Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method
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Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method Patrizia Donato1 and Iulian T¸en¸tea2* * Correspondence: [email protected] 2 Simion Stoilow Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei Street, Bucharest, 010702, Romania Full list of author information is available at the end of the article
Abstract We study an ε -periodic model of a medium with double porosity which consists of two components, one of them being connected. We assume that the elasticity of the medium in the inclusion is of order ε 2 and also, on the interface between the two components, we consider a jump of the displacement vector condition, proportional to the stress tensor which is continuous. The aim of the paper is to prove the convergence of the homogenization process using the periodic unfolding method. Keywords: homogenization; elasticity; double porosity; interface jumps
1 Introduction This paper deals with the homogenization of a double porosity model in elasticity describing a medium occupying an open set Ω in RN which consists of two components, one of them being connected and the second one disconnected. More exactly, we suppose that Ω is the union of two open subsets Ωε and Ωε and their common boundary Γ ε , and we consider the problem ⎧ ∂σ αε ⎪ – ∂xijj = gi in Ωαε , α ∈ {, }, ⎪ ⎪ ⎪ ⎨ ε σij nj = σijε nj on Γε , ε ⎪ ⎪ σijε nj = εhε (uε ⎪ i – ui ) on Γε , ⎪ ⎩ ε u = on ∂Ω, where n is the outward unit normal to Ωε . The set Ωε is a disconnected union of ε-periodic open sets. We suppose that the elasticity tensor, which defines the strains σ , is of order in Ωε and of order ε in the inclusions Ωε . The jump conditions on the surface model in fact a layer of a soft material that surrounds the particles; the layer is here modeled as a surface, thus not only the tangential, but also the normal component of the displacement can have a jump. Using the periodic unfolding method, we prove some convergence results and describe the homogenized problems. More precisely, first, in Theorem we describe the homogenized problem in the variables x and y, as is usually done. Then, in Theorem we identify the homogenized prob©2013 Donato and T¸en¸tea; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Donato and T¸en¸tea Boundary Value Problems 2013, 2013:265 http://www.boundaryvalueproblems.com/content/2013/1/265
lem in Ω, and we show that u˜ ε |Y | · u
weakly in L (Ω)N ,
u˜ ε |Y | · u + gl · ql
weakly in L (Ω)N ,
where, for α ∈ {, }, u˜ αε is the extension by zero of uαε to the whole Ω and u is the unique solution of the problem ⎧ ⎨–
∂uk ∂ (a∗ ) = gi ∂xj ijkh ∂xh
⎩u =
in Ω,
on ∂Ω.
The homogenized tensor A∗ is the same as that obtained for the usual problem s
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