Homogenization via unfolding in periodic layer with contact

Abstract

The elasticity problem for two domains separated by a heterogeneous layer of the thickness ε is considered. The layer has an ε-periodic structure, $ε ≪ 1$ , including a multiple cracks and the contact between the structural components. The inclusions are surrounded by cracks and can have rigid displacements. The contacts are described by the Signorini and Tresca-friction conditions. In order to obtain preliminary estimates, a modification of the Korn inequality for the ε-dependent periodic layer is performed.

An asymptotic analysis with respect to $ε \to 0$ is provided and the limit elasticity problem is obtained, together with the transmission condition across the interface. The periodic unfolding method is used to study the limit behavior.

Keywords

Korn inequality for contact with inclusions unfolding on the oscillating interface homogenization of contact linear elasticity

1. Introduction

The elasticity problem in a domain containing a thin periodic heterogeneous layer of the small thickness ε is studied in this paper. The layer is made of a soft material whose stiffness is of the order of ε and includes cracks and multiple micro-components. Examples of application of such structures include foam materials, used for damping, e.g., in car or plane seat cushions, as well as spacer fabrics, used in protection wear or as geo-textiles, drainages, where certain outer-plate stiffness is required.

The considered layer includes closed and open cracks. Closed cracks completely surround inclusions, which can have rigid displacements. The considered contact condition combines the Signorini problem with the Tresca-friction contact condition. Our aim is to investigate the influence of the local friction on the effective interface transmission condition between the domains.

There exists a large number of papers devoted to the problems with layers of different structure. Depending on relation between small parameters involved in geometry and stiffness of the layers, different limit problems can be obtained. In particular, [1,2,9,11,14,20] deal with Neumann sieves of different thickness and sizes of inclusions. [3,4] consider the case of a thin stiff layer. [15–17,19] consider the layers with the same stiffness as main domains. A case of a soft homogeneous layer is discussed in [12,18]. The elasticity problem for a heterogeneous domain with the Tresca-type friction condition on a microstructure (involving inclusions and cracks) was considered in [10,13] and Korn’s inequality for disconnected inclusions was obtained.

For the derivation of the limit problem we use the periodic unfolding method, which was first introduced in [6], later developed in [7] and was used for different types of problems, particularly, a contact problem in [10] and problems for thin layers in [9,20].

However, additional difficulties arise in proving uniform boundedness for the minimizing sequence of the solutions. The idea consists in controlling the norm on a domain by the trace on its boundary and, therefore, by the norm on the outside domain. Working this way we first obtain estimate for the Dirichlet domain, through which an estimate for the trace is obtained and, therefore, norm on the structural components of the layer. Two Korn’s inequalities are introduced: for the inclusions placed in the heterogeneous periodic layer (based on the results from [10]) and for the connected part of the layer. The main result of the study is an asymptotic model for the layer between elastic blocks.

The paper is organized as follows. Section 2 gives the geometric setting for the ε-periodic problem, including the unit cell. In Section 3 we establish a Korn inequality for the matrix of the layer and then obtain an estimate for the entire structure. Section 4 deals with the convergence result. In Section 5 the problem for fixed ε is introduced. At last, in Section 6 the limit problem is obtained and the case of the linearized contact conditions is considered.

1.1. Notations

Let $O$ be a bounded domain in $R^{3}$ with a Lipschitz boundary. For any $v \in H^{1} (O; R^{3})$ , the normal component of a vector field v on the boundary of $O$ is denoted $v_{ν} = v_{| \partial O} \cdot ν$ , while the tangential component $v_{| \partial O} - v_{ν} ν$ is denoted $v_{τ}$ (ν is the outward unit normal vector to the boundary).

The strain tensor of a vector field v is denoted by $e (v)$ , $\begin{matrix} e^{i j} (v) = \frac{1}{2} (\frac{\partial v_{i}}{\partial x_{j}} + \frac{\partial v_{j}}{\partial x_{i}}), (i, j) \in {1, 2, 3}^{2} . \end{matrix}$ The kernel of e in a connected domain is the finite dimensional space of rigid motions denoted by $R$ .

2. Geometric statement of the problem

In the Euclidean space $R^{2}$ consider a connected domain ω with Lipschitz boundary and let $L > 0$ be a fixed real number. Define: $\begin{array}{l} Ω^{b} = ω \times (- L, 0), \\ Ω^{a} = ω \times (0, L), \\ Σ = ω \times {0}, \\ Ω = Ω^{a} \cup Ω^{b} \cup Σ = ω \times (- L, L) . \end{array}$ The following notations are introduced to describe a structure with a layer: $\begin{array}{l} Ω_{ε}^{a} = ω \times (ε, L), \\ (2.1) & Ω_{ε}^{M} = ω \times (0, ε), \\ S_{ε}^{a} = ω \times {ε} . \end{array}$ Here ε is a small parameter corresponding to the thickness of the layer.

The assemblage is fixed on Γ, which is a non-empty part of $\partial Ω^{b}$ (Γ is a set where the Dirichlet condition will be prescribed). Furthermore, we assume that the external boundary of the layer $\partial ω \times [0, ε]$ is traction free.

Fig. 1.

The unit cell Y.

The layer $Ω_{ε}^{M}$ has a periodic in-plane structure. The unit cell is denoted Y $\begin{matrix} Y = {(0, 1)}^{3} \subset R^{3}, Y_{ε} = ε Y . \end{matrix}$ Additionally, $\begin{matrix} S_{Y}^{b} = {y \in Y : y_{3} = 0}, S_{Y}^{a} = {y \in Y : y_{3} = 1} \end{matrix}$ are the lower and upper boundaries of Y.

There are two kinds of cracks, the first ones $S^{1}, \dots, S^{m}$ (the “closed cracks”) are the closed boundaries of open Lipschitzian sets $Y^{j}$ , $j \in {1, \dots, m}$ . We assume that every $S^{j}$ , $j \in {1, \dots, m}$ , has only one connected component and $⋃_{j = 1}^{m} \overline{Y^{j}} \subset Y$ . The other cracks (the “open cracks”), whose union is denoted by $S^{0}$ , are the finite union of closed Lipschitz surfaces included in $Y ∖ ⋃_{j = 1}^{m} \overline{Y^{j}}$ (see Fig. 1).

We set $\begin{matrix} Y^{0} = Y ∖ (⋃_{j = 1}^{m} \overline{Y^{j}} \cup S^{0}) \end{matrix}$ and assume that there exists $t_{0} > 0$ such that $\begin{matrix} \forall x^{'} \in S^{0}, [x^{'} - t_{0} ν (x^{'}), x^{'} + t_{0} ν (x^{'})] ∖ {x^{'}} \subset Y^{0} . \end{matrix}$ We make the assumption that the set $Y^{0}$ satisfies the hypotheses of the paper on the unfolding for holes [5] (concerning the set which is denoted in [5] by $Y^{*}$ ). The results from [5] will be our essential tools in the present paper. As in [5], we require that $Y^{0}$ satisfy the Poincaré–Wirtinger inequality (for the space $H^{1}$ ), and that the union of $Y^{0}$ and its direct neighbors are connected.

Since the set of cracks $⋃_{j = 0}^{m} S^{j}$ is a closed subset strictly included in Y, there exists $η > 0$ such that $\begin{matrix} \forall x \in ⋃_{j = 0}^{m} S^{j}, dist (x, \partial Y) ⩾ η . \end{matrix}$ Denote $\begin{matrix} Y^{'} = {(0, 1)}^{2} . \end{matrix}$ Recall that in the periodic setting almost every point $z \in R^{3}$ (resp. $z^{'} \in R^{2}$ ) can be written as $\begin{array}{l} (2.2) & z = {[z]}_{Y} + {z}_{Y}, {[z]}_{Y} \in Z^{3}, {z}_{Y} \in Y, \\ (2.3) & (resp. z^{'} = {[z^{'}]}_{Y^{'}} + {z^{'}}_{Y^{'}}, {[z^{'}]}_{Y^{'}} \in Z^{2}, {z^{'}}_{Y^{'}} \in Y^{'}) . \end{array}$

Denote by

$Ξ_{ε}^{M} = {(ξ^{'}, 0) \in Z^{3} | ε (ξ^{'} + ε Y^{'}) \subset ω}$ ,

${\hat{ω}}_{ε} = interior (⋃_{ξ \in Ξ_{ε}^{M}} ε (ξ^{'} + \overline{Y^{'}}))$ , ${\hat{Ω}}_{ε}^{M} = interior (⋃_{ξ \in Ξ_{ε}^{M}} ε (ξ + \overline{Y})) = {\hat{ω}}_{ε} \times (0, ε)$ ,

$Λ_{ε} = ω ∖ \overline{{\hat{ω}}_{ε}}$ , $Λ_{ε}^{M} = Ω_{ε}^{M} ∖ \overline{{\hat{Ω}}_{ε}^{M}} = Λ_{ε} \times (0, ε)$ .

For $j = 1, \dots, m$ introduce the set $\begin{matrix} Ω_{ε}^{j} = {x \in {\hat{Ω}}_{ε}^{M} | ε {\frac{x}{ε}}_{Y} \in Y^{j}} . \end{matrix}$ The boundary $\partial Ω_{ε}^{j}$ is the set of “closed cracks” associated with $S^{j}$ , $\begin{matrix} \partial Ω_{ε}^{j} ≐ S_{ε}^{j} = {x \in {\hat{Ω}}_{ε}^{M} | ε {\frac{x}{ε}}_{Y} \in S^{j} = \partial Y^{j}} . \end{matrix}$ For $j = 0$ set $\begin{matrix} S_{ε}^{0} = {x \in {\hat{Ω}}_{ε}^{M} | ε {\frac{x}{ε}}_{Y} \in S^{0}} \end{matrix}$ and $\begin{matrix} Ω_{ε}^{0} ≐ Ω_{ε}^{M} ∖ (⋃_{j = 1, \dots, m} \overline{Ω_{ε}^{j}} \cup S_{ε}^{0}), {\hat{Ω}}_{ε}^{0} ≐ Ω_{ε}^{0} \cap {\hat{Ω}}_{ε}^{M} . \end{matrix}$ The union of all the cracks is denoted by $S_{ε}$ $\begin{matrix} S_{ε} = ⋃_{j = 0, 1, \dots, m} S_{ε}^{j} . \end{matrix}$ We define the sets $Ω_{ε}$ and $Ω_{ε}^{*}$ as follows $\begin{matrix} Ω_{ε} ≐ Ω ∖ S_{ε}, Ω_{ε}^{*} ≐ Ω ∖ (⋃_{j = 1, \dots, m} \overline{Ω_{ε}^{j}} \cup S_{ε}^{0}) . \end{matrix}$ Note that from these definitions it is clear that there are no cracks in the part of the layer $Λ_{ε}^{M}$ .

For a function v defined on $Ω_{ε}^{*}$ , for simplicity, we denote its restriction to $Ω_{ε}^{j}$ by $v^{j}$ $\begin{matrix} v^{j} ≐ v_{| Ω_{ε}^{j}} for j = 1, \dots, m . \end{matrix}$

In the following, for any bounded set $O$ and $φ \in L^{1} (O)$ , $M_{O} (φ)$ denotes the mean value of φ over $O$ , i.e. $\begin{matrix} M_{O} (φ) = \frac{1}{| O |} \int_{O} φ d y . \end{matrix}$

3. Some inequalities related to unfolding and the geometric domain

3.1. The boundary-layer unfolding operator $T_{ε}$ , $T_{ε}^{b l, j}$

Here we recall the definition and the properties of the boundary-layer unfolding operator (for more details see [9]).

Definition 3.1.
For φ Lebesgue-measurable on $Ω_{ε}^{M}$ (resp. on $Ω_{ε}^{j}$ , $j = 1, \dots, m$ ), the unfolding operator $T_{ε}$ is defined by $\begin{matrix} T_{ε} (φ) (x^{'}, y) = \{\begin{matrix} φ (ε {[\frac{x^{'}}{ε}]}_{Y^{'}} + ε y) & for a.e. (x^{'}, y) \in {\hat{ω}}_{ε} \times Y (resp. for a.e. (x^{'}, y) \in {\hat{ω}}_{ε} \times Y^{j}), \\ 0 & for a.e. (x^{'}, y) \in Λ_{ε} \times Y (resp. for a.e. (x^{'}, y) \in Λ_{ε} \times Y^{j}) . \end{matrix} \end{matrix}$ For φ Lebesgue-measurable on $S_{ε}^{j}$ , $j \in 1, \dots, m$ , the unfolding operator $T_{ε}^{b l, j}$ is defined by $\begin{matrix} T_{ε}^{b l, j} (φ) (x^{'}, y) = \{\begin{matrix} φ (ε {[\frac{x^{'}}{ε}]}_{Y^{'}} + ε y) & for a.e. (x^{'}, y) \in {\hat{ω}}_{ε} \times S^{j}, \\ 0 & for a.e. (x^{'}, y) \in Λ_{ε} \times S^{j} . \end{matrix} \end{matrix}$
Remark 3.1.
If $ϕ \in W^{1, p} (Ω_{ε}^{j})$ , $j = 1, \dots, m$ , $p \in [1, + \infty]$ , $T_{ε}^{b l, j} (φ)$ is just the trace of $T_{ε} (φ)$ on the inner side of $ω \times S^{j}$ .
Proposition 3.1 (Properties of the operators $T_{ε}$ , $T_{ε}^{b l, j}$ ).

