\documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 34, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/34\hfil Asymptotic behavior of solutions] {Asymptotic behavior of solutions on a thin plastic plate} \author[A. Ait Moussa, J. Messaho\hfil EJDE-2009/34\hfilneg] {Abdelaziz Ait Moussa, Jamal Messaho} % in alphabetical order \address{Abdelaziz Ait Moussa \newline D\'epartement de math\'ematiques et informatique\\ Facult\'e des sciences, Universit\'e Mohammed 1er\\ Oujda, Morocco} \email{a\_aitmoussa@yahoo.fr} \address{Jamal Messaho \newline D\'epartement de math\'ematiques et informatique\\ Facult\'e des sciences, Universit\'e Mohammed 1er\\ Oujda, Morocco} \email{j\_messaho@yahoo.fr} \thanks{Submitted July 30, 2008. Published February 23, 2009.} \subjclass[2000]{35B40, 82B24, 76M50} \keywords{Asymptotic behavior; plasticity problem; epiconvergence method; \hfill\break\indent limit problems} \begin{abstract} In the present work, we study the asymptotic behavior of solutions to a plasticity problem in a containing structure, a thin plastic plate of thickness that tends to zero. To find the limit problems with interface conditions we use the epiconvergence method. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newcommand{\norm}[1]{\|#1\|} \newcommand{\abs}[1]{|#1|} \section{Introduction} The study of the inclusion between two elastic bodies involves introducing a very thin third body between them. A very similar situation occurs when taking into account the effects of a thin layer which has been bonded onto the surface of a body to prevent wear caused by the contact with another solid. It is, therefore of interest to study the asymptotic behavior of thin layer between the two bodies, assuming various contact laws between them. In the case of a thin plate, the thermal conductivity problems were treated by Brillard et al and Sanchez-Palencia et al in \cite{brd1, sanch}. The elasticity problems, linear and nonlinear case, were widely studied by Ait Moussa et al, Ait moussa, Brillard et al, Geymonat et al and Lenci et al in \cite{aitmoussa, moussa, brd2, Gey, lenci}. In the case of an oscillating layer, we have treated the scalar case for a thermal conductivity problem in Messaho et al in \cite{messaho}. In the present work, we consider a structure containing a thin plastic plate of thickness depending on a parameter $\varepsilon$ intended to tend towards 0. The aim of this work is to study the asymptotic behavior of the solution of a plasticity problem posed on a such structure. This paper is organized in the following way. In section \ref{sec:2}, we express the problem to study, and we give some notation and we define functional spaces for this study in the section \ref{sec:3}. In the section \ref{sec:4}, we study the problem \eqref{pbmin}. The section \ref{sec:5} is reserved to the determination of the limits