  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 the layer
direction.
