Résumé : This talk revolves around two variational models in finite crystal plasticity where the possible deformations are restricted to plastic glide along one active slip system. In the first model for polycrystals, our focus lies on the set of attainable macroscopic strains, whose analysis is linked to the solvability of an inhomogeneous differential inclusion problem with affine boundary values. We discuss how to estimate this set by exploiting admissible boundary interaction, global compatibility, and the interplay between the slip mechanism and the polycrystalline texture. The second model describes high-contrast composites with periodically arranged layers and gives rise to energy functionals with non-convex differential constraints. We prove homogenization theorems via $\Gamma$-convergence in the Sobolev and BV setting and study the resulting limit models, addressing the uniqueness of minimizers and deriving necessary conditions. These results are joint work with Fabian Christowiak (University of Regensburg), Elisa Davoli (TU Vienna), Dominik Engl (KU Eichstätt-Ingolstadt), and Rita Ferreira (KAUST).

Keywords : relaxation; homogenization; Gamma-convergence; microstructure; differential inclusions; plasticity