For any $φ \in L^{1} (Ω_{ε}^{M})$ , $\begin{matrix} \int_{{\hat{Ω}}_{ε}^{M}} φ d x = ε \int_{ω \times Y} T_{ε} (φ) (x^{'}, y) d x^{'} d y . \end{matrix}$

For any $φ \in L^{2} (Ω_{ε}^{M})$ , $\begin{matrix} ‖ φ ‖_{L^{2} ({\hat{Ω}}_{ε}^{M})} = \sqrt{ε} {‖ T_{ε} (φ) ‖}_{L^{2} (ω_{ε} \times Y)} . \end{matrix}$

Let $φ \in H^{1} (Ω_{ε}^{M})$ . Then $\begin{matrix} \nabla_{y} (T_{ε} (φ)) = ε T_{ε} (\nabla φ) a.e. in ω \times Y . \end{matrix}$ In a similar way, for any $v \in H^{1} (Ω_{ε}^{M}; R^{3})$ $\begin{matrix} e_{y} (T_{ε} (φ)) = ε T_{ε} (e (φ)) a.e. in ω \times Y . \end{matrix}$

For any $ψ \in L^{p} (S_{ε}^{j})$ , $p \in [1, + \infty]$ $\begin{matrix} (3.1) & \int_{S_{ε}^{j}} ψ d σ (x) = \int_{ω_{ε} \times S^{j}} T_{ε}^{b l, j} (ψ) (x^{'}, y) d x^{'} d σ (y) \end{matrix}$ and $\begin{matrix} (3.2) & ‖ ψ ‖_{L^{p} (S_{ε}^{j})} = {‖ T_{ε}^{b l, j} (ψ) ‖}_{L^{p} (ω \times S^{j})} . \end{matrix}$

Proof.
Proofs for the properties 1–3 can be found in [9]. For the last property starting from the right-hand side we obtain $\begin{array}{l} \int_{ω \times S^{j}} T_{ε}^{b l, j} (ψ) (x^{'}, y) d x^{'} d σ (y) \\ = \int_{{\hat{ω}}_{ε} \times S^{j}} T_{ε}^{b l, j} (ψ) (x^{'}, y) d x^{'} d σ (y) = \sum_{ξ \in Ξ_{ε}^{M}} \int_{(ε ξ^{'} + ε Y^{'}) \times S^{j}} T_{ε}^{b l, j} (ψ) d x^{'} d σ (y) \\ = ε^{2} \sum_{ξ \in Ξ_{ε}^{M}} \int_{S^{j}} ψ (ε ξ^{'} + ε s) d σ (s) = \sum_{ξ \in Ξ_{ε}^{M}} \int_{(ε ξ^{'} + ε S^{j})} ψ (x) d σ (x) = \int_{S_{ε}^{j}} ψ (x) d σ (x) . \end{array}$ □

3.2. Unilateral Korn inequality

Let $O$ be a bounded open subset of $R^{3}$ . We denote $\begin{matrix} R = {r \in H^{1} (O; R^{3}) | r (x) = a + b \land x, (a, b) \in R^{3} \times R^{3}} . \end{matrix}$ Following the ideas from [10] we recall that a bounded domain $O$ satisfies the Korn–Wirtinger inequality if there exists a constant $C_{O}$ such that for every $v \in H^{1} (O; R^{3})$ there exists $r \in R$ such that $\begin{matrix} (3.3) & ‖ v - r ‖_{H^{1} (O; R^{3})} ⩽ C_{O} {‖ e (v) ‖}_{L^{2} (O; R^{3 \times 3})} . \end{matrix}$ A domain like $O$ is called a Korn-domain. We equip $H^{1} (O; R^{3})$ with the following scalar product $\begin{matrix} ⟨ u, v ⟩ = \int_{O} e (u) : e (v) d x + \int_{O} u \cdot v d x . \end{matrix}$ If $O$ is a Korn-domain, the associated norm is equivalent to the usual norm of $H^{1} (O; R^{3})$ .

Definition 3.2.
For a Korn-domain $O$ denote $\begin{matrix} W^{1} (O) ≐ {v \in H^{1} (O; R^{3}) | \int_{O} v (x) \cdot r (x) d x = 0 for all r \in R} . \end{matrix}$

Observe that there exists a constant C such that for every $v \in W^{1} (O)$ we get $\begin{matrix} ‖ v ‖_{H^{1} (O; R^{3})} ⩽ C {‖ e (v) ‖}_{L^{2} (O; R^{3 \times 3})} . \end{matrix}$ Considering the orthogonal decomposition $H^{1} (O; R^{3}) = W^{1} (O) \oplus R$ , every $v \in H^{1} (O; R^{3})$ can be written as $\begin{matrix} v = (v - r_{v}) + r_{v}, v - r_{v} \in W^{1} (O), r_{v} \in R . \end{matrix}$ The map $v \mapsto r_{v}$ is the orthogonal projection of v on $R$ . From (3.3) we get $\begin{matrix} (3.4) & ‖ v - r_{v} ‖_{H^{1} (O; R^{3})} ⩽ C_{O} {‖ e (v) ‖}_{L^{2} (O; R^{3 \times 3})} . \end{matrix}$ We also recall that if $O$ is a bounded Lipschitz domain, there exists a constant C such that $\begin{matrix} (3.5) & \forall v \in H^{1} (O; R^{3}), ‖ v ‖_{L^{2} (\partial O; R^{3})} ⩽ C ({‖ e (v) ‖}_{L^{2} (O; R^{3 \times 3})} + ‖ v ‖_{L^{2} (O; R^{3})}) . \end{matrix}$ We will use the following proposition from [10].
Proposition 3.2.
If $O$ is a bounded Lipschitz domain, then there exists a constant C such that $\begin{matrix} (3.6) & \forall r \in R, ‖ r ‖_{L^{1} (O; R^{3})} ⩽ C ({‖ {(r_{ν})}^{+} ‖}_{L^{1} (\partial O)} + ‖ r_{τ} ‖_{L^{1} (\partial O; R^{3})}) . \end{matrix}$

We denote $O^{j}$ the center of gravity of $Y^{j}$ , $j = 1, \dots, m$ .

Let u be in $H^{1} (Ω_{ε}^{j}; R^{3})$ and $r_{u}^{j}$ , $j = 1, \dots, m$ , the orthogonal projection of $u_{| Ω_{ε}^{j}}$ on $R$ . We write $\begin{matrix} r_{u}^{j} (x) = a^{j} (ε {[\frac{x^{'}}{ε}]}_{Y^{'}}) + b^{j} (ε {[\frac{x^{'}}{ε}]}_{Y^{'}}) \land (ε {\frac{x}{ε}}_{Y} - ε O^{j}), x \in Ω_{ε}^{j} . \end{matrix}$ We define the piecewise constant functions $a_{u}^{j}$ and $b_{u}^{j}$ by $\begin{matrix} (3.7) & \begin{matrix} a_{u}^{j} (x^{'}) = a^{j} (ε ξ), x^{'} \in ε ξ + ε Y^{'}, ξ \in Ξ_{ε}, \\ a_{u}^{j} (x^{'}) = 0, x^{'} \in Λ_{ε}, \\ b_{u}^{j} (x^{'}) = b^{j} (ε ξ), x^{'} \in ε ξ + ε Y^{'}, ξ \in Ξ_{ε}, \\ b_{u}^{j} (x^{'}) = 0, x^{'} \in Λ_{ε} . \end{matrix} \end{matrix}$

These functions belong to $L^{\infty} (ω; R^{3})$ and the associated rigid body field, still denoted $r_{u}^{j}$ , belongs to $L^{\infty} (ω; R)$ .

As a consequence of the Proposition 3.2 we get Proposition 3.3.
There exists a constant C (independent of ε) such that for every $j = 1, \dots, m$ and for every u in $H^{1} (Ω_{ε}^{j}; R^{3})$ , $\begin{matrix} (3.8) & \begin{matrix} {‖ u - r_{u}^{j} ‖}_{L^{2} (Ω_{ε}^{j}; R^{3})} + ε {‖ \nabla (u - r_{u}^{j}) ‖}_{L^{2} (Ω_{ε}^{j}; R^{3 \times 3})} ⩽ C ε {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{j}; R^{3 \times 3})}, \\ {‖ a_{u}^{j} ‖}_{L^{1} (ω; R^{3})} + ε {‖ b_{u}^{j} ‖}_{L^{1} (ω; R^{3})} ⩽ C ({‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} + {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j}; R^{3})}), \\ ‖ u ‖_{L^{1} (Ω_{ε}^{j}; R^{3})} ⩽ C ε^{3 / 2} {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{j}; R^{3 \times 3})} + C ε ({‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} + {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j}; R^{3})}) . \end{matrix} \end{matrix}$
Proof.
Applying (3.4) (after ε-scaling) gives $\begin{matrix} (3.9) & {‖ u - r_{u}^{j} ‖}_{L^{2} (ε ξ + ε Y^{j}; R^{3})}^{2} + ε^{2} {‖ \nabla (u - r_{u}^{j}) ‖}_{L^{2} (ε ξ + ε Y^{j}; R^{3 \times 3})}^{2} ⩽ C ε^{2} {‖ e (u) ‖}_{L^{2} (ε ξ + ε Y^{j}; R^{3 \times 3})}^{2} . \end{matrix}$ Then adding the above inequalities (with respect to ξ) yields (3.8)₁. Taking into account (3.6) (after ε-scaling), we get $\begin{matrix} (3.10) & | a_{u}^{j} (ε ξ) | ε^{2} + | b_{u}^{j} (ε ξ) | ε^{3} ⩽ C ({‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (ε ξ + ε S^{j})} + {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (ε ξ + ε S^{j}; R^{3})}) . \end{matrix}$ Adding these inequalities (with respect to ξ) gives $\begin{matrix} {‖ a_{u}^{j} ‖}_{L^{1} (ω; R^{3})} + ε {‖ b_{u}^{j} ‖}_{L^{1} (ω; R^{3})} ⩽ C ({‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} + {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j}; R^{3})}) . \end{matrix}$ Finally, estimate (3.8)₃ is an immediate consequence of (3.8)₁ and (3.8)₂. □
Remark 3.2.
Due to (3.6) and then (3.9), we also obtain $\begin{matrix} (3.11) & \begin{matrix} {‖ a_{u}^{j} ‖}_{L^{2} (ω; R^{3})} + ε {‖ b_{u}^{j} ‖}_{L^{2} (ω; R^{3})} ⩽ \frac{C}{ε} ({‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} + {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j}; R^{3})}), \\ ‖ u ‖_{L^{2} (Ω_{ε}^{j}; R^{3})} ⩽ C ε {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{j}; R^{3 \times 3})} + \frac{C}{ε^{1 / 2}} ({‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} + {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j}; R^{3})}), \\ ‖ \nabla u ‖_{L^{2} (Ω_{ε}^{j}; R^{3 \times 3})} ⩽ C {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{j}; R^{3 \times 3})} + \frac{C}{ε^{3 / 2}} ({‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} + {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j}; R^{3})}) . \end{matrix} \end{matrix}$ The constants do not depend on ε.

3.3. Korn inequality for a perforated layer

Now we want to derive the Korn inequality for the simply connected part of the layer.

Denote $\begin{matrix} H_{Γ}^{1} (Ω_{ε}^{*}) = {ϕ \in H^{1} (Ω_{ε}^{*}) | ϕ = 0 a.e. on Γ} . \end{matrix}$

Proposition 3.4.
There exists a constant C independent of ε such that for every u in $H_{Γ}^{1} (Ω_{ε}^{}; R^{3})$ $\begin{matrix} (3.12) & ‖ u ‖_{H^{1} (Ω_{ε}^{}; R^{3})} ⩽ C {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{}; R^{3 \times 3})} . \end{matrix}$ We also have* $\begin{array}{rcl} ‖ u ‖_{H^{1} (Ω_{ε}^{a}; R^{3})}^{2} + \frac{1}{ε} ‖ u ‖_{L^{2} (Ω_{ε}^{0}; R^{3})}^{2} + ε ‖ \nabla u ‖_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})}^{2} + ‖ u ‖_{H^{1} (Ω^{b}; R^{3})}^{2} \\ (3.13) & ⩽ C ({‖ e (u) ‖}_{L^{2} (Ω_{ε}^{a}; R^{3 \times 3})}^{2} + ε {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})}^{2} + {‖ e (u) ‖}_{L^{2} (Ω^{b}; R^{3 \times 3})}^{2}) . \end{array}$
Proof.
Step 1. First, we restrict u to a subset of the domain without cracks and then construct an “extension” of u. We set $\begin{matrix} Y_{η} ≐ {y \in Y | dist (y, \partial Y) < η / 2} . \end{matrix}$ The domain $Y_{η}$ is a bounded domain with a Lipschitz boundary. Moreover, observe that $\partial Y_{η} \subset \partial Y$ . Therefore there is an extension operator $P_{η}$ from $H^{1} (Y_{η})$ into $H^{1} (Y)$ and a constant C (which depends on η) such that (see [8]) $\begin{array}{rcl} \forall v \in H^{1} (Y_{η}), \\ (3.14) & {‖ P_{η} (v) ‖}_{L^{2} (Y)} ⩽ C ‖ v ‖_{L^{2} (Y_{η})} and {‖ \nabla_{y} P_{η} (v) ‖}_{L^{2} (Y; R^{3})} ⩽ C ‖ \nabla_{y} v ‖_{L^{2} (Y_{η}; R^{3})} . \end{array}$ Let $w \in H^{1} (Y_{η}; R^{3})$ and $r_{w}$ the projection of w on $R$ , we have $\begin{matrix} (3.15) & ‖ w - r_{w} ‖_{H^{1} (Y_{η}; R^{3})} ⩽ C {‖ e_{y} (w) ‖}_{L^{2} (Y_{η}; R^{3 \times 3})} . \end{matrix}$ The constant depends on η.

Now for every $w \in H^{1} (Y_{η}; R^{3})$ we define the extension $Q_{η} (w) ≐ P_{η} (w - r_{w}) + r_{w}$ of w. From (3.14) and (3.15) we get $\begin{matrix} (3.16) & Q_{η} (w) \in H^{1} (Y; R^{3}), {‖ e_{y} (Q_{η} (w)) ‖}_{L^{2} (Y; R^{3 \times 3})} ⩽ C {‖ e_{y} (w) ‖}_{L^{2} (Y_{η}; R^{3 \times 3})} . \end{matrix}$ Applying the above result to the restriction of the displacement $y \mapsto u (ε ξ + ε y)$ to the cell $Y_{η}$ , $ξ \in Ξ_{ε}^{M}$ allows to define an extension $\tilde{u}$ of u in the layer ${\hat{Ω}}_{ε}^{M}$ . Estimate (3.16) leads to $\begin{matrix} \tilde{u} \in H^{1} ({\hat{Ω}}_{ε}^{M}; R^{3}), {‖ e (\tilde{u}) ‖}_{L^{2} ({\hat{Ω}}_{ε}^{M}; R^{3 \times 3})}^{2} ⩽ C \sum_{ξ \in Ξ_{ε}^{M}} {‖ e (u) ‖}_{L^{2} (ε (ξ + Y_{η}; R^{3 \times 3}))}^{2} ⩽ C {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})}^{2} . \end{matrix}$ The constants do not depend on ε.

