1 Introduction, notations and setting of the problem

The aim of this work is the study of the asymptotic behavior of the solutions of the linearized system of elasticity posed in a cylindrical domain \( \Omega _\varepsilon :=\omega \times \varepsilon {\mathrm I}\) of \(\mathbb R^3\) which is the configuration domain of a composite material. The material is made up of two components with high contrast: the first one \(F_\varepsilon \) representing the stiff part of the material has an elasticity tensor with coefficients of order 1. The second component \(M_\varepsilon \) (the soft material) surrounds the first one and the coefficients of its elasticity tensor are of order \(\varepsilon ^2\).

Under an appropriate assumption on the volumes forces, we aim to approximate the behavior of the displacements and that of the associated tensors as the small parameter \(\varepsilon \) tends to zero. Hence the present work may be viewed as the \(3d-2d\) version of the study addressed in [14] where the \(3d-1d\) reduction of dimension problem was considered as well as the homogenization of a \(\varepsilon \)-periodic fibered medium inducing a local \(3d-1d\) reduction of dimension. It was proved in [14] that the homogenized problem in a such setting is a copy of the one-dimensional problem obtained in the \(3d-1d\) study. Although one can also consider here the \(\varepsilon \)-periodic homogenization problem of a medium containing \( \frac{1}{\varepsilon }\) cells which are the translates of \(\Omega _\varepsilon \) in such a way that the homogenization process leads to a local \(3d-2d\) reduction of dimension, for the sake of brevity we restrict ourselves to the \(3d-2d\) reduction of dimension problem arising in the single composite structure \(\Omega _\varepsilon \). We consider the critical case where the ratio between the elasticity coefficients of the two components is equal to \( \frac{1}{\varepsilon ^2}\) but other scalings may be considered as pointed out in [14], see also [11, 16].

On the other hand, we deal with general elasticity tensors including anisotropic materials, see also [4, 8, 13, 15]. Several studies on composite materials with hight contrast between their components have been performed during the last years, see for instance [3, 5,6,7, 10,11,12,13]. The founding work studying media with high contrasting properties is the reference [2]. It is known that for this kind of materials, the limit problem has in general a different structure than the starting problem. In particular, nonlocal phenomena can appear at the limit. We show here that the limit problem obtained after reduction of dimension \(3d-2d\) is indeed a nonlocal problem. Theorem 3.2 below, which gives the limit problem, shows that the displacements in the circular layer F are essentially of Kirchov-Love type and the associated equation may be obtained from system (29) by choosing \(\bar{z} = 0\); but to determine the limit displacements in the structure, another equation related to the matrix M (the outside of the layer) is necessary. That equation is obtained by choosing \(\bar{u} = \bar{v} = 0\) in (29). This nonlocal phenomenon is emphasized in Theorem 3.4 which gives the limit of the average of the displacements \(\hat{u}^\varepsilon \) in the three-dimensional structure \(\Omega _\varepsilon \). In particular convergences (33) and (34) show clearly that the limit displacements are determined after solving the two equations posed in F and in M respectively.

In terms of correctors, our result shows that the transversal displacements \(u^\varepsilon _\alpha \) behave as \( u_\alpha + \varepsilon v_\alpha + z_\alpha \) while the horizontal displacements \(u^\varepsilon _3\) behave as \(u_3 + \varepsilon ^2 v_3 + \varepsilon z_3. \)

We now make more precise the notations we will use throughout the paper.

A vector x in \({{\mathbb {R}}}^3\) is denoted by \(x=(x', x_3)\) where \(x_3\) denotes the vertical coordinate. Latin indices will usually range from 1 to 3 and Greek ones take values in \(\{1,2\}\); the summation convention applies whenever indices are repeated. We write \(\partial _{i}:=\frac{\partial }{\partial x_i}\) and \(\partial _{ij}:=\frac{\partial ^2}{\partial x_i\partial x_j}\). The gradient with respect to x and \(x'\) are denoted by \(\nabla \) and \(\nabla '\) respectively.

Given any \(\phi \in (\mathcal {D}'(\Omega ))^3\), we denote the strain of \(\phi \) by \(\mathbf { \mathtt E } \phi :=\textrm{sym}\nabla \phi =\frac{1}{2}\left( \nabla \phi +{\nabla \phi }^{\text{ T }} \right) \). We shall also use the following matrices notations

Let \(\omega \subset {{\mathbb {R}}}^{2}\) be an open, bounded, simply connected set with Lipschitz boundary \(\partial {\omega }\). Let \({\mathrm I}:=(-\tfrac{1}{2},\tfrac{1}{2})\), \({\mathrm J}:=(-\tfrac{r}{2},\tfrac{r}{2}),\ 0<r<\tfrac{1}{2}\), \(\Omega :=\omega \times {\mathrm I}\), \(F:=\omega \times {\mathrm J}\), \(M:=\omega \times \left( {\mathrm I}{\setminus } \overline{\mathrm J}\right) \), where \(\overline{\mathrm J}\) denotes the closure of \({\mathrm J }\). According to Remark 1.1 below, physically we can think of \(\Omega \) as the reference configuration of a rescaled thin plate \(\Omega _\varepsilon \) reinforced by the layer \(F_\varepsilon \) while its complement, the matrix \(M_\varepsilon \), is occupied by a soft material. For every \(\varepsilon >0\) we denote the diagonal matrix whose entries are 1, 1 and \(\varepsilon \) by \(R^\varepsilon :=\textrm{diag}(1,1,\varepsilon )\), then the scaled gradient \(\nabla ^\varepsilon \phi \) and the scaled strain \(\mathbf { \mathtt E } ^\varepsilon \phi \) are defined respectively by

(1)

and

(2)

We are now in position to state the problem.

