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Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method
Boundary Value Problems volume 2013, Article number: 265 (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.
More exactly, we suppose that Ω is the union of two open subsets and and their common boundary , and we consider the problem
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.
More precisely, first, in Theorem 11 we describe the homogenized problem in the variables x and y, as is usually done. Then, in Theorem 12 we identify the homogenized problem in Ω, and we show that
where, for , is the extension by zero of to the whole Ω and is the unique solution of the problem
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 presence of the tensor of order in the second component does not contribute to the homogenized tensor . However, it gives rise to the additional term in the limit of , where for and is the unique solution of the cell problem
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, .
In the following, the parameter takes its values in a sequence of real numbers which, in the homogenizing process, will tend to zero. For each , we define and , where . We also define, for each ε,
and we set (see Figure 1)
The boundary of will be denoted by and n will be the normal on exterior to .
We introduce now, for each ε, the jump factor and the fourth-order elasticity tensor defined by
where and the components are smooth, real, Y-periodic functions, with the property that there exists such that
for any and any symmetric tensor . We also assume the symmetry condition for the elasticity tensor, namely
Finally, for and a function defined on , we denote the stress tensors by , where represent the components of the deformation tensor defined for any function v by
One can see that due to the symmetry of the deformation tensor, we have .
3 The problem
Our goal is to describe the asymptotic behavior, as , of the problem
where are the components of a vector field which represent the given body forces together with the boundary conditions
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 .
We introduce the space equipped with the -norm of the gradient and the Hilbert space
endowed with the scalar product
where the elements of any are denoted by .
The norm generated by the scalar product (3.6) is given by
Remark 2 Due to the fact that the elasticity tensor is of order in the inclusion , the functional space and the norm defined by (3.7) are different from those used in . More precisely, in  the space used was and it was equipped with the norm
where γ was the order of ε in the boundary condition (3.3), the case studied in the present paper being .
For any , we also use the notation
and we introduce the following variational formulation of problem (3.1):
Find such that
Theorem 3 For any , problem (3.10) has a unique solution . Moreover, there exists a constant independent of ε such that, for and each , we have
Proof The result is proved by applying the Lax-Milgram theorem. The coerciveness of the form can be easily shown using definition (3.9) and the properties (2.4) which give
Let us prove now the continuity of the right-hand side of (3.10). Applying the Cauchy-Schwarz inequality, we obtain
From (3.13) we have
Using now (3.14) and (3.15), we obtain
In order to give now the a priori estimates for the solution of (3.10), we choose in (3.10). Using the coerciveness of , the Cauchy-Schwarz inequality and (3.17), we find independent of ε such that
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.
In the following, for , we denote by its integer part such that and set
Then, for almost every ,
We introduce now the sets (see Figure 2)
For a given set and , we also denote . If there is no place for confusion, we will simply use instead of . If v is a function defined on , , then will denote the extension by zero to the whole Ω. Furthermore, will be the characteristic function of for each , and we introduce the spaces
Definition 4 For any Lebesgue measurable function φ on , , we define the periodic unfolding operator by the formula
Remark 5 If φ is a function defined in Ω, then, for the sake of simplicity, we write instead of .
Proposition 6 For and , the operators are continuous from to . Moreover,
if φ and ψ are two Lebesgue measurable functions on , one has ,
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 .
Lemma 7 If is a sequence in , then, for each ,
Remark 8 From (3.11), (3.12) and Proposition 6(iii), it is easy to see that for each ,
Lemma 9 If and , then, for ε small enough, for each , we have
Theorem 10 Let be a bounded sequence in . Then there exist a subsequence (still denoted by ε), , and such that for each ,
where for almost every . Furthermore,
weakly in , where .
From Proposition 6(viii) and (4.2)2,
Hence is bounded in , i.e.,
which means that there exists such that (4.3)3 holds. Furthermore, it is easy to check that
and the fact that ends the proof of (4.3)4. □
5 Homogenization results
In the following we will use the notation
Theorem 11 If is the solution of problem (3.1), then
where the triplet with for a.e. is the unique solution of the problem
Proof Convergences (5.1)3-6 are a consequence of Theorem 10. From (5.1)3,4 and Proposition 6(vii), we derive that
Since is constant with respect to y, we deduce that
To obtain the limit problem, we take as test functions in (3.10)
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.
Using suggestive notations, this can be written as
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).
First, observe that strongly converges to zero in . Hence, using Proposition 6(v), we obtain
Moreover, since , for , it is easy to see that
strongly in , where (no summation).
Applying the unfolding operator to , we obtain that
thus, using (5.1)5 and (5.8), we get
Similarly, by unfolding, we have
and using (5.1)6 and (5.8) yields
Also, from Lemma 9, we get
so that from (5.1)2,3
On the other hand, again Lemma 9 yields
and from (4.2)3 we have . Consequently,
Finally, we apply the appropriate unfolding operator to , , and for we make use of (5.7). Thus,
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 .
To conclude the proof, observe that endowing the Hilbert space V with the norm
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). □
We introduce now, for , the unique solutions of the problems
and for , the unique solutions of the problems
and finally, we define the homogenized coefficients,
One can verify that if is the unique solution of (5.2), then
Theorem 12 If is the solution of (3.10), then
where is the unique solution of the problem
and the components of are given by .
Proof Using suggestive notations, the homogenized problem (5.2) can be written
We will study now each term of (5.23).
From (5.18) we have
and taking into account (5.17) formula and the variational formulation of (5.15), we get
For we use (5.19), thus
and an integration by parts yields
where is the normal on Γ exterior to . Using now (5.16) and taking into account that , becomes
Obviously, using (5.19) and the boundary condition from (5.16), we get
and, due to Gauss-Ostrogradsky formula and the fact that ,
Hence (5.23) becomes
which means that
and finally (5.22) is proved. The convergences (5.20) and (5.21) are given directly by (5.1)1,2. □
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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.
The authors declare that they have no competing interests.
Both authors fully participated in the completion of this work. Both authors read and approved the final manuscript.