they form it must be at this step with strain-hardening behaviors [START_REF] Davis | Conjugate Riedel deformation band shear zones[END_REF]. A decrease in shear resistance from yield is observed (i.e. the weakening mechanism).

•

Step 3: end of weakening. The main Riedel bands R are less prominent but B & Y bands are still developing and increasing again the weakening of fault gouge. R-bands progressively connect with the boundary bands.

•

Step 4: steady-state. This is a residual stage, where shear resistance has reached a stable steady-state friction with one or several parallel principal Y-shears lying in the slip direction. The Y-shears can be developed at the center of the gouge [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF] or within the Boundary shear zone [START_REF] Bedford | The Role of Grain Size and Effective Normal Stress on Localization and the Frictional Stability of Simulated Quartz Gouge[END_REF]. A stable frictional strength is reached at the point at which well-developed Y shears become prominent [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF]. The transition from stick-slip to stable sliding corresponds to the formation of boundary andshear bands [START_REF] Gu | Development of shear localization in simulated quartz gouge: Effect of cumulative slip and gouge particle size[END_REF]. Even though Riedel shear bands are a necessary condition for stick-slip instabilities to occur in gouges, they are not sufficient, as they were also observed for some stable slidings [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF]. It has been speculated that the 𝑅′ bands may inhibit gouge movement by preventing the connection between the 𝑅 and bands [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF], but the link between the orientation of the Riedel bands and the slip modes is not yet fully understood. As suggested by [ [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF], [START_REF] Gu | Development of shear localization in simulated quartz gouge: Effect of cumulative slip and gouge particle size[END_REF]], it also exists a critical angle 𝛼 𝑖 between the bands and the direction of shearing that indicates whether or not instabilities can occur, higher angles tending to destabilize the gouge. According to Coulomb-Mohr criteria and geometry, 𝛼 𝑖 could be equal to 𝜑 2 ⁄ for simple shear (with 𝜑 being the internal friction angle). However, this relation has to be used with care, as other factors can be involved in the 𝛼 𝑖 angle [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF]. Recently, it was confirmed that Riedel bands can differ from the center of the granular fault gouge to the rock boundaries. They appear to develop at lower angles to rock-gouge boundaries toward unstable slip [START_REF] Hirata | Shear zone development and frictional instabilities[END_REF], and mostly depend on normal stress and strain velocity in major and minor stress orientation axes. The link between the orientation of Riedel bands and slip patterns is not yet fully understood. One part of the thesis will be devoted to studying the temporality of Riedel bands as a function of gouge characteristics. For this, we will focus on two types of gouges containing infill material, described in the next section.

Research strategy #2: study of infill materials

Once considering a mature enough fault zone that has already been sheared, infill material can be observed within void spaces around granular particles. They bring additional heterogeneities to the fault gouge modifying the stress field [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF] and petrophysical properties [START_REF] Philit | Elaboration d'un modèle structural, pétrophysique et mécanique des failles en milieu gréseux poreux; implication pour la migration et le piégeage des fluides[END_REF]. Even though these materials are well described from a geological point of view, it is more difficult to find associated lab experiments or simulations. In this section, we gather the main characteristics of two types of infill material, matrix particles, and cement, and wonder what is their influence on fault zone behavior and stability. This distinction was made thanks to a bibliography of different gouges and infill materials available in Appendix 1.A.

Cement or matrix?

Within a mature fault gouge, the distinction between cement and matrix between breccia clasts is not easy to make and partly depends on brecciation processes in fault zones [ [START_REF] Wise | Fault -related rocks: Suggestions for terminology[END_REF], [START_REF] Sibson | Brecciation processes in fault zones: Inferences from earthquake rupturing[END_REF], [START_REF] Riedmüller | Engineering Geological Characterization of Brittle Faults and classification of fault rocks[END_REF]]. Many rock classifications have been proposed from structural geology and based on the parameters fabric, texture, clast size, and matrix percentage but they all depend on the scale of observation and resolution of the individual component with the naked eye, giving sometimes really different classifications [ [START_REF] Sibson | Fault rocks and fault mechanisms[END_REF], [START_REF] Riedmüller | Engineering Geological Characterization of Brittle Faults and classification of fault rocks[END_REF], [START_REF] Woodcock | Classification of fault breccias and related fault rocks[END_REF]]. Figure 24 proposes a classification for cohesive and incohesive fault core and associated infill material from [START_REF] Riedmüller | Engineering Geological Characterization of Brittle Faults and classification of fault rocks[END_REF] combined with explanations from [START_REF] Woodcock | Classification of fault breccias and related fault rocks[END_REF].

Figure 24. (a) Image created from [START_REF] Riedmüller | Engineering Geological Characterization of Brittle Faults and classification of fault rocks[END_REF], rock classification from [START_REF] Sibson | Fault rocks and fault mechanisms[END_REF] and [START_REF] Woodcock | Classification of fault breccias and related fault rocks[END_REF] d er ro e r o e ze e wee ' ' d ' r ' settled at 0.1mm, which is the resolution of an individual component with naked eye [START_REF] Wise | Fault -related rocks: Suggestions for terminology[END_REF]. (b) Blocks embedded in a matrix, random distribution of blocks varying in size and orientation, microscale (thin section), image from [START_REF] Riedmüller | Engineering Geological Characterization of Brittle Faults and classification of fault rocks[END_REF]. (c) Micrographs of thin sections showing sandstone with abundant quartz cement and some reduced primary pores plane (polarised light with crossed polarization filters, length of photomicrographs is 0.96 mm). Image from [START_REF] Molenaar | Quartz cementation mechanisms and porosity variation in Baltic Cambrian sandstones[END_REF]. Mineral cementation, generally evolving with burial, refers to a crystalline material coming from rock dissolution or precipitation [START_REF] Walderhaug | Precipitation rates for quartz cement in sandstones determined by fluid-inclusion microthermometry and temperature-history modeling[END_REF], partial melting (pseudotachylite, non-crystalline material), [START_REF] Toro | Pseudotachylytes and earthquake source mechanics[END_REF], [START_REF] Fondriest | Pseudotachylyte Alteration and the Rapid Fade of Earthquake Scars From the Geological Record[END_REF]], or other processes derived from previous slips. It fills remaining pore spaces between particles, reducing permeability and changing the global state of cohesion of the gouge material [ [START_REF] Woodcock | Classification of fault breccias and related fault rocks[END_REF], [START_REF] Philit | Clusters of cataclastic deformation bands in porous sandstones[END_REF]]. The behavior of this cement depends on the material composition [ [START_REF] Caine | Internal structure, fault rocks, and inferences regarding deformation, fluid flow, and mineralization in the seismogenic Stillwater normal fault, Dixie Valley, Nevada[END_REF], [START_REF] Philit | Elaboration d'un modèle structural, pétrophysique et mécanique des failles en milieu gréseux poreux; implication pour la migration et le piégeage des fluides[END_REF], [START_REF] Kaminskaite | Microstructure and petrophysical properties of deformation bands in high porosity carbonates[END_REF]], on the volume, and on the contact interface between cement and particles. By contrast, the term Matrix refers here to very fine cohesionless particles whether produced by local comminution or by the introduction of other sediments [ [START_REF] Lee | Comparison of structural features of the fault zone developed at different protoliths: Crystalline rocks and mudrocks[END_REF], [START_REF] Woodcock | Classification of fault breccias and related fault rocks[END_REF]]. The scale of observation and classifications allow us to distinguish a weak gouge matrix from fragments or clasts with a ze e wee ' ' d ' r ' e ed re o o o an individual component with naked eye (Wise et al., 1984)). Matrix particles are from parts similar to a granular material with high fractal dimension and will present the same mechanical consequences (denser granular zone with porosity reduction). The lithology of matrix or clasts particles also influences the frictional behavior. For example, an increase in the percentage of clay matrix drastically reduces the effective friction toward 0.1-0.2 [START_REF] Wojatschke | Geochemistry, Geophysics, Geosystems[END_REF].

Impact of cementation on rheology and contact stability

For cohesive materials such as cemented rocks, two types of situations are observed: the lowcohesive (or non-cohesive) materials, where external stress is preponderant on interparticle cohesion, and ultra-cohesive materials where cohesion controls particles interactions (that will tend to a brittle behavior). Macroscopic cohesion can either be expressed as a tensile, or a shearing resistance [START_REF] Topin | Matériaux granulaires cimentés : modélisation et application à l ' albumen de blé[END_REF]. From a particle point of view, cohesion can be similar to an adhesion force or a slip-rolling resistance between particles [START_REF] Estrada | Shear strength and force transmission in granular media with rolling resistance[END_REF]. The cohesion in the sense of Coulomb-Mohr is easy to measure (cf. section 1.2.3) and it was widely studied for intact rocks [ [START_REF] Handin | On the Coulomb-Mohr Failure Criterion[END_REF], [START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF], [START_REF] References Abdelmalak | Tectonophysics Description of new dry granular materials of variable o e o d r o oe e o or laboratory modeling of the brittle crust[END_REF], [START_REF] Roy | Effect of cohesion on local compaction and granulation of sheared soft granular materials[END_REF]]. Not surprisingly, cohesive rocks are also those with the highest friction compared to low cohesion rocks [ [START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF], [START_REF] References Abdelmalak | Tectonophysics Description of new dry granular materials of variable o e o d r o oe e o or laboratory modeling of the brittle crust[END_REF]]. Indeed, [START_REF] Lade | Cementation Effects in Frictional Materials[END_REF] showed that, for low confining pressures, the increase of cementation and the associated tensile strength lead to an enhancement of friction coefficient (as well as shear and bulk elastic moduli): both porosity and permeability of shears bands are also reduced [START_REF] Wangen | Modelling quartz cementation of quartzose sandstones[END_REF], [START_REF] Ogilvie | The petrophysical properties of deformation bands in relation to their microstructure[END_REF], [START_REF] Fisher | The permeability of faults within siliciclastic petroleum reservoirs of the North Sea and Norwegian Continental Shelf[END_REF], [START_REF] Molenaar | Quartz cementation mechanisms and porosity variation in Baltic Cambrian sandstones[END_REF], [START_REF] Al-Hinai | Laboratory measurements of the relative permeability of cataclastic fault rocks: An important consideration for production simulation modelling[END_REF]]. The process of cementation gives birth to a new, stronger granular material combining its history, the state of initial density (i.e. porosity within the sample), and cementation [START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF]. During granular fault shearing at low confining pressure, cement first increases the brittleness at failure and prevents grain rearrangement by increasing cohesion between particles [START_REF] Johansen | The impact of syn-faulting porosity reduction on damage zone architecture in porous sandstone: An outcrop example from the Moab Fault, Utah[END_REF]. For low cementation, the progressive breakage of cement bridges between particles enhances local kinematic freedom and all ows grains reorganization. This loss of cement contribution reduces the stability of the system with a global softening and dilatant behavior letting appear shear plane failures previously described [ [START_REF] Rad | Influence of Cementation on the Static and Dynamic Behavior of Sands[END_REF], [START_REF] Das | A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables. Part II -Validation and localization analysis[END_REF], [START_REF] Lade | Cementation Effects in Frictional Materials[END_REF], [START_REF] Menendez | Pergamon Micromechanics of brittle faulting and cataclastic flow in Berea sandstone[END_REF]]. For high cementation, breakage of cement is very sudden, enabling fracture instead of shear failure [ [START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF], [START_REF] Schmocker | Granular flow and Riedel band formation in water-rich quartz aggregates experimentally deformed in torsion[END_REF], [START_REF] Kettermann | Changes in structural style of normal faults due to failure mode transition: First results from excavated scale models[END_REF]].

From a geological point of view, fault gouge is seen as an incohesive material, so it would have been reasonable to call granular w e e " e e ed " owe er or e ke o we w e " o e" e granular materials tested, may they be cohesive or not. It appeared that many field observations exist on these infill materials, but they count only a few representations in tribological studies dedicated to the understanding of fault sliding mechanisms. This Thesis aims to add a new key of understanding on these materials at the fault scale through granular mechanics and tribology research.

Research strategy #3: Modeling fault gouges

Why numerical modeling?

-The first objective is to model a single fault patch with granular fault gouge inside, large enough to observe shear deformations. We also want to model phenomena occurring at the grain scale, by representing cement and matrix filling pores around gouge particles. Numerical modeling allows easy access to both scales without building a complex observation system. The Discrete Element Method can be useful to understand granular physics of fault gouge by studying grains interactions and their influence on slip mechanisms.

-The interest is also to start with identical initial granular configuration and to be able to easily vary the amount of matrix and cement present in our samples to test a wide range of different material behaviors. We also want to quantify and follow the evolution of the cohesive bonds created between the grains over time, since such data is impossible to see or to follow during a lab or in-situ experiment.

a. Discrete Element Modelling (DEM)

Discrete Element Modeling [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF]) is a powerful tool to model micrometric to centimeter scales and observe the grain-scale deformation. A granular material (i.e. fault gouge) can be explicitly represented as a collection of rigid bodies, allowing to study very local fault behavior (grains movements) and highlighting the links between the mechano-physical properties of the fault and its rheological behavior. In order to be representative of the local behavior, a high number of particles needs to be modeled, and the movement and dynamic of each particle are computed. Moreover, is it possible to test and vary a huge number of physical and numerical parameters as well as different contact laws.

DEM has been already used in fault mechanics to understand frictional and contact properties of a fault core [ [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF], [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 2. Effects of particle size distribution and interparticle friction on mechanical behavior[END_REF], [START_REF] Guo | Influence of normal stress and grain shape on granular friction: Results of discrete element simulations[END_REF], [START_REF] Aharonov | Stick-slip motion in simulated granular layers[END_REF], [START_REF] Da Cruz | Rheophysics of dense granular materials: Discrete simulation of plane shear flows[END_REF], [START_REF] Cho | Development of a shear zone in brittle rock subjected to direct shear[END_REF], [START_REF] Zhao | Particle mechanics model for the effects of shear on solute retardation coefficient in rock fractures[END_REF], [START_REF] Zhao | Gouge particle evolution in a rock fracture undergoing shear: A microscopic DEM study[END_REF], [START_REF] Ferdowsi | Three -dimensional discrete element modeling of triggered slip in sheared granular media[END_REF], (Dorostkar et al., 2017a), [START_REF] Dorostkar | On the role of fluids in stickslip dynamics of saturated granular fault gouge using a coupled computational fluid dynamics-discrete element approach[END_REF], [START_REF] Gao | Modeling of Stick-Slip Behavior in Sheared Granular Fault Gouge Using the Combined Finite-Discrete Element Method[END_REF], [START_REF] Ferdowsi | A Granular Physics-Based View of Fault Friction Experiments[END_REF], (Mollon et al., 2021b)], and also in some interesting granular mechanics researches not directly applicated to fault mechanics [ [START_REF] Jensen | DEM simulation of granular media -Structure interface: effects of surface roughness and particle shape[END_REF], [START_REF] Aharonov | Rigidity phase transition in granular packings[END_REF], [START_REF] Maeda | Influence of grain properties on macro mechanical behaviors of granular media by DEM[END_REF], [START_REF] Maeda | Stress-chain based micromechanics of sand with grain shape effect[END_REF], (Mollon et al., 2012), [START_REF] Zhou | DEM modeling of shear bands in crushable and irregularly shaped granular materials[END_REF], (Y. [START_REF] Lu | Methodology for Simulation of Irregularly Shaped Gravel Grains and Its Application to DEM Modeling[END_REF], (Mollon, 2018b)]. More recently, the DEM has been enhanced by additional coupling or plug-e "e e ded " o der thermodynamics in order to study the high-temperature rise and melting within the granular fault zone, using a mixture between compliant and rigid bodies, Figure 25 [ (Mollon et al., 2021a) & (Mollon et al., 2021b)], the Computational Fluid Dynamic-DEM (CFD-DEM) is designed to observe fluid interaction within particles [ [START_REF] Goren | The Mechanical Coupling of Fluid-Filled Granular Material Under Shear[END_REF], [START_REF] Kloss | Models, algorithms and validation for opensource DEM and CFD-DEM[END_REF], (Dorostkar et al., 2017a), [START_REF] Dorostkar | On the role of fluids in stickslip dynamics of saturated granular fault gouge using a coupled computational fluid dynamics-discrete element approach[END_REF], [START_REF] Dorostkar | Stick-slip dynamics in dry and fluid saturated granular fault gouge investigated by numerical simulations[END_REF], [START_REF] Tomac | Micromechanics of hydraulic fracturing and damage in rock based on DEM modeling[END_REF]] and FEM-DEM coupling enables to link granular fault to a continuous damage zone [ [START_REF] Munjiza | The combined finite-discrete element method for structural failure and collapse[END_REF], [START_REF] Dratt | Coupling of FEM and DEM simulations to consider dynamic deformations under particle load[END_REF]]. Grain fragmentation has also been modeled with DEM using different methods [ [START_REF] Zhao | Gouge particle evolution in a rock fracture undergoing shear: A microscopic DEM study[END_REF], (D. [START_REF] Wang | On the effect of grain fragmentation o n frictional instabilities in faults with granular gouge[END_REF]].

Figure 25. Setup model for the reproduction of laboratory earthquakes with a discrete-continuum model, software MELODY, image from (Mollon et al., 2021a).

Nowadays, the wide variety of modeling technics enables to study both 2D and 3D problems. 3D modeling is a considerable technical advance in numerical modeling as it provides access to new information in a third direction and more realistic interactions between particles [ [START_REF] Ferdowsi | Three -dimensional discrete element modeling of triggered slip in sheared granular media[END_REF], (Dorostkar et al., 2017a), (C. [START_REF] Wang | Ensemble Shear Strength , Stability , and Permeability of Mixed er o o e e o ered ro r r ode o r o eo e e r o d Earth[END_REF], [START_REF] Ferdowsi | A Granular Physics-Based View of Fault Friction Experiments[END_REF]]. However, the current 3D numerical experiments are not entirely satisfactory as they mostly propose a very thin bandwidth of granular layer: the behavior of tens of particles is not representative of the entire granular fault thickness (Dorostkar et al., 2017a). Besides, the calculation cost of 3D DEM is much higher than that of 2D modeling, leading to other restrictions in the length of the model, the number of particles, shapes of particles, interaction laws, etc. Another interesting aspect of computational modeling is the possibility to choose particle shapes. However, due to simplicity, convenience, and calculation costs in terms of contact detection and repulsive forces computation, most DEM studies use circular (2D) or spherical (3D) shapes of particles. This shape simplification: (i) is not very representative of fragmented particles found within mature granular fault (1.4.1), (ii) does not provide accurate predictions of granular behavior due to the different contact interaction [START_REF] Nouguier-Lehon | Influence of particle shape and angularity on the e o r o r r er er International[END_REF], [START_REF] Mair | Influence of grain characteristics on the friction of granular shear zones[END_REF], [START_REF] Mollon | Can friction replace roughness in the numerical simulation of granular materials?[END_REF]] and is unable to yield interlocking forces, (iii) needs some research and additional numerical parameters to change contact properties and balance the oversimplification.

Figure 26 displays the steady-state effective friction as a function of the topological dimension and shapes of particles based on results from [ [START_REF] Mair | Influence of grain characteristics on the friction of granular shear zones[END_REF], [START_REF] Frye | The effect of particle dimensionality on Granular friction in laboratory shear zones[END_REF]], and personal results with the software MELODY. Lower effective friction is observed for 2D experiments which do not account for out-of-plane contacts and circular particles compared to angular particles. According to the results, friction also varies systematically between numerical and experimental studies. These changes can be explained by different initial and laboratory conditions.

b. Cement and matrix modeling

Cemented materials can be represented as cohesive granular materials, both playing on the type of contact law used (friction, cohesion), properties of the contact network (force distributions), and the rheology (dilatancy and effective friction), [START_REF] Rognon | Dense flows of cohesive granular materials[END_REF]. The Finite Element Method (FEM) is not very suitable for cemented granular material, due to the very fine mesh needed to accurately represent the rupture of cemented elements. The Lattice Element Method (LEM), in between the DEM and the FEM, has been widely used within the framework of the statistical analysis of fracture or for cracking of concrete and ceramics [ [START_REF] Topin | Strength and failure of cemented granular matter[END_REF], [START_REF] Topin | Matériaux granulaires cimentés : modélisation et application à l ' albumen de blé[END_REF]]. This approach considers a medium reduced as a set of nodes belonging to a grid. However, in the present study, once the cemented links are broken, the granular fault core re o er r r e or w wo ' e o e w e That is why Discrete Element Modelling (DEM) also seems to be a good way to complete knowledge on cemented materials and their influence on slip mechanisms. The increase of global cohesion (in the sense of Coulomb-Mohr) generated by the increase of cementation (cf. section 1.4.2) tends to a brittle rock behavior that can be easily schematized with DEM by cohesive and breakable bonds between particles changing the contact kinematics. Bonded Particle Models (BPM) are often [START_REF] Potyondy | A bonded-particle model for rock[END_REF], [START_REF] Cho | A clumped particle model for rock[END_REF], [START_REF] Kazerani | Micromechanical parameters in bonded particle method for modelling of brittle material failure[END_REF], [START_REF] Potyondy | A flat-jointed bonded-particle material for hard rock[END_REF],] or to represent cementation in granular mechanics (contact bond model), [START_REF] Estrada | Simulation of cemented granular materials. I. Macroscopic stress-strain response and strain localization[END_REF], [START_REF] Jiang | An investigation on loose cemented granular materials via DEM analyses[END_REF], [START_REF] Das | A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables. Part II -Validation and localization analysis[END_REF]), (McBeck et al., 2019)]. This BPM method mostly uses circular particles in 2D (or spherical in 3D) connected at a point of contact and representing a compact and dense rock, Figure 27 (a). The objective is to be able to study the influence of the micro properties of grains, such as density, shape, distribution, and even grain stacking. The reader can access more information on the different DEM and BPM in the review made by [START_REF] Lisjak | A review of discrete modeling techniques for fracturing processes in discontinuous rock masses[END_REF]. A really recent study from [START_REF] Harmon | Modeling connected granular med r e o d w e e e e d re e e e e e od[END_REF] presents a new method to model cemented particles with what they call the Level Set Discrete Element Method for Bonded Particles (LS-DEM-BPM enabling to connect granular material with arbitrary particle shape existing in both 2D and 3D). They used their method to first model a granular fault gouge with rate-and-states friction laws and their results seem to be very promising for 3D modeling with real particle shapes. [START_REF] Cho | A clumped particle model for rock[END_REF], and flat-joint contact model showing the effective interface geometry (from [START_REF] Potyondy | A flat-jointed bonded-particle material for hard rock[END_REF]] redrawn). Image from (Lisjak & Grasselli, 2014) (b) Example of a mixture between clasts (i.e. gravel) and matrix (i.e. sand), image from (Y. [START_REF] Lu | Methodology for Simulation of Irregularly Shaped Gravel Grains and Its Application to DEM Modeling[END_REF].

The modeling of matrix particles only requires the classical DEM modeling without cementation (only frictional contacts), with a very dense packing of fine particles. (Y. [START_REF] Lu | Methodology for Simulation of Irregularly Shaped Gravel Grains and Its Application to DEM Modeling[END_REF] modeled Irregularly Shaped Gravel Grains surrounded by sand particles, using circular particles to represent the sand (which is a kind of matrix in this context). Circular particles however tend to an initial void ratio of about 30%, which is very important within the granular media, Figure 27 o e de er " r " re re e o oro oï e e o [ [START_REF] Mollon | Fourier-Voronoi-based generation of realistic samples for discrete modelling of References granular materials[END_REF]] e e o ' r ' e e e ro d r r er w o no initial porosity. In summary, the study of a modification in percentage of matrix particles within the granular fault core is similar, in some aspects, to a change in Particle Size Distribution, as done in [ [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 2. Effects of particle size distribution and interparticle friction on mechanical behavior[END_REF], [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF]].

In this Thesis, a 2D numerical modeling of a granular fault gouge is proposed with DEM to study the impact of cement or matrix particles on slip mechanisms. As this new material is somehow complex, we decided to restrain the study to a pure dry contact mechanism but chose to use angular shapes of particles. The slip mechanism we are studying takes place in a mature fault (i.e. after many successive slips in previous geological times) that may have been cemented, or filled with matrix particles, over time without slip (e.g. in an interseismic period). We can therefore say that we are studying the reactivation of a gouge-filled fault zone, without studying the fragmentation process which is our starting point. Most fault core numerical models do not consider the infill material within the fault gouge. Moreover, the studies considering these materials with mature fault are not with DEM, or they use circular particles. The present Thesis proposes to combine 2D explicit DEM modeling with fault mechanics, and to investigate the onset of seismic sliding and slip mechanisms in presence of infill materials. 2D modeling still has a lot to bring to granular fault behavior, to which we add the originality of using realistic angular and faceted particle shapes for gouge particles.

Chapter 2. Numerical model description and methods

Foreword

In the previous chapter, we described the fault core as a granular medium. Even though certain continuous methods are used to model granular fault gouge, they commonly face difficulties sticking the reality due to the continuity assumption. As an alternative to the classical Finite Element Method, the Discrete Element Method (DEM) describes the flow of a granular material where each particle has its own behaviors and interactions. In this chapter, schematized in Figure 28, section 2.2 presents the DEM techniques used within the research and details the general contact model, governing equations, and solver. Section 2.3 proposes a method to model the angular shape of particles and displays the different types of granular media chosen to model infill materials. Finally, section 2.4 introduces the numerical framework, models, and choices made for granular gouge modeling. This last section also discusses the criteria and methods employed to calibrate and control the simulations.

Discrete Element Modelling (DEM)

The code used for the simulations, MELODY 2D (Multibody ELement-free Open code for DYnamic simulation), is a C++ code allowing to simulate a broad variety of granular media [ [START_REF] Mollon | A multibody meshfree strategy for the simulation of highly deformable granular materials[END_REF], (Mollon, 2018a)] in 2D. This software is a multibody code, as it can represent

Method

in the same numerical framework first and third bodies with their deformation and dynamic (Chapter 1) keeping the discontinuity of the 3rd body. This subsection describes the DEM algorithms with the software MELODY.

General contact algorithm

DEM was first proposed by [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF] Figure 29 presents the two types of bodies that can be modeled with MELODY. The main difference is that rigid bodies are represented by nodes and segments, whereas compliant bodies are discretized in field nodes (this latter case is outside of the strict DEM framework, and uses a Multibody Meshfree Approach, [ [START_REF] Mollon | A multibody meshfree strategy for the simulation of highly deformable granular materials[END_REF], (Mollon, 2018a) and (Mollon, 2018b)].

Figure 29. Difference between the Compliant and rigid particle, image modified from (Mollon, 2018a).

For our research, we do not try to model highly compliant material such as clay or talc, and no high deformation is observed at the grain scale. Since the use of compliant bodies also highly increases the calculation cost, we decided to stick with rigid bodies for the granular fault core particles and rock walls.

Contact detection

Rigid bodies are characterized by their mass (located in the center of mass) and rotational inertia with 3 degrees of freedom (two in displacements and one in rotation for 2D modeling). The contour of particles is then a succession of contact nodes and segments allowing the application of specific contact laws. In order to have both accurate results, and reduce calculation costs, a RIGID BODY COMPLIANT BODY

Centre of mass

Contact segment Field nodes

Contact node

Mesh-free shape functions (for compliant bodies) compromise has to be found for node discretization. As an example, it was found that 32 nodes is an acceptable value to discretize each particle contour [START_REF] Mollon | Can friction replace roughness in the numerical simulation of granular materials?[END_REF], but it can be less or more according to the situation. The contact forces calculated (with a penalization method) are then transferred to the center of mass to solve the equations of motion.

Since the contour of each grain is discretized by a piecewise linear frontier with nodes and segments, each contact considered in the code concerns a given node from a grain A and a given segment from a grain B. From this node and this segment, it can be computed at any moment a normal gap 𝛿 𝑛 (obtained by projecting the node on the segment) and a tangential gap 𝛿 (integrated in time based on the history of the relative motions of the node and the segment in the tangential direction), Figure 30. 𝛿 𝑛 can be either negative (i.e. there is a small interpenetration between the grains, contact) or positive (i.e. there is a separation distance between the grains, no contact). The contact detection is bi-directional, meaning that each grain can adopt the roles of grains A and B at the same time. Edge-edge contacts hence do not need any special treatment in this framework, since they are automatically accounted for through their extremity nodes. To control overlapping, a contact stiffness is introduced in the normal and tangential directions (𝑘 𝑛 and 𝑘 ) as well as numerical damping (𝛾 𝑛 and 𝛾 ) in both directions. The normal and tangential gaps are extracted from particles geometry and kinematics. One of the interests of this code is its ability to represent realistic angular grain shapes, where most of the simulations reported in the literature use circular (2D)/spherical (3D) grains. The contact algorithm can handle particles with complex shapes and with anisotropic orientation. To do so, it is divided into three main steps, (Mollon, 2018a)

: i.
A broad proximity detection (Figure 31 (i)) -The objective of this first detection is to select pairs of close particles, considering as close any pair overlapping both when a projection is made on the -and -axis. This detection is based on a sweep-and-prune algorithm [START_REF] Cohen | I-COLLIDE: An interactive and exact collision detection system for large-scale environments[END_REF]. (Figure 31 (ii)) -This step defines potential contact between a node from particle A and a segment from particle B (considering a pair of close particles A and B). This detection is purely geometrical and considers that each segment and node have their own detection zone. Users can define the time period of this step, as it is not necessarily at each time step.

iii. A regular detection every time step (Figure 31 (iii)) -This last contact detection is called at each time step and only considers the actual contacting particles (i.e. some close proximity detections remain useless). To avoid a double detection (sharp particles) or loss of detection, a contact zone (i.e. contact box in the picture) is defined for each segment of particle A, considering only nodes of particle B inside this zone. For all the selected nodes (B j ) which have been detected to be in a proximity detection with the particle A (step i), classical mechanical and kinematics values are computed such as normal and tangential vectors, the projection of nodes B j in the segment A, and the normal and tangential gaps (𝛿 𝑛 and 𝛿 ) used for the different contact laws.

Contact laws

As our research focuses on the study of infill materials within a granular fault gouge, the objective is to be able to model matrix or cemented particles. We consider within this Thesis two contact laws: (i)

(ii) (iii)
The Damped-Mohr-Coulomb contact law (DMC)

In chapter 4, we will consider a fault zone that has already been sheared in previous slip episodes, and which is fragmented enough to observe matrix particles filling pores around big clasts. A classical friction law can be used to model a displacement between non-cohesive particles. This DMC law considers the typical geometrical and numerical parameters described in Figure 30 with interparticle friction only between particles in contact. Normal and tangential contact forces can be related with the following expression:

|𝜎 | ≤ 𝜇 𝑛 𝑚 𝜎 𝑛 (2.2)
With 𝜎 and 𝜎 𝑛 the tangential and normal stresses between two particles in contact, 𝜇 𝑛 𝑚 the interparticle friction. 𝜇 𝑛 𝑚 limits the tangential force for two particles sliding over each over.

𝜎 𝑛 and 𝜎 can then be computed thanks to the physical and numerical parameters of the contact:

𝑓 (𝛿 𝑛 > 0) 𝜎 𝑛 = 𝜎 = 0 (2.3) 𝑓 (𝛿 𝑛 < 0) { 𝜎 𝑛 = -𝑘 𝑛 𝛿 𝑛 -2𝛾 𝑛 𝑑𝛿 𝑛 𝑑𝑡 √ 𝑚 𝑎 𝑘 𝑛 𝐿 -𝑐 𝜎 = 𝑚 𝑛 (|𝑘 𝛿 2𝛾 𝑑𝛿 𝑑𝑡 √ 𝑚 𝑎 𝑘 𝐿 -𝑐 | , 𝜇 𝑛 𝑚 𝜎 𝑛 ) (2. ) 𝛿 = ± 𝜇 𝑛 𝑚 𝜎 𝑛 𝑘 (2.5)
𝐿 -𝑐 is the numerical contact length between two particles in contact.

The Bonded-Mohr-Coulomb contact law

In Chapter 3, we will consider a fault zone that has already been sheared in previous slip episodes, and which became aged (mature) enough to observe mineral cementation between particles. By aged we mean with no slip for a long-time, enabling cementation or precipitation processes.

e e e re e r er " o e e" e or [ [START_REF] Riedmüller | Engineering Geological Characterization of Brittle Faults and classification of fault rocks[END_REF], [START_REF] Wibberley | Recent advances in the understanding of fault zone internal structure: a review[END_REF]], this cementation can be explicitly simulated in DEM by considering breakable cohesive bonds between particles (Chapter 1).

e " o ded o r-o o " w d ed o o e r e ere ore ed contact law is similar to the classical law previously presented, except that a numerical cohesion is initially added as a pressure link between particles. This contact law is close to the Bonded-Particle-Model (BPM) from [START_REF] Potyondy | A bonded-particle model for rock[END_REF] and presents two main statuses (intact or broken) described below:

1. After compaction and before shearing, all contacts for which 𝛿 𝑛 < (𝛿 𝑑𝑒 𝑒𝑐 = 𝐶 𝑛 𝑚 /𝑘 𝑛 ), where 𝑘 𝑛 is the contact stiffness previously described and 𝐶 𝑛 𝑚 is the er o e o re e e e " " Figure 32 (a).

➢ e o

" " e o owing contact stresses are computed based on a purely elastic contact law: e o " roke " Figure 32 (b), either because it is a former intact bond or because it is newly created by grains motions at any time in the simulation, the following contact stresses are computed based on a purely frictional contact law: From these stresses, the associated contact forces (in the normal and tangential direction, as well as the associated torque) are computed on each grain, by considering that contact stresses act on a contact length 𝐿 -𝑐 (equal to the sum of half-lengths of the segments around contact nodes in grain A): The numerical cohesion 𝐶 𝑛 𝑚 considered in the code cannot be directly related to real cementation in rocks and does not bear much physical meaning by itself. A good way to associate a quantitative description to this numerical parameter is to define an associated level of cementation. It will be defined in Chapter 3.

𝜎 𝑛 = 𝑘 𝑛 𝛿 𝑛 (2.6) 𝜎 = 𝑘 𝛿 (2.7) ➢ If 𝜎 𝑛 ,
𝑓 (𝛿 𝑛 > 0) 𝜎 𝑛 = 𝜎 = 0 (2.8) 𝑓 (𝛿 𝑛 < 0) { 𝜎 𝑛 = 𝑘 𝑛 𝛿 𝑛 𝜎 = 𝑚 𝑛(𝑘 𝛿 , 𝜇 𝑛 𝑚 𝜎 𝑛 ) (2.9)

Solver and numerical parameters

The DEM methods are divided into two main categories based on the type of contact between bodies, the deformability representation of particles, the contact detection of particles, and the resolution of the equations of motion [START_REF] Lisjak | A review of discrete modeling techniques for fracturing processes in discontinuous rock masses[END_REF]): (i) Distinct Element Method (DEM) with explicit time-domain integration scheme for equations of motions (for both rigid and deformable bodies) [ [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF]) & codes from Itasca Consulting group]; (ii) the implicit DEM, or Discontinuous Deformation Analysis (DDA) method, which has almost disappeared, and (iii) the N "Nooo o " [START_REF] Jean | The non-smooth contact dynamics method[END_REF]. Very heavy in computing time, simulations are nevertheless limited in terms of number of particles or simulation time and it can be very advantageous to carry out parallelization of calculations. MELODY uses an explicit solver to integrate in time the deformation and motion of each particle (Mollon, 2018a).

Code resolution scheme

At the beginning of the simulation, it is possible to choose about using a constant time step or an adaptative time step ∆𝑡 (meaning that the value of the time step is automatically calculated from previous errors). In order to quantify the difference between adaptive time step and constant time step, studies have been carried out with mass scaling of particles (Appendix 2. A). It appears that the evolution of the effective friction (peak and steady-state) is quite different depending on whether we used an adaptive time step for the calculations or a constant time step. Indeed, the adaptive time step of the software MELODY has been coded for compliant particles, having no or very little numerical stiffness. If a high numerical stiffness is used within the contact, it will disturb the adaptive computation of the time step. It is, therefore, preferable to use a constant time step in these simulations with rigid bodies. The value of the time step used is provided in each corresponding chapter. The main algorithm is parallelized for each time step following the scheme provided in (Mollon, 2018a).

Choice of numerical parameters

In order to have meaningful particular contacts for particles considered as rigid, it is important to limit the interpenetration between two particles. This interpenetration is limited to a certain percentage (e.g. 1% or 0.1%) of the size of the smallest particle. In addition, the interparticle stiffness was also calibrated to have a consistent value of equivalent Young modulus regarding the other parameters used within each model (detailed value in each corresponding chapter). The damping value is constant and chosen equal to 0.2. According to the parameters chosen within the model and the equations of motion, it is then possible to compute the magnitude of the time step Δ𝑡 necessary for the simulation (and that has to correspond to Verlet schema):

Δ𝑡 = 𝑒 √ 𝑚 𝑚𝑖𝑛 𝑘 𝑛 𝐿 -𝑐 (2.12)
Where 𝑒 is a chosen parameter around 1/50, 𝑚 𝑚𝑖𝑛 the mass of the smallest particles, 𝑘 𝑛 the interparticle stiffness and 𝐿 -𝑐 the contact length. In most cases, although it is important to have a high enough stiffness to limit interpenetration, a too high stiffness is disadvantageous for the time step. In this assumption 𝑘 𝑛 is considered as a numerical parameter and if it increases too much it can lead to high numerical costs. In our simulation 𝑘 𝑛 is both a numerical and a physical parameter as it represents the characteristic deformability of the materials considered. In this case, there is no real cost-accuracy compromise but rather a physical calibration.

Granular fault zone and rock walls are considered piecewise-rigid with a constant numerical stiffness used to limit interpenetration between grains while mimicking the local deformation of the grains in the contact vicinity (value detailed in every model in chapters 3 and4). An explicit solver is used (Symplectic Euler scheme), to integrate in time the motion of each body.

Generation of a numerical gouge layer

In this thesis, we have chosen to represent angular-shaped gouge particles. For this purpose, we have presented in the previous section, the DEM code MELODY which allows managing the contacts between complex shapes of particles. This section presents a method to generate angular-shaped gouge samples.

How to create non-spherical particles?

Some authors highlighted the limitations of the software Particle Flow Code with circular particles (PFC 2D from ITASCA Consulting Group 1995) and suggest the use of a cluster of particles rather than single particles allowing for a more realistic grain shape than disks [START_REF] Cho | A clumped particle model for rock[END_REF]. Other researchers tried to represent a more realistic shape of particles, but they were mainly based on geometric simplifications and disc combinations (cf. Clump or Cluster Particle Model), [ [START_REF] Guo | Influence of normal stress and grain shape on granular friction: Results of discrete element simulations[END_REF], [START_REF] Cho | A clumped particle model for rock[END_REF], [START_REF] Ferellec | A simple method to create complex particle shapes for DEM[END_REF], [START_REF] Maeda | Stress-chain based micromechanics of sand with grain shape effect[END_REF], [START_REF] Härtl | Numerical investigation of particle shape and particle friction on limiting bulk friction in direct shear tests and comparison with experiments Top ring Normal load Bracket Loading stem References Shear force[END_REF], [START_REF] Zhou | DEM modeling of shear bands in crushable and irregularly shaped granular materials[END_REF]]. For these geometric shapes, contact parameters need to be adjusted to mimic a more realistic shape with pre-calibrated parameters such as inter-granular friction, stiffness, or viscosity. Only a few of them represent angular particle shapes [START_REF] Nouguier-Lehon | Influence of particle shape and angularity on the e o r o r r er er International[END_REF], [START_REF] Kawamoto | All you need is shape: Predicting shear banding in sand with LS-DEM[END_REF], (Mollon et al., 2021b)].

A relevant approach to generate realistic shapes of particles based on Fourier Descriptors has been developed in both 2D [START_REF] Mollon | Fourier-Voronoi-based generation of realistic samples for discrete modelling of References granular materials[END_REF]) and 3D [START_REF] Mollon | 3D generation of realistic granular samples based on random fields theory and Fourier shape descriptors[END_REF]. The authors identify statistical properties for unique particle shape characteristics gathered in a Fourier spectrum (projected contours of grains, Figure 33 (a)), with which they perform an inverse Fourier transform in order to obtain the real angular shapes, Figure 33 (b). The 2D DEM code called MELODY2D (Section 2.2) can deal with the contact interaction of these complex shapes in 2D, but the associated 3D-DEM code is not yet available.

The Matlab package Packing2D 1 is employed in the present work to create our granular samples. It is based on a Fourier-Voronoï method and generates a set of grains with userdefined size distribution and control on key morphological descriptors (such as elongation, circularity, and roundness). This control is performed by choosing a Fourier spectrum that quantifies the frequencies and amplitudes of the grain surface asperities. The advantage of this code is to create granular media with complex particle shapes, such as angular and faceted particles impossible to generate from the DEM code itself. More information is available in [START_REF] Mollon | Fourier-Voronoi-based generation of realistic samples for discrete modelling of References granular materials[END_REF] and [START_REF] Mollon | 3D generation of realistic granular samples based on random fields theory and Fourier shape descriptors[END_REF] for 3D particles.

Method for granular sample creation

The code Packing2D is divided into three main steps based on coupled Bounded Voronoï tessellation and Inverse Monte-Carlo Method:

1 Packing 2D is available for download at http://guilhem.mollon.free.fr The user chooses the size of the granular sample created (creation of the domain 𝑡ℎ 𝑖 x 𝐿 𝑔 in Figure 34) and the number of granular particles N. Then the type of distribution wanted, such as fractal, Gaussian, uniform, lognormal, bimodal, can also be specified. Finally, the Voronoi Tesselation evolves over several iterations, according to the target error allowed (specified by the user), until it reaches, as much as possible, the model characteristics chosen.

-Within Step 2, the user can define the shape of particles wanted, based on a Spectrum of morphologic descriptors, called Fourier descriptors, (complete procedure in [START_REF] Mollon | Fourier-Voronoi-based generation of realistic samples for discrete modelling of References granular materials[END_REF]). The code considers a contour of particle discretized by 𝑁 𝑖 points separated by an angle 𝑖 at a certain distance 𝑖 from the centre of the particle, Figure 35 Where 𝑜 is the average radius. Then each harmonic n allows defining a normalized amplitude 𝐷 𝑛 ed " o r er e r or"

𝐷 𝑛 = √𝐴 𝑛 2 𝑛 2 𝑜 (2.1 )
These normalized amplitudes, presented in Figure 35 (c), correspond to different morphologic descriptors of particle contours known in granular mechanics as Elongation, Roundness, Circularity, and Regularity, [START_REF] Mollon | Fourier-Voronoi-based generation of realistic samples for discrete modelling of References granular materials[END_REF].

The Fourier Descriptor 𝐷 2 essentially affects particle elongation (or the ratio between the smallest dimension of grain under a direction oriented in a particular angle), and has limited influence on the roundness of particles. Descriptors 𝐷 3 to 𝐷 8 define the shape modes or the importance of main irregularities within the grain contour. For example, a reduction of 𝐷 3 leads to rounder particles in Figure 36 (a) and (b). Finally, descriptors higher than 𝐷 8 characterize the roughness, considered as small irregularities within the contour increasing with 𝐷 8 value, Figure 36 (c). An inverse Fourier Transform on normalized amplitudes allows getting the real shape of a particle in a 2D domain as presented in Figure 35 (a). -The last step of the code, Figure 37, consists of randomly filling the initial Voronoï Tesselation (Figure 34), with the spectrum of granular particles created, Figure 35 (a). It is possible to both play on the initial solid fraction (percentage of filling of the Voronoi cell) and faceting of particles (more or less tangential to cell contours). Once the entire sample has been generated, it is possible to recover data such as the areas of grains generated , their position, and parameters as the inscribed and circumscribed radii, etc. [START_REF] Mollon | Fourier-Voronoi-based generation of realistic samples for discrete modelling of References granular materials[END_REF]. To have grains with flat facets, the optimization parameter 𝑁 𝑂 𝑖𝑚 can vary between 0 and 63, where 0 corresponds to a grain as defined in the spectrum and 63 a grain as much as possible with facets. Increasing this parameter is very helpful to have angular particle shapes, but it also increases a lot the calculation costs during sample generation. By increasing the target Solid fraction 𝐹 𝑆 (equal to ⁄ ), it decreases the compaction time of the granular sample. However, the final solid fraction of the sample also depends on how the grains will conform to the shapes of the Voronoi cells formed.

Three types of granular samples

This section presents the three different granular samples designed for the Thesis and used within the next chapters. It displays the type of granular samples, details numerical parameters, and shows a final representation of the numerical sample.

(i) Grains with angular shape (Chapter 3)

Mineral grains morphologies can be very diverse, but as mentioned in Chapter 1, granular gouges generated by comminution are expected to exhibit rather rough and angular shapes. To be closer to these shapes, angular and faceted particles are chosen to be modeled for the main granular sample of the Thesis. Since the morphological descriptors of the grains of granular gouges may vary significantly between faults, we calibrate our spectrum by visual comparison with published pictures of real gouges, Figure 38 (a) (more granular fault sample image library in Appendix 1. A). After several trials and variations of the Fourier descriptors, a suitable typical grain shape is selected, Figure 38 (b). The chosen numerical Fourier descriptors and numerical parameters are gathered in Table 1.

Figure 38. (a) Quartz, photomicrograph (crossed polars) of ATTL fault gouge from [START_REF] Muto | Fractal particle size distribution of pulverized fault rocks as a function of distance from the fault core[END_REF]. (b) Zoom on a granular sample generated with packing2D, with angular and faceted shapes, and compacted in MELODY2D.

Input parameters in table Table 1. 35).

𝐹 and 𝑁 𝑂 𝑖𝑚 comes from the cell filling step in the legend of Figure 37.

Fourier Spectrum properties

Cell filling

𝐷 2 𝐷 3 𝐷 8 𝜆 1 𝜆 2 𝐹 𝑆 𝑁 𝑂 𝑖𝑚 0.2 0.1 0.01 -2 -2 1 63
(ii) Matrix particles sample (Chapter 4)

For matrix particles, we consider fine particles in a very dense packing. To generate this kind of particle, a small code called Cvoro (C++) is used. In fact, this code is a simplified version of the previous method presented with a Voronoï tessellation algorithm, allowing to create a dense packing of polygonal particles.

(iii) Granular sample with matrix particles (Chapter 4)

This last type of granular sample is a combination of the two previous samples, Figure 40. The method first consists in identifying the overlapping zones between the matrix sample and angular particles. Then matrix cells that most fit the contour of angular particles are kept and matrix cells located in the geometrical domain of angular particles are removed. The final roughness of angular particles is not identical to the initially generated contour, but the choice of a small enough size of matrix cells enables to maintain the global shape of angular particles.

These three types of samples will be used all along the Thesis with some variations on physical and geometrical parameters and contact laws. More details are provided in each associated chapter.

Numerical framework 2.4.1. Direct shear modeling

Within this Thesis, direct shear modeling was chosen to simulate faulting and to allow comparison with Laboratory experiments. This subsection presents the different steps of a simulation, the assumptions, and the configuration of final models. Details on each model type will be discussed in the corresponding chapters.

Compaction step

Once the granular sample is generated with Packing 2D, it is then inserted between two rock walls in the pre-processing step. The first simulation step consists of compacting the sample with the chosen normal stress to obtain a mechanically stable packing of grains with a controlled granular density, Figure 41

Overview of the sheared granular fault model

Once the pre-processing step has been completed (application of contact laws and physical parameters), direct shear experiments can be settled. Figure 42 presents the 2D-DEM model of the simulated granular fault gouge, with rock walls at the top and bottom sides of the granular sample. In the following chapters, different granular materials are implemented as briefly described in the previous subsection. However, all the models present the same structure and external parameters: the lower wall is fixed, while normal stress ( -direction) and a sliding velocity ( -direction) are applied on the upper rock wall (defined in next section 2.4.2). Gravity is ignored in the model, assuming that the fault can be oriented in a wide range of directions, and that gravity forces are negligible compared to those related to normal and deviatoric stresses applied on the gouge. Periodic boundary conditions are present on both right-and left-hand sides of the sample to maintain the continuity of the movement at large slips, as a difference with [START_REF] Gao | Modeling of Stick-Slip Behavior in Sheared Granular Fault Gouge Using the Combined Finite-Discrete Element Method[END_REF] simulations. The movement of the upper rock wall in the -direction remains free to accommodate dilation or compaction of the sample.

A dry contact model is considered here, to investigate infill material influence, without fluid. Fault gouge and rock walls are considered rigid with a constant numerical stiffness used to limit interpenetration between grains (identical for normal and tangential directions) while mimicking the local deformation of the grains in the contact vicinity (section 2.2). The tangential and normal numerical stiffnesses are equal in this model and grain comminution is disregarded. In this study, we choose to simulate a density of 2600 kg/m 3 for particles, leading to an appropriate time step for these simulations of 10 -9 s. An explicit solver is used (Symplectic Euler scheme) to integrate in time the motion of each body.