Let \( {{\mathbb {A}}}\in L^\infty (\Omega )\) be a symmetric fourth-order tensor field. We assume that \( {{\mathbb {A}}}\) fulfills the following assumptions:

$$\begin{aligned} \left\{ \begin{array}{ll}{{{\mathbb {A}}}}_{ijkl}={{{\mathbb {A}}}}_{jikl}={{{\mathbb {A}}}}_{klij}, \ \mathrm {a.e.}\ \textrm{in}\ \Omega , \\ \exists \ C>0,\ {{{\mathbb {A}}}}_{ijkl}\xi _{kl}\xi _{ij}\ge C \xi _{ij}\xi _{ij},\ \forall \xi \in {{\mathbb {R}}}^{9}\ \mathrm {s.t.}\ \xi ^T=\xi .\end{array} \right. \end{aligned}$$
(3)

We shall assume that the plate is clamped at the lateral boundary of \(\Omega \) and subjected to body forces \(f\in L^2(\Omega ;{{\mathbb {R}}}^3)\), we thus set

$$\begin{aligned} H^1_L(\Omega ):=\{u\in H^1(\Omega ;{{\mathbb {R}}}^3): u=0\quad \textrm{on}\ \partial \omega \times {\mathrm I}\}. \end{aligned}$$

Consider the displacement field \(u^\varepsilon \) solution of the following system:

$$\begin{aligned} \left\{ \begin{array}{ll} u^\varepsilon \in H^1_L(\Omega ), \ \forall \phi \in H^1_L(\Omega ),\\ \displaystyle \int _{\Omega }\!\!\left( \chi _{F}+\varepsilon ^2 \chi _{M}\right) {{{\mathbb {A}}}} \mathbf { \mathtt E } ^\varepsilon (u^\varepsilon \!). \mathbf { \mathtt E } ^\varepsilon ({\phi }) dx = \int _\Omega f \phi dx. \end{array} \right. \end{aligned}$$
(4)

By virtue of the assumptions on the tensor \({{\mathbb {A}}}\) and the body forces F, for every \(\varepsilon >0\), the problem (4) admits a unique solution by the Lax-Milgram Theorem.

Remark 1.1

As in [9], the homothety along the vertical axis defined by \(r^\varepsilon (x):=(x',{\varepsilon }{x_3})\) transforms \(\Omega \), F and M respectively into \(\Omega _\varepsilon :=\omega \times \varepsilon {\mathrm I}\), \(F_\varepsilon :=\omega \times \varepsilon {\mathrm J}\) and \(M_\varepsilon :=\omega \times \varepsilon \left( {\mathrm I}{\setminus } \overline{\mathrm J}\right) \). Then, the problem (4) is the variational version in the fixed reference configuration \(\Omega \) of the elasticity problem (6) below posed in the variable thin domain \(\Omega _\varepsilon \). Indeed, for any \(v: \Omega \longmapsto {{\mathbb {R}}}^3\) we define \({\hat{v}}: \Omega _\varepsilon \longmapsto {{\mathbb {R}}}^3\) by

$$\begin{aligned} {\hat{v}}(x', \varepsilon x_3):=\left( (R^\varepsilon )^{-1}v\circ (r^\varepsilon )^{-1}\right) (x', \varepsilon x_3)=\left( v_\alpha (x), \tfrac{1}{\varepsilon }v_3(x)\right) . \end{aligned}$$
(5)

In addition

$$\begin{aligned} \nabla {\hat{v}}=\mathbf { \mathtt H } ^\varepsilon v\circ (r^\varepsilon )^{-1},\quad \mathbf { \mathtt E } {\hat{v}}=\mathbf { \mathtt E } ^\varepsilon v\circ (r^\varepsilon )^{-1}. \end{aligned}$$

With these new unknowns, the problem (4) may be rewritten as

$$\begin{aligned} \left\{ \begin{array}{ll} {\hat{u}}^\varepsilon \in H^1_L(\Omega _\varepsilon ), \ \forall \phi \in H^1_L(\Omega _\varepsilon ),\\ \displaystyle \int _{\Omega _\varepsilon }\!\!\left( \chi _{F_\varepsilon }+\varepsilon ^2 \chi _{M_\varepsilon }\right) {{{\mathbb {A}}}}^\varepsilon \mathbf { \mathtt E } ({\hat{u}}^\varepsilon \!). \mathbf { \mathtt E } ({\phi }) dx = \int _{\Omega _\varepsilon } f^\varepsilon \phi dx \end{array} \right. \end{aligned}$$
(6)

where \(H^1_L(\Omega _\varepsilon ):=\{u\in H^1(\Omega _\varepsilon ;{{\mathbb {R}}}^3): u=0\quad \textrm{on}\ \partial \omega \times {\varepsilon \mathrm I}\}\), \( {{{\mathbb {A}}}}^\varepsilon := {{{\mathbb {A}}}}\circ (r^\varepsilon )^{-1},\quad f^{\varepsilon }=R^\varepsilon f\circ (r^\varepsilon )^{-1}. \) Thus, the components of the loads are

$$\begin{aligned} f^{\varepsilon }=\left( f_\alpha \circ (r^\varepsilon )^{-1}, {\varepsilon } f_3\circ (r^\varepsilon )^{-1})\right) . \end{aligned}$$
(7)

We study the behavior as \( \varepsilon \rightarrow 0 \) of the sequence \(\{ u^\varepsilon \}\), solution of (4), through the forthcoming steps. That behavior will be described through Theorem 3.2 and then we will deduce the behavior of the sequence \(\hat{u}^\varepsilon \) solution of (6) in Theorem 3.4.

