Problèmes capacitaires en viscoplasticité avec effets de torsion

Bellieud, Michel

doi:10.1016/j.crma.2013.03.005

Cited by 5 publications

(4 citation statements)

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“…When linear elasticity is considered, this torsional contribution is characteristic of anisotropic media (see Remark 5 (v), (vi)), still may also appear in isotropic ones provided large applied body forces are considered (see Remark 8). Our results, partially announced in [9], can be generalized to non-periodic or random distributions of fibers (see Sections 5.1, 5.2), and apply to other types of high contrast homogenization problems (see Section 5.3). More precisely, we prove that the limit problem associated with (1.2) depends on the density function c f 8,p defined on R 3 ˆR in terms of the capacity (1.1) by c f 8,p pa a a, ζq :" lim εÑ0 1 ε 2 cap f 8,p pa a a, ζe e e 3 ; r ε S, R ε Dq,…”

Section: Introductionmentioning

confidence: 86%

A Notion of Capacity Related to Elasticity: Applications to Homogenization

Bellieud

2011

Arch Rational Mech Anal

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Abstract. We study a notion of capacity related to elasticity which proves convenient to analyze the concentrations of strain energy caused by the rigid displacements of some infinitesimal parts of an elastic body in two or three dimensions. By way of application, we investigate the behavior of solutions to initial boundary value problems describing vibrations of periodic elastic composites with rapidly varying elastic properties. More specifically, we analyze a two-phase medium whereby a set of heavy stiff tiny particles is embedded in a softer matrix. This task is set in the context of linearized elasticity.

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Section: Introductionmentioning

confidence: 86%

A Notion of Capacity Related to Elasticity: Applications to Homogenization

Bellieud

2011

Arch Rational Mech Anal

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“…Remark 1. (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.…”

Section: Setting Of the Problem And Resultsmentioning

confidence: 99%

“…(3.18) The case 0 < κ < +∞ is characterized by the emergence of fourth order derivatives with respect to x 1 , x 2 in the limit equations, revealing bending effects: (3.19) If κ = +∞, we get: Remark 1. (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.…”

Section: Setting Of the Problem And Resultsmentioning

confidence: 99%

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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“…Theorem 1 Assume (5) and (6), then for ε → 0 the solution u u u ε of (7) strongly converges in H 1 0 (Ω; R 3 )) to the unique solution u u u of (8) if 0 < m < 1, of (10) if m = 1, of (12) if 1 < m < 3, of (14) if m = 3, and of (17) if m > 3. The problem (7) has a unique solution u u u ε that realizes the minimum in…”

Section: Description Of the Problem And Statement Of The Resultsmentioning

confidence: 99%

Asymptotic Analysis of a Linear Isotropic Elastic Composite Reinforced by a Thin Layer of Periodically Distributed Isotropic Parallel Stiff Fibres

Bellieud

Geymonat

Krasucki

2015

J Elast

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Problèmes capacitaires en viscoplasticité avec effets de torsion

Cited by 5 publications

References 8 publications

A Notion of Capacity Related to Elasticity: Applications to Homogenization

A Notion of Capacity Related to Elasticity: Applications to Homogenization

Homogenization of Stratified Elastic Composites with High Contrast

Asymptotic Analysis of a Linear Isotropic Elastic Composite Reinforced by a Thin Layer of Periodically Distributed Isotropic Parallel Stiff Fibres

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