We provide in this paper homogenization results for the L2-topology leading to complete strain-gradient models and generalized continua. Actually, we extend to the L2-topology the results obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5, 259–288) using a topology adapted to minimization problems set in varying domains. Contrary to (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5, 259–288) we consider elastic lattices embedded in a soft elastic matrix. Thus our study is placed in the usual framework of homogenization. The contrast between the elastic stiffnesses of the matrix and the reinforcement zone is assumed to be very large. We prove that a suitable choice of the stiffness on the weak part ensures the compactness of minimizing sequences while the energy contained in the matrix disappears at the limit: the Γ-limit energies we obtain are identical to those obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5, 259–288).

While homogenization of periodic linear elastic structures is now a well-known procedure when the stiffness of the material varies inside fixed bounds, no homogenization formula is known which enables to compute the effective properties of highly contrasted structures. Examples have been given in which the effective energy involves the strain-gradient but no general formula provides this strain-gradient dependence. Some formulas have been proposed which involve such terms and provide a small correction to the classical effective energy still when the stiffness of the material varies inside fixed bounds. The goal of this paper is to check the applicability of these formulas for highly contrasted structures. To that aim we focus on structures whose limit energy is already known and we compare the energies given by (i) the convergence results, (ii) the corrective formulas and (iii) by a direct numerical simulation of the complete structure.

We introduce a model and several constraints for shape and topology optimization of structures, built by additive manufacturing techniques. The goal of these constraints is to take into account the thermal residual stresses or the thermal deformations, generated by processes like Selective Laser Melting, right from the beginning of the structural design optimization. In other words, the structure is optimized concurrently for its final use and for its behavior during the layer-by-layer production process. It is well known that metallic additive manufacturing generates very high temperatures and heat fluxes, which in turn yield thermal deformations that may prevent the coating of a new powder layer, or thermal residual stresses that may hinder the mechanical properties of the final design. Our proposed constraints are targeted to avoid these undesired effects. Shape derivatives are computed by an adjoint method and are incorporated into a level set numerical optimization algorithm. Several 2D and 3D numerical examples demonstrate the interest and effectiveness of our approach.

We study a model for visco-elasto-plastic deformation with fracture, in which fracture is approximated via a diffuse interface model. We show that a discretized (in time) quasistatic evolution, converges to a solution of the continuous (in time) evolution, proving existence of a solution to our model.

This paper deals with an existence theorem for a model describing an elasto-viscoplastic evolution of a 2D material with linear kinematic hardening and fracture where the Griffith fracture energy is regularized using a r -Laplacian.

Dans cette these nous nous interessons a la modelisation, analyse mathematique et simulation numerique d'une classe de modeles combinant differents phenomenes dissipatifs lies a la plasticite, rupture et dissipation visqueuse.Tout d'abord, nous construisons des modeles d'evolution contenant plasticite, viscoplasticite, ecrouissage cinematique lineaire et rupture. En particulier, nous montrons une inegalite thermodynamique de type Clausius-Duhem pour nos modeles. Ensuite, nous montrons l'existence d'evolutions pour deux modeles: celui d'elasto-visco-plasticite avec la rupture approchee via la fonctionnelle Ambrosio-Tortorelli et celui d'elasto-viscoplasticite avec ecrouissage cinematique lineaire et rupture approchee basee sur l'utilisation de la fonctionnelle d'Ambrosio-Tortorelli avec un r-Laplacien. Enfin, nous etudions numeriquement nos modeles en fonction de differents parametres mecaniques. Nous proposons aussi une extension de la methode numerique de backtracking aux materiaux a memoire. Au final, nous effectuons des comparaisons numeriques entre un de nos modeles et l'experience geophysique de plasticine de Peltzer et Tapponnier qui modelise la propagation des failles dans la croute terrestre.

We study a class of time evolution models that contain dissipation mech- anisms exhibited by geophysical materials during deformation: plasticity, viscous dissipation and fracture. We formally prove that they satisfy a Clausius-Duhem type inequality. We describe a semi-discrete time evolu- tion associated with these models, and report numerical 1D and 2D traction experiments, that illustrate that several dissipation regimes can indeed take place during the deformation. Finally, we report 2D numerical simulation of an experiment by Peltzer and Tapponnier, who studied the indentation of a layer of plasticine as an analogue model for geological materials.