2 A priori estimates

First, we shall recall a rescaled Korn’s inequality proved in [1]

$$\begin{aligned} \Vert \mathbf { \mathtt E } {\hat{v}}\Vert ^2_{L^2(F_\varepsilon )}\ge C\varepsilon ^2\left( \Vert {\hat{v}}\Vert ^2_{L^2(F_\varepsilon )}+\Vert \nabla {\hat{v}}\Vert ^2_{L^2(F_\varepsilon )} \right) ,\ \forall v\in H^1_L(F), \end{aligned}$$
(8)

where

$$\begin{aligned} H^1_L(F):=\{u\in H^1(F;{{\mathbb {R}}}^3): u=0\quad \textrm{on}\ \partial \omega \times {\mathrm J}\} \end{aligned}$$

and \({\hat{v}}=(R^\varepsilon )^{-1}v\circ (r^\varepsilon )^{-1}\). As a consequence, we deduce

$$\begin{aligned} \Vert \mathbf { \mathtt E } ^\varepsilon {v}\Vert ^2_{L^2(F)}\ge C\varepsilon ^2\left( \Vert {(R^\varepsilon )^{-1} v}\Vert ^2_{L^2(F)}+\Vert \nabla ^\varepsilon {v}\Vert ^2_{L^2(F)} \right) ,\ \forall v\in H^1_L(F). \end{aligned}$$
(9)

On the other hand, repeating the same arguments used in [14, Lemma 4.4], the following inequality holds true

$$\begin{aligned} \left( \Vert \nabla ^\varepsilon {v}\Vert ^2_{L^2(F)}+ \Vert \mathbf { \mathtt E } ^\varepsilon {v}\Vert ^2_{L^2(M)}\right) \ge C\Vert \nabla ^\varepsilon {v}\Vert ^2_{L^2(M)},\ \forall v\in H^1_L(\Omega ). \end{aligned}$$
(10)

For \(v\in H^1_L(\Omega )\) we set \(v':=(v_1,v_2)\), \(\nabla '{v'}:=\left( \partial _\beta v_\alpha \right) _{\alpha ,\beta =1,2}\) and

$$\begin{aligned} \mathcal {F}_\varepsilon (v):=\int _{\Omega }\!\!\left( \chi _{F}+\varepsilon ^2 \chi _{M}\right) {{{\mathbb {A}}}} \mathbf { \mathtt E } ^\varepsilon {v}. \mathbf { \mathtt E } ^\varepsilon {v} dx. \end{aligned}$$
(11)

By using the ellipticity assumption (3) we obtain, for all \(v\in H^1_L(\Omega )\)

$$\begin{aligned} \mathcal {F}_\varepsilon (v)\ge C\left( \Vert \mathbf { \mathtt E } ^\varepsilon {v}\Vert ^2_{L^2(F)}+\varepsilon ^2\Vert \mathbf { \mathtt E } ^\varepsilon {v}\Vert ^2_{L^2(M)} \right) \ge C \Vert \mathbf { \mathtt E } {v}\Vert ^2_{L^2(F)}\ge C \Vert {v}\Vert ^2_{H^1(F)}, \end{aligned}$$
(12)

where the last inequality follows from the Korn’s inequality. Furthermore, by virtue of (9) and (10) one has

$$\begin{aligned} \mathcal {F}_\varepsilon (v)\ge C\varepsilon ^2\left( \Vert \nabla ^\varepsilon {v}\Vert ^2_{L^2(F)}+\Vert \mathbf { \mathtt E } ^\varepsilon {v}\Vert ^2_{L^2(M)} \right) \ge C\varepsilon ^2 \Vert \nabla ^\varepsilon {v}\Vert ^2_{L^2(M)}. \end{aligned}$$
(13)

From (13), one has for \(\varepsilon \) small enough,

$$\begin{aligned} \mathcal {F}_\varepsilon (v)\ge C\varepsilon ^2\left( \tfrac{1}{\varepsilon ^2}\Vert \nabla '{v_3}\Vert ^2_{L^2(M)}+\tfrac{1}{\varepsilon ^4}\Vert \partial _3{v_3}\Vert ^2_{L^2(M)} \right) \ge C \Vert \nabla {v_3}\Vert ^2_{L^2(M)}. \end{aligned}$$
(14)

By means of the Poincaré’s inequality, one has

$$\begin{aligned} \mathcal {F}_\varepsilon (v)\ge C \Vert {v_3}\Vert ^2_{H^1(M)}. \end{aligned}$$
(15)

Likewise, it holds that

$$\begin{aligned} \mathcal {F}_\varepsilon (v)\ge C\left( {\varepsilon ^2}\Vert \nabla '{v'}\Vert ^2_{L^2(M)}+\Vert \partial _3{v'}\Vert ^2_{L^2(M)} \right) \ge C \Vert \partial _3 {v'}\Vert ^2_{L^2(M)}. \end{aligned}$$
(16)

From the following Poincaré’s type inequality

$$\begin{aligned} \Vert {v'}\Vert ^2_{L^2(M)}\le C\left( \Vert {v'}\Vert ^2_{L^2(F)}+\Vert \partial _3{v'}\Vert ^2_{L^2(M)} \right) , \end{aligned}$$

(16) and (12), we get the following inequality

$$\begin{aligned} \mathcal {F}_\varepsilon (v)\ge C \Vert {v'}\Vert ^2_{L^2(M)}. \end{aligned}$$
(17)

It follows from (15) and (17) that

$$\begin{aligned} \mathcal {F}_\varepsilon (v)\ge C\left( \Vert {v_3}\Vert ^2_{H^1(M)}+ \Vert {v'}\Vert ^2_{L^2(\omega ;H^1(\textrm{I}{\setminus }\overline{\textrm{J}}))}\right) . \end{aligned}$$
(18)

As an immediate consequence of the previous inequalities, the following apriori estimates hold true.