We set $\tilde{u} = u$ in $Ω ∖ \overline{{\hat{Ω}}_{ε}^{M}}$ . The displacement $\tilde{u}$ belongs to $H^{1} (Ω; R^{3})$ , it vanishes on Γ and satisfies $\begin{matrix} (3.17) & {‖ e (\tilde{u}) ‖}_{L^{2} (Ω_{ε}^{M}; R^{3 \times 3})} ⩽ C {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})}, {‖ e (\tilde{u}) ‖}_{L^{2} (Ω)} ⩽ C {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{}; R^{3 \times 3})} . \end{matrix}$ The constants do not depend on ε.

Step 2.* From the Korn’s inequality, the hypothesis that the measure of Γ is positive and (3.17)₂ we obtain $\begin{matrix} (3.18) & ‖ \tilde{u} ‖_{H^{1} (Ω; R^{3})} ⩽ C {‖ e (\tilde{u}) ‖}_{L^{2} (Ω; R^{3 \times 3})} ⩽ C {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{}; R^{3 \times 3})} . \end{matrix}$

Step 3.* We prove (3.12). Since by construction the domain $Y^{0}$ is a Korn-domain, there exists a constant $C > 0$ such that for every $v \in H^{1} (Y^{0}; R^{3})$ equal to zero on $\partial Y$ $\begin{matrix} ‖ v ‖_{H^{1} (Y^{0}; R^{3})} ⩽ C {‖ e_{y} (v) ‖}_{L^{2} (Y^{0}; R^{3 \times 3})} . \end{matrix}$ Applying the above result to the restriction of the displacement $y \mapsto (u - \tilde{u}) (ε ξ + ε y)$ to the cell $Y^{0}$ , $ξ \in Ξ_{ε}^{M}$ , gives $\begin{matrix} \begin{matrix} ‖ u - \tilde{u} ‖_{L^{2} ({\hat{Ω}}_{ε}^{M}; R^{3})}^{2} ⩽ C ε^{2} \sum_{ξ \in Ξ_{ε}} {‖ e (u - \tilde{u}) ‖}_{L^{2} (ε (ξ + Y^{0}; R^{3 \times 3}))}^{2}, \\ {‖ \nabla (u - \tilde{u}) ‖}_{L^{2} ({\hat{Ω}}_{ε}^{M}; R^{3 \times 3})}^{2} ⩽ C \sum_{ξ \in Ξ_{ε}^{M}} {‖ e (u - \tilde{u}) ‖}_{L^{2} (ε (ξ + Y^{0}; R^{3 \times 3}))}^{2} . \end{matrix} \end{matrix}$ The constants do not depend on ε. Hence, using the fact that $u - \tilde{u}$ vanishes in $Ω_{ε}^{0} ∖ \overline{{\hat{Ω}}_{ε}^{M}}$ and due to estimate (3.18), we obtain $\begin{matrix} (3.19) & ‖ u - \tilde{u} ‖_{L^{2} (Ω_{ε}^{0}; R^{3})} ⩽ C ε {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})}, {‖ \nabla (u - \tilde{u}) ‖}_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})} ⩽ C {‖ e (u) ‖}_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})} . \end{matrix}$ Combining the above inequalities and (3.18) gives (3.12).

Step 4. We prove (3.13). The Korn inequality and the trace theorem give $\begin{matrix} (3.20) & \begin{matrix} ‖ \tilde{u} ‖_{L^{2} (Ω^{b}; R^{3})} + ‖ \tilde{u} ‖_{L^{2} (Σ; R^{3})} + ‖ \nabla \tilde{u} ‖_{L^{2} (Ω^{b}; R^{3 \times 3})} ⩽ C {‖ e (\tilde{u}) ‖}_{L^{2} (Ω^{b}; R^{3 \times 3})}, \\ ‖ \tilde{u} ‖_{L^{2} (Ω_{ε}^{a}; R^{3})} + ‖ \nabla \tilde{u} ‖_{L^{2} (Ω_{ε}^{a}; R^{3 \times 3})} ⩽ C {‖ e (\tilde{u}) ‖}_{L^{2} (Ω_{ε}^{a}; R^{3 \times 3})} + C ‖ \tilde{u} ‖_{L^{2} (S_{ε}^{a}; R^{3})} . \end{matrix} \end{matrix}$ Besides we have $\begin{matrix} (3.21) & \begin{matrix} ‖ \tilde{u} ‖_{L^{2} (S_{ε}^{a}; R^{3})}^{2} ⩽ C (‖ \tilde{u} ‖_{L^{2} (Σ; R^{3})}^{2} + ε ‖ \nabla \tilde{u} ‖_{L^{2} (Ω_{ε}^{M}; R^{3 \times 3})}^{2}), \\ ‖ \tilde{u} ‖_{L^{2} (Ω_{ε}^{M}; R^{3})}^{2} ⩽ C (ε ‖ \tilde{u} ‖_{L^{2} (Σ; R^{3})}^{2} + ε^{2} ‖ \nabla \tilde{u} ‖_{L^{2} (Ω_{ε}^{M}; R^{3 \times 3})}^{2}) . \end{matrix} \end{matrix}$ Taking into account the above estimates (3.20)–(3.21) together with (3.18)–(3.19) we obtain (3.13). □

We set $\begin{array}{rcl} V_{ε} & ≐ & {v = (v, v^{1}, \dots, v^{m}, a_{v}, b_{v}) | v \in H_{Γ}^{1} (Ω_{ε}^{}; R^{3}) \times \prod_{j = 1}^{m} H^{1} (Ω_{ε}^{j}; R^{3}) \times {[Q^{0} (ω)]}^{m} \times {[Q^{0} (ω)]}^{m}, \\ (3.22) & v^{j} is orthogonal to the rigid displacements, j = 1, \dots, m}, \end{array}$ where $Q^{0}$ is the set of functions vanishing on $Λ_{ε}$ and constants on each cell $ε (ξ^{'} + Y^{'})$ , $ξ \in Ξ_{ε}^{M}$ . The rigid displacements $r_{v}^{j}$ are defined by $\begin{matrix} r_{v}^{j} (x) = a_{v}^{j} (ε {[\frac{x^{'}}{ε}]}_{Y^{'}}) + b_{v}^{j} (ε {[\frac{x^{'}}{ε}]}_{Y^{'}}) \land (ε {\frac{x}{ε}}_{Y} - ε O^{j}), x \in Ω_{ε}^{j}, j = 1, \dots, m . \end{matrix}$ We will denote by ${[v]}_{S_{ε}^{j}}$ the jump of the vector field across the surface $S_{ε}^{j}$ , $j = 0, \dots, m$ . More precisely, for $j = 1, \dots, m$ we set ${[v]}_{S_{ε}^{j}} = v_{| S_{ε}^{j}}^{j} + {(r_{v}^{j})}_{| S_{ε}^{j}} - v_{| S_{ε}^{j}}$ , ${[v_{ν}]}_{S_{ε}^{j}} = {[v]}_{S_{ε}^{j}} \cdot ν_{| S_{ε}^{j}}$ , ${[v_{τ}]}_{S_{ε}^{j}} = {[v]}_{S_{ε}^{j}} - {[v_{ν}]}_{S_{ε}^{j}}$ and we define ${[v]}_{S_{ε}^{0}}$ , ${[v_{ν}]}_{S_{ε}^{0}}$ , and ${[v_{τ}]}_{S_{ε}^{0}}$ by $\begin{array}{l} (3.23) & {[v]}_{S_{ε}^{0}} (x^{'}) = lim_{t \to 0, t > 0} v (x^{'} + t ν (x^{'})) - v (x^{'} - t ν (x^{'})), \\ (3.24) & {[v_{ν}]}_{S_{ε}^{0}} (x^{'}) = lim_{t \to 0, t > 0} (v (x^{'} + t ν (x^{'})) - v (x^{'} - t ν (x^{'}))) \cdot ν (x^{'}), for a.e. x^{'} \in S_{ε}^{0}, \\ (3.25) & {[v_{τ}]}_{S_{ε}^{0}} = {[v]}_{S_{ε}^{0}} - {[v_{ν}]}_{S_{ε}^{0}} . \end{array}$

The space $V_{ε}$ is usually equipped with the following norm (which is adjusted to the considered case where the layer and the inclusions are made of a soft material): $\begin{matrix} \forall v \in V_{ε}, v \to \sqrt{‖ \nabla v ‖_{L^{2} (Ω^{b}; R^{3 \times 3})}^{2} + ‖ \nabla v ‖_{L^{2} (Ω_{ε}^{a}; R^{3 \times 3})}^{2} + ε ‖ \nabla v ‖_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})}^{2} + \sum_{j = 1}^{m} ε {‖ v^{j} ‖}_{H^{1} (Ω_{ε}^{j}; R^{3})}^{2}} . \end{matrix}$ With the above norm, $V_{ε}$ is a Hilbert space. But the following norm over $V_{ε}$ is well adapted to the contact problem: $\begin{matrix} \begin{matrix} \forall v \in V_{ε}, ‖ v ‖_{V_{ε}} ≐ | v |_{V_{ε}} + ‖ a_{u} ‖_{{[L^{1} (ω; R^{3})]}^{m}} + ε ‖ b_{u} ‖_{{[L^{1} (ω; R^{3})]}^{m}}, \end{matrix} \end{matrix}$ where $\begin{array}{l} | v |_{V_{ε}} = \sqrt{{‖ e (v) ‖}_{L^{2} (Ω^{b}; R^{3 \times 3})}^{2} + {‖ e (v) ‖}_{L^{2} (Ω_{ε}^{a}; R^{3 \times 3})}^{2} + ε {‖ e (v) ‖}_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})}^{2} + \sum_{j = 1}^{m} ε {‖ e (v^{j}) ‖}_{L^{2} (Ω_{ε}^{j}; R^{3 \times 3})}^{2}}, \\ ‖ a_{u} ‖_{{[L^{1} (ω; R^{3})]}^{m}} = \sum_{j = 1}^{m} {‖ a_{u}^{j} ‖}_{L^{1} (ω; R^{3})}, ‖ b_{u} ‖_{{[L^{1} (ω; R^{3})]}^{m}} = \sum_{j = 1}^{m} {‖ b_{u}^{j} ‖}_{L^{1} (ω; R^{3})} . \end{array}$

Below we summarize the estimates for $v \in V_{ε}$ .
Proposition 3.5.
There exists a constant C independent of ε such that for all* $v = (v, v^{1}, \dots, v^{m}, a_{v}, b_{v}) \in V_{ε}$ $\begin{matrix} (3.26) & \begin{matrix} ‖ v ‖_{H^{1} (Ω_{ε}^{a}; R^{3})}^{2} + ε ‖ \nabla v ‖_{L^{2} (Ω_{ε}^{0}; R^{3 \times 3})}^{2} + ε \sum_{j = 1}^{m} {‖ \nabla v^{j} ‖}_{L^{2} (Ω_{ε}^{j}; R^{3 \times 3})}^{2} \\ + \frac{1}{ε} ‖ v ‖_{L^{2} (Ω_{ε}^{0}; R^{3})}^{2} + \frac{1}{ε} \sum_{j = 1}^{m} {‖ v^{j} ‖}_{L^{2} (Ω_{ε}^{j}; R^{3})}^{2} + ‖ v ‖_{H^{1} (Ω^{b}; R^{3})}^{2} ⩽ C | v |_{V_{ε}}^{2}, \\ ‖ a_{v} ‖_{{[L^{1} (ω; R^{3})]}^{m}} + ε ‖ b_{v} ‖_{{[L^{1} (ω; R^{3})]}^{m}} ⩽ C \sum_{j = 1}^{m} ({‖ {(r_{v}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} + {‖ {(r_{v}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j}; R^{3})}) . \end{matrix} \end{matrix}$
Proof.
The estimates (3.26) are the immediate consequences of (3.13) and (3.8). □
Remark 3.3.
From (3.11) we also obtain $\begin{matrix} (3.27) & \sum_{j = 1}^{m} ({‖ a_{v}^{j} ‖}_{L^{2} (ω; R^{3})} + ε {‖ b_{v}^{j} ‖}_{L^{2} (ω; R^{3})}) ⩽ \frac{C}{ε} ‖ v ‖_{V_{ε}} . \end{matrix}$ The constant does not depend on ε.

Further we will consider the following symmetric bilinear form on $V_{ε}$ : $\begin{matrix} \forall (u, v) \in {(V_{ε})}^{2}, A^{ε} (u, v) ≐ \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} a^{ε} e (u^{j}) : e (v^{j}) d x + \int_{Ω_{ε}^{*}} a^{ε} e (u) : e (v) d x, \end{matrix}$ where $\begin{matrix} (3.28) & a^{ε} (x) = \{\begin{matrix} a^{a} (x) & for a.e. x \in Ω_{ε}^{a}, \\ ε a_{ε}^{M} (x) & for a.e. x \in Ω_{ε}^{M}, \\ a^{b} (x) & for a.e. x \in Ω^{b} . \end{matrix} \end{matrix}$ The tensor fields $a_{ε}^{M}$ , $a^{a}$ , $a^{b}$ have the following properties.
Symmetry: $\begin{matrix} a^{ε} (x) η : ξ = a^{ε} (x) ξ : η a.e. x \in Ω, \forall ξ, η \in R^{3 \times 3} . \end{matrix}$

Boundedness: $a^{ε}$ belongs to $L^{\infty} (Ω; R^{3 \times 3 \times 3 \times 3})$ and $\begin{matrix} {‖ a_{ε}^{M} ‖}_{L^{\infty} (Ω_{ε}^{M}; R^{3 \times 3 \times 3 \times 3})} + {‖ a^{a} ‖}_{L^{\infty} (Ω^{a}; R^{3 \times 3 \times 3 \times 3})} + {‖ a^{b} ‖}_{L^{\infty} (Ω^{b}; R^{3 \times 3 \times 3 \times 3})} ⩽ C . \end{matrix}$ The constant does not depend on ε.

