Asymptotic analysis of a multimaterial with a thin piezoelectric interphase

Abstract

We study the electromechanical behavior of a multimaterial constituted by a linear piezoelectric transversely isotropic plate-like body with high rigidity embedded between two generic three-dimensional piezoelectric bodies by means of the asymptotic expansion method. After defining a small real dimensionless parameter $\varepsilon$, which will tend to zero, we characterize the limit model and the associated limit problem. We give also a mathematical justification of the model by means of a functional convergence argument. Moreover, we identify the non classical electromechanical transmission conditions between the two three-dimensional bodies.

Appendix

In the section we justify from mathematical point of view the obtained limit model through a variational convergence result. We prove that the solution of the physical problem strongly converges towards the solution of the limit model. The strong convergence corresponds to the convergence of the energies.

In the sequel we denote by $\Vert \cdot \Vert _{s,\Omega }$ the norm of the Sobolev space $H^s(\Omega ; \mathbb {R}^d)$ for all $d\ge 1$ and $|\cdot |_{0,\Omega }$ stands for the norm in $L^2(\Omega ; \mathbb {R}^d)$. Obviously, the same holds in $\Omega ^\pm$, $\Omega ^m$ and $\omega$.

To begin with, we introduce the following notation. We let

$$\begin{aligned} \begin{array}{ll} \mathbf{a} \cdot \mathbf{b}:=a_{i}b_{i},\ \ \ \ \mathbf{A} : \mathbf{B}:=a_{ij}b_{ij}, \end{array} \end{aligned}$$

for, respectively, all vectors $\mathbf{a}=(a_{i})$ and $\mathbf{b}=(b_{i})$, and for all symmetric second order matrices $\mathbf{A}=(a_{ij})$ and $\mathbf{B}=(b_{ij})$.

With the scaled electromechanical state $s(\varepsilon )=({\mathbf {u}}(\varepsilon ), \varphi (\varepsilon ))\in V\times \Psi$, we associate, respectively, the following scaled strain tensor field $\varvec{\kappa }(\varepsilon )=(\kappa _{ij}(\varepsilon ))$, with $\kappa _{ij}(\varepsilon )\in L^2(\Omega )$, and the scaled electric potential vector field $\varvec{\theta }(\varepsilon )=(\theta _i(\varepsilon ))$, with $\theta _i(\varepsilon )\in L^2(\Omega )$, defined by

$$\begin{aligned} \begin{array}{ll} \kappa _{ij}^\pm (\varepsilon ):=e_{ij}({\mathbf {u}}^\pm (\varepsilon )),\ \ \theta ^\pm _i(\varepsilon ):=\partial _i\varphi ^\pm (\varepsilon )\\ \kappa _{\alpha \beta }^m(\varepsilon ):=e_{\alpha \beta }({\mathbf {u}}^m(\varepsilon )),\ \ \kappa ^m_{\alpha 3}(\varepsilon ):=\frac{1}{\varepsilon }e_{\alpha 3}({\mathbf {u}}^m(\varepsilon )),\ \ \kappa ^m_{33}(\varepsilon ):=\frac{1}{\varepsilon ^2}e_{33}({\mathbf {u}}^m(\varepsilon )),\\ \theta ^m_\alpha (\varepsilon ):=\varepsilon \partial _\alpha \varphi ^m(\varepsilon ),\ \ \theta ^m_3(\varepsilon ):=\partial _3\varphi ^m(\varepsilon ). \end{array} \end{aligned}$$

With an arbitrary electromechanical state $r=({\mathbf {v}}, \psi )\in V\times \Psi$, we associate, respectively, the following tensor field $\varvec{\kappa }(\varepsilon ;{\mathbf {v}})=(\kappa _{ij}(\varepsilon ;{\mathbf {v}}))$ and vector field $\varvec{\theta }(\varepsilon ;\psi )=(\theta _i(\varepsilon ;\psi ))$. In particular, one has $\varvec{\kappa }(\varepsilon )=\varvec{\kappa }(\varepsilon ;{\mathbf {u}}(\varepsilon ))$ and $\varvec{\theta }(\varepsilon )=\varvec{\theta }(\varepsilon ;\varphi (\varepsilon ))$. Then the rescaled problem (2) can be rewritten in the following condensed form:

$$\begin{aligned} \left\{ \begin{array}{ll} \text {Find } s(\varepsilon )\in V\times \Psi \ \text { such\;that}\\ A^{-}(s(\varepsilon ),r)+A^{+}(s(\varepsilon ),r)+ A^{m} (s(\varepsilon ),r)=L(r),\quad \text {for\;all}\ \ r\in V\times \Psi , \end{array}\right. \end{aligned}$$