Measure and control during the simulation

To characterize the slip behavior within the simulation, different quantities and parameters are extracted, calculated, and observed during simulations. These results are then stored at various time intervals (~10 -6 -10 -8 s, depending on the model):

-A typical quantity investigated all other the manuscript is the effective friction 𝝁 = 𝝉 ⁄ (Chapter 1), Figure 43 (a). This effective friction is not to be confused with the Coulomb-Mohr coefficient of friction 𝜇 𝑓 or interparticle friction 𝜇 𝑛 𝑚 . Effective friction is recovered at the interface between the gouge and the upper rock wall, with 𝝉 depending on the normal stress applied on the rock wall, Figure 43 (b). The same quantity is used in other researches [START_REF] Grd Midi | On dense granular flows[END_REF], [START_REF] Azéma | Internal structure of inertial granular flows[END_REF], [START_REF] Berger | Scaling behaviour of cohesive granular flows[END_REF]]. It is also important to differentiate the effective friction peak μ p (maximum effective friction) and the steady-state effective friction 𝜇 which is the average value of the effective friction once the plateau zone is reached and for a constant applied shearing velocity, Figure 43 (a). -The dilation of the granular sample is also measurable and corresponds to the thickness variation ∆𝑡ℎ in the vertical direction, divided by the initial granular sample thickness

𝜺 𝒚 = ∆𝒕𝒉 𝒕𝒉 𝒊 ⁄ , ( s h upp w ll s d spl f h y -d o ).
-The initial surface percentage of porosity (𝑷 𝒑 𝒓𝒆 ) within the sample is the total surface of voids divided by the total surface (voids and particles) 𝑷 𝒑 𝒓𝒆 = 𝟎𝟎 (𝑺 𝒗 𝒊𝒅𝒔 𝑺 𝒈 𝒖𝒈𝒆 ). ⁄ It can also be expressed by the mean of the void ratio 𝐹 𝑣𝑜𝑖𝑑 and Solid fraction 𝐹𝑠, which is 𝐹 𝑣𝑜𝑖𝑑 = 1 -𝐹 𝑆 , (Appendix 2.C). -For each model, particles' granular flow needs to remain in a quasi-static regime in order to avoid inertial effects within the simulation. A way of quantifying this is to calculate the Inertial number , which is a dimensionless number used to measure the inertial effects for dry granular flow in-plane shearing [ [START_REF] Pouliquen | Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane[END_REF], [START_REF] Iordanoff | Granular Lubrication: Toward an Understanding of the Transition Between Kinetic and Quasi-Fluid Regime[END_REF], [START_REF] Da Cruz | Rheophysics of dense granular materials: Discrete simulation of plane shear flows[END_REF], [START_REF] Jop | A constitutive law for dense granular flows[END_REF], [START_REF] Pouliquen | Flow of dense granular material: Towards simple constitutive laws[END_REF]]:

= |𝛾̇|𝜙 𝑎𝑣 √𝜎 𝑁 𝜌 ⁄ (2.15)
With 𝜎 𝑁 being the applied stress, 𝜙 𝑎𝑣 the mean particle diameter (or it can be the local mean particle diameter defined by volume fraction of each size), 𝜌 the density of particles and the shearing rate 𝛾̇= ∆𝑣 𝑡ℎ ⁄ . A quasi-static regime is observed for ≤ 10 -3 , then characterizes a transitional regime until reaching 0.2 ≈ . For 0.2 < , the regime is considered as collisional with dynamics becoming dependent on the particle restitution coefficient. Moreover, the particle size distribution within the sample can highly modify the granular flow and its behavior [START_REF] Pouliquen | Granular Flow[END_REF]. Dedicated post-processing tools are also used in order to analyze the spatial distribution of the solid fraction, force chains, or velocity fields patterns during the onset of slip, in order to bring a new understanding of the physics observed from a granular point of view.

Main assumptions of the model

Due to the complexity of the parameters involved in the behavior of fault gouges (Chapter 1), it was decided to fix some physical parameters. In this section, we present the assumptions and choices that will be kept in the next chapters.

Normal stress

The model aims to simulate a sheared granular fault zone, subjected to a normal stress proportional to the depth of the fault. The pressure exerted by the rocks above the fracture is represented by the application of a normal stress to the wall in the -direction. The fault is modeled parallel to the surface, for the sake of simplicity, but it is a simplification, as it is clearly not the case everywhere [START_REF] Giorgetti | Experimental Insights Into Fault Reactivation in Gouge-Filled Fault Zones[END_REF]. The normal stress, as well as the shear rate, are non-negligible parameters of the gouge model (Chapter 1), that must be chosen with precision to have a meaningful model [START_REF] Aharonov | Stick-slip motion in simulated granular layers[END_REF]. Parametric studies were carried out with different normal stresses, 1, 5, 10, 40 & 100 𝑀𝑃𝑎, for a sheared granular gouge with 2500 angular particles (size of the 2D granular sample 2x10mm²). Figure 44 (a) presents the effective friction (peak and steady-state) and dilation as a function of the applied normal stress (𝑀 𝑎). For the cases studied, 0.5 ≤ 𝜇 ≤ 0.6 d de re w re e or re The 𝜇 value is similar and equal to ~ 0.45 for all cases except for the 1 𝑀𝑃𝑎 case, which is supposed not enough constrained relatively to the interparticle stiffness applied between particles. The variation amplitude of effective friction oscillations is decreasing towards the same behavior with increasing normal stress. The evolution of dilation is almost similar to effective friction peak evolution with a decrease at 100 𝑀 𝑎, meaning that the too high pressure prevents sample dilation. In our results, a clear quasi-static regime is only observed with a normal stress above 10 𝑀 𝑎 and begins to stabilize with 𝜎 𝑁 = 0 𝑀 𝑎, Figure 𝐹 𝑁 𝐹 𝑇 𝑡ℎ 44 (b). A possible interpretation is that in the range 5 -0 𝑀 𝑎, the 3 quantities (friction & dilation inertial number) are constant and considered as the domain of validity of your model. Outside this range, different effects disturb the results. For low stresses, it is certainly due to the inertial number (much higher than for other cases), and for high stresses, it may be caused by important interpenetrations between grains and to the collapse of the assumption of rigid grains. Increasing the normal stress until 100 𝑀𝑃𝑎 will imply adjusting numerical stiffnesses within the model to limit overlapping caused by the high-pressure increase, and will thus increase calculation costs. As normal stress is also known to control the shear modulus of the granular sample [START_REF] Lyu | On the mechanics of granular shear: The effect of normal stress and layer thickness on stick-slip properties[END_REF], it will imply a material with a nonphysical Young modulus in our case.

Regarding the stabilization of our results after 10 𝑀𝑃𝑎, the results from the literature, and trying to limit calculation costs, we decided to focus on a medium value of normal stress of 𝟒𝟎 𝑴𝒑𝒂 for all the Thesis experiments. This pressure corresponds to a depth of about 1.5 𝑘𝑚 which is the early beginning of the natural seismogenic zone and it is also an interesting depth for all geoengineering studies, such as enhanced Geothermal Systems.

Shear velocity

The shearing rate is also a difficult parameter to choose and mainly depends on the type of earthquake studied. The increase of velocity induces a modification in the contact law basically presenting a peak of effective friction and a weakening until a steady state. Small initial tests have been carried out to see the global influence of shearing velocity within the model of a sheared granular gouge with 2500 angular particles (size of the 2D granular sample 2x10 𝑚𝑚²).

Figure 45 presents effective friction, dilation, and inertial number as a function of different shearing velocities, 0.01, 0.1, and 1 𝑚/𝑠. The gouge has been sheared with the applied normal stress of 0 𝑀𝑃𝑎 for the first case and 10 𝑀𝑃𝑎 for the other ones. In all the simulations, the inertial number is observed to be bellow 10 -3 , Figure 45 (b), indicating that the granular flows observed remain in the quasi-static dense granular flow.

The obtained results are pretty similar in terms of effective friction and dilation. That is why, to have observable results in a reasonable computation time while avoiding disturbing inertial effects since the dimensionless inertial number, we chose a shearing velocity of 𝒎/𝒔 for the rest of the Thesis.

Effect of grain shape (circular vs angular particles)

To be closer to fault gouges observed within fault cores, angular and faceted particles are employed in this Thesis. However, to validate the importance of modeling these shapes of grains compared to circular shapes, and compare our model with literature, two samples were implemented: one with angular and faceted particles (the one used for the study) and another one with circular grains. This second sample is then discarded in the next chapters, but the results are presented in this subsection. The numerical and physical characteristics of both samples are presented in Appendix 2.D. Shearing velocity (𝑚/𝑠)

𝜎 𝑁 = 0𝑀𝑃𝑎 (a) 𝜇 𝜀 𝑦 (b)
The granular layer with circular shapes of grains is generated with the same parameters as the one with angular particles, Figure 46 (a) & (b). But due to the circular shape of particles, the fractal dimension is a bit lower with 𝐷 = 2.5 (vs 𝐷 = 2.6 for angular sample) and with a corresponding equivalent diameter in the range 20 -181 µ𝑚 (vs 28 -226 µ𝑚 for angular), Figure 46 (c). To have the same initial gouge thickness (1.7 𝑚𝑚) and a similar total surface of particles (26 mm² for circular particles and 29 𝑚𝑚² for angular particles), the gouge with circular particles is composed of 7515 grains.

The two samples (circular and angular) are sheared with the same parameters (Appendix 2.D) and with a simple friction law to consider a typical case with only frictional contacts. As in previous studies, the effective friction is smaller for circular grains and steady-state effective friction tends to 0.45 for angular grains and 0.3 for circular grains, Figure 47. Because of their invariance by rotation, smooth spherical shapes tend to roll to accommodate deformation of the grain assembly whereas interlocking between angular grains tends to promote dilation [START_REF] Anthony | Influence of particle characteristics on granular friction[END_REF]. The effective friction observed with our results is lower than for 3D experimental studies ( (Frye & Marone, 2002) and Chapter 1). However, gouge behaviors obtained fit well with previous 2D results with angular and circular shapes [ [START_REF] Jensen | DEM simulation of granular media -Structure interface: effects of surface roughness and particle shape[END_REF], [START_REF] Mair | Influence of grain characteristics on the friction of granular shear zones[END_REF], [START_REF] Nouguier-Lehon | Influence of particle shape and angularity on the e o r o r r er er International[END_REF], [START_REF] Guo | Influence of normal stress and grain shape on granular friction: Results of discrete element simulations[END_REF]]. It was also shown recently that mechanical effects of grains surface roughness can only be mimicked by intergranular friction to a certain extent, and that proper modeling of the shear behavior of granular samples requires realistic shapes [START_REF] Mollon | Can friction replace roughness in the numerical simulation of granular materials?[END_REF]. Here the large fractal dimension D promotes a stable sliding even for circular particles [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF]. The results confirm that the behavior of angular and circular grains, without cementation, is different, and using circular shapes is disadvantageous for the strength of gouge behavior and is not reflecting real contact interactions. The next chapters will thus deal with noncircular particles.

Representative Surface Element (RSE)

To justify that the size chosen for gouge samples is a RSE, three different lengths of models have been studied with identical numerical parameters and identical gouge thickness (~ 2 𝑚𝑚).

The different sizes of models (6x2 𝑚𝑚², 10x2 𝑚𝑚², and 20x2 𝑚𝑚²) induce a change in the total number of particles to keep the same equivalent diameter for particles (respectively 1500 grains, 2500 grains, and 5000 grains). By increasing the number of grains in the model, as well as the length of the gouge, similar behaviors of the sheared gouge layers are observed. The effective friction coefficient follows the same trend for the three model sizes. The effective friction peak 𝜇 is also of the same order of magnitude (0.73 -0.78), Figure 48. As the number of grains increases, there is, therefore, an increase in normal and tangential forces applied to the upper rock wall, but the higher number of grains makes it possible to maintain similar effective friction in the three models. Moreover, by comparing force chains in the gouge for the three models, we do not see significant changes, Figure 49 (a). They are oriented at 30 -5 ° relatively to the upper rock wall and normal forces seem oriented in this same direction, Figure 49 (b).

Dilation 𝜀 𝑦 Effective friction 𝜇 Slip distance (µm) Slip distance (µm) (a) (b)
Even though the three models showed similar behaviors, we selected the model with 5000 particles, in order to have a chance to observe local mechanisms and rupture patterns (more difficult with smaller length). It is therefore not necessary to represent a larger model for this type of micromechanical study. It would have been interesting to do the same kind of experiment by modifying the thickness of the gouge layer, as it is known as a really influent factor in the contact mechanisms (Chapter 1). However, by changing this thickness, we also modify the RSE in length, the model studied is therefore a compromise between length, gouge thickness and number of grains in the thickness. We judged that our models allow being in the case of a mature enough fault gouge to observe deformation patterns with a gouge thickness fixed at around 2 𝑚𝑚 (a value consistent with the literature [START_REF] Rice | Seismic Fault Rheology and Earthquake Dynamics[END_REF]).

Stability conditions

The granular literature related to fault mechanics often explores stick-slip instabilities, considering strain energy storage either at grain-scale [(J. [START_REF] Leeman | Stiffness evolution of granular layers and the origin of repetitive, slow, stick-slip frictional sliding[END_REF], [START_REF] Dorostkar | Stick-slip dynamics in dry and fluid saturated granular fault gouge investigated by numerical simulations[END_REF]] or in the loading system (Sathwik [START_REF] Kasyap | Micromechanical-based experimental and analytical studies on rate effects and stick-slip instability of smooth quartz surfaces in the presence of plastic and non-plastic gouges[END_REF]) and can thus be carried out in a stable or unstable way. Our model is not totally realistic as, the sliding is imposed by the Boundary Conditions, and the non-deformability of the rock does not allow energy storage within the fault system. Although such energy storage is not present in our simulation, it is to be kept in mind from the perspective of upscaling the frictional behaviors we report to actual fault dynamic systems.

Concluding remarks

This chapter has detailed the main principles of the Discrete Element Method (DEM) and how to apply it to a granular fault gouge containing infill materials such as mineral cementation or fine matrix particles. For that purpose, in the first part, the general algorithm has been detailed, as well as the contact laws which will be used in the following chapters. In a second part, we proposed a numerical tool to create a numerical gouge sample with angular and facetted shapes (Packing2D). This method allows creating three main types of granular samples: a fault gouge with angular particles in which cement will be added through contact laws, a fault gouge simply composed of polygonal matrix particles and a mixture between angular grains surrounded by matrices. The last section presented the 2D direct shear model simulated using DEM and the important parameters of the model. It also highlighted several preliminary analyses defining the physical and geometric parameters of numerical fault gouges (applied normal stress of 0 𝑀𝑃𝑎 and a shearing velocity of 1 𝑚/𝑠). It has been confirmed that the shape of the grains has a significant influence on the coefficient of friction and the shear behavior of the gouge, justifying the use of angular grains with facets.

However, there are some limitations to these numerical models, such as:

-The small wavelength used for rock walls, as opposed to a mature fault which may be very eroded with higher rock walls wavelength. -The choice of constant normal stress, shearing velocity, and gouge thickness implies results corresponding to a specific case of study. -The impossibility of a direct study of seismic or aseismic behavior, since the model cannot store energy.

The following chapters propose two different models of a sheared granular fault gouge with cemented material (Chapter 3) or matrix material (Chapter 4) with the associated contact algorithms detailed above. Both granular samples try to be as realistic as possible with angular d e ed r e or " " d e o e or r r e

Chapter 3. Slip behavior of a sheared cemented gouge

Foreword

Within a mature fault gouge, mineral cementation coming from rock dissolution, melting, or other processes derived from previous slips, can fill remaining pore spaces between particles and change the global state of cohesion [START_REF] Philit | Clusters of cataclastic deformation bands in porous sandstones[END_REF]. The objective of this chapter is to establish a link between the micro-mechanical and structural properties of a cemented gouge layer, and its rheological behavior under shearing, Figure 50. We also aim to understand the influence of interparticle bonds on slip mechanisms by employing the Discrete Element Method (Chapter 2). We consider a direct shear model without fluid in 2D, based on a granular sample with angular and faceted shapes. Section 3.2 presents the fault segment model in 2D involving two rough surfaces representing the rock walls separated by the cemented fault gouge. Section 3.3 describes and characterizes the cemented materials used for the numerical campaign.

Focusing on the physics of contacts inside the granular medium, we explore strength evolution, gouge kinematics, and force chains within the gouge in section 3.4. We aim in this section to study the effect of cementation and initial porosity on mechanical behaviors and kinematics of shear bands. The last section 3.5 offers a discussion on new insights and relations between cementation within the gouge, shear localization, and relevance of Coulomb-Mohr theory for fault models. This chapter is mostly based on the following paper: A 2mm-thick granular fault gouge is created before compaction, resulting in 1.7 𝑚𝑚 after compaction. A gouge of 2 x 20 𝑚𝑚² is found to be satisfactory (RSE, Chapter 2) and falls within the same order of magnitude as previous studies [ [START_REF] Ferdowsi | Discrete element modeling of triggered slip in faults with granular gouge Application to dynamic earthquake triggering[END_REF] & [START_REF] Dorostkar | On the role of fluids in stickslip dynamics of saturated granular fault gouge using a coupled computational fluid dynamics-discrete element approach[END_REF]]. A fractal size distribution is chosen to fit with the literature on granular gouge composition, [ [START_REF] Olgaard | The microstructure of gouge from a mining-induced seismic shear zone[END_REF], [START_REF] Blenkinsop | Cataclasis and Processes of Particle Size Reduction[END_REF], [START_REF] Billi | Fractal distribution of particle size in carbonate cataclastic rocks from the core of a regional strike-slip fault zone[END_REF], [START_REF] Billi | Grain size distribution and thickness of breccia and gouge zones from thin (<1 m) strike-slip fault cores in limestone[END_REF], [START_REF] Muto | Fractal particle size distribution of pulverized fault rocks as a function of distance from the fault core[END_REF]] with a fractal dimension factor D close to 2.

Two initial states

Once the granular sample is generated with Packing 2D, it is then inserted between two rock walls and the first simulation step consists in compacting the sample with 𝜎 𝑁 = 0 𝑀𝑃𝑎 to create a stabilized packing of granular material (Chapter 2). Thanks to a preliminary study on the influence of interparticle friction during compaction and shearing steps (Appendix 2.C), we chose to keep two initial porosities for this chapter, 𝑃 𝑜 𝑒 = 16 % (i.e. 𝐹 𝑆 = 0.8 ) and 𝑃 𝑜 𝑒 = 11 % (i.e. 𝐹 𝑆 = 0.89) re re e e ed " d-de e e " d "de e e ". When the granular media obtained is sheared (without cementation), the initial density has a considerable influence on the initial peak of effective friction and dilation phase with slip distance: the chosen samples correspond to two granular behaviors when the gouge is sheared without cementation, as observed in Figure 52. The mid-dense sample (𝐹 𝑆 = 0.89) shows a higher friction resistance than the mid-dense sample (𝐹 𝑆 = 0.8 ) that does not exhibit the same friction with a friction peak almost missing. The evolution of the gouge thickness Figure 52 (a) is also consistent with the initial density of both samples: the denser sample presents higher dilation. If we want to study the onset of sliding, it is important to know what is the initial state of the system, and more particularly the initial porosity of the granular material. Depending on the type of rock or the previous shearing in the gouge, the sample can have different initial states that will resist differently to shearing. Within the next sections, we will consider the two identified initial porosities, (𝑃 𝑜 𝑒 = 16 %) for mid-dense samples and, (𝑃 𝑜 𝑒 = 11 %) for dense samples.

Numerical setup for direct shear simulations

Introduction of cementation between particles

e e e re e r er " o e e" behavior [ [START_REF] Riedmüller | Engineering Geological Characterization of Brittle Faults and classification of fault rocks[END_REF], [START_REF] Wibberley | Recent advances in the understanding of fault zone internal structure: a review[END_REF] The numerical cohesion 𝐶 𝑛 𝑚 considered in the code cannot be directly related to real cementation in rocks and does not bear much physical signification by itself. A good way to associate a quantitative description to this numerical parameter is to define an associated level of cementation. In the initial state of our model, each cohesive bond between any pair of contacting grains requires a certain amount of mechanical energy for breaking. The energy of de-cementation 𝐸 𝐶𝑒𝑚 , represents the total energy needed by the system to break all cohesive bonds initially present within the gouge. It can be described by a relation between properties of the initial contact network (i.e. at the end of the compacted state):

𝐸 𝐶𝑒𝑚 = (∑ (∑ 𝐿 -𝑐𝑖𝑗 𝑛 𝑗=1 ) 𝑁 𝑖=1 ) ( 𝐶 𝑛 𝑚 2 ) 2 1 2 𝑘 𝑛 𝐿 𝑔 [𝐽. 𝑚 -2 ] (3.1)
With 𝐿 -𝑐𝑖𝑗 the contact length of one particle bond in one side of a particle, Figure 53 (a), N the total number of particles, 𝑛 the number of contact bonds for each particle, 𝑘 𝑛 the normal numerical stiffness, 𝐶 𝑛 𝑚 the initial numerical cohesion and 𝐿 𝑔 the length of the model.

The idea is then to be able to relate this energy to the physical characteristics of rocks. To do so, the total amount of energy that would be needed in order to break all the initial bonds between cemented particles is normalized with respect to a representative surface energy 𝐸 𝑆 . This energy is applied to the contour of each particle which is assumed to be cemented on its entire contour, (a unit length is considered in the third dimension for any necessary purpose).

The maximal energy for a fault patch of 1 𝑚² width (third direction) and cemented bonds covering the total external surface of all particles in the gouge, as in Figure 53 (b), could be written as:

𝐸 𝑚𝑎𝑥 = 2 𝐸 𝑆 ∑ 𝑃 𝑗 𝑁 𝑗 =1 𝐿 𝑔 [𝐽. 𝑚 -2 ] (3.2)
With 𝑃 𝑗 the er e er o r e "j" d 𝐸 𝑆 the surface energy of 62 𝐽/𝑚² which was reported for the Chilhowee quartzite and considered as an upper limit for rock surface energy by [START_REF] Friedman | Fracure-surface energy of rocks[END_REF].

Figure 53 e o e wee r r e e e r r r 'j' w o er e er 𝑃 𝑗 and total contact length of one grain 𝐿 -𝑐𝑗 (different from the total contact length of the sample 𝐿 -𝑐 ) (b) Theoretical view of a contact fully cemented (100 % cementation) where 𝐿 -𝑐𝑗 is equal to the perimeter of the grain.

𝑃 𝑗 𝐿 -𝑐

(a)

𝐿 -𝑐 = 𝑃 𝑗 Cement (b)
The next step is to determine which numerical cohesion corresponds to this state of total cementation Figure 53 (b), this value is called 𝐶 𝑛 𝑚-100% . We can thus express the decementation energy in our initial sample as a percentage of cementation 𝑃 𝑐𝑒𝑚 proportionally to the 100 % cemented material.

𝐸 𝐶𝑒𝑚 = 𝐸 𝑚𝑎𝑥 [𝐽. 𝑚 -2 ] (3.3) 𝐶 𝑛 𝑚-100% = √ 16 𝐸 𝑆 ( ∑ 𝑃 𝑗 𝑁 𝑗=1 ) 𝑘 (∑ (∑ 𝐿 -𝑐𝑖𝑗 𝑛 𝑖=1 ) 𝑁 𝑗=1 ) [𝑀𝑃𝑎] ( 3.
)

𝑃 𝑐𝑒𝑚 = 100 ( 𝐶 𝑛 𝑚 𝐶 𝑛 𝑚-100% ) [%] (3.5)
Hence, any initial state of cementation effectively introduced in the numerical samples by the means of the numerical cohesion 𝐶 𝑛 𝑚 can be defined as a certain percentage of this complete cementation. Thus, a simulation case with cementation of 𝑃 𝑐𝑒𝑚 = 20 % is to be interpreted in the sense that, in its initial state, the energy needed to break all its bonds is equal to 20 % of that of the fully cemented case. After compaction and stabilization, the contact law between grains is introduced with a percentage of cementation 𝑃 𝑐𝑒𝑚 from 0 % to 95 %.

Direct shear simulations

Figure 54 presents the DEM model of the cemented granular fault gouge, with rock walls at the top and bottom sides of the granular sample.

Contact surfaces of rock walls are sinusoidal to introduce a certain roughness and avoid wallslip effects since we want to ensure that slip accommodation takes place within the gouge. This is a pure modeling choice, and the rock surfaces should be considered as nominally flat.

Interparticle friction is equal to 1 at the contact interface between walls and particles to make sure that the motion is fully coupled at the wall-grains transition. Interparticle friction 𝜇 𝑛 𝑚 is set to 0.5 between particles in contact. Since the parametric study realized in Chapter 2, a normal stress of 40 𝑀𝑃𝑎 and a sliding velocity of 1 𝑚/𝑠 are applied on the upper rock wall.

To limit interpenetration between grains while mimicking the local deformation of the grains in the contact vicinity, a constant numerical stiffness of 10 1 𝑃𝑎/𝑚 is used (Appendix 3.A). This constant value was chosen to obtain the overall deformability of the sample of the same order of magnitude as the one for bulk granite or shearing modulus (≈ 10 -25 𝑀𝑃𝑎 depending on initial porosity, see next section). All the parameters are gathered in Table 3 and are reproducible (Appendix 3.B).

Figure 54. DEM model of a granular fault gouge, 4960 angular particles in a 1.7x20 𝑚𝑚 domain. As explained in Chapter 2, gravity is ignored in the model, assuming that the fault can be oriented in a wide range of directions, and that gravity forces are negligible compared to those related to normal and deviatoric stresses applied on the gouge. Periodic boundary conditions are present on both right-and left-hand sides of the sample to maintain the continuity of the movement at large slips. The movement of the upper rock wall in the -direction remains free to allow gouge dilatancy while the bottom rock wall is fixed. In this study, the chosen density of 2600 𝑘𝑔/𝑚 3 for particles, leads to an appropriate time step for these simulations of 10 -9 s.

Cemented materials characterization

Characterization under biaxial simulations

The strength of granular materials (may they be cemented or not) and rocks is generally considered to follow the Coulomb-Mohr criterion (Chapter 1). In this model, the effective stresses are equal to the total stresses as the material is dry.

To characterize our synthetic cemented gouges (in the sense of Coulomb-Mohr), we ran independent biaxial simulations of samples with the same characteristics as those used for our sheared fault gouge (in terms of grain shapes, size distributions, initial solid fraction, (b) Cohesion increases monotonously with cementation (as found in the literature by [START_REF] Wissa | Effective stress-strength behavior of cemented soils[END_REF] and [START_REF] Lade | Cementation Effects in Frictional Materials[END_REF]) and reaches much larger values for dense samples (up to 70 𝑀𝑃𝑎 for 𝑃 𝑐𝑒𝑚 ≈ 100 %) than for mid-dense samples (up to 20 𝑀𝑃𝑎). It can also be noted that a certain percentage of cementation is needed to have a measurable cohesion within the gouge (𝑃 𝑐𝑒𝑚 > 20 % for mid-dense samples and 𝑃 𝑐𝑒𝑚 > 10 % for dense samples).

(c) However, Elasticity moduli is rather unaffected by cementation and is close to 11 -12 𝐺𝑃𝑎 for mid-dense samples and to 25 -26 𝐺𝑃𝑎 for dense samples.

Comparison to real cemented material and rocks

The cementation considered in this chapter has a clear definition from a numerical point of view (3.2.2) but it is also interesting to characterize and compare our samples with real materials. For this reason, it is instructive to consider Coulomb-Mohr properties of cemented gouges, for with typical rock properties found in the literature [START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF], in terms of internal friction angle and cohesion.

We observe that our samples have adequate values of internal friction angle (especially in the mid-dense case) and are in the lower range of cohesions. The low cohesion is consistent with the fact that we are simulating cemented gouges instead of intact rock. Figure 57 Even though some friction angle and cohesion are close to some Coulomb-Mohr values, a fault gouge develops a complex Riedel band structure reflecting a heterogeneous stress field different from the one observed for intact rocks [START_REF] Lockner | Rock failure and Earthquakes[END_REF]. Future work might be undertaken to target the Coulomb-Mohr properties of a cohesive rock for which direct shear experimental results are available, in order to propose a direct comparison with our simulations. [START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF] listed in the table next to the graph.

Influence of the cementation on gouge kinematics

Simulations with various cementation levels are performed for both dense and mid-dense samples with the numerical setup for a direct-shear experiment (section 3.2.2). In the next sections, we will use the effective friction 𝜇 and the dilation 𝜀 𝑦 (Chapter 2) to analyze the mechanical behavior of gouges. It is important to keep in mind that cementation makes 𝜇 [START_REF] Handin | On the Coulomb-Mohr Failure Criterion[END_REF].

Rock type

Granite ( 1) Gabbro ( 2) Trachyte ( 3) Serpentinized olivine ( 4) Sandstone ( 5) Berea sandstone [START_REF]CHAPTER[END_REF] Weber sandstone (7)

Effective friction and dilation within dense samples

Friction and gouge strength Figure 58 (a) & (b) provides typical curves of the measured effective friction of the fault as a function of the horizontal displacement imposed to the upper wall. In all cases, the tangential force increases linearly with cementation until a maximum effective friction 𝜇 , demonstrating the maximal effort that the loading system must provide to overcome interlocking in the gouge and accommodate imposed shearing. The linear elastic part of the curves represents the stiffness of the granular material. It should be noted that this stiffness is only related to the gouge layer itself since no other compliance (related to the surrounding medium, for example) is considered in the simulations (Chapter 2). All simulations with initial dense samples follow the same elastic slope, and adding more cementation extends the elastic part before effective friction peak, Figure 58 (a).

The peak strength evolves from a smooth, delayed, and of moderate amplitude shape (poorly cemented cases) to a sharp, short, and intense shape (highly-cemented cases). After the peak, the effective friction decreases in all cases towards a plateau and does not evolve significantly anymore until the end of the simulation. All steady-state values oscillate around the same effective friction 𝜇 𝑆𝑆 , (averaged from the beginning of the plateau until the end of the simulation), close to 0.5 (ranging from 0.45 to 0.51). This value is in agreement with other numerical studies [START_REF] Rathbun | Numerical investigation of the interplay between wall geometry and friction in granular fault gouge[END_REF], but lower than typical 3D experimental values (which are usually above 0.6). This discrepancy is related to the 2D character of the simulations [START_REF] Frye | The effect of particle dimensionality on Granular friction in laboratory shear zones[END_REF]. More details on that subject can be found in Chapter 1.

Brittleness

According to the literature, shear strength is enhanced with cohesion, [START_REF] Wissa | Effective stress-strength behavior of cemented soils[END_REF], and thus with the increase of cementation, Figure 56 (b). Cohesive strength in granular materials is known to correlate with an increase in the brittleness [START_REF] Meng | Rock brittleness indices and their applications to different fields of rock engineering: A review[END_REF], of the material [START_REF] Das | A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables. Part II -Validation and localization analysis[END_REF], and this is confirmed by Figure 58 (c), which presents the evolution of brittleness within each sample (based on strength ratio [START_REF] Bishop | The Influence of Progressive Failure on the Choice of the Method of Stability Analysis[END_REF]).

Despite the cementation level of each simulation, all the dense samples exhibit an overconsolidation peak. A reduction of this peak may appear for higher applied normal stress as found by [START_REF] Haines | Shear zones in clay-rich fault gouge: A laboratory study of fabric development and evolution[END_REF]. A qualitative change in behavior is observed close to 𝑃 𝑐𝑒𝑚 = 10 %. This transition corresponds to the value where cementation starts to induce cohesion within the gouge (red line). From this moment a marked evolution in terms of peak strength is observed, with the emergence of a sharp peak strength increasing with cementation. o re re e ed ere similar to the case with % cementation in terms of friction and dilation variation. These results are not in the range of small deformations, as the total slip displacement (𝐻 = 2.5 𝑚𝑚) is higher than the gouge thickness (≈ 1.75 𝑚𝑚).

Dilation 𝜀 𝑦

Dilation

For 𝑃 𝑐𝑒𝑚 ≤ 10 %, dilation is similar to a case without cementation as the cohesion in the sense of Coulomb-Mohr is smaller than 1 𝑀𝑃𝑎 (i.e. negligible compared to the normal stress). A progressive dilation until the steady-e " " o er ed e re e o e e o or 𝑃 𝑐𝑒𝑚 ≥ 10 %) seems to accelerate the dilation of the gouge, Figure 58 (d), which appears earlier with slip distance. This dilation enhancement is possible because some frictional contacts begin to form with the breakage of cemented bonds. Observing the second part of simulations from " " o " " Figure 58 (e), three main trends are highlighted: materials with medium cementation (~10 % ≤ 𝑃 𝑐𝑒𝑚 ≤ 75 %) reach a stabilized dilated material whereas poorly cemented materials continue to dilate and highly cemented materials present a contractive behavior. The higher dilation observed for poorly cemented materials (𝑃 𝑐𝑒𝑚 ≤ 10 %) can be explained by the fact that these simulations present more frictional contacts than cohesive contacts, and that frictional particles are known to enhance dilation [START_REF] Roy | Effect of cohesion on local compaction and granulation of sheared soft granular materials[END_REF]

Interface failure modes within dense samples

Cohesive strength correlating with brittleness increase has a major influence on strain localization [START_REF] Maurer | How Porosity Controls Macroscopic Failure via Propagating Fractures and Percolating Force Chains in Porous Granular Rocks[END_REF]. From observations on friction and dilation, the simulated materials can be gathered in three different cementation regimes: poorly cemented (𝑃 𝑐𝑒𝑚 ≤ 10 %), cemented (10 % ≤ 𝑃 𝑐𝑒𝑚 ≤ 75 %), and highly-cemented (𝑃 𝑐𝑒𝑚 ≥ 75 %), corresponding to three modes of deformation zone.

The rupture of a cohesive bond is represented by the increase of damage to the concerned grains. This damage is set to 0 for each grain when cohesive bonds are first established (all the bonds re d e o e e e o d re e " roke " Chapter 2). It is thus a relative damage with respect to the initial state. The representation of the relative damage gives a picture of the state of cementation between grains and their location within the gouge, Figure 59, Figure 60 & Figure 61. As shear band formation and evolution vary with the level of bond strength [START_REF] Jiang | An investigation on loose cemented granular materials via DEM analyses[END_REF], the damage allows following the formation of failure patterns (Riedel cracks, shear bands, etc.) and their orientation. These clusters can lead to rather inhomogeneous behaviors inside the granular gouge, changing the whole geometry and particle size distribution. Similar observations were made by [ [START_REF] Cho | Development of a shear zone in brittle rock subjected to direct shear[END_REF] & [START_REF] Rognon | Dense flows of cohesive granular materials[END_REF]]. Cohesive agglomerates also participate in increasing the overall dilation before steady-state, Figure 58 (b), as also found by [START_REF] Lade | Cementation Effects in Frictional Materials[END_REF] for large cemented particles at low confining pressures. Even though dilation is needed to observe shear bands, this cemented material (presenting the highest dilation after peak strength) inhibits the persistence of shear bands until the end of the shearing. This may be due to the important breakage of cohesive links into frictional contacts after the end of the effective friction peak.

(iii) Increasing again 𝑃 𝑐𝑒𝑚 leads to a highly-cemented material (𝑃 𝑐𝑒𝑚 = 95 %) where most cohesive bonds stay intact during the entire simulation, Figure 61. The numbers of tensile forces are obviously higher than in previous materials, Figure 61 (i), and correspond to [START_REF] Lade | Cementation Effects in Frictional Materials[END_REF] 95 " e r k e reduction was also observed in other numerical studies with the increase of strain inside the model (Cundall, 1989). As previously supposed (Section 3.4.1), the progressive breakage of cohesive agglomerates (i.e. asperities) at the interface between gouge and rock boundary seems to explain the contractive response observed in Figure 58 on the simulation) and lets the majority of particles (still cemented) behave as a single solid-ke od e r or " e" zo e re e w d e

Influence of initial porosity with cementation

The same percentages of cementation have been tested on a second set of samples with a higher initial porosity 𝑃 𝑜 𝑒 = 16 % (termed as mid-dense in what follows, as opposed to the dense samples with 𝑃 𝑜 𝑒 = 11 %). The three types of cemented materials previously highlighted have been kept to compare the results between dense and mid-dense samples. The trend observed for mid-dense samples is similar to dense samples, but we note an influence of both initial porosity and cementation on the strength of cemented materials.

Friction & dilation

The increase of initial porosity is supposed to reduce the fracture strength [START_REF] Taylor | Fundamentals of Soil Mechanics[END_REF]. Our results present the same conclusion, but different shapes of effective friction peak are observed: the peak strength may be sharp, short, and intense (dense and highly cemented cases) or smooth, delayed and of moderate amplitude (mid-dense and moderately cemented cases), Figure 62 (a) & (b). Denser samples show higher dilation rates than mid-dense samples Figure 62 (d), as the initial gouge is initially more compacted [START_REF] Wood | Soil behaviour and critical state soil Mechanics[END_REF]. When the steady-state is reached, a maximum dilatancy is also observed for denser initial samples with twice the deformation obtained with mid-dense samples, for all initial cementations (𝜀 𝑦 = 8 % for dense samples vs 𝜀 𝑦 = % for mid-dense).

Brittleness

Figure 62 (c) gathers the brittleness as a function of cementation for the whole simulation campaign and gives more precise information on the transition zone. For samples where a measurable Coulomb cohesion exists, an increase of cementation increases the brittleness for both dense and mid-dense samples, and thus enhances the strength of the gouge, Figure 62 (a). However, the increase of initial porosity leads to a weaker material in terms of cohesive strength.

Without the effect of cementation, we can observe that the brittleness is almost four times higher for dense samples (i.e. ~ 0.4) than for mid-dense samples (i.e. ~ 0.1), meaning a smaller overconsolidation peak. The mid-dense material with the lower cementation is assumed to be here, the weaker material tested. When the cementation level reaches values close to its maximum (𝑃 𝑐𝑒𝑚 ≈ 100 %), the brittleness can reach almost 0.9 in the dense material and almost 0.66 in the mid-dense material. Such high values are likely related to the fact that the cementation of the sample is very high while the confining stress ( 0 𝑀𝑃𝑎) is rather low [START_REF] Lade | Cementation Effects in Frictional Materials[END_REF].

Failure patterns

Mid-dense samples also present higher relative damage with easier grains reorganizations, Figure 63, which agrees with the relatively low Coulomb cohesion involved. One high angle Riedel shear band 𝑅' in the 40 % case, and two high angle Riedel bands 𝑅' in the 95 %, are followed by a horizontal shear localization at the bottom of the granular gouge. Conjugate Riedel bands observed do not have the same orientation as those observed with dense materials and are less persistent during shearing. The more ductile character observed for mid-dense samples could be a reason for the different Riedel angles observed [START_REF] Misra | Formation of Riedel shear r re r r er Findings from analogue shear experiments and theoretical analyses[END_REF]. Moreover, the non-persistence of shear bands for mid-dense samples is in adequation with previous studies on sandstones [START_REF] Dunn | Porosity Dependence and Mechanism of Brittle Fracture in Sandstones[END_REF], [START_REF] Antonellini | Microstructure of deformation bands in porous sandstones at Arches National Park, Utah[END_REF]]. They have shown that gouges with high porosities enhance a distributed deformation that takes the form of cataclastic flow, in contrast with low porosity sandstones which better fail by localization and strain softening [START_REF] Hirth | The effects of pressure and porosity on the micromechanics of the brittle-ductile transition in quartzite[END_REF].

Influence of ductility with cementation and porosity

Evolution of ductility with cementation

Force chains are plotted between grains centers (for each contact) in Figure 64. They are commonly used in granular physics for load transfer observation [ [START_REF] Radjai | Force Distributions in Dense Two -Dimensional Granular Systems[END_REF], [START_REF] Radjai | Contact forces in a granular packing[END_REF], [START_REF] Majmudar | Contact force measurements and stress-induced anisotropy in granular materials[END_REF], [START_REF] Zhang | The role of force chains in granular materials: from statics to dynamics[END_REF]]. Similar information is coded in polar diagrams which provide the distribution of the orientation of contact normal vectors.

The poorly cemented materials (𝑃 𝑐𝑒𝑚 < 10 -25 %) show a ductile behavior, with a progressive particle reorganization, and no localized shear observed, Figure 59. They display a preferential orientation for force chains inclined at 45° from the upper rock wall, Figure 64 (a). Similar force chains orientations have been observed by [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF] for circular particles. The change of orientation of normal forces at 45° appears before the effective friction peak, showing that the gouge started to dilate before reaching the peak. The evolution of the granular flow gives way to simple contact law with interparticle friction only. Once the gouge has dilated, grains can reorganize to allow shearing, and the gouge tends towards a steady state of sliding. The combination of a limited cementation (mostly frictional contact, no cohesion in the sense of Coulomb-Mohr), gouge dilation (Figure 58), and preferred orientation of force chains confirm a typical granular Couette flow for the poorly cemented material [ [START_REF] Grd Midi | On dense granular flows[END_REF], [START_REF] Da Cruz | Rheophysics of dense granular materials: Discrete simulation of plane shear flows[END_REF]].

With the increase of cementation (𝑃 𝑐𝑒𝑚 = 38 %), the orientation at 5° is not yet present at r o e k " ", because no significant motion has occurred yet and grains are still in their initial configuration. Instead, there are a large number of smaller ramified force chains distributed in a homogenous distribution of force networks. The strength of the material is linked to the development of force chains [START_REF] Zhang | The role of force chains in granular materials: from statics to dynamics[END_REF], and our results confirm that the intensity of force chains is enhanced with cementation, as strength is also enhanced, Figure 64 (a). Combining force chains and relative damage makes it possible to identify from the force network the cemented agglomerates highlighted in section 3.4.2 (red arrows). Each cemented agglomerate is crossed by force chains, while isolated frictional particles are not, Figure 64 (a). These cemented clusters also modify the particle size distribution within the gouge, which is well-known to act on shear bands formation in addition to the initial density of the sample [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF].

For a dense sample with 95 % cementation, some particles are detached from the cohesive band at the end of friction peak, and operate alone (within the active or shearing zone), defining clear geometrical asperities. These asperities create contacts between the cemented layer and the rock wall, which concentrate the normal load. A steady-state is reached when enough of this tribological third body (Chapter 1) has been released to avoid any asperity contact and to produce a three-body sliding. Gouge first dilates thanks to the emergence of asperities (i.e. grains emerging from cohesive bond breakage) and clusters of grains, forming interparticle bridges, which releases gouge in the interface as the rupture is going on. Normal forces orientation follows the same trend as the dilation: a first increase at the first step (due to interparticle bridge formation) and a decrease back to the initial state, as the majority of cohesive bonds are still active, Figure 64 (i). Cementation also enhances local dilation during the strength weakening, increasing local porosity inside the gouge as found by [START_REF] Faqih | Flow -Induced Dilation of Cohesive Granular Materials[END_REF]. This difference is assumed to be related to the more brittle behavior observed for dense samples, for which no reorganization of the grains occurs before the peak. It preserves the initial homogeneous contact network, but makes the failure more sudden, in contrast with mid-dense materials where irreversible grains motions occur before the peak. The graph provides the numbers of contacts with a normal vector oriented in a given direction (using polar diagrams where the Theta-axis is the orientation and the R-axis is the number of contacts), and with a color evolving during the different stages of each simulation.

Rheology and weakening mechanisms

Critical dilation

Critical dilation can be defined as the necessary dilation for macroscopic shear failure formation [ [START_REF] Kranz | Critical Dilatant Volume of Rocks at the Onset of Tertiary Creep crack observation using the scanni0g electron microscope each quartzite[END_REF], [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF]]. In our simulations, it corresponds to the dilation 𝜀 𝑦 obtained at the effective friction peak (Figure 58 & Figure 66) which is different from the final dilation 𝜀 𝑦-𝑆𝑆 previously introduced. For cases with limited cementation (𝑃 𝑐𝑒𝑚 < 10 % 𝑑𝑒𝑛 𝑒 -25 % 𝑚𝑖𝑑𝑑𝑒𝑛 𝑒 )where 100 % frictional contact is quickly reached and no localized shear is observedthe critical dilation is similar for both dense and mid-dense samples and presents the highest values (~ 2 % in Figure 66). The initial porosity effect, without cementation, does not have a major influence on critical dilation. The high dilation rate presents in the pre-peak phase is consistent with the important dilation observed at friction peak.

For cases with moderate to high cementation (𝑃 𝑐𝑒𝑚 > 10 % 𝑑𝑒𝑛 𝑒 -25 % 𝑚𝑖𝑑𝑑𝑒𝑛 𝑒 ), two main trends are observed with the increase of cementation: 1) a decrease of the critical dilation for mid-dense samples, 2) an enhancement of the critical dilation for dense samples. It can be noted that all critical dilations observed in this zone are er zo e " " Figure 66. For cohesionless materials, dilation is supposed to be larger for a denser granular gouge, but the introduction of cementation creates the opposite effect. Similar results have been found with other types of dense cohesive materials experiments [START_REF] Faqih | Flow -Induced Dilation of Cohesive Granular Materials[END_REF]. The critical dilation is here smaller for dense samples than for mid-dense samples, a result that may be due to the real cohesion values involved within the sample (i.e. higher for dense sample).

Comparing shear band formation and critical dilation, we can deduce that with more than 1 % of critical dilation (for both dense and mid-dense samples), shear band formation is inhibited. Then, the persistence of shear bands until steady-state mostly depends on cementation, and occurs for highly-cemented materials, corresponding to a critical dilation between 0.08 % and 1 %, zo e " " Figure 66. [START_REF] Noel | Brittle faulting of ductile rock induced by pore fluid pressure build-up[END_REF] found that for Tavel limestone (11 % initial porosity corresponding to our dense sample), a critical dilation of about 1 % has to be reached to obtain shear failure of an initially ductile rock subjected to a pore fluid pressure increase. Their higher critical dilation could be explained by the fact that they study intact rocks, but also by the higher confining pressure used and the presence of fluid.

Evolution of weakening mechanisms

𝐷 value, similar to the critical distance of slipping, is the distance needed to weaken the fault from friction peak to the steady-state of sliding, it is also relevant for rupture energy calculation (Chapter 5).

Two main trends are highlighted for the evolution of 𝐷 , with cementation Figure 67 (c).

Initial porosity seems to affect the weakening length only for cases with limited cementation (𝑃 𝑐𝑒𝑚 < 10 % 𝑑𝑒𝑛 𝑒 -25 % 𝑚𝑖𝑑𝑑𝑒𝑛 𝑒) , mid-dense samples need a higher displacement than dense samples to reach their steady-state regime (810 vs 300 𝜇𝑚). When cementation is large enough (𝑃 𝑐𝑒𝑚 > 10 % 𝑑𝑒𝑛𝑠𝑒 -25% 𝑚 𝑑𝑑𝑒𝑛𝑠𝑒 ) e e o oro doe ' e additional influence on 𝐷 , decreasing similarly with 𝑃 𝑐𝑒𝑚 for both dense and mid-dense samples. The transition between these two trends corresponds to the mechanical limit between the cohesionless and cohesive model (i.e. formation of cohesion in the sense of Coulomb-Mohr theory) represented by the red line in Figure 67 (c). Adding more cementation within the gouge seems to diminish the porosity effect regarding the peak length. The evolution of the peak distance 𝐷 (i.e. distance reached at effective friction peak), is similar to the evolution of critical dilation, Figure 67 (b). Double weakening shapes are observed in some effective friction curves in Figure 67. This pattern seems to be a characteristic of dense and cemented materials, as mid-dense materials only show one weakening phase. When it exists, the first weakening period seems to follow a decreasing affine law and appears for dense samples with 𝑃 𝑐𝑒𝑚 ≥ 10%. Similar double weakening has been observed experimentally as reported in an incoming paper from [START_REF] Paglialunga | On the scale de e de e e d o r o r re o r re e er er ze-dependent breakdown work[END_REF]. With the separation into two weakening phases, it appears that Riedel bands only form at the beginning of the 2nd weakening phase (letter C in previous figures). In fact, the first weakening phase just allows movement within the gouge by breaking some isolated bonds, but actual sliding needs dilation to operate and thus is mostly observed within the 2nd weakening. Isolating contributions from each mechanism (bond breakage, dilation, friction) and through energetical considerations, Chapter 5 provides a detailed analysis of these two weakening periods.