Theorem 2.1

Let \(u_\varepsilon \) be the solution of problem (4). Then

$$\begin{aligned} \sup _\varepsilon \left( \Vert {u^\varepsilon }\Vert _{H^1(F)}+ \Vert \varepsilon \nabla ^\varepsilon {u^\varepsilon }\Vert _{L^2(F)}+ \Vert \mathbf { \mathtt E } ^\varepsilon {u^\varepsilon }\Vert _{L^2(F)} \right) \le C, \end{aligned}$$
(19)
$$\begin{aligned} \sup _\varepsilon \left( \Vert {u_3^\varepsilon }\Vert _{H^1(M)}+\Vert {u_\alpha ^\varepsilon }\Vert _{L^2(\omega ;H^1(\textrm{I}{\setminus }\overline{\textrm{J}}))}+ \Vert \varepsilon \nabla ^\varepsilon {u^\varepsilon }\Vert _{L^2(M)} \right) \le C, \end{aligned}$$
(20)

in particular

$$\begin{aligned} \sup _\varepsilon \left( \Vert {u_3^\varepsilon }\Vert _{H^1(\Omega )}+\Vert {u_\alpha ^\varepsilon }\Vert _{L^2(\omega ;H^1(\textrm{I}))}+ \Vert \varepsilon \nabla ^\varepsilon {u^\varepsilon }\Vert _{L^2(\Omega )} \right) \le C,\ \alpha =1,2. \end{aligned}$$
(21)

Proof

Taking \(\phi =u^\varepsilon \) in the problem (4), we derive

$$\begin{aligned} \mathcal {F}_\varepsilon (u^\varepsilon )\le \Vert f\Vert _{L^2(\Omega )} \Vert {u^\varepsilon }\Vert _{L^2(\Omega )}, \end{aligned}$$

while from (12), (15) and (17) we find

$$\begin{aligned} \mathcal {F}_\varepsilon (u^\varepsilon )\ge C \Vert {u^\varepsilon }\Vert _{L^2(\Omega )}. \end{aligned}$$

Thus

$$\begin{aligned} \sup _\varepsilon \mathcal {F}_\varepsilon (u^\varepsilon )\le C. \end{aligned}$$

Therefore, making use of (12)–(13) and then (13)–(18) we deduce (19) and (20) respectively.

3 Convergence results

We first define the following functional spaces:

$$\begin{aligned} H_{KL}(\Omega ):=\left\{ u\in H_L^1(\Omega ) : (\mathbf { \mathtt E } u)_{i3} =0, i=1,2,3\right\} \end{aligned}$$

which is the space of Kirchhoff-Love displacements on \(\Omega \); it can be characterized also as

$$\begin{aligned} \!\!H_{KL}(\Omega )\!=\!\left\{ u\in \!\! H_L^1(\Omega ),\exists (g_\alpha ,g_3)\in H_0^1(\omega )\!\times \!H_0^2(\omega )\!:u_\alpha \!=g_\alpha -x_3\partial _\alpha g_3, u_3=g_3\right\} , \end{aligned}$$

for the sake of brevity we set \({\mathcal {U}}:=H_{KL}(\Omega )\),

where \(H_m^1(\textrm{I}):=\left\{ \psi \in H^1(\textrm{I}) : \displaystyle \int _{\textrm{I}}\psi (t)dt=0 \right\} .\)

The following lemma is concerned with some convergence results.

Lemma 3.1

There exists \((u,v,z)\in \mathcal {U}\times \mathcal {V}\times \mathcal {Z}\), such that up to a subsequence, we have

Proof

Step 1: convergence of \((u^\varepsilon )_\varepsilon \).

From estimate (19) which implies that \(u_\alpha ^\varepsilon \) is bounded in \(H^1(F)\) and from estimate (20) which implies that \(u_\alpha ^\varepsilon \) is bounded in \(L^2(\omega ; H^1(I{\setminus } \overline{J}))\) for \(\alpha = 1,2,\) we conclude that there exist \(u_\alpha \in H^1(\Omega )\) such that for a subsequence

$$\begin{aligned} u_\alpha ^\varepsilon \rightharpoonup u_\alpha \ \textrm{in} \ H^1(F), \quad (\mathbf { \mathtt E } ^\varepsilon u^\varepsilon )_{\alpha \beta }\chi _F \rightharpoonup (\mathbf { \mathtt E } u)_{\alpha \beta }\chi _F \ \textrm{in} \ L^2(F). \end{aligned}$$
(22)

On the other hand, from (21) we get that up to a subsequence,

$$\begin{aligned} \left\{ \begin{array}{ll} u_\alpha ^\varepsilon \rightharpoonup \tilde{u}_\alpha \, \, \ \textrm{in} \ L^2(\omega ; H^1(\textrm{I})), \\ u_3^\varepsilon \rightharpoonup u_3 \, \, \ \textrm{in} \ H^1(\Omega ). \end{array} \right. \end{aligned}$$
(23)