problems and our main result. \section{Statement of the problem} \label{sec:2} We consider a structure constituted of two linear elastics bodies, joined together by a thin plastic plate of thickness $\varepsilon$, the latter obeys to a nonlinear plastic law of power type. More precisely the stress field is related to the displacement's field by \begin{align*} \sigma^{\varepsilon}&=& \lambda|e(u^{\varepsilon})|^{-1}e(u^{\varepsilon}),\quad \lambda> 0. \end{align*} The structure occupies the regular domain $\Omega=B_{\varepsilon}\cup\Omega_{\varepsilon}$, where $B_{\varepsilon}$ is given by $B_{\varepsilon}=\{x=(x',x_3)/\abs{x_3}<\frac{\varepsilon}{2}\}$, and $\Omega_\varepsilon= \Omega \setminus B_{\varepsilon}$ represent the regions occupied by the thin plate and the two elastic bodies (see figure \ref{fig1}). $\varepsilon$ being a positive parameter intended to approach 0. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig1} \caption{The domain $\Omega$.}\label{fig1} \end{center} \end{figure} The structure is subjected to a density of forces of volume $f$, $f\colon \Omega\to\mathbb{R}^3$, and it is fixed on the boundary $\partial\Omega$. Equations which relate the stress field $\sigma^{\varepsilon}$, $\sigma^{\varepsilon}\colon \Omega\to\mathbb{R}^9_{S}$, and the field of displacement $u^\varepsilon$, $u^\varepsilon\colon \Omega\to\mathbb{R}^3$ are $$\label{prob} \begin{gathered} \mathop{\rm div}\sigma^\varepsilon+f=0\quad \text{in }\Omega,\\ \sigma_{ij}^{\varepsilon}= a_{ijkh} e_{kh}(u^\varepsilon) \quad \text{in }\Omega_\varepsilon,\\ \sigma^{\varepsilon}=\lambda|e(u^{\varepsilon})|^{-1}e(u^{\varepsilon}) \quad\text{in }B_\varepsilon,\\ u^\varepsilon=0\quad \text{on }\partial \Omega. \end{gathered}$$ Where $a_{ijkh}$ are the elasticity coefficients and $\mathbb{R}^9_{S}$ the vector space of the square symmetrical matrices of order three. $e_{ij}(u)$ are the components of the linearized tensor of deformation $e(u)$. In the sequel, we assume that the elasticity coefficients $a_{ijkh}$ satisfy to the following hypotheses: \begin{gather} a_{ijkh}\in L^\infty(\Omega),\label{h1} \\ a_{ijkh}=a_{jikh}=a_{khij},\label{h2}\\ a_{ijkh}\tau_{ij}\tau_{kh}\ge C \tau_{ij}\tau_{ij},\quad \forall \tau\in \mathbb{R}^9_{S}\label{h3}. \end{gather} \section{Notation and functional setting}\label{sec:3} Here is the notation that will be used in the sequel:\\ $x=(x',x_3)$ where $x'=(x_1,x_2)$, $\tau\otimes\zeta= (\tau_{i}\zeta_{j})_{1\le i,j\le 3}$ and $\tau\otimes_{S}\zeta=\frac{\tau\otimes\zeta+\zeta\otimes\tau}{2}$ for all $\tau,\zeta\in \mathbb{R}^3$. In the following $C$ will denote any constant with respect to $\varepsilon$. Also, we use the convention $0.