As it is observed, a micro-scale friction variation appears during the 2 nd weakening phase, that could look like stick-slip instabilities which is a major subject of interest in fault system [ [START_REF] Byerlee | Stick slip, stable sliding, and earthquakes-Effect of rock type, pressure, strain rate, and stiffness[END_REF], [START_REF] Aharonov | Stick-slip motion in simulated granular layers[END_REF], [START_REF] Dorostkar | Stick-slip dynamics in dry and fluid saturated granular fault gouge investigated by numerical simulations[END_REF], (Sathwik [START_REF] Kasyap | Micromechanical-based experimental and analytical studies on rate effects and stick-slip instability of smooth quartz surfaces in the presence of plastic and non-plastic gouges[END_REF]]. However, earthquake sliding involves the release of strain energy stored in the surrounding rock, at the metric or kilometric scale. Such energy storage is not present in our simulation (no deformable rock walls or stiffness of the loading apparatus), that is why these variations should not be considered as stick-slip. The numerical model of [ [START_REF] Aharonov | Stick-slip motion in simulated granular layers[END_REF], [START_REF] Ferdowsi | Discrete element modeling of triggered slip in faults with granular gouge Application to dynamic earthquake triggering[END_REF]] studying stick-slip motion does consider a certain stiffness connected to their rock walls allowing to study stick-slip properly. l g h 𝐷 (𝜇𝑚), for both dense and mid-dense samples, as a function of the percentage of cementation within the gouge. 𝐷 is the distance between the peak of effective friction and the beginning of the steady-state

Cemented materials and Coulomb-Mohr theory

Cemented gouges have interesting behavior because they are not intact rock and neither are simple frictional gouges. In this section, we compare failure criteria applied to intact rock or fault gouge layer to the cemented gouges presented in the study in order to evaluate which criterion can best fit the failure of cemented gouges. [START_REF] Byerlee | Coulomb plasticity within the fault zone[END_REF]) also analyzed a fault gouge layer under shearing and observed that it behaves as a Coulomb material, with an elastic deformation until failure. However, (Marone et al., 1992) proposed another adapted criterion that is more suitable to thin fault gouge layer. The two failure criteria considered to support these assumptions are:

𝜏 𝜎 𝑁 ⁄ = 𝐶 𝜎 𝑁 ⁄ 𝜑 (3.6) 𝜏 𝜎 𝑁 ⁄ = 𝐶 os φ 𝜎 𝑁 ⁄ s φ (3.7)
The first criterion (3.6) stresses 𝜎 𝑁 and 𝜏 are not requested to apply on a specific crack. Rather, their orientation is related to that of the principal stresses that develop in the sheared domain. This is known as " o o " e ed n [START_REF] Marone | Fault zone strength and failure criteria[END_REF], the first criterion could be applied to faults with very thin layers of gouge, while the second one might be relevant to the case of thick layers of granular gouge where such rotation of the principal stresses could occur. In that case, Coulomb plasticity could lead to a reduction of the effective friction of the fault. This assumption was validated experimentally in a cohesionless material by [START_REF] Marone | Fault zone strength and failure criteria[END_REF].

It is instructive to determine to which extent these two expressions can predict the fault strength, o e ro e " r e" Coulomb-Mohr properties of the rock. This comparison can be done using our numerical results on fault direct shear tests and biaxial tests. Figure 68 thus presents the effective friction in the fault as obtained numerically with direct shear simulations in section 3.4 of the present work, as a function of the effective friction predicted by both failure criteria (3.6) and (3.7) based on our independent determination of 𝐶 and 𝜑. With this graph, the strength ratio obtained with both models can be compared (direct shear and biaxial test) realized with the same conditions and characteristics (cf. section 3.3.1). Indeed, gouges behaving as granular material (i.e. with no cohesion) present lower friction than intact rocks, as predicted by literature (points under the dotted line). However, cemented gouges present the opposite behavior once the percentage of cementation account for a small cohesion (i.e. for more than 10 % < 𝑃 𝑐𝑒𝑚 for dense samples and 25 % < 𝑃 𝑐𝑒𝑚 for mid-dense samples). The increase of cementation increases the effective friction of the cemented gouge compared to intact rock, and the reduction of porosity overstate the difference. It appears that the proposed adapted law (3.7) for fault gouges (Marone et al., 1992), does not fit very well with cemented gouges either, blue line. The initial stress ratio from the Coulomb criterion, equation (3.6), is closer to values found for our mildly cemented gouges but is not suitable as cementation increases, grey line.

Figure 68. Effective friction as measured with direct shear experiments (simulation campaign) as a function of effective friction as predicted by failure criterion (calculated with internal friction angle and cohesion found with Biaxial tests), for dense and mid-dense samples. The blue line (empty marker points) represents the effective friction given for sheared gouge with equation 3.7 (Marone et al., 1992) compared to the grey line (filled marker points), effective friction for intact rock with equation 3.6 [START_REF] Handin | On the Coulomb-Mohr Failure Criterion[END_REF]. One of the reasons explaining these differences could be the different boundary conditions used for the two kinds of tests (direct shear/ biaxial experiment). In particular, the presence of the two walls of the fault might act as barriers to the natural development of Riedel shear bands and increase the fault strength. Another key of understanding could be the heterogeneity of the stress field in the fault (in relation to the kinematic constraints mentioned above), which would prevent a simplistic use of the criteria (3.6) and (3.7). As further analyses and simulations are needed to confirm these hypotheses, these propositions will be discussed in future work comparing different gouges with cement or matrix particles.

Concluding remarks

A 2D granular fault gouge model has been implemented in the framework of DEM in order to establish a link between gouge properties (cement and initial porosity) and rheological behavior. One originality of this Chapter is the use of the angular and faceted shape of particles instead of circular ones. Our results confirm that both initial porosity and cementation influence gouge behavior. They play a role in the internal structure and geometry of the gouge and thus modify the rheological behavior of the fault. In the range of our numerical experiments (normal stress 40 𝑀𝑃𝑎 & slip rate 1𝑚/𝑠), the increase of cementation within the gouge leads to an increase of effective strength whereas an increase of initial porosity tends to a reduced strength.

(i)

The strength peak is sharp, short, and intense for dense and highly cemented materials and smooth, delayed and of moderate amplitude for mid-dense and moderately cemented materials. Brittleness is enhanced with cementation, especially in the case of dense materials. It evolves similarly with the internal cohesion (Coulomb-Mohr) within the gouge and leads to different failure patterns. The highly-cemented material with low initial porosity presents clear Riedel band formation evolving as in the theory of [START_REF] Tchalenko | Similarities between shear zones of different magnitudes[END_REF]. The high cohesion and internal friction angle values make this material very close to intact rock properties. In opposition, the weakest material is obtained with the highest porosity state and for no cementation: no failure pattern is observed and a typical granular Couette flow is highlighted.

(ii) Shear failures need a critical dilation to form. For poorly-cemented material, not presenting enough cohesion (i.e. 𝐶 < 1𝑀𝑃𝑎), the same critical dilation is observed for both dense and mid-dense samples. The initial porosity does not have a major influence on critical dilation for cohesionless materials. However, when cohesion is present within the particles, the critical dilation appears to be smaller for denser samples and increases with cementation, whereas mid-dense samples present the opposite behavior.

(iii) Effective friction curves present double weakening shapes for dense samples with enough cementation, triggered by different mechanisms (Chapter 5). The peak length (analogous to the 𝐷 𝑐 from rate and state laws) mainly decreases with the percentage of cementation.

(iv)

The increase of cementation increases the effective friction of the cemented gouge compared to intact rock, especially with dense cemented gouges. Effective friction, from the Coulomb failure criterion or the Coulomb plasticity criterion (Marone et al., 1992), does not predict very well our values for cemented fault gouges. The different boundary conditions and stress fields obtained for the direct shear experiment and the biaxial test could explain the difference, but further analyses are needed to confirm these hypotheses.

In future work, we will extend this numerical campaign to cover a wider range of normal stresses and gouge thickness, in order to derive an empirical slip-weakening friction law based on the micromechanical properties of the gouge. This law could then be implemented in dynamic rupture modeling at a larger scale for a dialogue with seismological data. An interesting line of work could also be to investigate the time scale of the cementation evolution in a fault and to evaluate to what extent the associated evolution of the fault strength interacts with its seismic cycle.

Chapter 4. Rheology and weakening of a sheared fault gouge with matrix particles

Main Foreword

In Chapter 3, we simulated a cemented granular fault zone with angular and faceted grains linked by cohesive bonds to model the cementation. Based on the previous description of infill material (Chapter 1), we now consider another brecciation process for which the observed infill er o e e er ed " r " r e d re er o er particles produced by previous fragmentation or introduction of sediments within the fault gouge [START_REF] Woodcock | Classification of fault breccias and related fault rocks[END_REF]).

e r

o je e o er o der d e " e / eo e r " ro e of matrix particles as opposed to the physical cementation (bonded law as cohesion) used in the previous chapter. Is a very dense granular fault gouge being as resistant to shearing as cemented cataclasites? The DEM approach is also suitable for this kind of granular-type filling and this chapter is divided into two main parts with two types of granular samples:

❖ Part A concerns granular samples with only hexagonal matrix particles. This granular media represents a highly dense packing of particles and reveals Riedel bands formation. This first part gives an understanding of localization patterns and evolution in this very dense packing of particles with different gouge characteristics.

❖ Part B presents some results on composite mixtures between angular and faceted grains surrounded by hexagonal cells representing a matrix of fines. It highlights similarities and differences with the previous model with only matrix particles (Part A) and with cementation (Chapter 3).

As said in Chapter 2, in these models the sliding is imposed by boundary conditions (imposed slip velocity), but this is not a realistic case. In real seismic faults, the sliding is driven by the elasticity of the surrounding medium, and can therefore take place in a stable or unstable way.

The models used here are local micromechanical studies of the behavior of fault gouges, which are designed to be inserted in larger-scale dynamic models.

This chapter is partly based on the following papers: The granular matrix is created with hexagonal cells to represent a highly dense packing of particles. As the shape is hexagonal, a Voronoï tessellation algorithm is used, in which the size of cells and their perturbation (i.e. the inhomogeneity of the hexagonal cells) are specified (Chapter 2). To realize parametric studies, a standard case (M-S or " rd rd" defined as a reference model. For this model, an almost constant size of cells is considered for all particles ( 𝑒𝑞 ≈ 20 𝜇𝑚), with a perturbation of 0.5, meaning that cells are not perfectly regular hexagons, Figure 70. The initial gouge thickness is equal to 2 𝑚𝑚 and the length is equal to 20 𝑚𝑚. As in the previous model, a Representative Elementary Surface is used here, as a result of the parametric study on the model length which is provided in Appendix A4.A.

Compaction and initial state

Once the granular sample is generated, it is then inserted between two rock walls. The first simulation step consists in compacting the sample with 𝜎 𝑁 = 0 𝑀𝑃𝑎 to create a stabilized packing of granular material (Chapter 2). As the generated sample is already very compact, Figure 70, this compaction stage is only influenced by the numerical stiffness and interparticle friction inserted between particles in contact. At the end of the compaction the sample is almost identical, but with a little interpenetration between particles (~ 0.1 % of the size of the particle) which provides mechanical equilibrium. An interesting point is that the initial porosity within the gouge is negligible here, with 𝑃 𝑜 𝑒 < 1 %, resulting in an extremely dense compacted sample.

Numerical setup for direct shear simulations

The granular matrix is then submitted to direct shear experiments, Figure 71. The lower rock wall is fixed and normal stress of 0 𝑀𝑃𝑎 and slip velocity of 1 𝑚/𝑠 are imposed on the upper rock wall. Periodic boundary conditions are present on both right-and left-hand sides of the sample to maintain the continuity of the movement at large slips, but the upper rock wall can freely move in the y-direction to allow dilatancy. A dry contact model with rigid bodies is considered, without gravity. A Coulomb friction contact law is also settled, meaning that every contact depends on the interparticle friction 𝜇 𝑛 𝑚 , numerical stiffnesses 𝑘 𝑛 (constant and equal to 10 1 𝑃𝑎/𝑚 in the reference case) and a stabilizing numerical damping 𝛾 (numerically set to 0.2). To avoid wall-slip effects with smooth boundaries, a certain cohesion is introduced at the interface between matrix and rock walls (500 𝑀𝑃𝑎). It enables the creation of a certain roughness by cementing particles to the top and bottom layer. It can be noted that interparticle friction 𝜇 𝑛 𝑚 is equal to 0.3 in the (M-S) reference case, which is lower than the one used in the model of Chapter 3. The objective is to test the property of matrix particles as a weaker material with limited friction. It should be noted that the boundary conditions, in particular the non-deformable walls, can induce differences in the results found with those of the literature studying stick-slip experiments (in particular, in the formation of the Riedel bands). All the parameters are gathered in Table 4 and are reproducible. 𝜎 𝑁 𝑉 Matrix between four rigid walls. The lower wall is fixed in displacement and the upper wall is submitted to a constant downwards velocity 𝑉 , and the lateral walls are submitted to a confining pressure 𝜎3, fixed in vertical and rotational motions, and free to move horizontally.

Three tests are performed on the standard case (M-S), with confining stresses of 10 𝑀𝑃𝑎, 0 𝑀𝑃𝑎, and 80 𝑀𝑃𝑎 and for two interparticle stiffnesses (𝑘 𝑛 = 10 15 , 10 16 𝑃𝑎/𝑚), Figure 72 In Figure 72, the slope of shear bands changes between the peak (e) and the plateau (h), this is because the effective friction also changes, corresponding to a reorganization of the failure pattern and a rotation of the stress field. Based on the observed results:

-Almost no cohesion is observed for matrix material (in the sense of Coulomb-Mohr), Figure 73. It is in adequation with the description of matrix material which is an incohesive material (Chapter 1). -A change of interparticle stiffness does not have much influence on the internal friction angle, but increases the effective friction peak observed, Table 5 and Figure 73. By multiplying by ten the numerical stiffness, the resulting Elasticity modulus is ten times 123 Chapter 4. Rheology and weakening of a sheared fault gouge with matrix particles higher. This will probably have an impact on gouge kinematics. This point will be discussed in section 4.3.

-The orientation angle of the major shear bands formed is growing from the effective friction peak to the steady-state zone, Figure 72 (c). [START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF] listed in the table next to the graph.

Matrix characteristics & gouge kinematics

In order to characterize the matrix material, several parametric studies are performed and presented in this section. The purpose of these studies is not only to observe the influence of each parameter on the mechanical behavior but also to observe the rheological behavior through Riedel band formation and evolution. The wide range of variations in Riedel bands deserves to be properly presented here as they can be a marker of the stability of the system [ [START_REF] Byerlee | Friction of rocks[END_REF], [START_REF] Gu | Development of shear localization in simulated quartz gouge: Effect of cumulative slip and gouge particle size[END_REF]]. If an unstable frictional sliding is observed, it will certainly show Riedel bands localization, although the reverse proposition is not necessarily true. Indeed, Riedel bands can be observed in stable sliding without any instabilities. The angle between the R-band and the fault core boundary could be a critical point to distinguish whether instabilities [START_REF] Handin | On the Coulomb-Mohr Failure Criterion[END_REF].

Rock type

Granite ( 1) Gabbro ( 2) Trachyte ( 3) Serpentinized olivine ( 4) Sandstone ( 5) Berea sandstone [START_REF]CHAPTER[END_REF] Weber sandstone (7) Tennessee sandstone (8) Lueders limestone ( 9) Solnhofen limestone ( 10 can be present or not [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF], [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF]], and this angle depends on physical and numerical properties of the model.

Qualitative and quantitative results for the reference case

This first subsection aims to present typical results obtained for the standard granular sample (M-S) as it will be used for comparison in the rest of the chapter. Typical effective friction 𝜇 and dilation 𝜀 𝑦 are presented in Figure 74, and Figure 75 shows deformation profiles observed at different times of the simulation, (with letters from A to E from Figure 74). We distinguish three main steps within direct shear experiments, according to the main stages of Riedel shears formation (Chapter 1):

-A pre-peak zone is first observed, which is composed of an elastic phase (0-A) involving the shear modulus of the gouge and the overall stiffness. Once the elastic zone is over, the sample starts to deform in a non-reversible way until the friction peak (A-B), for which the maximum shear strength of the gouge is reached (B). During all this part, the sample dilates (red curve), but the critical dilation 𝜀 𝑦-is still lower than the averaged steady-state dilation 𝜀 𝑦-𝑆𝑆 . As presented in Chapter 1, direct shear makes appear typical patterns of shear bands, called Riedel shear bands for a very dense granular sample [ [START_REF] Tchalenko | Similarities between shear zones of different magnitudes[END_REF], [START_REF] Marone | Laboratory-derived friction laws and their application to seismic faulting[END_REF], [START_REF] Bedford | The Role of Grain Size and Effective Normal Stress on Localization and the Frictional Stability of Simulated Quartz Gouge[END_REF]]. These patterns follow closely what is commonly observed in natural faults (Y. [START_REF] Katz | Geometry and kinematic evolution of Riedel shear structures, Capitol Reef National Park, Utah[END_REF]. Just before friction peak (B), five main Riedel bands have formed. They can be separated into two kinds of low R-bands: primary R-bands 𝑅 3 , 𝑅 𝜀 𝑦-𝑆𝑆 𝐴 𝜀 𝑦- Chapter 4. Rheology and weakening of a sheared fault gouge with matrix particles disappear in favor of the two main secondary R-bands 𝑅 1 and 𝑅 2 , which localize all the gouge deformation (Figure 75). A boundary shear 1 also starts growing at the top of the gouge layer. -Once the steady-state zone is reached (D to E), and without modification of numerical parameters, the gouge reaches a stationary state of evolution. The friction evolves very weakly around a mean value which is similar in most of the simulations and ranges between 0.4 and 0.5. From this moment, the dilation of the granular sample is maximum and stabilized. Even though the main R-bands do not evolve anymore, the 1 -band is fully formed at the top rock wall (D), and a secondary 2 -band is partly observed at the bottom rock wall. Conjugate Riedel bands 𝑅′ are also observed in the simulations in the slip weakening and steady-state phases. These are oriented at about 𝛼 ≈ 60° from the top wall, a higher angle than for simple tensile -bands (oriented in the direction of the major stress at 5°), but lower than that described for 𝑅′-bands observed in the literature, which approaches 75°[ [START_REF] Davis | Conjugate Riedel deformation band shear zones[END_REF], (Y. [START_REF] Katz | Geometry and kinematic evolution of Riedel shear structures, Capitol Reef National Park, Utah[END_REF]]. It is assumed that this is a rotation of the -bands tending towards 𝑅′-bands, as in the scheme presented by (Y. [START_REF] Katz | Geometry and kinematic evolution of Riedel shear structures, Capitol Reef National Park, Utah[END_REF]. Moreover, the theory of Riedel bands is based on the Coulomb-Mohr model, which is an idealized model that does not consider the latest shear variation. It is thus considered that the bands observed between 50 and 60° are indeed 𝑅 -bands according to their link with the observed Riedel structures. Although the solid fraction is an accurate and relevant quantity to follow in order to study the formation of Riedel bands, similar results can be observed with velocity profiles as presented in Figure 76. They show that the highest velocity discontinuities are concentrated within Riedel and boundary shear bands, especially in the middle of the weakening part (C). Once the gouge is stabilized, the -velocity discontinuity is localized in the -band in the upper part of the gouge and equal to the applied shear velocity of 1 𝑚/𝑠. Changing the length of the fault gouge sample does not inhibit the formation of shear bands but it modifies the rate of friction weakening with the slip distance (Appendix 4.A).

Influence of interparticle friction and particle shape

The standard model is used here with a modification of interparticle friction (𝜇 𝑛 𝑚 ) or of the initial shape of particles (with a variation in perturbation 𝑃 𝑖 ). Analyses of interparticle friction and particle shape are gathered in this section because they appear to have similar consequences on Riedel band formation. The different modified samples are gathered in Table 6 and the rest of the parameters are identical to those presented in section 4.2.2.

𝑅′

A modification in interparticle friction changes the main orientation of Riedel shear bands ( ≈ 16 ° from the horizontal shear direction with 𝜇 𝑛 𝑚 = 0.6 and ≈ 6 ° for 𝜇 𝑛 𝑚 = 0.1). These values are inferior to the orientation from Coulomb-Mohr theory, requiring an angle equal to 𝜑/2 for direct shear, which corresponds to 21.5° for the (M-S) case. Increasing interparticle friction also seems to increase the number of Riedel bands formed at the beginning of the friction peak ( 2 in Figure 77 (c)) and within the decreasing friction part (𝐶 2 in Figure 77 (d)). This augmentation results in reducing the averaged distance d between Riedel bands. Conjugate Riedels (> 5 ° from the horizontal shear direction) are also more pronounced with higher friction, (at 𝐶 2 in Figure 77 (c)).

The two types of Riedel bands described as low-angle (𝑅) and high angle (𝑅') do not have the same growth pattern. They both depend on porosity-weakening and strain-rate-weakening mechanisms (R. [START_REF] Katz | The dynamics of melt and shear localization in partially molten aggregates[END_REF]. The initial state of porosity is not modified here, but interparticle friction plays a role in the rheology of the granular media, as it changes the contact between every hexagonal cell. It may explain why high-angle Riedel bands are more pronounced with higher friction. The increasing number of Riedel bands with interparticle friction can be explained with contact mechanics [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 2. Effects of particle size distribution and interparticle friction on mechanical behavior[END_REF]. For the peak phase, when 𝜇 𝑛 𝑚 is low, slip may occur between hexagonal cells, and particles are mobilized within the shear zone, with minimal dilation. With a higher 𝜇 𝑛 𝑚 , sliding is partly inhibited and particles tend to separate leading to higher dilations, and thus in our case, higher number of Riedel bands (i.e. dilation in specific direction). However, at the end of weakening, Riedel bands reduce toward a stabilized form and thickness. The observed thickness of the 𝑅-bands within the weakening phase also seems to increase with increasing friction, Figure 77 (f). As shown in Figure 77 (a), an interparticle friction of 0.6 also leads to a more sudden post-peak weakening, which is prone to switch the fault behavior from a ductile aseismic response to a brittle seismic slip, depending on the stiffness of the surrounding medium.

ii.

Shape of particles and irregularities

The evolution of particle shape is settled at the creation of the matrix sample by varying the " er r o " r e er er ree d ere er r o d re ed with from almost perfect hexagonal cells (𝑃 0.2 ) to more angular and random hexagonal cells (𝑃 0.8 ). Figure 78 presents the evolution of effective friction and dilation as a function of the slip distance for these three different initial shapes of particles. (Mair et al., 2002) used 3D experimental data, and[START_REF] Nouguier-Lehon | Influence of particle shape and angularity on the e o r o r r er er International[END_REF] used 2D numerical modeling, in order to analyze the influence of grain shape and angularity on the behavior of granular materials. Both studies result in the same conclusion: the increase of angularity increases the resistance to friction, and thus the effective friction peak. Our grains shapes are not perfectly hexagonal here, and particles present a small variation in angularity according to the perturbation applied. This difference remains quite low as the global shape is mainly hexagonal, but can explain the small difference in friction peak observed in Figure 78 (a). Due to their homogeneity and less angular shape, the 𝑃 0.2 material also presents less dilation than the two other granular media, Figure 78 (b). Visual analysis shows that the 𝑃 0.2 material is pathological and too regular to represent the real granular samples observed in faults. However, its representation gives interesting patterns and rheological behavior concerning Riedel bands in Figure 79. At friction peak, the regularity of the shape of the cells seems to be quite important in the development of shear bands. In the same way as interparticle friction, an increase in angularity also implies a change in the number of Riedels. A higher angularity (similar to more disorder, 𝑃 0.8 ) results in a larger number of low-angle Riedels at friction peak, less marked, and of lower thickness. We also observe some possible high-angle Riedel bands 𝑅' (against no high-angle Riedel for 𝑃 0.2 ). When the steady-state is reached, all three samples have the same number of low-angle Riedels 𝑅 (secondary Riedels). These bands have a similar orientation 𝛼 𝑖 and thickness, meaning that the change in the angularity of particles is too small to affect Riedel's angle orientation at steady-state. To better understand how these mechanisms operate, two chains of grains were followed during the deformation of the gouge, Figure 80. Comparing both chain's evolution, we first note that the place where the chain breaks is composed of several grains belonging to the Riedel band. This result is different from the patterns of granular chain theory proposed by [START_REF] Anthony | Influence of particle characteristics on granular friction[END_REF], where the breaking point of the chain just gives two different chains sliding parallel to each other. However, in their case, no 𝑅-band was present within the granular sample. We conclude that the presence of low and high 𝑅-bands modifies the kinematic of particles chains. and (c) n°2). It explains the higher thickness observed in Figure 79 (a). The initialization and development of shear bands are often accompanied by an increase of the void ratio and particle rotations inside the shear band (Chapter 1). This phenomenon is specific to the shape of particles which needs to create spaces before particles movements. However, at the steady-state, both chains tend to have the same thickness, with 3-4 grains belonging to the Riedel, consistent with previous Figure 79 (b).

The results from this section are summarized in the following Figure 81 with the effective friction curve and Riedel structure formed at steady-state of sliding.

Influence of the number of particles within the gouge thickness and Global Stiffness

In this section, two ways of modifying the global stiffness of the gouge layer are compared: an increase in the interparticle stiffness or in the size of particles. The different mechanisms influence the formation and evolution of Riedel bands in the granular gouge. The corresponding samples variation are gathered in Table 7.

i. Interparticle stiffness modification

How interparticle stiffness acts on Riedel band formation? To be in the same conditions as for the first model (Chapter 3), and because the magnitude of particle diameter was the same, the standard case (M-S) was implemented with an interparticle stiffness of 10 1 P / (section 4.2.2) and then by varying 𝑘 𝑛 from two to ten times the reference value (respectively samples M-K-2 and M-K-10).

The increase of stiffness both increases the effective friction peak, the shear modulus (later discussed), and the total energy released by the weakening phase, Figure 82 (a) & (b). The pattern of the friction curve was predictable for the elastic part, as numerical stiffness is only supposed to play on the overlapping of particles which modifies the elastic part of the effective friction curve: an augmentation in interparticle stiffness increases the global stiffness and thus the shear modulus. But we would have thought that the friction weakening part would follow a similar slope for each granular sample. Such a discussion can only take place when the sliding scenario is well established, which will only be the case in the next Chapter 5: it is necessary to save explanations for later.

A difference in interparticle stiffness (i.e. modification in shear modulus) also seems to affect the angle orientation of Riedel bands, Figure 82 (c). In the case with higher stiffness, the orientation angle of the main Riedel band 𝛼 𝑖 is reduced from 12 ° to 7 ° and the number of Riedel bands at friction peak is reduced to one big Riedel band. The 𝑅'-bands rotate at a higher speed than the 𝑅-bands and usually disappear as they grow to give way for 𝑅-bands. 𝑅'-bands are rarely observed because of their short duration and limited thickness [(Morrow and Byerlee, 1989), (Logan et al., 1992), [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF]]. That is why in other studies [ [START_REF] Schmocker | Granular flow and Riedel band formation in water-rich quartz aggregates experimentally deformed in torsion[END_REF], (R. [START_REF] Katz | The dynamics of melt and shear localization in partially molten aggregates[END_REF]], the low-angle 𝑅bands are the only bands observed. In this study, high-angle Riedel bands 𝑅' are very pronounced in the case with higher stiffness, Figure 83 (a) & (c1). These 𝑅′-bands are present during the whole simulation and seem to be responsible for the difficulty of the material (M-K-10) to deform by shearing, inhibiting multiple 𝑅-bands formation.

The distribution of force chains in the gouge allows us to better understand the formation of 𝑅′bands. While the 𝑅-bands seem to correspond to an area free of force chains, the 𝑅′-bands show the opposite behavior with 𝑅′-bands located where force chains are connected within the 𝑅band, Figure 83 (c2). It explains why they are more pronounced for stiffer or stronger materials and with a higher magnitude of force chains. c1) Solid fraction between 0 and 1, (c2) Snapshot of the force-chains network (from 0 to 10 4 𝑁), (c3) Velocity profiles in the -direction (between 1 and -1 𝑚/𝑠). Force chains are defined according to the typical level of stress present in the system (here the applied normal stress) and thickness and color are evaluated between the minimum and maximum contact force (0 to 10 4 𝑁). A cut-off is made at 10 4 𝑁 to have the same visualization for all cases (higher force chains are observed for higher stiffness).

ii. Size of particles and equivalent stiffness

The constant initial gouge thickness (𝑡ℎ 𝑖 ) chosen in models implies that an increase in the size of particles also increases the global stiffness of the granular sample, Figure 84 (a), (c), (d) (full lines). Indeed, the places of elastic accommodation are much less numerous while having the same individual elastic response. The size augmentation reduces the total number of particles in the gouge thickness from 100 to 40 (respectively for (M-S) (𝜙 𝑒𝑞 = 20 𝜇𝑚) and (M-PS-50) (𝜙 𝑒𝑞 = 50 𝜇𝑚)), and leads to a similar behavior as the one observed for a modification of interparticle stiffness (Figure 82): an increase of effective friction peak and a longer slip weakening part. However, the steady-state effective friction is increasing from 0.4 to 0.5, Figure 84 (c), contrary to all the previous simulations observed.

Without the stiffness effect

In order to only study the influence of cell size variation, the interparticular stiffness is modified to keep the same initial shear modulus for all models, Figure 84 d o re o e e " e e e ". The secant shear modulus is selected similar to the one of the standard material (M-S) and calculated at 0 % of the maximum shear stress (36.5 𝑀𝑃𝑎). Then, a new stiffness 𝑘 𝑒𝑞 is calculated for each model with a diameter of cells 𝜙 𝑖 , as

𝑘 𝑒𝑞 𝑖 = 𝑘 𝑛 𝐺 𝑀-𝑆 𝐺 𝜙𝑖 ( .1)
where 𝑘 𝑛 is the interparticle stiffness of the standard case ( 10 The direct consequence of an increase in the size of particles (without stiffness effect) is a longer slip weakening duration, meaning that more energy is needed to deform the gouge and to reach a steady state. But it also means that this long and smoother slip weakening may reduce the probability of a big instability to occur (discussed in Chapter 5). The pre-peak section, on the other hand, is identical for all grain sizes, which validates the stiffness calibration of Eq. ( 4.1). The equivalent stiffness also allows keeping 𝜇 and 𝜇 𝑆𝑆 almost constant, dotted lines Figure 84 (c). However, the value of 𝜇 𝑆𝑆 obtained for (M-PS-50) is biased because it has not yet reached its stabilized steady-state of sliding in the picture.

From a purely kinematic point of view, by doubling the particle size, the steady-state dilatancy is not doubled for the same interparticle stiffness (full red line, Figure 84 (d)). But if one passes in equivalent stiffness, (i.e. only considering the change of particles size), a multiplication by two of particles size doubles the steady-state dilatancy value (from 0.0 to 0.08 %, Figure 84 (d), red dashed line).

Riedel angle

The angle of the main Riedel band at friction peak seems to depend on the shear modulus value, as we found the same angle for the standard case (a) in Figure 85 and the case (c) (which have the same shear modulus), whereas the case (b) presents a different Riedel angle (and a different shear modulus). At this step, it can be hypothesized that the size of particles does not change the rheology of the gouge but changes the temporality of Riedel bands formation and particularly the duration of slip weakening.

Conjugate Riedels (𝑹')

The effect on 𝑅'-bands is similar to the one observed for a change in interparticle stiffness. By increasing the size of particles, 𝑅'-bands become more persistent and may have consequences on the longer critical slip distance observed. The antithetic particle movement, due to the persistence of 𝑅'-band, appears to modify the entire kinematics with the creation of a Riedel structure [(Y. [START_REF] Katz | Geometry and kinematic evolution of Riedel shear structures, Capitol Reef National Park, Utah[END_REF], (Y. [START_REF] Katz | Strain localization in sandstone during embryonic stages of shear -zone evolution[END_REF]] during the slip weakening part observed in Figure 86. They are mainly caused by granular flow sub-parallel to the 𝑅-bands, which both induce grain reorganization and rotation of 𝑅' (Y. [START_REF] Katz | Geometry and kinematic evolution of Riedel shear structures, Capitol Reef National Park, Utah[END_REF]. Although iii.

Gouge thickness

To a certain extent, decreasing the thickness of the granular sample has similar frictional behavior as an increase in the size of cells. Indeed, Figure 87 (a) shows that the same number of particles within the gouge thickness (𝑛𝑏 1 = 𝑡ℎ 𝑖 / 𝑒𝑞 ) leads to the same effective friction at peak and steady-state for two kinds of study: the one with reduction of initial thickness but with the same size of particles (grey line) or the one with an increase in the size of particles (section ii), with the same initial thickness and without stiffness effect in yellow (equivalent stiffness previously described). However, the dilation behavior is not similar, Figure 87 The ratio of gouge thickness to grain size Two samples with the same ratio of gouge thickness to grain size are compared: (M-PS-40-Keq) and (M-Th-1), which both have 50 particles within the gouge thickness. The box in Figure 87 shows their mechanical characteristics, but to have a more precise comparison Figure 88 is added with the evolution of effective friction and dilation as a function of the fault slip distance. Although the resulting peak of effective friction is similar in both cases, Figure 88 (a), the friction evolution as a function of slip is very different. Indeed, sample (M-PS-40-Keq) needs almost twice more slip along the fault gouge to reach the resistance peak (18 µ𝑚 vs 83 µ𝑚) and then to reach the steady-state plateau zone (700 µ𝑚 vs 00 µ𝑚). These results are consistent with the results presented for equivalent stiffness. Similarly, the averaged dilation 𝜀 𝑆𝑆 in the steady-state zone of the sample (M-PS-40-Keq) is doubled (0.078 vs 0.035), Figure 88 (b).

However, results can also be presented with dimensionless parameters (right side pictures), considering: the fault slip distance as a dimensionless number 𝑛 𝑆𝑑 , number of particles (or ratio between the fault slip distance 𝑑 and the equivalent diameter of particles 𝜙 𝑒𝑞 ), and dilation as 𝑛 𝑌 which is the ratio between the -displacement of the rock wall divided by the equivalent diameter of particles 𝜙 𝑒𝑞 (number of particles). With this representation, it is observed that both friction and dilation curves are similar. These two simulations also have similar tangent and secant shear moduli (respectively 𝐺 𝑎𝑛 = 12.2 𝐺𝑃𝑎 et 𝐺 𝑒𝑐 = 6.2 -5.77 𝐺𝑃𝑎). Figure 89 compares Riedel bands for these two cases and displays the same type of deformation for both simulations at steady-state. The low-Riedel (𝑅) of both models are oriented at an angle equal to Following these first results, 𝑛𝑏 1 , the number of particles within the gouge thickness seems to control the magnitude of frictional data observed (both at effective friction peak and steady-state), and it may influence the orientation angle of 𝑅-bands. It also controls the dilation and slip distance observed through dimensionless numbers 𝑛 𝑌 and 𝑛 𝑆𝑑 .

Moreover, the shear modulus played on Riedel kinematics inside the sample, numbers of 𝑅bands, Riedel structure, temporality, partially presented in Figure 90.

Discussions

The orientation angle of main Riedel bands

In previous sections, it was observed that the orientation angle 𝛼 𝑖 of low 𝑅-bands, or angle between the direction of shearing and the 𝑅-band, was changing with the modification of some a) An increment in the interparticle friction between particles drastically increases the orientation angle of the main Riedel bands (from the beginning of the simulation and until steady-state), because it is linked to the internal friction 𝜑. This is a typical result of Coulomb-Mohr's theory [ [START_REF] Tchalenko | Similarities between shear zones of different magnitudes[END_REF], (Y. [START_REF] Katz | Geometry and kinematic evolution of Riedel shear structures, Capitol Reef National Park, Utah[END_REF]]. [START_REF] Gu | Development of shear localization in simulated quartz gouge: Effect of cumulative slip and gouge particle size[END_REF] plotted the evolution of Riedel angles 𝛼 𝑖 as a function of effective friction and also found an increase in 𝛼 𝑖 with the increase of internal friction angle (which is equal to 𝜇 with a cohesionless material as in this study).

b) An increment in 𝑛𝑏 1 also increases the orientation of the main Riedel bands, both by reducing the size of particles or increasing gouge thickness. This result was only partially discussed in the literature, in the sense that particle size evolution was found to be responsible for the 𝑅-band angle as well as the gouge thickness, but the link between these two parameters is less investigated. For an increase in the initial gouge thickness, a change in orientation is observed close to the boundaries, which may come from edge effects due to the larger sample. This 𝛼 𝑖 angle also evolves during the simulation before stabilizing at a steady-state. In the standard material (M-S), 𝛼 𝑖 variation with the slip distance is not very significant (~1 -2 °) and evolves similarly for each simulated material. It was not studied in the present study and could be been done properly at a different moment of the simulation (elastic phase, effective r o e k we ke r … w e e re e r work W re ore interesting are the shapes of these Riedel bands, which are most of the time not really linear and present some lower angle at the boundaries, as found by [START_REF] Hirata | Shear zone development and frictional instabilities[END_REF]. In some cases, the 𝑅′-bands present within the gouge modify the structure and slip temporality. These bands are unfortunately complex to follow during the simulation but could be the subject of further study. [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF]) also worked on the kinematics of sliding and compared sliding behavior and internal geometry of laboratory fault zones. They observed that stick-slip samples with the largest stress drops also had the higher Riedel angles. On the contrary, their samples with low Riedel angles corresponded to stable sliding. In our case, we are not able to compare our results to stick-slip as defined in [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF]), but the Riedel angles observed can be correlated with friction strength and to the critical distance to reach a steady-state of sliding. Considering the orientation angle of the different granular materials observed, the smaller orientation angles could be linked to gouge stability.

The combined effect of Shear modulus, number of particles in the gouge thickness, and interparticle friction on rheology and weakening mechanisms

From previous sections, it has been shown that a reduction of Riedel angle can be both due to a decrease in interparticle friction or b 1 , or, an increase in shear modulus. In this section, the objective is to look into the combined effect of a modification of the shear modulus (𝐺) with a different number of particles in the gouge thickness (𝑛𝑏 1 ) and/or a variation in interparticle friction (𝜇 𝑛 𝑚 ) between particles in contacts. Data were collected from sections 4.3.2 and 4.3.3 and gathered in Figure 92 and Figure 93 trying to connect possible interactions between parameters and comparing Riedel-bands evolution with their distinct elastic characteristics.

Zone (i): gouges with low G & low 𝝁 𝒖𝒎 or low

Materials within the identified zone (i) present a reduction of interparticle friction 𝜇 𝑛 𝑚 (or angularity via 𝑃 𝑖 ) or a reduction of 𝑛𝑏 1 , but they have a low shear modulus similar to the one of the standard material (M-S) (𝐺 𝑒𝑐 ~5.7 𝐺𝑃𝑎 and 𝐺 𝑎𝑛 ~12.2 𝐺𝑃𝑎). A decrease in the number of particles within the gouge thickness (𝑛𝑏 1 ) with stiffness modification (i.e. equivalent stiffness, (M-PS-Keq)) also belongs to this zone, as it is not linked to the increase in shear modulus, Figure 93 (e). At steady-state, these characteristics lower both the orientation angle 𝛼 𝑖 of secondary Riedels (cf. section 4.4) and the effective friction peak 𝜇 .

These mechanical changes enhance the life duration of 𝑅'-bands in their maximum size: explaining why more Riedel structures are observed, Figure 92 (a) and (c), and increasing the critical weakening length. Materials from this zone are prone to enhance stability within the gouge sample but can easily go into a slip. They can be related to weak gouges behavior [ [START_REF] Collettini | Beyond Byerlee friction, weak faults and implications for slip behavior[END_REF], [START_REF] Bedford | Fault rock heterogeneity can produce fault weakness and reduce fault stability[END_REF]] which are supposed to have a very small effective friction. On the contrary, an increase in interparticle friction increases a lot the maximum shear stress, and thus 𝜇 without changing the shear modulus. These materials lead to an increase of the number of Low Riedel bands within the gouge, and a higher angle of orientation at steady-state, Figure 92 (b). If interparticle friction and shear modulus both increased, we could expect to observe more numerous 𝑅-bands at friction peak and lower Riedel angle and Riedel structure creation during the slip weakening part. A reduction of the critical slip distance is observed, leading to a more sudden post-peak weakening, which is prone to switch the fault behavior from a ductile aseismic response to a brittle seismic slip, depending on the stiffness of the surrounding medium. Materials with high interparticle friction coefficient (and low stiffness) are also known to present greater slippage and a decrease in the recurrence time between two slip events (Dorostkar & Carmeliet, 2019).

Zone (iii): gouges with high G

As seen in the previous sections, the shear modulus increases due to an increase of interparticle stiffness or a decrease in the number of particles within the thickness. Materials with high shear modulus present a small orientation 𝛼 𝑖 of 𝑅-bands. As previously found, they also present some Riedel structures caused by a longer life duration of 𝑅'-bands, Figure 93 (f), and (g). It can be noticed that a very small interparticle friction (𝜇 𝑛 𝑚 = 0.1) leads to almost the same R-angle. Similar values of both shear modulus and b 1 within the gouge thickness (for two different samples) give similar kinematics of evolution of the Riedel structures (Figure 89) but they will not have the same slip temporality depending on the other characteristics of the model. The stiffness of the fault was found to depend on the ratio of the shear modulus to the size of the rupture nucleation zone. A fault with a very low shear modulus (and/or large regions) is more likely to tend towards slow slip nucleation (J. R. [START_REF] Leeman | Laboratory observations of slow earthquakes and the spectrum of tectonic fault slip modes[END_REF].

Zone (iv): gouges with low G & high

This area is specific to an increase in the 𝑛𝑏 1 number with no change in either the interparticle friction or the shear modulus. These materials present a large gouge thickness (20 % of the total fault patch) without changing the size of particles, Figure 93 (d). They are not very representative of reality, since an increase of the gouge necessarily induces the wear of the particles in the center of the sliding zone and thus tends towards a reduction of the size of certain particles. For these materials 𝜇 slightly decreases and 𝜇 have values similar to those of the (M-S) case, however, the orientation angle of Riedel bands increases. According to [START_REF] Scholz | Detailed studies of frictional sliding of granite and implications for the earthquake mechanism[END_REF], an accumulation of gouge layer tends to stabilize the system, and according to [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF])'s theory, low Riedel angles also tend to stabilize the system. In our case, we have an increase in the orientation angle of the 𝑅-bands, which is a bit in disagreement with the theory, maybe due to the non-realistic material observed. These results could also be related to the stick-slip phenomenon by [START_REF] Lyu | On the mechanics of granular shear: The effect of normal stress and layer thickness on stick-slip properties[END_REF] testing a variability of gouge thickness with different normal stress. They observe that an increase in layer thickness, (i.e. an augmentation of the number b 1 ), implied a reduction of stress drop and friction and has a significant effect on stick-slip.

Zone (v): extremely low

Some very thin gouges were also modeled (𝑡ℎ 𝑖 = 200 𝜇𝑚) to avoid any Riedel formation Figure 93. By reducing the number of particles within the gouge thickness to a value lower than 10-15, and for any change in other gouge characteristics, Riedel band formation is prevented.

The maximum shear stress also increases whereas the dilation is slightly reduced, as previously found by [START_REF] Biegel | The frictional properties of a simulated gouge having a fractal particle distribution[END_REF] for a reduction in fault gouge thickness.

The results can also be related to the observed dilation at steady-state as a function of the ratio

Zone (vi): gouges with similar size of particles

For granular samples with the same size of particles (𝜙 𝑒𝑞 = 20𝜇𝑚), values confirm that an increase in 𝑛𝑏 1 also increases the total averaged dilation at steady-state (from left to right in Figure 94). Similar results were found in the study of [START_REF] Lyu | On the mechanics of granular shear: The effect of normal stress and layer thickness on stick-slip properties[END_REF] for different normal stress. But an increase in gouge thickness (green line) seems to show a stabilization of dilation for a large ratio 𝑛𝑏 1 > 100.

Zone (vii): gouges with a decrease in size of particles

A decrease in particle size remains the most important factor in reducing granular samples dilation. A higher size of particles may also rise the amount of slip needed to weaken the fault. However, if the dilation is expressed as a dimensionless number, similar dilation is found for each material (Section 4.3.3). Markers in this zone also correlate with a reduction in Riedel angle orientation.

Figure 94. Averaged dilation at steady state, as a function of the number of particles within the gouge thickness for all the simulations. Zone (v), is a zone where the thickness of the gouge is too small (𝑡ℎ 𝑖 = 200𝜇𝑚) to present any Riedel band formation (even with the same parameter variation than previously described), zone (vi) is a zone where all the simulation have the same size of particles ( 𝑒𝑞 = 20𝜇𝑚) (vii) is a zone where a change in the size of particles is observed. The secant shear modulus at 40% of the maximum shear stress is calculated for all cases and plotted as colors corresponding to the color bar on the right-hand side. The standard case M-S is situated on the graph.

These simulations represent different behaviors that can occur within the gouge during a fault slip for different types of materials. For example, although it is not possible in our numerical code to represent grain fragmentation, the variation in particle size and the ratio of the number of particles in the thickness of the gouge, gives us a glimpse of the changes that fragmentation could have on the sliding behavior. (Kenigsberg et al., 2020)

Conclusion of Part A

Another 2D granular fault gouge model has been implemented in the framework of DEM in order to study the influence of matrix particles characteristics on the fault rheological behavior and Riedel bands evolution. This chapter first gave an overview of a wide range of Riedel bands kinematics for different gouge parameters. For a highly dense granular matrix, small variations in physical or numerical characteristics of the model change the kinematics of the gouge:

-Gouge materials with high interparticle friction present an increase in the effective friction peak (typical results from Coulomb-Mohr theory). This high friction also influences shear bands formation with two main consequences: low-angle Riedel bands R are mostly observed (few or no conjugate Riedels) and their number increases within the gouge.

Conversely, for materials with low interparticle friction, Riedel structures are formed accompanied by: a reduction of 𝛼 𝑖 and the formation of a single low-angle Riedel linked to the Y slip localization by conjugated Riedel bands. A modification in shape angularity and disorder have the same consequences, but with less amplitude of variation. As the orientation angle of Riedels can be seen as an indicator of stability for granular fault gouge [START_REF] Moore | Relationships between sliding behavior and internal geometry of laboratory fault zones and some creeping and locked strike-slip faults of California[END_REF], the reduction in interparticle friction is prone to switch fault gouge behavior from sudden and seismic slip to moderate and slow slip.

-Some parameters appear to modify the global stiffness of gouge materials by increasing the interparticle stiffness or the size of particles, leading to a rise of the shear modulus. A higher shear modulus is mainly responsible for a higher number of conjugate Riedels and Riedel structures observed: they inhibit the formation of low R-bands and slow down the arrival of the steady-state of the system. Even if the orientation angle of Riedel bands is mainly affected by a change in the ratio between the size of gouge thickness and size of particles, it is also influenced by the shear modulus.

-The direct consequence of an increase in the size of particles, without the stiffness effect, is a longer peak duration, meaning that more energy is needed to deform the gouge and to reach a steady state of sliding. The size of particles does not change the shape of the Riedel structure observed but changes the temporality of their formation and evolution, particularly the duration of the slip weakening part. But it also means that a longer and smoother slip weakening decreases the probability of instabilities to occur.

-For all the simulation campaigns, three main quantities are relevant in the formation and evolution of Riedel bands and main stability of the granular gouge: (a) the bulk shear modulus, (b) the ratio of gouge thickness to grain size (i.e. number of particles within the gouge thickness), and (c) the interparticle friction. They all participate in a modification in the orientation angle of Riedel bands and the entire Riedel structure, having a strong impact on gouge stability. The combined effects of these three parameters lead to different gouge materials more or less likely to generate stable or unstable slips.

Studies from next part B are dedicated to a mixture between hexagonal cells with angular grains representing cataclastic clasts surrounded by matrix particles (i.e. hexagonal cells). Different results are expected in terms of Riedel patterns as the internal structure will be less homogeneous.

Part B -Granular fault zone with a composite mixture of angular grains and matrix particles

Foreword

The objective of this section is to properly study the influence of matrix particles within a granular sample by combining the matrix previously studied with angular grains used in Chapter 3, Figure 95. The first section 4.7 presents the granular material with a composite mixture between matrix particles and angular grains. In the second section 4.8, a qualitative comparison is made between the kinematics and rheology of this composite mixture under shearing and previous gouge samples. From this preliminary study, only two of those samples are selected for direct shear simulations: (MG-33) 33 %, and (MG-56) 56 % of matrix particles. The behavior of the two selected samples (MG-33) and (MG-56) is then compared with a case with only granular particles (G-C-4), and with 𝑃 𝑐𝑒𝑚 = % (its behavior was similar to a case with no cementation in Chapter 3) and with a sample with only matrix particles (M-S), standard case simulation from Chapter 4 (Part A), considered with 𝑃 𝑚𝑎 𝑖𝑥 = 100 % and no angular particles.

In the two samples with only one kind of particles, the amount of initial porosity is measured: 𝑃 𝑜 𝑒 = 11 % for (G-C-4) and less than 1 % for (M-S). Samples with a composite re do o re e oro e r o r o e " er e " e ween matrix and angular particles removes any porosity initially present. There is, in fact, about as much porosity as in the matrix samples. This conceptualization was chosen to fit with cementation cases observed in literature where cementation can lead to nearly zero porosity [START_REF] Molenaar | Quartz cementation mechanisms and porosity variation in Baltic Cambrian sandstones[END_REF]. The main characteristics of the four samples are gathered in Table 8. Figure 97. (1-CDF), with CDF the Cumulative Distributive Function, as a function of the equivalent diameter of particles in mm, for different surface percentages of matrix particles (29 to 60 %). Samples are named from (MG-29) to (MG-60). Note that the remaining composite mixtures not used in this study are available for an extended analysis on this topic. Table 8. Information on the samples used for direct simulations and compared in this section. 𝑃 𝑚𝑎 𝑖𝑥 is the surface percentage of matrix particles, 𝜇 𝑛 𝑚 is the numerical interparticle friction, 𝑃 𝑜 𝑒 is the surface percentage of initial porosity within gouge samples, 𝜙 𝑒𝑞-𝑚𝑎 𝑖𝑥 is the equivalent diameter of matrix cells, 𝜙 𝑒𝑞-𝑔 𝑎𝑖𝑛 is the equivalent diameter of grains, 𝑁 𝑔 𝑎𝑖𝑛 is the total number of angular grains and 𝑁 𝑚𝑎 𝑖𝑥 is the total number of matrix cells.

Samples

Numerical setup for direct shear modeling

Apart from the shapes and sizes of the grains, the model used in this section is the same direct shear model as in Part A. Interparticle friction 𝜇 𝑛 𝑚 is equal to 0.3 between each particle (matrix-matrix, matrix-grains, and grains-grains).