Moreover, due to the estimate \(\displaystyle \Vert \tfrac{1}{\varepsilon }\partial _3 u_3^\varepsilon \Vert _{L^2(\Omega )}\le C\) which is a consequence of (21), we have

$$\begin{aligned} \partial _3 u_3^\varepsilon \rightarrow 0 \ \textrm{in} \ L^2(\Omega ) \end{aligned}$$

in such a way that

$$\begin{aligned} \partial _3 u_3=0. \end{aligned}$$

Thus, there exists \(g_3 \in H_0^1(\omega )\) such that

$$\begin{aligned} u_3(x)=g_3(x') \ \mathrm {for\ a. e.} \ x \in \Omega . \end{aligned}$$
(24)

Comparing the first convergences in (22) and (23), we derive the equality

$$\begin{aligned} \tilde{u}_\alpha = u_\alpha \quad \hbox {in} \, F. \end{aligned}$$
(25)

On the other hand, using (19) we infer

$$\begin{aligned} (\mathbf { \mathtt E } u^\varepsilon )_{\alpha 3} \rightharpoonup (\mathbf { \mathtt E } \tilde{u})_{\alpha 3} \, \ \textrm{in} \ L^2(F). \end{aligned}$$

Since

$$\begin{aligned} (\mathbf { \mathtt E } u^\varepsilon )_{i3} \rightarrow 0 \ \textrm{in} \ L^2(F), \end{aligned}$$

we are led to

$$\begin{aligned} (\tilde{u}_\alpha ,u_3)\in H_{KL}(F). \end{aligned}$$

By definition of the space \(H_{KL}(F)\), the component \(u_3\) belongs to \( H_0^2(\omega )\), so that equality (24) implies that \(g_3 \in H_0^2(\omega )\). Hence, there exist \(g_\alpha \in H_0^1(\omega )\) for \( \alpha =1,2,\) such that

$$\begin{aligned} \tilde{u}_\alpha (x)=g_\alpha (x')-x_3\partial _\alpha g_3(x'),\quad u_3(x)=g_3(x')\ \mathrm {for\ a.\ e.}\ x\in F. \end{aligned}$$

For a. e. \( x\in \Omega \), we set

$$\begin{aligned} u_\alpha (x):=g_\alpha (x')-x_3\partial _\alpha g_3(x')\in H^1_L(\Omega ),\\ z_\alpha (x):=\tilde{u}_\alpha (x)-u_\alpha (x)\in L^2(\omega ; H^1(\textrm{I})), \end{aligned}$$

so that \(z_\alpha =0, \ \mathrm {a.\ e.\ in} \ F\) as seen from (25) and \(u=(u_\alpha ,u_3)\in \mathcal {U} \) and

$$\begin{aligned} u_\alpha ^\varepsilon \rightharpoonup u_\alpha +z_\alpha , \ \textrm{in} \ L^2(\omega ; H^1(\textrm{I})). \end{aligned}$$

Step 2: convergence of \((\mathbf { \mathtt E } ^\varepsilon u^\varepsilon )_\varepsilon \).

To identify the limit of the sequence \((\mathbf { \mathtt E } ^\varepsilon u^\varepsilon )_{i3}\chi _F\), we introduce the following sequences:

$$\begin{aligned} \hat{u}_\alpha ^\varepsilon:= & {} \int _{-\tfrac{1}{2}}^{x_3}\tfrac{1}{\varepsilon }\partial _\alpha u^\varepsilon _3(x',t) dt, \end{aligned}$$
(26)
(27)

Obviously, on has

$$\begin{aligned} \partial _3 v_\alpha ^\varepsilon= & {} \tfrac{2}{\varepsilon }(\mathbf { \mathtt E } u^\varepsilon )_{\alpha 3},\\ \partial _3 v_3^\varepsilon= & {} \tfrac{1}{\varepsilon ^2}(\mathbf { \mathtt E } u^\varepsilon )_{33}. \end{aligned}$$

Since \(\tfrac{2}{\varepsilon }(\mathbf { \mathtt E } u^\varepsilon )_{\alpha 3}\) and \( \tfrac{1}{\varepsilon ^2}(\mathbf { \mathtt E } u^\varepsilon )_{33} \) are bounded in \(L^2(F)\), due to (19), and \(v^\varepsilon :=(v_\alpha ^\varepsilon ,v_3^\varepsilon )\) has mean-value zero with respect to \(x_3\), the sequence \(v^\varepsilon \) is bounded in \(L^2(\omega ; H_m^1(\textrm{J}))^3\), where

Therefore there exists some \(v\in L^2(\omega ; H_m^1(\textrm{J}))^3\) such that

$$\begin{aligned} v^\varepsilon \rightharpoonup v\ \textrm{in} \ L^2(\omega ; H_m^1(\textrm{J}))^3. \end{aligned}$$

In particular, one has \(v\in \mathcal {V}\),

$$\begin{aligned} (\mathbf { \mathtt E } ^\varepsilon u^\varepsilon )_{\alpha 3}\chi _F \rightharpoonup \tfrac{1}{2}\partial _3 v_\alpha \chi _F \ \textrm{in} \ L^2(\Omega ), \end{aligned}$$

and

$$\begin{aligned} (\mathbf { \mathtt E } ^\varepsilon u^\varepsilon )_{33}\chi _F \rightharpoonup \partial _3 v_3\chi _F \ \textrm{in} \ L^2(\Omega ). \end{aligned}$$

As a consequence, one obtains the following convergence

where \((u,v)\in \mathcal {U}\times \mathcal {V}\).

