Diese Arbeit liefert ein dreidimensionales numerisches Homogenisierungskonzept, welches beim Übergang von der Mikro- zur Makroskala einen Wechsel in der Kontinuumsbeschreibung beinhaltet. Während für die Beschreibung der Makroskala das verallgemeinerte Cosserat-Kontinuum verwendet wird, basiert die Mikroskala auf der klassischen Cauchy-Theorie. Um das homogene Cosserat-Ersatzmaterial im Rahmen numerischer Simulationen nutzen zu können, erfolgt die Implementierung geeigneter Finiter Elemente in das Programmsystem Abaqus und deren Verifikation. Neben der Diskussion der bei der Homogenisierung beobachteten Effekte werden anhand eines idealisierten Modells eines biaxialverstärkten Mehrlagengestrickes die Vorteile gegenüber der klassischen Herangehensweise aufgezeigt. / This contribution provides a threedimensional homogenization approach which includes the switch of the continuum theory during the scale transition. Whereas the microscopic scale is described in the framework of the classical Cauchy theory, the macroscopic scale is based on the generalized Cosserat continuum. In order to use the obtained homogeneous Cosserat material, suitable finite elements are implemented in the commercial program system Abaqus followed by an appropriate verification. Beside the discussion of the arising effects the advantages of this approach compared to the classical procedure are shown by means of an idealized model of a biaxial woven fabric.

The mechanical behaviour of geomaterials (e.g. soils, rocks and concrete) under plastic deformation is highly complex due to that fact that they are granular materials consisting of discrete non-uniform particles. Failure of geomaterials is often related to localisation of deformation (strain-localisation) with excessive shearing inside the localised zones. The microstructure of the material then dominates the material behaviour in the localised zones. The formation of the localised zone (shear band) during plastic deformation decreases the material strength (softening) significantly and initiates the failure of the material. There are two main approaches to the numerical modelling of localisation of deformation in geomaterials; discrete and continuum. The discrete approach can provide a more realistic material description. However, in the discrete approach, the modelling of all particles is complicated and computationally very expensive for a large number of particles. On the other hand, the continuum approach is more flexible, avoids modelling the interaction of individual particles and is computationally much cheaper. However, classical continuum plasticity models fail to predict the localisation of deformation accurately due to loss of ellipticity of the governing equations, and spurious mesh-dependent results are obtained in the plastic regime. Generalised plasticity models are proposed to overcome the difficulties encountered by classical plasticity models, by relaxing the local assumptions and taking into account the microstructure-related length scale into the models. Among generalised plasticity models, Cosserat (micropolar) and stain-gradient models have shown significant usefulness in modelling localisation of deformation in granular materials in the last few decades. Currently, several elastoplastic models are proposed based on Cosserat and strain-gradient theories in the literature. The individual formulation of the models has been examined almost always in isolation and are paired with specific materials in a mostly arbitrary fashion. Therefore, there is a lack of comparative studies between these models both at the theory level and in their numerical behaviour, which hinders the use of these models in practical applications. This research aims to enable broader adoption of generalised plasticity models in practical applications by providing both the necessary theoretical basis and appropriate numerical tools. A detailed comparison of some Cosserat and strain-gradient plasticity models is provided by highlighting their similarities and differences at the theory level. Two new Cosserat elastoplastic models are proposed based on von Mises and Drucker- Prager type yield function. The finite element formulations of Cosserat and strain-gradient models are presented and compared to better understand their advantages and disadvantages regarding numerical implementation and computational cost. The finite elements and material models are implemented into the finite element program ABAQUS using the user element subroutine (UEL) and an embedded user material subroutine (UMAT) respectively. Cosserat finite elements are implemented with different Cosserat elastoplastic models. The numerical results show how the Cosserat elements behaviour in the plastic regime depends on the models, interpolation of displacement and rotation and the integration scheme. The effect of Cosserat parameters and specific formulations on the numerical results based on the biaxial test is discussed. Two new mixed-type finite elements as well as existing ones (C1, mixed-type and penalty formulation), are implemented with different strain-gradient plasticity models to determine the numerical behaviour of the elements in the plastic regime. A detailed comparison of the numerical results of Cosserat and strain-gradient elastoplastic models is provided considering specific strain-localisation problems. Finally, some example problems are simulated with both the Cosserat and strain-gradient models to identify their applicability.

