Appendix
1.1An Auxiliary Optimization Problem
In this section, we solve the auxiliary optimization problem (6.5). We calculate the variation of the energy (6.5) at equilibrium to be minimized over\(c\in \mathbb {R}^3\) in order to determine the minimizer\(d^*\). For arbitrary increment\(\delta d^*\in \mathbb {R}^3\), we have
$$\begin{aligned}&\forall \;\; \delta d^*\in \mathbb {R}^3: \quad \bigl \langle {\textrm{D}}{W}_{\textrm{mp}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}),\nonumber \\&\qquad \overline{Q}_{e}^{\natural ,T}(0|0|\delta d^*)[(\nabla _x\Theta )^\natural ]^{-1}\bigr \rangle =0. \end{aligned}$$
(A.1)
By applying\({\textrm{D}}{W}_{\textrm{mp}}\), we obtain
$$\begin{aligned}&\bigl \langle 2\,\mu \,\Big ({\textrm{sym}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}-{{\mathbb {1}}}_3)\Big ),\overline{Q}_{e}^{\natural ,T}(0|0|\delta d^*)[(\nabla _x\Theta )^\natural ]^{-1}\bigr \rangle _{\mathbb {R}^{3\times 3}}\nonumber \\&\quad +\bigl \langle 2\,\mu _c\,\Big (\mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1})\Big ),\overline{Q}_{e}^{\natural ,T}(0|0|\delta d^*)[(\nabla _x\Theta )^\natural ]^{-1}\bigr \rangle _{\mathbb {R}^{3\times 3}}\nonumber \\&\quad +\lambda {\textrm{tr}}\Big ({\textrm{sym}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}-{{\mathbb {1}}}_3)\Big ) \langle {{\mathbb {1}}}_3, \overline{Q}_{e}^{\natural ,T}(0|0|\delta d^*)[(\nabla _x\Theta )^\natural ]^{-1} \rangle _{\mathbb {R}^{3\times 3}}=0. \end{aligned}$$
(A.2)
This is equivalent to
$$\begin{aligned}&\bigl \langle 2\,\mu \,\overline{Q}_{e}^\natural \Big ({\textrm{sym}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}-{{\mathbb {1}}}_3)\Big )[(\nabla _x\Theta )^\natural ]^{-T}e_3, \delta d^*\bigr \rangle _{\mathbb {R}^3}\nonumber \\&\quad +\bigl \langle 2\,\mu _c\,\overline{Q}_{e}^\natural \Big (\mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1})\Big )[(\nabla _x\Theta )^\natural ]^{-T}e_3,\delta d^*\bigr \rangle _{\mathbb {R}^3}\nonumber \\&\quad +\lambda {\textrm{tr}}\Big ({\textrm{sym}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}-{{\mathbb {1}}}_3)\Big )\,\nonumber \\&\quad \langle \overline{Q}_{e}^\natural [(\nabla _x\Theta )^\natural ]^{-T}e_3, \delta d^* \rangle _{\mathbb {R}^3}=0\,, \end{aligned}$$
(A.3)
and it gives
$$\begin{aligned}&\bigl \langle 2\,\mu \,\overline{Q}_{e}^\natural \Big ({\textrm{sym}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}-{{\mathbb {1}}}_3)\Big )n_0, \delta d^*\bigr \rangle _{\mathbb {R}^3}\nonumber \\&\qquad +\left\langle 2\,\mu _c\,\overline{Q}_{e}^\natural \Big (\mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1})\Big )n_0,\delta d^* \right\rangle _{\mathbb {R}^3}\nonumber \\&\qquad +\lambda {\textrm{tr}}\Big ({\textrm{sym}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}-{{\mathbb {1}}}_3)\Big )\langle \overline{Q}_{e}^\natural n_0, \delta d^* \rangle _{\mathbb {R}^3}=0. \end{aligned}$$
(A.4)
Recall that thefirst Piola–Kirchhoff stress tensor in the reference configuration\(\Omega _\xi \) is given by\(S_1(F_\xi ,\overline{R}_\xi ):={\textrm{D}}_{F_\xi } {W}_{\textrm{mp}}(F_\xi ,\overline{R}_\xi )\), while the\(\textit{Biot-type stress tensor}\) is\(T_{\text {Biot}}(\overline{U}_\xi ):={\textrm{D}}_{\overline{U}_\xi }W_{\textrm{mp}}(\overline{U}_\xi )\). Since\({\textrm{D}}_{F_\xi }\overline{U}_\xi \,.\,X=\overline{R}^T_\xi X\) and