We now seek for the limit of the sequence \(\varepsilon \mathbf { \mathtt E } ^\varepsilon u^\varepsilon \chi _M\). To that aim, we consider the following sequence

$$\begin{aligned} z_\alpha ^\varepsilon:= & {} u_\alpha ^\varepsilon + \hat{u}_\alpha ^\varepsilon -\int _{\textrm{I}}\left( u_\alpha ^\varepsilon + \hat{u}_\alpha ^\varepsilon \right) d{x_3},\nonumber \\ z^\varepsilon _3:= & {} \tfrac{1}{\varepsilon }u_3^\varepsilon -\int _{\textrm{I}}\tfrac{1}{\varepsilon }u_3^\varepsilon d{x_3}. \end{aligned}$$
(28)

where \(\hat{u}^\varepsilon \) is defined by (26). Since

$$\begin{aligned} \partial _3 z_\alpha ^\varepsilon= & {} {2}(\mathbf { \mathtt E } u^\varepsilon )_{\alpha 3}, \end{aligned}$$
(29)
$$\begin{aligned} \partial _3 z_3^\varepsilon= & {} \tfrac{1}{\varepsilon }(\mathbf { \mathtt E } u^\varepsilon )_{33}. \end{aligned}$$
(30)

Due to (21), \( {2}(\mathbf { \mathtt E } u^\varepsilon )_{\alpha 3}\) and \( \tfrac{1}{\varepsilon }(\mathbf { \mathtt E } u^\varepsilon )_{33} \) are bounded in \( L^2(\Omega )\) with mean-value zero with respect to \(x_3\); hence the sequence \(z^\varepsilon \) is bounded in \(L^2(\omega ; H_m^1(\textrm{I}))^3\). Therefore there exists some \(\hat{z}\in L^2(\omega ; H_m^1(\textrm{I}))^3\), such that

$$\begin{aligned} z^\varepsilon \rightharpoonup \hat{z}\ \textrm{in} \ L^2(\omega ; H_m^1(\textrm{I}))^3. \end{aligned}$$

We claim that \( \hat{z}_\alpha = {z}_\alpha \). Indeed,

$$\begin{aligned} \partial _3 u_\alpha ^\varepsilon \rightharpoonup \partial _3 u_\alpha +\partial _3 z_\alpha \ \textrm{in} \ L^2(\Omega ) \end{aligned}$$

and

$$\begin{aligned} \partial _3 z_\alpha ^\varepsilon \rightharpoonup \partial _3\hat{z}_\alpha \ \textrm{in} \ L^2(\Omega ). \end{aligned}$$

But

$$\begin{aligned} \partial _3 z_\alpha ^\varepsilon =\partial _3 u_\alpha ^\varepsilon +\partial _\alpha u_3^\varepsilon \rightharpoonup \partial _3u_\alpha + \partial _3z_\alpha +\partial _\alpha u_3 \ \textrm{in} \ L^2(\Omega ). \end{aligned}$$

Thus

$$\begin{aligned} \partial _3\hat{z}_\alpha =\partial _3u_\alpha + \partial _3z_\alpha +\partial _\alpha u_3. \end{aligned}$$

Since \(u\in \mathcal {U}\), we have

$$\begin{aligned} \partial _3u_\alpha +\partial _\alpha u_3=0, \end{aligned}$$

and therefore

$$\begin{aligned} \partial _3\hat{z}_\alpha = \partial _3z_\alpha \ \mathrm {i.e.}\ \hat{z}_\alpha (x)= z_\alpha (x)+c(x'),\ \mathrm {a.e.}\ x\in \Omega . \end{aligned}$$

In particular

$$\begin{aligned} \hat{z}_\alpha (x)=c(x'),\ \mathrm {a.e.}\ x\in F \end{aligned}$$

since

$$\begin{aligned} z_\alpha (x)=0,\ \mathrm {a.e.}\ x\in F. \end{aligned}$$

On the other hand one has by virtue of (20),

$$\begin{aligned} z^\varepsilon \rightarrow 0\ \textrm{in} \ L^2(F). \end{aligned}$$

Hence

$$\begin{aligned} c(x')=0,\ \mathrm {a.e.}\ x\in F \end{aligned}$$

and therefore

$$\begin{aligned} \hat{z}_\alpha (x)={z}_\alpha (x),\ \mathrm {a.e.}\ x\in \Omega . \end{aligned}$$

This proves our claim.

Finally, from (29) and (30) we deduce that

$$\begin{aligned} {\varepsilon }(\mathbf { \mathtt E } ^\varepsilon u^\varepsilon )_{\alpha 3}\chi _M \rightharpoonup \tfrac{1}{2}\partial _3 z_\alpha \chi _M,\ {\varepsilon }(\mathbf { \mathtt E } ^\varepsilon u^\varepsilon )_{3 3}\chi _M \rightharpoonup (\mathbf { \mathtt E } z)_{3 3}\chi _M\ \textrm{in} \ L^2(\Omega ), \end{aligned}$$

where \(z_3:=\hat{z}_3\).

