Two-Scale Homogenization of Piezoelectric Perforated Structures

Abstract

We are interested in the homogenization of the elastic-electric coupling equation with rapidly oscillating coefficients, in a periodically perforated piezoelectric body. The holes, whose size are supposed to tend to zero, are periodically distributed. We give a new approach, based on the two-scale convergence, and we justify the two first terms in the usual asymptotic development of the solution. A two-scale homogenized system is obtained as the limit of the periodic problem, and explicit formulae of elastic, piezoelectric and dielectric homogenized coefficients are reported. In the static limit, the method provides homogenized electroelastic coefficients coinciding with those deducted from alternative approaches.

1. Introduction

The piezoelectrical effect is the capacity exhibited by some materials to convert strain to electrical energy and electrical energy to strain. Piezoelectric materials have received significant attention due to their use in devices such as sensors, to measure strain or voltage, actuators and energy harvesters. Composites and perforated (lattice) materials are widely used in many practical applications, such as aircraft, civil engineering, electrotechnics, and many others. These are materials with a large number of heterogeneities (inclusions or holes), and in strong contrast to continum materials, their behavior is definitively influenced by micromechanical events. The increased application of composite and perforated (lattice) materials in ground-breaking Micro-Electro-Mechanical-Systems (MEMS) and Nano-Electro-Mechanical-Systems (NEMS) has stimulated a great interest in several studies (see Akdogan et al. [1], Miara et al. [2], Sirohi and Chopra [3,4,5,6]).

The composite with periodic microstructures, homogenization techniques represent a useful and advantageous method for providing a rigorous and synthetic description of the effects of microscopic phases on the overall properties of the materials. The application of these approaches makes it possible to avoid the challenging numerical computations required by computational modeling of heterogeneous media. To determine the effective properties of heterogeneous materials and structures, several nano-mechanical techniques and homogenization schemes have been developed and discussed in the literature (see Bensoussan et al. [7], Cioranescu and Donato [8], Oleinik et al. [9]). The rigorous modeling of piezoelectric composites goes back to the 1990s, in particular several works concerning the modelling of nonhomogeneous materials (see [10,11,12,13,14]) to obtain explicit formulae for the effective properties of laminated piezoelectric composite. Based on the $\mathrm{\Gamma }$-convergence Telega [15] to obtain the homogeneous tensors for the piezoelectric composites, the homogenization of this problem for of three-dimensional four-step piezoelectric braided composites has already been studied by Feng and Wu [16], and Ruan et al. [17] have studied the effective properties of 2-step braided composites with a polymeric matrix by the analytical approach.

There are many methods in the homogenization process (Cioranescu and Donato [8], Oleinik et al. [9]). Here, we are interested the two-scale convergence approach as introduced by Nguetseng [18] and Allaire [19]. The objective of this work is to illustrate the application of two-scale convergence for the determination of the effective coefficients in the case of heterogeneous piezoelectric media with rapidly oscillating coefficients in a periodically perforated domain. We obtain the limit homogeneous problem and we give a convergence results based upon the two-scale convergence approach, when the size of the microstructures (size of perforations) goes to zero. We give new convergence results concerning the same model by using the homogenization technique of two-scale convergence. We can deduce as a limit problem that the approximation of final state is altered by a constant named the volume fraction, which depends on the proportion of material in the perforated domain and which is equal to 1 when there are no holes. The second goal of this paper is to apply the technique of a formal asymptotic homogenization, to determine the effective elastic, piezoelectric and dielectric moduli of a periodic medium. The final formulae for the effective parameters are given in a relatively simplest closed-form.

2. Homogenization Problem

Throughout this paper, $L^2\left(\mathsf{\Omega }\right)$ is in the Sobolev space of real-valued functions that are measurable and square summable in $\mathsf{\Omega }$ with respect to the Lebesgue measure. We denote by $C_{♯}^{\infty }\left(Y\right)$ the space of infinitely differentiable functions in ${\mathbb{R}}^3$ that are periodic of $Y.$ The subscript ${}_{♯}$ stands for Y—periodic functions in the last variable.

2.1. Geometric of the Medium

Let $\mathsf{\Omega }\subset {\mathbb{R}}^3$ be a bounded three-dimensional domain with the boundary $\mathrm{\Gamma }=\partial \mathsf{\Omega }$. We denote by x the macroscopic variable and by $y=\frac{x}{\epsilon }$ the microscopic variable. Let us define ${\mathsf{\Omega }}_{\epsilon }$ of periodically perforated subdomains of a bounded open set $\mathsf{\Omega }$. The period of ${\mathsf{\Omega }}_{\epsilon }$ is $\epsilon Y^{*}$, where $Y^{*}$ is a subset of the unit cube $Y={(0,1)}^3$, which represented the solid or material domain, $S^{*}$ obtained by Y-periodicity from $Y^{*}$ is a smooth connected (the material is in one piece) open set in ${\mathbb{R}}^3$. Denoting by $\chi \left(y\right)$ the characteristic function of $S^{*}$ (Y-periodic), ${\mathsf{\Omega }}_{\epsilon }$ will be defined analytically by

$${\mathsf{\Omega }}_{\epsilon }=\left\{x\in \mathsf{\Omega },\chi \left(\frac{x}{\epsilon }\right)=1\right\}$$

2.2. Model Problem

We adopt the Einstein summation convention of repeated indices; we use Latin indices that are understood to be run from 1 to $3,$ we note by ${\mathbf{u}}^{\epsilon }$ the fields of displacement in elastic and by ${\phi }^{\epsilon }$ of electric potential. The equations of equilibrium and Gauss’s law of electrostatics in the absence of free charges are written as

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}-\mathbf{div}{\mathbf{œ}}^{\epsilon }({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })& =\hfill & \mathbf{f}\mathrm{in}{\mathsf{\Omega }}_{\epsilon },\hfill \\ -\mathbf{div}{\mathbf{D}}^{\epsilon }({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })& =\hfill & 0\mathrm{in}{\mathsf{\Omega }}_{\epsilon }.\hfill \end{array}\right.\end{array}$$

(1)

We complete the boundary conditions,

$$\left\{\begin{array}{cc}\hfill ({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })=(\mathbf{0},0)& \mathrm{on}\partial \mathsf{\Omega },\hfill \\ \hfill {\mathbf{œ}}^{\epsilon }({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })·n^{\epsilon }=0& \mathrm{on} \mathrm{the} \mathrm{boundary} \mathrm{of} \mathrm{holes}\partial {\mathsf{\Omega }}_{\epsilon }-\partial \mathsf{\Omega },\hfill \\ \hfill {\mathbf{D}}^{\epsilon }({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })·n^{\epsilon }=0& \mathrm{on} \mathrm{the} \mathrm{boundary} \mathrm{of} \mathrm{holes}\partial {\mathsf{\Omega }}_{\epsilon }-\partial \mathsf{\Omega },\hfill \end{array}\right.$$

(2)

where $\mathbf{f}\in {\mathbf{L}}^2\left({\mathsf{\Omega }}_{\epsilon }\right)$ (in fact $\mathbf{f}$ denotes the restriction of $\mathbf{f}$ in ${\mathsf{\Omega }}_{\epsilon }$). $n^{\epsilon }$ is the outer unit normal to the boundary of holes $\partial {\mathsf{\Omega }}_{\epsilon }-\partial \mathsf{\Omega }.$ The second-order stress tensor ${\sigma }^{\epsilon }=\left({\sigma }_{ij}^{\epsilon }\right)$ and the electric displacement vector ${\mathbf{D}}^{\epsilon }=\left(D_i^{\epsilon }\right)$ are linearly related to the second-order strain tensor $s_{kl}\left(\mathbf{u}\right)=\frac{1}{2}({\partial }_k{\mathbf{u}}_l+{\partial }_l{\mathbf{u}}_k)$ and the electric field vector ${\partial }_k{\phi }^{\epsilon }$ by the constitutive law

$$\left\{\begin{array}{cc}\hfill {\mathbf{œ}}_{ij}^{\epsilon }({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })=c_{ijkl}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+e_{kij}^{\epsilon }{\partial }_k{\phi }^{\epsilon }& \mathrm{in}{\mathsf{\Omega }}_{\epsilon },\hfill \\ \hfill D_i^{\epsilon }({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })=-e_{ikl}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+d_{ij}^{\epsilon }{\partial }_j{\phi }^{\epsilon }& \mathrm{in}{\mathsf{\Omega }}_{\epsilon }.\hfill \end{array}\right.$$

(3)

$$1\le i,j,k,l\le 3,$$

where ${\left(\mathbf{div}{\sigma }^{\epsilon }\right)}^j={\partial }_i{\sigma }_{ij}^{\epsilon },\mathbf{div}{\mathbf{D}}^{\epsilon }={\partial }_i{D}_i^{\epsilon },{\partial }_i=\frac{\partial }{\partial x_i},x=\left(x_i\right)\in \mathsf{\Omega }$. The material proprieties are given by the fourth-order stiffness tensor $c_{ijkl}^{\epsilon }$ measured at the constant electric field, and the elastic coefficients satisfy the following symmetries and ellipticity uniformly in $\epsilon$,

$$\left\{\begin{array}{c}{c}_{ijkl}^{\epsilon }\left(x\right)=c_{ijkl}(x,\frac{x}{\epsilon }),\hfill \\ c_{ijkl}^{\epsilon }=c_{jikl}^{\epsilon }=c_{klij}^{\epsilon }=c_{ijlk}^{\epsilon },\hfill \\ c_{ijkl}(x,y)\in L^{\infty }(\mathsf{\Omega };C_{♯}\left(Y\right)),\hfill \\ \exists {\alpha }_c\ne {\alpha }_c\left(\epsilon \right)>0:c_{ijkl}^{\epsilon }{X}_{ij}{X}_{kl}\ge {\alpha }_c{X}_{ij}{X}_{ij},\forall X_{ij}=X_{ji}\in \mathbb{R}.\hfill \end{array}\right.$$

(4)

The third-order piezoelectric tensor $e_{ijk}^{\epsilon }$ (the coupled tensor) verifies the following symmetries:

$$\left\{\begin{array}{c}{e}_{ijk}^{\epsilon }=e_{ijk}(x,\frac{x}{\epsilon }),\hfill \\ e_{ijk}^{\epsilon }=e_{ikj}^{\epsilon },\hfill \\ e_{ijk}(x,y)\in L^{\infty }(\mathsf{\Omega };C_{♯}\left(Y\right)).\hfill \end{array}\right.$$

(5)