$$\begin{aligned} \langle {\textrm{D}}_{F_\xi } {W}_{\textrm{mp}}(F_\xi ,\overline{R}_\xi ),X \rangle =\langle {\textrm{D}}_{\overline{U}_\xi }W_{\textrm{mp}}(\overline{U}_\xi ),{\textrm{D}}_{F_{\xi }} \overline{U}_{\xi } X \rangle ,\ \forall X\in \mathbb {R}^{3\times 3}, \end{aligned}$$
we obtain
$$\begin{aligned} {\textrm{D}}_{F_{\xi }} {W}_{\textrm{mp}}(F_{\xi },\overline{R}_{\xi })=\overline{R}_{\xi }\,{\textrm{D}}_{\overline{U}_{\xi }}W_{\textrm{mp}}(\overline{U}_{\xi })\,. \end{aligned}$$
(A.5)
Therefore,\(S_1(F_\xi ,\overline{R}_\xi )=\overline{R}_\xi \, T_{\text {Biot}}(\overline{U}_\xi )\) and\( T_{\text {Biot}}(\overline{U}_\xi )=\overline{R}^T_\xi \, S_1(F_\xi ,\overline{R}_\xi )\). Here, we have
$$\begin{aligned} T_{\text {Biot}}(\overline{U}_\xi )=2\,\mu \,{\textrm{sym}}(\overline{U}_\xi -{{\mathbb {1}}}_3)+2\,\mu _c\,\mathop {{\textrm{skew}}}\nolimits (\overline{U}_\xi -{{\mathbb {1}}}_3)+\lambda {\textrm{tr}}({\textrm{sym}}(\overline{U}_\xi -{{\mathbb {1}}}_3)){{\mathbb {1}}}_3\,, \end{aligned}$$
(A.6)
where\(\overline{U}_{\xi }(\Theta (x_1,x_2,x_3))=\overline{U}_e(x_1,x_2,x_3)\). Thus, we can express the first Piola–Kirchhoff stress tensor
$$\begin{aligned} S_1(F_\xi ,\overline{R}_\xi )&=\overline{R}_\xi \Big [2\,\mu \,{\textrm{sym}}(\overline{R}_\xi ^T F_\xi -{{\mathbb {1}}}_3)+2\,\mu _c\,\mathop {{\textrm{skew}}}\nolimits (\overline{R}_\xi ^T F_\xi -{{\mathbb {1}}}_3)\nonumber \\&\quad +\lambda {\textrm{tr}}({\textrm{sym}}(\overline{R}_\xi ^T F_\xi -{{\mathbb {1}}}_3)){{\mathbb {1}}}_3\Big ]\,, \end{aligned}$$
(A.7)
with\(\overline{R}_\xi (\Theta (x_1,x_2,x_3))=\overline{Q}_e(x_1,x_2,x_3)\) for the elastic microrotation\(\overline{Q}_e:\Omega _h\rightarrow \text {SO(3)}\). Hence, we must have
$$\begin{aligned} \forall \delta d^*\in \mathbb {R}^3:\qquad \langle S_1((\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1},\overline{Q}_{e}^\natural )n_0,\delta d^* \rangle _{\mathbb {R}^3}=0, \end{aligned}$$
(A.8)
implying
$$\begin{aligned} S_1((\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)&[(\nabla _x\Theta )^\natural ]^{-1},\overline{Q}_{e}^\natural )\,n_0=0 \qquad \forall \, \eta _3\in \left[ -\frac{1}{2},\frac{1}{2}\right] . \end{aligned}$$
(A.9)
In shell theories, the usual assumption is that the normal stress on the transverse boundaries are vanishing, that is
$$\begin{aligned} S_1(F_\xi ,\overline{R}_\xi )\big |_{\omega _\xi ^\pm }\, (\pm n_0)=0\,, \qquad \text {(normal stress on lower and upper faces is zero)}\,. \end{aligned}$$
(A.10)
We notice that the condition (A.9) is for all\(\eta _3\in \left[ -\frac{1}{2},\frac{1}{2}\right] \), while the condition (A.10) is only for\(\eta _3=\pm \frac{1}{2}\). Therefore, it is possible that the Cosserat-membrane type\(\Gamma \)-limit underestimates the real stresses (e.g., the transverse shear stresses). From the relation between the first Piola–Kirchhoff tensor and the Biot-stress tensor we obtain
$$\begin{aligned} T_{\text {Biot}}\Big (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}\Big ) n_0=0\,,\qquad \forall \,\eta _3\in [-\frac{1}{2},\frac{1}{2}]\,, \end{aligned}$$
(A.11)
or, equivalently,
$$\begin{aligned} T_{\text {Biot}}(\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*})\,n_0=0, \end{aligned}$$
(A.12)
where