(+\infty)=0$. \subsection*{Functional setting} First, we introduce the space \begin{align*} V^\varepsilon=\big\{& u\in L^{1}(\Omega,\mathbb{R}^3): e(u)\in L^2(\Omega_\varepsilon,\mathbb{R}^9_S),\; u\in BD(B_{\varepsilon}),\\ &[u]^{\varepsilon}=0 \text{ in }\Sigma_\varepsilon^{\pm} \text{ and }u=0 \text{ in }\partial\Omega \big\}, \end{align*} where $[u]^{\varepsilon}$ is the jump of $u$ on $\Sigma_\varepsilon^{\pm}$ defined by \begin{gather*} [u]^{\varepsilon} =\pm u_{|_{\Omega_{\varepsilon}^{\pm}}}\mp u_{|_{B_\varepsilon^{\pm}}},\\ BD(B_{\varepsilon})=\big\{ u\in L^{1}(\Omega,\mathbb{R}^3): e(u)\in M_{1}(B_\varepsilon,\mathbb{R}^9_S)\big\},\\ BD_0(\Omega)=\big\{ u\in BD(\Omega,\mathbb{R}^3): u= 0 \text{ in } \partial\Omega\big\}, \end{gather*} and $M_{1}(.)$ is a bounded measure space, for more information we can refer the reader to \cite{temam}. We show easily that $V^\varepsilon$ is a Banach space with the norm $$u\to \norm{e(u)}_{L^2(\Omega_{\varepsilon},\mathbb{R}^9_{S})} +\norm{e(u)}_{M_{1}(B_{\varepsilon},\mathbb{R}^9_{S})}.$$ Where $\norm{e(u)}_{M_{1}(B_{\varepsilon},\mathbb{R}^9_{S})} =\int_{B_\varepsilon}\abs{e(u)} =\sup_{\tau\in\mathcal{C}^{\infty}_0{(B_\varepsilon)},\, \abs{\tau(x)}\le 1.} \big\langle e(u), \tau \big\rangle.$ We remark that $V^\varepsilon\subset BD_0(\Omega)$. Our goal in this work is to study the problem (\ref{prob}), and its limit behavior when $\varepsilon$ tends to zero. \section{Study of problem \eqref{prob}}\label{sec:4} Problem (\ref{prob}) is equivalent to the minimization problem $$\label{pbmin} \inf_{v\in V^{\varepsilon}}\big\{\frac{1}{2} \int_{\Omega_\varepsilon} a_{ijhk}e_{hk}(v)e_{ij}(v)dx +\lambda\int_{B_\varepsilon}\abs{e( v)} -\int_{\Omega}fvdx\big\}$$ To study problem \eqref{prob}, we will study the minimization problem \eqref{pbmin}. The existence and uniqueness of solutions to \eqref{pbmin} is given in the following proposition. \begin{proposition}\label{existe} Under the hypotheses \eqref{h1}, \eqref{h2}, \eqref{h3} and for $f\in L^{\infty}(\Omega,\mathbb{R}^3)$, problem \eqref{pbmin} admits an unique solution $u^\varepsilon$ in $V^\varepsilon$. \end{proposition} \begin{proof} From \eqref{h1} and \eqref{h3}, we show easily that the energy functional in \eqref{pbmin} is weakly lower semicontinuous, strictly convex and coercive over $V^\varepsilon$. Since $V^\varepsilon$ is not reflexive, so we may not apply directly result given in Dacorogna \cite[theorem 1.1 p.48]{daco}, but we can follow our proof by using the compact imbedding of Sobolev for the BD space, for more information we can refer the reader to \cite{daco}. Indeed, let $u_n$ be a minimizing sequence for \eqref{pbmin}, to simplify the writing let $$\mathbb{F}^\varepsilon(u)=\frac{1}{2} \int_{\Omega_\varepsilon} a_{ijhk}e_{hk}(u)e_{ij}(u)dx +\lambda\int_{B_\varepsilon}\abs{e(u)} -\int_{\Omega}fudx,$$ so, we have $\displaystyle\mathbb{F}^\varepsilon(u_n)\to \inf_{v\in V^\varepsilon}\mathbb{F}^\varepsilon(v)$. Using the coercivity of $\mathbb{F}^\varepsilon$, we may then deduce that there exists a constant $C>0$, independent of $n$, such that $$\norm{u_n}_{V^\varepsilon}\le C,$$ according to the reflexivity of $H^1(\Omega_\varepsilon)$ and using the given result in \cite[p.158]{temam} for $BD(B_\varepsilon)$, then for a subsequence of $u_n$, still denoted by $u_n$, there exists $u_0\in V^\varepsilon$ such that $u_n\rightharpoonup u_0$ in $V^\varepsilon$. The weak lower semi-continuity and the strict convexity of $\mathbb{F}^\varepsilon$ imply then the result. \end{proof} \begin{lemma}\label{lemme1} Assuming that for any sequence $(u^\varepsilon)_{\varepsilon>0}\subset V^\varepsilon$, there exists a constant $C>0$ such that $\mathbb{F}^\varepsilon(u^\varepsilon)\le C$, under \eqref{h1}, \eqref{h3} and for $f\in L^{\infty}(\Omega,\mathbb{R}^3)$, $(u^\varepsilon)_{\varepsilon>0}$ satisfies \begin{gather} \norm{e(u^\varepsilon)}_{L^2(\Omega_\varepsilon, \mathbb{R}^9_{S})}^2\leq C, \label{assert:1}\\ \norm{e(u^\varepsilon)}_{M_1(B_\varepsilon, \mathbb{R}^9_{S})} \leq C,\label{assert:2} \end{gather} moreover $u^\varepsilon$ is bounded in $BD_0(\Omega,\mathbb{R}^3)$. \end{lemma} \begin{proof} Since $\mathbb{F}^\varepsilon(u^\varepsilon)\le C$, we have $$\frac{1}{2} \int_{\Omega_\varepsilon} a_{ijhk}e_{hk}(u^\varepsilon)e_{ij}(u^\varepsilon)dx +\lambda\int_{B_\varepsilon}\abs{e(u^\varepsilon)} -\int_{\Omega}fu^\varepsilon dx\le C\,.$$ Then $\frac{1}{2} \int_{\Omega_\varepsilon} a_{ijhk}e_{hk}(u^\varepsilon)e_{ij}(u^\varepsilon)dx +\lambda\int_{B_\varepsilon}\abs{e(u^\varepsilon)} \le C + \int_{\Omega}fu^\varepsilon dx \,.$ According to \eqref{h3}, H\"{o}lder and Young the inequalities, we have \begin{align*} \norm{e( u^\varepsilon)}^2_{L^2(\Omega_\varepsilon,\mathbb{R}^9_{S})} +\int_{B_\varepsilon}\abs{e(u^\varepsilon)} &\le C + C\int_{\Omega}fu^\varepsilon dx ,\\ &\le C + C\norm{e(u^\varepsilon)}_{L^2(\Omega_\varepsilon,\mathbb{R}^9_{S})} + \int_{B_\varepsilon}fu^\varepsilon dx, \end{align*} since $BD(\Omega)\hookrightarrow L^{q}(\Omega,\mathbb{R}^3)$ for all $q\in[1,\frac{3}{2}]$, (with a continuous imbedding, see for example \cite{temam}). In particular $BD(\Omega)\hookrightarrow L^{q_0}(\Omega,\mathbb{R}^3)$ with $10$ and $\Omega^\eta=\big\{ x\in\Omega: \abs{x_3}>\eta\big\}$, such that $\varepsilon<\eta$. From \eqref{assert:1}, we then have $\norm{e(u^\varepsilon)}_{L^2(\Omega^\eta, \mathbb{R}^9_{S})}^2\leq C,$ Therefore, $e(u^\varepsilon)$ is bounded in $L^2(\Omega^\eta,\mathbb{R}^9_S)$, so for a subsequence of $e(u^\varepsilon)$, still denoted by $e(u^\varepsilon)$, there exists $w\in L^2(\Omega^\eta,\mathbb{R}^9_S)$, such that $$e(u^\varepsilon)\rightharpoonup w \quad \text{in }L^2(\Omega^\eta,\mathbb{R}^9_S),$$ according \eqref{eq:a} and \eqref{eq:b} remains true in $\mathcal{C}^{\infty}_0(\Omega^\eta,\mathbb{R}^9_S)$, we then deduce $e(u^*)=w$, hence $e(u^*)\in L^2(\Omega^\eta,\mathbb{R}^9_S)$ for all $\eta>0$, so by passing to the