(7)

where the bilinear forms $A^{\pm }$ and $A^{m}$ are given by

$$\begin{aligned} A^{\pm }(s(\varepsilon ),r)&:=\int \limits _{\Omega ^{\pm }}\left\{ \mathsf A ^\pm \varvec{\kappa }^\pm (\varepsilon ):\varvec{\kappa }^\pm (\varepsilon ;{\mathbf {v}})+\mathsf H ^{\pm }\varvec{\theta }^\pm (\varepsilon )\cdot \varvec{\theta }^\pm (\varepsilon ;\psi )\right. \\&\quad \left. +\,\mathsf P ^{\pm } \varvec{\kappa }^\pm (\varepsilon ;{\mathbf {v}})\cdot \varvec{\theta }^\pm (\varepsilon )-\mathsf P ^{\pm } \varvec{\kappa }^\pm (\varepsilon )\cdot \varvec{\theta }^\pm (\varepsilon ;\psi )\right\} dx,\\ A^{m}(s(\varepsilon ),r)&:=\int \limits _{\Omega ^{m}}\left\{ \mathsf A ^m \varvec{\kappa }^m(\varepsilon ):\varvec{\kappa }^m(\varepsilon ;{\mathbf {v}})+\mathsf H ^{m}\varvec{\theta }^m(\varepsilon )\cdot \varvec{\theta }^m(\varepsilon ;\psi )\right. \\&\quad \left. +\,\mathsf P ^{m} \varvec{\kappa }^m(\varepsilon ;{\mathbf {v}})\cdot \varvec{\theta }^m(\varepsilon )-\mathsf P ^{m} \varvec{\kappa }^m(\varepsilon )\cdot \varvec{\theta }^m(\varepsilon ;\psi )\right\} dx. \end{aligned}$$

$\mathsf A =(A_{ijk\ell })$, $\mathsf P =(P_{ijk})$ and $\mathsf H =(H_{ij})$ represent, respectively, the elasticity tensor, the coupling tensor and the dielectric tensor.

The strong convergence result is stated in the following theorem.

Theorem 4

The sequence $(s(\varepsilon ))_{\varepsilon >0}=({\mathbf {u}}(\varepsilon ), \varphi (\varepsilon ))_{\varepsilon >0}$ strongly converges to $s^0=({\mathbf {u}}^0, \varphi ^0)$ in $H^1(\Omega ,\mathbb {R}^3)\times L^2(\Omega )$, the unique solution of (4).

Proof

For convenience, the proof is divided into three steps, numbered from (i) to (iii).

• (i) By letting $r=s(\varepsilon )$, i.e., ${\mathbf {v}}={\mathbf {u}}(\varepsilon )$ and $\psi =\varphi (\varepsilon )$, the rescaled problem (7) becomes, as customary,

  $$\begin{aligned} \begin{array}{ll} &\int \limits _{\Omega ^{\pm }}\left\{ \mathsf A ^\pm \varvec{\kappa }^\pm (\varepsilon ):\varvec{\kappa }^\pm (\varepsilon )+\mathsf H ^{\pm }\varvec{\theta }^\pm (\varepsilon )\cdot \varvec{\theta }^\pm (\varepsilon )\right\} dx\\ &\quad +\,\int \limits _{\Omega ^{m}}\left\{ \mathsf A ^m \varvec{\kappa }^m(\varepsilon ):\varvec{\kappa }^m(\varepsilon )+\mathsf H ^{m}\varvec{\theta }^m(\varepsilon )\cdot \varvec{\theta }^m(\varepsilon )\right\} dx=L(s(\varepsilon )). \end{array} \end{aligned}$$