Coercivity (with constant $\overline{α} > 0$ independent of ε): $\begin{matrix} (3.29) & \begin{matrix} \overline{α} η : η ⩽ a_{ε}^{M} (x) η : η for a.e. x \in Ω_{ε}^{M}, \\ \overline{α} η : η ⩽ a^{a} (x) η : η for a.e. x \in Ω^{a}, \\ \overline{α} η : η ⩽ a^{b} (x) η : η for a.e. x \in Ω^{b}, \end{matrix} \forall η \in R^{3 \times 3} . \end{matrix}$

4. Convergence results

Every $v \in H^{1} (Ω_{ε}^{a}; R^{3})$ is extended by reflexion in a displacement belonging to $H^{1} (ω \times (ε, L + ε); R^{3})$ . Denote $\begin{matrix} (4.1) & \begin{matrix} H_{per}^{1} (Y^{0}) = {ϕ \in H^{1} (Y^{0}) | ϕ (0, y_{2}, y_{3}) = ϕ (1, y_{2}, y_{3}) for a.e. (y_{2}, y_{3}) \in {(0, 1)}^{2}, \\ ϕ (y_{1}, 0, y_{3}) = ϕ (y_{1}, 1, y_{3}) for a.e. (y_{1}, y_{3}) \in {(0, 1)}^{2}}, \\ H_{Γ}^{1} (Ω^{b}) = {ϕ \in H^{1} (Ω^{b}) | ϕ = 0 a.e. on Γ} . \end{matrix} \end{matrix}$ Before giving the convergence results, we prove the following lemma of homogenization.

Lemma 4.1.
Let ${ϕ_{ε}}_{ε}$ be a sequence in $H^{1} (Ω)$ satisfying $\begin{matrix} ‖ ϕ_{ε} ‖_{H^{1} (Ω_{ε}^{a})}^{2} + ‖ ϕ_{ε} ‖_{H^{1} (Ω^{b})}^{2} + ε ‖ \nabla ϕ_{ε} ‖_{L^{2} (Ω_{ε}^{M}; R^{3})}^{2} + \frac{1}{ε} ‖ ϕ_{ε} ‖_{L^{2} (Ω_{ε}^{M})}^{2} ⩽ C, \end{matrix}$ where the constant C does not depend on ε. There exist a subsequence – still denoted ε – and $ϕ^{b} \in H^{1} (Ω^{b})$ , $ϕ^{a} \in H^{1} (Ω^{a})$ , $\hat{ϕ} \in L^{2} (ω; H_{per}^{1} (Y))$ such that $\begin{array}{l} ϕ_{ε} ⇀ ϕ^{b} weakly in H^{1} (Ω^{b}), \\ (4.2) & ϕ_{ε} (\cdot + ε e_{3}) ⇀ ϕ^{a} weakly in H^{1} (Ω^{a}), \\ T_{ε} (ϕ_{ε}) ⇀ \hat{ϕ} weakly in L^{2} (ω; H^{1} (Y)) . \end{array}$ Moreover, we have $\begin{matrix} (4.3) & ϕ^{b} (x^{'}, 0) = \hat{ϕ} (x^{'}, y_{1}, y_{2}, 0), ϕ^{a} (x^{'}, 0) = \hat{ϕ} (x^{'}, y_{1}, y_{2}, 1) for a.e. (x^{'}, y_{1}, y_{2}) \in ω \times Y^{'} . \end{matrix}$
Proof.
The function $ϕ_{ε}$ is extended by reflexion in a function belonging to $H^{1} (ω \times (0, L + ε))$ in order to obtain convergence (4.2)₂.

We only prove the first equation in (4.3), the second one is obtained in the same way. Consider the function defined by $\begin{matrix} {\overline{ϕ}}_{ε} (x_{1}, x_{2}, x_{3}) = ϕ_{ε} (x_{1}, x_{2}, x_{3}) - ϕ_{ε} (x_{1}, x_{2}, - x_{3}), x = (x_{1}, x_{2}, x_{3}) \in Ω_{ε}^{M} . \end{matrix}$ It satisfies $\begin{matrix} ‖ \nabla {\overline{ϕ}}_{ε} ‖_{L^{2} (Ω_{ε}^{M}; R^{3})}^{2} ⩽ ‖ \nabla ϕ_{ε} ‖_{L^{2} (Ω_{ε}^{M}; R^{3})}^{2} + ‖ \nabla ϕ_{ε} ‖_{L^{2} (Ω^{b}; R^{3})}^{2} ⩽ \frac{C}{ε}, {\overline{ϕ}}_{ε} = 0 on ω \times {0} . \end{matrix}$ Hence $‖ {\overline{ϕ}}_{ε} ‖_{L^{2} (Ω_{ε}^{M})}^{2} ⩽ C ε$ . Due to the convergences (4.2)₁ and (4.2)₃ we have $\begin{matrix} T_{ε} ({\overline{ϕ}}_{ε}) ⇀ \hat{ϕ} - ϕ_{| ω \times {0}}^{b} weakly in L^{2} (ω; H^{1} (Y)) . \end{matrix}$ Since the trace of the function $y \mapsto T_{ε} ({\overline{ϕ}}_{ε}) (x^{'}, y)$ on the face $Y^{'} \times {0}$ vanishes for a.e. $x^{'} \in ω$ the result is proved. □
Theorem 4.1.
Let ${v_{ε}}_{ε}$ , $v_{ε} = (v_{ε}, v_{ε}^{1}, \dots, v_{ε}^{m}, a_{v_{ε}}, b_{v_{ε}})$ , be a sequence in $V_{ε}$ satisfying $\begin{matrix} (4.4) & ‖ v_{ε} ‖_{V_{ε}} ⩽ C, \end{matrix}$ where the constant C does not depend on ε. There exist a subsequence – still denoted ε – and $v^{b} \in H_{Γ}^{1} (Ω^{b}; R^{3})$ , $v^{a} \in H^{1} (Ω^{a}; R^{3})$ , ${\hat{v}}^{0} \in L^{2} (ω; H_{per}^{1} (Y^{0}; R^{3}))$ , ${\hat{v}}^{j} \in L^{2} (ω; H^{1} (Y^{j}; R^{3}))$ , $a^{j} \in M (ω; R^{3})$ and $b^{j} \in M (ω; R^{3})$ ( $j = 1, \dots, m$ ), such that $\begin{matrix} (4.5) & \begin{matrix} v_{ε} ⇀ v^{b} weakly in H_{Γ}^{1} (Ω^{b}; R^{3}), \\ v_{ε} (\cdot + ε e_{3}) ⇀ v^{a} weakly in H^{1} (Ω^{a}; R^{3}), \\ T_{ε} (v_{ε}) ⇀ {\hat{v}}^{0} weakly in L^{2} (ω; H^{1} (Y^{0}; R^{3})), \\ ε T_{ε} (e (v_{ε})) ⇀ e_{y} ({\hat{v}}^{0}) weakly in L^{2} (ω \times Y^{0}; R^{3 \times 3}), \\ T_{ε} (v_{ε}^{j}) ⇀ {\hat{v}}^{j} weakly in L^{2} (ω; H^{1} (Y^{j}; R^{3})), \\ ε T_{ε} (e (v_{ε}^{j})) ⇀ e_{y} ({\hat{v}}^{j}) weakly in L^{2} (ω \times Y^{j}; R^{3 \times 3}), \\ a_{v_{ε}}^{j} ⇀ a_{V}^{j} weakly-* in M (ω; R^{3}), \\ b_{v_{ε}}^{j} ⇀ b_{V}^{j} weakly-* in M (ω; R^{3}) . \end{matrix} \end{matrix}$ Moreover we have $\begin{array}{rcl} v^{b} (x^{'}, 0) = {\hat{v}}^{0} (x^{'}, y_{1}, y_{2}, 0), v^{a} (x^{'}, 0) = {\hat{v}}^{0} (x^{'}, y_{1}, y_{2}, 1) \\ (4.6) & for a.e. (x^{'}, y_{1}, y_{2}) \in ω \times Y^{'} . \end{array}$ Furthermore, setting $\begin{matrix} \forall y \in Y^{j}, r_{V}^{j} (\cdot, y) = a_{V}^{j} + b_{V}^{j} \land (y - O^{j}) in M (ω; R^{3}) \end{matrix}$ we get $\begin{matrix} (4.7) & \begin{matrix} T_{ε}^{b l, j} ({[v_{ε}]}_{S_{ε}^{j}}) & ⇀ {\hat{v}}_{| S^{j}}^{j} + {(r_{V}^{j})}_{| S^{j}} - {\hat{v}}_{| S^{j}}^{0} weakly- * in M (ω \times S^{j}; R^{3}), j = 1, \dots, m . \end{matrix} \end{matrix}$
Proof.
The convergences (4.5) are the immediate consequences of the estimates (3.26). To prove (4.6) we apply the Lemma 4.1 with the fields of displacements ${\tilde{v}}_{ε}$ introduced in Step 1 of the Proposition 3.4. □

5. The contact problem for fixed ε

Let $K_{ε}$ be the convex set defined, for non negative functions $g_{ε}^{j}$ belonging to $L^{1} (S_{ε}^{j})$ , $j = 0, \dots, m$ , by $\begin{matrix} (5.1) & K_{ε} ≐ {v \in V_{ε}, {[v_{ν}]}_{S_{ε}^{j}} ⩽ g_{ε}^{j} on S_{ε}^{j}, j = 0, \dots, m} . \end{matrix}$ The vector fields $v \in V_{ε}$ are the admissible deformation fields with respect to the reference configuration $Ω_{ε}$ . By standard trace theorems, the jumps belong to $H^{1 / 2} (S_{ε}^{j}; R^{3})$ . The tensor field $\begin{matrix} σ^{ε} (v) ≐ a^{ε} e (v) in Ω_{ε} \end{matrix}$ is the stress tensor associated to the deformation v (not to be confused with the surface measures $d σ$ !).

The functions $g_{ε}^{0}$ and the $g_{ε}^{j}$ ’s are the original gaps (in the reference configuration), and the corresponding inequalities in the definition of $K_{ε}$ represent the non-penetration conditions. In case there is a contact in the reference configuration, these functions are just 0.

Consider also the family of convex maps $Ψ_{ε}^{j}$ , $0 ⩽ j ⩽ m$ , where $Ψ_{ε}^{j}$ is non negative, continuous on $L^{1} (S_{ε}^{j}; R^{3})$ and satisfies $\begin{array}{rcl} w \in L^{1} (S_{ε}^{j}; R^{3}), M_{ε}^{j} ‖ w ‖_{L^{1} (S_{ε}^{j}; R^{3})} - a_{ε}^{j} ⩽ Ψ_{ε}^{j} (w) ⩽ M_{ε}^{' j} ‖ w ‖_{L^{1} (S_{ε}^{j}; R^{3})} \\ (5.2) & for non negative real numbers M_{ε}^{' j}, M_{ε}^{j}, a_{ε}^{j}, M_{ε}^{j} \neq 0, M_{ε}^{' j} \neq 0 . \end{array}$

In case of Tresca friction, the maps $Ψ_{ε}^{j}$ are explicitly given by $\begin{matrix} (5.3) & Ψ_{ε}^{j} (w) ≐ \int_{S_{ε}^{j}} G_{ε}^{j} (x) | w (x) | d σ (x), G_{ε}^{j} \in L^{\infty} (S_{ε}^{j}), w \in L^{1} (S_{ε}^{j}; R^{3}) \end{matrix}$ with $G_{ε}^{j}$ bounded from below by $M_{ε}^{j} > 0$ for $j = 0, \dots, m$ .

Problem $P_{ε}$ .
Given $f_{ε} = (f, f_{ε}^{1}, \dots, f_{ε}^{m})$ in $L^{2} (Ω; R^{3}) \times L^{\infty} (Ω_{ε}^{1}; R^{3}) \times \dots \times L^{\infty} (Ω_{ε}^{m}; R^{3})$ find a minimizer over $K_{ε}$ of the functional $\begin{matrix} (5.4) & E_{ε} (v) ≐ \frac{1}{2} A^{ε} (v, v) + \sum_{j = 0}^{m} Ψ_{ε}^{j} ({[v_{τ}]}_{S_{ε}^{j}}) - \int_{Ω_{ε}} f_{ε} \cdot v d x, \end{matrix}$ where $\begin{matrix} \int_{Ω_{ε}} f_{ε} \cdot v d x = \int_{Ω_{ε}^{*}} f \cdot v d x + \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} f_{ε}^{j} \cdot (v^{j} + r_{v}^{j}) d x . \end{matrix}$

From the properties of convexity of the $Ψ_{ε}^{j}$ , $j = 0, \dots, m$ , the solutions of $P_{ε}$ are the same as that of the following problem:
Problem $P_{ε}^{'}$ .
Find $u_{ε} \in K_{ε}$ such that for every $v \in K_{ε}$ , $\begin{matrix} (5.5) & A^{ε} (u_{ε}, v - u_{ε}) + \sum_{j = 0}^{m} (Ψ_{ε}^{j} ({[v_{τ}]}_{S_{ε}^{j}}) - Ψ_{ε}^{j} ({[{(u_{ε})}_{τ}]}_{S_{ε}^{j}})) ⩾ \int_{Ω_{ε}} f_{ε} \cdot (v - u_{ε}) d x . \end{matrix}$

The strong formulation of the problem is (with $σ^{ε}$ for the stress tensor $σ^{ε} (u_{ε})$ ): $\begin{array}{l} (5.6) & \{\begin{matrix} \nabla \cdot σ^{ε} = - f_{ε} in Ω_{ε}, \\ σ^{ε} {(ν)}_{ν} ⩽ 0, \\ σ^{ε} {(ν)}_{ν} ({[{(u_{ε})}_{ν}]}_{S_{ε}^{j}} - g_{ε}^{j}) = 0, \\ σ^{ε} {(ν)}_{τ} \in \partial Ψ_{ε}^{j} ({[{(u_{ε})}_{τ}]}_{S_{ε}^{j}}) on S_{ε}^{j} for j = 0, \dots, m, \end{matrix} \end{array}$ where $\partial Ψ_{ε}^{j}$ denotes the subdifferential of $Ψ_{ε}^{j}$ .