$$\begin{aligned} T_{\text {Biot}}(\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*})&=2\,\mu \,{\textrm{sym}}(\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*}-{{\mathbb {1}}}_3)+2\,\mu _c\,\mathop {{\textrm{skew}}}\nolimits (\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*}-{{\mathbb {1}}}_3)\nonumber \\&\quad +\lambda {\textrm{tr}}({\textrm{sym}}(\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*}-{{\mathbb {1}}}_3)){{\mathbb {1}}}_3\,, \end{aligned}$$
(A.13)
and we have introduced the notation\(\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*}:=\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}\). With the help of the following decomposition
$$\begin{aligned} \overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*}-{{\mathbb {1}}}_3&=(\overline{Q}_{e}^{\natural ,T}\nabla _{(\eta _1,\eta _2)} \varphi ^\natural -(\nabla y_0)^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}\nonumber \\&\quad +(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1}\nonumber \\&= \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } +(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1}\,, \end{aligned}$$
(A.14)
with\(\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }=(\overline{Q}_{e}^{\natural ,T}\nabla _{(\eta _1,\eta _2)} \varphi ^\natural -(\nabla y_0)^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}\), and relations (A.29)-(A.31), the relation (A.13) can be expressed as
$$\begin{aligned} T_{\text {Biot}}(\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*})n_0&=\mu \,\Big ( \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\nonumber \\&\quad +[(\nabla _x\Theta )^\natural ]^{-T}(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)^Tn_0\Big )\nonumber \\&\quad +\mu _c\,\Big (- \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\nonumber \\&\quad -[(\nabla _x\Theta )^\natural ]^{-T}(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)^Tn_0\Big )\nonumber \\&\quad +\lambda \Big (\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,{{\mathbb {1}}}_3 \rangle n_0+(\overline{Q}_{e}^{\natural ,T}d^*-n_0)n_0\otimes n_0\Big )\nonumber \\&= (\mu \,+\mu _c\,)(\overline{Q}_{e}^{\natural ,T}d^*-n_0)+(\mu \,-\mu _c\,) \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0\nonumber \\&\quad +(\mu \,-\mu _c\,)((0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1})^Tn_0\nonumber \\&\quad +\lambda {\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0+\lambda (\overline{Q}_{e}^{\natural ,T}d^*-n_0)n_0\otimes n_0, \end{aligned}$$
(A.15)
and the condition (A.12) on\(T_{\text {Biot}}\) reads
$$\begin{aligned}&(\mu \,+\mu _c\,)(\overline{Q}_{e}^{\natural ,T}d^*-n_0)+(\mu \,-\mu _c\,)(\overline{Q}_{e}^{\natural ,T}d^*-n_0)n_0\otimes n_0\nonumber \\&\quad +\lambda (\overline{Q}_{e}^{\natural ,T}d^*-n_0)n_0\otimes n_0\nonumber \\&=-\Big [(\mu \,-\mu _c\,)\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^Tn_0+\lambda {\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\Big ], \end{aligned}$$
(A.16)
where\(((0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1})^Tn_0=(\overline{Q}_{e}^{\natural ,T}d^*-n_0)n_0\otimes n_0\). Before continuing the calculations, we introduce the tensor
$$\begin{aligned} \textrm{A}_{y_0}&:=(\nabla y_0|0)\,\,[(\nabla _x\Theta )(0) \,]^{-1}={{\mathbb {1}}}_3-n_0\otimes n_0\in \textrm{Sym}(3), \end{aligned}$$