  By virtue of the positive definiteness of $\mathsf A$ and $\mathsf H$ and by means of the Korn’s inequality and the Poincaré’s inequality, we derive that

  $$\begin{aligned} \begin{array}{ll} &\int \limits _{\Omega ^{\pm }}\left\{ \mathsf A ^\pm \varvec{\kappa }^\pm (\varepsilon ):\varvec{\kappa }^\pm (\varepsilon )+\mathsf H ^{\pm }\varvec{\theta }^\pm (\varepsilon )\cdot \varvec{\theta }^\pm (\varepsilon )\right\} dx\\ &\quad +\,\int \limits _{\Omega ^{m}}\left\{ \mathsf A ^m \varvec{\kappa }^m(\varepsilon ):\varvec{\kappa }^m(\varepsilon )+\mathsf H ^{m}\varvec{\theta }^m(\varepsilon )\cdot \varvec{\theta }^m(\varepsilon )\right\} dx\\ &\ge C\left\{ |\varvec{\kappa }^\pm (\varepsilon )|^2_{0,\Omega ^\pm }+|\varvec{\theta }^\pm (\varepsilon )|^2_{0,\Omega ^\pm }+|\varvec{\kappa }^m(\varepsilon )|^2_{0,\Omega ^m}+|\varvec{\theta }^m(\varepsilon )|^2_{0,\Omega ^m}\right\} \\ &\ge C\left\{ |\varvec{\kappa }^\pm (\varepsilon )|^2_{0,\Omega ^\pm }+|\varvec{\theta }^\pm (\varepsilon )|^2_{0,\Omega ^\pm }+ \sum \limits _{\alpha ,\beta }|e_{\alpha \beta }({\mathbf {u}}(\varepsilon ))|^2_{0,\Omega ^m}+\frac{1}{\varepsilon ^2} \sum \limits _{\alpha }|e_{\alpha 3}({\mathbf {u}}(\varepsilon ))|^2_{0,\Omega ^m}\right. \\ &\quad \left. +\,\frac{1}{\varepsilon ^4}|e_{33}({\mathbf {u}}(\varepsilon ))|^2_{0,\Omega ^m}+\varepsilon ^2 \sum \limits _{\alpha }|\partial _\alpha \varphi (\varepsilon )|^2_{0,\Omega ^m}+|\partial _3 \varphi (\varepsilon )|^2_{0,\Omega ^m}\right\} \\ &\ge \Bigg\{ \Vert {\mathbf {u}}(\varepsilon )\Vert ^2_{1,\Omega ^\pm }+|\varphi (\varepsilon )|^2_{0,\Omega ^\pm }+ \Vert {\mathbf {u}}(\varepsilon )\Vert ^2_{1,\Omega ^m}+\varepsilon ^2 \sum \limits _{\alpha }|\partial _\alpha \varphi (\varepsilon )|^2_{0,\Omega ^m}+|\partial _3 \varphi (\varepsilon )|^2_{0,\Omega ^m}\Bigg\} . \end{array} \end{aligned}$$

  On the other side, by virtue of the continuity of the linear form, one has:

  $$\begin{aligned} \begin{array}{ll} L(s(\varepsilon )) &\le C\left\{ \Vert {\mathbf {u}}(\varepsilon )\Vert _{1,\Omega ^\pm }+|\varphi (\varepsilon )|_{0,\Omega ^\pm }\right\} \\ &\le \left\{ \Vert {\mathbf {u}}(\varepsilon )\Vert ^2_{1,\Omega ^\pm }+|\varphi (\varepsilon )|^2_{0,\Omega ^\pm }+ \Vert {\mathbf {u}}(\varepsilon )\Vert ^2_{1,\Omega ^m}+\varepsilon ^2 \sum \limits _{\alpha }|\partial _\alpha \varphi (\varepsilon )|^2_{0,\Omega ^m}+|\partial _3 \varphi (\varepsilon )|^2_{0,\Omega ^m}\right\} ^{1/2}. \end{array} \end{aligned}$$

