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Motivated by the study of biological tissues, this work contributes to developing numerical tools to predict the mechanical response of nonlinear heterogeneous materials in which the energies of interfaces can no longer be ignored. First, a computational homogenization strategy combined with a model reduction technique based on the proper orthogonal decomposition is implemented in the cases of large elastic deformations and highly nonlinear conduction. The interfaces between the different phases of a composite are described by means of a coherent interface model and taken into account numerically by an extended finite element method in tandem with a level-set technique. Finally, experimental results of single cell spreading between two fixed parallel microplates are exploited through finite element modelling. Our two models show that the bilayer membrane and the actin cortex do not play a significant role in the cell mechanical response, Motivé par l'étude de tissus biologiques, ce travail contribue aux développements d'outils numériques permettant de prédire la réponse mécanique de matériaux hétérogènes non linéaires dans lesquels les énergies d'interfaces deviennent prépondérantes. Ainsi, une méthode d'homogénéisation multi échelle combinée à une technique de réduction de modèle basée sur la décomposition orthogonale aux valeurs propres est proposée dans un cadre thermique et hyperélastique. Les énergies d'interfaces entre les différentes phases des composites sont décrites par un modèle d'interface cohérent et prises en compte numériquement par une approche liant la méthode des éléments finis étendus et la méthode level-set. Une étude de l'étalement d'une cellule vivante entre deux lamelles fixes est ensuite réalisée. Les deux modèles utilisés pour les simulations montrent que l'assemblage cortex d'actine-membrane plasmique ne joue qu'un rôle minime dans la réponse mécanique cellulaire

International audience; The mechanical response of an elastoplastic polycrystalline aggregate is estimated by means of Taylor and Hill's incremental self-consistent models in conjunction with two rate-independent crystal plasticity laws: standard and regularized Schmid laws. With the Taylor model, both constitutive laws lead to the same result whereas for the self-consistent model, the standard Schmid law predicts softer effective response than the regularized Schmid law and higher strain heterogeneity within the polycrystal. It is found that these features are related to the decrease of the shear tangent moduli during deformation when the standard Schmid law is considered. The presented results with the regularized Schmid law are in agreement with previous findings for power law viscoplastic polycrystals. It is demonstrated that the description of the crystal plasticity law significantly affects the estimate delivered by the Hill's incremental self-consistent model for elastoplastic polycrystals.

The present paper deals with the problem of the determination of the in-plane behavior of masonry material. The masonry is considered as a composite material composed by a regular distribution of blocks connected by horizontal and vertical mortar joints. The overall constitutive relationships of the regular masonry are derived by a rational micromechanical and homogenization procedure. Linear elastic constitutive relationship is considered for the blocks, while a new special nonlinear constitutive law is proposed for the mortar joints. In particular, a mortar constitutive law, which accounts for the coupling of the damage and friction phenomena occurring during the loading history, is proposed; the developed model is based on an original micromechanical analysis of the damage process of the mortar joint. Then, an effective nonlinear homogenization procedure, representing the main novelty of the paper, is proposed; it is based on the transformation field analysis, using the technique of the superposition of the effects and the finite element method. The presented methodology is implemented in a numerical code. Finally, numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of masonry material.

The present Note concerns the formulation, implementation and a first application of a micromechanically based hyperelastic damage model. The approach is based on the second order homogenization method proposed by Lopez-Pamies and Ponte Castañeda (2000) for hyperelastic composites and recently developed by Lopez-Pamies and Ponte Castañeda (2007) in the case of porous elastomers. We first implement the method and proceed to its verification by comparison with Finite Element simulations on a unit cell. Taking advantage of this validation and by using standard thermodynamics arguments, we propose an hyperelastic damage model founded on voids growth phenomena. Finally, we provide an example of validation of the model by comparison with experimental data obtained on an EPDM/PP composite.

International audience; This work presents a means for extracting the statistics of the local fields in nonlinear composites from the effective potential of suitably perturbed composites. The idea is to introduce a parameter in the local potentials, generally a tensor, such that differentiation of the corresponding effective potential with respect to the parameter yields the volume average of the desired quantity. In particular, this provides a generalization to the nonlinear case of well-known formulas in the context of linear composites, which express phase averages and second moments of the local fields in terms of derivatives of the effective potential. Such expressions are useful since they allow the generation of estimates for the field statistics in nonlinear composites, directly from homogenization estimates for appropriately defined effective potentials. Here, use is made of these expressions in the context of the ‘variational’, ‘tangent second-order’ and ‘second-order’ homogenization methods, to obtain rigorous estimates for the first and second moments of the fields in nonlinear composites. While the variational estimates for these quantities are found to be identical to those proposed in previous works, the tangent second-order and second-order estimates are found be different. In particular, the new estimates for the first moments given in this work are found to be entirely consistent with the corresponding estimates for the macroscopic behaviour. Sample results for two-phase, power-law composites are provided in part II of this work.

