Spectral Homogenization Problems in Linear Elasticity with Large Reaction Terms Concentrated in Small Regions of the Boundary

Gómez, D.; Назаров, С. А.; Pérez‐Martínez, María‐Eugenia

doi:10.1007/978-3-030-48186-5_7

Cited by 7 publications

(10 citation statements)

References 24 publications

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“…On account of this result, ( 13) and (14), showing the convergence for the eigenpairs of (11) amounts to showing the convergence of solutions of associated stationary problems. Hence, it proves useful to introduce here the stationary homogenization problem:…”

Section: Some Backgroundmentioning

confidence: 98%

“…In this section, for the sake of completeness, we state all the stationary homogenized problems. They can be obtained as in [14], using the technique of matched asymptotic expansions, with minor modifications. We also state the local problems that allow us to describe the strange terms in the boundary conditions.…”

Section: The Homogenized Problemsmentioning

confidence: 99%

“…This dependence is due to both, the nonhomogeneous media filling Ω and the nonconstant Robin matrix M . A formal study of the problem, based on asymptotic expansions, has been addressed in [14] describing convergence as an open problem that we broach here. For the sake of completeness, we provide all the homogenized problems depending on the relations between ε, r ε and β(ε) (cf.…”

Section: Introductionmentioning

confidence: 99%

“…All these works belong to a large class of boundary homogenization problems for several operators, which have been studied for a long time: in this respect, we refer to [14] for an extensive annotated bibliography on vector and scalar problems. Below, we mention just some of the pioneering works in the literature, either because of the geometry or the key words here used.…”

Section: Introductionmentioning

confidence: 99%

“…We mention [15] and [33] in connection with the homogenization of spectral problems, for the Laplacian, with large parameters on the boundary conditions; the technique cannot be extended to the vectorial case here considered. [14] and the present work represent the first spectral boundary homogenization models with large parameters in elasticity theory; [14] contains the formal procedure which differs completely from the technique here used for justifications.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation

Gómez

Назаров

Pérez‐Martínez

2020

J Elast

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We consider a spectral homogenization problem for the linear elasticity system posed in a domain Ω of the upper half-space R 3+ , a part of its boundary Σ being in contact with the plane {x 3 = 0}. We assume that the surface Σ is traction-free out of small regions T ε , where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function M (x) and a reaction parameter β(ε) that can be very large when ε → 0. The size of the regions T ε is O(r ε ), where r ε ≪ ε, and they are placed at a distance ε between them. We provide all the possible spectral homogenized problems depending on the relations between ε, r ε and β(ε), while we address the convergence, as ε → 0, of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on Σ. New capacity matrices are introduced to define these strange terms.

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Section: Some Backgroundmentioning

confidence: 98%

Section: The Homogenized Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%