  The inequalities above imply that the norms $|\varvec{\kappa }^\pm (\varepsilon )|_{0,\Omega ^\pm }$, $|\varvec{\theta }^\pm (\varepsilon )|_{0,\Omega ^\pm }$, $|\varvec{\kappa }^m(\varepsilon )|_{0,\Omega ^m}$ and $|\varvec{\theta }^m(\varepsilon )|_{0,\Omega ^m}$ are bounded independently of $\varepsilon$ in $L^2(\Omega )$. This means that $\kappa _{ij}^\pm (\varepsilon )\rightharpoonup \kappa _{ij}^\pm$ in $L^2(\Omega ^\pm )$, $\theta _i^\pm (\varepsilon )\rightharpoonup \theta _i^\pm$ in $L^2(\Omega ^\pm )$, $\kappa _{ij}^m(\varepsilon )\rightharpoonup \kappa _{ij}^m$ in $L^2(\Omega ^m)$ and $\theta _i^m(\varepsilon )\rightharpoonup \theta _i^m$ in $L^2(\Omega ^m)$. Moreover, the norms $\Vert {\mathbf {u}}(\varepsilon )\Vert _{1,\Omega ^\pm }$, $|\varphi (\varepsilon )|_{0,\Omega ^\pm }$ and $\Vert {\mathbf {u}}(\varepsilon )\Vert _{1,\Omega ^m}$ are also bounded. Then, from definition of $\kappa _{ij}(\varepsilon )$ and $\theta _i(\varepsilon )$, there exists a constant $c$ such that

  $$\begin{aligned} \begin{array}{ll} |e_{ij}({\mathbf {u}}(\varepsilon ))|_{0,\Omega ^\pm }\le c, & \Vert {\mathbf {u}}^\pm (\varepsilon )\Vert _{1,\Omega ^\pm }\le c ,\\ |\partial _i\varphi ^\pm (\varepsilon )|_{0,\Omega ^\pm }\le c, & |\varphi ^\pm (\varepsilon )|_{0,\Omega ^\pm }\le c, \end{array} \end{aligned}$$

  (8)

  and, also,

  $$\begin{aligned} \begin{array}{ll} |e_{\alpha \beta }({\mathbf {u}}(\varepsilon ))|_{0,\Omega ^m}\le c, & |e_{\alpha 3}({\mathbf {u}}(\varepsilon ))|_{0,\Omega ^m}\le c\varepsilon ,\\ |e_{33}({\mathbf {u}}(\varepsilon ))|_{0,\Omega ^m}\le c\varepsilon ^2, & \Vert {\mathbf {u}}(\varepsilon )\Vert _{1,\Omega ^m}\le c, \\ |\partial _\alpha \varphi ^m(\varepsilon )|_{0,\Omega ^m}\le \frac{c}{\varepsilon }, & |\partial _3\varphi ^m(\varepsilon )|_{0,\Omega ^m}\le c. \end{array} \end{aligned}$$

  (9)

  From the first set of inequalities (8), we obtain that ${\mathbf {u}}^\pm (\varepsilon )\rightharpoonup {\mathbf {u}}^\pm$ in $H^1(\Omega ^\pm ;\mathbb {R}^3)$ and, consequently, $e_{ij}({\mathbf {u}}(\varepsilon ))\rightharpoonup e_{ij}({\mathbf {u}}^\pm )$ in $L^2(\Omega ^\pm )$, i.e., $\kappa ^\pm = e_{ij}({\mathbf {u}}^\pm )$; besides, since both $|\partial _i\varphi ^\pm (\varepsilon )|_{0,\Omega ^\pm }$ and $|\varphi ^\pm (\varepsilon )|_{0,\Omega ^\pm }$ are bounded in $L^2(\Omega ^\pm )$, we have that $\varphi ^\pm (\varepsilon ) \rightharpoonup \varphi ^\pm$ in $H^1(\Omega ^\pm )$.

  From the second set of inequalities (9), we get that ${\mathbf {u}}^m (\varepsilon ) \rightharpoonup {\mathbf {u}}^m$ in $H^1(\Omega ^m;\mathbb {R}^3)$ and $e_{i3}({\mathbf {u}}(\varepsilon ))\rightarrow 0$ in $L^2(\Omega ^m)$. This implies that $\partial _3 u^m_3(\varepsilon )\rightarrow 0$ and, thus, by continuity of the derivative operator, we obtain that $\partial _3 u^m_3=0$, i.e., $u^m_3(\tilde{x},x_3)=u^m_3(\tilde{x})$ is independent of $x_3$. We also have that $\partial _3 u^m_\alpha (\varepsilon )+\partial _\alpha u_3^m(\varepsilon )\rightarrow 0$, meaning that $\partial _3 u^m_\alpha =-\partial _\alpha u_3^m$, i.e., $u^m_\alpha (\tilde{x},x_3)=\bar{u}^m_\alpha (\tilde{x})-x_3\partial _\alpha u_3^m(\tilde{x})$. Consequently, ${\mathbf {u}}^m \in V_{KL}$. Moreover, we obtain that $e_{\alpha \beta }({\mathbf {u}}(\varepsilon )) \rightharpoonup e_{\alpha \beta }({\mathbf {u}}^m)$ in $L^2(\Omega ^m)$, $e_{\alpha 3}(\frac{{\mathbf {u}}(\varepsilon )}{\varepsilon }) \rightharpoonup \kappa ^m_{\alpha 3}$ in $L^2(\Omega ^m)$, $e_{33}(\frac{{\mathbf {u}}(\varepsilon )}{\varepsilon }) \rightarrow 0$ in $L^2(\Omega ^m)$ and also $e_{33}(\frac{{\mathbf {u}}(\varepsilon )}{\varepsilon ^2}) \rightharpoonup \kappa ^m_{33}$ in $L^2(\Omega ^m)$.