The corresponding explicit Tresca conditions on the interfaces $S_{ε}^{j}$ with the function $Ψ_{ε}^{j}$ given in (5.3) are as follows: $\begin{array}{l} \{\begin{matrix} | σ^{ε} {(ν)}_{τ} | < G_{ε}^{j} (x) \Rightarrow {[{(u_{ε})}_{τ}]}_{S_{ε}^{j}} = 0, \\ | σ^{ε} {(ν)}_{τ} | = G_{ε}^{j} (x) \Rightarrow \exists λ_{ε}^{j} \in S_{ε}^{j} s.t. | {[{(u_{ε})}_{τ}]}_{S_{ε}^{j}} | + λ_{ε}^{j} | σ^{ε} {(ν)}_{τ} | = 0 a.e. on S_{ε}^{j} . \end{matrix} \end{array}$

Our aim now is to study the behavior of the solutions $u_{ε}$ for small values of the parameter ε. We will do this by studying the asymptotic behavior of the sequence $u_{ε}$ for $ε \to 0$ . When ε tends to zero, the thin layer $Ω_{ε}^{M}$ approaches the interface Σ. The domain $Ω_{ε}^{a}$ tends to the domain $Ω^{a}$ .
5.1. A priori estimates and existence of solutions for the Problem $P_{ε}$

The first step in the proof of existence of the solution consists in obtaining a bound for minimizing sequences. Since for $v = 0$ we have $E_{ε} (v) = 0$ , without loss of generality we can assume that every field u of a minimizing sequence satisfies $E_{ε} (u) ⩽ 0$ .

Proposition 5.1 (Estimate for minimizing sequences of $E_{ε}$ ).

We assume that $\begin{matrix} (5.7) & C_{0}^{'} ε max_{j = 1, \dots, m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})} max_{k = 1, \dots, m} (\frac{1}{M_{ε}^{k}}) ⩽ \frac{1}{2} . \end{matrix}$ Then, there exists a constant C which does not depend on ε, such that for every field u satisfying $E_{ε} (u) ⩽ 0$ we have $\begin{matrix} (5.8) & ‖ u ‖_{V_{ε}} ⩽ C (ε max_{j = 1, \dots, m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})} + ‖ f ‖_{L^{2} (Ω; R^{3})} + \sum_{k = 0}^{m} ({‖ {(g_{ε}^{k})}^{+} ‖}_{L^{1} (S_{ε}^{k})} + a_{ε}^{k})) . \end{matrix}$

Proof.
At first, note that the assumption (5.7) means that the total load applied to the layer does not depend on ε.

For simplicity’s sake, we set $\begin{matrix} R_{u} = \sum_{j = 1}^{m} ({‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} + {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j}; R^{3})}) . \end{matrix}$

Let u be in $K_{ε}$ , such that $E_{ε} (u) ⩽ 0$ . We have $\begin{matrix} (5.9) & \overline{α} | u |_{V_{ε}}^{2} + \sum_{j = 0}^{m} Ψ_{ε}^{j} ({[u_{τ}]}_{S_{ε}^{j}}) ⩽ \int_{Ω_{ε}^{}} f \cdot u d x + \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} f_{ε}^{j} \cdot (u^{j} + r_{u}^{j}) d x . \end{matrix}$ Now we use (3.26)₂ to get $\begin{array}{rcl} \int_{Ω_{ε}^{}} f \cdot u d x + \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} f_{ε}^{j} \cdot u^{j} d x \\ ⩽ C ‖ f ‖_{L^{2} (Ω; R^{3})} (‖ u ‖_{L^{2} (Ω^{b}; R^{3})} + ‖ u ‖_{L^{2} (Ω_{ε}^{a}; R^{3})} + ‖ u ‖_{L^{2} (Ω_{ε}^{0}; R^{3})}) \\ + C \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{2} (Ω_{ε}^{j}; R^{3})} {‖ u^{j} ‖}_{L^{2} (Ω_{ε}^{j}; R^{3})} \\ (5.10) & ⩽ C (‖ f ‖_{L^{2} (Ω; R^{3})} + ε \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) | u |_{V_{ε}} . \end{array}$ Using (3.8)₂, the last term on the right-hand side of (5.9) is simply bounded as follows: $\begin{array}{rcl} \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} f_{ε}^{j} \cdot r_{u}^{j} d x & ⩽ & C ε \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})} ({‖ a_{u}^{j} ‖}_{L^{1} (ω; R^{3})} + ε {‖ b_{u}^{j} ‖}_{L^{1} (ω; R^{3})}) \\ (5.11) & ⩽ & C_{0}^{'} ε max_{j = 1, \dots, m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})} R_{u} . \end{array}$ Hence $\begin{array}{l} \int_{Ω_{ε}^{*}} f \cdot u d x + \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} f_{ε}^{j} \cdot (u^{j} + r_{u}^{j}) d x \\ ⩽ C (‖ f ‖_{L^{2} (Ω; R^{3})} + ε \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) | u |_{V_{ε}} + C_{0}^{'} (ε max_{j = 1, \dots, m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) R_{u} . \end{array}$ From the above inequality and (5.9) we derive $\begin{matrix} (5.12) & \begin{matrix} \overline{α} | u |_{V_{ε}}^{2} + \sum_{j = 0}^{m} Ψ_{ε}^{j} ({[u_{τ}]}_{S_{ε}^{j}}) \\ ⩽ C (‖ f ‖_{L^{2} (Ω; R^{3})} + ε \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) | u |_{V_{ε}} + C_{0}^{'} (ε max_{j = 1, \dots, m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) R_{u} . \end{matrix} \end{matrix}$ Assumption (5.2) gives $\begin{matrix} (5.13) & \sum_{j = 1}^{m} (M_{ε}^{j} {‖ {[u_{τ}]}_{S_{ε}^{j}} ‖}_{L^{1} (S_{ε}^{j})} - a_{ε}^{j}) ⩽ \sum_{j = 1}^{m} Ψ_{ε}^{j} ({[u_{τ}]}_{S_{ε}^{j}}) . \end{matrix}$ One has $\begin{matrix} (5.14) & {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j})} ⩽ {‖ {[u_{τ}]}_{S_{ε}^{j}} ‖}_{L^{1} (S_{ε}^{j})} + 2 {‖ u_{| S_{ε}^{j}}^{j} - u_{| S_{ε}^{j}} ‖}_{L^{1} (S_{ε}^{j})} . \end{matrix}$ Besides, from the trace theorem (after ε-scaling) and (3.26), we have $\begin{array}{rcl} \sum_{j = 1}^{m} {‖ u_{| S_{ε}^{j}}^{j} - u_{| S_{ε}^{j}} ‖}_{L^{1} (S_{ε}^{j}; R^{3})} & ⩽ & \frac{C}{\sqrt{ε}} \sum_{j = 0}^{m} ({‖ u^{j} ‖}_{L^{2} (Ω_{ε}^{j}; R^{3})} + ε {‖ \nabla u^{j} ‖}_{L^{2} (Ω_{ε}^{j}; R^{3 \times 3})}) \\ (5.15) & ⩽ & C (‖ f ‖_{L^{2} (Ω; R^{3})} + ε \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) | u |_{V_{ε}} . \end{array}$ Then, the above estimate, (5.14) and (5.13) lead to $\begin{array}{l} min_{j = 1, \dots, m} M_{ε}^{j} \sum_{j = 1}^{m} {‖ {(r_{u}^{j})}_{τ} ‖}_{L^{1} (S_{ε}^{j})} \\ (5.16) & ⩽ \sum_{j = 1}^{m} Ψ_{ε}^{j} ({[u_{τ}]}_{S_{ε}^{j}}) + \sum_{j = 1}^{m} a_{ε}^{j} + C (‖ f ‖_{L^{2} (Ω; R^{3})} + ε \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) | u |_{V_{ε}} . \end{array}$ By definition of $K_{ε}$ (see (5.1)) we have $\begin{matrix} (5.17) & {(r_{u}^{j})}_{ν}^{+} ⩽ {(g_{ε}^{j})}^{+} + | {(u^{j} \cdot ν)}_{| S_{ε}^{j}} - {(u \cdot ν)}_{| S_{ε}^{j}} | a.e. S_{ε}^{j}, j = 1, \dots, m . \end{matrix}$ Then $\begin{matrix} {‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} ⩽ {‖ {(g_{ε}^{j})}^{+} ‖}_{L^{1} (S_{ε}^{j})} + {‖ u_{| S_{ε}^{j}}^{j} - u_{| S_{ε}^{j}} ‖}_{L^{1} (S_{ε}^{j}; R^{3})} . \end{matrix}$ The above estimate together with (5.15) yields $\begin{matrix} \sum_{j = 1}^{m} {‖ {(r_{u}^{j})}_{ν}^{+} ‖}_{L^{1} (S_{ε}^{j})} ⩽ \sum_{j = 1}^{m} {‖ {(g_{ε}^{j})}^{+} ‖}_{L^{1} (S_{ε}^{j})} + C (‖ f ‖_{L^{2} (Ω; R^{3})} + ε \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) | u |_{V_{ε}} . \end{matrix}$ Inequality (5.16) and the above one lead to $\begin{matrix} \begin{matrix} min_{j = 1, \dots, m} M_{ε}^{j} R_{u} ⩽ & \sum_{j = 1}^{m} Ψ_{ε}^{j} ({[u_{τ}]}_{S_{ε}^{j}}) + \sum_{j = 1}^{m} {‖ {(g_{ε}^{j})}^{+} ‖}_{L^{1} (S_{ε}^{j})} + \sum_{j = 1}^{m} a_{ε}^{j} \\ + C (‖ f ‖_{L^{2} (Ω; R^{3})} + ε \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) | u |_{V_{ε}} . \end{matrix} \end{matrix}$ Now the above estimate and (5.12) give $\begin{array}{rcl} \overline{α} | u |_{V_{ε}}^{2} + min_{j = 1, \dots, m} M_{ε}^{j} R_{u} & ⩽ & C (‖ f ‖_{L^{2} (Ω; R^{3})} + ε \sum_{j = 1}^{m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) | u |_{V_{ε}} \\ (5.18) & + C_{0}^{'} (ε max_{j = 1, \dots, m} {‖ f_{ε}^{j} ‖}_{L^{\infty} (Ω_{ε}^{j}; R^{3})}) R_{u} + \sum_{j = 1}^{m} {‖ {(g_{ε}^{j})}^{+} ‖}_{L^{1} (S_{ε}^{j})} + \sum_{j = 1}^{m} a_{ε}^{j} . \end{array}$ Then under assumption (5.7) and (3.26)₃ we obtain (5.8). □
Proposition 5.2 (Existence of solutions for $P_{ε}$ ).

Under the assumption ( 5.7 ) there exists at least one global minimizer for the functional $E_{ε}$ .

Proof.
Let ${u_{η}}_{η > 0}$ be a minimizing sequence for $P_{ε}$ . Due to (5.8) and (3.27) there exists a constant C which does not depend on η (but which depends on ε) such that $\begin{array}{l} u_{η} = (u_{η}, u_{η}^{1}, \dots, u_{η}^{m}, a_{u_{η}}, b_{u_{η}}) \in V_{ε}, \\ (5.19) & ‖ u_{η} ‖_{H^{1} (Ω_{ε}^{*}; R^{3})} + \sum_{j = 1}^{m} {‖ u_{η}^{j} ‖}_{H^{1} (Ω_{ε}^{j}; R^{3})} + \sum_{j = 1}^{m} ({‖ a_{u_{η}}^{j} ‖}_{L^{2} (ω; R^{3})} + ε {‖ b_{u_{η}}^{j} ‖}_{L^{2} (ω; R^{3})}) ⩽ C . \end{array}$

Since $E_{ε}$ is bounded from below, convex and weakly lower semi-continuous as a sum of weakly lower semi-continuous functions, there exists at least a minimizer $u_{ε} \in K_{ε}$ for $E_{ε}$ . □
Remark 5.1.
Let $u_{ε} = (u_{ε}, u_{ε}^{1}, \dots, u_{ε}^{m}, a_{u_{ε}}, b_{u_{ε}}) \in K_{ε}$ , $u_{ε}^{'} = (u_{ε}^{'}, u_{ε}^{' 1}, \dots, u_{ε}^{' m}, a_{u_{ε}^{'}}, b_{u_{ε}^{'}}) \in K_{ε}$ be two minimizers of $P_{ε}$ , both fields satisfy (5.5). Hence $\begin{matrix} u_{ε} = u_{ε}^{'}, u_{ε}^{j} = u_{ε}^{' j}, j = 1, \dots, m . \end{matrix}$

6. Main result

In this section we only consider the case of Tresca friction.

6.1. Hypotheses on $g_{ε}^{j}$ , $G_{ε}^{j}$ and $f_{ε}^{j}$

To pass to the limit in the homogenization process, we need structural assumptions concerning the Tresca data which are more precise than those of Section 5. Hence we assume that there exist

$g^{j} \in L^{1} (S^{j})$ , $j = 0, \dots, m$ , such that $\begin{matrix} g_{ε}^{j} = g^{j} ({\frac{\cdot}{ε}}_{Y}), \end{matrix}$ and therefore $\begin{matrix} T_{ε}^{b l, j} (g_{ε}^{j}) (x^{'}, y) = g^{j} (y) for a.e. (x^{'}, y) \in ω \times S^{j} . \end{matrix}$

$G^{j} \in C_{c} (ω \times S^{j})$ , $j = 0, \dots, m$ , such that $\begin{array}{l} T_{ε}^{b l, j} (G_{ε}^{j}) \to G^{j} strongly in L^{\infty} (ω \times S^{j}), \\ G^{j} (x^{'}, y) ⩾ M^{j} for any (x^{'}, y) \in ω \times S^{j}, M^{j} > 0 . \end{array}$ This choice of the parameter corresponds to the Tresca friction coefficient $G^{j} (x^{'}, {\frac{x}{ε}}_{Y})$ .

