Homogenization of Stratified Elastic Composites with High Contrast

Abstract

Abstract. We determine the asymptotic behavior of the solutions to the linear elastodynamic equations in a stratified media comprising an alternation of possibly very stiff layers with much softer ones, when the thickness of the layers tends to zero. The limit equations may depend on higher order terms, characterizing bending effects. A part of this work is set in the context of non-periodic homogenization and an extension to stochastic homogenization is presented.Key words. homogenization, elasticity, non-local effects AMS subject classifications. 35B27, 35B40, 35R60, 74B05, 74Q101. Introduction. In this paper we analyze the asymptotic behavior of the solution to the linear elastodynamic equations in a composite material wherein, at a microscopic scale, possibly very "stiff" layers alternate with a much "softer" medium. Stratified composite media have been intensively investigated over the last decades, especially in the context of diffusion equations [18,27,29,30,31,32,39,52,54]. As regards linear elasticity, layered elastic composites have been studied in [26,28,33,38] under assumptions of uniform boundedness and uniform definite positiveness of the elasticity tensor guaranteeing that the effective equation is a standart linear elasticity equation. When these assumptions break down, as for instance in the so-called "high contrast case", the limit equilibrium equation may be of a quite different type: it may correspond, in theory, to the Euler equation associated to the minimization of any lower semi-continuous quadratic form on L 2 vanishing on rigid motions [20]. In particular, it may be non-local and depend on higher order derivatives of the displacement. Elastic media with high contrast have been studied under various geometrical assumptions. Composites with stiff grain-like inclusions have been investigated in [7,8,45], stiff fibered structures in [8,12,13,46,50], and stiff media with holes filled with a soft material in [22,24,47]. Our aim is to complement this body of work in the context of stratified media. Our approach is based on the two-scale convergence method [3,5,19,23,40,41], which yields the convergence to an effective solution. It also yields a first order corrector result in L 2 (see Remark 3.14), but not the rigorous error estimates of higher order with respect to small parameters provided by the asymptotic expansions method [1,2,6,15,16,21,43,44,45,48,49].For a given bounded smooth open subset Ω of R 3 , we consider a linear elastodynamic problem like (3.5). We assume that the Lamé coefficients take possibly large values in a subset B ε of Ω and much smaller values elsewhere. The set B ε consists of a non-periodic distribution of parallel disjoint homothetic layers of thickness r ε , whose median planes are orthogonal to e 3 and separated by a minimal distance ε, where ε, r ε are positive reals converging to zero (see fig. 3.1). The effective volume fraction of the stiff phase B ε is characterized by the parameter ϑ defined by (3.10). Both cases ϑ = 0 and 0 < ϑ < 1 are investigated. The ord...

Homogenization of periodic graph-based elastic structures

“…However very few explicit examples have been given in which such a phenomenon appears. In [47], [12], [21], [14], [13], [11] the homogenized material becomes a second gradient one: the elastic energy depends on the second gradient of the displacement instead of the first one only. However all these results fall under the framework of couple-stress theory, [54], [55], [40], [42]: the dependence with respect to the second gradient of the displacement is limited to dependence on the gradient of the skew-symmetric part of the gradient of the displacement only.…”

Stochastic homogenisation of high-contrast media

“…Various aspects of multiscale analysis of high-contrast media in the stochastic context have been looked at in a handful of papers, e.g. [BMP03], [BBM15], [Bel17].…”

Asymptotic Analysis of Stratified Elastic Media in the Space of Functions with Bounded Deformation

“…The general high-contrast theory of elliptic problems is still lacking, particularly in the case of non-periodic composites where some preliminary work has been done in the case of stratified materials, i.e. scalar elliptic partial differential equations with coefficients that depend only on one variable, see [8], [14], [16] and references therein. This article further develops the theory in the direction of elliptic systems.…”