The application of polynomial reproducing methods has been explored in the context of linear and non linear problems. Of specific interest is the application of a recently developed reproducing scheme, referred to as the error reproducing kernel method (ERKM), which uses non-uniform rational B-splines (NURBS) to construct the basis functions, an aspect that potentially helps bring in locall support, convex approximation and variation diminishing properties in the functional approximation. Polynomial reproducing methods have been applied to solve problems coming under the class of a simplified theory called Cosserat theory. Structures such as a rod which have special geometric properties can be modeled with the aid of such simplified theories. It has been observed that the application of mesh-free methods to solve the aforementioned problems has the advantage that large deformations and exact cross-sectional deformations in a rod could be captured exactly by modeling the rod just in one dimension without the problem of distortion of elements or element locking which would have had some effect if the problem were to be solved using mesh based methods. Polynomial reproducing methods have been applied to problems in fracture mechanics to study the propagation of crack in a structure. As it is often desirable to limit the use of the polynomial reproducing methods to some parts of the domain where their unique advantages such as fast convergence, good accuracy, smooth derivatives, and trivial adaptivity are beneficial, a coupling procedure has been adopted with the objective of using the advantages of both FEM and polynomial reproducing methods. Exploration of SMW (Sherman-Morrison-Woodbury) in the context of polynomial reproducing methods has been done which would assist in calculating the inverse of a perturbed matrix (stiffness matrix in our case). This would to a great extent reduce the cost of computation. In this thesis, as a first step attempts have been made to apply Mesh free cosserat theory to one dimensional problems. The idea was to bring out the advantages and limitations of mesh free cosserat theory and then extend it to 2D problems.

Les matériaux soumis à de grandes déformations présentes pour la plupart l’apparition de déformations inélastiques. Ce phénomène est souvent accompagné d’une localisation des déformations dans une zone étroite, précurseur de la rupture. Un cas particulier, mais très fréquent, est les bandes de cisaillement qui apparaissent pour beaucoup de géomatériaux. Ces bandes peuvent être rencontrées à des échelles allant de l’échelle kilométrique pour les zones de subduction à l’échelle micrométrique à l’intérieur des zones de faille. Etudier et modéliser la création de ces zones d’instabilité est fondamental pour décrire la rupture des géomatériaux et des phénomènes associés comme les glissements sismiques dans les zones de faille mature de la lithosphère. Les conditions de pression, de température, l’interaction de l’eau interstitielle avec un matériau finement fracturé conduisent à l’apparition de multiples processus physiques impliqués dans les glissements sismiques. Dans ce travail, nous nous attachons à modéliser la création de bandes de cisaillement à l’intérieur des gouges de faille en prenant en compte l’effet de la microstructure par l’intermédiaire des milieux continus de Cosserat, ainsi que les couplages thermo-hydro-mécanique. L’utilisation de la théorie de Cosserat permet non seulement de régulariser le problème de localisation des déformations par l’introduction d’une longueur interne dans les lois constitutives, mais en même temps de prendre en compte l’effet de la microstructure. Deux approches sont employées pour étudier le système d’équations couplées aux dérivées partielles non linéaires : L’analyse de stabilité linéaire et la méthode des éléments finis. L’analyse de stabilité linéaire permet d’examiner les conditions d’apparitions d’instabilités pour un système mécanique avec des couplages multi-physiques. Par ailleurs, des considérations sur les perturbations appliquées au système permettent aussi de déterminer l’épaisseur de la zone de cisaillement, un paramètre clé pour la compréhension du mécanisme mécanique des failles. Ces estimations sont confirmées par l’intégration numérique pour des déformations restant dans une gamme donnée. Elles sont confrontées aux observations expérimentales et in situ et présentent une bonne corrélation. D’autre part, les simulations numériques permettent d’obtenir la réponse mécanique de la gouge de faille et de donner des informations sur l’influence des différents couplages dans le budget énergétique d’un tremblement de terre / When materials are subjected to large deformations, most of them experience inelastic deformations. It is often accompanied by a localization of these deformations into a narrow zone leading to failure. One particular case of strain localization is the formation of shear bands which are the most common patterns observed in geomaterials. In geological structures, they appear at very different scales, from kilometer scale for subduction zones, to micrometric scale inside fault cores. Studying their occurrence and evolution is of key importance to describe the failure of geomaterials and model seismic slip for mature crustal faults. The pressure and temperature conditions in these faults and the interaction with the pore water inside a highly fractured materials highlight the importance of different physical processes involved in the nucleation of earthquakes. In this thesis, we study the occurrence and evolution of shear bands inside fault gouges taking into account the material microstructure by resorting to elastoplastic Cosserat continua and also the effect of thermo-hydro mechanical couplings. The use of Cosserat theory introduces information about the gouge microstructure, namely the grain size, and permits to regularize the mathematical problem of in the post-localization regime by introducing an internal length into the constitutive equations. Two approaches are used to study the coupled non-linear partial differential set of equations: linear stability analysis and finite element simulations. Linear stability analysis allows to study the occurrence of localized deformation in a mechanical system with multi-physical couplings. Considerations on the dominant wave length of the perturbations permit also to determine the width of the localized zone. This shear band thickness is confirmed by numerical integration in the post-localization regime for a certain range of deformation. The obtained widths of the localized zone are key parameters for understanding fault behavior, are in agreement with experimental and field observations. Moreover, numerical finite element computations enable to model the mechanical response of a fault gouge during seismic slip and give insights into the influence of various physical couplings on the energy budget