$F^{j} \in C_{c} (ω \times Y^{j}; R^{3})$ , $j = 1, \dots, m$ , such that $\begin{matrix} (6.1) & f_{ε}^{j} (x) = \frac{1}{ε} F^{j} (ε {[\frac{x^{'}}{ε}]}_{Y^{'}}, ε {\frac{x}{ε}}_{Y}), for a.e. x \in Ω_{ε}^{j} . \end{matrix}$

The Assumption (5.7) becomes

\begin{matrix} (6.2) & C_{0}^{'} max_{j = 1, \dots, m} ({‖ F^{j} ‖}_{L^{\infty} (ω \times Y^{j}; R^{3})}) max_{j = 1, \dots, m} (\frac{1}{M^{j}}) ⩽ \frac{1}{2}, \end{matrix}

which is fulfilled for small enough

C_{0}^{'}

6.2. The limit problem

We equip the product space $\begin{array}{rcl} H & = & H_{Γ}^{1} (Ω^{b}; R^{3}) \times H^{1} (Ω^{a}; R^{3}) \times L^{2} (ω; H_{per}^{1} (Y^{0}; R^{3})) \\ \times \prod_{j = 1}^{m} L^{2} (ω; H^{1} (Y^{j}; R^{3})) \times {[M (ω; R^{3})]}^{m} \times {[M (ω; R^{3})]}^{m} \end{array}$ with the norm $\begin{matrix} ‖ V ‖_{H} = {‖ v^{b} ‖}_{H^{1} (Ω^{b}; R^{3})} + {‖ v^{a} ‖}_{H^{1} (Ω^{a}; R^{3})} + \sum_{j = 0}^{m} {‖ {\hat{v}}^{j} ‖}_{L^{2} (ω; H^{1} (Y^{j}; R^{3}))} + ‖ a_{V} ‖_{{[M (ω; R^{3})]}^{m}} + ‖ b_{V} ‖_{{[M (ω; R^{3})]}^{m}}, \end{matrix}$ where $V = (v^{b}, v^{a}, {\hat{v}}^{0}, \dots, {\hat{v}}^{m}, a_{V}, b_{V}) \in H$ .

We set $\begin{array}{rcl} V & = & {V \in H | v^{b} (x^{'}, 0) = {\hat{v}}^{0} (x^{'}, y_{1}, y_{2}, 0), v^{a} (x^{'}, 0) = {\hat{v}}^{0} (x^{'}, y_{1}, y_{2}, 1) \\ for a.e. (x^{'}, y_{1}, y_{2}) \in ω \times Y^{'}, {\hat{v}}^{j} is orthogonal to the rigid displacements, j = 1, \dots, m} . \end{array}$ For any $V \in V$ , as in Section 5, we define ${[V]}_{S^{j}} = {\hat{v}}_{| S^{j}}^{j} - {\hat{v}}_{| S^{j}}^{0}$ , ${[V_{ν}]}_{S^{j}} = {[V]}_{S^{j}} \cdot ν_{| S^{j}}$ , ${[V_{τ}]}_{S^{j}} = {[V]}_{S^{j}} - {[V_{ν}]}_{S^{j}}$ , $j = 1, \dots, m$ , and $\begin{array}{l} {[V]}_{S^{0}} (x^{'}, y) = lim_{s \to 0, s > 0} ({\hat{v}}^{0} (x^{'}, y + s ν (y)) - {\hat{v}}^{0} (x^{'}, y - s ν (y))), \\ {[V_{ν}]}_{S^{0}} (x^{'}, y) = lim_{s \to 0, s > 0} ({\hat{v}}^{0} (x^{'}, y + s ν (y)) - {\hat{v}}^{0} (x^{'}, y - s ν (y))) \cdot ν (y), \\ {[V_{τ}]}_{S^{0}} (x^{'}, y) = {[V]}_{S^{0}} (x^{'}, y) - {[V_{τ}]}_{S^{0}} (x^{'}, y) for a.e. (x^{'}, y) \in ω \times S^{0} . \end{array}$ Now we introduce the closed convex set $K$ $\begin{array}{rcl} K & = & {V \in V | {[V_{ν}]}_{S^{0}} ⩽ g^{0} in ω \times S^{0}, {[V_{ν}]}_{S^{j}} + r_{V}^{j} \cdot ν ⩽ g^{j} in ω \times S^{j},^{1} \\ where \forall y \in Y^{j} r_{V}^{j} (\cdot, y) = a_{V}^{j} + b_{V}^{j} \land (y - O^{j}) in M (ω; R^{3}), j = 1, \dots, m} . \end{array}$

¹
This condition means $\begin{array}{l} \forall ϕ \in C_{c} (ω \times S^{j}), s.t. ϕ (x^{'}, y) ⩾ 0 on ω \times S^{j}, \\ {⟨ ϕ, r_{V}^{j} \cdot ν ⟩}_{C_{c} (ω \times S^{j}), M (ω \times S^{j})} ⩽ \int_{ω \times S^{j}} (g^{j} - {[V_{ν}]}_{S^{j}}) ϕ d x^{'} d σ (y), j = 1, \dots, m . \end{array}$

Theorem 6.1.

Assume $f \in L^{2} (Ω; R^{3})$ and $f_{ε}^{j}$ , $g_{ε}^{j}$ , $G_{ε}^{j}$ satisfy the hypotheses of Section 6.1 and also assume that ( 6.2 ) is fulfilled. Suppose that the following assumption holds: there exists a tensor $a^{M}$ such that, as $ε \to 0$ , $\begin{matrix} (6.3) & T_{ε} (a_{ε}^{M}) \to a^{M} a.e. in ω \times Y . \end{matrix}$ Let $u_{ε}$ be the solution of Problem ( 5.5 ), then there exist a subsequence – still denoted ε – and $U = (u^{b}, u^{a}, {\hat{u}}^{0}, \dots, {\hat{u}}^{m}, a_{U}, b_{U}) \in K$ such that ( $j = 1, \dots, m$ ) $\begin{array}{rcl} u_{ε} ⇀ u^{b} weakly in H_{Γ}^{1} (Ω^{b}; R^{3}), \\ u_{ε} (\cdot + ε e_{3}) ⇀ u^{a} weakly in H^{1} (Ω^{a}; R^{3}), \\ T_{ε} (u_{ε}) ⇀ {\hat{u}}^{0} weakly in L^{2} (ω; H^{1} (Y^{0}; R^{3})), \\ ε T_{ε} (e (u_{ε})) ⇀ e_{y} ({\hat{u}}^{0}) weakly in L^{2} (ω \times Y^{0}; R^{3 \times 3}), \\ T_{ε} (u_{ε}^{j}) ⇀ {\hat{u}}^{j} weakly in L^{2} (ω; H^{1} (Y^{j}; R^{3})), \\ ε T_{ε} (e (u_{ε}^{j})) ⇀ e_{y} ({\hat{u}}^{j}) weakly in L^{2} (ω \times Y^{j}; R^{3 \times 3}), \\ a_{u_{ε}}^{j} ⇀ a_{U}^{j} weakly-* in M (ω; R^{3}), \\ b_{u_{ε}}^{j} ⇀ b_{U}^{j} weakly-* in M (ω; R^{3}), \end{array}$ where $M (ω; R^{3})$ is the space of bounded measures on ω with values in $R^{3}$ .

The limit field U satisfies the following unfolded problem: $\begin{array}{l} \int_{Ω^{b}} a^{b} e (u^{b}) : e (v^{b} - u^{b}) d x + \int_{Ω^{a}} a^{a} e (u^{a}) : e (v^{a} - u^{a}) d x \\ + \sum_{j = 0}^{m} \int_{ω \times Y^{j}} a^{M} e_{y} ({\hat{u}}^{j}) : e_{y} ({\hat{v}}^{j} - {\hat{u}}^{j}) d x^{'} d y \\ + \sum_{j = 0}^{m} {⟨ G^{j}, | {[V_{τ}]}_{S^{j}} + {(r_{V}^{j})}_{τ} | - | {[U_{τ}]}_{S^{j}} + {(r_{U}^{j})}_{τ} | ⟩}_{C_{c} (ω \times S^{j}), M (ω \times S^{j})} \\ ⩾ \int_{Ω^{b}} f \cdot (v^{b} - u^{b}) d x + \int_{Ω^{a}} f \cdot (v^{a} - u^{a}) d x + \sum_{j = 1}^{m} \int_{ω \times Y^{j}} F^{j} \cdot ({\hat{v}}^{j} - {\hat{u}}^{j}) d x^{'} d y \\ (6.4) & + \sum_{j = 1}^{m} {⟨ F^{j}, r_{V}^{j} - r_{U}^{j} ⟩}_{C_{c} (ω \times Y^{j}; R^{3}), M (ω \times Y^{j}; R^{3})}, \forall V \in K, \end{array}$ where $\begin{matrix} r_{V}^{0} = r_{U}^{0} = 0, {(r_{V}^{j})}_{τ} = r_{V}^{j} - (r_{V}^{j} \cdot ν) ν, {(r_{U}^{j})}_{τ} = r_{U}^{j} - (r_{U}^{j} \cdot ν) ν, j = 1, \dots, m . \end{matrix}$

Proof.

Based on the Proposition 5.1 and the assumptions of Section 6.1, the solution $u_{ε}$ of Problem $P_{ε}$ is uniformly bounded with respect to ε. Hence up to a subsequence of ε (still denoted ε), the convergences of Theorem 4.1 hold. Therefore, we should show that the limits furnished by the Theorem 4.1 satisfy a homogenized limit problem. The main point now is to pass to the limit in (5.5).

Let $V = (v^{b}, v^{a}, {\hat{v}}^{0}, \dots, {\hat{v}}^{m}, a_{V}, b_{V})$ be in $\begin{array}{l} K \cap H_{Γ}^{1} (Ω^{b}; R^{3}) \times H^{1} (Ω^{a}; R^{3}) \times C_{c}^{\infty} (ω; H_{per}^{1} (Y^{0}; R^{3})) \\ \times \prod_{j = 1}^{m} C_{c}^{\infty} (ω; H^{1} (Y^{j}; R^{3})) \times {[C_{c}^{\infty} (ω; R^{3})]}^{m} \times {[C_{c}^{\infty} (ω; R^{3})]}^{m} . \end{array}$ We use the following test function $v_{ε} \in K_{ε}$ : $\begin{matrix} (6.5) & v_{ε} (x) = \{\begin{matrix} v_{ε} (x) = v^{b} (x) & in Ω^{b}, \\ v_{ε} (x) = v^{a} (x - ε e_{3}) & in Ω_{ε}^{a}, \\ v_{ε} (x) = {\hat{v}}^{0} (x^{'}, {\frac{x}{ε}}_{Y}) & in Ω_{ε}^{0}, \\ v_{ε}^{j} (x) = {\hat{v}}^{j} (x^{'}, {\frac{x}{ε}}_{Y}) & in Ω_{ε}^{j}, \\ r_{v_{ε}}^{j} (x) = a_{V}^{j} (x^{'}) + b_{V}^{j} (x^{'}) \land ({\frac{x}{ε}}_{Y} - O^{j}) & in Ω_{ε}^{j} . \end{matrix} \end{matrix}$ By construction we have ( $j = 1, \dots, m$ ) $\begin{matrix} (6.6) & \begin{matrix} v_{ε} = v^{b} in Ω^{b}, v_{ε} (\cdot + ε e_{3}) = v^{a} in Ω^{a}, \\ T_{ε} (v_{ε}) ⟶ {\hat{v}}^{0} strongly in L^{2} (ω; H^{1} (Y^{0}; R^{3})), \\ ε T_{ε} (e (v_{ε})) ⟶ e_{y} ({\hat{v}}^{0}) strongly in L^{2} (ω \times Y^{0}; R^{3 \times 3}), \\ T_{ε} (v_{ε}) ⟶ {\hat{v}}^{j} strongly in L^{2} (ω; H^{1} (Y^{j}; R^{3})), \\ ε T_{ε} (e (v_{ε})) ⟶ e_{y} ({\hat{v}}^{j}) strongly in L^{2} (ω \times Y^{j}; R^{3 \times 3}), \\ T_{ε} (r_{v_{ε}}^{j}) ⟶ a_{V}^{j} + b_{V}^{j} \land (y - O^{j}) strongly in L^{1} (ω \times Y^{j}; R^{3}) . \end{matrix} \end{matrix}$