The second-order electric tensor $d_{ij}^{\epsilon }$ (dielectric permittivity), measured at constant strain, verifies the conditions of symmetric and ellipticity uniformly by $\epsilon$,

$$\left\{\begin{array}{c}{d}_{ij}^{\epsilon }=d_{ij}(x,\frac{x}{\epsilon }),\hfill \\ d_{ij}^{\epsilon }=d_{ji}^{\epsilon },\hfill \\ d_{ij}(x,y)\in L^{\infty }(\mathsf{\Omega };C_{♯}\left(Y\right)),\hfill \\ \exists {\alpha }_d\ne {\alpha }_d\left(\epsilon \right)>0:d_{ij}^{\epsilon }{X}_i{X}_j\ge {\alpha }_d{X}_i{X}_i,\forall X_i\in \mathbb{R}.\hfill \end{array}\right.$$

(6)

2.3. Variational Problem

The two allowable Hilbert spaces are introduced:

$${\mathbf{V}}_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)=\{\mathbf{v}\in {\mathbf{H}}^1\left({\mathsf{\Omega }}_{\epsilon }\right),\mathbf{v}=\mathbf{0}\mathrm{on}\partial \mathsf{\Omega }\}$$

$$W_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)=\{\psi \in H^1\left({\mathsf{\Omega }}_{\epsilon }\right),\psi =0\mathrm{on}\partial \mathsf{\Omega }\}$$

With two norms: ${\Vert .\Vert }_{{\mathbf{V}}_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)}={\Vert .\Vert }_{{\mathbf{H}}^1\left(\mathsf{\Omega }\right)}{,\Vert .\Vert }_{W_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)}={\Vert .\Vert }_{H^1\left(\mathsf{\Omega }\right)}$. The variational problem is defined by:

$$\left\{\begin{array}{c}\mathrm{Find}({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })\in {\mathbf{V}}_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)\times W_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right),\mathrm{such} \mathrm{that}\hfill \\ \\ a_{\epsilon }(({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon }),(\mathbf{v},\psi ))=L_{\epsilon }(\mathbf{v},\psi )\forall (\mathbf{v},\psi )\in {\mathbf{V}}_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)\times W_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right),\hfill \end{array}\right.$$

(7)

where

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{\displaystyle a_{\epsilon }(({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon }),(\mathbf{v},\psi ))}\hfill & =& {\displaystyle {\int }_{{\mathsf{\Omega }}_{\epsilon }}\{[c_{ijkl}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+e_{kij}^{\epsilon }{\partial }_k{\phi }^{\epsilon }]s_{ij}\left(\mathbf{v}\right)}\hfill \\ \\ & {\displaystyle +}& [-e_{ikl}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+d_{ij}^{\epsilon }{\partial }_j{\phi }^{\epsilon }]{\partial }_i\psi \}dx\hfill \\ {\displaystyle L_{\epsilon }(\mathbf{v},\psi )={\int }_{{\mathsf{\Omega }}_{\epsilon }}{f}_i{v}_{i}dx}\hfill \end{array}\right.\end{array}$$

(8)

It is pointed out that, under assumptions (4)–(6), the variational problem (7) and (8) have a unique solution $({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })\in {\mathbf{V}}_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)\times W_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)$, corresponding to the saddle point of this functional (see [20]):

$$(\mathbf{v},\psi )\to \frac{1}{2}{\int }_{{\mathsf{\Omega }}_{\epsilon }}(c^{\epsilon }(\mathbf{v},\mathbf{v})+2e^{\epsilon }(\mathbf{u},\psi )-d^{\epsilon }(\psi ,\psi ))dx-{\int }_{{\mathsf{\Omega }}_{\epsilon }}\mathbf{f}\mathbf{v}dx,$$

where

$$\left\{\begin{array}{ccc}{c}^{\epsilon }(\mathbf{u},\mathbf{v})\hfill & =& c_{ijkl}^{\epsilon }{s}_{ij}\left(\mathbf{u}\right)s_{kl}\left(\mathbf{v}\right)\hfill \\ e^{\epsilon }(\mathbf{u},\psi )\hfill & =& e_{ikl}^{\epsilon }{s}_{kl}\left(\mathbf{u}\right){\partial }_i\psi \hfill \\ d^{\epsilon }(\psi ,\psi )\hfill & =& d_{ij}^{\epsilon }{\partial }_i\psi {\partial }_j\psi \hfill \end{array}\right.$$

2.4. A Priori Estimates

In order to prove the main convergence results of this paper, we use the notion of two-scale convergence, which was introduced in [18] and developed further in [19]. The idea of this convergence is to first obtain a priori estimates for the displacement field and for the electric potential. Next, we use the relatively compact propriety with the classical procedure of prolongation (which is the extension by 0 from ${\mathsf{\Omega }}_{\epsilon }$ to $\mathsf{\Omega }$). Finally, we pass the limit $\epsilon \to 0$, in order to obtain the homogenized and the local problems at the same time.

Proposition 1.

Use the two equivalent norms of ${\mathbf{V}}_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right),W_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)$, for any sequence of solution ${({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })}_{\epsilon }\subset {\mathbf{V}}_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)\times W_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)$ of variational problems (7) and (8). Then, this solution is bounded, and we have this a priori estimate uniformly by ε

$$\Vert {\mathbf{u}}^{\epsilon }{\Vert }_{{\mathbf{H}}^1\left({\mathsf{\Omega }}_{\epsilon }\right)}+{\Vert {\phi }^{\epsilon }\Vert }_{H^1\left({\mathsf{\Omega }}_{\epsilon }\right)}\le C,$$

(9)

where C is constant, strictly positive and independent by ε.

Proof. 

By choosing $\mathbf{v}={\mathbf{u}}^{\epsilon }$ and $\psi ={\phi }^{\epsilon }$ in variational formulation (7) and (8), and by using the Korn’s and Poincaré’s inequalities in perforated domains (see Oleinik et al. [9] for the Korn’s inequality and Allaire–Murat [21] for Poincaré’s inequality), we see that ${\mathbf{u}}^{\epsilon }$ and ${\phi }^{\epsilon }$ are bounded, by a constant that does not depend on $\epsilon$. For other details, see [20]. □

3. Two-Scale Convergence Results

We denote by $\stackrel{\sim }{.}$ the extension by zero in the holes $\mathsf{\Omega }-{\mathsf{\Omega }}_{\epsilon }$. The sequence of solution ${({\mathbf{u}}^{\epsilon },{\phi }^{\epsilon })}_{\epsilon }\subset {\mathbf{V}}_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)\times W_{\epsilon }\left({\mathsf{\Omega }}_{\epsilon }\right)$ of variational problem (7) and (8) verify (9) and, in this case, by adding the relatively compact property and elementary properties of two-scale convergence, imply

Lemma 1.

1 

There exists $\mathbf{u}\left(x\right)\in {\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)$ and $\phi \left(x\right)\in H_0^1\left(\mathsf{\Omega }\right)$ such that the two sequences ${\left(\stackrel{\sim }{{\mathbf{u}}^{\epsilon }}\right)}_{\epsilon },{\left(\stackrel{\sim }{{\phi }^{\epsilon }}\right)}_{\epsilon }$ two-scale converge to $\chi \left(y\right)\mathbf{u}\left(x\right),\chi \left(y\right)\phi \left(x\right)$, respectively.

2 

There exists ${\mathbf{u}}_1(x,y)\in {\mathbf{L}}^2[\mathsf{\Omega };{\mathbf{H}}_{♯}^1\left(Y^{*}\right)/\mathbb{R}],{\phi }_1(x,y)\in L^2[\mathsf{\Omega };H_{♯}^1\left(Y^{*}\right)/\mathbb{R}]$ such that,

$$\stackrel{\sim }{\nabla }{\mathbf{u}}^{\epsilon }\to \chi \left(y\right)[{\nabla }_x\mathbf{u}\left(x\right)+{\nabla }_y{\mathbf{u}}_1(x,y)]in two\text{-}scale sense$$

$$\stackrel{\sim }{\nabla }{\phi }^{\epsilon }\to \chi \left(y\right)[{\nabla }_x\phi \left(x\right)+{\nabla }_y{\phi }_1(x,y)]in two\text{-}scale sense$$

3 

We have

$$\stackrel{\sim }{s}\left({\mathbf{u}}^{\epsilon }\right)\to \chi \left(y\right)[s_x\left(\mathbf{u}\left(x\right)\right)+s_y\left({\mathbf{u}}_1(x,y)\right)]in two\text{-}scale sense$$

index x or y means that the derivatives are with respect to the variable.

Proof. 

For details, see [18,19,20,22]. □

Corollary 1.

The sequence ${\left(\stackrel{\sim }{{\mathbf{u}}^{\epsilon }}\right)}_{\epsilon >0}$$(resp.{\left(\stackrel{\sim }{{\phi }^{\epsilon }}\right)}_{\epsilon >0})$ converges weakly to a limit $\theta \mathbf{u}$$(resp.\theta \phi )$ in ${\mathbf{L}}^2\left(\mathsf{\Omega }\right)$$(resp.L^2\left(\mathsf{\Omega }\right))$.

Remark 1.

Let $\rho \in L_{♯}^2\left(Y\right)$, define ${\rho }^{\epsilon }\left(x\right)=\rho \left(\frac{x}{\epsilon }\right)$, and ${\left(v^{\epsilon }\right)}_{\epsilon }\subset L^2\left(\mathsf{\Omega }\right)$ two-scale converge to a limit $v\in L^2(\mathsf{\Omega }\times Y)$. Then, ${\left({\rho }^{\epsilon }{v}^{\epsilon }\right)}_{\epsilon }$ two-scale converges to a limit $\rho v$ (see [20]).

From these last results, we can state the next theorem

Theorem 1.