The goal of this research is to develop a robust, continuum-based approach for a three-dimensional, Finite Element Method (FEM) simulation of layered geomaterials. There are two main approaches to the numerical modeling of layered geomaterials; discrete or discontinuous techniques and an equivalent continuum concept. In the discontinuous methodology, joints are explicitly simulated. Naturally, discrete techniques provide a more accurate description of discontinuous materials. However, they are complex and necessitate care in modeling of the interface. Also, in many applications, the definition of the input model becomes impractical as the number of joints becomes large. In order to overcome the difficulties associated with discrete techniques, a continuum-based approach has become popular in some application areas. When using a continuum model, a discrete material is replaced by a homogenized continuous material, also known as an 'equivalent continuum'. This leads to a discretization that is independent of both the orientation and spacing of layer boundaries. However, if the layer thickness (i.e., internal length scale of the problem) is large, the classical continuum approach which neglects the effect of internal characteristic length can introduce large errors into the solution. In this research, a full 3D FEM formulation for the elasto-plastic modeling of layered geomaterials is proposed within the framework of Cosserat theory. The effect of the bending stiffness of the layers is incorporated in the matrix of elastic properties. Also, a multi-surface plasticity model, which allows for plastic deformation of both the interfaces between the layers and intact material, is introduced. The model is verified against analytical solutions, discrete numerical models, and experimental data. It is shown that the FEM Cosserat formulation can achieve the same level of accuracy as discontinuous models in predicting the displacements of a layered material with a periodic microstructure. Furthermore, the method is capable of reproducing the strength behaviour of materials with one or more sets of joints. Finally, due to the incorporation of layer thickness into the constitutive model, the FEM Cosserat formulation is capable of capturing complicated failure mechanisms such as the buckling of individual layers of material which occur in stratified media.