limit ($\eta\to 0$), we then have $e(u^*)\in L^2(\Omega\setminus\Sigma,\mathbb{R}^9_S)$. According to the classical result \cite[proposition 1.2, p. 16]{temam}, we have $u^*\in H^1(\Omega\setminus\Sigma,\mathbb{R}^3)$. \end{proof} In the following, let \begin{gather*} \mathbb{H}^{1}_0=\big\{u\in H^1(\Omega\setminus\Sigma,\mathbb{R}^3): u=0 \text{ on }\partial\Omega\big\}. \\ \mathbb{C}^{\infty}_0=\big\{u\in \mathcal{C}^\infty(\Omega\setminus\Sigma,\mathbb{R}^3) : u=0 \text{ on }\partial\Omega\big\}. \end{gather*} \begin{remark}\label{remark1} \rm Proposition \ref{prop2} remains valid for any weak cluster point $u$ of a sequence $u_\varepsilon$ in $V^\varepsilon$,that satisfies \eqref{assert:1} and \eqref{assert:2}. \end{remark} To study the limit behavior of the solution of the problem \eqref{pbmin}, we will use the epiconvergence method, (see Annex, definition \ref{def:epi}). \section{Limit behavior}\label{sec:5} Let $$\label{F0} \begin{gathered} F^{\varepsilon}(u)= \begin{cases} \displaystyle\frac{1}{2} \int_{\Omega_\varepsilon} a_{ijkh} e_{kh}(u)e_{ij}(u)dx +\lambda\int_{B_\varepsilon}\abs{e( u)} &\text{if }u\in V^\varepsilon,\\ +\infty &\text{if }u\in BD_0(\Omega)\setminus V^\varepsilon. \end{cases} \\ G(u)=-\int_{\Omega}fu dx, \quad \forall u\in BD_0(\Omega). \end{gathered}$$ We design by $\tau_f$ the weak topology on the space ${BD}_0(\Omega)$. In the sequel, we shall characterize, the epi-limit of the energy functional given by (\ref{F0}) in the following theorem. \begin{theorem}\label{cal:epi} Under \eqref{h1}, \eqref{h2}, \eqref{h3} and for $f\in L^{\infty}(\Omega,\mathbb{R}^3)$, there exists a functional $F\colon BD_0(\Omega)\to \mathbb{R}\cup\{+\infty\}$ such that $\tau_f-lim _{e}F^{\varepsilon}{=}F\quad \text{in }BD_0(\Omega),$ where $F$ is given by $F(u)= \begin{cases} \displaystyle\frac{1}{2} \int_{\Omega} a_{ijkh} e_{kh}(u)e_{ij}(u)+\lambda\int_{\Sigma}\abs{[u]\otimes_Se_3} &\text{if } u\in \mathbb{H}^{1}_0,\\ +\infty &\text{if }u\in BD_0(\Omega)\setminus\mathbb{H}^{1}_0. \end{cases}$ \end{theorem} \begin{proof} (a) We are now in position to determine the upper epi-limit.\\ Let $u\in \mathbb{H}^{1}_0\subset BD_0(\Omega)$, so there exists a sequence $(u^n)$ in $\mathbb{C}^{\infty}_0$ such that $u^n \to u$ in $\mathbb{H}^{1}_0\text{when } n\to +\infty$, so $u^n \rightharpoonup u$ weakly in $BD_0(\Omega)$. Let us consider the sequence $u^{\varepsilon,n}=\begin{cases} u^{n}(x',x_3)& \text{if }\abs{x_3}>\frac{\varepsilon}{2},\\[3pt] \frac{1}{2}\big(u^{n}(x',\frac{\varepsilon}{2})+u^{n}(x',-\frac{\varepsilon}{2})\big)\\ +\frac{x_3}{\varepsilon} \big(u^{n}(x',\frac{\varepsilon}{2})-u^{n}(x',-\frac{\varepsilon}{2})\big) & \text{if }\abs{x_3}<\frac{\varepsilon}{2}. \end{cases}$ We have $u^{\varepsilon,n}\in V^\varepsilon$ and we prove easily that $u^{\varepsilon,n}\rightharpoonup u^n$ in $\mathbb{H}^{1}_0$ when $\varepsilon\to 0$. As \begin{align*} F^\varepsilon(u^{\varepsilon,n})&=& \frac{1}{2} \int_{\Omega_\varepsilon} a_{ijkh}e_{kh}(u^{\varepsilon,n})e_{ij}(u^{\varepsilon,n}) +\lambda\int_{B_\varepsilon}\abs{ e(u^{\varepsilon,n})}. \end{align*} It implies that $F^\varepsilon(u^{\varepsilon,n}) = \frac{1}{2} \int_{\Omega_\varepsilon} a_{ijkh}e_{kh}(u^{n})e_{ij}(u^{n}) +\lambda \int_{B_\varepsilon}\abs{e( u^{\varepsilon,n})} =: S_1+S_2.