The sequences ${\left(\stackrel{\sim }{{\mathbf{u}}^{\epsilon }}\right)}_{\epsilon }$, ${\left(\stackrel{\sim }{s}\left({\mathbf{u}}^{\epsilon }\right)\right)}_{\epsilon }$, ${\left(\stackrel{\sim }{{\phi }^{\epsilon }}\right)}_{\epsilon }$ and ${\left(\stackrel{\sim }{\nabla }{\phi }^{\epsilon }\right)}_{\epsilon }$ two-scale converge to $\chi \left(y\right)\mathbf{u}\left(x\right)$, $\chi \left(y\right)[s_x\left(\mathbf{u}\right)+s_y\left({\mathbf{u}}_1\right)]$, $\chi \left(y\right)\phi \left(x\right)$ and $\chi \left(y\right)[{\nabla }_x\phi +{\nabla }_y{\phi }_1]$ respectively, where $(\mathbf{u}\left(x\right),{\mathbf{u}}_1(x,y),\phi \left(x\right)$, ${\phi }_1(x,y))$ are the unique solutions in ${\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathbf{L}}^2[\mathsf{\Omega };{\mathbf{H}}_{♯}^1\left(Y^{*}\right)/\mathbb{R}]\times H_0^1\left(\mathsf{\Omega }\right)\times L^2[\mathsf{\Omega };H_{♯}^1\left(Y^{*}\right)/\mathbb{R}]$ of the following two-scale homogenized system:

$$\left\{\begin{array}{c}{\displaystyle -\frac{\partial }{\partial x_j}\left[{\int }_{Y^{*}}\{c_{ijkl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+e_{kij}(x,y)[{\partial }_{k,x}\phi +{\partial }_{k,y}{\phi }_1]\}dy\right]=\theta f_i\left(x\right) in \mathsf{\Omega },}\hfill \\ \\ \\ {\displaystyle -\frac{\partial }{\partial x_i}\left[{\int }_{Y^{*}}\{-e_{ikl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+d_{ij}(x,y)[{\partial }_{j,x}\phi +{\partial }_{j,y}{\phi }_1]\}dy\right]= 0 in \mathsf{\Omega },}\hfill \\ \\ \\ {\displaystyle -\frac{\partial }{\partial y_j}\{c_{ijkl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+e_{kij}(x,y)[{\partial }_{k,x}\phi +{\partial }_{k,y}{\phi }_1]\}= 0 in \mathsf{\Omega }\times Y^{*},}\hfill \\ \\ \\ {\displaystyle -\frac{\partial }{\partial y_i}\{-e_{ikl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+d_{ij}(x,y)[{\partial }_{j,x}\phi +{\partial }_{j,y}{\phi }_1]\}= 0 in \mathsf{\Omega }\times Y^{*},}\hfill \end{array}\right.$$

(10)

and we have these boundary conditions

$$\left\{\begin{array}{cccc}\mathbf{u}\left(x\right)\hfill & =& \mathbf{0}\hfill & on\partial \mathsf{\Omega },\hfill \\ \\ \phi \left(x\right)\hfill & =& 0\hfill & on\partial \mathsf{\Omega },\hfill \\ \\ \{c_{ijkl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+e_{kij}(x,y)[{\partial }_{k,x}\phi +{\partial }_{k,y}{\phi }_1]\}.n_j\hfill & =& 0\hfill & on\partial Y^{*}-\partial Y,\hfill \\ \\ \{-e_{ikl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+d_{ij}(x,y)[{\partial }_{j,x}\phi +{\partial }_{j,y}{\phi }_1]\}.n_i\hfill & =& 0\hfill & on\partial Y^{*}-\partial Y.\hfill \end{array}\right.$$

(11)

$$\left\{\begin{array}{cc}y\to {\mathbf{u}}_1(x,y)\hfill & isY-periodic,\\ \\ y\to {\phi }_1(x,y)\hfill & isY-periodic,\end{array}\right.$$

(12)

where θ is the volume fraction of material ($i.e.$, $\theta =<\chi >={\int }_Y\chi \left(y\right)dy=\mid Y^{*}\mid$), $\mid Y\mid$ denotes a measure of Y.

The equations (10)–(12) are referred to as the two-scale homogenized system.

Proof. 

From the idea of Nguetseng [18], the test functions in (7) and (8) are chosen on the form

$${\mathbf{v}}^{\epsilon }\left(x\right)=\mathbf{v}(x,\frac{x}{\epsilon })={\mathbf{v}}^0\left(x\right)+\epsilon {\mathbf{v}}^1(x,\frac{x}{\epsilon }),$$

$${\psi }^{\epsilon }\left(x\right)=\psi (x,\frac{x}{\epsilon })={\psi }^0\left(x\right)+\epsilon {\psi }^1(x,\frac{x}{\epsilon }),$$

where ${\mathbf{v}}^0\in {\mathbf{C}}_0^{\infty }\left(\mathsf{\Omega }\right),{\psi }^0\in C_0^{\infty }\left(\mathsf{\Omega }\right),{\mathbf{v}}^1\in {\mathbf{C}}_0^{\infty }(\mathsf{\Omega };{\mathbf{C}}_{♯}^{\infty }\left(Y\right))\mathrm{and} {\psi }^1\in C_0^{\infty }(\mathsf{\Omega };C_{♯}^{\infty }\left(Y\right))$, we obtain

$$\begin{array}{ccc}& {\displaystyle {\int }_{{\mathsf{\Omega }}_{\epsilon }}\{[c_{ijkl}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+e_{kij}^{\epsilon }{\partial }_k{\phi }^{\epsilon }][s_{ij,x}\left({\mathbf{v}}^0\right)\left(x\right)+\{s_{ij,y}\left({\mathbf{v}}^1\right)+\epsilon s_{ij,x}\left({\mathbf{v}}^1\right)\}(x,\frac{x}{\epsilon })]}& \hfill \\ & {\displaystyle -[-e_{kij}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+d_{ij}^{\epsilon }{\partial }_j{\phi }^{\epsilon }][{\partial }_{i,x}{\psi }^0\left(x\right)+\{{\partial }_{i,y}{\psi }^1+\epsilon {\partial }_{i,x}{\psi }^1\}(x,\frac{x}{\epsilon })]\}dx}& \hfill \\ & {\displaystyle ={\int }_{{\mathsf{\Omega }}_{\epsilon }}{f}_i\left(x\right)[v_i^0\left(x\right)+\epsilon v_i^1(x,\frac{x}{\epsilon })]dx}\end{array}$$

Under the precedent hypotheses, and passing to the two-scale limit, yields

$$\begin{array}{ccc}& {\displaystyle {\int }_{\mathsf{\Omega }}{\int }_Y[c_{ijkl}(x,y)\chi \left(y\right)(s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right))+e_{kij}(x,y)\chi \left(y\right)({\partial }_{k,x}\phi +{\partial }_{k,y}{\phi }_1)]\chi \left(y\right)}& \hfill \\ & {\displaystyle [s_{kl,x}\left({\mathbf{v}}^0\right)+s_{kl,y}\left({\mathbf{v}}^1\right)]dxdy}& \hfill \\ & {\displaystyle -{\int }_{\mathsf{\Omega }}{\int }_Y[-e_{ikl}(x,y)\chi \left(y\right)(s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right))+d_{ij}(x,y)\chi \left(y\right)({\partial }_{j,x}\phi +{\partial }_{j,y}{\phi }_1)]\chi \left(y\right)}& \hfill \\ & {\displaystyle [{\partial }_{i,x}{\psi }^0+{\partial }_{i,y}{\psi }^1]dxdy}& \hfill \\ & {\displaystyle ={\int }_{\mathsf{\Omega }}{\int }_Y{f}_i\left(x\right)\chi \left(y\right)v_i^0\left(x\right)dxdy.}\end{array}$$

(13)

By a definition of $\chi$, we have

$$\begin{array}{ccc}& {\displaystyle {\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}[c_{ijkl}(x,y)(s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right))+e_{kij}(x,y)({\partial }_{k,x}\phi +{\partial }_{k,y}{\phi }_1)]}& \hfill \\ & {\displaystyle [s_{kl,x}\left({\mathbf{v}}^0\right)+s_{kl,y}\left({\mathbf{v}}^1\right)]dxdy}& \hfill \\ & {\displaystyle +{\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}[-e_{ikl}(x,y)(s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right))+d_{ij}(x,y)({\partial }_{j,x}\phi +{\partial }_{j,y}{\phi }_1)]}& \hfill \\ \hfill {\displaystyle }& [{\partial }_{i,x}{\psi }^0+{\partial }_{i,y}{\psi }^1]dxdy& \hfill \\ & {\displaystyle =\theta {\int }_{\mathsf{\Omega }}{f}_i\left(x\right)v_i\left(x\right)dx,}& \hfill \end{array}$$

(14)

where $\theta ={\int }_Y\chi \left(y\right)dy$, by densite of spaces from when we chose the test functions, Equation (14) holds true for any ${\mathbf{v}}^0\in {\mathbf{H}}_0^1\left(\mathsf{\Omega }\right),{\psi }^0\in H_0^1\left(\mathsf{\Omega }\right)$, and for any ${\mathbf{v}}^1\in {\mathbf{L}}^2[\mathsf{\Omega };{\mathbf{H}}_{♯}^1\left(Y^{*}\right)/\mathbb{R}]$, ${\psi }^1\in L^2[\mathsf{\Omega };H_{♯}^1\left(Y^{*}\right)/\mathbb{R}]$, Integration by parts shows that (14) is a variational formulation associated with the two-scale homogenized system

$$\left\{\begin{array}{c}{\displaystyle -{\partial }_j\left[{\int }_{Y^{*}}\{c_{ijkl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+e_{kij}(x,y)[{\partial }_{k,x}\phi +{\partial }_{k,y}{\phi }_1\}dy\right]=\theta f_i\left(x\right)}\hfill \\ \\ {\displaystyle -{\partial }_i\left[{\int }_{Y^{*}}\{-e_{ikl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+d_{ij}(x,y)[{\partial }_{j,x}\phi +{\partial }_{j,y}{\phi }_1]\}dy\right]=0}\hfill \end{array}\right.$$

(15)

We complete (15) by the boundary conditions (11) and (12). To prove existence and uniqueness in (14), by application of the Lax–Milgram lemma, let us focus on the coercivity in ${\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathbf{L}}^2[\mathsf{\Omega };{\mathbf{H}}_{♯}^1\left(Y^{*}\right)/\mathbb{R}]\times H_0^1\left(\mathsf{\Omega }\right)\times L^2[\mathsf{\Omega };H_{♯}^1\left(Y^{*}\right)/\mathbb{R}]$ of the bilinear form defined by the left-hand side of (14) (For a complete demonstration, see [20]). □

Remark 2.

It is evident that the two-scale homogenized problem (10)–(12) is a system of four equations, four unknowns $(\mathbf{u},{\mathbf{u}}_1,\phi ,{\phi }_1)$, each dependent on both the two space variables x and y (i.e., the macroscopic and microscopic scales) that are mixed. Although it seems to be complicated, it is a well-posed system of equations. In addition, it is clear that the two-scale homogenized problem has the same form as the original equation.

The objective of this paragraph shall now give another form of a theorem that is more suitable for further physical interpretations. Indeed, we shall eliminate the microscopic variable y (one does not want to solve the small scale structure), and decouple the two-scale homogenized problem (10)–(12) in homogenized and cell equations. However, it is preferable from a physical or numerical point of view (see [20]).