The goal of this research is to develop a robust, continuum-based approach for a three-dimensional, Finite Element Method (FEM) simulation of layered geomaterials. There are two main approaches to the numerical modeling of layered geomaterials; discrete or discontinuous techniques and an equivalent continuum concept. In the discontinuous methodology, joints are explicitly simulated. Naturally, discrete techniques provide a more accurate description of discontinuous materials. However, they are complex and necessitate care in modeling of the interface. Also, in many applications, the definition of the input model becomes impractical as the number of joints becomes large. In order to overcome the difficulties associated with discrete techniques, a continuum-based approach has become popular in some application areas. When using a continuum model, a discrete material is replaced by a homogenized continuous material, also known as an 'equivalent continuum'. This leads to a discretization that is independent of both the orientation and spacing of layer boundaries. However, if the layer thickness (i.e., internal length scale of the problem) is large, the classical continuum approach which neglects the effect of internal characteristic length can introduce large errors into the solution. In this research, a full 3D FEM formulation for the elasto-plastic modeling of layered geomaterials is proposed within the framework of Cosserat theory. The effect of the bending stiffness of the layers is incorporated in the matrix of elastic properties. Also, a multi-surface plasticity model, which allows for plastic deformation of both the interfaces between the layers and intact material, is introduced. The model is verified against analytical solutions, discrete numerical models, and experimental data. It is shown that the FEM Cosserat formulation can achieve the same level of accuracy as discontinuous models in predicting the displacements of a layered material with a periodic microstructure. Furthermore, the method is capable of reproducing the strength behaviour of materials with one or more sets of joints. Finally, due to the incorporation of layer thickness into the constitutive model, the FEM Cosserat formulation is capable of capturing complicated failure mechanisms such as the buckling of individual layers of material which occur in stratified media.

Une structure ordonnée, périodique et non cohésive de billes sphériques monodisperses forme un cristal phononique granulaire. L'objet de ce travail de thèse porte sur la propagation d'une onde acoustique à l'intérieur de ce type de structure. Lors de la propagation d'une onde acoustique à travers un cristal granulaire, les propriétés des cristaux phononiques (dispersion due à la périodicité géométrique) sont combinées avec les propriétés de propagation dans les milieux granulaires (non-linéarités, degrés de liberté de rotation). Dans ce travail de thèse, les relations de dispersion des modes de volumes se propageant dans une structure granulaire hexagonale compacte sont déterminées théoriquement. La structure est modélisée par une structure masses- ressorts. Deux ressorts modélisent les interactions au niveau du contact entre deux billes, alors considérées comme des masses rigides, suivant la théorie de Hertz-Mindlin, avec un ressort pour les interactions normales et un ressort pour les interactions transverses. Dans un premier temps, le spectre des modes de volume est déterminé dans le cas où les interactions transverses sont négligées, les relations de dispersion des modes de translation se propageant dans le cristal sont obtenues. Dans un deuxième temps, les rigidités transverses sont prises en compte. L'existence de l'interaction transverse ainsi que la distance non nulle séparant les contacts des centres de billes nécéssitent la prise en compte des degrés de liberté de rotation de chaque bille. La modélisation de la structure s'inscrit alors dans le cadre d'une théorie généralisée de l'élasticité appelée théorie de Cosserat ou théorie micropolaire. Ceci conduit à la prédiction de modes de translation, de modes de rotation et de modes couplés de rotation-translation. Les modes induits par la théorie de Cosserat (modes de rotation et modes couplés translation-rotation) n'ont auparavant pas été observés expérimentalement dans les milieux granulaires. Après l'étude théorique de la propagation des ondes dans une direction particulère, l'observation expérimentale de modes couplés rotation-translation dans cette direction dans un cristal phononique granulaire est reportée, validant ainsi les effets de la théorie de Cosserat dans les milieux granulaires. Finalement, la non-réciprocité de l'effet non linéaire classique d'auto-démodulation par rapport à la direction de propagation de l'onde acoustique dans un cristal granulaire soumis à la gravité est démontrée.

