Asymptotic Behavior of Crystal Plasticity with One Slip System in the Limit of Rigid Elasticity

Abstract

We consider a family of models in elastoplasticity describing crystals with one active slip system and linear hardening in two spatial dimensions. In particular, our studies focus on the asymptotic behavior of the system in the limit of diverging elastic coefficients. We determine explicitly the $\Gamma$-limit of the energy functionals underlying the variational model and show that it coincides with the relaxation of a variational problem with rigid elasticity. The analysis combines a priori estimates that reflect the anisotropic growth of the energy density, a subtle generalization of the classical div–curl lemma to recover the constraint of incompressibility in the limit, and a careful construction of the recovery sequence, which involves laminates with a position-dependent period.