(A.17)
and we notice that, identically as in the proof of Lemma 4.3 in Ghiba et al. (2020a), we can show that
$$\begin{aligned} \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \textrm{A}_{y_0}=\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \qquad \Longleftrightarrow \qquad \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0\otimes n_0=0. \end{aligned}$$
(A.18)
Actually, for an arbitrary matrix\(X=(*|*|0)\,[ \nabla _x \Theta (0)]^{-1}\), since\(\textrm{A}_{y_0}^2=\textrm{A}_{y_0}\in \textrm{Sym}(3)\) and\(X\textrm{A}_{y_0}=X\), we have
$$\begin{aligned} \bigl \langle ({{\mathbb {1}}}_3-\textrm{A}_{y_0}) \,X , \textrm{A}_{y_0} \,X\bigr \rangle = \bigl \langle (\textrm{A}_{y_0}-\textrm{A}_{y_0}^2) \,X , \,X\bigr \rangle =0, \end{aligned}$$
but also
$$\begin{aligned} ({{\mathbb {1}}}_3-\textrm{A}_{y_0}) \,X^T=\big (X({{\mathbb {1}}}_3-\textrm{A}_{y_0})\big )^T=\big (X-X\textrm{A}_{y_0}\big )^T=0, \end{aligned}$$
(A.19)
and consequently
$$\begin{aligned} \bigl \langle X^T ({{\mathbb {1}}}_3-\textrm{A}_{y_0}) , \textrm{A}_{y_0} \,X\bigr \rangle = 0 \qquad \text {as well as} \qquad \bigl \langle X^T ({{\mathbb {1}}}_3-\textrm{A}_{y_0}) ,({{\mathbb {1}}}_3-\textrm{A}_{y_0}) \,X\bigr \rangle = 0. \end{aligned}$$
In addition, since\(\textrm{A}_{y_0}={{\mathbb {1}}}_3-(0|0|n_0)\,(0|0|n_0)^T=\,{{\mathbb {1}}}_3-n_0\otimes n_0\), the following equalities holds
$$\begin{aligned} \Vert ({{\mathbb {1}}}_3-\textrm{A}_{y_0})\,X\Vert ^2&= \bigl \langle \,X,({{\mathbb {1}}}_3-\textrm{A}_{y_0})^2\,X\bigr \rangle = \bigl \langle \,X,({{\mathbb {1}}}_3-\textrm{A}_{y_0})\,X\bigr \rangle \nonumber \\&= \bigl \langle \,X,(0|0|n_0)\,(0|0|n_0)^T\,X\bigr \rangle \nonumber \\&= \bigl \langle \,(0|0|n_0)^T X,(0|0|n_0)^T\,X\bigr \rangle \nonumber \\&=\Vert X\,(0|0|n_0)^T\Vert ^2 =\Vert X^T\,(0|0|n_0)\Vert ^2=\Vert X^T\,n_0\Vert ^2. \end{aligned}$$
(A.20)
We have the following decomposition
$$\begin{aligned} (\overline{Q}_{e}^{\natural ,T}d^*-n_0)&={{\mathbb {1}}}_3(\overline{Q}_{e}^{\natural ,T}d^*-n_0)=(A_{y_0}+n_0\otimes n_0)(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\nonumber \\&= A_{y_0}(\overline{Q}_{e}^{\natural ,T}d^*-n_0)+n_0\otimes n_0(\overline{Q}_{e}^{\natural ,T}d^*-n_0). \end{aligned}$$
(A.21)
By using that
$$\begin{aligned}&n_0\otimes n_0 (\overline{Q}_{e}^{\natural ,T}d^*-n_0)=n_0\langle n_0,(\overline{Q}_{e}^{\natural ,T}d^*-n_0) \rangle =\langle (\overline{Q}_{e}^{\natural ,T}d^*-n_0),n_0 \rangle n_0\nonumber \\&\quad =(\overline{Q}_{e}^{\natural ,T}d^*-n_0) n_0\otimes n_0, \end{aligned}$$
(A.22)
and with (A.16), we get
$$\begin{aligned}&(\mu \,+\mu _c\,)A_{y_0}(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\nonumber \\&\qquad +(\mu \,+\mu _c\,)n_0\otimes n_0(\overline{Q}_{e}^{\natural ,T}d^*-n_0)+(\mu \,-\mu _c\,)n_0\otimes n_0(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\nonumber \\&\qquad +\lambda \, n_0\otimes n_0(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\nonumber \\&\quad =-\Big [(\mu \,-\mu _c\,) \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+\lambda {\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\Big ]. \end{aligned}$$
(A.23)
Therefore,