4. Derivation of the Homogenized Coefficients

Due to the linearity of the original problem, and assuming the regularity in variation of the coefficients, we take

$${\mathbf{u}}_1(x,y)=s_{mh,x}\left(\mathbf{u}\left(x\right)\right){\mathbf{w}}^{mh}\left(y\right)+\frac{\partial \phi \left(x\right)}{\partial x_n}{\mathbf{q}}^n\left(y\right),$$

(16)

$${\phi }_1(x,y)=s_{mh,x}\left(\mathbf{u}\left(x\right)\right){\phi }^{mh}\left(y\right)+\frac{\partial \phi \left(x\right)}{\partial x_n}{\psi }^n\left(y\right),$$

(17)

where ${\mathbf{w}}^{mh},{\phi }^n,{\mathbf{q}}^{mh}$ and ${\psi }^n$ are $Y^{*}$-periodic functions in y, independent of x, with solutions of these two locals problems in $Y^{*}$

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\frac{\partial }{\partial y_j}\left\{c_{ijkl}(x,y)\left[{\tau }_{mh}^{kl}+s_{kl,y}\left({\mathbf{w}}^{mh}\right)\right]+e_{kij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle -\frac{\partial }{\partial y_i}\left\{-e_{ikl}(x,y)\left[{\tau }_{mh}^{kl}+s_{kl,y}\left({\mathbf{w}}^{mh}\right)\right]+d_{ij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_j}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle {\mathbf{w}}^{mh},{\phi }^{mh}}\hfill & Y^{*}-\mathrm{periodics},\end{array}\right.\end{array}$$

(18)

where

$${\tau }_{mh}^{kl}=\frac{1}{2}[{\delta }_{km}{\delta }_{lh}+{\delta }_{kh}{\delta }_{lm}]1\le k,m,l,h\le 3.$$

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\frac{\partial }{\partial y_j}\left\{c_{ijkl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+e_{kij}(x,y)\left[{\delta }_{kn}+\frac{\partial {\psi }^n}{\partial y_k}\right]\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle -\frac{\partial }{\partial y_i}\left\{-e_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+d_{ij}(x,y)\left[{\delta }_{jn}+\frac{\partial {\psi }^n}{\partial y_j}\right]\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle {\phi }^n,{\psi }^n}\hfill & Y^{*}-\mathrm{periodics}.\end{array}\right.\end{array}$$

(19)

However, in general, relations (16)–(17) like this do not exist, if we do not have linearity of the problem.

Now, substituting the expansions (16) and (17) in this equation

$$-\frac{\partial }{\partial y_j}\{c_{ijkl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+e_{kij}(x,y)[{\partial }_{k,x}\phi +{\partial }_{k,y}{\phi }_1]\}=0in\mathsf{\Omega }\times Y^{*},$$

we obtain

$$\begin{array}{ccc}& -& {\displaystyle \frac{\partial s_{mh,x}\left(\mathbf{u}\right)}{\partial y_j}\left\{c_{ijkl}(x,y)\left[{\tau }_{mh}^{kl}+s_{kl,y}\left({\mathbf{w}}^{mh}\right)\right]+e_{kij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}\right\}}\hfill \\ & +& {\displaystyle \frac{\partial \phi }{\partial x_n}\left\{c_{ijkl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+e_{kij}(x,y)\left[{\delta }_{kn}+\frac{\partial {\psi }^n}{\partial y_k}\right]\right\}=0.}\hfill \end{array}$$

(20)

Call ${\tau }_{mh}$ the basics of symmetrical second order tensors ${\tau }_{mh}^{kl}=\frac{1}{2}[{\delta }_{km}{\delta }_{lh}+{\delta }_{kh}{\delta }_{lm}]$, where ${\delta }_{ij}$ is the Kronecker symbol.

Analogously, we substitute the expansions (16) and (17) in this equation

$$-\frac{\partial }{\partial y_i}\{-e_{ikl}(x,y)[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)]+d_{ij}(x,y)[{\partial }_{j,x}\phi +{\partial }_{j,y}{\phi }_1]\}=0in\mathsf{\Omega }\times Y^{*},$$

we obtain

$$\begin{array}{ccc}& -& {\displaystyle \frac{\partial s_{mh,x}\left(\mathbf{u}\right)}{\partial y_i}\left\{-e_{ikl}(x,y)\left[{\tau }_{mh}^{kl}+s_{kl,y}\left({\mathbf{w}}^{mh}\right)\right]+d_{ij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_j}\right\}}\hfill \\ & +& {\displaystyle \frac{\partial \phi }{\partial x_n}\left\{-e_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+d_{ij}(x,y)\left[{\delta }_{jn}+\frac{\partial {\psi }^n}{\partial y_j}\right]\right\}=0.}\hfill \end{array}$$

(21)

From the relation (20) and (21), after lengthy calculations, we arrive at the homogenized (effective) coefficients:

$${\displaystyle c_{ijmh}^H=⟨c_{ijkl}(x,y)\left[{\tau }_{mh}^{kl}+s_{kl,y}\left({\mathbf{w}}^{mh}\right)\right]+e_{kij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}⟩,}$$

(22)

$${\displaystyle e_{nij}^H=⟨c_{ijkl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+e_{kij}(x,y)\left[{\delta }_{kn}+\frac{\partial {\psi }^n}{\partial y_k}\right]⟩,}$$

(23)

$${\displaystyle f_{imh}^H=⟨e_{ikl}(x,y)\left[{\tau }_{mh}^{kl}+s_{kl,y}\left({\mathbf{w}}^{mh}\right)\right]-d_{ij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_j}⟩,}$$

(24)

$${\displaystyle d_{in}^H=⟨-e_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+d_{ij}(x,y)\left[{\delta }_{jn}+\frac{\partial {\psi }^n}{\partial y_j}\right]⟩,}$$

(25)

where $⟨h⟩={\int }_{Y^{*}}h\left(y\right)dy$ is measured on $Y^{*}$ of function h.

Now, we give results concerning some properties of elasticity homogenized tensor.

Proposition 2.

The coefficients of elasticity homogenized tensor ${\mathcal{C}}^{\mathcal{H}}=\left(c_{ijkl}^H\right)$ defined by (22) satisfy:

(a) $c_{ijkl}^H=c_{klij}^H=c_{ijlk}^H=c_{jilk}^H,\forall 1\le i,j,k,l\le 3,$

(b) There exists ${\alpha }_c^H>0$, such that, for all ξ, symmetric tensor (${\xi }_{ij}={\xi }_{ji}$),

$$c_{ijkl}^H{\xi }_{ij}{\xi }_{kl}\ge {\alpha }_c^H{\xi }_{ij}{\xi }_{ij}$$

Proof. 

The part of the symmetry of these coefficients is evident

$$c_{ijmh}^H=c_{jimh}^H=c_{jihm}^H.$$

We are interested in the proof of

$$c_{ijmh}^H=c_{mhij}^H$$

Following the ideas, we transform the above expression to obtain a symmetric form.

If we define the second order tensor $\sum$, by ${\sum }^{kl}=\frac{1}{2}(y_k{\overrightarrow{e}}_l+y_l{\overrightarrow{e}}_k)$, and we define the $3\times 3$ $H^{kl}$ matrix by $H^{kl}=s_y\left({\sum }^{kl}\right)$, it is obvious that the coefficients of this matrix are defined by

$${\left[H^{kl}\right]}_{mh}={\tau }_{mh}^{kl}=\frac{1}{2}[{\delta }_{km}{\delta }_{lh}+{\delta }_{kh}{\delta }_{lm}]1\le k,l,m,h\le 3.$$

If we use this new notation, we can rewrite problem (18), as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\frac{\partial }{\partial y_j}\left\{c_{ijkl}(x,y)s_{kl,y}\left({\sum }^{mh}+{\mathbf{w}}^{mh}\right)+e_{kij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle -\frac{\partial }{\partial y_i}\left\{-e_{ikl}(x,y)s_{kl,y}\left({\sum }^{mh}+{\mathbf{w}}^{mh}\right)+d_{ij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_j}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle {\mathbf{w}}^{mh},{\phi }^{mh}}\hfill & Y^{*}-\mathrm{periodics}.\end{array}\right.\end{array}$$

(26)

We introduce the problem functions $({\mathbf{w}}^{ij},{\mathbf{q}}^{ij})$ solution of this problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\frac{\partial }{\partial y_j}\left\{c_{kj\alpha \beta }(x,y)s_{\alpha \beta ,y}\left({\sum }^{ij}+{\mathbf{w}}^{ij}\right)+e_{\alpha kj}(x,y)\frac{\partial {\phi }^{ij}}{\partial y_{\alpha }}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle -\frac{\partial }{\partial y_k}\left\{-e_{k\alpha \beta }(x,y)s_{\alpha \beta ,y}\left({\sum }^{ij}+{\mathbf{w}}^{ij}\right)+d_{kj}(x,y)\frac{\partial {\phi }^{ij}}{\partial y_j}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle {\mathbf{w}}^{ij},{\phi }^{ij}}\hfill & Y^{*}-\mathrm{periodics}.\end{array}\right.\end{array}$$

(27)

The coefficients of elasticity tensor can be rewritten as

$$c_{ijmh}^H={\int }_{Y^{*}}{c}_{ijkl}(x,y)s_{kl,y}\left({\sum }^{mh}+{\mathbf{w}}^{mh}\right)dy+{\int }_{Y^{*}}{e}_{kij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}dy.$$

(28)

The second integral of the right-hand side of precedent expression is evaluated as follows:

$$\begin{array}{ccc}\hfill {\int }_{Y^{*}}{e}_{kij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}dy& =& {\int }_{Y^{*}}{e}_{k\alpha \beta }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}{\delta }_{\alpha i}{\delta }_{\beta j}dy,\hfill \\ & =& \frac{1}{2}{\int }_{Y^{*}}{e}_{k\alpha \beta }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}\left({\delta }_{\alpha i}{\delta }_{\beta j}+{\delta }_{\alpha j}{\delta }_{\beta i}\right)dy,\hfill \\ & =& -{\int }_{Y^{*}}{e}_{k\alpha \beta }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}{s}_{\alpha \beta ,y}\left({\mathbf{w}}^{ij}\right)dy\hfill \\ & +& {\int }_{Y^{*}}{e}_{k\alpha \beta }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}{s}_{\alpha \beta ,y}({\sum }^{ij}+{\mathbf{w}}^{ij})dy.\hfill \end{array}$$

(29)

We use the variational formulation of the first equation of problem (26), and choosing the test function $\mathbf{v}={\mathbf{w}}^{ij}$, we obtain

$${\int }_{Y^{*}}{e}_{k\alpha \beta }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}{s}_{\alpha \beta ,y}\left({\mathbf{w}}^{ij}\right)dy={\int }_{Y^{*}}{c}_{\alpha \beta kl}(x,y)s_{kl,y}({\sum }^{mh}+{\mathbf{w}}^{mh})s_{\alpha \beta ,y}\left({\mathbf{w}}^{ij}\right)dy.$$

(30)