$ So that $\lim_{\varepsilon\to 0}S_1= \frac{1}{2} \int_{\Omega}a_{ijkh}e_{kh}(u^{n})e_{ij}(u^{n}).$ we have $$\label{s3} S_2=\lambda\int_{B_\varepsilon}\abs{e(u^{\varepsilon,n})},$$ As in \cite{moussa} we show that $%\label{s3} \lim_{\varepsilon\to 0} \int_{B_\varepsilon} | e(u^{\varepsilon,n}) -\frac{1}{\varepsilon}[u^n]\otimes_{S}e_{3} |=0.$ Consequently, $\limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon,n}) =\frac{1}{2} \int_{\Omega} a_{ijkh}e_{kh}(u^{n})e_{ij}(u^{n})+ \lambda\int_{\Sigma} \abs{[u^n]\otimes_{S}e_{3}}.$ Since $u^{n}\to u$ in $\mathbb{H}^{1}_0$ when $n\to+\infty$, therefore according to a classic result, diagonalization's lemma, (see, \cite[Lemma 1.15 p. 32]{attouch}), there exists a function $n(\varepsilon):\mathbb{R}^+\to\mathbb{N}$ increasing to $+\infty$ when $\varepsilon\to 0$ such that $u^{\varepsilon,n(\varepsilon)}\rightharpoonup u$ in $\mathbb{H}^{1}_0$ when $\varepsilon\to 0$. and while $n\to+\infty$, consequently we have \begin{align*} \limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon,n(\varepsilon)}) &\leq \limsup_{n\to +\infty}\limsup_{\varepsilon\to 0} F^\varepsilon(u^{\varepsilon,n}),\\ &\leq \frac{1}{2} \int_{\Omega} a_{ijkh}e_{kh}(u)e_{ij}(u)+ \lambda\int_{\Sigma} \abs{[u]\otimes_{S}e_{3}}. \end{align*} For $u\in BD_0(\Omega,\mathbb{R}^3)\setminus\mathbb{H}^{1}_0$, so for any sequence $u^\varepsilon\rightharpoonup u$ in $BD_0(\Omega)$,we obtain $$\limsup_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon})\leq +\infty.$$ (b) We are now in position to determine the lower epi-limit. Let $u\in \mathbb{H}^{1}_0$ and $(u^\varepsilon)\subset V^\varepsilon$ such that $u^\varepsilon{\rightharpoonup}u$ in $BD_0(\Omega)$. If $\liminf_{\varepsilon\to 0}F^\varepsilon(u^\varepsilon)=+\infty$, there is nothing to prove, because $$\frac{1}{2} \int_{\Omega} a_{ijkh}e_{kh}(u)e_{ij}(u)+\lambda \int_{\Sigma}\abs{[u]\otimes_{S}e_{3}}\leq +\infty.$$ otherwise, $\liminf_{\varepsilon\to 0}F^\varepsilon(u^\varepsilon)<+\infty$, there exists a subsequence of $F^\varepsilon(u^\varepsilon)$, still denoted by $F^\varepsilon(u^\varepsilon)$ and a constant $C>0$, such that $F^\varepsilon(u^\varepsilon)\leq C$, which implies that \begin{gather*} \norm{e(u^\varepsilon}_{L^2(\Omega_\varepsilon,\mathbb{R}^9_S)} \leq C,\\ \int_{B_{\varepsilon}}\abs{e(u^\varepsilon)} \leq C, \end{gather*} then $\chi_{{\Omega_{\varepsilon}}}e( u^\varepsilon)$ is bounded in $L^2(\Omega,\mathbb{R}^9_S)$, so