$$\begin{aligned}&\Big ((\mu \,+\mu _c\,)A_{y_0}+(2\,\mu \,+\lambda )n_0\otimes n_0\Big )(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\nonumber \\&\quad =-\Big [(\mu \,-\mu _c\,) \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+\lambda {\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\Big ]. \end{aligned}$$
(A.24)
Direct calculation shows
$$\begin{aligned}&\Big ((\mu \,+\mu _c\,)A_{y_0}+(2\,\mu \,+\lambda )n_0\otimes n_0\Big )^{-1}:=\Big (\frac{1}{\mu \,+\mu _c\,}A_{y_0}+\frac{1}{2\,\mu \,+\lambda }n_0\otimes n_0\Big )\,. \end{aligned}$$
(A.25)
Next, by using
$$\begin{aligned}&A_{y_0}n_0=({{\mathbb {1}}}_3-n_0\otimes n_0)n_0=n_0-n_0\langle n_0,n_0 \rangle =n_0-n_0=0,\nonumber \\&n_0\otimes n_0 \,\mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0 = (0|0|n_0)(0|0|n_0)^T \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0\nonumber \\&=(0|0|n_0)\Big ((\overline{Q}_{e}^{\natural ,T}\nabla _{(\eta _1,\eta _2)} \varphi ^\natural -(\nabla y_0)^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}(0|0|n_0)\Big )^T n_0\nonumber \\&\quad =(0|0|n_0)\Big ((\overline{Q}_{e}^{\natural ,T}\nabla _{(\eta _1,\eta _2)} \varphi ^\natural -(\nabla y_0)^\natural |0)(0|0|e_3)\Big )^T n_0=0\,, \end{aligned}$$
(A.26)
eq. (A.24) can be written as
$$\begin{aligned} \overline{Q}_{e}^{\natural ,T}d^*-n_0&=-\Big [\frac{1}{\mu \,+\mu _c\,}A_{y_0}+\frac{1}{2\,\mu \,+\lambda }n_0\otimes n_0\Big ]\nonumber \\&\quad \times \Big [(\mu \,-\mu _c\,) \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+\lambda {\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\Big ]\nonumber \\&=-\Big [\frac{\mu \,-\mu _c\,}{\mu \,+\mu _c\,}A_{y_0} \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0\nonumber \\&\quad +\frac{\mu \,-\mu _c\,}{2\,\mu \,+\lambda }\,n_0\otimes n_0\, \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+\frac{\lambda }{\mu \,+\mu _c\,}{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )A_{y_0}n_0\nonumber \\&\quad +\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )(n_0\otimes n_0) n_0\Big ]\nonumber \\&=-\Big [\frac{\mu \,-\mu _c\,}{\mu \,+\mu _c\,}A_{y_0} \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\Big ]\,. \end{aligned}$$
(A.27)
Simplifying (A.27), we obtain
$$\begin{aligned} d^*&=\Big (1-\frac{\lambda }{2\,\mu \,+\lambda }\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,{{\mathbb {1}}}_3 \rangle \Big ) \overline{Q}_{e}^\natural n_0+\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,}\;\overline{Q}_{e}^\natural \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0. \end{aligned}$$
In terms of\(\overline{Q}_{e}^\natural =\overline{R}^\natural Q_0^{\natural ,T}\) we obtain the following expression for\(d^*\)
$$\begin{aligned} d^*&=\Big (1-\frac{\lambda }{2\,\mu \,+\lambda }\langle (Q_0^{\natural }\overline{R}^{\natural ,T}\nabla _{(\eta _1,\eta _2)} \varphi ^\natural -(\nabla y_0)^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1},{{\mathbb {1}}}_3 \rangle \Big ) \overline{R}^{\natural }Q_0^{\natural ,T}n_0\nonumber \\&\quad +\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,}\;\overline{R}^{\natural }Q_0^{\natural ,T}\Big ((Q_0^{\natural }\overline{R}^{\natural ,T}\nabla _{(\eta _1,\eta _2)} \varphi ^\natural -(\nabla y_0)^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}\Big )^Tn_0. \end{aligned}$$
(A.28)
1.2Calculations for the\(T_{\text {Biot}}\) Stress
Here, we present the lengthy calculation related to the\(T_{\text {Biot}}\) stress tensor in expression (A.13). We have
