TY - JOUR

T1 - Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation

AU - Gómez, Delfina

AU - Nazarov, Sergei A.

AU - Pérez-Martínez, María Eugenia

N1 - Funding Information: This work has been partially supported by Russian Foundation on Basic Research grant 18-01-00325, Spanish MICINN grant PGC2018-098178-B-I00 and the Convenium Banco Santander - Universidad de Cantabria 2018. Publisher Copyright: © 2020, Springer Nature B.V. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - We consider a spectral homogenization problem for the linear elasticity system posed in a domain Ω of the upper half-space R3 +, a part of its boundary Σ being in contact with the plane { x3= 0 }. We assume that the surface Σ is traction-free out of small regions Tε, 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 Tε is O(rε) , where rε≪ 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, rε 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.

AB - We consider a spectral homogenization problem for the linear elasticity system posed in a domain Ω of the upper half-space R3 +, a part of its boundary Σ being in contact with the plane { x3= 0 }. We assume that the surface Σ is traction-free out of small regions Tε, 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 Tε is O(rε) , where rε≪ 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, rε 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.

KW - Boundary homogenization

KW - Capacity matrices

KW - Critical relations

KW - Elasticity

KW - Spectral perturbations

KW - Winkler foundation

KW - ELASTICITY

KW - DOMAINS

KW - HOMOGENIZATION

UR - http://www.scopus.com/inward/record.url?scp=85091375805&partnerID=8YFLogxK

U2 - 10.1007/s10659-020-09791-8

DO - 10.1007/s10659-020-09791-8

M3 - Article

AN - SCOPUS:85091375805

VL - 142

SP - 89

EP - 120

JO - Journal of Elasticity

JF - Journal of Elasticity

SN - 0374-3535

IS - 1

ER -