Multiplying the second equation by ${\phi }^{mh}$, and integrating by parts, we have

$${\int }_{Y^{*}}{e}_{k\alpha \beta }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}{s}_{\alpha \beta ,y}({\sum }^{ij}+{\mathbf{w}}^{ij})dy={\int }_{Y^{*}}{d}_{k\alpha }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}\frac{\partial {\phi }^{ij}}{\partial y_{\alpha }}dy.$$

(31)

Regrouped this results, and using the definition (28), we derive

$$\begin{array}{ccc}\hfill c_{ijmh}^H& =& {\int }_{Y^{*}}{c}_{ijkl}(x,y)s_{kl,y}({\sum }^{mh}+{\mathbf{w}}^{mh})dy+{\int }_{Y^{*}}{e}_{kij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}dy,\hfill \\ & =& {\int }_{Y^{*}}{c}_{ijkl}(x,y)s_{kl,y}({\sum }^{mh}+{\mathbf{w}}^{mh})dy\hfill \\ & +& {\int }_{Y^{*}}{c}_{\alpha \beta kl}(x,y)s_{kl,y}({\sum }^{mh}+{\mathbf{w}}^{mh})s_{\alpha \beta ,y}\left({\mathbf{w}}^{ij}\right)dy+{\int }_{Y^{*}}{d}_{k\alpha }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}\frac{\partial {\phi }^{ij}}{\partial y_{\alpha }}dy,\hfill \\ & =& {\int }_{Y^{*}}{c}_{\alpha \beta kl}(x,y)s_{kl,y}({\sum }^{mh}+{\mathbf{w}}^{mh})s_{\alpha \beta ,y}\left({\sum }^{ij}\right)dy+{\int }_{Y^{*}}{d}_{k\alpha }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}\frac{\partial {\phi }^{ij}}{\partial y_{\alpha }}dy\hfill \\ & +& {\int }_{Y^{*}}{c}_{\alpha \beta kl}(x,y)s_{kl,y}({\sum }^{mh}+{\mathbf{w}}^{mh})s_{\alpha \beta ,y}\left({\mathbf{w}}^{ij}\right)dy,\hfill \\ & =& {\int }_{Y^{*}}{c}_{\alpha \beta kl}(x,y)s_{kl,y}({\sum }^{mh}+{\mathbf{w}}^{mh})s_{\alpha \beta ,y}({\sum }^{ij}+{\mathbf{w}}^{ij})dy\hfill \\ & +& {\int }_{Y^{*}}{d}_{k\alpha }(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}\frac{\partial {\phi }^{ij}}{\partial y_{\alpha }}dy.\hfill \end{array}$$

(32)

It is immediate from the above form (32) that the coefficients of elasticity tensor satisfy

$$c_{ijmh}^H=c_{mhij}^H$$

This is the end of proof of the first section proposing the symmetry.

We now study the ellipticity of the coefficients of the elasticity tensor, we note that $c_{ijkl}^H$ is elliptic, if, for all the second order tensor $X_{ij}$ symetric ($X_{ij}=X_{ji}$), we have

$$\exists {\alpha }_c^H>0,c_{ijkl}^H{X}_{ij}{X}_{kl}\ge {\alpha }_c^H{X}_{ij}{X}_{ij}$$

Considering expression (22) of the tensor $c_{ijmh}^H$, we have

$$\begin{array}{ccc}\hfill c_{ijmh}^H{X}_{ij}{X}_{mh}& =& {\int }_{Y^{*}}{c}_{ijmh}(x,y)X_{ij}{X}_{mh}dy+{\int }_{Y^{*}}{c}_{ijkl}(x,y)s_{kl,y}\left({\mathbf{w}}^{mh}\right)X_{ij}{X}_{mh}dy\hfill \\ & +& {\int }_{Y^{*}}{e}_{kij}(x,y)\frac{\partial {\phi }^{mh}}{\partial y_k}{X}_{ij}{X}_{mh}dy,\hfill \\ & =& {\int }_{Y^{*}}{c}_{ijmh}(x,y)X_{ij}{X}_{mh}dy+{\int }_{Y^{*}}{c}_{ijkl}(x,y)s_{kl,y}\left({\mathbf{w}}^{mh}{X}_{mh}\right)X_{ij}dy\hfill \\ & +& {\int }_{Y^{*}}{e}_{kij}(x,y)\frac{\partial \left({\phi }^{mh}{X}_{mh}\right)}{\partial y_k}{X}_{ij}dy,\hfill \\ & =& {\int }_{Y^{*}}{c}_{ijkl}(x,y)\left(s_{kl,y}\left(\mathbf{w}\right)+P_{kl}\right)P_{ij}dy+{\int }_{Y^{*}}{e}_{kij}(x,y)\frac{\partial \zeta }{\partial y_k}{P}_{ij}dy,\hfill \end{array}$$

where $\mathbf{w}={\mathbf{w}}^{mh}{X}_{mh},\zeta ={\phi }^{mh}{X}_{mh}$ and $P_{ij}={\tau }_{mh}^{ij}{X}_{mh}=X_{ij}$. Therefore, the couple $(\mathbf{w},\zeta )$ is a saddle point of the functional J defined by

$$(\mathbf{v},\psi )\to J(\mathbf{v},\psi )$$

$$J(\mathbf{v},\psi )=\frac{1}{2}{\int }_{Y^{*}}\left\{c_{ijkl}\left(s_{ij,y}\left(\mathbf{v}\right)+P_{ij}\right)\left(s_{kl,y}\left(\mathbf{v}\right)+P_{kl}\right)+2e_{kij}\frac{\partial \psi }{\partial y_k}\left(s_{ij,y}\left(\mathbf{v}\right)+P_{ij}\right)-d_{ij}\frac{\partial \psi }{\partial y_i}\frac{\partial \psi }{\partial y_j}\right\}dy,$$

By definition of the saddle point, we have

$$J(\mathbf{w},\psi )\le J(\mathbf{w},\zeta )\le J(\mathbf{v},\zeta )\forall (\mathbf{v},\psi )\mathrm{periodics} \mathrm{functions}.$$

or

$$J(\mathbf{w},0)=\frac{1}{2}{\int }_{Y^{*}}{c}_{ijkl}(x,y)\left(s_{ij,y}\left(\mathbf{w}\right)+P_{ij}\right)\left(s_{kl,y}\left(\mathbf{w}\right)+P_{kl}\right)dy.$$

However, if we use the first equation of system (18), we obtain

$$\begin{array}{ccc}\hfill c_{ijmh}^H{X}_{ij}{X}_{mh}& =& 2J(\mathbf{w},\zeta ),\hfill \\ & \ge & J(\mathbf{w},0),\hfill \\ & =& \frac{1}{2}{\int }_{Y^{*}}{c}_{ijkl}(x,y)\left(s_{ij,y}\left(\mathbf{w}\right)+P_{ij}\right)\left(s_{kl,y}\left(\mathbf{w}\right)+P_{kl}\right)dy,\hfill \\ & >& 0,\hfill \end{array}$$

We have that the homogenized elasticity tensor ${\mathcal{C}}^H=\left(c_{ijkl}^H\right)$ is elliptic.

Now, we give results concerning some properties of the dielectric homogenized tensor. □

Proposition 3.

The coefficients of dielectric homogenized tensor ${\mathcal{D}}^H=\left(d_{in}^H\right)$ defined by (25) satisfy:

(a) $d_{in}^H=d_{ni}^H,\forall 1\le i,n\le 3,$

(b) There exists ${\alpha }_d^H>0$, such that, for all vector ξ,

$$d_{in}^H{\xi }_i{\xi }_n\ge {\alpha }_d^H{\xi }_i{\xi }_i$$

Proof. 

Analogously to the idea, we now transform these coefficients to obtain a symmetric form

$$d_{in}^H=d_{ni}^H$$

Problem (19) can also be rewritten as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\frac{\partial }{\partial y_j}\left\{c_{ijkl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+e_{kij}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_k}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle -\frac{\partial }{\partial y_i}\left\{-e_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+d_{ij}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_j}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle {\mathbf{q}}^n,{\psi }^n}\hfill & Y^{*}-\mathrm{periodics}.\end{array}\right.\end{array}$$

(33)

Introducing $({\mathbf{q}}^i,{\psi }^i)$, the solution of this local problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\frac{\partial }{\partial y_j}\left\{c_{ijkl}(x,y)s_{kl,y}\left({\mathbf{q}}^i\right)+e_{kij}(x,y)\frac{\partial (y_i+{\psi }^i)}{\partial y_k}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle -\frac{\partial }{\partial y_i}\left\{-e_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^i\right)+d_{ij}(x,y)\frac{\partial (y_i+{\psi }^i)}{\partial y_j}\right\}}\hfill & =0\mathrm{in}{Y}^{*},\\ \\ {\displaystyle {\mathbf{q}}^i,{\psi }^i}\hfill & Y^{*}-\mathrm{periodics}.\end{array}\right.\end{array}$$

(34)

We can rewritten these coefficients of the electric tensor as the form

$$d_{in}^H={\int }_{Y^{*}}-e_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)dy+{\int }_{Y^{*}}{d}_{ij}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_j}dy.$$

(35)

The first term of the second integral of precedent expression is evaluated as follows:

$$\begin{array}{ccc}\hfill {\int }_{Y^{*}}-e_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)dy& =& {\int }_{Y^{*}}-e_{\alpha kl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right){\delta }_{\alpha i}dy,\hfill \\ & =& {\int }_{Y^{*}}-e_{\alpha kl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)\frac{\partial y_i}{\partial y_{\alpha }}dy,\hfill \\ & =& {\int }_{Y^{*}}-e_{\alpha kl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)\frac{\partial {\psi }^i}{\partial y_{\alpha }}dy\hfill \\ & -& {\int }_{Y^{*}}-e_{\alpha kl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)\frac{\partial (y_i+{\psi }^i)}{\partial y_{\alpha }}dy.\hfill \\ \hfill \end{array}$$

(36)

Using the variational formulation of the second equation of system (33), and choosing a test function $\phi ={\psi }^i$, we obtain

$${\int }_{Y^{*}}-e_{\alpha kl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)\frac{\partial {\psi }^i}{\partial y_{\alpha }}dy={\int }_{Y^{*}}{d}_{\alpha j}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_j}\frac{\partial {\psi }^i}{\partial y_{\alpha }}dy.$$

Let us now consider the second integral in (36). Multiplying the first equation of system (34) by ${\varphi }^n$, and integrating by parts, we have

$${\int }_{Y^{*}}-e_{\alpha kl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)\frac{\partial (y_i+{\psi }^i)}{\partial y_{\alpha }}dy=-{\int }_{Y^{*}}{c}_{kl\alpha \beta }(x,y)s_{\alpha \beta ,y}\left({\mathbf{q}}^i\right)s_{kl,y}\left({\phi }^n\right)dy.$$