To avoid wall-slip effects with smooth boundaries, a certain cohesion is introduced at the interface between matrix and rock walls and between angular grains and rock walls (500 𝑀𝑃𝑎).

It enables the creation of a certain roughness by cementing particles to the top and bottom layer.

All the other physical and geometrical parameters are identical to those used for Part A, but for the sake of clarity the numerical setup and properties are gathered in

Effective friction

The addition of some matrix particles within the initial sample (G-C-4) densifies the sample by removing the initial porosity. The effective friction peak observed in Figure 98 (b) is much higher for the two composite mixtures compared to both 0 % (G-C-4) and 100 % (M-S) matrix samples where some porosity is observed (respectively 𝑃 𝑜 𝑒 = 11 % and ~1 %). The effect of initial porosity (or increase in initial density) on effective friction is in adequation with our previous Chapter 3 and with literature [START_REF] Taylor | Fundamentals of Soil Mechanics[END_REF]. In their study, (McBeck et al., 2019) confirm numerically that an increase in the initial porosity tends to reduce the maximum failure stress or peak of effective friction. Although the initial porosity of the granular gouge is a key parameter it needs to be combined with gouge fabric. Indeed, an augmentation of the percentage of matrix particles reduces this effective friction peak (from MG-33 to M-S). The insertion of matrix particle corresponds to a reduction of the broader grain size distribution (that strengthens the gouge), weakening the gouge for the same reason of increasing particles of simil ar sizes (i.e. reducing the fractal dimension D), as observed by [START_REF] Sammis | The kinematics of gouge deformation[END_REF], [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF]].

As usual, the steady-e e e e r o er ed e " e " zo e o r with the four different samples (𝜇 𝑆𝑆 ≈ 0.5). But minor differences can be discerned for the sample without matrix with 𝜇 𝑆𝑆-(𝐺-𝐶-4) > 𝜇 𝑆𝑆-(𝑀-𝑆) , because the interparticle friction used within the model from Chapter 3 is slightly higher than the one used in this chapter (𝜇 𝑛 𝑚-(𝐺-𝐶-4) = 0.5 > 𝜇 𝑛 𝑚-(𝑀-𝑆) = 0.3). The augmentation of matrix particles also lowers the steady-state friction from 0.54 to 0.40 from the sample (MG-33) to (M-S). It is consistent with the change of particles shapes present within the sample (reduction of broad particles with angularities and augmentation of small hexagonal particles).

The (MG-33) sample presents a higher number of angular particles and a non-homogeneous particle size distribution within the sample enhancing again the shearing resistance.

Homogeneity and dilation

The elastic part of the effective friction curve is similar for a percentage of matrix particles of 0 % (G-C-4) or 100 % (M-S), which are composed of only one sort of particle, whereas the two samples with composite mixture have a similar elastic part too. This evolution in the early beginning of the friction curve is mostly due to the initial presence of porosity which modifies the shear modulus and global rigidity of the gouge.

In terms of dilation, the more matrix particle is added, the less dilation is observed (Figure 99), which seems logical because the shape of the matrix particles (hexagonal) allows less expansion than angular grains. The (G-C-4) sample expands a lot because the whole gouge localizes the shear (Chapter 3), which is consistent with a high fractal dimension [START_REF] Morgan | Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization[END_REF].

On the other hand, in the material with only matrix (M-S), smaller cells imply finer displacement of particles and localization in the form of R-bands. The two inhomogeneous materials (MG-33) and (MG-56) are a composition of these two behaviors, with progressive decrease of dilation (at steady state).

𝜇

Riedel bands formation and evolution

As described in the previous Part A, it is possible to study Riedel band formation and evolution thanks to the evolution of solid fraction with shearing. For material (𝑀𝐺 -33), Riedel bands form before friction peak (as for (M-S) sample in previous Part A). Letters A to D, Figure 100 (a), refer to different states of the gouge with slip distance. The observed behavior is a mix between the sample used in Chapter 3 (with angular particles and cementation) and Chapter 4, part A (with matrix particles only):

(i)
From the beginning of shearing and until effective friction peak (images A to B), the observed kinematic behavior is comparable to the one observed with 100 % matrix particles (M-S), where small Riedel bands progressively form. The granular sample behaves as a very dense packing. (ii)

During the weakening part (images B to D), the gouge rheology evolves and R-bands are less homogeneous. Two secondary R-bands (𝑅 1 and 𝑅 2 ) tend to concentrate the total kinematic deformation but they do not reach the boundary shears. (iii) At the end of the weakening part (images D to the end), in contrast with the behavior observed for (M-S), the R-bands disappear in favor of a global shear zone (Y) at the center of the composite mixture. This behavior is very close to the one observed for the cemented model with less than 75 % of cementation (Chapter 3), except that the shear zone is localized in the middle zone of the granular gouge. Agglomerates of particles are also observed. An explanation is that the Y-band cannot be formed because of the lack of R-bands, interrupted by the presence here and there of large unbreakable grains.

The only way to shear the gouge is not by localization, but by shearing in the mass of the sample, thus necessarily by localizing on a thicker band. Normally, from an energetic point of view, the system always goes towards the solution that is the least costly. Here, there is no "easy" solution because each beginning of the previously formed R-band collides with some very large and rigid angular grains on their trajectory.

The system is thus obliged to use the most expensive solution which consists in Dilation Void ratio

𝑃 𝑚𝑎 𝑖𝑥 (%)
𝐹 𝑣-𝑆𝑆 𝜀 𝑦-𝑆𝑆 expanding the whole sample. So necessarily there is more energy spent and a higher friction coefficient at the peak as observed in Figure 98. By increasing the percentage of matrix particles, we tend towards a system less expensive energetically, and the small matrix particles also result in reducing fault resistance, lowering magnitudes of coseismic slip, [START_REF] Fournier | Insights to slip behavior on rough faults using discrete element modeling[END_REF].

Figure 100. Solid fraction snapshots for the sample with 𝑃 𝑚𝑎 𝑖𝑥 = 33 %, (𝑀𝐺 -33). Letters correspond to different steps in the effective friction curve, from A to D, the beginning of the plateau zone. The solid fraction is plotted between 0.7 and 1.

Influence of fabric and ratio of matrix to angular grains

Qualitative comparison between the different models

Although porosity reduction is commonly recognized to have a dominant effect on the mechanical and elastic properties of sediments and other granular materials [ [START_REF] Schöpfer | The impact of porosity and crack density on the elasticity, strength and friction of cohesive granular materials: Insights from DEM modelling[END_REF]), (McBeck et al., 2019), [START_REF] Casas | DEM Analyses of Cemented Granular Fault Gouges at the Onset of Seismic Sliding: Peak Strength, Development of Shear Zones and Kinematics[END_REF]], the proportion of matrix, or conversely the proportion of clasts into the matrix can also have a non-negligible effect on kinematics and shear strength of granular materials. Considering previous results of section 4.8.1, and in order to compare results with a different distribution of Matrix-Grains particle, another granular sample with the same characteristics as (M-S) was modeled. This is a bi-disperse mixture with the addition of 50 angular particles homogeneously distributed within the matrix model. The sample is called (MG-H-86), Table 10 and Figure 101.

In this section, we not only consider two particle sizes (matrix vs grains), but also different particle shapes (angular vs hexagonal) and different granular distributions (fractal distribution vs homogeneous distribution), meaning that we can only conclude on global tendency and qualitative comparisons. These are preliminary results and more experiments need to be done in order to analyze in detail the full parametric space of this sort of simulation.

Shear modulus

The combined evolution of maximum shear stress and shear modulus is displayed as a function of the percentage of matrix particles, for all samples from Part B, Figure 102. Matrix percentage modifies the elastic properties of the gouge, which is both logical and an artifact of our simulation: materials with larger grains present fewer sites of elastic accommodation, and the sample is automatically stiffer with more clasts. On the other hand, the material with only angular grains (G-C-4) has a much less rigid behavior with a smaller shear modulus, because its initial porosity is much higher (𝑃 𝑜 𝑒 = 11 %).

It can be observed in Figure 102 that both the maximum shear stress and the shear modulus are higher for the bi-disperse and composite mixtures, zone (ii), compared to samples with only matrix or only grains, zone (i). An augmentation of matrix content, from 33 % to 100 %, reduces the initial rigidity of the sample and thus the shear modulus. A gouge material with a high shear modulus does not present the same consequences on gouge rheology from Part A: although it increases the shearing resistance, the presence of large grains disturbs the initial mechanism and prevents the formation of a successful Riedel structure. Assuming that sample (MG-33) (with no initial porosity) and sample (M-S) (with less than 0.1 % of initial porosity) are in the same initial state of porosity, the difference between both slip behaviors is very important and mainly caused by the surface percentage of grains and matrix.

An increase in matrix content weakens the gouge until material (M-S), where a minimum shear stress and shear modulus are observed. The ratio of matrix particles content, or any different type of infill particles content, needs to be identified in each granular material and has to be Chapter 4. Rheology and weakening of a sheared fault gouge with matrix particles linked with the homogeneity of particle distribution within the gouge. Although it is done for most Lab experiment samples, it is not obviously considered in numerical simulations where mostly mono-disperse representations of gouge particles are simulated.

Focus on clasts

In Figure 103 and Figure 104, three granular samples are compared: the standard sample (M-S) with only matrix particles, the sample (MG-H-86) with 50 big angular particles, and the sample with 56 % of matrix particles and fractal distribution of angular particles (MG-56). The three granular materials considered here have similar initial porosity (𝑃 𝑜 𝑒 ~ 0 %) but different distribution and size of clasts content (angular grains). The first remark is that, whatever the angular grain distribution, an introduction of larger angular grains within the matrix model reinforces the shear strength and shear modulus. In general, the presence of large particles within shear bands limits the deformation observed [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF]. The consequences of the addition of bigger clasts within the gouge can be separated into the three slip phases, Figure 103: A.

Pre-peak phase. The composition of the granular sample and the proportion of angular particles within the gouge has a huge effect on the pre-peak stage. Both shear modulus or strength peak increase with an increase of angular particles content within the model, which makes the granular media much more resistant to shearing. [START_REF] Wojatschke | Geochemistry, Geophysics, Geosystems[END_REF] found similar results with their synthetic fault gouge mixing Clay matrix (size< 63𝜇𝑚) and Clasts (size> 63𝜇𝑚). Moreover, the slip distance needed to reach the peak of effective friction is smaller when the proportion of angular particles increases.

B.

Weakening phase. When the effective friction peak is reached, three kinds of weakening are observed. In contrast with (M-S) weakening slope which is almost linear from 𝜇 to 𝜇 𝑆𝑆 , the slope of material (MG-H-86) is very abrupt which suggests a more sudden post-peak weakening, reducing the slip weakening distance 𝐷 𝑐 from the one observed for (M-S). This behavior is prone to switch the fault behavior from a ductile aseismic response to a brittle seismic slip, (depending on the stiffness of the surrounding medium, which is infinite in our case). Contrarily to what we could expect, the weakening observed for material (MG-56), with more angular particles, does not follow the sudden slip-weakening of material (MG-H-86), and presents a non-linear weakening. The fractal distribution of particles in the gouge (MG-56) may be responsible for this different weakening slope. Despite the mechanical difference between sample (M-S) and (MG-H-86) in terms of strength peak, the rheological behavior at effective friction peak does not seem to be disturbed, Figure 104

C.

Steady-state zone. Steady-state effective friction tends to a similar value for the three gouges. Samples (M-S) and (MG-H-86) present the same two Secondary Riedels, meaning that in the sample (MG-H-86), the addition of big clasts is not sufficient to change the rheology of the gouge. However, differences are observed near clasts, leading to small and very localized conjugate Riedels, Figure 104 (i). The last gouge (MG-56) has a completely different rheology with no Riedel bands observed (as for MG-33 described in 4.8.1). This granular material presents agglomerates of particles surrounded by porosity with a clearly identified and unique shearing zone at the center The oscillations observed in the effective friction curves for samples (MG-H-86) and (MG-56) could look similar to oscillations from stick-slip experiments, but here they are simply due to variation in friction created by changing contact interactions at the interface. Granular materials from type (MG-i) seem to be the most optimal configuration to increase the shearing resistance, and the absence of Riedel bands at steady-state reduces the possibility of stick-slip to happen. On the other hand, very loose material or with a low diversity of material (homogeneous material) will lead to break much more easily with sudden stress drop, as material (MG-H-86). However, these results do not consider clay materials, for example, which have different elastic and frictional properties that can also lead to a loss of gouge strength and thus a change in stability. In their studies [START_REF] Collettini | Fault zone fabric and fault weakness[END_REF] show for example that the sliding behavior of rocks is weakened by the presence of phyllosilicate (talc and smectite). They concluded that fault weakening strongly depends on rock fabric and on the distribution of weak phases within a fault zone. (Kenigsberg et al., 2020) proposed that, depending on matrix or particle content (smectite in that paper), fabric and shear plane development also play a key role in the evolution of these properties with shearing. In lab experiments or real faults, fragmentation of particles is also observed [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF], decreasing the porosity within the sample with shearing [START_REF] Kenigsberg | The Effects of Shear Strain, Fabric, and Porosity Evolution on Elastic and Mechanical Properties of Clay-Rich Fault Gouge[END_REF]

Conclusion of Part B

Direct shear tests were conducted on the composite mixture, clearly showing that the surface percentages of both matrix and angular particles have a huge influence on the mechanics and rheology of fault gouge. The existence of angular particles or initial porosity also affects the shape and thickness of shear bands in the specimens. Inserting angular particles within matrix cells or matrix cells within a granular sample strengthens the gouge and gives different weakening behaviors depending on the type of particle distribution within the sample.

-For a gouge with a small matrix percentage, the friction peak shape is sharper, and slip weakening is more sudden with higher stress drop, also implying the possibility to switch the fault behavior to a more brittle seismic slip. The high number of angular particles and the non-homogeneous particle size distribution within the sample enhance again the shearing resistance. The energy needed to weaken the fault is much higher: Riedel bands cannot join the rock walls and form boundary shears, leading to a deformation of the entire gouge thickness. -Inhomogeneous and fractal particle distribution seems to favor a smoother slip weakening than homogeneously distributed clasts. -The temporality of Riedel bands formation depends on the percentage of the infill matrix material.

More results are needed to analyze the different behaviors observed such as a modification of the intrinsic deformability of matrix material, to represent clay content for example. Chapter 5. From energy budget to simplified models, towards dynamic simulations

Chapter 5. From energy budget to simplified models, towards dynamic simulations

Main foreword

This last chapter is both a synthesis of previous chapters and an opening for the continuation of the thesis. Starting from the results of Chapters 3 and 4, and focusing on friction laws and energy budget, the objective is to define friction laws characteristic of each type of infill material that can be injected into a large-scale dynamic fault model. For that we will distinguish three completely independent parts:

❖ Part A is a comparison of numerical frictional responses to linear slip weakening models from the previous models of Chapters 3 and 4. The linear slope of slip weakening, the associated fracture energy, and the nucleation length are extracted, compared and discussed.

❖ Part B restarts from the results of Chapter 3 on cemented models. A new partitioning of breakdown energy is discussed, allowing to differentiate three energy contributions from Coulomb friction, dilation and cementation failure process. This representation allows the construction of a simplified macroscopic friction model which has the final objective to be injected into dynamic failure models at a higher scale. The fracture energy and slip behavior of the fault gouge appears to be controlled by the intensity and temporality of these three mechanisms.

❖ Part C is built on the results of Chapter 4, Part A, and presents a Spatio-temporal analysis of Riedel bands activations and their implication in the global behavior of fault gouges. We connect the mechanical behavior inside each band to the entire gouge kinematics by the means of an energetic approach.

This chapter will lead to two upcoming papers: In order to study more deeply the results obtained in the previous chapters (cement and matrix), it may be interesting to first interpret them according to a very simplified model such as the linear slip weakening model (Chapter 1). Thanks to this representation it is possible to extract the slope associated to slip weakening models, which gives information on the stability of the system. The rupture energy and critical nucleation length associated with each model are calculated to relate to the intensity of seismic events that may occur. In the first section 5.2, the method used to recover and compute the different parameters is explained. Section 5.3 compares and discusses the results obtained for the different models, with a focus on the critical nucleation length associated with simulations and on the influence of each of the tested materials, section 5.3.2. This section gathers the data obtained for each simulation used during the Thesis work as sketched in Figure 105.

Method and theory

Breakdown energy

Throughout an earthquake, energy is dissipated during fault sliding by the means of different mechanisms, gathered in the total energy budget ∆𝑊, the sum of a fracture energy 𝐸 𝐺 , a radiated energy 𝐸 𝑅 and a frictional energy 𝐸 𝐻 (Chapter 1). In this section, the focus is made on postpeak rupture energy called 𝐸 𝐺 (𝐽/𝑚 2 ), or Breakdown Energy, which is the potential energy needed to weaken the fault. It can be calculated from previous numerical results of Chapters 3 and 4, as the area under the friction-slip curve, corrected by subtracting the steady-state friction (eq. ( 1.11) of Chapter 1).

Critical weakening slope

One possibility to determine the laboratory rock failure stability is to compare the stiffness of the fault with its weakening rate. The objective here is thus to calculate the equivalent critical slope 𝑐 of the slip weakening models (Chapter 1, eq. (1.9)), for all simulation cases with different infill materials (cementation and matrix).

Slip instabilities may occur if the loading stiffness of the fault K (i.e. stiffness of the loading system) appears to be lower than 𝑐 ( 𝑐 > ). Apart from being an essential parameter for RSF laws, 𝑐 is thus an essential quantity in terms of stability. It is expressed here in 𝐺𝑃𝑎/𝑚 and can be extracted from each of the friction curves presented all over the manuscript, provided that we accept to fit a linear weakening model on the post-peak frictional response of each of our faults, Figure 106. The Breakdown energy 𝐸 𝐺 (𝐽/𝑚²) is first computed as the area under the shear stress curve from the stress peak 𝜏 to the averaged steady-state shear stress 𝜏 𝑆𝑆 . The critical sliding distance 𝐷 𝑐 , is determined to obtain an energy 𝐸 = (𝜏 -𝜏 ) 𝐷 𝑐 /2, similar to the 𝐸 𝐺 previously calculated (yellow area, Figure 106). In Figure 106, 𝑐 fits well with the shape of the decreasing shear stress, but it is not as satisfactory for all the models tested (Appendix 5.A). From energy budget to simplified models, towards dynamic simulations stiffness is defined by the ratio of stress to displacement = 𝜏 𝑈 ⁄ . The fault stiffness is found to be proportional to the ratio 𝐺 𝐿 ⁄ , with G the shear modulus and L the length of the slipping fault section, meaning that the loading stiffness decreases as the slipping fault section increases, Figure 107 (a).

If the slipping region is treated as an elliptical crack [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF], the critical nucleation length before instabilities can be calculated as

𝐿 𝑐 = 𝐸 2(1 -𝜈 2 ) 𝑐 (5.1)
With E a relevant young modulus (or equivalent elasticity modulus), 𝜈 the Poisson ratio and 𝐿 𝑐 the critical length of the slipping region. The fault is supposed to be unstable when the slipping fault section exceeds the critical nucleation length 𝐿 𝑐 . Figure 107 (b) illustrates the dynamic behavior observed when 𝐿 𝑐 is reached.

e o je e ere o o e e "e " e o 𝐿 𝑐 , but to observe the global trend and to put forward a dependence on certain gouge characteristics. The choice was made to compare all the results with the same Young modulus and Poisson coefficient as if the same fault rock material was compared. Medium values for granite material are considered, 𝐸~60 𝐺𝑃𝑎, and 𝜈 = 0.3, with an approximative 𝑚𝑒𝑎𝑛 ~900 𝐺𝑃𝑎/𝑚 (considering the 20 𝑚𝑚length gouge).

Comparison between the different gouge materials

Evolution of the slope of slip weakening and breakdown energy

In this section, all the results from the previous chapters are gathered in terms of Breakdown energy 𝐸 𝐺 , Figure 108, and linear weakening slope 𝑐 , Figure 109. From a first glance at the results, it can be observed that the percentage of infill material, either matrix or cement (image (b) and (c) in each figure) has much more influence on these two quantities than the inner characteristics of the gouge that show a smaller difference between the extrema obtained.

𝐿 𝜎 𝑁 𝜏

Breakdown energy

The observed breakdown energy E G ranges between 1 𝑘𝐽/𝑚² and 10 𝑘𝐽/𝑚², values in the range to those of laboratory and seismological estimated fracture energy, ranging between 1 𝐽/𝑚² and 10 J/𝑚² for minerals and rocks [ [START_REF] Abercrombie | o er o o e r ke o r we ke ?[END_REF], [START_REF] Nielsen | Scaling in natural and laboratory earthquakes[END_REF], [START_REF] Scholz | The Mechanics of Earthquakes and Faulting 3rd Edition[END_REF]]. According to [START_REF] Nielsen | Scaling in natural and laboratory earthquakes[END_REF] and large data set from our simulations, the total associated slip observed for our fault gouge would be ranged between 1 and 10 mm.

The evolution of breakdown energy appears to mostly depend on the ratio of infill material within the granular sample, Figure 108 (b) and (c), with values los o 10 4 𝐽/𝑚². While increasing the percentage of cementation promotes higher fracture energy, increasing the percentage of matrix decreases the fracture energy: a matrix percentage higher than 80 % shows breakdown energy around 10 3 𝐽/𝑚², as well as mid-dense and low cemented models. On the other hand, for dense materials (either very cemented or with a very little percentage of matrix particles), energies are displayed around 10 4 𝐽/𝑚², suggesting much more intense seismic events in case of slip initiation. Changing cells perturbation, Figure 108 (a4), does not have a huge influence on the total breakdown energy. However, an increase in breakdown energy is observed when changing the interparticle friction from 0.1 to 0.6, Figure 108

𝑒𝑞 (mm) 𝑛 (Pa/m) 𝑃 𝑐𝑒𝑚 (%) 𝑃 𝑚𝑎 𝑖𝑥 (%) 𝑃 𝑖 𝜇 𝑛 𝑚 𝐸 𝐺 (J/m²) (a2) (a3) (a4) (c)
𝐸 𝐺 (J/m²) Chapter 5. From energy budget to simplified models, towards dynamic simulations

Linear weakening slope

Even though the inner characteristics of the gouge have more influences on the Linear weakening slope 𝑐 , they are still negligible compared to the variation in the surface percentage of matrix or cement, Figure 109. In line with the results of Chapter 4, Part A, a significant influence of the interparticle friction coefficient and the particle size is observed on the 𝑐 value. Increasing friction increases the inclination of the weakening slope, which enhances the probability of generating an instability while the increase in the size of the particles has the opposite effect when it is cumulated with an equivalent stiffness allowing to have a constant shear modulus.

Shear modulus, friction, and number of particles within the gouge thickness, Figure 110

As interparticle friction, the shear modulus, and number of particles within the gouge thickness selected in Chapter 4, were playing on the Riedel structure observed, they also influence 𝐸 and 𝑐 . Going back to the matrix materials dataset, a link between the breakdown energy and the ratio 𝑛𝑏 1 is observed: a slight increase in 𝐸 𝐺 is observed when decreasing 𝑛𝑏 1 with the opposite effect on 𝑐 . However, an increase of 𝑐 , even for a low energy, can be quite penalizing, as it also implies a reduction of the sliding distance necessary to weaken the fault, which could tend to rather an unstable sliding.

These results agree with the Riedel structure observed in Chapter 4: materials with lower 𝑐 presented lower Riedel angles and a highly visible Riedel structure, promoting slip stability. A high value of initial tangent shear modulus increases both 𝑐 and 𝐸 𝐺 , although its influence is less important than 𝑛𝑏 1 or 𝜇 𝑛 𝑚 .

Dense

Mid-dense (a1) (b) (a2) (a3) (a4) (c)
𝑒𝑞 (mm) 𝑛 (Pa/m) 𝑃 𝑐𝑒𝑚 (%) 𝑃 𝑚𝑎 𝑖𝑥 (%) 𝑃 𝑖 𝜇 𝑛 𝑚 𝑐 (𝐺𝑃𝑎/𝑚) 𝑐 (𝐺𝑃𝑎/𝑚)
In the case where there is both an increase of shear modulus and a decrease in 𝑛𝑏 1 , the two effects compensate with almost no variation in both 𝑐 and 𝐸 𝐺 , as observed for an increase in the size of particles, Figure 109 (a2).

Figure 110. Evolution of the linear slope of weakening 𝑐 (𝐺𝑃𝑎/𝑚) as a function of (a) the interparticle friction 𝜇 𝑛 𝑚 , (b) the initial tangent shear modulus 𝐺 𝑎𝑛 (𝐺𝑃𝑎) and (c) 𝑛𝑏 1 the ratio of gouge thickness on particle size. Evolution of the breakdown energy 𝐸 𝐺 (𝐽/𝑚²) as a function of (a) the interparticle friction 𝜇 𝑛 𝑚 , (b) the initial tangent shear modulus 𝐺 𝑎𝑛 (𝐺𝑃𝑎) and (c) 𝑛𝑏 1 the ratio of gouge thickness on particle size. Matrix samples (M-… w o e parameters variation: particle size with equivalent stiffness (dark blue circles), particle size (orange circles), the initial thickness of the gouge (green squares), interparticle stiffness (pink diamonds), interparticle friction (red hexagrams) and perturbation shape (black asterisks). Linear regression line in black.

Critical nucleation length and stability consequences

A slip within fault can become problematic when a change from slow slip to fast stick-slip is observed, which is mainly due to an alteration of the elastic stiffness of the crust (for crustal faults) or the loading apparatus (for laboratory experiments) or a change within gouge material properties. According to several recent studies, it has been shown that the evolution of the gouge fabric is of major influence in the observed slip patterns with shearing.

Simulations from this Thesis work do not allow to conclude on the stability of the system, as shearing is perfectly displacement-controlled and no global stiffness is present. Although we cannot calculate the total nucleation length of the system, it is possible to calculate for each experiment a critical nucleation length from which, depending on the medium stiffness, the fault could switch to fast stick-slip and thus instabilities.

The evolution of the critical nucleation length 𝐿 is computed according to the previous method (Section 5.2). The obtained curves, Figure 111, are directly the inverse curves of those observed for , as a direct consequence from equation (5.1). These results can then be interpreted in terms of instabilities and linked to higher-scale faults. The last Figure 112 presents the critical nucleation length as a function of the breakdown energy for each simulation. The majority of critical nucleation lengths are observed in between 0.1 and 1 m but some off-center zones are noticed:

(i) High 𝑳 and low 𝑬 𝑮 -The mid-dense samples with nearly no cementation ( 𝑃 𝑐𝑒𝑚 < 25 %) show a material with a large critical nucleation zone (𝐿 𝑐 > 3 𝑚) and very low breakdown energy (𝐸 𝐺 ≤ 10 3 𝐽/𝑚²). By increasing the critical nucleation length, more flexibility of movement is allowed within the sliding zone before going into instabilities, Figure 107, leading to a less rigid fault. This implies that these materials are not very susceptible to become unstable. Even if instability were to occur, the energy released by slip would be small compared to other materials tested. These middense materials without cementation have characteristics that are rather susceptible to generate slow sliding.

(ii) Medium 𝑳 and medium to large 𝑬 𝑮 -This second zone gathers most of the samples studied. For 𝐸 𝐺 > 10 3 𝐽/𝑚², a linear relation describes the link between the observed fracture energy and its associated critical nucleation length (black line). Although the range of observed 𝐿 𝑐 is similar, the gradient of fracture energy observed is more important. As previously observed in section 5.3.1, higher fracture energy mostly corresponds to higher resisting strength of material: high cementation, a small percentage of matrix particles, or high interparticle friction. Increasing the size of Dense Mid-dense

𝑒𝑞 (mm) 𝑛 (Pa/m) 𝑃 𝑐𝑒𝑚 (%) 𝑃 𝑚𝑎 𝑖𝑥 (%) 𝑃 𝑖 𝜇 𝑛 𝑚 𝐿 𝑐 (m) 𝐿 𝑐 (m) (a1) (b) (a2) (a3) (a4) (c)
particles (top right corner) both increases the 𝐿 𝑐 and the 𝐸 𝐺 resulting in delaying earthquake nucleation. (iii) Low 𝑳 and high 𝑬 𝑮 -In contrast, this last zone is characteristic of highly dense and cemented materials (𝑃 𝑐𝑒𝑚 > 75 %). For a very small critical nucleation zone, there is not much mobility allowed for the system and more risks that the model goes into instability with a high-stress release. It also depends on the length of the fault which will add rigidity to the global model (but here all samples have the same length). As a difference with the ductile material presented in (i), these materials are almost brittle, the consequences in terms of instabilities and energy release are mostly related to this brittle behavior.

The main behavior differences appear for the two extremes cemented materials observed in Chapter 3, which seems consistent since they represent a rather ductile behavior (mid-dense, with no cementation), which tends to give a rather stable sliding and its brittle opposite (dense and highly cemented) which tends to rapid changes of behavior, leading to a rather unstable sliding. Another lecture of the data is possible by looking at the linear regression lines corresponding to each of the four types of materials tested. The two cemented materials (dense and mid-dense) follow similar evolutions with a shift due to the initial porosity of the sample.

In the same way, the two materials with matrix particles follow the same evolution with a shift in the nucleation length, this time due to the presence of large grains in the sample which decreases the critical nucleation length. Samples with matrices follow a different slope than those without: the presence of matrix particles changes the scaling. It would be interesting to compare these data with in-situ or lab experiments. Analyses could be improved by studying these same materials with the presence of water.

Besides, in the proposed definition of 𝐿 𝑐 , time dependence is not considered. However, according to a recent study [START_REF] Mclaskey | Earthquake Initiation From Laboratory Observations and Implications for Foreshocks[END_REF], 𝐿 𝑐 has to be carefully used for rough and heterogeneous faults which need an intrinsic time dependence law such as power density (strain energy released per unit time per unit area). It could be another comparison to make with our samples. Then, there are two very important aspects that we almost didn't address here: the influence of normal stress (because the results presented here are only valid for 0 𝑀𝑃𝑎) and the gouge thickness, probably inoperative in some cases (dense and very cemented materials that fracture horizontally) and important in other cases (inclined Riedel of matrix cases that have more or less time to develop). It is obvious that a change in the applied normal stress will affect these parameters. It would be interesting to repeat this study for cases where the confining pressure is much higher, in order to judge the evolution of these parameters under such stresses.

Even though we can compute and compare the critical nucleation lengths with literature, it is not possible, at this step, to say whether or not the system is stable or unstable. For this reason, friction laws observed need to be inserted in dynamic modeling considering the global stiffness of the fault and a dynamic rupture process. This is expected to be a natural perspective of the present thesis.

Conclusion of Part A

Infill materials, which are playing on gouge fabric, have a major role in the critical nucleation length and fracture energy released. The cementation within the gouge seems to be the most influent parameter tested and an initially dense cemented material will tend to a very short slip distance before instabilities occur (small 𝐿 𝑐 ) as well as a high breakdown energy (high 𝐸 𝐺 ).

Even though they take place in a specific range of experiments (constant imposed pressure, constant slip velocity, and absence of water), our results raise the question of the slip weakening shape. It appears that only mid-dense cemented samples and some matrix samples can be easily modeled with the classical linear form (Appendix 5.A). Dense cemented samples display a bilinear decreasing shape that can be compared to a model proposed by [START_REF] Abercrombie | o er o o e r ke o r we ke ?[END_REF]. Exponential weakening models are used by [START_REF] Sone | Frictional resistance of faults during accelerating and decelerating earthquake slip[END_REF] in their highvelocity friction experiments. Although their results cannot be directly compared with those presented here, the friction curve described by an exponential fit is consistent with some of our mid-dense cemented samples but not for dense cemented samples. A reason that could explain this difference is that a simple exponential weakening considers that all the physics is acting at the same time, which is maybe not the case for very cemented material.

Simple slip weakening models are intended to approximate behaviors to make them more tractable to use for understanding fault weakening with displacement (Chapter 1). Real fracture is well known to be much more complex, but complex models are difficult to upscale. A balance must be found between a too simple and a too complex model. For this reason, we propose other friction and energetic laws linking local rheology and gouge mechanics in the next two sections.

Part B -Energetic study of cemented gouges: a way to friction laws

Foreword

Previous simulations from Chapter 3 show that a modification of the initial porosity (i.e. density), or of the cementation within the sample could induce an important change in slip mechanisms. In order to isolate the main mechanisms acting on fracture processes, a partitioning of the total macroscopic friction generated during the shearing of a granular gouge sample is proposed. Based on the physical content of the numerical model used, three factors are identified as playing a role in the total effective friction and energy partitioning:

-

Definition of breakdown energy

In this section, the focus is also made on post-peak rupture energy called 𝐸 𝐺 (𝐽/𝑚 2 ), for Breakdown Energy, previously calculated on Part A. It is calculated from previous numerical results (Chapter 3), as the area under the friction-slip curve, corrected by subtracting the steadystate effective friction value:

E G = ∫ (𝜇 (𝑈) -𝜇 ss )𝜎 𝑁 𝑑 𝑈 𝑈 𝑒𝑛𝑑 𝑈 𝑝 (5.2)
With 𝜇 (𝑈) the current effective friction, 𝑈 the -displacement at effective friction peak, 𝑈 𝑒𝑛𝑑 the -displacement at the end of effective friction peak, and 𝑈 the current slip distance.

Calculation of the different energies

As the model does not present any radiated or heat energy consumption, it is admitted that the total energy of the model, to which the Coulomb friction part (𝐸 𝑓𝑐 ) and a potential pre-peak part would have been subtracted, will be similar to the breakdown energy previously presented.

The objective here is to calculate each term of the Breakdown energy contribution previously described, meaning the energy posterior to the effective friction peak: the dilation energy 𝐸 𝐷𝑖 -, the frictional energy 𝐸 𝑓-and the Damage energy 𝐸 𝐷𝑎𝑚-(pp for post-peak). The total energy of the sheared gouge model 𝐸 𝑜 (𝐽/𝑚²) is first calculated for every slip distance U( ) (i.e. at every time step i of the simulation), knowing the total tangential force 𝐹 𝑇 acting on the upper rock wall ( -direction):

𝐸 𝑜 ( ) = 𝐸 𝑜 ( -1) (𝑈( ) -𝑈( -1))(𝐹 𝑇 ( ) 𝐹 𝑇 ( -1)) 2 𝑔 (5.3)
The surface 𝑔 is the surface perpendicular to the applied normal stress 𝜎 𝑁 , due to the (2D) simulation, with a width of 1 𝑚 with 𝑔 = 𝐿 𝑔 1 2 . Parameters are shown in Figure 115.

Figure 115. Schema of the model with the different parameters used. V is the shearing velocity applied on the upper rock wall (𝑚/𝑠), 𝜎 𝑁 is the applied normal stress on the upper rock wall (𝑀𝑃𝑎), 𝑔 is the surface perpendicular to the applied normal stress (𝑚²), 𝐿 𝑔 is the gouge length (m), 𝐹 𝑇 and 𝐹 𝑁 are respectively the total tangential and normal forces acting on the upper rock wall ( -direction), (N).

The total dilation energy 𝐸 𝐷𝑖 (J/m²) is the product between the vertical distance covered by the upper rock wall Δ ℎ (the -displacement is allowed in the model) and the total normal force 𝐹 𝑁 acting on this rock wall (y-direction):

𝐸 𝐷𝑖 ( ) = Δ ℎ ( )𝐹 𝑁 ( ) 𝑔 ( 5.
)

It is also possible to calculate an averaged post-peak dilation energy 𝐸 𝐷𝑖 -𝑎𝑣 e o [𝑡ℎ 𝑆𝑆 -𝑡ℎ ] 𝜎 𝑁 With 𝑡ℎ 𝑆𝑆 the gouge thickness at steady-state, 𝑡ℎ the gouge thickness at friction peak and 𝜎 𝑁 the normal stress.

Finally, the Damage energy 𝐸 𝐷𝑎𝑚 (𝐽/𝑚²) is calculated considering the contact parameters of each particle as follows S g = 0.02 m²

L g = 0.02 m 𝑉 𝐸 𝐷𝑎𝑚 ( ) = ∑ 𝐿 -𝑐 ( ) 2 ( 𝐶 𝑛 𝑚 2 ) 2 1
𝑘 𝑛 1 𝐿 𝑔 (5.5) With ∑ 𝐿 -𝑐 ( ) the total contact length of contacting grains evolving during shearing, 𝐶 𝑛 𝑚 e er o e o de r ed er d 𝑘 𝑛 e er r e e . Frictional energies 𝐸 𝑓 is the remaining energy of the model: 𝐸 𝑓 = 𝐸 𝑜 -𝐸 𝐷𝑖 -𝐸 𝐷𝑎𝑚 (5.6) As an example, Figure 116 displays the evolution of energy released for a granular sample with 𝑃 𝑐𝑒𝑚 = 57 % (G-C-57), as a function of the slip distance. Figure 116 (a) presents the evolution of each energy contribution through slip distance evolution. The sum of the 3 contributing energies (Dilation, Damage, and Friction) gives the black curve which is the total energy released during the sliding (pre and post-peak), increasing with slip distance. By derivation of each energy term, an associated friction can be calculated, allowing to obtain the specific frictional contribution of each mechanism presented above, Figure 116 (b). These friction contributions will be used to propose simplified models in section 5.6.2. Figure 116 (c) helps analyze the percentage of energy consumed in pre-peak or post-peak phases, where we see for this sample that pre-peak energy dissipation can be neglected (although this conclusion does not stand for all samples). The surface percentage of cementation (presence of cohesion) remaining in the sample with the increasing slip distance can be observed in Figure 116 (d), it represents the damage observed in Chapter 3. Similar data are extracted from each experiment of Chapter 3, both for dense and mi-dense samples.

A way to friction laws

Now that each friction contribution has been identified and calculated, the objective is to obtain a simplified friction law that accounts for the three main mechanisms (dilation, friction and damage) and that takes as input the physical parameters of the gouge. Once this friction law is determined, it will be possible to simulate the mechanical and energetic behavior of any cemented gouge (within the framework of the hypotheses of the thesis) by simply changing the percentage of cementation (𝑃 𝑐𝑒𝑚 ) present in the sample.

Depending on the percentage of cementation and thus on the three different types of cemented materials previously identified in Chapter 3 (poorly cemented, cemented, and highly cemented), the simplified model can give the evolution of the effective friction as a function of the slip distance with a well-detailed slope of slip weakening. The following section explains the relationship between each parameter of the gouge and the steps leading to the final friction laws.

The method aims to find a simplified model for each frictional contribution, as observed in Figure 116. -The dilation friction evolution 𝜇 𝑑𝑖 is illustrated employing three parameters, Ψ the maximum dilatancy angle [START_REF] Rowe | The stress-dilatancy relation for static equilibrium of an assembly of particles in contact[END_REF], 𝑈 𝑑 the slip distance needed to reach the dilation peak, and ∆𝐻 𝑑 , the gain in gouge thickness at the end of the dilation phase, Figure 117 (c). These parameters do not follow linear laws and both poorly and ultra-cemented samples present constant values for Ψ and ∆𝐻 𝑑 , meaning that major evolution occurs for mildly cemented materials, Figure 118 (a) and (c). As Ψ is increasing with the percentage of cementation, the slip distance 𝑈 𝑑 associated with maximum dilation is reducing, which is consistent with results from Chapter 3. ∆𝐻 𝑑 gives the shape of the Gaussian model. Even though a lognormal law seemed to better fit the dilation friction contribution than a Gaussian law, it was not satisfactory for the total friction law. The 𝑈 𝑑 is higher for mid-dense materials and decreases with higher cementation, Figure 118 andB2). The interesting parameters are: the effective friction peak at the end of the elastic phase 𝜇 𝑓 , the post peak friction 𝜇 𝑓 and the characteristic distance of the exponential decay ∆𝑈 𝑓 . B1 corresponds to the granular sample studied with 𝜇 𝑓 > 𝜇 𝑆𝑆 and B2 is an example of the second type of behavior, observed for the 30 % mid-dense case with 𝜇 𝑓 ≤ 𝜇 𝑆𝑆 .

Behavior laws of shape parameters

(c) Zoom in on dilation contribution 𝜇 𝑑𝑖 , simulation data in red and simplified model in dark with: the maximum dilatancy angle Ψp, the slip distance corresponding to the maximum dilation 𝑈 𝑑 and the gain in gouge thickness ∆𝐻 𝑑 at the end of the dilation phase. (d) Zoom in on the damage contribution 𝜇 𝑑𝑎𝑚 with the characteristic distance of the exponential decay ∆𝑈 𝑑𝑎𝑚 and the maximum friction induced by breakage of cohesive links 𝜇 𝑑 . It can be noted that k, μ , and ∆𝑈 𝑓 do not depend on the cementation and are taken as constant.

-Friction contribution 𝜇 𝑓 is more complex to model because two different behaviors are -Damage contribution 𝜇 𝑑𝑎𝑚 presents a peak 𝜇 𝑑 with rising linear law from 0 to 100 % cementation, Figure 118 (f), consistent with previous results since increasing cementation increased the resistance of bonded particles to shearing (Chapter 3).

These parameters are chosen and adjusted to picture the global trend of friction contributions.

The separation zone between the three distinct regimes described in Chapter 3 is marked by a vertical dotted line. The objective of these graphs is not to reproduce the exact location of dilation friction peak, but to model a global and consistent shape evolution from one case to another. For the sake of simplicity, ∆𝑈 𝑓 (distance between the slip distance at 𝜇 𝑓 and the slip distance at 𝜇 𝑓 ), 𝑘 (the elasticity slope) and 𝜇 𝑆𝑆 are independent of cementation and taken as constant. The distance ∆𝑈 𝑓 is equal to 8 µm for dense samples and 0 µ𝑚 for mid-dense samples. The gouge layer stiffness 𝑘 is proposed equal to 140 𝑘𝑁/𝑚 for dense samples and to 80 𝑘𝑁/𝑚 for mid-dense samples. Details on the equations allowing to obtain these curves are presented in Appendix 5.B. • Dilation contribution (𝝁 𝒅𝒊𝒍 ). The three dilation parameters (Ψ , 𝑈 𝑑 and ∆𝐻 𝑑 ) are introduced in a Gaussian-like law with maximum dilatancy friction of (Ψ p ) and a squared exponential decrease as a function of slip:

𝜇 𝑑𝑖 (𝑈 ) = 𝑡𝑎𝑛(Ψ ) • 𝑒 -log(2)( 𝑈 -𝑈 𝑑𝑝 𝑈 𝑑 ℎ𝑎𝑙𝑓 -𝑈 𝑑 𝑝 ) 2 (5.7)
U is the current displacement of the upper rock wall and 𝑈 𝑑ℎ𝑎 𝑓 is the distance between 𝑈 𝑑 𝑝 and the slip distance for a half reduced dilatancy-related friction. This last parameter is determined by solving numerically:

𝑈 𝑑ℎ𝑎 𝑓 -𝑈 𝑑 2 • (Ψ ) • √ 𝜋 log(2) • (1 𝑒 𝑓 ( 𝑈 𝑑ℎ𝑎 𝑓 𝑈 𝑑ℎ𝑎 𝑓 -𝑈 𝑑 • √log(2))) = ∆𝐻 𝑑
• Friction contribution (𝝁 𝒇 ). The Coulomb contribution is more complex due to the two possible behaviors B1 and B2, Figure 117. The peak behavior where the effective friction peak 𝜇 𝑓 is higher than 𝜇 𝑆𝑆 exhibits a first linear elastic part, characterized by a stiffness 𝑘 (gouge The second type with 𝜇 𝑓 ≤ 𝜇 𝑆𝑆 displays the succession of an elastic part from 0 to 𝜇 𝑓 followed by an asymptotic exponential behavior until 𝜇 𝑆𝑆 , Figure 117 (b) B2. This case is especially observed in mid-dense samples or for mildly cemented materials:

𝜇 𝑓-2 (𝑈 ) = { 𝑘𝑈 , 𝑈 ≤ 𝑈 𝑓 𝜇 𝑆𝑆 (𝜇 𝑓 -𝜇 𝑆𝑆 ) • 𝑒 - 𝑈 -𝑈 𝑓𝑝 ∆𝑈 𝑓 , 𝑈 𝑓 ≤ 𝑈 (5.9)
With 𝜇 𝑓 = 𝜇 𝑓 and 𝑈 𝑓 = 𝜇 𝑓𝑝 𝑘 .

• Damage contribution (𝝁 𝒅𝒂𝒎 ). The last contribution is the damage, characterized by an absence of contribution during elastic loading, and a maximum friction peak 𝜇 𝑑 followed by an exponential decrease until 0, Figure 117 (e):

𝜇 𝑑𝑎𝑚 (𝑈 ) = { 0, 𝑈 ≤ 𝑈 𝑑𝑎𝑚 𝜇 𝑑 • 𝑒 - 𝑈 -𝑈 𝑑𝑎𝑚 ∆𝑈 𝑑𝑎𝑚 , 𝑈 𝑑𝑎𝑚 ≤ 𝑈 (5.10)
With ∆𝑈 𝑑𝑎𝑚 , a characteristic distance of the exponential decay, and ∆𝑈 𝑑𝑎𝑚 = ∆𝑈 𝑓 .

Validation of the model

Figure 119 presents the simplified models and the initial effective friction evolution for dense and mid-dense samples for the three types of cemented materials.

𝑃 𝑐𝑒𝑚 = 95%
These six chosen cases are presented here to validate the experimental friction curves with the friction decomposition proposed: one case for each cemented material described in Chapter 3 (poorly cemented, cemented and ultra-cemented) and for each type of compacted sample (dense and mid-dense). Despite its simplicity, the empirical model fits all the materials quite remarkably and can be implemented with confidence in large-scale rupture models. The middense and poorly cemented patterns are the least well represented by the model and may require some improvement. The rest of the models and validation are presented in Appendix 5.C. Considering every friction contribution 𝜇 𝑑𝑖 , 𝜇 𝑓 d 𝜇 𝑑𝑎𝑚 , this new model gives a simplified representation of the effective friction coefficient 𝜇 as a function of the initial percentage of cementation. The proposed functional expressions and parameters evolutions of this model remain theoretical and might be modified or improved based on future findings, but allow to enrich slip-weakening laws by proposing a more general model that could be implemented in dynamic rupture modeling.

Results & Discussions

Breakdown energy evolution

Thanks to the previous method, the breakdown energy 𝐸 𝐺 as well as its three contributed parts (𝐸 𝐷𝑖 -, 𝐸 𝑓-and 𝐸 𝐷𝑎𝑚-) are calculated with previous equations, Figure 120. As the energies calculated with the previous models depend on simulation data, the observed results are of the same order of magnitude as those observed in Chapter 3 and Part A of this chapter.

The energetic behavior can be analyzed regarding the three distinct cemented materials previously identified:

-Poorly cemented materials (𝑃 𝑐𝑒𝑚 ≤ 25 %) present a low and almost constant energy release (although larger for denser samples) remaining relatively small compared to the two other cemented materials: 10 3 𝐽/𝑚² for mid-dense materials and around 3. -For cemented materials (25 < 𝑃 𝑐𝑒𝑚 < 75 %), 𝐸 𝐺 is increasing with the percentage of cementation, characterized by the chaotic character of the gouge observed with particles agglomerates. These large agglomerates are a hindrance to the stabilization of the gouge and explain the need for more energy to reach a stabilized state.

-Ultra-cemented materials (𝑃 𝑐𝑒𝑚 ≥ 75 %) display an almost constant energy released, but more than twice higher than for the poorly cemented materials. Due to the rather brittle nature of the highly cemented materials (Chapter 3), the frictional resistance strongly increases, which amplifies the energy required to weaken the friction to its steady-state value. As the critical weakening distance also decreases, a stabilization of the energy is reached for the highly cemented models. Chapter 5. From energy budget to simplified models, towards dynamic simulations A difference is observed between the total Energy, considering the pre-peak energy part (dashed lines), and the breakdown energy (full lines). For mid-dense and poorly cemented samples, the pre-peak part is very important and is probably caused by the early beginning of dilation due to the more ductile behavior of this material. On the other hand, dense and highly cemented samples also present higher 𝐸 𝑜 , the pre-peak energy is increasing with the increase of friction peak. [START_REF] Ohnaka | A constitutive scaling law and a unified comprehension for frictional slip failure , shear fracture of intact rock , and earthquake rupture[END_REF] The grey diamonds represent the breakdown energy directly extracted from simulations. The dense simplified models reproduce well the energetic behavior of the cemented gouge, but further improvements need to be done for mid-dense and highly cemented materials, Figure 120 (a). The difference found can be explained in three ways: (i) the shape parameters of middense simplified friction can still be improved, (ii) the original curves of mid-dense samples are very noisy, which explains why the energy calculated in Part A is not exactly the same as the one obtained with the simplified model, (iii) the steady-state value is equal for all simplified ode w ere e zo e re o er " e r" or e re r o r e e d to resistance to fracture) and the initial porosity of the sample. This is consistent with previous results showing higher dilation for denser samples, Chapter 3. As cementation increases, both damage (E D -pp ) and frictional energy (E f -pp ) increase, but they are still negligible compared to dilation energy which represents almost 70 % to 80 % of the total energy and breakdown energy. Damage energy is almost independent of the initial state of porosity as the energy needed to break a cohesive link rather depends on the contact length and the initial percentage of cementation.