Moreover, since \(\varepsilon u_\alpha ^\varepsilon \longrightarrow 0\) in \(L^2(M)\), we get \({\varepsilon }(\mathbf { \mathtt E } ^\varepsilon u^\varepsilon )_{\alpha \beta }\chi _M \rightharpoonup 0\ \textrm{in} \ L^2(\Omega ).\) We conclude that

This ends the proof of Lemma 3.1.

In the following theorem, we state our main result.

Theorem 3.2

There exists \((u,v,z)\in \mathcal {U}\times \mathcal {V}\times \mathcal {Z}\), such that the sequence of solution \(u^\varepsilon \) of the problem (4) fulfills the following strong convergences

where the limit (u, v, z) is the unique solution of the problem

(31)

Proof

We first prove the convergence of the sequence of energies to the energy associated to the limit problem, assuming that the limit problem is the system (31). To that aim we will use the weak convergences proved in Lemma 3.1. We follow the argument already used in [14, Proof of Theorem 3.1].

Choosing \((\bar{u},\bar{v},\bar{z})=(u,v,z)\) in (31) and passing to the limit thanks to Lemma 3.1, we get

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \displaystyle \int _{\Omega }\!\!\left( \chi _{F}+\varepsilon ^2 \chi _{M}\right) {{{\mathbb {A}}}} \mathbf { \mathtt E } ^\varepsilon (u^\varepsilon \!). \mathbf { \mathtt E } ^\varepsilon ({u^\varepsilon \!}) dx= & {} \lim _{\varepsilon \rightarrow 0} \int _\Omega f u^\varepsilon \! dx\\= & {} \displaystyle \int _{\Omega }\Bigl ( f_\alpha \left( u_\alpha + z_\alpha \right) + f_3 u_3 \Bigr )dx\\= & {} \int _\Omega \left( AE_F.E_F\chi _F+ AE_M.E_M\chi _M\right) dx, \end{aligned}$$

where

Using the equivalence of the norms

$$\begin{aligned} L^2(\Omega )\ni E\longmapsto \int _\Omega AE.Edx, L^2(\Omega )\ni E\longmapsto \int _\Omega E.Edx \end{aligned}$$

we deduce that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\left\| \left( \chi _{F}+\varepsilon \chi _{M}\right) \mathbf { \mathtt E } ^\varepsilon (u^\varepsilon \!)\right\| _{ L^2(\Omega )}= \left\| \chi _FE_F+\chi _ME_M\right\| _{ L^2(\Omega )}. \end{aligned}$$

Hence the strong convergence of the tensor sequences is proved. Now, by using the last strong convergence together with the following Korn’s inequality

$$\begin{aligned} \left\| \mathbf { \mathtt E } ^\varepsilon (u^\varepsilon \!)\right\| _{ L^2(F)}\ge \left\| \mathbf { \mathtt E } (u^\varepsilon \!)\right\| _{ L^2(F)}\ge C \left\| u^\varepsilon \!\right\| _{ L^2(F)}, \end{aligned}$$

we deduce that \(u^\varepsilon \) is a Cauchy sequence in \(H^1(F)\).

On the other hand, using (9) and (10) one has

$$\begin{aligned} \Vert \mathbf { \mathtt E } ^\varepsilon {u^\varepsilon \!}\Vert ^2_{L^2(F)}\ge C\varepsilon ^2\left( \Vert {(R^\varepsilon )^{-1} u^\varepsilon \!}\Vert ^2_{L^2(F)}+\Vert \nabla ^\varepsilon {u^\varepsilon \!}\Vert ^2_{L^2(F)} \right) \end{aligned}$$

and

$$\begin{aligned} \left( \Vert \nabla ^\varepsilon {u^\varepsilon \!}\Vert ^2_{L^2(F)}+ \Vert \mathbf { \mathtt E } ^\varepsilon {u^\varepsilon \!}\Vert ^2_{L^2(M)}\right) \ge C\Vert \nabla ^\varepsilon {u^\varepsilon \!}\Vert ^2_{L^2(M)}. \end{aligned}$$

Hence, \(\varepsilon \nabla ^\varepsilon {u^\varepsilon \!}\) is also a Cauchy sequence in \(L^2(\Omega )\). It remains to prove that \(u_\alpha ^\varepsilon \) and \(u_3^\varepsilon \) are two Cauchy sequences in \(L^2(\omega ; H^1(\textrm{I}))\) and \(H^1(M)\) respectively. To that aim, it suffices to use (18) to obtain

$$\begin{aligned} \left\| \left( \chi _{F}+\varepsilon \chi _{M}\right) \mathbf { \mathtt E } ^\varepsilon (u^\varepsilon \!)\right\| ^2_{ L^2(\Omega )}= \mathcal {F}_\varepsilon (u^\varepsilon )\ge C\left( \Vert {u^\varepsilon _3}\Vert ^2_{H^1(M)}+ \Vert {u_\alpha ^\varepsilon }\Vert ^2_{L^2(\omega ;H^1(\textrm{I}{\setminus }\overline{\textrm{J}}))}\right) \end{aligned}$$

then we can argue as previously to complete the proof. \(\square \)

We now prove that the limit problem is nothing but the system (31).