Finally, we regrouped these lasts results, and using the definition (35), we obtain

$$\begin{array}{ccc}\hfill d_{in}^H& =& {\int }_{Y^{*}}-e_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)dy+{\int }_{Y^{*}}{d}_{ij}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_j}dy,\hfill \\ & =& {\int }_{Y^{*}}{d}_{\alpha j}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_j}\frac{\partial {\psi }^i}{\partial y_{\alpha }}dy-{\int }_{Y^{*}}{c}_{kl\alpha \beta }(x,y)s_{\alpha \beta ,y}\left({\mathbf{q}}^i\right)s_{kl,y}\left({\mathbf{q}}^n\right)dy\hfill \\ & +& {\int }_{Y^{*}}{d}_{ij}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_j}dy,\hfill \\ & =& {\int }_{Y^{*}}{d}_{\alpha j}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_j}\frac{\partial {\psi }^i}{\partial y_{\alpha }}dy-{\int }_{Y^{*}}{c}_{kl\alpha \beta }(x,y)s_{\alpha \beta ,y}\left({\mathbf{q}}^i\right)s_{kl,y}\left({\mathbf{q}}^n\right)dy\hfill \\ & +& {\int }_{Y^{*}}{d}_{\alpha j}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_j}\frac{\partial y_i}{\partial y_{\alpha }}dy,\hfill \\ & =& {\int }_{Y^{*}}{d}_{\alpha j}(x,y)\frac{\partial (y_n+{\psi }^n)}{\partial y_j}\frac{\partial (y_i+{\psi }^i)}{\partial y_{\alpha }}dy-{\int }_{Y^{*}}{c}_{kl\alpha \beta }(x,y)s_{\alpha \beta ,y}\left({\mathbf{q}}^i\right)s_{kl,y}\left({\mathbf{q}}^n\right)dy.\hfill \\ \hfill \end{array}$$

(37)

It is immediate from the above form that the coefficients of electric tensor is symmetric. Now, we are interested in the ellipticity of this tensor, and note that $d_{in}^H$ is elliptic, if for all vectors $X_i$, we have

$$\exists {\alpha }_d^H>0,d_{in}^H{X}_i{X}_n\ge {\alpha }_d^H{X}_i{X}_i.$$

We consider the expression (25) of the tensor $d_{in}^H$, and derive

$$\begin{array}{ccc}\hfill d_{in}^H{X}_i{X}_n& =& {\int }_{Y^{*}}{d}_{in}(x,y)X_i{X}_{n}dy-{\int }_{Y^{*}}{e}_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)X_i{X}_{n}dy\hfill \\ & +& {\int }_{Y^{*}}{d}_{ij}(x,y)\frac{\partial {\psi }^n}{\partial y_j}{X}_i{X}_{n}dy,\hfill \\ & =& {\int }_{Y^{*}}{d}_{in}(x,y)X_i{X}_{n}dy-{\int }_{Y^{*}}{e}_{ikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n{X}_n\right)X_{i}dy\hfill \\ & +& {\int }_{Y^{*}}{d}_{ij}(x,y)\frac{\partial \left({\psi }^n{X}_n\right)}{\partial y_j}{X}_{i}dy,\hfill \\ & =& {\int }_{Y^{*}}{d}_{ij}(x,y)\left(Q_j+\frac{\partial \xi }{\partial y_j}\right)Q_{i}dy-{\int }_{Y^{*}}{e}_{ikl}(x,y)s_{kl,y}\left(\varsigma \right)Q_{i}dy,\hfill \end{array}$$

where $\xi ={\psi }^n{X}_n,\varsigma ={\phi }^n{X}_n$ and $Q_i={\delta }_{in}{X}_n=X_i$. However, the couple $(\xi ,\varsigma )$ is a saddle point of the functional G defined by:

$$(\mathbf{v},\psi )\to G(\mathbf{v},\psi )$$

$$G(\mathbf{v},\psi )=\frac{1}{2}{\int }_{Y^{*}}\left\{c_{ijkl}{s}_{ij,y}\left(\mathbf{v}\right)s_{kl,y}\left(\mathbf{v}\right)+2e_{kij}{s}_{ij,y}\left(\mathbf{v}\right)\left(Q_k+\frac{\partial \psi }{\partial y_k}\right)-d_{ij}\left(Q_i+\frac{\partial \psi }{\partial y_i}\right)\left(Q_j+\frac{\partial \psi }{\partial y_j}\right)\right\}dy.$$

By definition of the saddle point, we have

$$G(\xi ,\psi )\le G(\xi ,\varsigma )\le G(\mathbf{v},\varsigma )\forall (\mathbf{v},\psi )\mathrm{periodics} \mathrm{functions}.$$

or

$$G(0,\varsigma )=-\frac{1}{2}{\int }_{Y^{*}}{d}_{ij}(x,y)(Q_i+\frac{\partial \varsigma }{\partial y_i})(Q_j+\frac{\partial \varsigma }{\partial y_j})dy$$

However, if we use the second equation of system (19), we have

$$\begin{array}{ccc}\hfill G(0,\varsigma )& =& -\frac{1}{2}{\int }_{Y^{*}}{d}_{ij}(x,y)\left(Q_j+\frac{\partial \varsigma }{\partial y_j}\right)Q_{i}dy-\frac{1}{2}{\int }_{Y^{*}}{d}_{ij}(x,y)\left(Q_j+\frac{\partial \varsigma }{\partial y_j}\right)\frac{\partial \varsigma }{\partial y_i}dy,\hfill \\ & =& -\frac{1}{2}{\int }_{Y^{*}}{d}_{ij}(x,y)\left(Q_j+\frac{\partial \varsigma }{\partial y_j}\right)Q_{i}dy+\frac{1}{2}{\int }_{Y^{*}}{e}_{ikl}(x,y)s_{kl,y}\left(\varsigma \right)X_{i}dy,\hfill \\ & =& -\frac{1}{2}{d}_{in}^H{X}_i{X}_n,\hfill \\ & <& 0.\hfill \end{array}$$

We have that the dielectric homogenized tensor is elliptic.

Now, we give results concerning some properties of piezoelectric homogenized tensors. □

Proposition 4.

The coefficients of a piezoelectric homogenized tensor ${\mathcal{E}}^H=\left(e_{nij}^H\right)$ defined by (23) satisfy:

$$e_{nij}^H=e_{nji}^H$$

Furthermore, we have the identity

$$e_{nij}^H=f_{nij}^H$$

Proof. 

By definition of coefficients $e_{nij}^H$ (using the fact that $c_{ijkl}(x,y)=c_{jikl}(x,y)$, $e_{kji}(x,y)=e_{kij}(x,y)$), we have

$$\begin{array}{ccc}\hfill e_{nij}^H& =& {\int }_{Y^{*}}\left\{c_{ijkl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+e_{kij}(x,y)\left({\delta }_{kn}+\frac{\partial {\psi }^n}{\partial y_k}\right)\right\}dy,\hfill \\ & =& {\int }_{Y^{*}}\left\{c_{jikl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+e_{kji}(x,y)\left({\delta }_{kn}+\frac{\partial {\psi }^n}{\partial y_k}\right)\right\}dy,\hfill \\ & =& e_{nji}^H.\hfill \end{array}$$

(38)

We can rewrite the coefficients $e_{nij}^H$, as the form

$$\begin{array}{ccc}\hfill e_{nij}^H& =& {\int }_{Y^{*}}\left\{c_{ijkl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)+e_{kij}(x,y)\left({\delta }_{kn}+\frac{\partial {\psi }^n}{\partial y_k}\right)\right\}dy,\hfill \\ & =& {\int }_{Y^{*}}\left\{e_{nij}(x,y)+e_{kij}(x,y)\frac{\partial {\psi }^n}{\partial y_k}+c_{ijkl}(x,y)s_{kl,y}\left({\mathbf{q}}^n\right)\right\}dy.\hfill \end{array}$$

(39)

In the same way, we can rewrite the coefficients $f_{nij}^H$, as the form

$$\begin{array}{ccc}\hfill f_{nij}^H& =& {\int }_{Y^{*}}\left\{e_{nkl}(x,y)\left({\tau }_{ij}^{kl}+s_{kl,y}\left({\mathbf{w}}^{ij}\right)\right)-d_{nt}(x,y)\frac{\partial {\phi }^{ij}}{\partial y_t}\right\}dy,\hfill \\ & =& {\int }_{Y^{*}}\left\{e_{nij}(x,y)+e_{nkl}(x,y)s_{kl,y}\left({\mathbf{w}}^{ij}\right)-d_{nt}(x,y)\frac{\partial {\phi }^{ij}}{\partial y_t}\right\}dy.\hfill \end{array}$$

(40)

Using the two variational formulations corresponding to problems (18) and (19), and choosing the test functions conveniently, we prove directly as $e_{nij}^H=f_{nij}^H$. Finally, using the tree lasts propositions, we can state the alternative form of the principal convergence theorem. □

Theorem 2.

(the alternative form) ] Setting the $(\mathbf{u},\phi )$ solution of the two-scale homogenized problem (10)–(12), then $(\mathbf{u},\phi )$ is defined by that the solution of this homogenized problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cccc}{\displaystyle -\mathbf{div}{\sigma }^H(\mathbf{u},\phi )}\hfill & =& \theta \mathbf{f}\hfill & in\mathsf{\Omega },\hfill \\ \\ {\displaystyle -\mathbf{div}{D}^H(\mathbf{u},\phi )}\hfill & =& 0\hfill & in\mathsf{\Omega },\hfill \end{array}\right.\end{array}$$

(41)

where the boundary conditions

$$\left\{\begin{array}{ccc}\mathbf{u}\left(x\right)\hfill & =\mathbf{0}& on\partial \mathsf{\Omega },\hfill \\ \\ \phi \left(x\right)\hfill & =0& on\partial \mathsf{\Omega },\hfill \end{array}\right.$$

(42)

${\sigma }_{ij}^H$ and $D_i^H$ are defined by the homogenized constitutive law

$$\left\{\begin{array}{ccc}{\displaystyle {\sigma }_{ij}^H(\mathbf{u},\phi )}\hfill & =& {\displaystyle c_{ijmh}^H{s}_{mh,x}\left(\mathbf{u}\right)+e_{nij}^H\frac{\partial \phi }{\partial x_n},}\hfill \\ \\ {\displaystyle D_i^H(\mathbf{u},\phi )}\hfill & =& {\displaystyle -e_{imh}^H{s}_{mh,x}\left(\mathbf{u}\right)+d_{in}^H\frac{\partial \phi }{\partial x_n},}\hfill \end{array}\right.$$

(43)

the homogenized coefficients $c_{ijkl}^H$, $e_{nij}^H$ and $d_{ij}^H$ are defined respectively by (22), (23) and (25).

