Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method
© Donato and Ţenţea; licensee Springer. 2013
Received: 28 August 2013
Accepted: 7 November 2013
Published: 3 December 2013
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 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.
This paper deals with the homogenization of a double porosity model in elasticity describing a medium occupying an open set Ω in which consists of two components, one of them being connected and the second one disconnected.
where n is the outward unit normal to . The set is a disconnected union of ε-periodic open sets. We suppose that the elasticity tensor, which defines the strains σ, is of order 1 in and of order in the inclusions .
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.
The homogenized tensor is the same as that obtained for the usual problem stated in the perforated domain with a homogeneous Neumann boundary condition.
The periodic unfolding method was introduced by Cioranescu, Damlamian and Griso in  (see  for a general presentation) and later it was extended to periodically perforated domains by Cioranescu, Damlamian, Donato, Griso and Zaki in  and . In  Donato et al. use the periodic unfolding method for a two-component domain similar to the one considered in this paper. Hence, they introduce two unfolding operators. The first one is denoted by and deals with functions defined on which is the same operator introduced in . The second one concerns the functions defined on and is denoted by . Similar properties for and also the relations between the two operators and, in particular, the properties of their traces on the common boundary are investigated.
Here, we adapt the ideas therein to our problem and, because the elasticity tensor in the inclusion is of order , we define a suitable functional space endowed with an adequate norm (see Remark 2). Also, when we apply the unfolding method, we need to construct proper test functions, which is a key step in obtaining the homogenized problem.
The pioneering paper for the heat diffusion in a two-component domain with similar jump conditions on the interface is due to Auriault and Ene  (see also , where the results were proved using asymptotic expansions). Later, in , Ene and Poliševski gave a rigorous proof of the convergence for one of the cases studied in  using the two-scale convergence method introduced in  and developed in . Several cases of the same problem were treated by Monsurrò in  and Donato and Monsurrò in . Successively, Donato et al. in  treated the problem by the periodic unfolding method. In  Poliševski added to the diffusion matrix in the inclusions.
In the present paper, we extend the results of  to the case of the linearized elasticity. The main difficulty when applying the periodic unfolding method consists in finding suitable test functions adapted to our elasticity tensor and to the interface term appearing in the variational formulation. Let us mention that a similar model is treated via asymptotic expansions by Smyshlyaev in , where, in the inclusions, the tensor is of the form with B being only non-negative. Other related elasticity problems were studied by Damlamian, Cioranescu and Orlik in  and .
For classical results about the homogenization of the linearized elasticity system in composites media, see, for instance, [17–20] and references therein. For the case of periodically perforated domains, we refer to Léné  (see also ).
2 The domain
Let Ω be an open bounded subset of () with a Lipschitz continuous boundary ∂Ω and the unit cube in . We suppose that is a subset of Y such that and its boundary Γ is also Lipschitz continuous. Moreover, we define . One can see that repeating Y by periodicity, the union of all is a connected domain in which will be denoted by . Furthermore, .
One can see that due to the symmetry of the deformation tensor, we have .
3 The problem
Remark 1 Everywhere in this paper we use the Einstein summation convention, except for the cases where we mention that some product does not follow this convention. Also, if there is no mention of the contrary, the writing will follow the same convention since .
where the elements of any are denoted by .
where γ was the order of ε in the boundary condition (3.3), the case studied in the present paper being .
and we introduce the following variational formulation of problem (3.1):
Moreover, the definition of the norm in implies that for , estimates (3.11) and (3.12) hold. □
4 Periodic unfolding operators in two-component domain
In this section we present the definitions of the unfolding operators for a two-component domain introduced by Donato et al. in  and their main properties. We will prove only the results that bring some other properties than those already proved in the paper quoted above. The important characteristic of these operators is that they map functions defined on the oscillating domains and into functions defined on the fixed domains and respectively.
Remark 5 If φ is a function defined in Ω, then, for the sake of simplicity, we write instead of .
if φ and ψ are two Lebesgue measurable functions on , one has ,
- (ii)for every ,
for every ,
strongly in for ,
if is a sequence such that strongly in , then strongly in ,
if is Y-periodic and , then strongly in ,
if satisfies and weakly in , then weakly in ,
if , then and belongs to .
weakly in , where .
and the fact that ends the proof of (4.3)4. □
5 Homogenization results
with no summation of repeated index where , and (). Everywhere in the sequel, the product will not follow the Einstein summation convention and will represent the i th component of a vector field.
We will now apply the appropriate unfolding operator to each term of (5.6) and pass to the limit as in order to obtain the limit problem (5.2).
strongly in , where (no summation).
The last thing to do in order to obtain the limit problem is to sum up (5.10)-(5.13) and use the density of in , that of in and the density of in .
the uniqueness of the solution of problem (5.2) can be proved applying the Lax-Milgram theorem with the norm (5.14) and the bilinear form defined by the left-hand side of (5.2). □
and the components of are given by .
We will study now each term of (5.23).
and finally (5.22) is proved. The convergences (5.20) and (5.21) are given directly by (5.1)1,2. □
The second author’s work was accomplished during his visit to the Raphaël Salem Laboratory of Mathematics of the University of Rouen and it was supported by the Sectorial Operational Programme Human Resources Development (SOP HRD), financed from the European Social Fund and by the Romanian Government under the contract number SOP HRD/107/1.5/S/82514. He would like to express his special thanks for the invitation to the laboratory and he is also very grateful to Prof. Horia I Ene for useful discussions regarding the mechanical model studied in the present paper.
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