We rewrite (5.5) in the form $\begin{array}{l} ε \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} a_{ε}^{M} e (u_{ε}^{j}) : e (v_{ε}^{j}) d x + ε \int_{Ω_{ε}^{0}} a_{ε}^{M} e (u_{ε}) : e (v_{ε}) d x + \int_{Ω^{b}} a^{b} e (u_{ε}) : e (v_{ε}) d x \\ + \int_{Ω^{a}} a^{a} (\cdot + ε e_{3}) e (u_{ε} (\cdot + ε e_{3})) : e (v^{a}) d x \\ + \sum_{j = 0}^{m} \int_{S_{ε}^{j}} G_{ε}^{j} (x) | {[{(v_{ε})}_{τ}]}_{S_{ε}^{j}} | d σ (x) - \int_{Ω_{ε}} f_{ε} \cdot v_{ε} d x \\ ⩾ ε \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} a_{ε}^{M} e (u_{ε}^{j}) : e (u_{ε}^{j}) d x + ε \int_{Ω_{ε}^{0}} a_{ε}^{M} e (u_{ε}) : e (u_{ε}) d x + \int_{Ω^{b}} a^{b} e (u_{ε}) : e (u_{ε}) d x \\ + \int_{Ω^{a}} a^{a} (\cdot + ε e_{3}) e (u_{ε} (\cdot + ε e_{3})) : e (u_{ε} (\cdot + ε e_{3})) d x \\ + \sum_{j = 0}^{m} \int_{S_{ε}^{j}} G_{ε}^{j} (x) | {[{(u_{ε})}_{τ}]}_{S_{ε}^{j}} | d σ (x) - \int_{Ω_{ε}} f_{ε} \cdot u_{ε} d x . \end{array}$ Therefore, by unfolding and due to the convergences in Theorem 4.1 and the equations and convergence in (6.6), we have $\begin{array}{l} lim_{ε \to 0} (ε \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} a_{ε}^{M} e (u_{ε}^{j}) : e (v_{ε}^{j}) d x + ε \int_{Ω_{ε}^{0}} a_{ε}^{M} e (u_{ε}) : e (v_{ε}) d x) \\ = lim_{ε \to 0} (ε^{2} \sum_{j = 1}^{m} \int_{ω \times Y^{j}} T_{ε} (a_{ε}^{M}) T_{ε} (e (u_{ε}^{j})) : T_{ε} (e (v_{ε}^{j})) d x^{'} d y \\ + ε^{2} \int_{ω \times Y^{j}} T_{ε} (a_{ε}^{M}) T_{ε} (e (u_{ε})) : T_{ε} (e (v_{ε})) d x^{'} d y) \\ = \sum_{j = 0}^{m} \int_{ω \times Y^{j}} a^{M} e_{y} ({\hat{u}}^{j}) : e_{y} ({\hat{v}}^{j}) d x^{'} d y, \\ lim_{ε \to 0} \sum_{j = 0}^{m} \int_{S_{ε}^{j}} G_{ε}^{j} | {[{(v_{ε})}_{τ}]}_{S_{ε}^{j}} | d σ (x) = lim_{ε \to 0} \sum_{j = 0}^{m} \int_{ω \times S^{j}} T_{ε}^{b l, j} (G_{ε}^{j}) T_{ε}^{b l, j} (| {[{(v_{ε})}_{τ}]}_{S_{ε}^{j}} |) d x^{'} d σ (y) \\ = \sum_{j = 0}^{m} \int_{ω \times S^{j}} G^{j} | {[V_{τ}]}_{S^{j}} + {(r_{V}^{j})}_{τ} | d x^{'} d σ (y) . \end{array}$ By the lower semi-continuity with respect to weak (or weak-*) convergences, we first obtain $\begin{array}{rcl} \underset{ε \to 0}{lim inf} \sum_{j = 0}^{m} \int_{Ω_{ε}^{j}} ε a_{ε}^{M} e (u_{ε}^{j}) : e (u_{ε}^{j}) d x & = & \underset{ε \to 0}{lim inf} ε^{2} \sum_{j = 0}^{m} \int_{ω \times Y^{j}} T_{ε} (a_{ε}^{M}) T_{ε} (e (u_{ε}^{j})) : T_{ε} (e (u_{ε}^{j})) d x^{'} d y \\ ⩾ & \sum_{j = 0}^{m} \int_{ω \times Y^{j}} a^{M} e_{y} ({\hat{u}}^{j}) : e_{y} ({\hat{u}}^{j}) d x^{'} d y, \end{array}$ then unfolding the term corresponding to the Tresca friction, passing to the limit and making use of lower semi-continuity give $\begin{matrix} \underset{ε \to 0}{lim inf} \sum_{j = 0}^{m} \int_{S_{ε}^{j}} G_{ε}^{j} | {[{(u_{ε})}_{τ}]}_{S_{ε}^{j}} | d σ (x) ⩾ \sum_{j = 0}^{m} {⟨ G^{j}, | {[U_{τ}]}_{S^{j}} + {(r_{U}^{j})}_{τ} | ⟩}_{C_{c} (ω \times S^{j}), M (ω \times S^{j})} . \end{matrix}$ Considering the terms corresponding to the applied forces we get $\begin{array}{l} lim_{ε \to 0} \int_{Ω^{b} \cup Ω_{ε}^{a}} f \cdot u_{ε} d x = lim_{ε \to 0} (\int_{Ω^{b}} f \cdot u_{ε} d x + \int_{Ω^{a}} f (\cdot + ε e_{3}) \cdot u_{ε} (\cdot + ε e_{3}) d x) \\ = \int_{Ω^{b}} f \cdot u^{b} d x + \int_{Ω^{a}} f \cdot u^{a} d x, \\ lim_{ε \to 0} \int_{Ω_{ε}^{0}} f \cdot u_{ε} d x = lim_{ε \to 0} ε \int_{ω \times Y^{0}} T_{ε} (f) \cdot T_{ε} (u_{ε}) d x^{'} d y = 0, \\ lim_{ε \to 0} \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} f_{ε}^{j} \cdot u_{ε}^{j} d x = lim_{ε \to 0} ε \sum_{j = 1}^{m} \int_{ω \times Y^{j}} T_{ε} (f_{ε}^{j}) \cdot T_{ε} (u_{ε}^{j}) d x^{'} d y \\ = \sum_{j = 1}^{m} \int_{ω \times Y^{j}} F^{j} \cdot {\hat{u}}^{j} d x^{'} d y + \sum_{j = 1}^{m} {⟨ F^{j}, r_{V}^{j} ⟩}_{C_{c} (ω \times Y^{j}; R^{3}), M (ω \times Y^{j}; R^{3})} . \end{array}$ In a similar way $\begin{array}{l} lim_{ε \to 0} \int_{Ω_{ε}^{*}} f \cdot v_{ε} d x = \int_{Ω^{b}} f \cdot v^{b} d x + \int_{Ω^{a}} f \cdot v^{a} d x, \\ lim_{ε \to 0} \sum_{j = 1}^{m} \int_{Ω_{ε}^{j}} f_{ε}^{j} \cdot v_{ε}^{j} d x = \sum_{j = 1}^{m} \int_{ω \times Y^{j}} F^{j} \cdot ({\hat{v}}^{j} + r_{V}^{j}) d x^{'} d y . \end{array}$ Using the established convergences we obtain $\begin{array}{l} \int_{Ω^{b}} a^{b} e (u^{b}) : e (v^{b}) d x + \int_{Ω^{a}} a^{a} e (u^{a}) : e (v^{a}) d x + \sum_{j = 0}^{m} \int_{ω \times Y^{j}} a^{M} e_{y} ({\hat{u}}^{j}) : e_{y} ({\hat{v}}^{j}) d x^{'} d y \\ + \sum_{j = 0}^{m} {⟨ G^{j}, | {[V_{τ}]}_{S^{j}} + {(r_{V}^{j})}_{τ} | ⟩}_{C_{c} (ω \times S^{j}), M (ω \times S^{j})} - \int_{Ω^{b}} f \cdot v^{b} d x - \int_{Ω^{a}} f \cdot v^{a} d x \\ - \sum_{j = 1}^{m} {⟨ F^{j}, {\hat{v}}^{j} + r_{V}^{j} ⟩}_{C_{c} (ω \times S^{j}; R^{3}), M (ω \times S^{j}; R^{3})} \\ ⩾ \int_{Ω^{b}} a^{b} e (u^{b}) : e (u^{b}) d x + \int_{Ω^{a}} a^{a} e (u^{a}) : e (u^{a}) d x + \sum_{j = 0}^{m} \int_{ω \times Y^{j}} a^{M} e_{y} ({\hat{u}}^{j}) : e_{y} ({\hat{u}}^{j}) d x^{'} d y \\ + \sum_{j = 0}^{m} {⟨ G^{j}, | {[U_{τ}]}_{S^{j}} + {(r_{U}^{j})}_{τ} | ⟩}_{C_{c} (ω \times S^{j}), M (ω \times S^{j})} - \int_{Ω^{b}} f \cdot u^{b} d x - \int_{Ω^{a}} f \cdot u^{a} d x \\ - \sum_{j = 1}^{m} {⟨ F^{j}, {\hat{u}}^{j} + r_{U}^{j} ⟩}_{C_{c} (ω \times Y^{j}; R^{3}), M (ω \times Y^{j}; R^{3})} . \end{array}$ Subtracting the terms on the right-hand side from the left-hand side we get (6.4). By a density argument, (6.4) holds for any $V \in K$ . □

Remark 6.1.

Since the functional $\begin{array}{rcl} E (V) & ≐ & \frac{1}{2} (\int_{Ω^{b}} a^{b} e (v^{b}) : e (v^{b}) d x + \int_{Ω^{a}} a^{a} e (v^{a}) : e (v^{a}) d x \\ + \sum_{j = 0}^{m} \int_{ω \times Y^{j}} a^{M} e_{y} ({\hat{v}}^{j}) : e_{y} ({\hat{v}}^{j}) d x^{'} d y) \\ + \sum_{j = 0}^{m} {⟨ G^{j}, | {[V_{τ}]}_{S^{j}} + {(r_{V}^{j})}_{τ} | ⟩}_{C_{c} (ω \times S^{j}), M (ω \times S^{j})} - \int_{Ω^{b}} f \cdot v^{b} d x - \int_{Ω^{a}} f \cdot v^{a} d x \\ (6.7) & - \sum_{j = 1}^{m} {⟨ F^{j}, {\hat{v}}^{j} + r_{V}^{j} ⟩}_{C_{c} (ω \times Y^{j}; R^{3}), M (ω \times Y^{j}; R^{3})}, V = (v^{b}, v^{a}, {\hat{v}}^{0}, \dots, {\hat{v}}^{m}, a_{V}, b_{V}) \in K, \end{array}$ is weakly lower semi-continuous and convex over $K$ , Problem (6.4) is equivalent to finding a minimizer over $K$ of the functional $E$ . The field U obtained in Theorem 6.1 is a global minimizer of this functional over $K$ and every limit point of the sequence ${u_{ε}}$ is a global minimizer of $E$ .

Lemma 6.1.

For every $M ⩾ 0$ there exists a constant $C (M)$ such that for any $V \in K$ satisfying $E (V) ⩽ M$ we have $\begin{array}{l} {‖ v^{a} ‖}_{H^{1} (Ω^{a}; R^{3})} + {‖ v^{b} ‖}_{H^{1} (Ω^{b}; R^{3})} + \sum_{j = 0}^{m} {‖ {\hat{v}}^{j} ‖}_{L^{2} (ω; H^{1} (Y^{j}; R^{3}))} + ‖ a_{V} ‖_{{[M (ω; R^{3})]}^{m}} + ‖ b_{V} ‖_{{[M (ω; R^{3})]}^{m}} \\ (6.8) & ⩽ C (M) . \end{array}$

Proof.

From the expression (6.7) of $E (V)$ , we first deduce that $E (V) ⩽ M$ implies $\begin{array}{l} \frac{\overline{α}}{2} ({‖ e (v^{a}) ‖}_{L^{2} (Ω^{a}; R^{6})}^{2} + {‖ e (v^{b}) ‖}_{L^{2} (Ω^{b}; R^{6})}^{2} + \sum_{j = 0}^{m} {‖ e_{y} ({\hat{v}}^{j}) ‖}_{L^{2} (ω \times Y^{j}; R^{6})}^{2}) \\ + \sum_{j = 0}^{m} M^{j} {‖ {[V_{τ}]}_{S^{j}} + {(r_{V}^{j})}_{τ} ‖}_{M (ω \times S^{j}; R^{3})} \\ - ‖ f ‖_{L^{2} (Ω^{a}; R^{3})} {‖ v^{a} ‖}_{L^{2} (Ω^{a}; R^{3})} - ‖ f ‖_{L^{2} (Ω^{b}; R^{3})} {‖ v^{b} ‖}_{L^{2} (Ω^{b}; R^{3})} \\ (6.9) & - \sum_{j = 1}^{m} {‖ F^{j} ‖}_{L^{2} (ω \times Y^{j}; R^{3})} {‖ {\hat{v}}^{j} ‖}_{L^{2} (ω \times Y^{j}; R^{3})} - \sum_{j = 1}^{m} {‖ F^{j} ‖}_{C_{c} (ω \times Y^{j}; R^{3})} {‖ r_{V}^{j} ‖}_{M (ω \times Y^{j}; R^{3})} ⩽ M . \end{array}$ From the Korn inequality and the trace theorem we get $\begin{array}{l} {‖ v^{a} ‖}_{H^{1} (Ω^{a}; R^{3})} + {‖ v^{b} ‖}_{H^{1} (Ω^{b}; R^{3})} + \sum_{j = 0}^{m} {‖ {\hat{v}}^{j} ‖}_{L^{2} (ω; H^{1} (Y^{j}; R^{3}))} \\ ⩽ C ({‖ e (v^{a}) ‖}_{L^{2} (Ω^{a}; R^{6})}^{2} + {‖ e (v^{b}) ‖}_{L^{2} (Ω^{b}; R^{6})}^{2} + \sum_{j = 0}^{m} {‖ e_{y} ({\hat{v}}^{j}) ‖}_{L^{2} (ω \times Y^{j}; R^{6})}^{2}), \end{array}$ where constant C depends on the geometry of the sets $Ω^{a}$ , $Ω^{b}$ , ω, $S^{0}$ and $Y^{j}$ ( $j = 1, \dots, m$ ).

Then the above inequality and (6.9) yield $\begin{array}{l} \overline{β} ({‖ v^{a} ‖}_{H^{1} (Ω^{a}; R^{3})} + {‖ v^{b} ‖}_{H^{1} (Ω^{b}; R^{3})} + \sum_{j = 0}^{m} {‖ {\hat{v}}^{j} ‖}_{L^{2} (ω; H^{1} (Y^{j}; R^{3}))}) + \sum_{j = 0}^{m} M^{j} {‖ {(r_{V}^{j})}_{τ} ‖}_{M (ω \times S^{j}; R^{3})} \\ - \sum_{j = 1}^{m} {‖ F^{j} ‖}_{C_{c} (ω \times Y^{j}; R^{3})} {‖ r_{V}^{j} ‖}_{M (ω \times Y^{j}; R^{3})} ⩽ M, \end{array}$ where $\overline{β} > 0$ .

Now, in the case $a_{V} \in {[L^{1} (ω; R^{3})]}^{m}$ and $b_{V} \in {[L^{1} (ω; R^{3})]}^{m}$ , from Proposition 3.2, there exist constants C which depend on $Y^{j}$ ( $j = 1, \dots, m$ ) such that $\begin{matrix} ‖ a_{V} ‖_{{[L^{1} (ω; R^{3})]}^{m}} + ‖ b_{V} ‖_{{[L^{1} (ω; R^{3})]}^{m}} ⩽ C \sum_{j = 1}^{m} ({‖ {(r_{V}^{j})}_{τ} ‖}_{L^{1} (ω \times S^{j}; R^{3})} + {‖ {({(r_{V}^{j})}_{ν})}^{+} ‖}_{L^{1} (ω \times S^{j})}) . \end{matrix}$ Due to the definition of $K$ we have $‖ {({(r_{V}^{j})}_{ν})}^{+} ‖_{L^{1} (ω \times S^{j})} ⩽ ‖ {(g^{j})}^{+} ‖_{L^{1} (S^{j})} + ‖ [V_{ν}] ‖_{L^{1} (ω \times S^{j})}$ . Proceeding as in the proofs of Propositions 3.3 and 5.1 and thanks to the condition (6.2) and the above inequalities, we deduce the existence of a constant $C (M)$ such that (we recall that $‖ r_{V}^{j} ‖_{M (ω \times S^{j}; R^{3})} = ‖ r_{V}^{j} ‖_{L^{1} (ω \times S^{j}; R^{3})}$ ) $\begin{matrix} {‖ v^{a} ‖}_{H^{1} (Ω^{a}; R^{3})} + {‖ v^{b} ‖}_{H^{1} (Ω^{b}; R^{3})} + \sum_{j = 0}^{m} {‖ {\hat{v}}^{j} ‖}_{L^{2} (ω; H^{1} (Y^{j}; R^{3}))} + ‖ a_{V} ‖_{{[L^{1} (ω; R^{3})]}^{m}} + ‖ b_{V} ‖_{{[L^{1} (ω; R^{3})]}^{m}} ⩽ C (M) . \end{matrix}$ Then for general elements $V \in K$ , using regularization by convolution in $x^{'}$ and this last estimate we obtain (6.8). □

Independently of Remark 6.1, using the above lemma we can easily prove the existence of at least one global minimizer for the functional $E$ . Remark 6.2.