E f -pp presents a special behavior with a zero contribution for poorly cemented materials and a small increasing contribution for the other materials, Figure 120 (a) &(b). An interesting point is that this energy contribution is the only one that may lead to heat creation in the system (in addition or subtraction of the steady-state friction) since dilation energy is nothing but a mechanical work and damage energy is related to surface creation. The constant part of friction energy E f c (not considered in E G ) appears to be twenty times higher than E G for the total slip considered here. This high difference explains why (J. [START_REF] Chester | Fracture surface energy of the Punchbowl fault, San Andreas system[END_REF] consider that the Fracture energy is negligible in comparison to other energies presented in the theoretical energy budget, Chapter 1. However, the knowledge of E G is crucial for the understanding of the onset of sliding.

The role of dilation energy

Why is dilatancy so important? Dilation is assumed to be very influential in the way a fracture grows. [ [START_REF] Rice | Earthquake Precursory Effects Due to Pore Fluid Stabilization of a Weakening Fault Zone[END_REF], [START_REF] Rice | Constitutive Relations for fault slip and earthquake instabilities[END_REF], [START_REF] Caniven | The Role of Along-Fault Dilatancy in Fault Slip Behavior[END_REF], [START_REF] Aben | Dilatancy stabilises shear failure in rock[END_REF]] studied the fault-zone models where dilatancy is assumed to occur within the fault zone during the nucleation process (with or without fluid). In gouge or breccia materials, dilation appears with shearing and constitutes the basis of the well-known slip-weakening models. According to the theory, and in the presence of fluid, dilation has two major effects, the reduction of pore pressure and an increase in frictional resistance due to dilatancy hardening mechanism. As a result, a potential stabilization of the fault can be observed thanks to the delay or even a suppression of the nucleation process.

The energy partitioning previously presented gives some additional understanding of this phenomenon from a micro-mechanical point of view (and without fluid). It helps to identify the slip stages where the dilation energy is higher and how it interacts with frictional and damage energy, Figure 121. Comparing the scale of each energy in Figure 120, dilation is indeed the most influential factor, while the other energies do not exceed 2 kJ/m². This is not a new observation as dilation is well understood as a primary factor influencing the energy balance in granular shear.

In the representation of [START_REF] Caniven | The Role of Along-Fault Dilatancy in Fault Slip Behavior[END_REF], the expansion of dilation energy implies the storage of more elastic strain energy increasing the radiated energy, which may lead to fast earthquakes and instabilities. According to them, high roughness and asperities within the fault slip plane obviously reduce the contact area between the two walls, enhancing dilation and an important fault strength reduction with instability conditions ( 𝑐 > ). In our results a Chapter 5. From energy budget to simplified models, towards dynamic simulations correlation between the inner fault plane roughness and asperities, the rapidity of stress drop (𝐷 ), and the dilation energy was observed (Chapter 3 and Part A). For mid-dense materials, increasing the cementation increases the dilation and total breakdown energies and decreases the critical distance of sliding 𝐷 (i.e. increase 𝑐 ). Through energy observations and previous dilation results of Chapter 3, behaviors of dense materials can be summarized in Figure 122. The observed cohesive agglomerates formed within the dense cemented samples create large contact asperities within the fault plane, drastically enhancing the dilation energy and reducing the critical slip distance needed for the stress drop. These materials could enhance regular to fast earthquakes, based on the representation of [START_REF] Caniven | The Role of Along-Fault Dilatancy in Fault Slip Behavior[END_REF], due to its higher dilation energy that is likely to give a much higher slip velocity response to sliding. However, for dense and ultra-cemented samples, the contact plane was closer to a particle-plane asperity contact (a highly cohesive band with micro-roughness), the actual dilation energy is thus decreasing in comparison with cemented samples as observed in Figure 120 (b). This is also consistent with the global dilation/contraction ratio of the gouge [START_REF] Caniven | The Role of Along-Fault Dilatancy in Fault Slip Behavior[END_REF]. For this cemented material, the ratio is decreasing at the end of the observed slip as a very contractive behavior was observed. Even though the effective friction peak is higher for these ultra-cemented materials, the resulting breakdown energy and, by deduction, the elastic energy storage is smaller than for cemented materials, probably due to the change of roughness created by cemented particles. In comparison, for poorly cemented materials, the dilation begins in the pre-e k e or " re-e k" dilation energy, Figure 121 (a.1) & (b.1). In this case, the system has already been set into motion before reaching the breaking point, leading to a drastic reduction of the breakdown energy (post-peak) dissipated by the system which does not need much dilation energy to slide. Frictional energy is also reduced (with some negative values, Appendix 5.C) as all particles slide together with no significant shearing resistance. The observed variation of effective friction, Figure 119 (a) is diffuse and oscillating as observed by [START_REF] Caniven | The Role of Along-Fault Dilatancy in Fault Slip Behavior[END_REF] for this kind of asperity contact leading to a rather slow sliding. The recent study by [START_REF] Caniven | The Role of Along-Fault Dilatancy in Fault Slip Behavior[END_REF] is relevant to this topic, although it does not consider gouge material. Here the cementation adds a different mechanism that slightly modifies the rheological behaviors they observed. Hence, even though dilation can prevent an earthquake to happen [START_REF] Aben | Dilatancy stabilises shear failure in rock[END_REF], it is also related to the unlocking of an asperity and can From energy budget to simplified models, towards dynamic simulations drastically accelerate the slip when the fault is relocking. With more time, it would have been interesting to observe these phenomena in detail and to make similar simulations with compliant walls to represent locally the dilatancy changes in the model. Another perspective would be to study the different asperities, the agglomerates formed, due to the cementation and track their shape and evolution as a function of slip to be able to observe their link with the dilation energy.

Double weakening shape and stress excess

What are the mechanisms acting on the two weakening peaks? Chapter 3 showed cemented and ultra-cemented materials presenting a double weakening shape. In this section, another understanding of this double shape is brought thanks to energetic studies Figure 121 (b2) and ( b3).

-The first weakening period depends on both interparticle friction, cement breakage, and dilation, and generates high frictional contacts between the few grains which are not cohesive anymore. The energy generated by this first peak strongly increases with the cementation which reduces the number of particles that can move in relation to each other, and mostly corresponds to frictional contact energy, Figure 121 (b). If there was temperature control within the simulations we could have observed higher temperature for this slip, which could be able to change the fabric inside the fault and modify gouge behavior. But the question remains, is the first weakening important within the behavior of the fault? The e r re kdow work o ed " re e e " [START_REF] Kanamori | The physics of earthquakes[END_REF]. This first weakening is important in the nucleation of the fault since it controls the initiation of the rupture, but its energy is, most of the time, not considered because the amount of slip during this high-stress stage is small so that little energy is involved. Similar results were found for the cemented models: the value of the peak frictional resistance is significant, however, its energy value is low compared to that of the second weakening.

-The second weakening period is mainly due to dilation mechanisms and more precisely corresponds to the weakening part of the dilation friction. It ends with the end of the dilation e er r o orre o d o e " e " zo e o o e d o phenomenon that produces this weakening but rather its progressive disappearance. However, the existence of this second peak needs a previous dilation. This energetic separation in two phases adds a new key of understanding for weakening mechanisms.

This double weakening has already been observed in shears with dynamic weakening, a first weakening due to mechanics and friction and a second one which can be generated by thermal effects. However, our models do not consider the temperature, this mechanism is purely due to the addition of cementation, combined with a very dense initial model.

Recent research on the fracture process tries to understand how and to which extent the frictional rupture can be viewed as an ordinary fracture process [ [START_REF] Barras | Emergence of Cracklike Behavior of Frictional Rupture: The Origin of Stress Drops[END_REF], [START_REF] Barras | The emergence of crack-like behavior of frictional rupture: Edge singularity and energy balance[END_REF], [START_REF] Paglialunga | On the scale de e de e e d o r o r re o r re e er er ze-dependent breakdown work[END_REF]]. According to them, the friction scaling laws previously presented in Chapter 1 need to be used with care: (i) the residual stress 𝜏 1 (assumed to be an interfacial property) o ed and has been shown to depend on the external applied stress, on the properties of the bodies surrounding the interface, and on the rate dependence of the frictional behavior, (ii) the effective fracture energy G c, calculated from crack-like theory, is not representative for all the breakdown energy dissipation E G (in excess compared to G c ). This result could explain the two-step weakening process: sharp stress drop that takes place over a relatively small slip followed by a slower and longer-term process bringing the friction to its residual value.

Conclusion of Part B

This part takes up the mechanical results obtained in Chapter 3. The objective was to study the behavior of cemented gouges through an energetic approach as it appears that cementation was the most influential parameter on breakdown energy in Part A. To do so, we decomposed the friction and energy variables into three main mechanisms: dilation, friction between the particles and breakage of the cemented links. Through this decomposition, the first objective was to propose a simplified friction model 𝜇 , as the sum of the three contributions: 𝜇 = 𝜇 𝑑𝑖 𝜇 𝑑𝑎𝑚 𝜇 𝑓 . The advantage of this decomposition is to be able to locate the different mechanisms over time and see their implication in the evolution of the fault gouge. This same decomposition was performed from an energetic point of view, on the breakdown energy E G . This breakdown energy is mainly composed of the energy related to the dilation of the sample, which always has a preponderant role in the mechanisms of rupture and cracks propagation. We could observe that, according to the initial porosity of the sample or the percentage of cementation, the influence of the dilation is not similar and does not induce the same mechanical behavior. The mid-dense materials have a rather linear increase of the breakdown energy (and of the energy necessary to dilate the sample) whereas the behavior of the dense materials is more complex. Indeed, the material most likely to generate sliding instabilities appears to be dense cemented materials (between 25 % and 75 % cementation). The agglomerates of cohesive particles generated in the gouge make the fault rather heterogeneous and increase significantly the roughness in the fault, which enhances the dilation energy more significantly and implies a much higher slip velocity response.

A double weakening was observed for dense and highly cemented material, although no temperature variation is present. The first weakening is a mixture of the three mechanisms (dilation, friction and cement breakage), but with an increase in frictional energy with the increase in cementation. The reduction of the number of particles in contact, coupled with the shear rate applied to the gouge increases the friction between particles. On the other hand, the observation leads to a second weakening mainly due to the decrease of dilation, which progressively weakens the effective friction.

Simplified models proposed are pretty convincing about the final shape and could be adapted to a variable imposed stress and velocity if properly fed with additional simulations. However, the characterization of mid-dense cemented models could be improved because the total breakdown energy does not fit very well with the data observed.

Part C -Time and space evolution of R-bands in a dense granular material 5.9. Foreword

The previous Chapter 4 revealed many interesting behaviors of fault gouges, but further analyses can be done on Riedel bands and their link to deformation evolution and stability of the system, Figure 123. First of all, it is possible to observe both global and local variations of Riedel band activation. The orientation angle of these bands can be different depending on whether the observed zone is in the center of the gouge or close to rock walls (Section 5.10.1). Then we can wonder about the temporal and spatial evolution of sliding in the Riedel bands, focusing on the local variation of each band. When, where, and how does each band appear? (Section 5.10.2). Finally, we will investigate the role that these bands can have on the entire behavior of the gouge and try to connect the mechanical behavior inside each band to the entire gouge kinematic (Section 5.11). Section 5.11.3 is a proposed energetic interpretation composed by the behavior of each R and Y-band. progressively until steady-state). 𝑅 represents the conjugate Riedel bands appearing during the slip, but are only indirectly quantified here.

Method and first results

The objective of the proposed method is to recover precisely the time evolution of R-bands thanks to the opening of each shear band in real simulation time. It consists in extracting particles' positions from each side of a R or Y band, and deriving displacement jumps in the directions perpendicular and tangential to the band orientation, for different total slip values (i.e. different time steps). Particles A and B are two particles selected from each side of the band 𝑅 1 , Figure 125.

The variation in the 𝑅 -direction, ∆Y op n , characterizes the opening of each band and corresponds to the relative displacement between particle A and B through time and space. It remains different from the actual thickness of the R-band, 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 , observed on the pictures. However, ∆Y op n and 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 are linked together and determining ∆Y op n is an easy way to understand the evolution of 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 . The method is detailed in Appendix 5.D. The horizontal and vertical displacements of each band (in the local reference frame ( 𝑅 ⃗⃗⃗⃗ , 𝑅 ⃗⃗⃗⃗ )) are spatially interpolated at different times of the simulation on each side of the band (Appendix 5.D). The ∆Y op n is then calculated for the interpolated couples of points along each R-band. It can be observed that each band evolves differently from the others in terms of space and time (i.e. total slip distance), but they can be gathered in three groups, Figure 126 and Figure 127: -For Primary R-bands 𝑅 3 and 𝑅 4 , the ∆ 𝑖 (total slip within each band), and the ∆Y op n (total opening of each band) mostly increase before the effective friction peak of the entire gouge (dark blue lines). Past this point, both quantities seem to reduce until a final state in red. Interestingly, a reduction of ∆ 𝑖 implies a negative 𝑖 which means that grains are moving backward in the weakening part within these bands. The 𝑅 bands, not represented here, evolves similarly.

-Secondary R-bands 𝑅 1 and 𝑅 2 evolve differently from the primary bands. Both the ∆ 𝑖 and the ∆Y op n increase until steady-state where they stabilize. The total slip ∆ 𝑖 observed within the secondary R-band is almost five times higher than the one observed for primary bands and corresponds to the size of five particles (~100 𝜇𝑚). It means that particle sliding within these bands is more important than inside the Primary R-bands, which may explain their longer duration and higher importance within the entire slip. The total -Y-boundary shears present interesting patterns with a slightly higher ∆ 𝑖 for 1 than for the other bands. This is because this is the principal shear plane and we consider the upper rock wall as a reference, which is moving with a constant slip velocity of 1 𝑚/𝑠. Local behaviors also appear on each ∆Y op n variation as a really nonlinear opening. Besides, there is clearly a progressive activation of Y-bands starting from the point of connection of 𝑅bands: the very regular slopes appearing in figure Figure 127 (a) & (b) for ∆ 𝑖 correspond to a characteristic deformation, which can be related to stress through Young's modulus, or simply mentioned as a maximum compression deformation. 1 presents two major slopes between the top connection points of 𝑅 1 and 𝑅 2 (slope of about 1.5 %) whereas in the ∆ 𝑖 of 2 we can clearly identify four connection points of primary and secondary R-bands (slope of about -0.5 %). In biaxial experiments, the maximum principal stress was observed for a vertical shortening equal to 2% for a confining stress of 40MPa, meaning that for both Y-bands the rupture of the material is not reached. Results are close to the Coulomb-Mohr rupture criterion in the vicinity of the R-band (and to the rupture within it). Chapter 5. From energy budget to simplified models, towards dynamic simulations For the 𝑅 -bands evolution, we also suppose that the peaks or troughs observedwhen the final value is smaller than the previous value during the simulationcorrespond to a change of orientation of Riedel bands. Indeed, in Chapter 4, for the two secondary 𝑅bands 𝑅 1 and 𝑅 2 , we remark that the orientation angle was changing at the boundaries, meaning that the 𝑅 1 band has at least three different orientations. The proximity to rock boundaries seems to diminish the Riedel band angle that becomes part of the next Y-bands formed. This assumption seems interesting, but a validation of this theory, also mentioned in [START_REF] Hirata | Shear zone development and frictional instabilities[END_REF], would need further research.

Time evolution of R-bands

Opening of R-bands through time

Thanks to the two variables ∆ 𝑜 𝑒𝑛 and ∆ 𝑖 , it is now possible to know the variation of each band in space (local evolution on the total band thickness) and time (evolution as a function of the total slip of the entire gouge). In order to see the evolution through time of the opening of each band, an average value ∆ 𝑜 𝑒𝑛-𝑎𝑣 (of ∆ 𝑜 𝑒𝑛 ) is computed on the total length of each band. A component of ∆ 𝑜 𝑒𝑛-𝑎𝑣 represents the average opening for a specific slip distance of the total fault. The evolution of ∆Y op n-v (in meter), Figure 128 -Primary bands (𝑅 3 , 𝑅 4 and 𝑅 ) grow faster than secondary R-bands from A to A'. They are in their maximum opening just before the effective friction peak (A' to B) and then start to disappear giving way to the secondary R-bands which are still opening. Their orientation angle is also higher (~15° for primary bands and ~11° for secondary bands). Primary bands 𝑅 3 & 𝑅 4 clearly present the smaller ∆ 𝑜 𝑒𝑛-𝑎𝑣 , confirming their less important role in the gouge history. In Chapter 4, we postulated that only secondary R-bands and Y-bands were present in the steady-state, however, this representation lets us think that although they are almost not visible on our previous images, primary bands are still present in the model at steady-state, Figure 128. -The effective friction peak seems to be reached when the opening of each secondary band (𝑅 1 d 𝑅 2 ) exceeds the maximum opening of the primary bands. However, the maximum opening of secondary R-bands appears during the slip weakening phase, where the dilation of the gouge has reached its maximum value (grey curve). Then they slightly diminish until the steady-state where they stabilize. -Finally, the evolution of the Y-d or o d we " e e r zo e" described in [START_REF] Bedford | The Role of Grain Size and Effective Normal Stress on Localization and the Frictional Stability of Simulated Quartz Gouge[END_REF], is quite different as they slowly grow from the end of the elastic partwhere the still inconspicuous R-bands connect to the rock wallsto the steady-state, where all bands opening stabilize. While 1 tends to a ∆ 𝑜 𝑒𝑛-𝑎𝑣 similar to secondary bands, 2 approaches the behavior of primary R-bands. This different behavior is probably due to the one side shearing.

Three remarkable points indicate a change between active (i.e. visible or with high opening) and inactive bands (i.e. non-visible or with low opening): point B, where primary bands become active (friction peak), point C, where 1 become active opening more than primary Riedels (small friction plateau), and points E where the opening of 1 as reached the opening of 𝑅 1 (steady-state). As previously presented, high angle Riedel bands R' exist, but they are too thin and too numerous to be measured and quantified with this method (Chapter 4). Chapter 5. From energy budget to simplified models, towards dynamic simulations

We notice that at steady-state, the sum of each ∆ 𝑜 𝑒𝑛-𝑎𝑣 is not equal to the total ∆ 𝑜 𝑒𝑛 (i.e. dilation of the gouge), this is not surprising as the orientation of the opening of each band is different, which makes this addition not directly possible. Moreover, there are certain areas of contracting behavior that counteract these values, and these are "averaged" values that do not reflect the heterogeneity of each band. This representation of Riedel band opening is pretty new in the literature, as it is complicated to measure, quantify and follow R-bands through lab experiments. Moreover, the distinction into two kinds of R-bands (primary, secondary) is not very common but makes sense after analyzing the ∆Y op n evolution and images of each R-band. The R-bands evolution in the prepeak zone is scarcely investigated in the literature [ [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF], [START_REF] Kenigsberg | The Effects of Shear Strain, Fabric, and Porosity Evolution on Elastic and Mechanical Properties of Clay-Rich Fault Gouge[END_REF], [START_REF] Bedford | The Role of Grain Size and Effective Normal Stress on Localization and the Frictional Stability of Simulated Quartz Gouge[END_REF]], but thanks to this representation, it is possible to see that these two types of bands (R and Y) start to appear just after the end of the elastic part and before the effective friction peak. The absence from the literature of Riedel bands within the pre-peak zone can firstly be explained by the fact that the effective friction peak is not very sudden (contrary to other studies) and allows a gradual growth of the band opening. Moreover, at 𝜇 the 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 is not as its maximum, and it is possible that R-bands existence before the peak was not detected in other studies because of this low thickness, and simply neglected in the rest of the study. [START_REF] Kenigsberg | The Effects of Shear Strain, Fabric, and Porosity Evolution on Elastic and Mechanical Properties of Clay-Rich Fault Gouge[END_REF] studied a Clay-rich fault gouge with different material characteristics which also may lead to a different evolution of R-band within the gouge that the one observed here.

Opening and Riedel thickness

As the ∆ 𝑜 𝑒𝑛 does not provide the real thickness 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 of R-bands, it has been measured manually with the ImageJ software (Appendix 5.D), giving an overall trend similar for both variables. The 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 values are not very precise due to artifacts between the filter chosen for the solid fraction observed, image saturation, and the difficulty to measure manually such a small distance. The values found for 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 (100 -300𝜇𝑚) are consistent with values found in previous studies for a similar thickness of initial gouge [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF]. A factor of ten was also observed between the values of ∆ 𝑜 𝑒𝑛 and 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 measured on the images, but additional measurements on other gouges would be necessary to confirm if this order of magnitude is always true. We did not explore these values, but as the global trend is consistent with ∆ 𝑜 𝑒𝑛-𝑎𝑣 , it confirms that ∆ 𝑜 𝑒𝑛 gives a consistent evolution of the opening of R-bands.

e o o w e w e e d e er o μ (M-PS-40) were also carried out. The maximum opening for one band corresponds approximately to the size of one particle, and we do know that there is a link between the number of particles within the thickness and global shearing and dilatancy behavior (Chapter 4). The size of particles does not have a huge relative impact on the ∆ 𝑜 𝑒𝑛 at steady state both on total gouge opening or secondary Rbands opening, Figure 129 (a). For both sizes of particles, the opening of one R-band represents between 30-40% of the total opening of the gouge at steady-state, except for the primary bands that are only 15% of the total opening at steady-state, Figure 129

Friction into R-bands

To link the mechanical behavior of each band and that of the entire gouge, we decided to go further in the understanding of both local and global mechanisms. This section briefly presents the first analyses and observations leading to the objective of this chapter on simplified models: to isolate specific laws for each R and Y-band (friction/dilatancy) and to try to relate it to the global behavior of the entire gouge.

To isolate the o e or o e d e de w r o ode "Thin Bands" o gouge material, with a thickness similar to the one observed in R and Y-bands. The objective of these TB is to simulate the mechanical behavior (friction/deformation) of a R-band within the gouge. The chosen thickness must be low enough to avoid any development of other inclined Riedel bands, but not too small to avoid an excessive increase in friction (because if it is narrower than a natural shear band, friction is overestimated). Previously observing that the thickness of a Y-band (or R-band) was composed of around 10 particles, we simulated thin gouges of a 10-particles thickness (similar to an initial band thickness 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 = 200 𝜇𝑚) with the same physical and numerical characteristics as the entire gouge, to have the local mechanical and kinematic behavior of one single band, Figure 130. However, the TB model is displacement-driven, meaning that the whole length of this gouge model slides simultaneously. This is not the case of the individual shear bands previously described (either R or Y) in the full gouge model, which develops spontaneously following complex histories, with a sliding distance varying along with their spatial extension at a given time. Hence, the TB model should be considered as an average approximation of the local behavior (i.e. at a given point) of any R or Y band. The friction curve and dilation of all thin bands generated can be observed in Appendix 5.E. resulting thin band opening is consistent with the ∆ 𝑜 𝑒𝑛 variation of the Riedel bands found in the previous part (~ 2.10 -𝑡𝑜 3.10 -𝑚). This result confirms that the thickness chosen for the thin gouge is appropriate and gives good hope to find some convolution between all the Rbands and the overall behavior of the gouge. It raises two interesting questions: if the w hole fault is concerned with the sliding of a certain number of R and Y bands, why is the maximum fault friction lower than that of each shear band? And why is its breakdown energy larger?

Figure 131. (a) Effective friction 𝜇 as a function of the slip distance for the standard case (black pointed curve) and the associated thin gouge (blue line). (b) Opening of the gouge in the -direction (∆ 𝑜 𝑒𝑛 for the entire gouge) as a function of the slip distance (m) for the standard case (grey pointed curve) and the associated thin gouge (blue line).

Trajectories and opening of R-bands

We superimpose the Y-opening observed in Figure 131 The Y-band has a major role in global behavior with an opening almost similar to the one of the thin gouges. We can add to this the evolution of the bulk deformation (grey cross) which is the deformation of the sheared granular flow, following the slope of the "total" opening of the gouge (which is the displacement of the upper rock wall in the -direction, direction perpendicular to the fault gouge). (Y. [START_REF] Katz | Strain localization in sandstone during embryonic stages of shear -zone evolution[END_REF] suggested that the granular flow is the dominant deformation mechanism at the early stage of the movement, and progressively gives way to Riedel structure as the dominant mechanism of deformation. Our x 10 - Chapter 5. From energy budget to simplified models, towards dynamic simulations results seem to confirm this theory but will require separating the dilation and friction from each mechanism to understand their role.

To

Energetic interpretations

Now that the mechanical behavior of each band observed in the model is known, we propose to gather all their contributions into a single energetic interpretation. This small exploratory part is also based on energetic distributions.

Proposed method

Based on the previously defined rupture energy, the total energy 𝐸 𝑜 spent by the fault at a given sliding distance 𝑈 can be linked to the observed friction and the normal stress applied to the fault:

𝐸 𝑜 (𝑈) = ∫ 𝜎 𝑁 𝑈 0 𝜇 (𝑢) d𝑢 (5.11) It can also be written as a small increment of energy d𝐸 𝑜 spent for a sliding increment d𝑢, such that: d𝐸 𝑜 = 𝜎 𝑁 𝜇 (𝑢) d𝑢, with 𝜇 (𝑢) the current effective friction of the fault gouge for a current slip 𝑈 (in the -direction of the fault), and 𝜎 𝑁 the normal stress applied to the fault.

𝑈(µm)

TB-S

M-S

𝑜 𝑒𝑛 (m)

1 𝑅 2 𝑅 1 𝑅 3 , 𝑅 4 𝑅 𝑢 𝑘
All these data have been exploited or calculated in Chapter 4 and Part A of this chapter so that they are all known.

The objective here is to reconstruct the effective friction 𝜇 observed on the entire fault with the 𝜇 𝑇 extracted from thin bands simulations. This reconstruction is done at each instant of the simulation, i.e. for each slip increment of the fault.

If we start from the small energy increment of the complete fault, we can write that the small increment of energy spent d𝐸 𝑜 is also the sum of all the incremental energies d𝐸 𝑇 spent in the Y and R bands present within the fault integrated on the total band length, plus some possible energy spent in the bulk of the gouge outside of the localization sites:

d𝐸 𝑜 (𝑈) = ∑ ∫ d𝐸 𝑇 (𝑈 𝑇 )d
𝐿 𝑇𝐵 𝑁 𝑏𝑎𝑛𝑑𝑠 (5.12) With 𝑁 𝑏𝑎𝑛𝑑 the total number of R and Y bands selected for the model (here 𝑅 1 , 𝑅 2 , 𝑅 3 , 𝑅 4 , 𝑅 , 1 and 2 ), 𝐿 𝑇 the length of each band and 𝑈 𝑇 the current slip inside each band.

d𝐸 𝑜 (𝑈) = d𝐸 𝑅1 (𝑈) d𝐸 𝑅2 (𝑈) ⋯ d𝐸 𝑌2 (𝑈) d𝐸 𝑘 (𝑈) (5.13) For example, the spent incremental energy on 𝑅 1 is

d𝐸 𝑅1 (𝑈) = ∫ (𝜎 𝑁 𝜇 𝑅 1 (𝑈 𝑅 1 ( )) d𝑈 𝑅 1 ( ))d 𝐿 𝑅 1
with 𝜇 𝑅 1 (𝑈 𝑅 1 ( )) the current effective friction on 𝑅 1 at the abscissa 𝑈 𝑅 1 and d𝑈 𝑅 1 which is the incremental slip on 𝑅 1 (corresponding to the previous ∆ 𝑖 calculated at each time step and for the different spatial locations). The d𝑈 𝑅 1 values come from previous calculations on each Rband and the 𝜇 𝑅 1 (𝑈 𝑅 1 ) comes from the effective friction extracted from Thin Bands simulations presented in 5.11.1.

First results

Figure 133 presents the effective friction of the entire gouge as a function of the slip distance and the energetic contribution of each band during the shearing, represented by a colored area under the friction curve. When the energetic contribution of each band is summed, it is observed that energy is missing in the pre-peak part. This energy is considered as a bulk energy from internal gouge deformation. This bulk energy is mainly active before effective friction peak and progressively disappears as deformation localizes within the different shear bands (mainly Rand Y-bands).

The energetic behavior obtained seems to be consistent with the previous results observed and reveals the importance of each band within the model, (numeration on Figure 133): i.

The energetic contribution of primary bands is observed at the beginning of the friction curve in the pre-peak part and is small compared to other contributions. Chapter 5. From energy budget to simplified models, towards dynamic simulations ii. Then comes the energy contribution of secondary bands 𝑅 1 and 𝑅 2 with maximum energy spent just after friction peak iii. At the middle of the weakening part, the energetic contribution of Y-bands becomes dominant (both for 1 and 2 ) and progressively reduces and stabilizes until the steadystate zone. iv.

The green energetic peaks may come from contractive bulk response which is not considered here. Further improvement could be done on that. v.

When the steady-state plateau is reached, the two boundary shears total all the spent energy of the system, suggesting that the rest of the gouge is stabilized. The secondary bands' activity (𝑅

Internal mechanisms

As for Part B, it is possible to calculate the part of released energy related to the dilation of the gouge (dilation energy 𝐸 𝐷𝑖 ), for the whole gouge, as well as for the different bands, Figure 134 (a). As there is no cementation in these models, the total remaining energy is considered as . Contrary to the behavior observed for cemented materials there is first a dilation contribution and then a frictional contact contribution. This seems logical as the very dense matrix gouge first dilates and as dilation decreases, frictional contacts are enhanced. As mentioned above, it is possible that areas of contraction, not considered here, can compensate for the excess energy caused by the expansion of the 1 and 2 bands observed, Figure 134 (a) zone i. These energetic models seem to confirm the importance of Y-bands in the development of fault zones. The sum of the energetic contributions of all bands is higher than the bulk energy participation. We also note the non-equal and non-simultaneous contributions of the primary and secondary bands: From energy budget to simplified models, towards dynamic simulations -The primary bands participate in the pre-peak phase for both dilation and frictional contributions. If the R'-bands were considered, it is assumed that they would play in this same zone. These bands are a precursory event in the life of a fault and could characterize the type of gouge material studied. Interestingly these primary bands did not appear for cemented material. It could be explained by the longer pre-peak zone observed here, and we suppose that they are characteristic of dense and non-cohesive material. Cemented materials with their brittle character directly switch to secondary R-bands. -The secondary bands and Y-bands seem to always form for dense materials (cemented or not) starting from the pre-peak zone and growing until the beginning of the weakening zone.

Comments on unanswered questions

-Why is the effective friction peak of the complete fault lower than that of the individual thin bands? It is a question of lack of synchronicity. Each R-or Y-band reaches its friction peak at different times, from one band to another, but also within each band individually (by the spatial variability of slip on each of them). As they do not synchronize to reach their own peak together, the peak of the entire gouge is thus lower than the individual peaks. -Why is the energy of the complete fault much higher than that of the thin bands? It is here a question of chronology. If there was only the 1 -band, there would be a perfect equality between the two energies. However, within the real simulated gouge, before activating this

1 -band and sliding on it alone, the fault takes lots of energetic dead ends: primary bands, then the secondary bands, the second 2 -band, and the bulk deformation. All these detours consume energy unnecessarily and contribute to increase the breakdown energy of the fault. In the end, these patterns (i.e. Riedel bands) lead to (i) reduce the intensity of the friction peak and (ii) increase the breakdown energy: consequently, it reduces considerably the weakening rate (i.e. the downward slope of the friction curve): it has, therefore, a powerful stabilizing effect on the fault.

In future studies, we plan to perform these same studies on several models presented in Chapter 4. The idea is to be able to observe different energetic behaviors depending on the parameters of the gouge and insert these models in dynamic modeling. Interparticle friction or stiffness may have a non-negligible influence on the link between local and global behavior. Another good reason to expand this study is to understand why the fault takes these different kinematic paths and be able to model them in order to write a friction law of the whole fault based on the energetic decomposition proposed.

Conclusion of Part C

This last section goes further in the analysis of Riedel bands and their link to deformation evolution and energy release of the system. First of all, it is possible to observe both global and local variations of Riedel band opening and sliding. The orientation angle of these bands can be different depending on whether the observed zone is in the center of the gouge or close to rock walls. Then, we observed the temporal evolution of Riedel bands with slip distance, focusing on the global opening variation of each band. Primary bands and secondary bands appear together, but primary bands grow faster, giving spaces to secondary bands at friction Finally, we wondered about the role that these bands might have on the global behavior of the gouge and tried to connect the mechanical behavior inside each band to the entire gouge kinematic. An overview of the method is proposed with energetic description and proper relations between each band and the gouge. The energetic contribution of each band agrees with their time and space location but gives another way to observe Riedel mechanisms and to conclude on their role and activity.

Although some simplifications and research are still missing to have a better energetic model, this little simplified model is very promising to study the Energy budget and the role of each Riedel band within the weakening mechanism as a function of the characteristics of each infill material.

Chapter 6. Conclusions & Perspectives

Conclusions

The fault gouge, coming from the wear of previous slips, acts on friction stability and plays a key role in the sudden energy release. A large part of slip mechanisms is influenced, if not controlled, by fault gouge characteristics and environment. During this thesis work, 2D Discrete Element Modelling was performed for a better understanding of (i) how infill materials (matrix or cement) participate in the strengthening or weakening of the fault, (ii) the rheological behavior observed highlighting shear band formation, and its link with gouge characteristics, (iii) the contribution of each of these mechanisms in the observed breakdown energy and friction laws. Three main types of gouge samples were generated to simulate a mature fault gouge with infill material and then inserted between two rock walls to realize direct shear experiments using the software MELODY2D (Chapter 2). A dry contact model was considered to investigate mechanisms without fluid (displacement-driven and under constant confining pressure). Numerical simulations were performed and the associated analyses provided answers to the questions raised in the General Introduction.

• What role plays cementation in the weakening of mature fault gouges? Do cemented materials promote slips in faults? A first fault gouge model was implemented with a cohesive law to mimic the cementation within the gouge (Chapter 3). Both initial porosity and surface percentage of cementation influence gouge weakening and slip behaviors, playing a role in the gouge strength (brittleness, cohesion) and granular flow (particles agglomerates, Riedel bands). High cementation within materials increases the shearing resistance, limiting in some way the number of slips that could take place in the fault, given that high stress is needed to observe a nucleation. However, for these materials, the observed stress drop is much larger and also steeper (sharp, short, and intense peak), which means that if the shear stress peak is reached, the observed slip is more likely to occur in an unstable manner and with high energy release. Both the critical distance of slipping and the nucleation length decrease with an increase in cementation (Chapter 5), confirming that not much mobility is allowed within the system, and increasing the risks of instabilities to happen. Riedel bands, which are key elements in the understanding of stability in faults, are significantly affected by a change in contact laws between particles (cohesion or friction) occurring with cement breakage. They only appear for dense materials when a threshold of cementation is reached and are directly linked to the importance of the dilation phase since the majority of the gouge remains cemented even after the weakening phase. Low cemented materials follow a rather ductile behavior without observation of apparent Riedel bands inhibited by a critical dilation higher than 1%, whatever the tested samples are dense and mid-dense. The whole gouge area is sheared with low fracture energy and a low dilatancy.

Although Riedel bands are not sufficient to detect unstable behavior, they are nevertheless necessary conditions, leading to the conclusion that dense and highly cemented materials are more likely to generate unstable behaviors, while mid-dense or dense materials with very low cementation will rather tend towards a ductile, and stable behavior.

How can the energy balance of a cemented gouge be decomposed and what mechanisms remain most influential?

The effective friction was decomposed into three main mechanisms: gouge dilation, friction between particles, and breakage of the cemented bonds (Chapter 5, Part B). Using this decomposition, a simplified friction model is proposed as the sum of these three contributing frictions. The advantages of the proposed model are its ability to locate the different mechanisms over the fault displacement and quantify their relative contribution to fault gouge weakening. The same decomposition was performed from an energetic point of view, on the breakdown energy. Results obtained during simulations agree with previous research and highlight that the dilation phenomenon is the main mechanism involved in the breakdown energy, even for cemented gouge. Although mid-dense materials have a rather linear increase in both the breakdown energy and the necessary energy to dilate the sample, the behavior of dense materials is more complex. As previously stated, the increase in cementation has a general tendency to promote unstable behaviors, but materials most likely to generate sliding instabilities are not the most cemented material but the mildly-cemented ones (between 25% and 75% cementation). Indeed, the agglomerates of cohesive particles generated within the gouge make the fault rather heterogeneous and increase the roughness within the slip surface, which has the effect of significantly increasing the dilation energy. This could generate a much higher slip velocity response during sliding.

The energetic decomposition performed also allowed us to understand the reasons for a double weakening observed for dense and cemented materials. Effective friction curves present double weakening shapes for dense samples with a sufficiently high percentage of cementation (to create an internal cohesion higher than 1 MPa or 2.5 % of the applied normal stress). The first weakening is a combination of the three mechanisms (dilation, friction, and cement breakage), and the first corresponding stress peak increases with the increase in cementation (proportionally to the peak of frictional energy). The reduction of the number of particles in contact, coupled with the shear rate applied to the gouge increases contact friction for these particles. According to our observations, a second weakening is observed, mainly due to the decrease of dilation energy, which progressively weakens the effective friction.

• What role plays matrix particles in the weakening behavior of mature fault gouges?

How important are the different physical and geometric characteristics of matrix particle s in the slip behavior and in Riedel bands formation and evolution? To answer this question, another gouge model was implemented in DEM, with a very dense sample composed of hexagonal-shaped particles representing matrix grains. The variation of the different geometrical (shape, size, and distribution of the grains) or physical (stiffness and interparticle friction) properties of the matrix represents different rock lithologies, allowing to observe a large variety of rheological behavior (Chapters 4 and 5). It appeared that all these lithologic variations could be summarized into only three parameters: (a) the bulk shear modulus, (b) the ratio of gouge thickness to grain size (i.e. number of particles within the gouge thickness), and (c) the interparticle friction.

A fault gouge with a small interparticle friction coefficient, a low number of grains in the gouge thickness, or a high shear modulus will, in all three cases, tend to decrease the orientation angle of Riedel bands. This small angle is partly due to the presence of a Riedel structure with a large number of R' -bands (conjugate Riedels) inhibiting the formation of the R-band and delaying the arrival of a steady-state. However, their consequences on gouge strength, nucleation length, and breakdown energy are not similar and lead to different types of sliding behavior. Materials with both a low shear modulus and low interparticle friction, also show a reduction in the maximum frictional resistance. These materials are close to those defined as weak gouge materials [ [START_REF] Collettini | Beyond Byerlee friction, weak faults and implications for slip behavior[END_REF], [START_REF] Bedford | Fault rock heterogeneity can produce fault weakness and reduce fault stability[END_REF]], with a rather ductile behavior that tends to increase both the critical slip distance and the critical nucleation length. Greater mobility is thus allowed within the fault, and the associated breakdown energy remains low, as for weak cemented materials previously studied. The very low shear modulus combined with large regions of nucleation is more likely to tend towards slow slip nucleation [START_REF] Leeman | Laboratory observations of slow earthquakes and the spectrum of tectonic fault slip modes[END_REF]. On the contrary, for low shear modulus with increasing interparticle friction (ex. mineral transfor o e r r e d r e ro e … o e or e o angle of Riedel bands and the number of Riedels observed are increasing. The critical slip distance is reduced (as the critical nucleation zone), leading to a more sudden weakening, which is prone to switch the fault behavior from a ductile aseismic response to a brittle seismic slip, depending on the stiffness of the surrounding medium. No more Riedel structure is observed. Materials with high shear modulus (ex. increase in interparticle stiffness) produce a higher resistance to slipping, giving a shorter nucleation length and slightly higher fracture energy. A rather strong, but unstable behavior is observed. For a reduction in the number of particle s in the gouge thickness (ex. increase in the size of particles, reduction of the gouge thickness) , the observed breakdown energy increases as well as the critical nucleation length. The mechanical behavior tends to the one of weak materials, but with a material that struggles to stabilize over time, explaining the higher fracture energy observed. But it also means that this long and smooth slip weakening decreases the probability of instabilities to occur.

To which extent is there an influence on the proportion of matrix/grain and their distribution in the fault gouge samples? Direct shear tests conducted on a composite mixture between matrix and angular particles showed that the percentages of both matrix and angular particles, and initial porosity participate in the mechanics and rheology of fault gouge (Chapters 4, and 5 Part A). Mixing angular grains with matrix grains strengthens the gouge regardless of the granular distribution in the sample. Indeed, gouges composed of a single type of material have a lower frictional resistance, partly due to a higher initial porosity, but also to a change in the distribution of contacts at the granular scale.

The most slip-resistant material appears to be the one with the lowest percentage of the matrix but in which the porosity is close to zero. It is therefore a material with a very high shear modulus, implying a reduction of the orientation angle of the Riedel bands and the formation of a more important Riedel structure. An increase in fracture energy is observed with decreasing matrix percentage as angular particles inhibit Riedels to join the Boundaries of the gouge and to form Y-bands, leading to a deformation of the entire gouge. However, inhomogeneous and fractal grains distribution (angular particles) seems to favor a smoother and stable slip weakening than homogeneously distributed clasts which promote a very sudden friction drop, very likely to generate seismic instabilities.

How the behavior of each Riedel shear band observed in the fault gouge can be related to the overall behavior of the gouge? To answer this question, the evolution of each Riedel band' thickness was followed over time and space (Chapter 5, Part C). For this purpose, the variation of the band opening was measured during the shearing process at different points of the Riedel band, giving a glimpse of both global and local variations of the band opening. Two ed " e o d r ede " (with slightly higher angle orientation). When secondary bands reach their maximum thickness, they de-activate the primary bands, which are no longer visible. These secondary bands persist during the whole shearing process. The Y bands, on the other hand, evolve progressively from the end of the elastic phase and until the steady-state is reached, where they become the only "active" bands (localizing particle displacements).

Finally, the mechanical behavior inside each band has been connected to the entire gouge kinematic. To do so, the mechanical behavior of a single Riedel band was needed (friction and deformation). By shearing a gouge with tens of particles, inhibiting other Riedel formations in the Riedel, we model a one Riedel band behavior enabling to recover the associated friction and dilation. Each friction recovered from a single Riedel can be transformed into an energetical contribution to the whole gouge. By adding each energetic contribution as a function of the fault slip distance, an energy distribution map is drawn as a function of the total slip distance. The energetical contribution of each band is in adequation with their time and space location but gives another way to observe Riedel mechanisms and to conclude on their role and activity. This energetic model is still to be improved but allows a new form of decomposition of the breakdown energy according to the different rheological mechanisms that take place during the slip.

• What friction laws can be used to model fault gouges with infill materials in dynamic models?

Classical slip weakening models were first applied to the friction curves obtained, intending to approximate behaviors by making them more tractable to use for understanding fault weakening (Chapter 5, Part A). The linear slip weakening model is not valid for all the granular samples tested, since it is not accurate enough for some granular materials, such as dense and highly cemented fault gouges that exhibit double frictional weakening. However, similar difficulties would have been encountered for a non-linear slip weakening model. A balance must be found between a too simple and a too complex model. For this reason, Chapter 5 proposes other friction and energetic laws linking local rheology and gouge mechanics. Simplified models proposed for cemented gouges (Part B) are pretty convincing about the final friction shape and evolution and the different mechanisms involved in gouge shearing. These friction laws could be adapted to a large range of imposed stress and velocity if properly fed with additional simulations. The energetic model used in Part C also intends to be easily integrated into a larger scale dynamic model in which one could specify the number of Riedel bands present and their category and thus be able to test a multitude of deformation cases at larger scales and especially with a thorough study of the seismic or aseismic behavior generated by these slip localizations.

These friction laws are primarily intended to be integrated into dynamic models at larger scales and are proposing a different view of friction laws that can be completed by rate-and-state laws widely used in fault mechanics.

Perspectives

Following the work done, several perspectives are proposed to this thesis either to improve our numerical models, to propose the next research steps, or to answer questions emerging from the Thesis work.

Improvement in the present research

-For several reasons explained in Chapters 1 and 2, the research carried out focused on a certain set of numerical experiments under a normal stress of 0 𝑀𝑃𝑎, a sliding velocity of 1 𝑚/𝑠, and an initial gouge thickness of 2 𝑚𝑚. It might be interesting to extend the numerical campaign to cover a wider range of normal stresses, velocities, and gouge thicknesses, in order to derive an empirical friction law based on the micromechanical properties of the gouge. -The friction law obtained for cemented gouge (Chapter 5, Part B) can also be improved by changing the parameters law for mid-dense and poorly cemented materials. Further simulations might be undertaken to fulfill this objective. Cement evolution mostly depends on reaction-diffusion equations mimicking the transport and precipitation of minerals (Magnus [START_REF] Wangen | Modeling porosity evolution and cementation of sandstones[END_REF]). An interesting line of work could be the investigation of the characteristic time of cementation in a fault and to evaluate to what extent the associated evolution of the fault strength interacts with its seismic cycle. -The detailed analysis of the evolution of Riedel bands during sliding (Chapter 5, Part C) has only been performed for one gouge sample. It would be interesting to perform the same post-treatment study on the other simulations with different kinematics (variation of friction or interparticle stiffness, grain size, gouge thickness...). The results will allow us to observe how each characteristic of the gouge will influence the opening and activity of the different Riedel bands. It will also enrich the proposed energetic model according to the characteristics of the gouge.

Study of the seismic and aseismic character of the tested granular fault gouge with dynamic modeling and earthquakes recurrence models

Now that we have proposed energetical models for the different fault gouge studied, it would be interesting to insert them into larger-scale dynamic models to study the seismic or aseismic behavior of these different gouge materials. These models would be very instructive to complete the information learned about infill materials (cement and matrix) at the millimeter scale and especially to test our friction laws on larger dynamic scales. This part will be the subject of an upcoming study.

The well-known Burridge-Knopoff model [ [START_REF] Burridge | Model and theoretical seismicity[END_REF], [START_REF] Stefanou | Control instabilities and incite slow-slip in generalized Burridge-Knopoff models[END_REF]], is a good starting point to quantify the seismic character of a fault. This model consists of an Ndegrees of freedom spring-slider system with specific friction laws at the sliding interface between blocks and sliding walls, Figure 135 (a). This model is very simple and allows a quick overview of the stability of the studied system as well as the prediction in time of slip events that can occur at the scale of the whole fault, Figure 135 (b). As a better perspective, we would like to turn to physics-based dynamic models of earthquake rupture that couple friction laws and the elastodynamics of coseismic off-fault damage and their effect on rupture dynamics, radiation and the global energy balance (Okubo et al., 2019a). The idea is to implement our different friction and energetic laws (Chapter 5) into such models and see if these laws have an impact on the entire behavior of the fault.

Heterogeneous cemented faults

Heterogeneous faults are known for their important role in friction weakening [START_REF] Collettini | Beyond Byerlee friction, weak faults and implications for slip behavior[END_REF]. A recent study by [START_REF] Bedford | Fault rock heterogeneity can produce fault weakness and reduce fault stability[END_REF] confirmed this result showing that heterogeneous faults (composed of quartz and clay) have a higher frictional weakening behavior and are less stable than their equivalent homogeneous fault gouges (composed of the same percentage of materials). In line with their work and our previous work on matrices and cement, it would be interesting to model a gouge composed of cemented and non-cemented materials

Weak gouge representation as a mixture of hard and soft particles

In our work, completely rigid grains are assumed for both matrix and angular particles. However, it is obvious that weak materials with deformability properties, such as clays or talc, have a significant influence on the weakening behavior of faults [ [START_REF] Haines | Clay fabric intensity in natural and artificial fault gouges: Implications for brittle fault zone processes and sedimentary basin clay fabric evolution[END_REF], [START_REF] Orellana | Do scaly clays control seismicity on faulted shale rocks?[END_REF], [START_REF] Kenigsberg | The Effects of Shear Strain, Fabric, and Porosity Evolution on Elastic and Mechanical Properties of Clay-Rich Fault Gouge[END_REF]]. It would therefore be interesting to start again from the composite mixture of grains and matrix simulation and to replace the matrix grains with deformable particles.

The software MELODY used for DEM modeling allows to deal with mixtures of both deformable and rigid grains and has already shown very interesting results for studies in the field of granular materials [ (Mollon, 2018b), [START_REF] Bouillanne | Detecting vorticity in cohesive deformable granular material[END_REF]]. These studies could be compared with results already observed in Laboratory earthquakes. They would also provide insight into the behavior of weak or deformable mineral inclusions in gouge rheology and their implication in slow slip events: the proportion of weak minerals going into Riedel band localization, their deformability rate, and also the stresses observed within each compliant particle.

Wear evolution during gouge shearing

Another interesting approach is to observe the influence of a change in wall roughness due to the wear of the rock during sliding. For this, it is possible to model a fault gouge in three parts: a continuous and rigid part representing the host rock, a gouge material already presents in the fault and a degradable part (pre-cut) which may separate into wear particles (i.e. rock grains), Figure 137. This model can be realized with a (CZM)-contact law, often used in tribology applications to consider the effect of the wear of a material. Thus, it will be possible to observe several interesting phenomena such as a variability in the wavelength of the walls and an increase in the gouge during the sliding which will be likely to modify the slip weakening behavior. The addition of deformable rock walls also allows to observe the associated stress field, but this will be very costly in computing time.

