Asymptotic Rigidity of Layered Structures and Its Application in Homogenization Theory

Abstract

In the context of elasticity theory, rigidity theorems allow 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.MSC (2010): 49J45 (primary); 74Q05, 74B20

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

“…Analogous arguments to those in [14,Section 3] show that a function f ∈ BV (Ω; R d ) satisfying D 1 f = 0 is locally one-dimensional in the e 2 -direction. The following geometrical requirement is the counterpart of [14, Definitions 3.6 and 3.7] in our setting.…”

“…Under this geometrical assumption, the notions of locally and globally one-dimensional functions in the e 2 -direction coincide. We refer to [14,Section 3] for an extended discussion on the topic, as well as for some explicit geometrical examples.…”

“…These results have been extended to general dimensions, to energy densities with p-growth for 1 < p < +∞, and to the case with non-trivial elastic energies, which allows treating very stiff (but not necessarily rigid) layers, see [14,12].…”

“…A simple modification of the proofs of Propositions 3.1 and 3.2 shows that these results hold for any such rectangle. Then, Theorem 1.1 follows by extension and exhaustion arguments in the spirit of[14, Lemma A.2].…”

On static and evolutionary homogenization in crystal plasticity for stratified composites

“…In [11], these techniques have been carried forward to a model for plastic composites without linear hardening in the spirit of [9], which leads to a variational limit problem on the space of functions of bounded variation. Natural generalizations of these models to three (and higher) dimensions, where the material heterogeneities are either layers or fibers are studied [6] and [15], respectively. Note that these two references, which are formulated in context of nonlinear elasticity, use energy densities with p-growth for 1 < p < +∞, Ω ⊂ R 2 ε Figure 1.…”