$$\begin{aligned}&2{\textrm{sym}}(\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*}-{{\mathbb {1}}}_3)n_0\nonumber \\&\quad = \Big (2{\textrm{sym}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )+2{\textrm{sym}}((0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1})\Big )n_0\nonumber \\&\quad = \Big ( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } + \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \Big )n_0+\Big ((0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1}\nonumber \\&\qquad +[(\nabla _x\Theta )^\natural ]^{-T}(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)^T\Big )n_0\nonumber \\&\quad =\underbrace{ \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0}_{=0}+ \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1}n_0\nonumber \\&\qquad +[(\nabla _x\Theta )^\natural ]^{-T}(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)^Tn_0\nonumber \\&\quad = \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)e_3+[(\nabla _x\Theta )^\natural ]^{-T}(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)^Tn_0\nonumber \\&\quad = \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+(\overline{Q}_{e}^{\natural ,T}d^*-n_0)+[(\nabla _x\Theta )^\natural ]^{-T}(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)^Tn_0, \end{aligned}$$
(A.29)
and
$$\begin{aligned} 2\mathop {{\textrm{skew}}}\nolimits (\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*}&-{{\mathbb {1}}}_3)n_0= \Big (2\mathop {{\textrm{skew}}}\nolimits ( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )+2\mathop {{\textrm{skew}}}\nolimits ((0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)\nonumber \\&\quad [(\nabla _x\Theta )^\natural ]^{-1})\Big )n_0\nonumber \\&= \Big ( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } - \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \Big )n_0+\Big ((0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1}\nonumber \\&\quad -[(\nabla _x\Theta )^\natural ]^{-T}(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)^T\Big )n_0\nonumber \\&=- \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0+(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\nonumber \\&\quad -[(\nabla _x\Theta )^\natural ]^{-T}(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)^Tn_0. \end{aligned}$$
(A.30)
Calculating the trace of\(T_{\text {Biot}}\) gives
$$\begin{aligned} {\textrm{tr}}({\textrm{sym}}(\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*}-{{\mathbb {1}}}_3))n_0&=\langle {\textrm{sym}}(\overline{U}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural },d^*}-{{\mathbb {1}}}_3),{{\mathbb {1}}}_3 \rangle n_0\nonumber \\&=\Big (\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,{{\mathbb {1}}}_3 \rangle +\langle (0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1},{{\mathbb {1}}}_3 \rangle \Big )n_0\nonumber \\&=\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,{{\mathbb {1}}}_3 \rangle n_0+(\overline{Q}_{e}^{\natural ,T}d^*-n_0)n_0\otimes n_0, \end{aligned}$$
(A.31)
where we have used that\(\langle (0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1},{{\mathbb {1}}}_3 \rangle _{\mathbb {R}^{3\times 3}}\,n_0 =\langle (\overline{Q}_{e}^{\natural ,T}d^*-n_0),n_0 \rangle _{\mathbb {R}^3}\,n_0=(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\,n_0\otimes n_0\).