  As already shown, the vector $\varvec{\theta }^m(\varepsilon ):=(\varepsilon \partial _1 \varphi ^m(\varepsilon ), \varepsilon \partial _2 \varphi ^m(\varepsilon ), \partial _3 \varphi ^m(\varepsilon )) \rightharpoonup (\theta ^m_1,\theta ^m_2, \theta ^m_3)$ in $L^2(\Omega ^m;\mathbb {R}^3)$. Thanks to the $H^1$-boundedness of the sequences $(\varphi ^\pm (\varepsilon ))_{\varepsilon >0}$ and by virtue of the continuity of the trace operator, from expression

  $$\begin{aligned} \varphi ^m(\varepsilon )(\tilde{x}, x_3)=\varphi ^m(\varepsilon )|_{S^-}+\int \limits _{-h}^{x_3} \partial _3 \varphi ^m(\varepsilon )(\tilde{x},\xi )d\xi , \end{aligned}$$

  it follows that there exist two constants, $c_1$ and $c_2$, such that $|\varphi ^m(\varepsilon )|_{0,\Omega ^m}\le c_1\{|\varphi ^-(\varepsilon )|_{0,S^-}+|\partial _3\varphi ^m(\varepsilon )|_{0,\Omega ^m}\}\le c_2$. This implies that $\varphi ^m(\varepsilon )\rightharpoonup \varphi ^m$ in $L^2(\Omega ^m)$ and, thus, $(\varepsilon \partial _1 \varphi ^m(\varepsilon ), \varepsilon \partial _2 \varphi ^m(\varepsilon ), \partial _3 \varphi ^m(\varepsilon ))\rightharpoonup (0, 0,\partial _3 \varphi ^m)$.

  Hence, the weak limit $({\mathbf {u}}, \varphi )$ belongs to $\widetilde{V}\times \widetilde{\Psi }$.

• (ii) Now we characterize the expression of the weak limit. Let us multiply the rescaled problem (7) by $\varepsilon ^2$ and let $\varepsilon$ tends to zero. If we choose $\psi =0$, we find that

  $$\begin{aligned} \int \limits _{\Omega ^m} \left( \tau _1\kappa ^m_{33}+\tau _2e_{\sigma \sigma }({\mathbf {u}}^m)+\delta _3\partial _3\varphi ^m\right) e_{33}({\mathbf {v}})dx=0, \end{aligned}$$

  for all ${\mathbf {v}}\in V$, which is satisfied when $\kappa ^m_{33}=-\frac{1}{\tau _1}(\tau _2e_{\sigma \sigma }({\mathbf {u}}^m)+\delta _3\partial _3\varphi ^m)$ in $L^2(\Omega ^m)$. By multiplying by $\varepsilon$ and by choosing test functions such that $v_3=\psi =0$, we obtain

  $$\begin{aligned} \int \limits _{\Omega ^m}2\eta \kappa ^m_{\alpha 3}\partial _3 v_\alpha dx=0, \end{aligned}$$

  for all $v_\alpha \in V$, which implies that $\kappa ^m_{\alpha 3}=0$ and, thus, $e_{\alpha 3}(\frac{{\mathbf {u}}(\varepsilon )}{\varepsilon }) \rightarrow 0$ in $L^2(\Omega ^m)$. Let us choose test functions $r\in \widetilde{V}\times \widetilde{\Psi }$ in the rescaled problem (7) and let $\varepsilon$ tends to zero. We get

  $$\begin{aligned} \begin{array}{ll} A^\pm (s,r)&+\int \limits _{\Omega ^m} \left\{ 2\mu e_{\alpha \beta }({\mathbf {u}}^m)e_{\alpha \beta }({\mathbf {v}})+(\lambda e_{\sigma \sigma }({\mathbf {u}}^m)+\,\delta _1\partial _3\varphi ^m+\tau _2\kappa ^m_{33})e_{\tau \tau }({\mathbf {v}})\right. \\ &\quad +\,\left. (-\delta _1e_{\sigma \sigma }({\mathbf {u}}^m)+\,\gamma _2 \partial _3\varphi ^m-\delta _3 \kappa ^m_{33})\partial _3\psi \right\} =L(r). \end{array} \end{aligned}$$