5. Correctors Result

Corrector results are easily obtained with the two-scale convergence method. The objective of the next theorem is to rigorously justify the two first terms in the usual asymptotic expansion of the solution.

Following Allaire [19], we introduce the following definition:

Definition 1.

We call $\psi (x,y)$ an admissible test function, if it is Y-periodic, and satisfies the following relation:

$$\underset{\epsilon \to 0}{lim}{\int }_{\mathsf{\Omega }}\psi {(x,\frac{x}{\epsilon })}^{2}dx={\int }_{\mathsf{\Omega }}{\int }_Y\psi {(x,y)}^{2}dxdy$$

(44)

Here, we recall Allaire’s lemma

Lemma 2

(Allaire [19]). Set the function $\psi (x,y)$ in $L^2(\mathsf{\Omega };C\left(Y\right))$, in which $\psi (x,y)$ is an admissible test function.

Using this lemma, we have the following proposition:

Proposition 5.

The two functions $s_{ij,y}\left({\mathbf{u}}_1(x,y)\right)$ and ${\partial }_{i,y}{\phi }_1(x,y)$ are admissible test functions in Definition 1 sens.

Proof. 

By definition, we have

$${\mathbf{u}}_1(x,y)=s_{mh,x}\left(\mathbf{u}\left(x\right)\right){\mathbf{w}}^{mh}\left(y\right)+\frac{\partial \phi \left(x\right)}{\partial x_n}{\phi }^n\left(y\right),$$

$${\phi }_1(x,y)=s_{mh,x}\left(\mathbf{u}\left(x\right)\right){\mathbf{q}}^{mh}\left(y\right)+\frac{\partial \phi \left(x\right)}{\partial x_n}{\psi }^n\left(y\right),$$

and we obtain

$$s_{ij,y}\left({\mathbf{u}}_1(x,y)\right)=s_{mh,x}\left(\mathbf{u}\left(x\right)\right)s_{ij,y}\left({\mathbf{w}}^{mh}\left(y\right)\right)+\frac{\partial \phi \left(x\right)}{\partial x_n}{s}_{ij,y}\left({\phi }^n\left(y\right)\right),$$

$${\partial }_{i,y}{\phi }_1(x,y)=s_{mh,x}\left(\mathbf{u}\left(x\right)\right){\partial }_{i,y}{\mathbf{q}}^{mh}\left(y\right)+\frac{\partial \phi \left(x\right)}{\partial x_n}{\partial }_{i,y}{\psi }^n\left(y\right).$$

Using the Lemma 2, $s_{ij,y}\left({\mathbf{u}}_1(x,y)\right)$ and ${\partial }_{i,y}{\phi }_1(x,y)$ are admissible test functions in Definition 1 sens. □

Theorem 3.

We have these two strong convergence results

$$\left\{\begin{array}{cc}{\displaystyle \stackrel{\sim }{{\mathbf{u}}^{\epsilon }}\left(x\right)-\chi \left(\frac{x}{\epsilon }\right)[\mathbf{u}\left(x\right)+{\mathbf{u}}_1(x,\frac{x}{\epsilon })]\to 0}\hfill & strongly in{\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)\\ {\displaystyle \stackrel{\sim }{{\phi }^{\epsilon }}\left(x\right)-\chi \left(\frac{x}{\epsilon }\right)[\phi \left(x\right)+{\phi }_1(x,\frac{x}{\epsilon })]\to 0}\hfill & strongly in{H}_0^1\left(\mathsf{\Omega }\right)\end{array}\right.$$

Proof. 

We consider the variational formulation under the following form:

$${\int }_{{\mathsf{\Omega }}_{\epsilon }}\{[c_{ijkl}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+e_{kij}^{\epsilon }{\partial }_k{\phi }^{\epsilon }]s_{ij}\left(\mathbf{v}\right)+[-e_{ikl}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+d_{ij}^{\epsilon }{\partial }_j{\phi }^{\epsilon }]{\partial }_i\psi \}dx={\int }_{{\mathsf{\Omega }}_{\epsilon }}{f}_i\left(x\right)v_i\left(x\right)dx$$

(45)

Choosing in the variational formulation (45) $\mathbf{v}={\mathbf{u}}^{\epsilon } \mathrm{and} \psi ={\phi }^{\epsilon }$, we obtain

$${\int }_{{\mathsf{\Omega }}_{\epsilon }}\{[c_{ijkl}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+e_{kij}^{\epsilon }{\partial }_k{\phi }^{\epsilon }]s_{ij}\left({\mathbf{u}}^{\epsilon }\right)+[-e_{ikl}^{\epsilon }{s}_{kl}\left({\mathbf{u}}^{\epsilon }\right)+d_{ij}^{\epsilon }{\partial }_j{\phi }^{\epsilon }]{\partial }_i{\phi }^{\epsilon }\}dx={\int }_{{\mathsf{\Omega }}_{\epsilon }}{f}_i\left(x\right)u_i^{\epsilon }\left(x\right)dx$$

After some simplifications, we obtain

$${\int }_{{\mathsf{\Omega }}_{\epsilon }}\{c_{ijkl}^{\epsilon }\left(x\right)s_{ij}\left({\mathbf{u}}^{\epsilon }\right)\left(x\right)s_{kl}\left({\mathbf{u}}^{\epsilon }\right)\left(x\right)+d_{ij}^{\epsilon }\left(x\right){\partial }_i{\phi }^{\epsilon }\left(x\right){\partial }_j{\phi }^{\epsilon }\left(x\right)\}dx={\int }_{{\mathsf{\Omega }}_{\epsilon }}{f}_i\left(x\right)u_i^{\epsilon }\left(x\right)dx$$

(46)

By applying (46), we can write

$$\begin{array}{ccc}& {\displaystyle {\int }_{\mathsf{\Omega }}}& c_{ijkl}^{\epsilon }\left\{{\stackrel{\sim }{s}}_{ij}\left({\mathbf{u}}^{\epsilon }\right)-\chi \left(\frac{x}{\epsilon }\right)\left[s_{ij,x}\left(\mathbf{u}\right)+s_{ij,y}\left({\mathbf{u}}_1\right)\right]\right\}\left\{{\stackrel{\sim }{s}}_{kl}\left({\mathbf{u}}^{\epsilon }\right)-\chi \left(\frac{x}{\epsilon }\right)\left[s_{kl,x}\left(\mathbf{u}\right)+s_{kl,y}\left({\mathbf{u}}_1\right)\right]\right\}dx\hfill \\ \\ & {\displaystyle +{\int }_{\mathsf{\Omega }}}& d_{ij}^{\epsilon }\left(x\right)\left\{\stackrel{\sim }{{\partial }_i}{\phi }^{\epsilon }\left(x\right)-\chi \left(\frac{x}{\epsilon }\right)\left[{\partial }_{i,x}\phi +{\partial }_{i,y}{\phi }_1\right]\right\}\left\{\stackrel{\sim }{{\partial }_j}{\phi }^{\epsilon }\left(x\right)-\chi \left(\frac{x}{\epsilon }\right)\left[{\partial }_{j,x}\phi +{\partial }_{j,y}{\phi }_1\right]\right\}dx\hfill \\ \\ & =& {\int }_{\mathsf{\Omega }}{f}_i\left(x\right)\stackrel{\sim }{u_i^{\epsilon }}\left(x\right)dx\hfill \\ \\ & +& {\int }_{\mathsf{\Omega }}{c}_{ijkl}^{\epsilon }\left(x\right)\chi \left(\frac{x}{\epsilon }\right)\left[s_{ij,x}\left(\mathbf{u}\left(x\right)\right)+s_{ij,y}\left({\mathbf{u}}_1(x,\frac{x}{\epsilon })\right)\right]\left[s_{kl,x}\left(\mathbf{u}\left(x\right)\right)+s_{kl,y}\left({\mathbf{u}}_1(x,\frac{x}{\epsilon })\right)\right]dx\hfill \\ \\ & +& {\int }_{\mathsf{\Omega }}{d}_{ij}^{\epsilon }\left(x\right)\chi \left(\frac{x}{\epsilon }\right)\left[{\partial }_{i,x}\phi \left(x\right)+{\partial }_{i,y}{\phi }_1(x,\frac{x}{\epsilon })\right]\left[{\partial }_{j,x}\phi \left(x\right)+{\partial }_{j,y}{\phi }_1(x,\frac{x}{\epsilon })\right]dx\hfill \\ \\ & -& 2{\int }_{\mathsf{\Omega }}{c}_{ijkl}^{\epsilon }\left(x\right)\chi \left(\frac{x}{\epsilon }\right)\stackrel{\sim }{s_{ij}}\left({\mathbf{u}}^{\epsilon }\right)\left[s_{kl,x}\left(\mathbf{u}\left(x\right)\right)+s_{kl,y}\left({\mathbf{u}}_1(x,\frac{x}{\epsilon })\right)\right]dx\hfill \\ \\ & -& 2{\int }_{\mathsf{\Omega }}{d}_{ij}^{\epsilon }\left(x\right)\chi \left(\frac{x}{\epsilon }\right)\stackrel{\sim }{{\partial }_i}{\phi }^{\epsilon }\left(x\right)\left[{\partial }_{j,x}\phi \left(x\right)+{\partial }_{j,y}{\phi }_1(x,\frac{x}{\epsilon })\right]dx.\hfill \end{array}$$

