Calculus of Variations and Geometric Measure Theory

E. Davoli - R. Ferreira - C. Kreisbeck

Homogenization in $BV$ of a model for layered composites in finite crystal plasticity

created by davoli on 31 Jan 2019
modified on 04 Sep 2020


Published Paper

Inserted: 31 jan 2019
Last Updated: 4 sep 2020

Journal: Adv. Calc. Var.
Year: 2019
Doi: 10.1515/acv-2019-0011


In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from these modeling assumptions are of integral form, featuring linear growth and non-convex differential constraints. We approach this non-standard homogenization problem via Gamma-convergence. A crucial first step in the asymptotic analysis is the characterization of rigidity properties of limits of admissible deformations in the space $BV$ of functions of bounded variation. In particular, we prove that, under suitable assumptions, the two-dimensional body may split horizontally into finitely many pieces, each of which undergoes shear deformation and global rotation. This allows us to identify a potential candidate for the homogenized limit energy, which we show to be a lower bound on the Gamma-limit. In the framework of non-simple materials, we present a complete Gamma-convergence result, including an explicit homogenization formula, for a regularized model with an anisotropic penalization in layer direction.