  By means of the expression of $\kappa ^m_{33}$, we obtain, as customary, the limit problem (4) and, hence, we can identify the weak limit with the limit electromechanical state $s^0=({\mathbf {u}}^0, \varphi ^0)$. Besides, by integrating by parts we can obtain the same characterization (5) for $\varphi ^m$, already shown in Sect. 3, which means that $\varphi ^m\in H^1(-h,h;L^2(\omega ))$. Since the variational equation (4) has a unique solution, not only a subsequence but the whole family $(s(\varepsilon ))_{\varepsilon >0}=({\mathbf {u}}(\varepsilon ),\varphi (\varepsilon ))_{\varepsilon >0}$ weakly converges to $s^0=({\mathbf {u}}^0, \varphi ^0)$ in $H^1(\Omega ,\mathbb {R}^3)\times L^2(\Omega )$.

• (iii) To show that $(s(\varepsilon ))_{\varepsilon >0}=({\mathbf {u}}(\varepsilon ),\varphi (\varepsilon ))_{\varepsilon >0}$ strongly converges to $s^0=({\mathbf {u}}^0, \varphi ^0)$ in $H^1(\Omega ,\mathbb {R}^3)\times L^2(\Omega )$, it suffices to show that both $(\kappa _{ij}(\varepsilon ))_{\varepsilon >0}=(e_{ij}({\mathbf {u}}(\varepsilon )))_{\varepsilon >0}$ and $(\theta _i(\varepsilon ))_{\varepsilon >0}=\partial _i\varphi (\varepsilon ))_{\varepsilon >0}$, respectively, strongly converges to $\kappa _{ij}=e_{ij}({\mathbf {u}})$ and $\theta _i=\partial _i\varphi$ in $L^2(\Omega )$, as a consequence of Korn’s and Poincaré’s inequalities with boundary conditions. Using the variational problem (7), we infer that

  $$\begin{aligned} &c\Big\{ |\varvec{\kappa }^\pm (\varepsilon )-\varvec{\kappa }^\pm |^2_{0,\Omega ^\pm }+|\varvec{\theta }^\pm (\varepsilon )-\varvec{\theta }^\pm |^2_{0,\Omega ^\pm }\\&\quad+|\varvec{\kappa }^m(\varepsilon)-\varvec{\kappa }^m|^2_{0,\Omega ^m}+|\varvec{\theta }^m(\varepsilon )-\varvec{\theta }^m|^2_{0,\Omega ^m}\Big\}\\&\le \int \limits _{\Omega ^{\pm }}\Big\{ \mathsf A ^\pm ( \varvec{\kappa }^\pm (\varepsilon )-\varvec{\kappa }^\pm ):(\varvec{\kappa }^\pm (\varepsilon )-\varvec{\kappa }^\pm )\\&\quad+\,\mathsf H ^{\pm }(\varvec{\theta }^\pm (\varepsilon )-\varvec{\theta }^\pm )\cdot (\varvec{\theta }^\pm (\varepsilon )-\varvec{\theta }^\pm )\Big\} dx\\ &\quad +\,\int \limits _{\Omega ^{m}}\Big\{ \mathsf A ^m (\varvec{\kappa }^m(\varepsilon )-\varvec{\kappa }^m):(\varvec{\kappa }^m(\varepsilon )-\varvec{\kappa }^m)\\&\quad+\mathsf H ^{m}(\varvec{\theta }^m(\varepsilon )-\varvec{\theta }^m)\cdot (\varvec{\theta }^m(\varepsilon )-\varvec{\theta }^m)\Big\} dx\\ &=\,\int \limits _{\Omega ^{\pm }}\Big\{ \mathsf A ^\pm \varvec{\kappa }^\pm :(\varvec{\kappa }^\pm -2\varvec{\kappa }^\pm (\varepsilon ))\\&\quad+\mathsf H ^{\pm }\varvec{\theta }^\pm\cdot (\varvec{\theta }^\pm -2\varvec{\theta }^\pm (\varepsilon ))\Big\} dx+L(s(\varepsilon ))\\ &\quad +\,\int \limits _{\Omega ^{m}}\Big\{ \mathsf A ^m \varvec{\kappa }^m:(\varvec{\kappa }^m-2\varvec{\kappa }^m(\varepsilon ))\\&\quad+\mathsf H ^{m}\varvec{\theta }^m\cdot (\varvec{\theta }^m-2\varvec{\theta }^m(\varepsilon ))\Big\} dx. \end{aligned}$$