1.3Calculations for the Homogenized Membrane Energy
In this part, we do the calculations for obtaining the minimizer separately. By inserting\(d^*\) in the membrane part of the relation (4.10), we have
$$\begin{aligned} \Vert {\textrm{sym}}(\overline{U}_h^{\natural }-{{\mathbb {1}}}_3)\Vert ^2&=\Vert {\textrm{sym}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}-{{\mathbb {1}}}_3)\Vert ^2\nonumber \\&=\Vert {\textrm{sym}}\Big (\underbrace{\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural -[\nabla y_0]^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}}_{= \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } }\nonumber \\&\quad +(0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1}\Big ))\Vert ^2\nonumber \\&=\Vert {\textrm{sym}}\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \Vert ^2+ \Vert {\textrm{sym}}((0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1})\Vert ^2\nonumber \\&\quad +2\left\langle {\textrm{sym}}\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,{\textrm{sym}}((0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1}) \right\rangle \nonumber \\&= \Vert {\textrm{sym}}\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \Vert ^2+\Vert {\textrm{sym}}\Big (\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,} \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ^Tn_0\otimes n_0\nonumber \\&\quad -\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\otimes n_0\Big )\Vert ^2\nonumber \\&\quad +2\left\langle {\textrm{sym}}\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,{\textrm{sym}}\Big (\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,} \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ^Tn_0\otimes n_0 -\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\otimes n_0\Big ) \right\rangle . \end{aligned}$$
(A.32)
We have
$$\begin{aligned}&\Vert {\textrm{sym}}\Big (\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,} \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ^Tn_0\otimes n_0\nonumber -\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\otimes n_0\Big )\Vert ^2\nonumber \\&\quad =\frac{(\mu _c\,-\mu \,)^2}{(\mu _c\,+\mu \,)^2}\Vert {\textrm{sym}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0)\Vert ^2\nonumber \\&\qquad +\frac{\lambda ^2}{(2\,\mu \,+\lambda )^2}{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )^2\Vert n_0\otimes n_0\Vert ^2\nonumber \\&\qquad -2\,\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,}\;\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )\left\langle {\textrm{sym}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ^Tn_0\otimes n_0),n_0\otimes n_0 \right\rangle \nonumber \\&\quad = \frac{(\mu _c\,-\mu \,)^2}{(\mu _c\,+\mu \,)^2}\left\langle {\textrm{sym}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0),{\textrm{sym}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0) \right\rangle \nonumber \\&\qquad +\frac{\lambda ^2}{(2\,\mu \,+\lambda )^2}{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )^2\nonumber \\&\qquad -\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,}\;\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )\left\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0,n_0\otimes n_0 \right\rangle \nonumber \\&\qquad -\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,}\;\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )\left\langle n_0\otimes n_0\, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,n_0\otimes n_0 \right\rangle \nonumber \\&\qquad = \frac{(\mu _c\,-\mu \,)^2}{4(\mu _c\,+\mu \,)^2}\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0 \rangle \nonumber \\&\qquad +\frac{(\mu _c\,-\mu \,)^2}{4(\mu _c\,+\mu \,)^2}\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0,n_0\otimes n_0 \, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \rangle \nonumber \\&\qquad +\frac{(\mu _c\,-\mu \,)^2}{4(\mu _c\,+\mu \,)^2}\langle n_0\otimes n_0\, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } , \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0 \rangle \nonumber \\&\qquad +\frac{(\mu _c\,-\mu \,)^2}{4(\mu _c\,+\mu \,)^2}\langle n_0\otimes n_0\, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,n_0\otimes n_0\, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \rangle \nonumber \\&\qquad +\frac{\lambda ^2}{(2\,\mu \,+\lambda )^2}{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )^2=\frac{(\mu _c\,-\mu \,)^2}{2(\mu _c\,+\mu \,)^2}\Vert \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\Vert ^2 +\frac{\lambda ^2}{(2\,\mu \,+\lambda )^2}{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )^2. \end{aligned}$$
(A.33)
Since, using (A.18) we have\(\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ^T n_0\otimes n_0,n_0\otimes n_0 \rangle = \langle n_0\otimes n_0,\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0\otimes n_0 \rangle =0\),
