Asymptotic Rigidity of Layered Structures and Its Application in Homogenization Theory

Abstract

In the context of elasticity theory, rigidity theorems allow one to derive global properties of a deformation from local ones. This paper presents a new asymptotic version of rigidity, applicable to elastic bodies with sufficiently stiff components arranged into fine parallel layers. We show that strict global constraints of anisotropic nature occur in the limit of vanishing layer thickness, and give a characterization of the class of effective deformations. The optimality of the scaling relation between layer thickness and stiffness is confirmed by suitable bending constructions. Beyond its theoretical interest, this result constitutes a key ingredient for the homogenization of variational problems modeling high-contrast bilayered composite materials, where the common assumption of strict inclusion of one phase in the other is clearly not satisfied. We study a model inspired by hyperelasticity via Γ -convergence, for which we are able to give an explicit representation of the homogenized limit problem; it turns out to be of integral form with its density corresponding to a cell formula.