Temperature control and dynamic weakening

Temperature control in fault gouges simulations is a major asset since it allows to highlight melting zones that are responsible for both a change in material behavior and rheology as well as a weakening of the gouge [ [START_REF] Otsuki | Fluidization and melting of fault gouge during seismic slip: Identification in the Nojima fault zone and implications for focal earthquake mechanisms[END_REF], [START_REF] Niemeijer | Frictional melting of gabbro under extreme experimental conditions of normal stress , acceleration , and sliding velocity[END_REF]]. Recently temperature control has been instrumented in the DEM software MELODY, allowing to follow temperature evolution during sliding by temperature increment in grains (Mollon et al., 2021b).

It could be considered to add a module allowing to transform the initially rigid grains into deformable grains when the temperature becomes sufficiently high. The idea is to be able to observe the dynamic weakening effect, or secondary weakening observed with thermal activation.

Non-degradable rock wall (continuum) Degradable rock wall (discrete) Granular gouge

𝜎 𝑁

Damage

Appendices to chapter 1 A. Collection of Gouges

For the creation of the numerical granular samples, we relied on numerous images of fault gouge from natural faults [ [START_REF] Olgaard | The microstructure of gouge from a mining-induced seismic shear zone[END_REF], [START_REF] Rutter | Comparative microstructures of natural and experimentally produced clay-bearing fault gouges[END_REF], [START_REF] Sammis | The kinematics of gouge deformation[END_REF], [START_REF] Blenkinsop | Cataclasis and Processes of Particle Size Reduction[END_REF][START_REF] Chester | Intemal Structure and Weakening Mechanisms of the San Andreas Fault[END_REF], [START_REF] Antonellini | Microstructure of deformation bands in porous sandstones at Arches National Park, Utah[END_REF], [START_REF] An | Particle Size Distribution of Cataclastic Fault Materials from Southern or d Pure and Applied Geophysics[END_REF], (Antonellini & Mollema, 2002), [START_REF] Billi | The damage zone-fault core transition in carbonate rocks: Implications for fault growth, structure and permeability[END_REF], [START_REF] Billi | Fractal distribution of particle size in carbonate cataclastic rocks from the core of a regional strike-slip fault zone[END_REF], [START_REF] Billi | Grain size distribution and thickness of breccia and gouge zones from thin (<1 m) strike-slip fault cores in limestone[END_REF], [START_REF] Wibberley | Recent advances in the understanding of fault zone internal structure: a review[END_REF], [START_REF] Muto | Fractal particle size distribution of pulverized fault rocks as a function of distance from the fault core[END_REF]] and experimental results [ [START_REF] Sammis | Self-similar cataclasis in the formation of fault gouge[END_REF], [START_REF] Biegel | The frictional properties of a simulated gouge having a fractal particle distribution[END_REF], [START_REF] Marone | Particle-size distribution and microstructures within simulated fault gouge[END_REF], [START_REF] Marone | Frictional behavior and constitutive modeling of simulated fault gouge[END_REF], [START_REF] Beeler | Frictional behavior of large displacement experimental faults[END_REF], [START_REF] Mair | Friction of simulated fault gouge for a wide range of velocities and normal stresses[END_REF], [START_REF] Mair | Influence of grain characteristics on the friction of granular shear zones[END_REF], etc] as well as on the work done by Heiarii Pons during its internship, who gathered the following images and descriptions of granular fault gouges in Figure 138. This graph gave us an idea of the wide variability of matrix percentages found in the literature. The numbers on the graph correspond to the images below N° [X]. The particles are angular in shape, with fractal distribution ranging between 1.86 ≤ 𝐷 ≤ 2.92 and sizes between 17.5 ≤ ≤ 80 µ𝑚. Friable rocks and lack of cohesion. Images from [START_REF] Muto | Fractal particle size distribution of pulverized fault rocks as a function of distance from the fault core[END_REF]. [START_REF] Trouw | Atlas of Mylonites-and related microstructures[END_REF].), [N°22] Cenozoic cataclasis of granite, from the Cajon Pass dritlhole, southern California, with (b) Particle Size Distributions with values of D equal to 2.32, depth 701 m. Images from [START_REF] Blenkinsop | Cataclasis and Processes of Particle Size Reduction[END_REF].

[N°23] Backscattered electron micrograph with trail development in experimentally deformed clay-silt gouge, through the intersection of an R1 shear, included ruffle particle in white, Betic Zone of southeastern Spain. Image from [START_REF] Rutter | Comparative microstructures of natural and experimentally produced clay-bearing fault gouges[END_REF] [START_REF] Heilbronner | Grain size and grain shape analysis of fault rocks[END_REF], [N°33] Hematitic cataclasite fracture-filling separated by shear surface from microbrecciated and locally cataclastically foliated granite. Image from [START_REF] Snoke | Fault-related rocks: a photographic atlas[END_REF]. [N°34] Cataclasite fracture-fillings in granite. (b) [N°35] Cataclasites from a wide fault. Grains of quartz lied in a matrix of sericite and feldspar. Images from [START_REF] Snoke | Fault-related rocks: a photographic atlas[END_REF].

Appendices to Chapter 2 A. Time step (adaptative vs constant)

The evolution of the peak and steady-state effective friction (average friction value within the plateau zone) is quite different depending on whether an adaptive time step or a constant time step is used for calculation. In order to quantify this difference between adaptive time step and constant time step, several studies have been carried out by varying the density of bodies. With a very small constant time step (Δ = 10 -9 𝑠), a progressive increase of the friction peak (orange) is observed, whereas the evolution of the friction peak with an adaptive time step is varying a lot, Figure 141. At constant time step, the steady-state friction keeps the same average value of 0.46. However, for an adaptive time step the density is much more taken into account in the physics of the model. Indeed, the value of the steady-state friction increases as a function of the density of the bodies, until reaching a value of 0.56. The inertial effects are also much more marked in the case of an adaptive time step (blue area around friction curve), characteristics of the consideration of the increase in particle density. It appears that the adaptive ti me step of the software MELODY has been coded for deformable grains, having no or very little digital stiffness. The fact that a great numerical stiffness is used (𝑘 𝑛 = 10 14 10 16 Pa/m) disturbs the adaptive computation of the time step. It is, therefore, preferable to use a constant time step in these simulations with non-deformable rigid bodies.

B. Generation of matrix particles

The code Cvoro (C++) is a simplified version of the method presented with a Voronoï tessellation algorithm (Packing2D), allowing the creation of a dense packing of polygonal homogeneity/regularity of the hexagonal cells). Figure 142 presents different equivalent diameters of particles merged with angular grains created with Packing2D. As expected, the smaller matrix better fits the grain-matrix contour, keeping more information on the roughness of particles. If the grains are too close, we lose some information when a too high size matrix is used ( 𝑒𝑞 = 25 𝜇𝑚). In order to limit calculation costs due to the higher number of particles, we chose a 𝑒𝑞 = 20 𝜇𝑚 for the rest of the Thesis work, small enough to observe a matrix layer between all big particles. However, in the case of a mixture between angular particles and matrix, some roughness information will be lost or modified.

For more information on how works « Cvoro » and how to use it, you can either contact Guilhem Mollon or Nathalie Casas.

C. Initial porosity and state of density

The solid fraction 𝐹 is the ratio of the surface occupied by the grains ( 𝑎 𝑖𝑐 𝑒 ) to the apparent surface of the sample ( 𝑔𝑜 𝑔𝑒 ). Since we are here in 2D, we will take an area rather than a volume, considering that the 3 rd dimension is the same in the numerator and denominator. In the associated manuscript, we mostly talk about the percentage of porosity (i.e. the ratio of the surface occupied by voids ( 𝑣𝑜𝑖𝑑 ) to the apparent surface of the sample). The solid fraction and porosity in a sample can therefore be calculated as follows, with S a surface area: 500 𝜇𝑚 𝑘 𝑛 giving a quite dense packing of particles (𝐹 𝑆 = 0.89). On the opposite, with 𝜇 𝑛 𝑚 = 1, friction reduces possible movements and the resulting solid fraction is smaller, meaning much more porosity within the compacted state (𝐹 𝑆 = 0.83). Similar results have been found on the influence of interparticle friction for the initial packing of particles [START_REF] Shi | Steady state rheology of homogeneous and inhomogeneous cohesive granular materials[END_REF]. The idea is, therefore, to see the influence of this initial state on gouge shearing. The compaction of samples was carried out on a sample of 4960 grains generated according to a fractal distribution law (dimension factor 𝐷 = 2.6), i.e. with the same initial sample before the compaction step. Parameters used for sample compaction and shearing are written in Table 11. Figure 143 displays the solid fraction and gouge thickness (𝑚𝑚) for each sample generated with a different interparticle friction coefficient. We observe that gouge thickness decreases with the increas e of solid fraction. The highest solid faction (0.89) corresponds to a very dense sample in which there is very little porosity between grains. In contrast, the lowest solid fraction (0.83) corresponds to a slightly less dense sample, with a gouge thickness of 0.12 𝑚𝑚 greater, (5 % of the total thickness).

Samples used in chapter 3 are sample 10 (𝐹 𝑆 = 0.89 or 𝑃 𝑜 𝑒 = 11 %) and 15 (𝐹 𝑆 = 0.8 or 𝑃 𝑜 𝑒 =16 %).

Shearing step (for Chapter 3) -All the different initial compacted states have been sheared with an imposed velocity of 1m/s (with frictional contact law only) in order to quantify the influence of this initial state in the slip behavior of the gouge. Figure 144 shows that, to a certain extent, the initial state of compaction of the sample does influence the first steps of the gouge behavior. Comparing the evolution of effective friction as a function of the initial solid fraction, we observe that the denser initial sample (𝐹 𝑆 = 0.89 or 𝑃 𝑜 𝑒 = 11 %) presents the higher friction strength with a friction peak 𝜇 = 0.75. e e e or o er ed between 0.86 ≤ 𝐹 𝑆 ≤ 0.89, with an increasing peak strength with the increase of solid fraction. Particles organized themselves to upgrade packing and reduce voids between them. The increase of initial density of the granular sample (i.e. higher solid fraction) significates that packings of 𝑡ℎ 𝑖 grains become more difficult to unpack, and moving the entire sample asks for more energy to the system to reach a steady state. For lower solid fraction (0.86 ≥ 𝐹 𝑆 ), the compaction state does not seem to be an obstacle to shearing and the effective friction progressively increases until a plate where we suppose the system is in steady-state (cf. inertial number). The mid-dense sample (𝐹 𝑆 = 0.8 or 𝑃 𝑜 𝑒 =16 %), has a friction peak almost missing. Even though the effective peak depends on the initial solid fraction, if the shearing proceeds for a certain time of the experiment, all the samples exhibit the same mechanical behavior with the same steadystate friction 𝜇

𝜇 𝜇

D. Circular vs angular particles

The numerical and physical characteristics of both samples (circular and angular particles) used in Chapter 2 are presented in Table 12.

Appendices to Chapter 3 A. Numerical stiffness influence

As explained in Chapter 2, numerical stiffnesses are not completely physical nor numerical but make it possible to quantify the rate of interpenetration between the bodies in contact. A stiffness that tends towards infinity will give grains that will be tangent, while a weak stiffness will make that the bodies in contact can be allowed to overlap. Tests were carried out with different stiffnesses from 10 14 𝑃𝑎/𝑚 to 10 17 𝑃𝑎/𝑚, in order to quantify the interpenetrations between the particles in contact. Figure 145 presents the evolution of effective friction as a function of the numerical stiffness between particles. Changing the numerical stiffness does not have a major influence on the effective steady-state friction coefficient which has the same averaged value 𝜇 ≈ 0. 5 On the other hand, we note that an increase in numerical stiffness augments the standard deviation from the average friction value. In other words, the dynamic effects are greater with higher stiffness, since the interpenetrations which could also serve as damping are reduced. It is interesting to be able to choose both a fairly high stiffness, allowing to have realistic modeling concerning the contacts between the bodies, but also about the dynamics of the system. A stiffness of 10 14 𝑃𝑎/𝑚 reveals too high interpenetrations, whereas a stiffness of 10 16 or 10 17 𝑃𝑎/𝑚 gives too many spaces for dynamic effects in the numerical shearing. The stiffness 10 1 𝑃𝑎 /𝑚 allows us to obtain more precise results than 10 14 𝑃𝑎/𝑚, while remaining in models with a standard deviation from the average steady-state friction lower than 10 %.

Besides, it is also possible to compute the equivalent Young modulus 𝐸 for a typical rock as granite for our cemented materials, thanks to the parameter of a cemented contact between particles: It was thus chosen to continue the simulation campaign with a numerical stiffness of 10 1 𝑃𝑎/𝑚 which is numerically and physically consistent.

B. Reproducibility

In order to validate the model and results obtained within packing 2D and the DEM code MELODY, reproducibility tests have been carried out with the same parameters as the materi al of Chapter 2. Four granular samples were generated randomly with the same initial characteristics (with Packing2D) meaning that the granular samples are globally identical, but locally different (each sample is unique in terms of grain position and shapes). They are all composed of 5000 grains distributed randomly according to a fractal granular distribution in the defined area of 2x20 mm² and their initial solid fraction is equal to 0.89. The four samples are named from 1 to 4, Figure 147. By comparing the results from the different samples, it can be seen that the effective friction peak, the averaged values at steady-state of friction and dilation remain constant over the four samples: steady-state friction is around 0.47 with a relative margin of 2 % from this value, and dilation is also similar with an averaged value at steady-state ≈ 0.092. Although steady-state values remain very close, each sample represents a different stack of grains distinct local evolutions of the gouge with upper wall slip distance. Common rocks

This small study validates that a granular sample generated and sheared with the same initial conditions (i.e. same global geometry, numerical and physical boundary conditions), produces the same mechanical results. The resulting kinematic is also similar. More data and results files are available at: "Casas, Nathalie; Mollon, Guilhem (2022), "Cohesion and Initial Porosity of Granular Fault Gouges", Mendeley Data, V3, doi: 10.17632/7c3dcj7spw.3" The representation of the relative damage gives a picture of the state of cementation between grains and their location within the gouge. This damage is set to 0 when cohesive bonds are first established (all the bonds are intact) and may evolve until 1 if all these bonds reach the " roke " er re e d e w re e o e e following movies illustrate the evolution of the gouge state as a function of the slip distance:

C. Results tables for cemented materials

Appendices to Chapter 4 A. Representative Surface Element (RSE)

Different lengths of matrices are created to evaluate the influence of this parameter on shear localization and granular behavior. In order to observe the influence of the length of the granular third body, five samples are generated: L 2 (2 × 2 𝑚𝑚²), L 4 ( × 2 𝑚𝑚²), L 10 (10 × 2 𝑚𝑚²), L 20 (20 × 2 𝑚𝑚²), & L 40 ( 0 × 2 𝑚𝑚²) with an interparticle friction 𝜇 𝑛 𝑚 equal to 0.3 between cells. In Figure 148, from 0 to point 𝐴 1 , a first elastic linear part is observed with the same behavior for all samples. At point 1 (friction peak), almost all samples behave in the same way except for the L 2 sample, which has a specific behavior due to its particularly small size. Point 𝐶 1 is very interesting, this is the decreasing friction point located at the middle distance between friction peak and steady-state friction. An evolution of suddenness of friction weakening is observed (i.e. slip distance needed for friction to drop from 𝜇 to 𝜇 ) and this area is also the place of maximum dilatancy. Increasing the length of the matrix softens the transition between 𝜇 to 𝜇 until a certain limit. It also expands the dilation of the sample, Figure 148 (b). Friction transition evolution stops evolving from after L 20 (i.e. Representative Surface Element obtained for this gouge thickness, see the path of L 40 in Figure 148 (a). The same behavior is obtained for dilation, increasing until L 20 . However, changing the gouge thickness or size of particles modifies the global stiffness of the granular media and thus could influence the RSE. Between 𝜇 and 𝜇 , crack opening takes place in a preferred Riedel band R at 𝐶 1 , where a maximum dilatancy is observed (i.e. minimum solid fraction inside the Riedel shear). The augmentation of the length of the matrix does not seem to change the formation of observed Riedel shear bands. However, this increase creates a larger observation window displaying more Riedel bands within the sample. These bands are always inclined with the same orientation of ≈ 10 ° from the main shear direction. They grow from the top surface to the bottom rock wall in the opposite direction of shearing. Steady-state is reached with the formation of a localized horizontal shear band S at the top or bottom part of the granular media.

B. Gouge thickness

Figure 149 presents the effective friction and dilation as a function of the effective friction for two different gouge thicknesses, 1 and 3 mm. All the other parameters are similar. The effective friction peak is almost similar for both cases, as the steady-state friction. But the dilation is higher with a more important gouge layer, which is consistent with the theory.

C. Shear modulus calculi and graphs

The shear modulus, Figure 150, can be calculated as where 𝜏 is the shearing stress, 𝑡ℎ 𝑖 the initial gouge width and ∆ 𝑥 the slip distance at the secant point. We compute both the secant shear modulus (at 40% of the maximum shear stress) and the initial tangent shear modulus. The results for all the models are presented in Table 15. In the following graphs, the slope of slip weakening is extracted from each model previously presented thanks to the method described in Chapter 5, (Figure 152, Figure 153, Figure 154, Figure 155, Figure 156, Figure 157).

D. Mechanical behavior of a bi-disperse mixture

❖ Cemented materials

❖ Matrix & grains models

B. Empirical parametric laws (Part B)

The laws used to represent the evolution of the shape parameters are presented below. These are the behavior laws of each of the 7 parameters considered in the simplified models. Constant and numerical variables are displayed in Table 16.

• Parameters used for dilation law:

Maximum dilatancy angle Ψp (°): Figure 162. Schema of the method used to measure the Riedel band opening. We first select points on each part of the interested band and measure the vertical displacement perpendicular to the band directio n, called ∆∆ 𝑖 . The same method is applied on the displacement in the band direction, ∆∆ 𝑖 .

Ψ (𝑃 𝑐𝑒𝑚 ) = { Ψ -
• Measure of the 𝑹 𝒕𝒉𝒊 𝒌 𝒆𝒔𝒔 with ImageJ

With the software ImageJ, the thickness of each Riedel band was measured manually. The main steps are the following:

-Recover Field images from MELODY, showing Solid fraction evolution at each time step.

-Application of a filter to threshold only a certain solid fraction value in the software Paraview. For example, we only extracted zones of high solid fraction (from 0.95 to 1). (Figure 163)

E. Thin bands results (Part C)

This appendix presents the friction and dilation results for the small gouges generated for Chapter 5. Data are also used with the graph of Chapter 4 to represent a gouge layer without shear localization. Figure 165 gathers all the results in a graph of the effective friction as a function of the interparticle friction. Figure 166, Figure 167, Figure 168 present the total effective friction curve with the associated dilation for all the thin gouges tested, listed in table Figure 165.

Nomenclature

The indicated chapter concerns the first chapter where the parameter was defined, but it can be used in other chapters.

𝐴

  e d' d er e rô e que joue la cimentation dans l'affaiblissement des gouges de failles matures, en regardant notamment si les matériaux cimentés favorisent ou non les glissements dans les failles (et proposer une loi de frottement pour décrire le comportement de ces matériaux cimentés (Chapitre 5, Partie B). Un deuxième modèle numérique permet cette fois de focaliser sur le rôle que jouent les particules de matrices dans le comportement d'affaiblissement des gouges de failles matures et plus particulièrement sur l'importance des différentes caractéristiques physiques et géométriques de la matrice dans le comportement de glissement observé (Chapitre 4, Partie A). Il est également étudié dans quelle mesure la ro or o de r e o de r re e r e o or e e d' e e de faille (Chapitre 4, Partie B).

Figure 1 .

 1 Figure 1. Schéma principal de la thèse. La colonne « Matériau de remplissage » décrit les deux matériaux de remplissage étudiés « ciment » et « matrice ». La colonne « Echantillon de grains » décrit les trois échantillons générés pour la thèse et ensuite insérés dans le modèle numérique. La colonne « Type de simulation » présente les 2 types de lois de contact utilisées et la colonne « e d' e » ce qui a été étudié pour chaque combinaison d' o o Chapitre 1 E de 'A e S e de e e e

Figure 2 .

 2 Figure 2. Schéma de représentation de la théorie du troisième corps et lubrification par poudre. (a) Surface de glissement avec gouge adhérente et couche de gouge, provenant du granite Sierra White en cisaillement rotatif. Image tirée de (Reches & Lockner, 2010), (b) Schéma de la théorie du troisième corps appliquée au système de la faille.

  est utilisé avec la méthode des éléments discrets (DEM) en 2D. Le premier objectif est de modél er e e e or o de e o o e d'

Figure 3 .

 3 Figure 3. (a) Zoom sur un échantillon granulaire généré avec Packing2D, avec des formes angulaires et à facettes. (b) Zoom sur un échantillon de matrice généré avec Cvoro_2D, avec des formes hexagonales modifiées (c) Zoom sur l'échantillon fusionné entre les cellules matricielles et les particules angulaires. Pour ce dernier type d'échantillon, les cellules matricielles ont un diamètre équivalent 𝜙 𝑒𝑞 = 20 𝜇𝑚 et une perturbation 𝑃 0.𝑖 = 0.5.

Figure 4 .

 4 Figure 4. Schéma du modèle DEM d'une gouge de faille granulaire cisaillée. Un modèle de contact sec est considéré ici. La gouge de faille et les parois rocheuses sont considérées comme rigides avec une raideur numérique constante et utilisée pour limiter l'interpénétration entre les grains. Dans cette étude, une densité de 2600 𝑘𝑔/𝑚 3 est utilisée pour les particules, ce qui conduit à un pas de temps approprié pour ces simulations de 10 -9 𝑠. Un solveur explicite est utilisé.

Figure 5 .

 5 Figure 5. Courbes de frottement macroscopique en fonction de la distance de glissement et images des chaînes de force pour des échantillons denses au pic de frottement [B] et à la fin de la simulation [H] pour 3 cas de cimentation (4%, 40% et 95%). Les chaînes de force sont définies en fonction du niveau de contrainte présent dans le système (ici la contrainte normale appliquée) et l'épaisseur est évaluée entre 0 et 1 entre la force minimale et maximale (0 à 10 4 𝑁 e d' r de o e r re Le matériau fortement cimenté avec une faible porosité initiale présente une formation claire de bandes de Riedel évoluant comme dans la théorie. Les valeurs élevées de cohésion et d'angle de frottement interne rendent ce matériau très proche des propriétés des roches intactes. En revanche, le matériau le plus faible est obtenu avec l'état de porosité le plus élevé et pour une absence de cimentation : aucun mode de rupture n'est observé et un écoulement de Couette granulaire est mis en évidence.

Figure 6 .

 6 Figure 6. Frottement effectif 𝜇 en fonction de la distance de glissement (µ𝑚) pour deux échantillons granulaires : l'échantillon de gouge standard avec uniquement des particules de matrice et la gouge avec des particules de matrice et une distribution fractale de particules angulaires. H représente une distance de glissement total de la faille de 600 µ𝑚 e o de ' r e r de o e o o er e r e à r o o de d ' o 𝐻 1 (matrice seule) et 𝐻 2 (matrice + grains). Les essais de cisaillement direct menés sur les mélanges composites montrent clairement que les pourcentages de particules de matrice et de grains angulaires ont une énorme influence sur e e r o o e de o e de e r e e d' e orosité initiale affecte

  , 𝐻 2 . La temporalité de formation des bandes de Riedel d

Figure 7 .

 7 Figure 7. (a) Illustration de la contrainte de cisaillement 𝜏 (Pa) en fonction de la distance de glissement (m) pour o de o e d rd e e d' e e K = (𝜏 𝑆𝑆 -𝜏 p )/𝐷 𝑐 est calculé pour avoir la même énergie en 𝐽/𝑚² o r e r e r r e de e e e o r ' er e de r re 𝐸 𝐺 (en gris). Avec 𝜏 p la contrainte de cisaillement maximale (Pa), 𝜏 ss la contrainte de cisaillement steady-state (Pa) et 𝐷 𝑐 la distance critique de glissement (m). (b) Modèle proposé pour les gouges cimentées avec la décomposition de l'énergie de rupture 𝐸 𝐺 . 𝐸 𝐺 regroupe une énergie de dilatation 𝐸 𝐷𝑖 𝑎 𝑖𝑜𝑛 , une énergie 𝐸 𝐷𝑎𝑚𝑎𝑔𝑒 provenant de la rupture de la cimentation (énergie de création de surface) et 𝐸 𝑓 e r e de ' er e de ro e e de o o 𝐸 𝐻 . (c) Modèle proposé pour les gouges avec matrices avec la décomposition de l'énergie de rupture 𝐸 𝐺 : 𝐸 𝐺 regroupe une énergie de dilatation 𝐸 𝐷𝑖 𝑎 𝑖𝑜𝑛 et une énergie du aux frottements 𝐸 𝑓 .

  Part C -Time and space evolution of R-bands in a dense granular material ................ 5.9. Foreword.............................................................................................................. 5.10. Opening of R-bands ........................................................................................ 5.10.1.Method and first results ............................................................................ 5.10.2.Time evolution of R-bands ....................................................................... 5.11. The inner mechanical behavior of R-bands: towards energetic models ......... 5.11.1.Friction into R-bands ................................................................................ 5.11.2.Trajectories and opening of R-bands ........................................................ 5.11.3.Energetic interpretations ........................................................................... 5.12. Conclusion of Part C ....................................................................................... CHAPTER 6. CONCLUSIONS & PERSPECTIVES....................................................... 6.1. Conclusions ......................................................................................................... 6.2. Perspectives ......................................................................................................... of Gouges ........................................................................................... APPENDICES TO CHAPTER 2 ........................................................................................ A. Time step (adaptative vs constant) ...................................................................... B. Generation of matrix particles .............................................................................

  for the model. It ends with a global description of the 2D direct shear fault models used and the main assumptions and choices we made for these models.• Chapter 3 focuses on the modeling of a sheared granular fault with cementation. A link is established between the micro-mechanical and structural properties of a cemented gouge layer, and its rheological behavior under shearing. The last section offers a discussion on new insights and relations between cementation within the gouge, shear localization, and the relevance of Coulomb-Mohr theory for fault models. • Chapter 4 focuses on the modeling of a sheared granular fault with matrix particles. study the evolution of shear localization (Ridel bands) as a function o the importance of the matrix percentage within the fault gouge, especially from a rheological point of view. • Chapter 5 represents a second objective of this research, which is to link micromechanical models at the grain scale (what we model) and dynamic models at the fault scale (what we want to participate in). Thanks to the numerical models carried out, the extracted friction laws, and the energy balance of the studied fault gouge patch, we generate simplified friction and energetic models which can then be injected into largerscale dynamic models.

Figure

  Figure 8. Main schema of the Thesis.

Figure 9 .

 9 Figure 9. World seismicity map: ISC-GEM catalog of about 42,000 earthquakes (version 8.0). The earthquake selection is based on magnitude Mw 5.5 and above, between 1904 and 2017. The earthquakes recorded take place at the boundaries betwee r d e o e r ' r e o ro (Agnew, 2014) and image from ISC-GEM (Di Giacomo et al., 2018).

Figure 10 .

 10 Figure 10. (a) Induced seismicity: the different types of industrial activities. Image from (Grigoli et al., 2017) (b) Location of the induced earthquakes within the Earth with their associated magnitude, (source: Davies et al.[2013]), updated until August 2016, (each type of industrial activity is represented by a particular color). The picture presents declustered seismic sequences with only the maximum magnitude of the sequence (if ML >1.5). Image from[START_REF] Grigoli | Current challenges in monitoring, discrimination, and management of induced seismicity related to underground industrial activities: A European perspective[END_REF].

Figure 12 .

 12 Figure 12. Typical fault zone structures. (a) Schema of fault zone for strike-slip faulting. The core zone consists of fault gouge and fault breccia. The damage zone consists of foliated and non-foliated cataclasite and fractured host rocks. The red arrows show the shear stress acting on the host rock. Image modified from[START_REF] Lin | Spatial variations in damage zone width along strike-slip faults: An example from active faults in southwest Japan[END_REF]. (b) A narrow zone of fault breccia and cataclasite in a quartzitic parent rock. Buzios, Rio de Janeiro States SE Brazil. Width of view 4 mm. PPL. From[START_REF] Trouw | Atlas of Mylonites-and related microstructures[END_REF]. (c) Photomicrographs in plane-polarized light from the internal structure of the Keumwang fault zone, images from[START_REF] Lee | Comparison of structural features of the fault zone developed at different protoliths: Crystalline rocks and mudrocks[END_REF]

  ]: Mode ed ' e e ode' e o e ode ode e ' o d ode' e -plane shearing), and mode III is the transverse shear (i.e. antiplane shear), Figure 13. However, the equations related to each mode present a nonphysical behavior: infinite stresses at the crack tip. To face this problem, the general concept of fracture energy 𝐸 𝐺 was introduced by

  Figure14. (a) Schema of stresses applied on a representative volume (biaxial (2D) compression test), with major stress 𝜎 1 , minor stress 𝜎 3 applied on the rock sample and resulting tangential and normal stresses 𝜏 d 𝜎 𝑁 applied on the fault plane. 𝛽 is the angle between the principal stress and the fault plane and 𝜓 the angle between the principal stress and the normal stress (b) Mohr-circle with stress representation and angle relations, shear stress 𝜏 as a function of the principal stresses applied on the volume.

Figure 15 .

 15 Figure 15. Effective friction properties of different rocks, gouge, fabric, and phyllosilicate -rich faults materials. e r re o o or re r e r r o er ee' r e e ro (Collettini et al., 2019).

Figure 21 .

 21 Figure 21. Third body theory representation schema and powder lubrication. (a) Sliding surface with adhered gouge and gouge layer, from Sierra White granite in rotary shear. Image from (Reches & Lockner, 2010), (b) TEM observations on post-mortem fault sections at 180 MPa, triaxial experiment. Image from (Jérôme Aubry, 2019).(c) schema of the third-body theory applied to the fault system. More information on the third body concept are available in [[START_REF] Iordanoff | Numerical study of a thin layer of cohesive particles under plane shearing[END_REF],[START_REF] Fillot | Wear modeling and the third body concept[END_REF]].

Figure 22 .

 22 Figure 22. Schematic diagram of a sheared gouge zone, showing the geometrical relationships of the main structural elements and the orientations of the principal stresses with the shear zone.

Figure 23 .

 23 Figure 23. (a) Riedel band formation within the granular fault gouge illustrated with stress ratio as a function of shear displacement. Image modified from (Marone, 1998). (b) Backscatter electron (BSE) images of 15 µm quartz gouge layers sheared σ n = 80 P . Stops 1-3 correspond to the different displacements where shearing is stopped, Stop 1: just after the yield (slip=1.5 mm), R Riedel shears within the layer. Stop 2: at the onset of stickslip instabilities (slip=3.2 mm), R Riedel shears and incipient B-shears are observed. Stop 3: end of the experiment after multiple stick-slip events (slip=6 mm), B-shears are observed at the top and bottom of the layer, with discrete Y-shear planes within the B-shear-zone. Images from[START_REF] Bedford | The Role of Grain Size and Effective Normal Stress on Localization and the Frictional Stability of Simulated Quartz Gouge[END_REF].

Figure 26 .

 26 Figure 26. Steady-state effective friction 𝜇 𝑆𝑆 as a function of the topological dimensions of the model from 1D to 3D for angular or circular grains, (Frye & Marone, 2003) and personal results with the software MELODY.

  the fragmentation process of rock (parallel bond rock model), [

Figure 27 .

 27 Figure 27. Different types of BPM (a) Clustered particles vs. clumped particles, image redrawn from (Cho et al., 2007), and flat-joint contact model showing the effective interface geometry (from (Potyondy, 2012)] redrawn). Image from (Lisjak & Grasselli, 2014) (b) Example of a mixture between clasts (i.e. gravel) and matrix (i.e. sand), image from (Y. Lu et al., 2017).

Figure 28 .

 28 Figure 28. Main schema of Chapter 2.

Figure 30 .

 30 Figure 30. (a) Sketch of a typical contact between two grains A and B with corresponding normal and tangential gaps 𝛿 𝑛 and 𝛿 used to control interpenetrations between particles. (b) Schema of the different parameters acting on a typical frictional contact, 𝑘 𝑛 and 𝑘 are respectively the normal stiffness and the tangential stiffness and 𝛾 𝑛 and 𝛾 normal and tangential damping fixed within the model.

Figure 31 .

 31 Figure 31. Three main steps of the contact detection algorithm are (i) Broad proximity detection, (ii) Close proximity detection, and (iii) regular detection at each time step. Images from (Mollon, 2018a).

  (a) a Damped-Mohr-Coulomb contact law (used in chapter 4 for matrix particles), which is a classical DEM law with interparticle friction only between contacting particles, and (b) a Bonded Mohr-Coulomb with interparticle friction and cohesion. This contact law is similar to the Bonded Particle Model (used in chapter 3 for cemented particles).

  or 𝑎𝑏𝑠(𝜎 ), exceeds the prescribed value of cohesion 𝐶 𝑛 𝑚 , the status of the o d ed o " roke " Figure 32 (b).

Figure 32 .

 32 Figure32. (a) Intact bond. Initialization of the contact law. A cohesive law links all grains in contact. The bond corresponds to a constant pressure to maintain particles in contact (Pa). (b) Broken bond. When the force applied to the particles becomes higher than the cohesive strength, the bond is broken. The contact becomes cohesionless and follows a classical Coulomb-Mohr law with interparticle friction only. Broken contacts cannot be cohesive again and this induces an augmentation of broken bonds during the shearing. Normal interpenetration is exaggerated in the figure for the sake of clarity.

  These laws are completed by a classical viscous dashpot to dissipate kinetic energy by contact damping and to stabilize the simulation (𝛾 𝑛 & 𝛾 ), Figure30.

Figure 33 .

 33 Figure 33. (a) Example of particles from a selected spectrum: first a random sampling of phase angles, then the computation of the random signals r(θ), and finally plotting of the corresponding particles contours by varying elongation or shape of particles. Image modified from (Mollon & Zhao, 2012) (b) 3D cell-filling algorithm: a. Polyhedral cell to be filled; b. Same cell expressed as a geodesic structure; c to i. Particle filled in a cell with increasing fitting parameter from 0 to 2562. Image from[START_REF] Mollon | 3D generation of realistic granular samples based on random fields theory and Fourier shape descriptors[END_REF].

-

  The first step generates a constrained Voronoi Tessellation, Figure34, with polygonal subdomains following a target size distribution and cells orientation.

Figure 34 .

 34 Figure 34. Example of Voronoi Tesselation generated with Packing2D. Fractal distribution of particles. 𝑡ℎ 𝑖 is the initial thickness of the numerical gouge and 𝐿 𝑔 the length of the numerical gouge, both in meters.

  (a). According to Discrete Fourier Transform (DFT), the signal ( ), Figure35 (b), can be written as the sum of N harmonics:

Figure 35 .

 35 Figure 35. Process of grain shape characterization. (a) The contour of a particle discretized by 𝑁 𝑖 points, each point is associated to a radius and position angle in polar coordinates (b) Signal described by the position of the 𝑁 𝑖 points in polar coordinates (c) Fourier descriptors as a function of the mode. The two decays observed within shape modes and roughness modes are described by: 𝛥 1 = 2 𝜆 1 log 2 ( 𝑛 3 )+log 2 (𝐷 3 ) (𝑓𝑜 3 < 𝑛 < 8) and 𝛥 2 = 2 𝜆 2 log 2 ( 𝑛 8)+log 2 (𝐷 8 ) (𝑓𝑜 𝑛 > 8). With 𝜆 1 and 𝜆 2 two constants equal to -2.

Figure 36 .

 36 Figure 36. Example grain samples with different Descriptor variation. (a) 𝐷3 = 0.1 (b) 𝐷3 = 0.03, (c) particle with different D8 for 𝐷2 = 𝐷3 = 0 (i.e. circular particle), last image from (Mollon & Zhao, 2012).

Figure 37 .

 37 Figure37. Voronoi cell (red) and grains generated inside the cell (blue), image modified from[START_REF] Mollon | Fourier-Voronoi-based generation of realistic samples for discrete modelling of References granular materials[END_REF]. To have grains with flat facets, the optimization parameter 𝑁 𝑂 𝑖𝑚 can vary between 0 and 63, where 0 corresponds to a grain as defined in the spectrum and 63 a grain as much as possible with facets. Increasing this parameter is very helpful to have angular particle shapes, but it also increases a lot the calculation costs during sample generation. By increasing the target Solid fraction 𝐹 𝑆 (equal to ⁄ ), it decreases the compaction time of the granular sample. However, the final solid fraction of the sample also depends on how the grains will conform to the shapes of the Voronoi cells formed.

  It is possible to choose the size of particles 𝑒𝑞 (equivalent diameter of particles, calibrated on o re d o e " er r o r e er " 𝑃 0.𝑖 (i.e. the homogeneity/regularity of the hexagonal cells). Figure 39 displays the three types of shape of matrix particles used in Chapter 4, with a variation in perturbation parameter. The equivalent diameter of particles 𝑒𝑞 was chosen equal to 20 𝜇𝑚 (App d x 2. B). The initial porosity, or void fraction 𝐹 𝑣 = 𝑣𝑜𝑖𝑑 ( 𝑎 𝑖𝑐 𝑒 𝑣𝑜𝑖𝑑 ) ⁄within the sample is negligible, with less than 1%, which leads to a state of compaction very close to intact rocks.

Figure 39 .

 39 Figure39. Change in the perturbation of particles, from 𝑃 0.2 to 𝑃 0.8 modifying the final cell shape, from hexagonal cells to angular polygonal cells.

Figure 40 .

 40 Figure 40. Zoom in on the merged sample with matrix cells and angular particles. For this last sample type, matrix cells have an equivalent diameter 𝑒𝑞 = 20 𝜇𝑚 and a perturbation 𝑃 0.𝑖 = 0.5.

  (a). We observe that the interparticle friction used during compaction controls the initial porosity within the sample (Appendix 2. C). When the granular sample is compacted and mechanically stable, the appropriate mechanical laws between particles in contact are implemented, Figure41(b).

Figure 41 .

 41 Figure 41. Compaction of the granular sample (a) Step 0, before compaction of the sample, (b) Step 1, just after compaction. 𝜎 𝑁 is the normal stress applied on the upper rock wall and 𝑡ℎ 𝑖 is the initial gouge thickness at the end of the compaction step.

Figure 42 .

 42 Figure 42. Schema of the DEM model of a granular fault gouge

Figure 43 .

 43 Figure 43. (a) Typical friction curve as a function of slip displacement of the upper rock wall. 𝜇 is the effective friction peak, i.e. maximum effective friction, and 𝜇 is the steady-state effective friction, which is an average value of the effective friction once the plateau zone is reached (also called residual friction). (b) Schema of the sheared gouge with the total normal and tangential forces (𝐹 𝑁 & 𝐹 𝑇 ) applied on the upper rock wall.

Figure 44 .

 44 Figure 44. (a) Effective friction 𝜇 and dilation 𝜀 𝑦 as a function of normal stress 𝜎 𝑁 (MPa) (b) Steady-state inertial number as a function of normal stress.

Figure 45 .

 45 Figure 45. (a) Effective friction 𝜇 (at peak and steady-state) and dilation 𝜀 𝑦 (at steady-state) as a function of the shearing velocity (m/s), (b) Steady-state inertial number as a function of shearing velocity (m/s)

Figure 46 .Figure 47 2 .

 46472 Figure 46. (a) Granular sample generated with packing2D, with circular shapes, and compacted in MELODY2D (b) Granular sample generated with packing2D, with angular and faceted shapes, and compacted in MELODY2D. (c) Log-log graphs showing the cumulated number of particles (Y-axis) plotted against particle equivalent diameters (X-axis) from the numerical gouge sample with CDF (Cumulative Distribution Function). The fractal dimension D equals 2.65 for angular grains (blue line) and fractal dimension D equals 2.5 for circular grains (green line). They have almost the same total surface of particles (26 mm² vs 29 mm²).

Figure 48 .

 48 Figure48. Effective friction 𝜇 (at peak and steady-state) and dilation 𝜀 𝑦 (at steady-state) as a function of the number of grains in the model. A slightly higher 𝜇 is observed for the case with 2500 grains, probably due to a slightly different stacking of the particles during the sample compaction. The effective friction curve obtained with the 1500-grains sample is noisier because the dynamic effects are more noticeable with fewer grains.

Figure 49

 49 Figure 49. (a) Force chains magnitude in Newton for 1500 grains, 2500 grains, and 5000 grains -(b) Number of contacts with a normal vector oriented in a given direction (using polar diagrams where the Theta-axis is the orientation and the R-axis is the number of contacts). Normal forces orientation as a function of the model, at steady state. Small sample with 1500 particles in red, the middle sample with 2500 particles in green, and the bigger sample in blue.

Figure 50 .

 50 Figure 50. Main schema of Chapter 3.

  Casas, N.,[START_REF] Casas | DEM Analyses of Cemented Granular Fault Gouges at the Onset of Seismic Sliding: Peak Strength, Development of Shear Zones and Kinematics[END_REF]. DEM Analyses of Cemented Granular Fault Gouges at the Onset of Seismic Sliding: Peak Strength, Development of Shear Zones and Kinematics. Pure and Applied Geophysics. https://doi.org/10.1007/s00024-021-02934-The software Packing2D (Chapter 2) is employed to create a realistic granular sample with angular and faceted shapes of particles. Since the morphological descriptors of the grains of granular gouges may vary significantly between faults, the spectrum was calibrated by visual comparison with published pictures of real gouge d re e "Grains with angular e" o Chapter 2, Figure 51 (a).

Figure 51

 51 Figure 51. (a) Zoom on a granular sample generated with packing2D, with angular and faceted shapes, and compacted in MELODY2D (b) Particle size distribution of the synthetic sample, fractal dimension with 𝐷 = 2.65.

  6. The gouge with angular grains is composed of 4960 particles with a corresponding equivalent diameter in the range of 28 -226 µ𝑚 er e e o μ d 50 ) equal to 70 𝜇𝑚, Figure 51 (b).

Figure 52 .

 52 Figure 52. (a) Gouge thickness (m) as a function of the slip of the upper rock wall (m) -for dense and mid dense sample (b) Effective friction as a function of of the slip of the upper rock wall (m), results from the shearing of granular samples with the following parameters (𝜎 𝑛 = 0 𝑀𝑃𝑎, 𝑉 = 1 𝑠 ⁄ , 𝜌 = 2600 Kg 3 ⁄ , 𝑘 = 10 1 𝑁. 𝑚 -3 ,𝜇 𝑛 𝑚 = 0.5 (during shearing)).

  𝑉interparticle cementation, and interparticle friction). Rectangular granular samples (4 mm wide and 10 mm high) are placed between four rigid walls, Figure55(a). The lower wall is fixed in displacement, the upper wall is submitted to a constant downwards velocity 𝑉 𝑦 , and the lateral walls are submitted to a confining pressure 𝜎 3 , fixed in vertical and rotational motions, and free to move horizontally.For each cementation level and each initial porosity, three tests are performed with confining stresses of 10 𝑀𝑃𝑎, 40 𝑀𝑃𝑎, and 80 𝑀𝑃𝑎. The vertical stress 𝜎 1 is monitored during vertical compression (series of pictures with relative damage, Figure55 (a)). Biaxial simulations are only used for the cemented material characterization in this section in order to obtain the internal friction angle 𝜑 and the cohesion 𝐶. These parameters are not to be confused with the interparticle cohesion (𝐶 𝑛 𝑚 ) and friction (𝜇 𝑛 𝑚 ) that are introduced in the DEM contact laws, since they represent a collective response of the granular material to shearing.

Figure 55 .

 55 Figure 55. Illustrative results for a numerical cohesion 𝐶 𝑛 𝑚 = 1000 𝑀𝑃𝑎 (a) Views of experiment and damage evolution for different percentage of vertical shortening under confining stress of 80 𝑀𝑃𝑎 -(b) Major principal stress as a function of vertical shortening and associated Mohr circles, for three confining stresses (10 𝑀𝑃𝑎, 0 𝑀𝑃𝑎, and 80 𝑀𝑃𝑎) for both dense and mid-dense samples.

Figure 55

 55 Figure 55 (b) and (c) provide illustrative results for a numerical cohesion 𝐶 𝑛 𝑚 = 1000 𝑀𝑃𝑎 (corresponding to 𝑃 𝑐𝑒𝑚 = 38 % in the dense case and 𝑃 𝑐𝑒𝑚 = 0 % in the mid-dense case). Graphs show that dense samples quickly reach a peak strength before softening towards a plateau value, while mid-dense samples are less stiff and reach the same plateau (approximately

Figure 56 .

 56 Figure 56. (a) Internal friction angle (°), (b) Cohesion (MPa), and (c) Elasticity moduli (GPa) as a function of the Cementation (%) for dense and mid-dense samples.

  are available. Figure57compares the different synthetic materials tested in this chapter (with different initial percentages of cementation and porosities)

  confirms that our samples have similar Coulomb-Mohr properties to some real rocks: (a) dense poorly cemented samples present similar Coulomb-Mohr properties to Quartz sand, (b) the dense gouge with 𝑃 𝑐𝑒𝑚 = 75 % ook ke ro k " " e e ee d o e e e ed o e (mid-de e r ro er e o ro k " " r e

Figure 57 .

 57 Figure 57. Internal friction angle as a function of Coulomb-Mohr cohesion for the simulated materials, as characterized by numerical biaxial tests (Figure55). Dense samples are marked by squares and mid-dense samples by diamonds. It can be noted that the samples used are large enough to avoid any size effect or any thickness dependence. The discs correspond to experimental characterizations of different rocks referenced in[START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF] listed in the table next to the graph.

  References: 1: (Jaeger and Cook, 1976) / 2: (Raleigh and Paterson, 1965) / 3: (Twiss and Moores, 1992) / 4:

Figure 58 .

 58 Figure 58. (a) Effective friction curve for different surface percentages of cementation for dense samples as a function of the slip distance (μm), (b) Zoom in on the peak, (c) Brittleness as a function of cementation, the brittleness is expressed as the ratio of (𝜇 -𝜇 ) divided by 𝜇 , (d) Dilation variation 𝜀 𝑦 as a function of the slip distance (μm) for dense samples -(e) Zoom out on the dilation curve, higher slip distance. Letters correspond to different steps in curves presented: [A] is the initial state before shearing and is identical in all cases. [B] is the peak location, [C] is the end of the first peak (only appears for dense samples with a lot of cementation), [D] is half-peak, [E] is the end of the major dilation phase, [F] is the observed end of the effective friction peak, [G] is a common state for beginning of steady-state and [H] is the end of simulation with 𝐷𝑠 = 2500 µ𝑚 re e or or o e e " e e o "o re re e ed ere similar to the case with % cementation in terms of friction and dilation variation. These results are not in the range of small deformations, as the total slip displacement (𝐻 = 2.5 𝑚𝑚) is higher than the gouge thickness (≈ 1.75 𝑚𝑚).

  (i)For poorly-cemented materials (𝑃 𝑐𝑒𝑚 ≤ 10 %), the interparticle cementation introduced between grains is not sufficient to maintain cohesive bonds during shearing. The simulation for 𝑃 𝑐𝑒𝑚 = %, Figure 59, reveals very few tensile forces consistent with the low percentage of cementation within the sample. As soon as the upper rock wall is set into motion, almost all cohesive bonds break and only a few of them resist until r o e k " 4 " o o d o e e e r d d r o z o[START_REF] Mead | The Geologic Rôle of Dilatancy[END_REF], and the limited dilation in the first stages of shearing is consistent with the absence of shear band formation.

Figure 59 .Figure 60 .

 5960 Figure 59. Relative damage snapshot for 𝑃 𝑐𝑒𝑚 = % in the dense sample (entire granular gouge). Letters correspond to different steps in the curves presented in Figure 58 " " o d e d o-d e " " maximum damage and slip distance of 1mm). (i) Tensile contact force orientations allow getting a grasp on the way load is transferred through a granular sample. This information is coded in a polar diagram which provides the distribution of the orientation of tensile contacts. (ii) The second type of material, Figure 60, corresponds to a cemented granular material (𝑃 𝑐𝑒𝑚 = 38 %). In contrast with the first case, this material presents clear augmentation o e e or e or o e " 38 " d rk e d e red Figure 60 e e o e red e ro e e e r o e k " 38 " o e e d o e e k " 38 " w e re k e o o e e o d owe er o e cohesive bonds re e o e e or r o e k " 38 " e r o o movement within the gouge, highlighting a preferential localization of cohesive bonds rupture (white arrow in Figure 60 is considered as the first Riedel deformation (Tchalenko, 1970) r re de e o e e e o e o " 38 to H 38 " w er r o ede d 𝑅 (oriented in the sheared direction, ∼12° from the upper wall). The progression of the Riedel band towards a shear band increases until the end of the effective friction peak, where it is no longer detectable among the damage zone. The different Riedel geometries are associated with different shear deformation degrees. Conjugate Riedel bands 𝑅 are not visible in the numerical results, but tensile fractures T are observed. This is in line with findings reported in (Cho et al., 2008), who highlighted that tensile fracturing is very important in the development of a shear zone, particularly at low normal stress. From frictio e k " 38 " d e e o o o e re e e o o e e agglomerates formed by intact bonds within the gouge (ex: Ag 1 ), with size decreasing with time. In contrast with poorly cemented material where contacts take place between two particles, contacts occur here between clusters of cohesive grains.

Figure 61 .