We choose a test function \(\phi ^\varepsilon \) in (4) in the following form

$$\begin{aligned} \left\{ \begin{array}{cc} \phi ^\varepsilon _\alpha =\bar{u}_\alpha +\varepsilon \bar{v}_\alpha +\bar{z}_\alpha \\ \phi ^\varepsilon _3=\bar{u}_3+\varepsilon ^2\bar{v}_3+\varepsilon \bar{z}_3 \end{array}\right. \end{aligned}$$
(32)

where \((\bar{u},\bar{v},\bar{z})\in \mathcal {U}\times \mathcal {D}\left( \omega ; H_m^1(J)\right) ^3\times \left\{ \mathcal {D}\left( \omega ;H_m^1(I) \right) ^3: z=0 ,\ \mathrm {a.e.}\ x\in F\right\} .\) Since, from the definition of \( \mathcal {U}\), one has \((E{\bar{u}})_{i3}=0\), for all \(i=1,2,3\), an elementary calculation shows that

$$\begin{aligned} \left\{ \begin{array}{ll} (\mathbf { \mathtt E } ^\varepsilon {\phi ^\varepsilon })_{\alpha \beta }= (\mathbf { \mathtt E } {\bar{u}})_{\alpha \beta }+\varepsilon (\mathbf { \mathtt E } {\bar{v}})_{\alpha \beta }+ (\mathbf { \mathtt E } {\bar{z}})_{\alpha \beta },\\ 2 (\mathbf { \mathtt E } ^\varepsilon {\phi ^\varepsilon })_{\alpha 3}=\partial _3{\bar{v}_\alpha }+\varepsilon \partial _\alpha {\bar{v}_3}+ \tfrac{1}{\varepsilon }\partial _3\bar{z}_\alpha +\partial _\alpha {\bar{z}_3},\\ (\mathbf { \mathtt E } ^\varepsilon {\phi ^\varepsilon })_{33}= (\mathbf { \mathtt E } {\bar{v}})_{33}+ \tfrac{1}{\varepsilon }(\mathbf { \mathtt E } {\bar{z}})_{33}. \end{array}\right. \end{aligned}$$

Using the following strong convergences

we can pass to the limit in (4) as \(\varepsilon \) goes to zero to get (31) thanks to the density of \(\mathcal {D}\left( \omega ; H_m^1(J)\right) ^3\) and \(\left\{ z\in \mathcal {D}\left( \omega ;H_m^1(I) \right) ^3: z_i=0 ,\ \mathrm {a.e.}\ x\in F,\ i=1,2,3\right\} \) in \(\mathcal {V}\) and \(\mathcal {Z}\) respectively.

Remark 3.3

The space \(\mathcal {U}\times \mathcal {V}\times \mathcal {Z}\) is a Hilbert space for the following norm

$$\begin{aligned} \left\| (u,v,z) \right\| ^2:=\sum _{\alpha ,\beta =1,2} \left\| {(\mathbf { \mathtt E } u)_{\alpha \beta }} \right\| ^2_{L^2(\Omega )}+\sum _{i=1}^3\left[ \left\| \partial _3v_i \right\| ^2_{L^2(\Omega )} +\left\| \partial _3z_i \right\| ^2_{L^2(\Omega )}\right] . \end{aligned}$$

Therefore, using the Lax-Milgram Theorem we obtain the well-posedness of the limit problem (31) since the tensor A is coercive by virtue of (3) and \(f\in L^2(\Omega ;{{\mathbb {R}}}^3)\).

We now turn back to the sequence of solutions of problem (6) to derive the following theorem from Theorem 3.2.

Theorem 3.4

Let \((u,v,z)\in \mathcal {U}\times \mathcal {V}\times \mathcal {Z}\) be the solution of the problem (31). Let

Then the sequence of solutions \({\hat{u}}^\varepsilon \) of the problem (6) fulfills the following convergences

(33)

Moreover, \(U_\alpha \) may be written as

(34)

where \(m_\alpha ^{0}\) and \(m_\alpha ^{00}\) are given by

$$\begin{aligned} m_\alpha ^{0}(x'):=\int _{\textrm{I}{\setminus }\textrm{J}}z_\alpha ^0(x',x_3)d{x_3},\ m_\alpha ^{00}(x'):=\int _{\textrm{I}{\setminus }\textrm{J}}z_\alpha ^{00}(x',x_3)d{x_3}, \end{aligned}$$

and \(z^{0}\), \(z^{00}\) are respectively the solutions of the following problems

(35)
(36)

Proof

The convergences (33) follow from the corresponding convergences stated in Theorem 3.2 by the change of variables (5).

Let us prove for instance the third convergence arising in (33) by the use of the first convergence arising in Theorem 3.2. Setting \( y_3 = \varepsilon x_3\) for \(x_3 \in I\) and bearing in mind that \( \hat{u}^\varepsilon \) is defined according to (5) by \(u^\varepsilon _\alpha (x',x_3) = \hat{u}^\varepsilon (x', \varepsilon x_3)\) for \((x', x_3) \in \omega \times I\) and that the length of the interval I is equal to 1, we get thanks to the Cauchy-Schwarz inequality (with respect to \(x_3\))

(37)

To prove (34), we first notice that one can derive the following equation satisfied by \(z_\alpha \) by choosing \({\bar{u}}={\bar{v}}=0\) in (31),

(38)

Taking advantage from the linearity of (38), one can check easily by a uniqueness argument that \(z_\alpha \) may be identified as \({z}_{\alpha }={z}_{\alpha }^1+{z}_{\alpha }^2\) where \({z}_{\alpha }^1\) and \({z}_{\alpha }^2\) are solutions of (38) but with right hand sides defined respectively by

By linearity and using once again a uniqueness argument, we conclude that

where \(z^{0}\) and \(z^{00}\) are the solutions of (35) and (36) respectively. Hence, \(U_\alpha \) defined in the third convergence (33) may be written as (34). This completes the proof of Theorem 3.4.