  Since we already established in part (i) and (ii) the weak convergences

  $$\begin{aligned} \begin{array}{ll} \kappa _{ij}^\pm (\varepsilon )\rightharpoonup \kappa _{ij}^\pm ,\ \ \ \theta _i^\pm (\varepsilon )\rightharpoonup \theta _i^\pm \ \text {in}\ L^2(\Omega ^\pm ),\\ \kappa _{ij}^m(\varepsilon )\rightharpoonup \kappa _{ij}^m,\ \ \ \theta _i^m(\varepsilon )\rightharpoonup \theta _i^m\ \text {in}\ L^2(\Omega ^m), \end{array} \end{aligned}$$

  the right-hand side of the last inequality converges to

  $$\begin{aligned} \begin{array}{ll} \Lambda &:=- \int \limits _{\Omega ^{\pm }}\left\{ \mathsf A ^\pm \varvec{\kappa }^\pm :\varvec{\kappa }^\pm +\mathsf H ^{\pm }\varvec{\theta }^\pm \cdot \varvec{\theta }^\pm \right\} dx\\ &\quad -\int \limits _{\Omega ^{m}}\left\{ \mathsf A ^m \varvec{\kappa }^m:\varvec{\kappa }^m+\mathsf H ^{m}\varvec{\theta }^m\cdot \varvec{\theta }^m\right\} dx+L(s) \end{array} \end{aligned}$$

  as $\varepsilon$ tends to zero. From relations $\kappa ^\pm _{ij}=e_{ij}({\mathbf {u}}^\pm )$, $\theta _i^\pm =\partial _i\varphi ^\pm$, $\kappa ^m_{\alpha \beta }=e_{\alpha \beta }({\mathbf {u}}^m)$, $\kappa ^m_{\alpha 3}=0$, $\kappa ^m_{33}=-\frac{1}{\tau _1}(\tau _2e_{\sigma \sigma }({\mathbf {u}}^m)+\delta _3\partial _3\varphi ^m)$, $\theta ^m_\alpha =0$ and $\theta ^m_3=\partial _3\varphi ^m$, we obtain that

  $$\begin{aligned} &\int \limits _{\Omega ^{\pm }}\left\{ \mathsf A ^\pm \varvec{\kappa }^\pm :\varvec{\kappa }^\pm +\mathsf H ^{\pm }\varvec{\theta }^\pm \cdot \varvec{\theta }^\pm \right\} dx\\&\quad+\int \limits _{\Omega ^{m}}\left\{ \mathsf A ^m \varvec{\kappa }^m:\varvec{\kappa }^m+\mathsf H ^{m}\varvec{\theta }^m\cdot \varvec{\theta }^m\right\} \\ &= \int \limits _{\Omega ^{\pm }}\left\{ A^\pm _{ijk\ell } e_{k\ell }({\mathbf {u}}^\pm )e_{ij}({\mathbf {u}}^\pm )+H^\pm _{ij}\partial _j \varphi ^\pm \partial _i \varphi ^\pm \right\} dx\\ &\quad +\,\int \limits _{\Omega ^{m}}\Big\{ 2\mu e_{\alpha \beta }({\mathbf {u}}^m)e_{\alpha \beta }({\mathbf {u}}^m)+\mathcal {B}e_{\sigma \sigma }({\mathbf {u}}^m)e_{\tau \tau }({\mathbf {u}}^m)\\&\quad+\,\mathcal {D}\partial _3\varphi ^m\partial _3\varphi ^m\big\} dx\\ &= L(s) \end{aligned}$$

  by step (ii). Hence $\Lambda =0$, as was to be proved.

$\square$