 61 Figure 61. Relative damage snapshot for 𝑃 𝑐𝑒𝑚 = 95 % in the dense sample (entire granular gouge). Letters correspond to different steps in the curves presented in Figure 58 " " o d e d o d e " " slip distance of 2.5 𝑚𝑚). R represents the Riedel shear bands and Y the horizontal shear localization. (i) Tensile contact force orientations allow getting a grasp on the way load is transferred through a granular sample. This information is coded in a polar diagram which provides the distribution of the orientation of tensile contacts.

  (e), from " 95 " o " 95 " e e r o ze o e o o or o r o e o e de e d

Figure 62 .

 62 Figure 62. (a) Effective friction curve, comparison between dense and mid-dense samples as a function of the slip distance (μm) for the three different cemented materials highlighted. (b) Zoom in on the peak for mid-dense samples. (c) Brittleness as a function of cementation for both porosity states, the brittleness is expressed as the ratio of (𝜇 -𝜇 ) divided by 𝜇 .(d) Dilation variation as a function of the slip distance (μm) for dense and middense samples. Letters correspond to different steps in curves presented: [A] is the initial state before shearing and is identical in all cases. [B] is the peak location, [C] is the end of the first peak (only appears for dense samples with a lot of cementation), [D] is half-peak, [E] is the end of the major dilation phase, [F] is the observed end of the effective friction peak, [G] is a common state for beginning of steady-state and [H] is the end of the simulation.

Figure 63 .

 63 Figure 63. (a) Relative damage snapshots for mid-dense sample (entire granular gouge). For both peak [B] and end [F] of effective friction peak. Letters correspond to different steps in the curves presented in Figur e 7. R2 represents high angle Riedel shear bands and Y the horizontal shear localization. Evolution of ductility with cementation and porosity. (b) Tensile contact forces orientation allows to get a grasp on the way load is transferred through a granular sample. This information is coded in a polar diagram which provides the distribution of the orientation of tensile contacts.

Force

  chains are key elements to understand the kinematics of the gouge layer and illustrate changes in ductility behavior inside the granular gouge as a function of cementation or porosity.o d e o ed e re e o e "d " o e der ood e e e o granular mechanics: it is related to the suddenness or slowness of the post-peak frictional weakening but does not involve any visco-plastic phenomenon.

Figure 64 .

 64 Figure 64. (a) Snapshot of the force-chains network for dense samples at friction peak [B] and at the end of simulation [H].Force chains are defined according to the typical level of stress present in the system (here the applied normal stress) and thickness evaluates between 0 and 1 between the minimum and maximum strength (0 to 10 4 N). A cut-off is made at 10 4 N to have the same visualization for all cases (higher force chains are observed for a high percentage of cementation). The norm of the contact force is coded by the thickness and color of each chain. In the case of edge-edge contacts, handled through the nodal contacts at the extremities of the contact segment, the total resulting force is considered and coded in a unique force chain. It allows to get a grasp on the way load is transferred through a granular sample. (b) Zoom in on the material with 𝑃 𝑐𝑒𝑚 = 95 %, snapshot of the force-chains network at the end of friction peak [F] and the end of the simulation [H]-(i) Tensile contact forces orientation allows getting a grasp on the way load is transferred through a granular sample. This information is coded in a polar diagram which provides the distribution of the orientation of tensile contacts.

  forces mostly oriented perpendicularly to rock walls, Figure65 (b). At effective friction peak, initially denser samples favor a more homogenous distribution of force chains, Figure65(a), passing through almost all particles. Results are consistent with the previous section showing that the strength increased with a highly cemented and dense sample.

Figure 65 .

 65 Figure 65. (a) Snapshots of the force-chains networks at the effective friction peaks [B] for six chosen cases (0 to 10 4 N). For 3 -, 0 -38, and 95 % of cementation. In the pictures, only a quarter of the total gouge is displayed, and even if the global behavior is similar to one quarter, force chains are not homogeneously distributed within the gouge after friction peak. (b)The graph provides the numbers of contacts with a normal vector oriented in a given direction (using polar diagrams where the Theta-axis is the orientation and the R-axis is the number of contacts), and with a color evolving during the different stages of each simulation.

Figure 66

 66 Figure 66. (a) Total dilation as un function of slip distance, the marked point locates the critical dilation for the 3 cemented materials (b) Critical dilation for macroscopic failure for both dense and mid-dense samples as a function of the percentage of cementation.

  Dilation as a function of slip distance -Zoomed from Fig. 11 (b) Critical dilation

Figure 67 .

 67 Figure 67. (a) Effective friction curves as a function of the slip distance. (b) Peak distance 𝐷 (𝜇𝑚), (c) P l g h 𝐷 (𝜇𝑚), for both dense and mid-dense samples, as a function of the percentage of cementation within the gouge. 𝐷 is the distance between the peak of effective friction and the beginning of the steady-state

  is dedicated to intact rocks, the normal and tangential stresses 𝜎 𝑁 and 𝜏 re ed o e r r k w e r or e o ed " o o re" d ro de e " r e" Coulomb-Mohr properties 𝐶 and 𝜑 of the intact rock. The second criterion (3.7) is dedicated to a domain where shearing occurs in a distributed and continuous way and the

  Casas, N.,Mollon, G., & Daouadji, A. (2021). Shear bands in dense fault gouge. EPJ Web of 249,[1][START_REF] Wilson | HiQuake: The human-induced earthquake database[END_REF][3][4] https://doi.org/10.1051/epjconf/202124911006 Casas, N.,[START_REF] Casas | DEM Analyses of Cemented Granular Fault Gouges at the Onset of Seismic Sliding: Peak Strength, Development of Shear Zones and Kinematics[END_REF] The role of matrix particles within granular fault gouge through Riedel bands observations. (in prep) studies the kinematics and strength of a sheared granular gouge with matrix particles alone. The first section 4.2 presents the numerical model and granular materials used within the study. As for the previous Chapter 3, a sample characterization is made with biaxial simulation on the same granular material used for direct shear experiments. The second section 4.3 reviews the different behaviors of a sheared matrix gouge obtained for variations in the initial parameters of the gouge. The role of the shear modulus, the interparticle friction, and the ratio between the thickness of the gouge on the size of particles will be discussed. Finally, the last section 4.5 concludes on the main elements highlighted through the entire part A. The main schema of part A, Chapter 4 is shown in Figure69.

Figure

  Figure 69. Main schema of Chapter 4, Part A 4.2. Numerical framework and sample generation 4.2.1. Granular matrix samples

Figure 70 .

 70 Figure 70. Matrix shape and size, equivalent diameter 20 𝜇𝑚. Left, zoom in on a 500 𝜇𝑚 x 500 𝜇𝑚 square part, right, zoom in on this sample.

Figure 71 .

 71 Figure71. DEM model of a direct shear experiment with granular fault gouge composed of matrix particles. V is the applied shearing velocity (𝑚/𝑠), 𝜎 𝑁 is the normal stress (𝑀𝑃𝑎), 𝑡ℎ 𝑖 is the initial gouge thickness (𝑚) and 𝐿 𝑔 is the gouge length (𝑚).

  (a) and (b). The vertical stress 𝜎1 is monitored during vertical compression, providing a series of pictures of cumulated slip (vertical displacement minus the elastic part), Figure72 (c).

Figure 72 .

 72 Figure 72. Illustrative results for a standard material (M-S) with different initial interparticle stiffness (a) Major principal stress as a function of vertical shortening, and (b) associated Mohr circles, for three confining stresses (10 𝑀𝑃𝑎, 0 𝑀𝑃𝑎, and 80 𝑀𝑃𝑎) for samples with 𝑘 𝑛 = 10 1 𝑃𝑎/𝑚 and 𝑘 𝑛 = 10 16 𝑃𝑎/𝑚. (c) Snapshots of cumulated slip (vertical displacement minus the elastic part) for biaxial experiment with an elastic membrane, for an applied stress 𝜎 𝑛 = 10 𝑀𝑃𝑎 for a standard case with 𝑘 𝑛 = 10 1 𝑃𝑎/𝑚. Each point in the curve corresponds to an image above. The evolution of shear bands can be observed and the orientation angle increases with vertical shortening (𝛼 ℎ > 𝛼 𝑒 ).

Figure 73 .

 73 Figure 73. Internal friction angle as a function of Coulomb-Mohr cohesion for the simulated materials, as characterized by numerical biaxial tests, the same graph is used in Chapter 3. Matrix samples are marked by hexagons with two different interparticle stiffness, 𝑘 𝑛 = 10 1 𝑃𝑎/𝑚 (green) and 𝑘 𝑛 = 10 16 𝑃𝑎/𝑚 (red). It can be noted that the samples used are large enough to avoid any size effect or any thickness dependence. The discs correspond to experimental characterizations of different rocks referenced in[START_REF] Schellart | Shear test results for cohesion and friction coe ffi cients for di ff erent granular er o or e r e o e ode Tectonophysics[END_REF] listed in the table next to the graph.

  :(Jaeger and Cook, 1976) / 2: (Raleigh and Paterson, 1965) / 3: (Twiss and Moores, 1992) / 4:

Figure 74 .

 74 Figure74. Effective friction 𝜇 (grey curve) and dilation 𝜀 𝑦 (red curve) as a function of the slip distance for the standard case (M-S). A is the end of the elastic zone. The zone from zero to B describes the pre-peak zone, followed by the slip weakening zone (B to D) and then by the steady-state zone (D to E). 𝜀 𝑦-represents the critical dilation.

  4 , and 𝑅 (appearing first) and secondary 𝑅-bands 𝑅 1 and 𝑅 2 with slightly higher Riedel angle orientation 𝛼 𝑖 .-Once the friction peak has passed, we enter the slip weakening zone, or post-peak phase (B to D), where the friction gradually decreases until it reaches a friction stabilization plateau at D. The dilation is not yet at its maximum. In this zone, primary 𝑅-

Figure 75 .

 75 Figure 75. Solid fraction snapshots for standard sample (M-S). The solid fraction is used here to observe the rheology and deformations within the granular gouge and underline zones of slip and shear concentration. Letters correspond to different steps in the effective friction curve Figure 74. The solid fraction is plotted between 0.8 and 1. 𝑅 1 and 𝑅 2 are secondary R-bands, 𝑅 3 , 𝑅 4 , d 𝑅 are primary R-bands, orientated at angle 𝛼 𝑖 from the direction of shearing. 𝑅 are conjugate Riedels orientated at angle 𝛼 𝑖 from the direction of shearing. Y is the slip localisation at the boundary: 1 is the main localisation zone occupying all the slip surface and 2 is a partially boundary zone at the opposite surface. [MOVIE]

Figure 76 .

 76 Figure 76. Velocity profiles in the -direction for standard sample (M-S). Letters correspond to different steps in the effective friction curve Figure 74. The velocity profile is plotted between -1 and 1 𝑚/𝑠. [MOVIE]

  Chapter 4. Rheology and weakening of a sheared fault gouge with matrix particles i. Interparticle friction Numerical experiments are difficult to calibrate, as input data injected in the model are poorly documented in real granular media (i.e. interparticle friction). A change in interparticle friction coefficient from 0.1 to 0.6 can yield significant differences in friction peak values, Figure 77 (a). But all numerical experiments tend to the same averaged friction at a steady-state (𝜇 = 0. 2).

Figure 77 .

 77 Figure 77. (a) Effective friction 𝜇 as a function of the slip distance (𝜇𝑚) for different interparticle friction (𝜇 𝑛 𝑚 g g f o 0.1 o 0.6). (b) Dilation 𝜀 𝑦 as a function of the slip distance (𝜇𝑚) for the same materials. The solid fraction is displayed for three different interparticle frictions (𝜇 𝑛 𝑚 = 0.1, 0.3 & 0.6), (c) at friction peak B, (d) at slip distance C, (e) at slip distance D. (f) Zoom in on the Weakening zone, point C. The grey and dashed curve presents the (M-S) case.

Figure 78

 78 Figure 78. (a) Effective friction as a function of the slip distance (𝜇𝑚) for different initial shape perturbations (P i = 0.2, 0.5 & 0.8). (b) Dilation as a function of the slip distance (𝜇𝑚) for the same cases. The grey and dashed curve presents the (M-S) case.

Figure 79 .

 79 Figure 79. Solid fraction at two different times of experiments, for the three materials (P i = 0.2, 0.5 & 0.8). (a) At friction peak B, (b) at steady state E. These two different moments are displayed in Figure 78 by points B and E.

Figure 80 . 4 .

 804 Figure 80. The figure displays the evolution of two chains of particles within the principal low-angle 𝑅-band at several instants of the simulation for the two extreme perturbation cases (𝑃 0.2 and 𝑃 0.8 ). (a) Effective friction curves with instants taken (b) Chain 1, one chain of the 𝑃 0.2 case at five different moments. (c) Chain 2, one chain of the 𝑃 0.8 case at five different moments.

Figure 81 .

 81 Figure 81. A simplified view of the influence of the interparticle friction 𝜇 𝑛 𝑚 and the angularity of the particles 𝑃 𝑖 on gouge strength and rheology. 𝜇 is the effective friction during the slip of the gouge. The effective friction peak 𝜇 and is the angle between the direction of shearing and the main Riedel band 𝛼 𝑖 are both increasing with an increase in 𝜇 𝑛 𝑚 or 𝑃 𝑖 .

Figure 82 .

 82 Figure 82. (a) Effective Friction as a function of the interparticle stiffness at friction peak and averaged value for the steady-state zone. The samples are M-S, M-K-2, M-K-5, and M-K-10 (respectively 10 1 P / , 2.10 1 P / , 5. 10 1 P / and 10 16 P / ), for two samples lengths (L=4mm and 20 mm) (b) Peak zone for the two extreme stiffness cases M-S and M-K-10 (10 1 P / and 10 16 P / ) with peak location. (c) The solid fraction between 0.7 and 1 for the two cases in (b) at B.Even after reaching the steady state, it can be observed that the increase in stiffness has completely changed the kinematics inside the gouge, Figure83 (a). The (M-K-10) material has a single final 𝑅-band that runs the length of the gouge and appears to have two slopes with different orientations. In this simulation, the -localization occurred on the lower part of the gouge, as the entire upper part moves as one unit, as seen in the velocity map, Figure83(c3), unlike the (M-S) material which localizes at the top, due to the left Riedel band which only allows part of the gouge to move along the -axis.

  Figure 83. (a) The solid fraction between 0 and 1 for the two samples at E, Figure 82 (b). (b) Details on gouge M-S (b1) Solid fraction between 0 and 1, (b2) Snapshot of the force-chains network (from 0 to 10 4 𝑁 ), (b3) Velocity profiles in the -direction (between 1 and -1 𝑚/𝑠); (c) Details on gouge M-K-10 (c1) Solid fraction between 0 and 1, (c2) Snapshot of the force-chains network (from 0 to 10 4 𝑁), (c3) Velocity profiles in the -direction (between 1 and -1 𝑚/𝑠). Force chains are defined according to the typical level of stress present in the system (here the applied normal stress) and thickness and color are evaluated between the minimum and maximum contact force (0 to 10 4 𝑁). A cut-off is made at 10 4 𝑁 to have the same visualization for all cases (higher force chains are observed for higher stiffness).

  e er o 20 µ𝑚, and 𝐺 𝜙𝑖 the shear modulus corresponding to each size variation (Appendix A4.C). The equivalent stiffness 𝑘 𝑒𝑞 𝑖 is then inserted in the numerical model, leading to the results presented in, Figure 84 (b) and with dotted lines in Figure 84 (c) & (d), corresponding to samples (M-PS-30-Keq), (M-PS-40-Keq), and (M-PS-50-Keq).

Figure 84 .

 84 Figure84. This figure presents results from two simulation campaigns related to global stiffness variation: (a) Effective friction curves for a variation in cells size only (from 20 to 50 µ𝑚 with an interparticle stiffness of 10 1 𝑃𝑎/𝑚) and, (b) Effective friction curves for a variation of both cells size and interparticle stiffness to have the same secant shear modulus for each simulation (interparticle stiffness of 10 1 P / , 6.8 10 14 P / , .7 10 14 P / , 10 14 P / ), (c) Effective friction results as a function of the size of the cells for both simulation campaigns (d) Steady-state averaged dilation (%) as a function of the size of the cells for both simulation campaigns. The grey curve presents the (M-S) case.

Figure 85 .

 85 Figure 85. The solid fraction between 0 and 1 at friction peak B for three samples: (a) 𝜙 𝑒𝑞 = 20 𝜇𝑚 (𝑏) 𝜙 𝑒𝑞 = 50 𝜇𝑚 for the standard stiffness and (c) 𝜙 𝑒𝑞 = 50 𝜇𝑚 with equivalent stiffness.

4 .

 4 Rheology and weakening of a sheared fault gouge with matrix particles Riedel structures are more visible when the size of particles is increased, in Figure86 (d), we suppose that all gouge models evolve as a kind of Riedel structure. The reason why they are more visible with larger particles is that increasing the size of particles, increases the interparticle stiffness and thus force chains between particles.

Figure 86 .

 86 Figure 86. The solid fraction between 0.7 and 1 at point E (steady-state of sliding), for the four samples with equivalent stiffness ((a) 𝜙 𝑒𝑞 = 20 𝜇𝑚 (𝑏) 𝜙 𝑒𝑞 = 30 𝜇𝑚 (c) 𝜙 𝑒𝑞 = 0 𝜇𝑚 and (d) 𝜙 𝑒𝑞 = 50 𝜇𝑚 ). The angle 𝛼 𝑖 represents the orientation of the 𝑅-bands from the direction of shearing and decrease with increasing particle size: 𝛼 1-𝑎 > 𝛼 1-𝑏 > 𝛼 1-𝑐 > 𝛼 1-𝑑 .

  (b).

Figure 87 .

 87 Figure 87. (a) Effective friction as a function of the number of particles within the gouge thickness 𝑛𝑏 1 , (b) averaged dilation at steady-state, obtained for variations in gouge thickness (full markers with a thick line, grey) and variations of the size of particles (yellow markers, study with equivalent stiffness). The pointed rectangles surround the two cases with the same number of particles within the gouge thickness (𝑛𝑏 1 =50 particles) for the sample (M-Th-1) with an initial thickness of 1 mm and equivalent cells diameter of 𝜙 20 and the sample (M-PS-40-Keq) with an initial thickness of 2 𝑚𝑚 and equivalent cells diameter of 𝜙 40 .

Figure 88 .

 88 Figure 88. (a) Effective friction 𝜇 as a function of slip distance 𝑑 (𝜇𝑚). (b) Dilation 𝜀 𝑦 (%) as a function of slip distance 𝑑 (𝜇𝑚). (c) Effective friction 𝜇 as a function of 𝑛 𝑆𝑑 , which is the ratio between the fault slip distance 𝑑 and the equivalent diameter of particles 𝜙 𝑒𝑞 . (d) Dimensionless dilation 𝑛 𝑌 (number of particles) as a function of 𝑛 𝑆𝑑 . The two cases in the hyphen box in Figure 87 are displayed: a simulation with a gouge thickness of 2 mm, an equivalent cell diameter of 𝜙 40 and an equivalent stiffness of .7 10 14 P / (M-PS-40-Keq), and a simulation of 1 mm thickness, with an equivalent cell diameter of 𝜙 20 and the standard stiffness of 10 1 P / , (M-Th-1).

  and similarly, the high-Riedel angles (𝑅') are both oriented at an angle equal to 58 -63 °. From this comparison, a conclusion is that the same number of particles within the gouge thickness (𝑛𝑏 1 = 𝑡ℎ 𝑖 / 𝑒𝑞 ) leads to similar 𝑅'-bands and Riedel structures with the gouge.

Figure 89 .

 89 Figure89. The solid fraction between 0.7 and 1 at steady-state for the two samples in the hyphen box in Figure87: (M-PS-40-Keq) (a simulation with a gouge thickness of 2mm, an equivalent cell diameter of 𝜙 40 and an equivalent stiffness of .7 10 14 P / ) and (M-Th-1) (a simulation of 1 mm thickness, with an equivalent cell diameter of 𝜙 20 and the standard stiffness of 10 1 P / ). They both have 50 particles within the gouge thickness.iv. Influence of shear modulusFigure90gathers all the simulations presented in this section in a graph of the secant shear modulus as a function of 𝑛𝑏 1 (number of particles within the gouge thickness). As previously presented, an increase in the global stiffness of the gouge (interparticle stiffness or size of particles) increases both the shear modulus and the maximum shear stress (orange and purple lines). The results with an augmentation of the size of particles, but with equivalent stiffness, almost delete the effect on shear modulus (blue line). However, a modification in the gouge thickness does not influence the shear modulus nor the maximum shear stress (green line).

Figure 90 .

 90 Figure 90. Secant shear modulus (𝐺𝑃𝑎) as a function of the number of particles within the gouge thickness 𝑛𝑏 1 for different values of maximum shear stress (𝑀𝑃𝑎) (colored marker). The color of arrows corresponds to the different parameters increased and the color of the diamonds to the maximum shear stress (𝑀𝑃𝑎).

  the 𝑅-bands at steady-state. It is interesting to note that almost all the tested parameters have an influence on the orientation angle of Low Riedel bands except the angularity of the cells which does not allow to conclude on a significant modification. The three relevant quantities in a change of principal orientation angle of low 𝑅-bands at steady-state are: (a) interparticle friction, Figure 91 (a), (b) the geometry of the model through the number of particles within the gouge thickness 𝑛𝑏 1 previously investigated, Figure 91 (b), and (c) the shear modulus.

c)

  At steady-state, a change in shear modulus also has a huge influence on Riedel bands orientation, Figure91(a) orange and purple lines. A drastic decrease in Riedel orientation is observed with an increase in Shear modulus.

Figure 91 .

 91 Figure 91. The orientation angle of 𝑅-bands at steady-state (averaged value for different 𝑅-bands), for all the simulations, as a function of the (a) effective friction peak 𝜇 , (b) the number of particles within the gouge thickness 𝑛𝑏 1 (i.e. thickness of the gouge divided by the size of particles). The sense of the arrow indicates the increase of the different parameters and the names correspond to the ones presented in the previous sections. In figure (a) the interparticle friction 𝜇 𝑛 𝑚 evolves from left to right.

Figure 92 .

 92 Figure 92. Schematic view of the previous Figure 91 (a) of the orientation of R-bands as a function of the effective friction peak, with distinction between 4 major zones of interest: (i) reduction of interparticle friction 𝜇 or 𝑛𝑏 1 (without modification of the shear modulus), (ii) Increase in 𝜇 , (iii) Decrease of 𝑛𝑏 1 with increase in shear modulus (iv) Increase in gouge thickness. The sense of the arrow represents an augmentation of all the parameters previously presented as detailed within the legend. Main Riedel structure at steady-state are represented below the figure and corresponds to the two extreme cases of interparticle friction study (a) and (b) and of thickness study (c) and (d). Details on Shear modulus values can be found in Appendix 4.C.

Figure 93 .

 93 Figure93. Schematic view of the shear modulus G (𝐺𝑃𝑎) as a function of the ratio 𝑛𝑏 1 , with distinction between 4 major zones of interest: (i) reduction of interparticle friction 𝜇 or 𝑛𝑏 1 (without modification of the shear modulus), previously presented in Figure92;(iii) Decrease of 𝑛𝑏 1 with increase in shear modulus (iv) Increase in gouge thickness, (v) Thin gouges with no Riedel formation. The sense of the arrow represents an augmentation of all the parameters previously presented as detailed within the legend. Main Riedel structure at steady-state are represented below the figure and corresponds to the extreme cases of thickness study (d), variation of particle size (e) and (f), and variation of interparticle stiffness (g). Details on Shear modulus values can be found in Appendix 4.C.

  importance of these factors in governing the elastic and mechanical properties a change in the mechanical property of the gouge on its shear and sliding behavior. But the main conclusion remains the importance of a co-evolution between mechanical and elastic gouge properties from one side, and kinematics of shearing from the other side.

Figure 1 .

 1 Figure 95. Main schema of Chapter 4, Part B 4.7. Model and numerical framework of a composite mixture 4.7.1. Generation of the gouge sampleA new granular sample with both angular particles and matrix particles is created. For the sake of simplicity and to easily compare results with previous parts, the model of Chapter 3 with angular grains is partly reused. This granular sample presents a gouge already sheared with a fractal distribution of particles settled to 𝐷 = 2.6. To create the new composite mixture, the granular sample generated in Chapter 3 is kept, and a certain amount of the smallest angular grains is replaced by matrix particles. The first step consists in removing a certain percentage of the smallest particles within the selected sample, Figure96 (a). In a second time, this material (a) is merged within matrix particles sample (M-S) to form the new composite mixture, Figure

Figure 96 .

 96 Figure 96. (a) Granular material from Chapter 3 with a certain number of small particles removed, (b) Mixture between angular particles and matrix particles.The ratio between the area of matrix particles and the area of angular particles gives the surface percentage of matrix particles within the sample 𝑃 𝑚𝑎 𝑖𝑥 :

Figure 98 .

 98 Figure 98. (a) Effective friction 𝜇 as a function of slip distance (𝜇𝑚) for the different surface percentages of matrix particles (P ix = 0, 33, 56 & 100 %) respectively corresponding to samples (G-C-4), (MG-33), (MG-56) and (M-S). (b) Peak of effective friction as a function of the percentage of matrix particles (triangles) and averaged value for the steady-state zone (diamonds). Sample (G-C-4) also has a different initial thickness of 1.7 𝑚𝑚, as opposed to 2 𝑚𝑚 for the other cases and a different interparticle friction 𝜇 𝑛 𝑚 = 0.5.

Figure 99 .

 99 Figure 99. Dilation 𝜀 𝑦-𝑆𝑆 and void ratio 𝐹 𝑣-𝑆𝑆 (porosity) at steady state as a function of the matrix percentage 𝑃 𝑚𝑎 𝑖𝑥 within the gouge. (Dilation curves in Appendix 4.D)

Figure 101 .

 101 Figure 101. New sample MG-H-86 with a bi-disperse mixture between matrix particles and 50 angular grains homogeneously distributed and of the same equivalent diameter ( 𝑒𝑞 = 315 𝜇𝑚). (MG-H-86, for Matrix-Grains sample with Homogeneous distribution and 86% of matrix particles)

Figure 102 .

 102 Figure 102. Maximum shear stress 𝜏 𝑚𝑎𝑥 (𝑀𝑃𝑎) and Initial tangent shear modulus 𝐺 𝑎𝑛 (𝐺𝑃𝑎) as a function of the percentage of matrix particles 𝑃 𝑚𝑎 𝑖𝑥 within the sample. (i) Zone with one type of granular material (only grains or only matrix), (ii) zone with composite granular samples (grains and matrix).

  (a) & (b) image 1. The same Riedel bands are observed for both samples. Gouge (MG-56) have different Riedel bands at effective friction peaks, which are more numerous and homogeneously distributed within the gouge, maybe due to the non-homogeneity of the er k d o "de e r red e work" w ere o r ede d manages to develop sufficiently.

Figure 103 .

 103 Figure 103. Effective friction 𝜇 as a function of slip distance (µ𝑚) for three granular samples: (a) the standard gouge sample (M-S) with only matrix particles, (b) the gouge (MG-H-86), and (c) the gouge with matrix particles and fractal distribution of angular particles (MG-56). The three granular materials are considered to have identical initial porosity (𝑃 𝑜 𝑒 ~ 0 %). Zone A is the pre-peak zone, zone B is the weakening zone and zone C is the steadystate zone.

  . It is supposed that the shearing zone will be extended to the entire gouge thickness for a longer time of the experiment. These results seem consistent with those of (Y.[START_REF] Lu | Methodology for Simulation of Irregularly Shaped Gravel Grains and Its Application to DEM Modeling[END_REF] increasing percentage of clasts contents, or[START_REF] Wojatschke | Geochemistry, Geophysics, Geosystems[END_REF] increasing clay content within a fault gouge mixture of clay matrix and clasts: by increasing clast content the development of Riedel shears is less evident.[START_REF] Wojatschke | Geochemistry, Geophysics, Geosystems[END_REF] also noticed that the percentage of clasts controlled the degree of orientation of the strain localization.

Figure 104 .

 104 Figure 104. Riedel bands evolution through solid fraction representation (0.7 to 1) for the three different samples: (a) the standard sample (M-S) with only matrix particles, (b) the new sample (MG-H-86), and (c) the sample with matrix particles and fractal distribution of angular particles (MG-56). (i) Zoom in on the steady-state of the sample (MG-H-86) and (ii) zoom in on the steady-state zone of the sample (MG-56).

  Casas, N., Mollon, G., & Daouadji, A. (2022) Energetic study and simplified models for cemented granular fault gouges. (in prep) Casas, N., Mollon, G., & Daouadji, A. (2022) Time and space evolution of Riedel bands in a dense granular material, relation to the evolution of the entire gouge. (in prep) Chapter 5. From energy budget to simplified models, towards dynamic simulations

Figure 105 .

 105 Figure 105. Main schema of Chapter 5, Part A. All the data are collected from each model.

Figure 106 . 2 .

 1062 Figure 106. Illustration of the two parameters previously described for the standard case M-S (Chapter 4) where the shear stress variation (Pa) is observed as a function of the slip distance (m). The slope K = |𝜏 𝑆𝑆 -𝜏 p | 𝐷 𝑐 is calculated to have the same energy in 𝐽/𝑚² for the triangle generated by the slope line than the 𝐸 𝐺 (colored in yellow). Breakdown energy can be calculated as E G = ∫ (𝜏 (𝑈) -𝜏 ) 𝑑 𝑈 𝐷 𝑐 0

Figure 107 .

 107 Figure 107. (a) A schematic figure showing the nucleation length 𝐿 on a frictional fault plane, (figure redrawn from (Kanamori & Brodsky, 2004)). (b) Initiation of dynamic rupture from a smooth nucleation model, (figure redrawn from (McLaskey, 2019)).

  (a1), the size of particles from 20 to 50 𝑚𝑚, Figure 108 (a2), or even the interparticle stiffness from 10 1 to 10 16 𝑃𝑎/𝑚, Figure 108 (a3).

Figure 108 .

 108 Figure 108. Evolution of the breakdown energy (𝐽/𝑚²) as a function of different parameters. (a) Matrix samples (M-… w o e parameters variation: (a1) as a function of the interparticle friction 𝜇 𝑛 𝑚 , (a2) as a function of the size of particles 𝑒𝑞 (mm), (a3) as a function of the interparticle stiffness in 𝑘 𝑛 (Pa/m), (a4) as a function of the cell perturbation 𝑃 𝑖 . (b) Composite mixture between matrix and grains (MG-samples) with variation of matrix percentage. (c) Cemented samples (G-samples) with variation in the percentage of cement for both dense and middense samples.

Figure 109 .

 109 Figure 109. Evolution of the linear slope of weakening 𝑐 (𝐺𝑃𝑎/𝑚) as a function of different parameters. (a) Matrix samples (M-… w o e parameters variation: (a1) as a function of the interparticle friction 𝜇 𝑛 𝑚 , (a2) as a function of the size of particles 𝑒𝑞 (mm), (a3) as a function of the interparticle stiffness in 𝑘 𝑛 (Pa/m), (a4) as a function of the cell perturbation 𝑃 𝑖 . (b) Composite mixture between matrix and grains (MG-samples) with variation of matrix percentage. (c) Cemented samples (G-samples) with variation in the percentage of cement for both dense and mid-dense samples.

  Figure 111. Evolution of the critical nucleation length 𝐿 𝑐 (m) as a function of different parameters. (a) Matrix samples (M-… w o e parameters variation: (a1) as a function of the interparticle friction 𝜇 𝑛 𝑚 , (a2) as a function of the size of particles 𝑒𝑞 (mm), (a3) as a function of the interparticle stiffness in 𝑘 𝑛 (Pa/m), (a4) as a function of the cell perturbation 𝑃 𝑖 . (b) Composite mixture between matrix and grains (MG-samples) with variation of matrix percentage. (c) Cemented samples (G-samples) with variation in the percentage of cement for both dense and mid-dense samples.

Figure 112 .

 112 Figure 112. Evolution of the critical nucleation length 𝐿 𝑐 (m) as a function of the breakdown energy (𝐽/𝑚²) for different parameters. Matrix samples (M) with matrix parameters variation are colored in dark blue, (b) Matrixgrains samples (MG) composite mixture) with a variation of surface percentage of matrix particles are colored in light blue. Cemented gouge samples (G) with a variation of the surface percentage of cement for both dense (colored in pink) and mid-dense samples (colored in red). The black line represents the linear regression of the data set for the selected zone. The three different zones are explained within the text.

  The d o o e o e or deformation in the -direction perpendicular to the sliding e d e coming from the breakage of cohesive bonds due to cementation -The friction generated by sliding contact interactions These contributions will be calculated thanks to global potential energy recovered at each time step, from the data set presented in Chapter 3 and computations details presented in the next sections, Figure 113. The o je e o r o o "re-d o er" e e involved in fracture mechanics, but above all to be able to clearly locate and quantify these three mechanisms and their interaction with fault rheology.

Figure 113 .

 113 Figure 113. Main schema of Chapter 5, Part B. All the data are collected from cemented granular gouge model (Chapter 3).

Figure 114 .

 114 Figure114. Proposed model (function shape strongly varies depending on cementation and initial porosity), with the decomposition of the breakdown energy 𝐸 𝐺 . 𝐸 𝐺 gathers a dilation energy 𝐸 𝐷𝑖 linked to sample dilation (mechanical energy), a Damage energy 𝐸 𝐷𝑎𝑚 coming from the breakage of cementation (surface creation energy) and 𝐸 𝑓 a part of Coulomb energy 𝐸 𝑓 . 𝜏 is the maximum shear stress, 𝜏 is the steady-state shear stress, or dynamic shear stress. 𝐷 is the duration of the slip weakening part (post-peak), similar to the critical weakening distance. Coulomb energy 𝐸 𝑓 collects all the Coulomb energy that is not considered in the constant part 𝐸 𝑓𝑐 = 𝜏 𝐷 . Note that in the sketch, 𝐸 and 𝐸 𝑓𝑐 are in Joule and not in J/m² as in the following equations.

Figure 116 .

 116 Figure 116. Dense sample with 57 % of cementation (a) Energy budget (𝐽/𝑚 2 ) as a function of slip distance (m) for the different contributions. (b) Effective friction 𝜇 extracted from energy consumption. (c) Energy budget (𝐽/𝑚 2 ) in pre-peak and post-peak phase; (d) Cementation spent and remaining expressed as cohesive energy (𝐽/𝑚 2 ) in the sample as a function of the slip distance (m).

Figure 117 (

 117 Figure117 (a) shows the evolution of effective friction (black curve) and its different contributions as a function of the slip distance, Figure116(b), for this granular sample: the green curve is the frictional or Coulomb contribution 𝜇 𝑓 , the dilation contribution 𝜇 𝑑𝑖 appears in red and the damage contribution 𝜇 𝑑𝑎𝑚 in blue. The first step consists in selecting and recovering all the essential parameters from these three contributive frictions and making appear a global tendency of evolution for each of them. Figure117 (b), (c), and (d) present an example with friction contributions and simplified models, highlighting the main parameters selected to reproduce a reliable empirical simplified model. These parameters evolve as a function of the surface percentage of cementation and initial porosity within the gouge, Figure118:

  (b).

Figure 117 .

 117 Figure117. As an example of the method, a dense sample with 30 % of cementation (G-C-30) is used. (a) Frictionslip curve with different contributions (dilatancy in red, Coulomb in green, and damage in blue), dense sample with 𝑃 𝑐𝑒𝑚 = 30 %. The black curve is the sum of all contributions. (b) Zoom in on the Coulomb contribution with simulation data in green and simplified model in dark, two behaviors are observed (B1 and B2). The interesting parameters are: the effective friction peak at the end of the elastic phase 𝜇 𝑓 , the post peak friction 𝜇 𝑓 and the characteristic distance of the exponential decay ∆𝑈 𝑓 . B1 corresponds to the granular sample studied with 𝜇 𝑓 > 𝜇 𝑆𝑆 and B2 is an example of the second type of behavior, observed for the 30 % mid-dense case with 𝜇 𝑓 ≤ 𝜇 𝑆𝑆 . (c) Zoom in on dilation contribution 𝜇 𝑑𝑖 , simulation data in red and simplified model in dark with: the maximum dilatancy angle Ψp, the slip distance corresponding to the maximum dilation 𝑈 𝑑 and the gain in gouge thickness ∆𝐻 𝑑 at the end of the dilation phase. (d) Zoom in on the damage contribution 𝜇 𝑑𝑎𝑚 with the characteristic distance of the exponential decay ∆𝑈 𝑑𝑎𝑚 and the maximum friction induced by breakage of cohesive links 𝜇 𝑑 . It can be noted that k, μ , and ∆𝑈 𝑓 do not depend on the cementation and are taken as constant.

  observed in Figure117(b), B1 and B2. The first one has a peak behavior where the effective friction peak 𝜇 𝑓 is higher than 𝜇 𝑆𝑆 , whereas the second one has an asymptotic behavior with 𝜇 𝑓 ≤ 𝜇 𝑆𝑆 , meaning no friction peak. Pushing up the cementation increases both Coulomb friction peak 𝜇 𝑓 and post-peak friction 𝜇 𝑓 following a linear law, Figure118Coulomb contribution 𝜇 𝑓 Chapter 5. From energy budget to simplified models, towards dynamic simulations (d), with a characteristic distance ∆𝑈 𝑓 remaining almost constant for dense samples, and decreasing for mid-dense samples as a function of cementation, Figure118 (e

Figure 118 .

 118 Figure 118. Behavior laws of each of the 7 parameters considered in the simplified models. It can be noted that k, 𝜇 𝑆𝑆 , and ∆𝑈 𝑓 do not depend on cementation and are taken as constant. (a) Maximum dilatancy angle Ψp (°), the difference observed between prediction and real values comes from a compromise between the friction peak value and a consistent friction sh e ro e d o We o e " er e" e o Ψ ow o obtain a consistent shape. (b) Slip distance corresponding to the maximum dilatancy 𝑈 𝑑 (𝜇𝑚). (c) Gain in gouge thickness at the end of the dilatancy phase ∆𝐻 𝑑 (𝜇𝑚). (d) Peak friction at the end of the elastic phase 𝜇 𝑓 and post peak friction 𝜇 𝑓 . (e) Characteristic distance of the exponential decay ∆𝑈 𝑓 (𝜇𝑚). (f) Maximum friction induced by rupture of cementation 𝜇 𝑑 . Dots are experimental data derived from DEM modeling with error bars. Some are not aligned with numerical modeling as some peak values are difficult to identify on raw data and the error can be important. ∆𝐻 𝑑 and ∆𝑈 𝑓 do ' e e er e d e re o e e re ed or e ode

  ' w o er ed e re o Figure 118, it is now possible to propose a simplified model for each friction contribution 𝜇 𝑑𝑖 , 𝜇 𝑓 d 𝜇 𝑑𝑎𝑚 .

  layer stiffness) from 0 to 𝜇 𝑓 . Then friction decreases linearly until a post-peak value called 𝜇 𝑓 . Friction contribution finally diminishes exponentially until 𝜇 𝑆𝑆 taken equal to 0.45 to simplify the model. Each part is modeled by a specific equation: With ∆𝑈 𝑓 a characteristic distance of the exponential decay, 𝑈 𝑓 = 𝜇 𝑓𝑝 𝑘 , ∆𝑈 𝑓 the slip distance between 𝑈 𝑓 and 𝑈 𝑓 and 𝑈 𝑓 = 𝑈 𝑓 ∆𝑈 𝑓 .

Figure 119 .

 119 Figure 119. Mid-dense and dense granular samplesvalidation of simplified models. Effective friction as a function of the percentage of cementation. Effective friction contributions are colored (dilation in red, Coulomb in green, and damage in blue) and the black curve is the sum of all contributions. (a) Poorly cemented materials, where the observed pre-peak phase is important. (b) Cemented materials with a diminution of the pre-peak phase with the increase of cementation. (c) Ultra cemented materials with almost no pre-peak phase. Dashed lines re re e re ro o d e o d e e r ed ed w ed ode ' e o

  10 3 𝐽/𝑚² for dense materials, Figure 120 (a) & (b). It is characteristic of the ductile trend highlighted in Chapter 3 with progressive particle reorganization and no localized shear requiring less energy from the system.

Figure 120 .

 120 Figure 120. Breakdown energy 𝐸 (𝐽/𝑚²) coming from the different contributions as a function of the percentage of cementation within the gouge. (a) Mid-dense samples (b) Dense samples. 𝐸 (in black) gathers a dilation energy 𝐸 𝑑𝑖 -(in red) linked to sample dilation (mechanical energy), a Damage energy 𝐸 𝑑𝑎𝑚-(in blue) coming from the breakage of cohesive links (surface creation energy) and 𝐸 𝑓 -(in green), a part of the total Coulomb energy 𝐸 𝑓 . Dashed lines represent the sum of the breakdown energy and the pre-peak energy for each contribution.

Figure 121 .

 121 Figure 121. Simplified model for each cemented material, the dashed line represents the effective friction peak. (a) Mid-dense samples: (a.1) Poorly cemented materials, where the observed pre-peak phase is important, (a.2) Cemented materials with a diminution of the pre-peak phase with the increase of cementation, (a.3) Ultra cemented materials with no pre-peak phase. (b) Dense samples:(b.1) poorly cemented materials with a diminution of the prepeak phase with the increase of cementation, (b.2) Cemented materials with a negligible pre-peak phase but an important influence of friction peak, (b.3) Ultra cemented materials with almost not pre-peak phase and a very high Coulomb friction peak.

Figure 122 .

 122 Figure 122. Schematic view of the different mechanisms operating within dense materials for different surface percentages of cementation. Poorly cemented materials (left) have low peak strength, small breakdown energy and high critical nucleation length. Ultra cemented materials (right) have very high peak strength, high breakdown energy and a very small critical nucleation length. Cemented materials (center) have high peak strength, very high breakdown energy and a medium critical nucleation length.

  < 𝐿 𝑐 < 1𝑚 𝐿 𝑐 < 0.1𝑚 Chapter 5.

Figure 123 .

 123 Figure 123. The main schema of chapter 5, Part C. All the data are collected from matrix gouge models, Chapter 4.

Figure 124 .

 124 Figure 124. Schema of the Riedel and shear bands observed during the shearing of the M-S standard case used in Chapter 4. (a) Friction-Slip curve with the three different zones presented in Chapter 4 (pre-peak, weakening, steady-state), associated with pictures of the evolution of Riedel and boundary shear bands. (b) Name and position of the different types of bands, the primary R-bands 𝑅 3 , 𝑅 4 and 𝑅 , the secondary R-bands 𝑅 1 and 𝑅 2 , and boundary shears 1 and 2 . 𝑅 represents the conjugate Riedel bands. [video of the simulation] 5.10. Opening of R-bands 5.10.1.Method and first results

  Figure 125. (a) Snapshot of the model at steady-state, where Riedel band 𝑅 1 can be observed (only half part of the model here). (b) Sketch of the method used to measure the Riedel band opening. We first select points on each side of the interested band and measure the vertical displacement perpendicular to the band direction, called ∆Y op n = ∆ 𝑖 -∆ 0 (for more details, see appendix 5.D). The same method is applied on the displacement in the band direction, ∆ 𝑖 = ∆ 𝑖 -∆ 0 .

  ∆Y op n at the end of the shearing is twice the one of primary bands and corresponds to the size of two particles (~ 0 𝜇𝑚).

Figure 126 .

 126 Figure 126. Opening ∆Y op n (𝑚) and total slip ∆ 𝑖 (𝑚) of the R-bands as a function of the 𝑅 -direction of the Riedel band for different total slips during the simulation (m), (a) for the 𝑅 1 band, (b) for the 𝑅 2 band, (c) for the 𝑅 3 band, (d) for the 𝑅 4 band. The four different bands are presented in Figure 124.

Figure 127 .

 127 Figure 127. Opening ∆Y op n (𝑚) and total slip ∆ 𝑖 (𝑚) of the Y-bands as a function of the -direction of the gouge for different total slips during the simulation (m), (a) for the 1 band, and (b) for the 2 band. The two different bands are presented in Figure 124.

  (b), for the relevant R and Ybands presented in Figure124, gives the same trends of evolution that have been presented in chapter 4. The observed effective friction peak is not very sudden, and 𝑅-bands have time to grow between the end of the elastic part (A) and the friction peak (B) (letters are from effective friction curve Figure128 (a). The three different behaviors (primary and secondary Riedels and Y bands) presented in the previous section are now evident. ∆Y op n-v allows to localize the important events in the life of the different bands and their interactions with each other:

Figure 128 .

 128 Figure 128. (a) Effective friction for the standard case (M-S), letters A to E are remarkable points and changes in Riedel band evolution. (b) Opening of the three presented bands, ∆ 𝑜 𝑒𝑛-𝑎𝑣 (m) (average value over the entire length of the band). The primary bands (𝑅 3 ,𝑅 4 and 𝑅 ) appear in red, the secondary bands (𝑅 1 and 𝑅 2 ) appear in bleu, and the boundary shears ( 1 and 2 ) appear in green. Both variables 𝜇 and ∆ 𝑜 𝑒𝑛-𝑎𝑣 are presented as a function of the total slip distance (m) of the gouge.

  (b).

Figure 129 .

 129 Figure 129. (a) Opening of the total gouge ∆𝑡ℎ 𝑜 -𝑆𝑆 at steady state (empty circles) and opening of the secondary and Y-bands ∆ 𝑜 𝑒𝑛-𝑆𝑆 at steady state for the two sizes of particles 20 and 40 . (b) Opening ratio of the bands to the total gouge ∆𝑡ℎ 𝑜 -𝑆𝑆 , for each band and for the two simulations with equivalent cells diameter of 20 and 40 , at the steady-state (letters E from Figure 124). The blue dotted circle selects the secondary and Y -bands, while the green circle surrounds the Primary bands (𝑅 2 and 𝑅 3 ).

Figure 130 .

 130 Figure 130. (a) Entire gouge with Riedel bands. The Riedel thickness 𝑅 ℎ 𝑖 of about 10 particles with 𝜙 𝑒𝑞 = 20 𝜇𝑚. (b) Simulated thin band with the same thickness as the observed 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 , 𝑅 ℎ 𝑖 = 200𝜇𝑚, and sheared within the same conditions as the previous models.For the standard case (M-S), the total opening of the entire gouge, as well as the one of the thin bands associated with the model, is presented in Figure131. Effective friction is obviously higher for the thin band presenting a thickness ten times smaller. The order of magnitude of the

  (b) with the different dilation-sliding trajectories corresponding to the main R-bands and Y-bands present within the gouge (i.e. ∆ 𝑜 𝑒𝑛-𝑎𝑣 = 𝑓(∆ 𝑖 )), Figure 132. It can be seen that each R-band seems to follow the initial slope of the thin band opening (until 𝑖 = 1 10 -4 𝑚), implying a similarity in behavior and giving us information on when and to which point the bands participate in the global behavior.

  each value of slip distance from the total behavior of the gouge, we can zoom in on each Y and Rsection presents the results and assumptions used to link local and global behavior in order to get a simplified frictional model of a gouge with matrix particles from these results.

Figure 132 .

 132 Figure 132. Opening of the entire gouge [ 𝑜 𝑒𝑛 = 𝑓(𝑈)] for the standard case (20 × 2 𝑚𝑚², dashed grey line) and the associated thin band (20 × 0.2 𝑚𝑚², dashed blue line), and dilation-sliding trajectories of each band [∆ 𝑜 𝑒𝑛-𝑎𝑣 = 𝑓(∆ 𝑖 )]. For (M-S) and (TB-S), U is the actual slip distance, and for the R-bands, U is the respective displacement within each band (called ∆ 𝑖 ) and total slip distance for the dashed grey line.

Figure 133 .

 133 Figure 133. (a) Effective friction on the entire gouge (black curve) as a function of the slip distance. Note that the slip distance is expressed as a dimensionless quantity which is the ratio between the slip distance (m) under the size of particles (m), giving an equivalent number of particles. The energetic contribution of each band during the shearing is colored under the friction curve such as the legend. Zone from (i) to (v) are points of interest explained within the text. (b) Schema of the standard gouge (M-S) with the principal Riedels and boundary shears observed.

  energy 𝐸 𝑓 = 𝐸 𝑜 -𝐸 𝐷𝑖 , Figure 134 (b)

Figure 134 .

 134 Figure 134. (a) Dilation contribution to the effective friction of the entire gouge (black curve) as a function of the slip distance. The energetic contribution of each band to dilation energy is colored under the friction curve such as the legend. Zone (i) is a point of interest explained within the text. (b) Frictional contribution to the effective friction of the entire gouge (black curve) as a function of the slip distance. The energetic contribution of each band to frictional energy is colored under the friction curve such as the legend. Note that the slip distance is expressed as a dimensionless quantity which is the ratio between the slip distance (m) under the size of particles (m), giving an equivalent number of particles.

  de -activate the primary bands. The Y bands evolve progressively until steady-e w ere e e o e e o " e" d representation of Riedel bands is really promising to realize microanalyses of the inside of each R-and Y-bands.

Figure 135. 2 -

 2 Figure 135. 2-blocks model with RS friction. (a) Schema of the model, (b) Displacement versus time for block 1. Arrows indicate cycles when both blocks slipped simultaneously. (From Abe and Kato, 2013 and (Scholz, 2019)), (c) Image of dynamic rupture models with mesh discretization and the off-fault fractures. Every interface betweenelements is regarded as a potential failure plane. Typical friction laws can be implemented within the code, and image from(Okubo et al., 2019a).

  distributed, Figure136. This could induce locking points within the fault, changes in dilatancy, and slip localization zones.

