A Notion of Capacity Related to Elasticity: Applications to Homogenization

Bellieud, Michel

doi:10.1007/s00205-011-0448-5

Cited by 8 publications

(9 citation statements)

References 37 publications

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“…Our results apply as well to the case of equilibrium equations and to multiphase composites (see remarks 6,7,10,11). The paper is organised as follows: in Section 2 we specify the notations and in Section 3 we state our main results.…”

Section: Preliminary Version 1 Introductionsupporting

confidence: 67%

“…(i) When stiff fibers [9,13] (resp. grain-like inclusions [7]) embedded in a matrix of stiffness of order 1 are considered, the fibers (resp. the inclusions) disappear from the limit problem if r ε exp − 1 ε 2 (resp.…”

Section: Setting Of the Problem And Resultsmentioning

confidence: 99%

“…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.…”

Section: Preliminary Version 1 Introductionmentioning

confidence: 99%

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Homogenization of Stratified Elastic Composites with High Contrast

Bellieud¹

2017

SIAM J. Math. Anal.

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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...

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Section: Preliminary Version 1 Introductionsupporting

confidence: 67%

Section: Setting Of the Problem And Resultsmentioning

confidence: 99%

Section: Preliminary Version 1 Introductionmentioning

confidence: 99%

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Homogenization of Stratified Elastic Composites with High Contrast

Bellieud¹

2017

SIAM J. Math. Anal.

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“…For (a a a, α) ∈ R 3 × R, the extended real c f (a a a, α) represents the limit of the sum of the images of the connected components of the cross section of the fibers under cap f (., Ω; a a a, α) per unit surface, where, for any open subsets U, V of R 2 such that U is bounded and U ⊂ V , denoting by x x x U the geometrical center of gravity of U , the application cap f is defined by (11). Similar notions of capacity are considered in [2], [10]. Equivalently, c f (a a a, α) is given by (12).…”

Section: Abridged English Versionmentioning

confidence: 99%

Problèmes capacitaires en viscoplasticité avec effets de torsion

Bellieud

2013

Comptes Rendus. Mathématique

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“…The relationship between such models and the study of traditional composites has been extensively explored. At the turn of the century, increasingly exotic composite materials which pertain to non-local, memory [11,15,25], or higher-order effects [5,6,10], negative effective density [33] [20], and novel wave effects such as directional propagation and cloaking [1], [26], were studied. These works explore particular examples of 'high-contrast' periodic composites where the ratio between the constituent's material parameters is unbounded in ε.…”

Section: Introductionmentioning

confidence: 99%

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

Bellieud

Cooper

2017

SIAM J. Math. Anal.

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We consider a heterogeneous elastic structure which is stratified in some direction. We derive the limit problem under the assumption that the Lamé coefficients and their inverses weakly* converge to Radon measures. Our method applies also to linear second-order elliptic systems of partial differential equations and in particular, for the case d = 1, this addresses the previously open problem of determining the asymptotic behaviour in this context for the general anisotropic heat equation.

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A Notion of Capacity Related to Elasticity: Applications to Homogenization

Cited by 8 publications

References 37 publications

Homogenization of Stratified Elastic Composites with High Contrast

Homogenization of Stratified Elastic Composites with High Contrast

Problèmes capacitaires en viscoplasticité avec effets de torsion

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

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