Let U and $U^{'}$ be two global minimizers of the functional $E$ . From (6.4) we deduce that $\begin{matrix} u^{a} = u^{' a}, u^{b} = u^{' b}, {\hat{u}}^{j} = {\hat{u}}^{' j}, j = 0, \dots, m \end{matrix}$ and $\begin{array}{l} \sum_{j = 0}^{m} {⟨ G^{j}, | {[U_{τ}]}_{S^{j}} + {(r_{U}^{j})}_{τ} | ⟩}_{C_{c} (ω \times S^{j}), M (ω \times S^{j})} - \sum_{j = 1}^{m} {⟨ F^{j}, r_{U}^{j} ⟩}_{C_{c} (ω \times Y^{j}; R^{3}), M (ω \times Y^{j}; R^{3})} \\ = \sum_{j = 0}^{m} {⟨ G^{j}, | {[U_{τ}]}_{S^{j}} + {(r_{U^{'}}^{j})}_{τ} | ⟩}_{C_{c} (ω \times S^{j}), M (ω \times S^{j})} - \sum_{j = 1}^{m} {⟨ F^{j}, r_{U^{'}}^{j} ⟩}_{C_{c} (ω \times Y^{j}; R^{3}), M (ω \times Y^{j}; R^{3})} . \end{array}$

Lemma 6.2.

Let $u_{ε}$ be a minimizer of $P_{ε}$ over $K_{ε}$ , $\begin{matrix} m_{ε} = min_{v \in K_{ε}} E_{ε} (v) = E_{ε} (u_{ε}) . \end{matrix}$ The whole sequence ${m_{ε}}$ converges and we have $\begin{matrix} m = min_{V \in K} E (V) = E (U) = lim_{ε \to 0} m_{ε} = lim_{ε \to 0} E_{ε} (u_{ε}) ⩽ 0 . \end{matrix}$

Proof.

As a consequence of Theorem 6.1 we have (with the subsequence of ε introduced in this theorem) $\begin{matrix} m = E (U) ⩽ \underset{ε \to 0}{lim inf} m_{ε} = \underset{ε \to 0}{lim inf} E_{ε} (u_{ε}) . \end{matrix}$ Now, let V be in $K$ and let ${V^{η}}_{η}$ be a sequence of fields in $\begin{array}{l} K \cap H_{Γ}^{1} (Ω^{b}; R^{3}) \times H^{1} (Ω^{a}; R^{3}) \times C_{c}^{\infty} (ω; H_{per}^{1} (Y^{0}; R^{3})) \\ \times \prod_{j = 1}^{m} C_{c}^{\infty} (ω; H^{1} (Y^{j}; R^{3})) \times {[C_{c}^{\infty} (ω; R^{3})]}^{m} \times {[C_{c}^{\infty} (ω; R^{3})]}^{m} \end{array}$ strongly converging to V in $V$ (that is possible using regularization by convolution in $x^{'}$ ). We fix η and for every $V^{η}$ we consider the test field belonging to $K_{ε}$ defined by (6.5) and here denoted $v_{ε}^{η}$ . Using the subsequence of ε introduced in Theorem 6.1, we can pass to the limit ( $ε \to 0$ ). Since $m_{ε} ⩽ E_{ε} (v_{ε}^{η})$ , due to (6.6) we obtain $\begin{matrix} \underset{ε \to 0}{lim sup} m_{ε} ⩽ \underset{ε \to 0}{lim sup} E_{ε} (v_{ε}^{η}) = lim_{ε \to 0} E_{ε} (v_{ε}^{η}) = E (V^{η}) . \end{matrix}$ Now η goes to 0; that gives $\begin{matrix} \forall V \in K, \underset{ε \to 0}{lim sup} m_{ε} ⩽ E (V) . \end{matrix}$ Then $\begin{matrix} \underset{ε \to 0}{lim sup} m_{ε} ⩽ E (U) ⩽ min_{V \in K} E (V) . \end{matrix}$ Finally we get $m = {lim}_{ε \to 0} m_{ε}$ and this result holds for the whole sequence ${ε}$ . □

6.3. Computation of the effective outer-plane properties for a case of heterogeneous layer without contact

Solving (6.4) numerically is difficult because of the presence of non-differentiable non-linear term $| {[U_{τ}]}_{S^{j}} |$ . Also a complete scale separation is impossible for a non-linear problem.

In this subsection due to application purposes we want to consider a case without a contact. Thus, we assume that the open sets $Y^{j}$ , $j = 1, \dots, m$ , are holes. For this case we can give a linear problem whose solution is the couple $(u^{a}, u^{b})$ with Robin-type condition on the interface. The result obtained will be similar to [9,12,19].

The space $V$ is replaced by $\begin{array}{rcl} V^{'} & = & {V = (v^{b}, v^{a}, {\hat{v}}^{0}) \in H^{'} | v^{b} (x^{'}, 0) = {\hat{v}}^{0} (x^{'}, y^{'}, 0), v^{a} (x^{'}, 0) = {\hat{v}}^{0} (x^{'}, y^{'}, 1) \\ for a.e. (x^{'}, y^{'}) \in ω \times Y^{'}}, \end{array}$ where $H^{'} = H_{Γ}^{1} (Ω^{b}; R^{3}) \times H^{1} (Ω^{a}; R^{3}) \times L^{2} (ω; H_{per}^{1} (Y^{0}; R^{3}))$ . We endowed this Hilbert space with the norm $\begin{matrix} ‖ V ‖_{V^{'}} = {‖ v^{b} ‖}_{H^{1} (Ω^{b}; R^{3})} + {‖ v^{a} ‖}_{H^{1} (Ω^{a}; R^{3})} + {‖ {\hat{v}}^{0} ‖}_{L^{2} (ω; H^{1} (Y^{0}; R^{3}))} . \end{matrix}$ Now, the closed convex set $K$ is replaced by the space $V^{'}$ . Since for any $V \in V^{'}$ , $- V$ also belongs to $V^{'}$ , the problem (6.4) becomes $\begin{array}{l} Find U = (u^{b}, u^{a}, {\hat{u}}^{0}) \in V^{'} s.t. \\ \int_{Ω^{b}} a^{b} e (u^{b}) : e (v^{b}) d x + \int_{Ω^{a}} a^{a} e (u^{a}) : e (v^{a}) d x + \int_{ω \times Y^{0}} a^{M} e_{y} ({\hat{u}}^{0}) : e_{y} ({\hat{v}}^{0}) d x^{'} d y \\ (6.10) & = \int_{Ω^{b}} f \cdot v^{b} d x + \int_{Ω^{a}} f \cdot v^{a} d x, \forall V = (v^{b}, v^{a}, {\hat{v}}^{0}) \in V^{'} . \end{array}$ Introduce the Hilbert space $\begin{matrix} {\hat{V}}_{0} = {\hat{w} \in H_{per}^{1} (Y^{0}; R^{3}) | \hat{w} (y_{1}, y_{2}, 0) = \hat{w} (y_{1}, y_{2}, 1) = 0 for a.e. (y_{1}, y_{2}) \in Y^{'}} . \end{matrix}$ We consider the 3 corrector displacements ${\hat{χ}}^{i} \in L^{\infty} (ω; H_{per}^{1} (Y^{0}; R^{3}))$ , $i = 1, 2, 3$ , defined by $\begin{matrix} (6.11) & \{\begin{matrix} {\hat{χ}}^{i} (x^{'}, y_{1}, y_{2}, 1) = e_{i}, {\hat{χ}}^{i} (x^{'} y_{1}, y_{2}, 0) = 0 & for a.e. (x^{'}, y_{1}, y_{2}) \in ω \times Y^{'}, \\ \int_{Y^{0}} a^{M} e_{y} ({\hat{χ}}^{i}) : e_{y} (ψ) d y = 0 & for a.e. x^{'} \in ω, \forall ψ \in {\hat{V}}_{0} . \end{matrix} \end{matrix}$ The displacements $e_{i} - {\hat{χ}}^{i} \in L^{\infty} (ω; H_{per}^{1} (Y^{0}; R^{3}))$ , $i = 1, 2, 3$ , satisfy $\begin{matrix} \{\begin{matrix} (e_{i} - {\hat{χ}}^{i}) (x^{'}, y_{1}, y_{2}, 1) = 0, (e_{i} - {\hat{χ}}^{i}) (x^{'}, y_{1}, y_{2}, 0) = e_{i} & for a.e. (x^{'}, y_{1}, y_{2}) \in ω \times Y^{'}, \\ \int_{Y^{0}} a^{M} e_{y} (e_{i} - {\hat{χ}}^{i}) : e_{y} (ψ) d y = 0 & for a.e. x^{'} \in ω, \forall ψ \in {\hat{V}}_{0} . \end{matrix} \end{matrix}$ Below we give the variational problem satisfied by $(u^{a}, u^{b})$ .

Theorem 6.2.

Let $U = (u^{b}, u^{a}, {\hat{u}}^{0})$ be the solution of ( 6.10 ). We have $\begin{matrix} (6.12) & {\hat{u}}^{0} (x^{'}, y) = \sum_{i = 1}^{3} {\hat{χ}}^{i} (x^{'}, y) u_{i | Σ}^{a} (x^{'}) + \sum_{i = 1}^{3} (e_{i} - {\hat{χ}}^{i} (x^{'}, y)) u_{i | Σ}^{b} (x^{'}) for a.e. (x^{'}, y) \in ω \times Y^{0} \end{matrix}$ and the couple $(u^{a}, u^{b})$ is the solution of the following variational problem: $\begin{matrix} (6.13) & \{\begin{matrix} Find (u^{a}, u^{b}) \in H^{1} (Ω^{a}; R^{3}) \times H_{Γ}^{1} (Ω^{b}; R^{3}) s.t. \\ \int_{Ω^{b}} a^{b} e (u^{b}) : e (v^{b}) d x + \int_{Ω^{a}} a^{a} e (u^{a}) : e (v^{a}) d x + \int_{ω} H (u_{| Σ}^{a} - u_{| Σ}^{b}) \cdot (v_{| Σ}^{a} - v_{| Σ}^{b}) d x^{'} \\ = \int_{Ω^{b}} f \cdot v^{b} d x + \int_{Ω^{a}} f \cdot v^{a} d x, \\ \forall (v^{a}, v^{b}) \in H^{1} (Ω^{a}; R^{3}) \times H_{Γ}^{1} (Ω^{b}; R^{3}), \end{matrix} \end{matrix}$ where H is the $3 \times 3$ symmetric matrix with coefficients in $L^{\infty} (ω)$ defined by $\begin{matrix} (6.14) & H_{i j} : = \int_{Y^{0}} a^{M} e_{y} ({\hat{χ}}^{i}) : e_{y} ({\hat{χ}}^{j}) d y, i, j = 1, 2, 3 . \end{matrix}$ Matrix H is the homogenized tensor of the effective outer-plane stiffness and the ${\hat{χ}}^{j}$ ’s ( $j = 1, 2, 3$ ) are the solution of the cell-problem ( 6.11 ).

Proof.

Take $\hat{w} \in {\hat{V}}_{0}$ as test-displacement in (6.10). That gives $\begin{matrix} \forall \hat{w} \in {\hat{V}}_{0}, \int_{ω \times Y^{0}} a^{M} e_{y} ({\hat{u}}^{0}) : e_{y} (\hat{w}) d x^{'} d y = 0 . \end{matrix}$ Hence, using the corrector displacements ${\hat{χ}}^{i}$ , $i = 1, 2, 3$ , we express ${\hat{u}}^{0}$ .

Let $(v^{a}, v^{b})$ be in $H^{1} (Ω^{a}; R^{3}) \times H_{Γ}^{1} (Ω^{b}; R^{3})$ , we set ${\hat{v}}^{0} = \sum_{i = 1}^{3} {\hat{χ}}^{i} v_{i | Σ}^{a} + \sum_{i = 1}^{3} (e_{i} - {\hat{χ}}^{i}) v_{i | Σ}^{b}$ . We consider $V = (v^{b}, v^{a}, {\hat{v}}^{0})$ as a test function in (6.10). Then we develop $\int_{ω \times Y^{0}} a^{M} e_{y} ({\hat{u}}^{0}) : e_{y} ({\hat{v}}^{0}) d x^{'} d y$ and we obtain (6.13). □

Footnotes

Acknowledgement

This work was supported by Deutsche Forschungsgemeinschaft (Grants No. OR 190/4-2 and OR 190/6-1).

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Homogenization via unfolding in periodic layer with contact

Abstract

Keywords

1. Introduction

1.1. Notations

2. Geometric statement of the problem

3.1. The boundary-layer unfolding operator T ε , T ε b l , j

Proposition 5.1 (Estimate for minimizing sequences of E ε ).

6.1. Hypotheses on g ε j , G ε j and f ε j

6.2. The limit problem

1 This condition means ∀ ϕ ∈ C c ( ω × S j ) , s.t. ϕ ( x ′ , y ) ⩾ 0 on ω × S j , ⟨ ϕ , r V j · ν ⟩ C c ( ω × S j ) , M ( ω × S j ) ⩽ ∫ ω × S j ( g j − [ V ν ] S j ) ϕ d x ′ d σ ( y ) , j = 1 , … , m .

Footnotes

Acknowledgement

References

3.1. The boundary-layer unfolding operator $T_{ε}$ , $T_{ε}^{b l, j}$

Proposition 5.1 (Estimate for minimizing sequences of $E_{ε}$ ).

6.1. Hypotheses on $g_{ε}^{j}$ , $G_{ε}^{j}$ and $f_{ε}^{j}$