Figure 136 .

 136 Figure 136. Schema of a granular fault gouge with two different materials non-homogeneously distributed.𝜏 is the applied shearing stress and 𝜎 𝑁 the normal stress.

Figure 137 .

 137 Figure 137. Image modified from (Quacquarelli, 2021) with CZM -o w w " e" w

Figure 138 .

 138 Figure 138. Size of gouge particles (average) in (µ𝑚) as a function of the percentage of the matrix with the granular sample. Graph from Heiarii Pons and created thanks to the images below and description in the associated papers.

Figure 139 .

 139 Figure 139. (a) & (b) Gneiss (15% chlorite), from San Gabriel Mountains, California, (Sammis et al., 1987); Optical photomicrograph showing the microtexture of (c) granitic cataclasite from the Itoigawa-Shizuoka tectonic line at Shimotsutaki Village, (d) the cataclastically deformed granitic conglomerate, Tanakura, Japan. Images from (Monzawa & Otsuki, 2003). (quartz et feldspar) (e) [N°21] Photomicrograph (crossed polars) of ATTL fault gouge.The particles are angular in shape, with fractal distribution ranging between 1.86 ≤ 𝐷 ≤ 2.92 and sizes between 17.5 ≤ ≤ 80 µ𝑚. Friable rocks and lack of cohesion. Images from[START_REF] Muto | Fractal particle size distribution of pulverized fault rocks as a function of distance from the fault core[END_REF].

Figure 141 .

 141 Figure 141. Effective friction 𝜇 (peak in orange and stady-state in blue) as a function of particle density (𝑘𝑔/𝑚 3 ), for constant time step and adaptative time step.

  particles. It is possible to play on the size of particles 𝑒𝑞 (equivalent diameter of particles, r ed o o re d o o e " er r o r e er " 𝑃 0.𝑖 (i.e. the

Figure 142 .

 142 Figure 142. Variation of the equivalent size of matrix particles with angular grains. From 𝑒𝑞 = 10 𝜇𝑚 to 𝑒𝑞 = 25 𝜇𝑚

Figure 143 .

 143 Figure 143. Evolution of solid fraction 𝐹 𝑆 (in grey) and gouge thickness 𝑡ℎ 𝑖 (in orange) as a function of the interparticle friction 𝜇 𝑛 𝑚 used to compact samples.

Figure 144 .

 144 Figure144. Effective friction 𝜇 as a function of the initial solid fraction 𝐹 𝑆 (at the end of the compacted state), results from the shearing of granular samples with the following parameters (𝜎 𝑁 = 0 𝑀𝑃𝑎, 𝑉 = 1𝑚 𝑠 ⁄ , 𝜌 = 2600 𝑘𝑔 𝑚3 ⁄ , 𝑘 = 10 1 𝑃𝑎/𝑚, 𝜇 𝑛 𝑚 = 0.5).

Figure 145 .

 145 Figure 145. Effective friction 𝜇 as a function of numerical stiffness 𝑘 𝑛 (𝑃𝑎/𝑚).

With

  𝜎 the normal stress applied on a contact (Pa), 𝜀 the deformation of the contact, 𝐶 𝑛 𝑚 the numerical cohesion implemented in the cemented materials (Pa), 𝑘 𝑛 the numerical stiffness (Pa/m), 𝜙 𝑔 the equivalent particle diameter (m).

Figure 146

 146 Figure 146 displays the equivalent Young modulus 𝐸 as a function of the interparticle stiffness for different equivalent grain diameter 𝜙 𝑔 and comparing values to those observe in real rocks (blue band). Using an interparticle stiffness of 10 1 𝑃𝑎/𝑚 gather more values of young modulus from common rocks than the other values.

Figure 146 .

 146 Figure 146. Equivalent Young modulus (GPa) as a function of the interparticle stiffness 𝜇 𝑛 𝑚 for the maximum equivalent grain diameter 𝜙 𝑚𝑎𝑥 and the minimum grain diameter 𝜙 𝑚𝑖𝑛 . The blue band represents the range of Young modulus values for common rocks.

Figure 147 .

 147 Figure 147. (a) Effective friction peak 𝜇 and steady-state effective friction 𝜇 (respectively in grey and black) and dilation at steady-state 𝜀 𝑦-(in red) for the four samples randomly generated with the same initial conditions.

Movie S1 .

 S1 Comparison of the evolution of relative damage with slip distance for dense samples (entire granular gouge), between 8% cementation (mildly cemented material) and 38% cementation (cemented material). From zero imposed slip [A] to the beginning of steady-state [G]. Movie S2. Comparison of the evolution of relative damage with slip distance for dense samples (entire granular gouge), between 38% cementation (cemented material) and 95% cementation (ultra-cemented material). From zero imposed slip [A] to the beginning of steady-state [G]. The following movies present the evolution of solid fraction as a function of the slip distance for the three regimes highlighted in the paper. Another way to observe Riedel bands and cracks within the gouge. Movie S3. The solid fraction in the dense sample (entire granular gouge) as a function of the slip distance for 8% cementation (mildly cemented material). From zero imposed slip [A] to the beginning of steadystate [G]. Movie S4. The solid fraction in the dense sample (entire granular gouge) is a function of the slip distance for 38% cementation (cemented material). From zero imposed slip [A] to the beginning of steady-state [G]. Movie S5. The solid fraction in the dense sample (entire granular gouge) as a function of the slip distance for 95% cementation (ultra-cemented material). From zero imposed slip [A] to the beginning of steadystate [G].

Figure 148 .

 148 Figure 148. (a) Effective friction 𝜇 as a function of the slip distance (mm) for different sizes of models, 𝐿 2 (2x2 mm²), 𝐿 4 (4x2 mm²), 𝐿 10 (10x2 mm²), 𝐿 20 (20x2 mm²), 𝐿 40 (40x2 mm²). 𝜇 is the peak friction and 𝜇 the steady state friction; (b) Dilation as a function of the slip distance (mm) for the same sizes of models.

Figure 149 .

 149 Figure 149. Effective friction 𝜇 and dilation 𝜀 𝑦 as a function of the effective friction for two different gouge thicknesses, 1 and 3 mm.

Figure 150 .

 150 Figure 150. Automatic calculation with Matlab 2020a. Sketch of the shear stress as a function of the shear strain with the secant shear modulus at 40 % of the maximum shear stress (red) and the initial tangential shear modulus (blue). Example realized for the thin gouge of the standard case (size 20 x 0.2 𝑚𝑚², 𝜇 𝑛 𝑚 = 0.3, 𝑘 𝑛 = 10 1 𝑃𝑎/𝑚, P0.5, = 20𝜇𝑚).

Figure 151

 151 Figure 151 presents friction and dilation curves as a function of the slip distance for the four samples detailed in chapter 4, part B: (G-C-4), with 𝑃 𝑚𝑎 𝑖𝑥 = 0 %, (MG-33) with 𝑃 𝑚𝑎 𝑖𝑥 = 33 %, (MG-56) with 𝑃 𝑚𝑎 𝑖𝑥 = 56 % and (M-S) with 𝑃 𝑚𝑎 𝑖𝑥 = 100 %.

Figure 151 .

 151 Figure 151. (a) Effective friction as a function of the slip distance (mm) for different percentages of matrix particles (P ix = 0, 33, 56 & 100%). (b) Dilation (%) as a function of the slip distance (mm) for the same cases.The first points for simulation with 0% of matrix particles or only angular particles have been made with interparticle friction of 0.5 which is higher than the one used in the other studies.

Figure 152 .Figure 153 .Figure 154 .

 152153154 Figure 152. Shear stress (Pa) as a function of the slip distance (m) for dense cemented models (Chapter 3). Percentage of cementation from 𝑃 𝑐𝑒𝑚 = 95%. The red slope is the linear slip weakening slope needed to equal the breakdown energy 𝐸 𝐺 .

Figure 155 .

 155 Figure 155. Shear stress (Pa) as a function of the slip distance (m) for matrix models (chap. 4, part A). (a) to (d): Variation of the size of particle from 𝑒𝑞 = 20 50 𝜇𝑚 . (e) to (h): Variation of the size of particle from 𝑒𝑞 = 20 50 𝜇𝑚 with equivalent stiffness. The red slope is the linear slip weakening slope needed to equal the breakdown energy 𝐸 𝐺 .

Figure 157 .

 157 Figure 157. Shear stress (Pa) as a function of the slip distance (m) for matrix models (chap. 4, part A). Variation of the interparticle stiffness from 𝑃 𝑚𝑎 𝑖𝑥 = 33 86%. The red slope is the linear slip weakening slope needed to equal the breakdown energy 𝐸 𝐺 .

Figure 158 .

 158 Figure 158. Proposed simplified models -Evolution of friction contributions as a function of the initial percentage of cementation for mid-dense materials.

•

  Measure of the 𝚫𝒀 𝒑𝒆 opening of each R or Y -band -Collection of particles around the R-band (~ 10 to 20 on each side) with the software r the R-band (angle compared to the measured data), Figure 160 (a). -Projection of the position of selected cells in the reference mark of the R-band (R B = (0, x R ⃗⃗⃗⃗ , y R ⃗⃗⃗⃗ )), with an offset in the y-direction to take the displacement of the center of the reference mark 𝑡ℎ 𝑖 into account, Figure 160 (b). -Addition of linear interpolation to have an x-alignment of the points on each side of the fault w ' ' o o Figure 161. -Calculation of the needed values from the collected data. ∆∆ is the relative displacement in the 𝑅 -direction for each time step. It is calculated as ∆Y op n = ∆∆ = ( 𝑖 -𝐴 𝑖 ) -( 0 -𝐴 0 ) (𝐴. 5.9) to consider the displacement of both particles within the gouge. With the same method, we calculate the relative displacement in the 𝑅 -direction, ∆X slip = ( 𝑖 -𝐴 𝑖 ) -( 0 -𝐴 0 ) (𝐴. 5.10) Both variables are observed in Figure 162. The graph of the ∆Y op n as a function of ∆X slip will give the trajectory of each band.

Figure 160 .Figure 161 .

 160161 Figure 160. (a) Selected points in the reference mark 𝑅 0 = (0, 0 ⃗⃗⃗⃗ , 0 ⃗⃗⃗⃗ ). (b) Selected points projected in the R-band mark 𝑅 = (0, 𝑅 ⃗⃗⃗⃗ , 𝑅 ⃗⃗⃗⃗ ). Points color coded with their position with the fault slip distance from 0 to 0.5 mm.

  This is a long way method, but it enables us to have the first idea on R-band temporality for three different simulations and to have an average value of the width 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 of R and Ybands.

Figure 163 .

 163 Figure 163. Image of the M-S sample with a filter on the solid fraction to threshold the solid fraction from 0.95 to 1.

Figure 165 .

 165 Figure 165. Effective friction 𝜇 as a function of the interparticle friction for all the thin bands. Cross for the peak of effective friction and stars for steady-state effective friction. In the table, sample names and associated modification with standard case (M-SG-S). Size of the samples 20 × 0.2 𝑚𝑚².

Figure 166 .

 166 Figure 166. Effective friction 𝜇 and dilation 𝜀 𝑦 (%) as a function of the slip distance (m) for different values of the interparticle stiffness 𝑘 𝑛 , for thin band models (20 × 0.2 𝑚𝑚²).

Figure 167 .Figure 168 .

 167168 Figure 167. Effective friction 𝜇 and dilation 𝜀 𝑦 (%) as a function of the slip distance (m) for different values of perturbation 𝑃 𝑖 , for thin band models (20 × 0.2 𝑚𝑚²).

4 )0

 4 𝑃 𝑜 𝑒 Surface percentage of porosity within the sample = 100 𝐹 𝑣 , (Chap. 2) R Riedel shear bands in the R-orientation, (Chap. 1) 𝑅′ Antithetic (or conjugate) Riedel shears in the R' orientation, (Chap. 1) 𝑅 ℎ𝑖𝑐𝑘𝑛𝑒 Thickness of a Riedel band (m), (Chap. 5) Averaged grain radius used (m), (Chap. 2) Signal of the grain radius, (Chap. 2) Total Voronoï cell area (𝑚 2 ), (Chap. 2) 𝑑 Slip distance (m), (Chap. 4)𝑔Gouge area perpendicular to the applied normal stress (𝑚 2 ), ( 𝑔 = 𝐿 𝑔 × 1 (2D))𝑔 𝑎𝑖𝑛 -𝑒𝑞 𝑖𝑣 Total equivalent angular grains area within the gouge (𝑚 2 ), (Chap. 4) 𝑚𝑎 𝑖𝑥-𝑒𝑞 𝑖𝑣 Total matrix particles area within the gouge (𝑚 2 ), (Chap. 4) Total area of one particle (𝑚 2 ), (Chap. 2) 𝑎 𝑖𝑐 𝑒 Total particles area within the gouge (𝑚 2 ), (Chap. 2) 𝑣𝑜𝑖𝑑 Total voids area within the gouge (𝑚 2 ), (Chap. 2) th Thickness of the numerical granular gouge model (m), (Chap. 2) th i Initial thickness of the numerical gouge sample (m), (Chap. 2) 𝑈 𝑑 Slip distance corresponding to the maximum dilatancy (𝑚), (Chap. 5) 𝑈 𝑒𝑛𝑑 Slip distance at the end of the simulation (𝑚), (Chap. 5) 𝑈 Slip distance at the effective friction peak (𝑚), (Chap. 5) V Initial shearing velocity applied on the upper rock wall (m/s), (Chap. 2) 𝑉 𝑦 Downwards velocity used for biaxial numerical experiments (m/s), (Chap. 3) strain 𝛾 𝑥𝑦 = Δ 𝑥 /𝑡ℎ 𝑖 , (Chap. 5) 𝛾̇ Shearing rate Δ DEM current and constant time step (s), (Chap. 2) Δ -𝑐𝑜𝑛 𝑎𝑐 Proximity updating period (s), (Chap. 2) ∆𝑡ℎ y displacement of the upper rock wall (m), (Chap. 2) Δ 𝑥 x displacement of the upper rock wall or slip distance (m), (Chap. 5) ∆𝐻 𝑑 Gain in gouge thickness (m), (Chap. 5) ∆𝑈 𝑓 Characteristic distance of the exponential decay, (Chap. 5) ∆𝑊 Total energy budget (J/m²), (Chap. 5) ∆ 𝑖The variation in the 𝑅 -direction, Slip within each band (m), (Chap. 5)∆ 𝑜 𝑒𝑛The variation in the 𝑅 -direction, opening of each band (m), (Chap. 5)∆ 𝑜 𝑒𝑛-𝑎𝑣Average opening for a specific slip distance of the total fault (m), (Chap. 5) ratio of the shear stress under normal stress), (Chap. 2) 𝜇 𝑎 Apparent friction (From Riedel shears (Lockner & Byerlee, 1993)), (Chap. 3) 𝜇 𝑑𝑎𝑚 Damage contribution of effective friction, (Chap. 5) 𝜇 𝑑𝑖 Dilation contribution of effective friction, (Chap. 5) 𝜇 𝑑 Maximum friction induced by rupture of cementation, (Chap. 5) 𝜇 𝑓 Frictional contribution of effective friction, (Chap. 5) 𝜇 𝑓 Peak friction at the end of the elastic phase, (Chap. 5) 𝜇 𝑓 Coefficient of friction (From Mohr-Coulomb definition), (Chap. 1) Maximum principal stress (Pa), (Chap. 1) 𝜎 3 Minimum principal stress (Pa), (Chap. 1) 𝜎 𝑒𝑓𝑓 Effective stress, 𝜎 𝑒𝑓𝑓 = 𝜎 𝑛 -𝑃 𝑓 𝜎 𝑁 Normal stress applied on the upper rock wall (MPa), (Chap. 2) 𝜎 𝑛 Contact normal stress between two particles (MPa), (Chap. 2) 𝜎 Contact tangential stress between two particles (MPa), (Chap. 2) 𝜏 Shear stress between the gouge and the upper rock (MPa), (Chap. 2) 𝜏 Maximum shear stress (MPa), (Chap. 5) 𝜏 Steady-state shear stress (MPa), (Chap. 5)𝜑Internal friction angle in the sense of Mohr-Coulomb, (Chap. 1) de transformations physico-chimiques telles que la dissolution ou encore la fusion à haute température. Ces dernières peuvent donner naissance à des matériaux de remplissage se logeant dans les pores de la gouge et impactant le comportement mécanique et rhéologique de la faille. Au cours de ce travail de thèse, des modèles 2D (Méthode des Eléments Discrets) de gouges cisaillées ont été proposés pour mieux comprendre : (i) comment les matériaux de re à travers la formation de bandes de cisaillement et son lien avec les caractéristiques physiques et mécaniques des gouges, (iii) la contribution de chacune de ces propriétés dans l'énergie de rupture et les lois de frottement observées. Un premier modèle met en avant trois types de matériaux cimentés montrant des comportements rhéologiques et de résistance aux glissements différents en fonction du pourcentage r dilatance de la faille, les frottements de Coulomb et la rupture des liaisons cimentées. Ensuite, gouge est évaluée dans un 3 ème modèle. Nous détaillons finalement une méthode énergétique er e de re er ' o o de e de de d or on avec le comportement de la gouge entière. Les lois de frottement issues de ces modèles pourront être utilisées dans des modèles dynamiques à plus grande échelle.MOTS-CLÉS: Fault mechanics, Granular materials, Tribology, Modelling & Simulations Laboratoire (s) de recherche : Laboratoire de mécanique des Contacts et des Structures (LaMCoS) et Laboratoire de Géomécanique, Matériaux, Structures (GEOMAS), INSA Lyon Directeur de thèse : Pr. Ali Daouadji (directeur) et Dr. Guilhem Mollon (co-directeur)

1.2.3. Notions in rock and fracture mechanics

							fracture,
	er e r k… re	o	re	o e r o w	e	er	e r
	to seismic nucleation

' e

For any real loaded material, imperfections can exist within the microstructure (i.e.

  It is important to specify that this link existed before the Coulomb-Mohr criterion and does not consider a possible cohesion of the material. Combining equations and o o ' w we re e o e r w we the effective friction 𝝁 different from the coefficient of friction 𝜇 𝑓 used for Coulomb-Mohr (cf. eq. 1.4):

	(a) o o ' r o 𝜎 1 𝜏 𝛽 intact rock.	𝜓	𝜎 𝑁 d e e e 𝜎 3	(b) 𝜏 𝜇 = e r o re 𝜑 𝜏 𝜎 𝑁 =	𝐶 𝜎 𝑁	2𝛽 𝜇 𝑓 r	e	2𝜓	𝜎 Triaxial-biaxial compression e o zero o e o w	(1.7) e
			𝜎 𝑁 tension compression 𝜎 3	𝜎 1 𝜎 3 2	𝜎 1
				nd	o o '	w
				𝜏 = 𝜇 𝜎 𝑁					(1.6)

Dividing the two forces by the nominal contact area gives a relation between the shear stress 𝜏 and the normal stress 𝜎 𝑁 , known as the 2

1.5. Concluding remarks

	This chapter provided fundamental concepts and reviewed recent advances in understanding
	earthquakes, fault mechanics, and the role of granular fault gouge. The main objective of this
	Thesis work is to study the rheology and mechanics of a geological third body, more commonly
	ed " r	r	o e"	e	w	o	o	er

Table 1 .

 1 Input Parameters in Packing 2D to generate the sample. 𝐷 2 to 𝐷 8 are Fourier descriptors, 𝜆 1 and 𝜆 2 are the two constants needed to calculate the two decays 𝛥 1 and 𝛥 2 (formulas in Figure

			y
				Generated grain
	Voronoï			with Fourier
	cell	𝑚𝑎𝑥	𝑖	𝑖	spectrum
				x

Table 3 .

 3 Numerical setup and properties for the experimental campaign.

	Property	Associated variable		Value
	Normal stress	𝜎 𝑁		0 𝑀𝑃𝑎
	Shear velocity	V		1 𝑚/𝑠
	Rock density	𝜌	2600 𝑘𝑔/𝑚 3
	Contact stiffness	𝑘 𝑛 & 𝑘	10 1 𝑃𝑎/𝑚
	Cohesive bond strength	𝐶 𝑛 𝑚	0	2500 𝑀𝑃𝑎
	Percentage of cementation	𝑃 𝑐𝑒𝑚	0 %	100 %
	Percentage of initial porosity	𝑃 𝑜 𝑒	11 % & 16 %
	Interparticle friction	𝜇 𝑛 𝑚	0.5 (grains -grains) 1 (grains -walls)
	Initial sample size	h i x 𝐿 𝑔	1.7 x 20 𝑚𝑚
	Particle equivalent diameter	𝜙 𝑚𝑖𝑛 -𝜙 𝑚𝑎𝑥	28 -226 𝜇𝑚
	Number of particles	𝑁		4960
	DEM time step	Δ	10 -9 s
	Proximity updating period	Δ -𝑐𝑜𝑛 𝑎𝑐	10 -7 s

Table 4 .

 4 Numerical setup and properties for the experimental campaign (set for standard cases). The top part of the table (grey color) concerns properties that do not vary within the simulation campaign.

	Property	Associated variable	Value
	Normal stress	𝜎 𝑁	0 𝑀𝑃𝑎
	Shear velocity	V	1 𝑚/𝑠
	Rock density	𝜌	2600 k𝑔/𝑚 3
	Interparticle cohesion	𝐶 𝑛 𝑚	500 𝑀𝑃𝑎 at boundaries
	Interparticle friction	𝜇 𝑛 𝑚	0.3
	Contact stiffness	𝑘 𝑛 & 𝑘	10 1 𝑃𝑎/𝑚
	Perturbation	𝑃 𝑖	0.5
	Sample size	𝑡ℎ 𝑖 x 𝐿𝑔	2 x 20 𝑚𝑚
	Particle equivalent diameter	𝜙 𝑒𝑞	20 𝜇𝑚
	Number of particles	𝑁	115 825
	DEM time step	Δ	5.10 -10 s
	Proximity updating period	Δ -𝑐𝑜𝑛 𝑎𝑐	10 -8 s
	4.2.3.

Characterization of matrix material with biaxial tests

	𝑡ℎ 𝑖
	L g
	Periodic boundary conditions

Using the same Coulomb-Mohr representation as in Chapter 3, independent biaxial simulations of samples were run with the characteristics previously presented for sheared fault gouge (in terms of grain shapes, size distributions, initial solid fraction, interparticle cementation, and interparticle friction). Rectangular granular samples ( 𝑚𝑚 wide and 10 𝑚𝑚 high) are placed Rock walls

Table 5 .

 5 Numerical results of the biaxial campaign for two interparticle stiffnesses.

	Property	𝒌 = 𝟎 𝟓 𝑷𝒂/𝒎	𝒌 = 𝟎 𝟔 𝑷𝒂/𝒎
	Effective friction peak 𝜇	0.9	1.17
	Internal friction angle 𝜑	43.3°	49.5°
	Cohesion C	6.4 MPa	2.0 MPa
	Elasticity Modulus E	18.96 GPa	118.39 GPa

Table 6 .

 6 Sample names and associated modifications with standard case (M-S) presented in section4.3.1.

	Section Sample Name Modified variable	Numerical value
	i	M-F01 M-F02	Interparticle friction	0.1 0.2
		M-F06	𝜇 𝑛 𝑚	0.6
	ii	M-P-02 M-P-08	Perturbation shape 𝑃 𝑖	0.2 0.8

Table 7 .

 7 Sample names and associated modifications with standard case (M-S) presented in section4.3.1.

	Section	Sample Name	Modified variable	Numerical value
	i	M-K-2 M-K-5 M-K-10	Interparticle stiffness 𝑘 𝑛	2.10 1 P / 5.10 1 P / 1.10 16 P /
	ii	M-PS-30 M-PS-40 M-PS-50 M-PS-30-Keq	Size of particles 𝜙 𝑒𝑞 Size of particles and	30 μ 40 μ 50 μ 30 μ -6.8 10 14 P /
		M-PS-40-Keq	equivalent stiffness	40 μ -.7 10 14 P /
		M-PS-50-Keq	𝜙 𝑒𝑞 and 𝑘 𝑒𝑞	50 μ -10 14 P /
		M-Th-1	Thickness of the	1 mm
	iii	M-Th-3	gouge	3 mm
		M-Th-4	𝑡ℎ 𝑖	4 mm

Table 9 .

 9 Table 9 below. Numerical setup and properties for the composite gouge simulation.

		𝑃 𝑚𝑎 𝑖𝑥
		29 %
		33 %
		40 %
		47 %
		52 %
		56 %
		60 %
	Matrix particles	Angular particles
		Equivalent diameter 𝒆 (mm)

Rheology of the composite mixture 4.8.1. Mechanics and Kinematics

Table 10 .

 10 Information 𝑃 𝑚𝑎 𝑖𝑥 is the surface percentage of matrix particles, 𝜇 𝑛 𝑚 is the numerical interparticle friction, 𝜙 𝑒𝑞-𝑚𝑎 𝑖𝑥 is the equivalent diameter of matrix cells, 𝜙 𝑒𝑞-𝑔 𝑎𝑖𝑛 is the equivalent diameter of grains, 𝑁 𝑔 𝑎𝑖𝑛 is the total number of angular grains and 𝑁 𝑚𝑎 𝑖𝑥 is the total number of matrix cells.

	Sample name	𝑃 𝑚𝑎 𝑖𝑥	𝜇 𝑛 𝑚	𝜙 𝑒𝑞-𝑚𝑎 𝑖𝑥	𝜙 𝑒𝑞 -𝑔 𝑎𝑖𝑛	𝑁 𝑔 𝑎𝑖𝑛	𝑁 𝑚𝑎 𝑖𝑥
	MG-H-86	86 %	0.3	20 𝜇𝑚	315 𝜇𝑚	50	104 274

on the sample with 50 angular particles surrounded by matrix particles. The name MG-H-86 comes from e ' ' rr e ' ' or o o e eo d r o d ' ' or e w % of matrix particles.

  . It modifies the total contact of particles and plays a role in fabric evolution, and so on effective friction and elastic properties,

	e o e o	we do '	e ere	o r	e	o	o dered

  1 and 𝑅 2 ) has died out, although they are still physically present and

	e	e o e	er	e o "	e"	d	e d	e	e 1 boundary
	shear.

  bearing granodiorite (upper part) cut by a cataclastic fault zone (lower part). San Juan, Argentina. Width of view 23 mm. CPL, [N°14] Cataclastic to protomylonitic granite cut by a zone of cataclasite (subhorizontal through the center). Leiden collection. Width of view 15 mm. PPL, [N°15] Cataclasite developed in an ultramylonite. Virginia, southern Minas Gerais State, SE Brazil. Width of view 20 mm. PPL, [N°16] A very low-grade mylonite, transitional to cataclasite, derived from hornblende syenite. Twijfelfontein, Kaokoveld, NW Namibia. Width of view 14 mm. CPL, [N°17] Example of a mosaic-r e ed e d r or ro ow-r de o e derived from granite. St. Barthélemy Massif, French Pyrenees. Width of view 4 mm. CPL, [N°18] Low-grade mylonite derived from granite. St. Barthélemy Massif, French Pyrenees. Width of view 12 mm. CPL (Images from

  The initial interparticle friction coefficient (i.e. numerical friction 𝜇 𝑛 𝑚 ) is used during the compaction phase of the gouge, to control the initial porosity of the sample (noted 𝑃 𝑜 𝑒 and defined as the ratio of the area occupied by voids to the apparent area of the sample). Eleven samples were compacted with different initial interparticle friction 𝜇 𝑛 𝑚 and the solid fraction was measured at the end of the compaction step. With 𝜇 𝑛 𝑚 = 0 between each particle in contact, the only contact parameter between grains is the numerical stiffness

	𝜙 𝑒𝑞 = 10𝜇𝑚	𝜙 𝑒𝑞 = 15𝜇𝑚	𝜙 𝑒𝑞 = 20𝜇𝑚	𝜙 𝑒𝑞 = 25𝜇𝑚
	Compaction step -500 𝜇𝑚	𝐹 𝑣 =	𝐹 = = 1 -𝐹 𝑔𝑜 𝑔𝑒 = 𝑣𝑜𝑖𝑑 𝑎 𝑖𝑐 𝑒 𝑔𝑜 𝑔𝑒 𝑣𝑜𝑖𝑑 𝑔𝑜 𝑔𝑒 𝑃 𝑜 𝑒 = 𝐹 𝑣 100 𝑎 𝑖𝑐 𝑒	(𝐴. 2.1) (𝐴. 2.2) (𝐴. 2.3)

Table 11 .

 11 Numerical setup and properties for the experimental campaign

	Effective friction
	1
	0.1
	0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90
				Solid fraction
	Property	Associated variable	Value
	Normal stress	𝜎 𝑁		40 𝑀𝑃𝑎
	Shear velocity	V (for shearing only)	1 𝑚/𝑠
	Rock density	𝜌		2600 grains 𝑘𝑔/𝑚 3
	Contact stiffness	𝑘 𝑛 & 𝑘	10 1 𝑃𝑎/𝑚
	Interparticle friction (compaction)	𝜇 𝑛 𝑚	From 0 to 1
	Interparticle friction (shearing)	𝜇 𝑛 𝑚	0.5 (grains -grains) / 1 (grains -walls)
	Initial Sample size	𝑡ℎ 𝑖 × 𝐿 𝑔	2 x 20 mm
	Particle equivalent diameter	𝜙 𝑚𝑖𝑛	𝜙 𝑚𝑎𝑥	28	226 𝜇𝑚
	Number of particles	𝑁		4960
	DEM time step Proximity updating period	Δ Δ -𝑐𝑜𝑛 𝑎𝑐	10 -9 s (∆𝑡 = cst) 10 -7 s

Table 12 .

 12 Data of both models with circular or angular particles. The granular layer with circular shapes of grains is generated with the same parameters as the one with angular particles.

	Property	Angular particles	Circular particles
	Normal stress, 𝜎 𝑁		40 𝑀𝑃𝑎
	Shear velocity, 𝑉		1 𝑚/𝑠
	Rock density, 𝜌	2600 grains 𝑘𝑔/𝑚3
	Contact stiffness, 𝑘 𝑛 & 𝑘		10 1 𝑃𝑎/𝑚
	Interparticle friction, 𝜇 𝑛 𝑚	0.5 (grains -grains) / 1 (grains -walls)
	Sample size, 𝑡ℎ 𝐿 𝑔		1.7 x 20 mm
	Particle equivalent diameter, 𝜙 𝑚𝑖𝑛 -𝜙 𝑚𝑎𝑥	28 -	μ	20 -181 µm
	Number of particles, 𝑁	4960		7515
	DEM time step, Δ		10 -9 s
	Proximity updating period, Δ -𝑐𝑜𝑛 𝑎𝑐		10 -7 s

Table 13 .

 13 Results for dense samples (𝑃 𝑜 𝑒 = 11% or solid fraction SF=0.89). With ∆τ (Pa) the stress drop from the effective friction peak to the plateau, 𝜇 𝑆𝑆 the steady-state effective friction, 𝜇 the effective friction peak, - the steady-state solid fraction, 𝑃 𝑜 𝑒-𝑆𝑆 the steady-state porosity, h the steady-state gouge thickness.

		Effective friction					Dilation
		1							0.1
		0.8	𝜇				𝜀 𝑦-	0.08
		0.6	𝜇						0.06
		0.4							0.04
		0.2							0.02
		0							0
		1		2 N°granular sample 3		4
	𝑪 𝒖𝒎	100	200	500	800	1000	1200	1500	2000	2500
	𝑷 𝒆𝒎 (%)	4	8	19	30	38	46	57	76	95
	∆𝝉 (Pa)	1.1+07 1.1E+07	1.8E+07	3.0E+07 3.6E+07 4.2E+07 5.3E+07 6.8E+07 8.8E+07
	𝝁 𝑺𝑺	0.481	0.482	0.490	0.477	0.496	0.484	0.475	0.473	0.431
	𝝁 𝒑	0.771	0.786	0.994	1.406	1.695	1.938	2.376	3.094	3.838
	𝑺-𝑺𝑺	0.869 0.867	0.876	0.877	0.869	0.87	0.884	0.895	0.889
	𝑷 𝒑 𝒓𝒆-𝑺𝑺 0.131 0.133	0.124	0.123	0.131	0.13	0.116	0.105	0.111
	𝒕𝒉 𝑺𝑺 (mm)	1.72	1.726	1.707	1.705	1.72	1.719	1.691	1.671	1.681

Table 14 .

 14 Results for mid-dense samples (𝑃 𝑜 𝑒 = 16% or initial solid fraction SF=0.84) With ∆τ (Pa) the stress drop from the effective friction peak to the plateau, 𝜇 𝑆𝑆 the steady-state effective friction, 𝜇 the effective friction peak, -the steady-state solid fraction, 𝑃 𝑜 𝑒-𝑆𝑆 the steady-state porosity, h the steady-state gouge thickness.

	𝑪 𝒖𝒎	100	200	500	800	1000	1200	1500	2000
	𝑷 𝒆𝒎 (%)	3	6	18	31	40	50	66	95
	∆𝝉 (Pa) 3.0E+06 2.6E+06 3.7E+06 9.1E+06 1.4E+07 1.9E+07 2.6E+07 3.6E+07
	𝝁 𝑺𝑺	0.461	0.466	0.460	0.458	0.469	0.479	0.476	0.455
	𝝁 𝒑	0.536	0.531	0.552	0.684	0.815	0.959	1.130	1.346
	𝑺-𝑺𝑺	0.855	0.854	0.852	0.85	0.851	0.857	0.864	0.858
	𝑷 𝒑 𝒓𝒆-𝑺𝑺 0.145	0.146	0.148	0.15	0.149	0.143	0.136	0.142
	𝒕𝒉 𝑺𝑺 (mm)	1.749	1.752	1.756	1.766	1.757	1.745	1.73	1.742

Table 15 :

 15 Secant shear modulus and tangential shear modulus for the different cases of study.

		Size	Friction	Stiffness	Diameter	Gsec	Gtan	P
		𝐿 𝑔 × 𝑡ℎ 𝑖	𝜇 𝑖	𝑘 𝑛	𝑖	𝐺	𝐺	𝑃 𝑖
	name	(𝑚𝑚)		(𝑃𝑎/𝑚)	(𝑚𝑚)	(𝐺𝑃𝑎)	(𝐺𝑃𝑎)
	M-S	20x2	0.3	10 1	0.02	5.52	10.4	0.5
	M-S-bis	20x2	0.3	10 1	0.02	5.73	12.2	0.5
	M-RES-3	10x2	0.3	10 1	0.02	5.54	10.4	0.5
	M-RES-2	4x2	0.3	10 1	0.02	5.5	10.3	0.5
	M-RES-1	2x2	0.3	10 1	0.02	5.53	10.9	0.5
	M-RES-4	40x2	0.3	10 1	0.02	5.49	10.4	0.5
	M-RES-3-F	2x2	0.6	10 1	0.02	4.76	9.17	0.5
	M-RES-2-F	4x2	0.6	10 1	0.02	4.71	8.89	0.5
	M-RES-1-F	10x2	0.6	10 1	0.02	5.11	9.07	0.5
	M-F01	20x2	0.1	10 1	0.02	5.38	12.4	0.5
	M-F02	20x2	0.2	10 1	0.02	5.71	12.8	0.5
	M-F06	20x2	0.6	10 1	0.02	5.05	9.04	0.5
	M-RES-4-F	40x2	0.6	10 1	0.02	4.98	9.02	0.5
	M-PS-30	20x2	0.3	10 1	0.03	8.36	16.1	0.5
	M-PS-30-K eq	20x2	0.3	6.8.10 14	0.03	6.22	12.2	0.5
	M-PS-40	20x2	0.3	10 1	0.04	11.2	23.3	0.5
	M-PS-40-K eq	20x2	0.3	.7.10 14	0.04	6.2	12.2	0.5
	M-PS-50	20x2	0.3	10 1	0.05	14.1	32.4	0.5
	M-PS-50-K eq	20x2	0.3	.10 14	0.05	6.86	13.6	0.5
	M-Kx2	20x2	0.3	2. 10 1	0.02	10.5	24	0.5
	M-Kx10	20x2	0.3	10 16	0.02	41.9	170	0.5
	M-P-02	20x2	0.3	10 1	0.02	5.87	9.83	0.2
	M-P-08	20x2	0.3	10 1	0.02	5.02	10.9	0.8
	M-P-08-Kx2	20x2	0.3	2. 10 1	0.02	9.39	25.4	0.8
	M-Th-1	20x1	0.3	10 1	0.02	5.77	12.2	0.5
	M-Th-3	20x3	0.3	10 1	0.02	5.41	9.015	0.5
	M-Th-4	20x4	0.3	10 1	0.02	5.35	9.15	0.5
	M-SG-P02	20x0.2	0.3	10 1	0.02	6.07	9.12	0.2
	M-SG-S	20x0.2	0.3	10 1	0.02	5.85	9.83	0.5
	M-SG-k10	20x0.2	0.3	10 16	0.02	46.6	85.7	0.5
	M-SG-P08	20x0.2	0.3	10 1	0.02	5.17	10.6	0.8
	M-SG-k2	20x0.2	0.3	2. 10 1	0.02	10.7	18.7	0.5
	M-SG-k5	20x0.2	0.3	5. 10 1	0.02	24.5	44.8	0.5
	M-SG-F06	20x0.2	0.6	10 1	0.02	5.6	9.7	0.5

used for Coulomb friction law:

  𝑚𝑖𝑛 , 𝑃 𝑐𝑒𝑚 ≤ 𝐶 in Ψ -𝑚𝑖𝑛 (𝑃 𝑐𝑒𝑚 -𝐶 in ) (Ψ -𝑚𝑎𝑥 -Ψ -𝑚𝑖𝑛 ) (𝐶 x -𝐶 in ) , 𝐶 in < 𝑃 𝑐𝑒𝑚 < 𝐶 xSlip distance corresponding to the maximum dilatancy 𝑈 𝑑 (𝜇𝑚):Gain in gouge thickness at the end of the dilatancy phase ∆𝐻 𝑑 (𝜇𝑚):Peak friction at the end of the elastic phase 𝜇 𝑓 and post-peak friction 𝜇 𝑓 :μ 𝑓 (𝑃 𝑐𝑒𝑚 ) = {𝜇 𝑓 _𝑣𝑎 (1) 𝑃 𝑐𝑒𝑚 100 (𝜇 𝑓 𝑣𝑎𝑙 (3) -𝜇 𝑓 𝑣𝑎𝑙 (1)) (𝐴. 5. ) μ 𝑓 (𝑃 𝑐𝑒𝑚 ) = {𝜇 𝑓 _𝑣𝑎 (1) 𝑃 𝑐𝑒𝑚 100 (𝜇 𝑓 _𝑣𝑎 (3) -𝜇 𝑓 _𝑣𝑎 (1)) (𝐴. 5.5) 𝑊 𝑡ℎ U fp (𝑃 𝑐𝑒𝑚 ) = μ 𝑓 /𝑘 Characteristic distance of the exponential decay ∆𝑈 𝑓 (𝜇𝑚): Maximum friction induced by rupture of cementation 𝜇 𝑑 : Characteristic distance of the exponential decay ∆𝑈 𝑑𝑎𝑚 (𝜇𝑚): Δ𝑈 𝑑𝑎𝑚 (𝑃 𝑐𝑒𝑚 ) = Δ𝑈 𝑓 (𝑃 𝑐𝑒𝑚 ) (𝐴. 5.8)

						Δ𝑈(1) ← 𝑃 𝑐𝑒𝑚 ≤ 𝐶 in
	Δ𝑈 𝑓 (𝑃 𝑐𝑒𝑚 ) =	111 𝑃 𝑐𝑒𝑚 2			222 𝑃 𝑐𝑒𝑚	333 ← 𝐶 in < 𝑃 𝑐𝑒𝑚 < 𝐶 x	(𝐴. 5.6)
	{					Δ𝑈(3),		𝑃 𝑐𝑒𝑚 ≥ 𝐶 x
	W h 𝑜𝑒𝑓 3 = (	𝐶 in 2 𝐶 x 2 2𝐶 x-2	Ψ -𝑚𝑎𝑥 , 𝐶 in 1 1 0 𝐶 x 1 ) ( ⁄	𝑃 𝑐𝑒𝑚 ≥ 𝐶 x Δ𝑈(1) 0 Δ𝑈(2) ) = [ 111 , 222 , 333 ]	(𝐴. 5.1)
					𝑈 𝑑 𝑣𝑎𝑙 (1) ← 𝑃 𝑐𝑒𝑚 = 𝐶 in-2
	𝑈 𝑑 (𝑃 𝑐𝑒𝑚 ) =	1 𝑃 𝑐𝑒𝑚 2	2 𝑃 𝑐𝑒𝑚 𝑈 𝑑 𝑣𝑎𝑙 (3), μ 𝑑 (𝑃 𝑐𝑒𝑚 ) =	3 ← 𝐶 in-2 < 𝑃 𝑐𝑒𝑚 < 𝐶 x 𝑃 𝑐𝑒𝑚 ≥ 𝐶 x 𝑃 𝑐𝑒𝑚 100 𝜇 𝑑 _𝑚𝑎𝑥	(𝐴. 5.2) (𝐴. 5.7)
		{
	W h 𝑜𝑒𝑓 1 = (	𝐶 in-2 2 𝐶 x-2 2 2𝐶 x-2	𝐶 in-2 1 1 0 𝐶 x-2 1	𝑈 𝑑 𝑣𝑎𝑙 (1) 𝑈 𝑑 𝑣𝑎𝑙 (3) ) ( 𝑈 𝑑 𝑣𝑎𝑙 (2) ) ⁄	= [ 1 , 2 , 3 ]
						𝐻 𝑑 (1) ← 𝑃 𝑐𝑒𝑚 ≤ 𝐶 in
	Δ𝐻 𝑑 (𝑃 𝑐𝑒𝑚 ) =	11 𝑃 𝑐𝑒𝑚 2			22 𝑃 𝑐𝑒𝑚	33 ← 𝐶 in < 𝑃 𝑐𝑒𝑚 < 𝐶 x	(𝐴. 5.3)
		{				𝐻 𝑑 (3),		𝑃 𝑐𝑒𝑚 ≥ 𝐶 x
	W h 𝑜𝑒𝑓 2 =	( ( 𝐶 x 𝐶 𝑚𝑖𝑛 𝐶 in 2 2 𝐶 x 2	) 2	𝐶 in 𝐶 x 𝐶 𝑚𝑖𝑛 ( 2 𝐶 x	1 1 ) ) 1	𝐻 𝑑 (1) 𝐻 𝑑 (3) ( 𝐻 𝑑 (2) ) ⁄	= [ 11 , 22 , 33 ]

• Parameters

• Parameters used for damage law:

Table 16 .

 16 Numerical variables are used to calculate the different parameters. 𝐻 𝑑 is the change in height dilatancy at the end of the Gaussian. 𝐻 𝑑-𝑓𝑖𝑛𝑎 is the total change in height dilation. k, 𝜇 𝑆𝑆 , and ∆𝑈 𝑓 do not depend on cementation and are taken as constant.

	Variable	Value
		DENSE	MID-DENSE
	[𝐶 in , 𝐶 x ]	[25 %, 75 %]
	[𝐶 in-2 , 𝐶 x-2 ]	[1 %, 75 %]
	𝜇	0.45
	Ψ p-in	20°	8°
	Ψ p-x	0°	25°
	𝑈 𝑑 𝑣𝑎𝑙	[100, 50, 50] (𝜇𝑚)	[250, 50, 50] (𝜇𝑚)
	𝐻 𝑑	[140, 200, 150] (𝜇𝑚)	[70, 95, 120] (𝜇𝑚)
	𝐻 𝑑-𝑓𝑖𝑛𝑎	[140, 0, 80] (𝜇𝑚)	[70, 0, 60] (𝜇𝑚)
	𝑘	140 000 kN/m	80 000 kN/m
	𝜇 𝑓 _𝑣𝑎	[0.2,3]	[0.12, 0.6]
	𝜇 𝑓 _𝑣𝑎	[0.45,0.95]	[0.2, 0.25]
	ΔU fpp	8 𝜇𝑚	0 𝜇𝑚
	ΔU = ΔU f = Δ𝑈 𝑑𝑎𝑚	[100, 50] 𝜇𝑚	[50,50] 𝜇𝑚
	𝜇 𝑑 _𝑚𝑎𝑥	0.5	0.4

  𝑛 Pair Fourier coefficient, morphologic descriptors, (Chap. 2) 𝑛 Odd Fourier coefficient, morphologic descriptors, (Chap. 2) 𝐶 Cohesion in the sense of Mohr-Coulomb, (Chap. 1) 𝐶 𝑛 𝑚 Numerical interparticle cohesion (Pa), (Chap. 2) 𝐶 𝑛 𝑚-100% Numerical interparticle cohesion (Pa) for complete cementation, (Chap. 3) 𝐷 Fractal Dimension factor, (Chap. 1) 𝐷 𝑐 Critical slip distance (m), (Chap. 1) 𝐷 𝑛 Normalized amplitude for Fourier descriptors, (Chap. 2) 𝐷 Total displacement before friction peak (m), from zero displacement, (Chap 3.) 𝐷 Slip weakening part (m), similar to Dc, (Chap 3.) 𝐷 End of slip displacement (m), (Chap 5.) 𝑑 -displacement of the upper rock wall (m), (Chap. 4) 𝐸 Elasticity moduli 𝐸 (Pa), (Chap. 3) 𝐸 𝐶𝑒𝑚 The energy of de-cementation (J/m²), (Chap. 3) Ratio between 𝑡ℎ 𝑖 (m) and equivalent diameter 𝑒𝑞 (m) of particles, (Chap. 4) 𝑛 𝑆𝑑 Ratio between 𝑑 (m) and 𝑒𝑞 (m), (Chap. 4) 𝑛 𝑌 Ratio between 𝑑 (m) and 𝑒𝑞 (m) of particles. (Chap. 4) P Primary shear, (Chap. 1) 𝑃 𝑖 Perturbation parameter (regularity of shape of particles), (Chap. 2) 𝑃 𝑐𝑒𝑚 Surface percentage of cementation within the gouge sample (%), (Chap. 3) 𝑃 𝑔 𝑎𝑖𝑛 Surface percentage of angular grains within the gouge sample, (Chap. 4) 𝑃 𝑗 Per e er o r e "j" 𝑃 𝑚𝑎 𝑖𝑥 Surface percentage of matrix particle within the gouge sample, (Chap.

	𝑘 𝑒𝑞		e	er r e	e	/
	𝑘 𝑛	Nor	er r e	e	/	(Chap. 2)
	𝑘	e		er r e	e	/	(Chap. 2)
	𝐿 𝑐	Critical nucleation length (m), (Chap. 5)
	𝐿 𝑔	Length of the numerical gouge sample (m), (Chap. 2)
	𝐿 -𝑐	Contact length between two particles (m), (Chap. 2)
	𝑚 𝐴	Mass of a particle from a grain A (kg), (Chap. 2)
	𝑚 𝑚𝑖𝑛	Mass of the smallest particle (kg), (Chap. 2)
	N	N	er o r e w	e numerical gouge sample, (Chap. 2)
	𝑁 𝑖	N	er o o	w	e o o r o	r e (Chap. 2)
	𝑁 𝑂 𝑖𝑚	Optimization parameter between 0 and 63 (Chap. 2)
	𝑛	Number of contact bonds for each particle, (Chap. 3)
	𝑛𝑏 1
	𝐸 𝐷𝑎𝑚		e e er		/ ² suffix -pp for post-peak,
	𝐸 𝐷𝑖	o e er		/ ² suffix -pp for post-peak,
	𝐸 𝐷𝑖 -𝑎𝑣	er e o		e k d	o e er	/ ² ,
	𝐸 𝑓	r o e er	/ ² suffix -pp for post-peak,
	𝐸 𝑓𝑐	o	r o e r o	e er	/ ² o o	e er ,
	𝐸 𝑓	e	r o e er	/ ²
	𝐸 𝐺	Breakdown energy, energy needed to weaken the fault (J/m²),
	𝐸 𝐻	Heat or frictional energy (J/m²),
	𝐸 𝑅	Radiated energy (J/m²) ,
	𝐸 𝑆	Representative surface energy (J/m²), (Chap. 3)
	𝐸 𝑜	Total energy generated by gouge sliding (J/m²),
	𝑒	Chosen parameter around 1/50 for time step calculation, (Chap. 2)
	𝐹 𝐴 ⃗⃗⃗⃗	Resultant of forces acting on a particle A, (Chap. 2)
	𝐹 𝑁	Total normal force (N) on the upper rock wall ( -direction), (Chap. 2)
	𝐹	Solid fraction, 𝑎 𝑖𝑐 𝑒 ( 𝑎 𝑖𝑐 𝑒 ⁄	𝑣𝑜𝑖𝑑 )
	𝐹 𝑇	Total tangential force (N) on the upper rock wall ( -direction), (Chap. 2)
	𝐹 𝑣	Voids fraction, 𝑣𝑜𝑖𝑑 ( 𝑎 𝑖𝑐 𝑒 ⁄	𝑣𝑜𝑖𝑑 )
	G	Shear modulus (GPa), (Chap. 4)
		Inertial number, (Chap. 2)
		Stiffness of the loading system (GPa/m), (Chap. 5)
	𝑐	r	we ke	o e (GPa/m), (Chap. 1)
	𝑘	Gouge layer stiffness, (

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