This dissertation proposes a finite element procedure for evaluating the high order strain derivatives in nonlocal computational mechanics. The superconvergent second derivative recovery methods used are proven to be effective in evaluating the Laplacian of the equivalent strain based on low order (linear) elements. Current nonlocal finite element techniques with linear elements are limited to structured meshes, while the new technique can deal with unstructured meshes with various element types. Other superconvergent patch recovery (SCP) based nonlocal approaches, such as the patch projection techniques only utilize nodal based patches to evaluate the first derivatives of the strain. The SCP technique has not yet been used for recovery of higher order strain derivatives. The proposed technique is capable of evaluating the Laplacian of the equivalent strain and has the potential for even higher order derivative recovery. The same patches can be easily utilized for error estimation and adaptive meshing for nonlocal problems. We employ two super-convergent patch options: the element based patch with all neighbors or only facing neighbors. The nonlocal strain derivatives can be recovered through either a mesh nodal averaging process or directly at the patch element quadrature points after the patch least square fitting problems are solved. Numerical examples for both strain gradient damage mechanics and strain gradient plasticity problems are given. In summary, the new finite element nonlocal computational technique based on the superconvergent second derivative recovery methods is proven to be robust in evaluating the high order strain derivatives with low order element unstructured meshes.

Les matériaux polymères sont largement utilisés pour des applications structurelles dans le secteur automobile et leurs comportements complexes nécessitent des modèles précis pour la simulation éléments finis. Les polymères possèdent un comportement dépendant du temps et de la vitesse. La dépendance à la vitesse peut être observée par un accroissement de la rigidité et de la limite élastique en fonction de la vitesse de déformation. Le long temps nécessaire pour retrouver des contraintes nulles après sollicitation du matériau met en évidence la dépendance du temps sur le comportement. De plus, particulièrement pour les polymères chargés, le phénomène de cavitation se traduisant par la création et la croissance de micro-cavités et de microfissures conduit à un changement de volume durant la déformation. Dans ce travail, un modèle de comportement est développé pour un polymère semi-cristallin chargé de talc utilisé dans l’industrie automobile. Un modèle constitutif viscoélastique-viscoplastique non-associatif avec endommagement non-local est proposé dans le but de simuler les phénomènes observés expérimentalement. Dans le modèle développé, une surface de charge non symétrique est utilisée pour prendre en compte la pression hydrostatique. La viscoplasticité non-associative couplée avec l’endommagement conduit aux déformations viscoplastiques non-isochoriques caractérisées expérimentalement. Les paramètres du modèle proviennent d’essais expérimentaux réalisés sous différentes conditions et `a différentes vitesses de déformation. Pour ces essais, plusieurs techniques de mesure, telles que la corrélation d’images et l’extensommetrie optique sont utilisées pour les mesures de champs de déplacements. La bonne corrélation entre les données expérimentales et les simulations numériques mettent en évidence la précision du modèle développé afin de modéliser le comportement des matériaux polymères semi-cristallins. / Polymer materials are widely used for structural applications in the automotive sector and their behaviours are complex and require accurate models for finite element simulations. Polymer materials exhibit rate and time dependent behaviours. The rate dependency can be observed by an increase of the stiffness and the yield stress at increasing strain rate. The long time to recover the zero stress after solicitation of the material highlight the time dependent behaviour. Furthermore, particularly for filled polymers, the cavitation phenomenon cause the creation and growth of micro-voids and microcracks called damage and leads to volume change during the deformation. In this work, a behaviourmodel for mineral filled semi-crystalline polymer used in automotive industry is developed. A constitutive viscoelastic-viscoplastic non-associated model coupled with nonlocal damage is proposed in order to simulate the phenomena observed experimentally. In the constitutive model, a non symmetric yield surface is used to take the hydrostatic pressure into account. The non associated viscoplasticity coupled with damage leads to the non-isochoric viscoplastic deformation characterised experimentally. The material parameters arise from experimental tests carried out under various loadings and strain rates. For these experimental tests, different measurement techniques like Digital Image Correlation and optical extensometry are used for the displacements and the strain field measurements. The good agreement between the experimental data and the numerical simulations highlights the accuracy of the developed model for polymer modelling.

To enhance material performance at different length scales, this study strives to develop a reliable analytical and computational tool with the help of internal state variables spanning micro and macro-level behaviors. First, the practical relevance of a nonlocal damage integral added to an internal state variable (BCJ) model is studied to alleviate numerical instabilities associated within the post-bifurcation regime. The characteristic length scale in the nonlocal damage, which is mathematical in nature, can be calibrated using a series of notch tensile tests. Then the same length scale from the notch tests is used in solving the problem of a high-velocity (between 89 and 107 m/s) rigid projectile colliding against a 6061-T6 aluminum-disk. The investigation indicates that incorporating a characteristic length scale to the constitutive model eliminates the pathological mesh-dependency associated with material instabilities. In addition, the numerical calculations agree well with experimental data. Next, an effort is made rather to introduce a physically motivated length scale than to apply a mathematical-one in the deformation analysis. Along this line, a dislocation based plasticity model is developed where an intrinsic length scale is introduced in the forms of spatial gradients of mobile and immobile dislocation densities. The spatial gradients are naturally invoked from balance laws within a consistent kinematic and thermodynamic framework. An analytical solution of the model variables is derived at homogenous steady state using the linear stability and bifurcation analysis. The model qualitatively captures the formation of dislocation cell-structures through material instabilities at the microscopic level. Finally, the model satisfactorily predicts macroscopic mechanical behaviors - e.g., multi-strain rate uniaxial compression, simple shear, and stress relaxation - and validates experimental results.

Les travaux présentés dans ce mémoire s'inscrivent dans l'étude et l'amélioration des modèles d'endommagement continus régularisés (non locaux), l'objectif étant d'étudier la transition entre un champ d'endommagement continu défini sur l'ensemble d'une structure et un modèle discontinu de fissuration macroscopique.La première étape consiste en l'étude semi-analytique d'un problème unidimensionnel (barre en traction) visant à identifier une famille de lois d'interface permettant de basculer d'une solution non homogène obtenue avec un modèle continu à gradient d'endommagement vers un modèle discontinu de fissuration cohésive. Ce passage continu / discontinu est construit de telle sorte que l'équivalence énergétique entre les deux modèles soit assurée, et reste exacte quelque soit le niveau de dégradation atteint par le matériau au moment où cette transition est déclenchée.Cette stratégie est ensuite étendue au cadre 2D (et 3D) éléments finis dans le cas de la propagation de fissures rectilignes (et planes) en mode I. Une approche explicite basée sur un critère de dépassement d'une valeur « critique » de l'endommagement est proposée afin de coupler les modèles continus et discontinus au sein d'un même calcul quasi-statique par éléments finis. Enfin, plusieurs résultats de simulations menées avec cette approche couplée sont présentés. / The present work deals with the study and the improvement of regularized (non local) damage models. It aims to study the transition from a continuous damage field distributed on a structure to a discontinuous macroscopic failure model.First, an analytical one-dimensional study is carried out (on a bar submitted to tensile loading) in order to identify a set of interface laws that enable to switch from an inhomogeneous solution obtained with a continuous gradient damage model to a cohesive zone model. This continuous / discontinuous transition is constructed so that the energetic equivalence between both models remains ensured whatever the damage level reached when switching.This strategy is then extended to the bi-dimensional (and tri-dimensional) case of rectilinear (and plane) crack propagation under mode I loading conditions, in a finite element framework. An explicit approach based on a critical damage criterion that allows coupling both continuous and discontinuous approaches is then proposed. Finally, results of several simulations led with this coupled approach are presented.