Les modèles continus de plasticité cristalline appliqués aux calculs de polycristaux métalliques sont efficaces pour voir le comportement mécanique global du polycristal, à partir des lois de comportement du monocristal. On obtient ainsi également les champs de contraintes et déformations locaux dans les grains. Ceci est rendu possible à l'aide de simulations par éléments finis sur un volume élémentaire représentatif d'agrégat et les modèles d'homogénéisation numérique. Ces approches échouent néanmoins pour décrire les effets d'échelle, classiquement observés en métallurgie physique et dont l'archétype est l'effet de taille de grain. Un modèle de plasticité cristalline de Cosserat appliqué dans le cas du comportement élastoplastique d'aciers IF ferritiques est proposé dans ce travail. Il introduit dans sa formulation une loi de durcissement supplémentaire associé à la courbure de réseau. La simulation d'agrégats polycristallins permet de reproduire numériquement un effet analogue à la loi de Hall-Petch. Une autre limite de la plasticité cristalline est liée à l'étape clé de validation expérimentale locale. L'information expérimentale est, en général, disponible à la surface de l'éprouvette.<br />La géométrie des grains sous la surface est cependant inconnue. Cette information est le plus souvent introduite dans les calculs par extension des joints de grains perpendiculairement à la surface. L'écart, fréquemment observé entre les résultats du calcul et les résultats expérimentaux, peut être expliqué par l'erreur qu'introduit ce choix. On donne ici un minorant de cette erreur en considérant plusieurs agrégats ayant la même morphologie granulaire à la surface libre mais des morphologies tridimensionnelles distinctes. En élasticité la dispersion des contraintes, en un point donné de la surface, avec différentes morphologies de grains sous-jacents est de l'ordre de 30%. En élastoplasticité la dispersion peut aisément atteindre 50% de la valeur de la contrainte, ce qui amène à considérer avec prudence l'identification d'une loi de comportement à partir des seules mesures de surface.

Biomechanics research investigating the knee kinematics of youth participants is very limited. The most accurate method of measuring knee kinematics utilizes invasive procedures such as bone pins. However, various experimental techniques have improved the accuracy of gait kinematic analyses using minimally invasive methods. In this study, gait trials were conducted with two participants between the ages of 11 and 13 to obtain the knee flexion-extension (FE), adduction-abduction (AA) and internal-external (IE) rotation angles of the right knee. The objectives of this study were to (1) conduct pilot experiments with youth participants to test whether any adjustments were necessary in the experimental methods used for adult gait experiments, (2) apply a Triangular Cosserat Point Element (TCPE) analysis for Soft-Tissue Artifact (STA) correction of knee kinematics with youth participants, and (3) develop a code to conduct a Principal Component Analysis (PCA) to find the PCA-defined flexion axis and calculate knee angles with both STA and PCA-correction for youth participants. The kinematic results were analyzed for six gait trials on a participant-specific basis. The TCPE knee angle results were compared between uncorrected angles and another method of STA correction, Procrustes Solution, with a repeated measures ANOVA of the root mean square errors between each group and a post-hoc Tukey test. The PCA-corrected results were analyzed with a repeated measures ANOVA of the FE-AA correlations from a linear regression analysis between TCPE, PS, PCA-TCPE and PCA-PS angles. The results indicated that (1) youth experiments can be conducted with minor changes to experimental methods used for adult gait experiments, (2) TCPE and PS analyses did not yield statistically different knee kinematic results, and (3) PCA-correction did not reduce FE-AA correlations as predicted.

Biomechanics research investigating gait and balance of ultimate frisbee players is an unexplored topic. Ultimate requires a wide range of motions that could improve balance and is also a sport prone to frequent injury. This study explores the impact of playing ultimate on gait parameters associated with balance and knee angles associated with joint injury. Gait trials were conducted on 8 ultimate players and 8 control participants between the ages of 18 and 23 to obtain total double support time, stance phase time, single support time, load response time, abduction-adduction (AA) angles, internal-external (IE) rotation angles, and flexion angles of the dominant leg’s knee. Knee angles were obtained through the application of a Triangular Cosserat Point Element (TCPE) analysis for Soft-Tissue Artifact (STA) correction of knee kinematics. The gait parameters and knee angles were compared between ultimate players and control group participants using two-sample t tests. The results indicated that (1) playing ultimate may be used to improve balance, and (2) playing ultimate may reduce the range of IE rotation angles.