for a subsequence of $\chi_{{\Omega_{\varepsilon}}}e( u^\varepsilon)$, still denoted by $\chi_{{\Omega_{\varepsilon}}}e(u^\varepsilon)$, we then show easily, like in the proof of the above proposition, that $$\chi_{{\Omega_{\varepsilon}}}e(u^\varepsilon) \rightharpoonup e( u)\quad \text{in }L^2(\Omega,\mathbb{R}^9_S)) \label{eq:liminf2}$$ From the subdifferentiability's inequality of $u\to \frac{1}{2} \int_{\Omega_\varepsilon} a_{ijkh}e_{kh}(u)e_{ij}(u)$, and passing to the lower limit, we obtain $\liminf_{\varepsilon\to 0}\frac{1}{2} \int_{\Omega_\varepsilon} a_{ijkh}e_{kh}(u^\varepsilon)e_{ij}(u^\varepsilon) \geq \frac{1}{2} \int_{\Omega} a_{ijkh}e_{kh}(u)e_{ij}(u).$ For $\eta<\varepsilon/2$, let us set $$B^\eta=\big\{x\in\Omega: \abs{x_3}<\eta\big\}.$$ According to the diagonalization's lemma \cite[Lemma 1.15 p. 32]{attouch}, there exists a function $\eta(\varepsilon):\mathbb{R}^+\to\mathbb{R}^+$ decreasing to $0$ when $\varepsilon\to 0$ such that $$\liminf_{\varepsilon\to 0}\int_{B^{\eta(\varepsilon)}}\abs{e(u^\varepsilon)} \ge\liminf_{\eta\to 0}\liminf_{\varepsilon\to 0}\int_{B^{\eta}} \abs{e(u^\varepsilon)}.\label{ddiag}$$ Since $$\int_{B^{\eta}}\abs{e(u^\varepsilon)}\ge \int_{B^{\eta}}\phi \big(e(u^\varepsilon)-e(u)\big)+\int_{B^{\eta}}\phi e(u) ,\quad \forall \phi\in \mathcal{C}^\infty_0(B^\eta,\mathbb{R}^9_S),$$ it follows that $$\liminf_{\varepsilon\to 0}\int_{B^{\eta}}\abs{e(u^\varepsilon)}\ge\int_{B^{\eta}}\phi e(u) ,\quad \forall \phi\in \mathcal{C}^\infty_0(B^\eta,\mathbb{R}^9_S).$$ Therefore, $$\liminf_{\varepsilon\to 0}\int_{B^{\eta}}\abs{e(u^\varepsilon)} \ge\int_{B^{\eta}}\abs{e(u)}.$$ According to a classic result \cite[Lemma 2.2 p. 145]{temam}), we then have $$\liminf_{\varepsilon\to 0}\int_{B^{\eta}}\abs{e(u^\varepsilon)} \ge\int_{B^{\eta}}\phi e(u)+\int_{\Sigma}\phi [u]\otimes_{S}e_{3}dx', \quad \forall \phi\in \mathcal{C}^\infty_0(\Omega,\mathbb{R}^9_S).$$ By passing to the limit, ($\eta\to 0$), we have $$\liminf_{\eta\to 0}\liminf_{\varepsilon\to 0}\int_{B^{\eta}}\abs{e(u^\varepsilon)} \ge\int_{\Sigma}\abs{[u]\otimes_{S}e_{3}}dx'.$$ According to the definition of $B^{\eta}$ and \eqref{ddiag}, we deduce that $$\liminf_{\varepsilon\to 0}\int_{B_{\varepsilon}}\abs{e(u^\varepsilon)} \ge\int_{\Sigma}\abs{[u]\otimes_{S}e_{3}}dx'.$$ Hence $$\liminf_{\varepsilon\to 0}F^\varepsilon(u^\varepsilon)\ge\frac{1}{2} \int_{\Omega} a_{ijkh}e_{kh}(u)e_{ij}(u)+\int_{\Sigma} \abs{[u]\otimes_{S}e_{3}}dx'.$$ For $u\in BD_0(\Omega)\setminus\mathbb{H}^{1}_0$ and $u^\varepsilon\in V^\varepsilon$, such that $u^\varepsilon\rightharpoonup u$ in $BD_0(\Omega)$. Assume that $\liminf_{\varepsilon\to 0}F^\varepsilon(u^{\varepsilon})<+\infty.$ So there exists a constant $C>0$ and a subsequence of $F^\varepsilon(u^{\varepsilon})$, still denoted by $F^\varepsilon(u^{\varepsilon})$, such that \label{eqliminf333} F^\varepsilon(u^{\varepsilon})