Abstract
Abstract 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.
Highlights
Metamaterials are artificially engineered composites whose heterogeneities are optimized to improve structural performances
In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites
We consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system
Summary
Metamaterials are artificially engineered composites whose heterogeneities are optimized to improve structural performances. An important first step towards identifying the limit behavior of the energies (Eε)ε (in the sense of Γ-convergence) is the proof of a general statement of asymptotic rigidity for layered structures in the context of functions of bounded variation. There has been an effort towards weakening higher-order regularizations: It is shown in [7] that the full norm of the Hessian can be replaced by a control of its minors (gradient polyconvexity) in the context of locking materials; for solid-solid phase transitions, an anisotropic second-order penalization is considered in [23] Along these lines, we introduce the regularized energies in (1.13) that penalize the variation of deformations only in the layer direction.
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