International audience; Part I of this work provided a methodology for extracting the statistics of the local fields in nonlinear composites, from the effective potential of suitably perturbed composites. In particular, exact relations were given for the first and even moments of the fields in each constituent phase. In this part, use is made of these exact relations in the context of the ‘variational’, ‘tangent second-order’ and ‘second-order’ nonlinear homogenization methods to generate estimates for the phase averages and second moments of the fields for two-phase, power-law composites with isotropic and transversely isotropic microstructures. The accuracy of these estimates is assessed by confronting them against corresponding exact results for sequentially laminated composites. Among the nonlinear homogenization estimates considered in this work, the second-order estimates are found to be, in general, the most accurate, especially for large heterogeneity contrast and nonlinearity. Thus, these estimates are able to capture, for example, the strong anisotropy in the strain fluctuations that can develop inside nonlinear porous and rigidly reinforced composites.

International audience; A porous medium, which matrix is a perfectly plastic solid, is considered. This paper proposes a method to determine the macroscopic admissible stress states. The method is based on a homogenization technique which takes advantage of the equivalence, under certain conditions, between a problem of limit analysis and a ficticious nonlinear elastic problem. The particular case of a Drucker–Prager solid matrix is considered. The method provides an analytical expression for the complete macroscopic strength criterion.; On propose une méthode de détermination des états de contrainte situés sur la frontière du domaine des contraintes macroscopiques admissibles pour un milieu poreux à matrice solide parfaitement plastique. On utilise une technique d'homogénéisation basée sur l'équivalence, dans certaines conditions, entre le problème de calcul à la rupture et un problème d'élasticité non linéaire bien choisi. On se penche en particulier sur le cas où la matrice solide a un comportement de Drucker–Prager, pour lequel on donne une détermination analytique de la frontière du critère macroscopique.

The ‘second–order’ homogenization procedure of Ponte Castaneda is used to propose new estimates of the self–consistent type for the effective behaviour of viscoplastic polycrystals. This is accomplished by means of appropriately generated estimates of the self–consistent type for the relevant ‘linear thermoelastic comparison composite’, in the homogenization procedure. The resulting nonlinear self–consistent estimates are the only estimates of their type to be exact to second order in the heterogeneity contrast, which, for polycrystals, is determined by the grain anisotropy. In addition, they satisfy the recent bounds of Kohn and Little for two–dimensional power–law polycrystals, which are known to be significantly sharper than the Taylor bound at large grain anisotropy. These two features combined, suggest that the new self–consistent estimates, obtained from the second–order procedure, may be the most accurate to date. Direct comparison with other self–consistent estimates, including the classical incremental and secant estimates, for the special case of power–law creep, appear to corroborate this observation.

The variational asymptotic method for unit cell homogenization (VAMUCH) has emerged as a general-purpose micromechanics code capable of predicting the effective properties of heterogeneous materials and recovering the local fields. The objective of this paper is to propose a micromechanics approach enabling VAMUCH to homogenize elasto-viscoplastic heterogeneous materials. An affine formulation of the constitutive relations for an elasto-viscoplastic constituent, which exhibits viscoplastic anisotropy and combined isotropic–kinematic hardening, is derived. The weak form of the problem is derived using an asymptotic method, discretized using finite elements, and implemented into VAMUCH. The new features of VAMUCH are validated with examples such as homogenizing binary, fiber-reinforced, and particle-reinforced composites. VAMUCH is found to be capable of handling various microstructure, complex material models, complex loading conditions, and complex loading paths. More sophisticated material models can be implemented into it.

International audience

International audience; The present work is devoted to the microbuckling analysis of long fiber composites. A multiscale finite element method (FE2) is combined with the asymptotic numerical method (ANM) to study the elastoplastic instability which may occur in structures at both macroscopic and microscopic scales. The fiber is described by a linear material constitutive law, while the matrix phase is described by a nonlinear Ramberg-Osgood relationship. The stress field is then obtained via the total mechanical strain without any history dependence. Large strains are considered, which induce geometrical nonlinearities in both cases. The ANM framework allows obtaining complex response curves involving limit points in loading and displacement to be obtained. In the present path following procedure, adjustment of the step length is naturally automatic because the validity range of the asymptotic solution is a posteriori estimated depending on the local nonlinearity of the response branches. Numerical examples show the effectiveness of the proposed approach by investigating microscopic and macroscopic instabilities of long fiber composite structures in compression.