Using the ellipticity property of the elasticity $\left(c_{ijkl}^{\epsilon }\right)$ and electric $\left(d_{ij}^{\epsilon }\right)$ tensors, we obtain

$$\begin{array}{ccc}& {\alpha }_c& \Vert {\stackrel{\sim }{s}}_{ij}\left({\mathbf{u}}^{\epsilon }\right)-\chi \left(\frac{x}{\epsilon }\right)s_{ij,x}\left(\mathbf{u}\left(x\right)\right)-\chi \left(\frac{x}{\epsilon }\right)s_{ij,y}\left({\mathbf{u}}_1(x,\frac{x}{\epsilon })\right){\Vert }_{{\mathbf{L}}^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \\ \hfill +& {\alpha }_d& \Vert \stackrel{\sim }{{\partial }_i}{\phi }^{\epsilon }\left(x\right)-\chi \left(\frac{x}{\epsilon }\right){\partial }_{i,x}\phi \left(x\right)-\chi \left(\frac{x}{\epsilon }\right){\partial }_{i,y}{\phi }_1(x,\frac{x}{\epsilon }){\Vert }_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \\ & \le & {\int }_{\mathsf{\Omega }}{f}_i\left(x\right)\stackrel{\sim }{u_i^{\epsilon }}\left(x\right)dx\hfill \\ \\ & +& {\int }_{\mathsf{\Omega }}{c}_{ijkl}^{\epsilon }\left(x\right)\chi \left(\frac{x}{\epsilon }\right)\left[s_{ij,x}\left(\mathbf{u}\left(x\right)\right)+s_{ij,y}\left({\mathbf{u}}_1(x,\frac{x}{\epsilon })\right)\right]\left[s_{kl,x}\left(\mathbf{u}\left(x\right)\right)+s_{kl,y}\left({\mathbf{u}}_1(x,\frac{x}{\epsilon })\right)\right]dx\hfill \\ \\ & +& {\int }_{\mathsf{\Omega }}{d}_{ij}^{\epsilon }\left(x\right)\chi \left(\frac{x}{\epsilon }\right)\left[{\partial }_{i,x}\phi \left(x\right)+{\partial }_{i,y}{\phi }_1(x,\frac{x}{\epsilon })\right]\left[{\partial }_{j,x}\phi \left(x\right)+{\partial }_{j,y}{\phi }_1(x,\frac{x}{\epsilon })\right]dx\hfill \\ \\ & -2& {\int }_{\mathsf{\Omega }}{c}_{ijkl}^{\epsilon }\left(x\right)\chi \left(\frac{x}{\epsilon }\right)\stackrel{\sim }{s_{ij}}\left({\mathbf{u}}^{\epsilon }\right)\left[s_{kl,x}\left(\mathbf{u}\left(x\right)\right)+s_{kl,y}\left({\mathbf{u}}_1(x,\frac{x}{\epsilon })\right)\right]dx\hfill \\ \\ & -2& {\int }_{\mathsf{\Omega }}{d}_{ij}^{\epsilon }\left(x\right)\chi \left(\frac{x}{\epsilon }\right)\stackrel{\sim }{{\partial }_i}{\phi }^{\epsilon }\left(x\right)\left[{\partial }_{j,x}\phi \left(x\right)+{\partial }_{j,y}{\phi }_1(x,\frac{x}{\epsilon })\right]dx.\hfill \end{array}$$

Using the fact that $s_{ij,y}\left({\mathbf{u}}_1(x,y)\right)$ and ${\partial }_{i,y}{\phi }_1(x,y)$ are admissible test functions and taking the limit in the sense of two-scale convergence on the second right-hand side of the inequality, we obtain

$$\begin{array}{ccc}& {\displaystyle {\alpha }_c\underset{\epsilon \to 0}{lim}{\Vert \stackrel{\sim }{s_{ij}}\left({\mathbf{u}}^{\epsilon }\right)-\chi \left(\frac{x}{\epsilon }\right)\{s_{ij,x}\left(\mathbf{u}\left(x\right)\right)-s_{ij,y}\left({\mathbf{u}}_1(x,\frac{x}{\epsilon })\right)\}\Vert }_{{\mathbf{L}}^2\left(\mathsf{\Omega }\right)}^2}& \hfill \\ & {\displaystyle +{\alpha }_d\underset{\epsilon \to 0}{lim}{\Vert \stackrel{\sim }{{\partial }_i}{\phi }^{\epsilon }\left(x\right)-\chi \left(\frac{x}{\epsilon }\right)\{{\partial }_{i,x}\phi \left(x\right)-{\partial }_{i,y}{\phi }_1(x,\frac{x}{\epsilon })\}\Vert }_{L^2\left(\mathsf{\Omega }\right)}^2}& \hfill \\ & {\displaystyle \le {\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}{f}_i\left(x\right)u_i\left(x\right)dxdy}& \hfill \\ & {\displaystyle -{\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}{c}_{ijkl}(x,y)[s_{ij,x}\left(\mathbf{u}\left(x\right)\right)+s_{ij,y}\left({\mathbf{u}}_1(x,y)\right)][s_{kl,x}\left(\mathbf{u}\left(x\right)\right)+s_{kl,y}\left({\mathbf{u}}_1(x,y)\right)]dxdy}& \hfill \\ & {\displaystyle -{\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}{d}_{ij}(x,y)[{\partial }_{i,x}\phi \left(x\right)+{\partial }_{i,y}{\phi }_1(x,y)][{\partial }_{j,x}\phi \left(x\right)+{\partial }_{j,y}{\phi }_1(x,y)]dxdy}\end{array}$$

(47)

Paying attention to two-scale homogenized problems (10)-(11)-(12), we observe that the right-hand side of the inequality (47) is also zero; therefore, we obtain

$${\displaystyle \underset{\epsilon \to 0}{lim}{\Vert \stackrel{\sim }{s_{ij}}\left({\mathbf{u}}^{\epsilon }\right)-\chi \left(\frac{x}{\epsilon }\right)\{s_{ij,x}\left(\mathbf{u}\left(x\right)\right)-s_{ij,y}\left({\mathbf{u}}_1(x,\frac{x}{\epsilon })\right)\}\Vert }_{{\mathbf{L}}^2\left(\mathsf{\Omega }\right)}=0}$$

and

$${\displaystyle \underset{\epsilon \to 0}{lim}{\Vert \stackrel{\sim }{{\partial }_i}{\phi }^{\epsilon }\left(x\right)-\chi \left(\frac{x}{\epsilon }\right)\{{\partial }_{i,x}\phi \left(x\right)-{\partial }_{i,y}{\phi }_1(x,\frac{x}{\epsilon })\}\Vert }_{L^2\left(\mathsf{\Omega }\right)}=0.}$$

 □

6. Asymptotic Behavior of the Energy

In this section, we are interested in the energetic aspect of our problem and give the result for the description of the asymptotic behaviour of the elastic, electric and total energies. The elastic and electric energies are given respectively by the following formulas:

$${\displaystyle E_1^{\epsilon }=\frac{1}{2}{\int }_{{\mathsf{\Omega }}_{\epsilon }}{c}_{ijkl}^{\epsilon }\left(x\right)s_{ij}\left({\mathbf{u}}^{\epsilon }\right)\left(x\right)s_{kl}\left({\mathbf{u}}^{\epsilon }\right)\left(x\right)dx,}$$

(48)

$${\displaystyle E_2^{\epsilon }=\frac{1}{2}{\int }_{{\mathsf{\Omega }}_{\epsilon }}{d}_{ij}^{\epsilon }\left(x\right){\partial }_i{\phi }^{\epsilon }\left(x\right){\partial }_j{\phi }^{\epsilon }\left(x\right)dx.}$$

(49)

We worked analogously with the article of Allaire [19], and we use the two-scale convergence for this description; then, it is easy, and we obtain the result

Proposition 6.

We have the asymptotic limits of mechanical and electrical energies

(i) 

From the mechanical deformation energy,

$$\begin{array}{ccc}\hfill E_1^0& =& \underset{\epsilon \to 0}{lim}{E}_1^{\epsilon }\hfill \\ & =& \frac{1}{2}{\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}{c}_{ijkl}(x,y)\left(s_{ij,x}\left(\mathbf{u}\right)\left(x\right)+s_{ij,y}\left({\mathbf{u}}_1\right)(x,y)\right)\left(s_{kl,x}\left(\mathbf{u}\right)\left(x\right)+s_{kl,y}\left({\mathbf{u}}_1\right)(x,y)\right)dxdy.\hfill \end{array}$$

(ii) 

From the electrical energy,

$$\begin{array}{ccc}\hfill E_2^0& =& \underset{\epsilon \to 0}{lim}{E}_2^{\epsilon }\hfill \\ & =& \frac{1}{2}{\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}{d}_{ij}(x,y)\left({\partial }_{i,x}\phi \left(x\right)+{\partial }_{i,y}{\phi }_1(x,y)\right)\left({\partial }_{j,x}\phi \left(x\right)+{\partial }_{j,y}{\phi }_1(x,y)\right)dxdy.\hfill \end{array}$$

The index x or y indicated the derivation variable.

Remark 3.

If we consider the second variational problem (7) and (8), then the total energy associated the initial problem is given by

$$\begin{array}{ccc}\hfill E^{\epsilon }& =& \frac{1}{2}\left\{{\int }_{{\mathsf{\Omega }}_{\epsilon }}{c}_{ijkl}^{\epsilon }\left(x\right)s_{ij}\left({\mathbf{u}}^{\epsilon }\right)\left(x\right)s_{kl}\left({\mathbf{u}}^{\epsilon }\right)\left(x\right)dx+{\int }_{{\mathsf{\Omega }}_{\epsilon }}{d}_{ij}^{\epsilon }\left(x\right){\partial }_i{\phi }^{\epsilon }\left(x\right){\partial }_j{\phi }^{\epsilon }\left(x\right)dx\right\}\hfill \\ & +& 2{\int }_{{\mathsf{\Omega }}_{\epsilon }}{e}_{ijk}^{\epsilon }\left(x\right)s_{ij}\left({\mathbf{u}}^{\epsilon }\right)\left(x\right){\partial }_k{\phi }^{\epsilon }\left(x\right)dx.\hfill \end{array}$$

Similarly, for Proposition 6 and applying the two-scale technique, we obtain the asymptotic convergence

$$\begin{array}{ccc}\hfill E& =& \underset{\epsilon \to 0}{lim}{E}^{\epsilon },\hfill \\ & =& \frac{1}{2}{\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}{c}_{ijkl}(x,y)\left[s_{ij,x}\left(\mathbf{u}\right)\left(x\right)+s_{ij,y}\left({\mathbf{u}}_1\right)(x,y)\right]\left[s_{kl,x}\left(\mathbf{u}\right)\left(x\right)+s_{kl,y}\left({\mathbf{u}}_1\right)(x,y)\right]dxdy\hfill \\ & +& \frac{1}{2}{\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}{d}_{ij}(x,y)\left[{\partial }_{i,x}\phi \left(x\right)+{\partial }_{i,y}{\phi }_1(x,y)\right]\left[{\partial }_{j,x}\phi \left(x\right)+{\partial }_{j,y}{\phi }_1(x,y)\right]dxdy\hfill \\ & +& 2{\int }_{\mathsf{\Omega }}{\int }_{Y^{*}}{e}_{ijk}(x,y)\left[s_{ij,x}\left(\mathbf{u}\right)\left(x\right)+s_{ij,y}\left({\mathbf{u}}_1\right)(x,y)\right]\left[{\partial }_{k,x}\phi \left(x\right)+{\partial }_{k,y}{\phi }_1(x,y)\right]dxdy.\hfill \end{array}$$

7. Conclusions

In this work, we provided new convergence results, and we rigorously established the limiting equations modelling the behaviour of a piezoelectric periodically perforated structure, i.e., we have explicitly described forms of elastic, piezoelectric and dielectric homogenized coefficients. The two-scale convergence applied to our problem yielded a strong convergence result for the correctors. This technique of two-scale convergence can also handle homogenization problems in media with a periodic structure—for example, the laminated piezocomposite materials or fiber materials ([13,16,17,20,22]).