and since we have used the matrix expression\( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } =(*|*|0)[(\nabla _x\Theta )^\natural ]^{-1}\) and\(n_0\otimes n_0=(0|0|n_0)[(\nabla _x\Theta )^\natural (0) ]^{-1}\), we deduce
$$\begin{aligned}&\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0 \rangle \nonumber \\&\quad =\bigl \langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T (0|0|n_0)[(\nabla _x\Theta )^\natural (0) ]^{-1}, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T (0|0|n_0)[(\nabla _x\Theta )^\natural (0) ]^{-1}\bigr \rangle \nonumber \\&\quad =\bigl \langle (0|0| \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0)^T(0|0| \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0), [(\nabla _x\Theta )^\natural (0) ]^{-1}[(\nabla _x\Theta )^\natural (0) ]^{-T}\bigr \rangle \nonumber \\&\quad =\langle (0|0| \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0)^T(0|0| \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0),(\widehat{\textrm{I}}_{y_0})^{-1} \rangle \nonumber \\&\quad =\left\langle \begin{pmatrix} 0&{}0&{}0\\ 0&{}0&{}0\\ {} &{} \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0 \end{pmatrix}(0|0| \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0),\begin{pmatrix} *&{}*&{}0\\ *&{}*&{}0\\ 0&{}0&{}1 \end{pmatrix} \right\rangle \nonumber \\&\quad =\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0 \rangle =\Vert \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\Vert ^2. \end{aligned}$$
(A.34)
On the other hand,
$$\begin{aligned} 2&\left\langle {\textrm{sym}}\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,{\textrm{sym}}(\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,}\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^Tn_0\otimes n_0-\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\otimes n_0) \right\rangle \nonumber \\&=\frac{1}{2}\left\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } + \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T,\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,}\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^Tn_0\otimes n_0+\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,}n_0\otimes n_0\; \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }\right. \nonumber \\&\quad \left. -\frac{2\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )n_0\otimes n_0 \right\rangle \nonumber \\&=\frac{\mu _c\,-\mu \,}{2(\mu _c\,+\mu \,)}\left\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } , \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0 \right\rangle +\frac{\mu _c\,-\mu \,}{2(\mu _c\,+\mu \,)}\left\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,n_0\otimes n_0\; \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \right\rangle \nonumber \\&\quad -\frac{\lambda }{(2\,\mu \,+\lambda )}{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,n_0\otimes n_0 \rangle +\frac{\mu _c\,-\mu \,}{2(\mu _c\,+\mu \,)}\left\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0 \right\rangle \nonumber \\&\quad +\frac{\mu _c\,-\mu \,}{2(\mu _c\,+\mu \,)}\left\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T,n_0\otimes n_0\, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \right\rangle -\frac{\lambda }{(2\,\mu \,+\lambda )}{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T,n_0\otimes n_0 \rangle \nonumber \\&\quad =\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,}\Vert \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\Vert ^2, \end{aligned}$$
(A.35)
due to (A.20). Therefore, (A.32) can be reduced to
$$\begin{aligned}&\Vert {\textrm{sym}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |c)[(\nabla _x\Theta )^\natural ]^{-1}-{{\mathbb {1}}}_3)\Vert ^2 =\Vert {\textrm{sym}}\mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \Vert ^2+\frac{(\mu _c\,-\mu \,)^2}{2(\mu _c\,+\mu \,)^2}\Vert \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\Vert ^2\nonumber \\&\qquad +\frac{\lambda ^2}{(2\,\mu \,+\lambda )^2}{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )^2+\frac{\mu _c\,-\mu \,}{(\mu _c\,+\mu \,)}\Vert \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\Vert ^2. \end{aligned}$$
(A.36)
Now we continue the calculations for the skew symmetric part,
$$\begin{aligned}&\Vert \mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1})\Vert ^2\nonumber \\&\quad =\Vert \mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1})\Vert ^2+ \Vert \mathop {{\textrm{skew}}}\nolimits ((0|0|\overline{Q}_{e}^{\natural ,T}d^*)[(\nabla _x\Theta )^\natural ]^{-1})\Vert ^2\nonumber \\&\quad +2\bigl \langle \mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}), \mathop {{\textrm{skew}}}\nolimits ((0|0|\overline{Q}_{e}^{\natural ,T}d^*)[(\nabla _x\Theta )^\natural ]^{-1})\bigr \rangle . \end{aligned}$$
(A.37)
In a similar manner, we calculate the terms separately. Since\(n_0\otimes n_0\) is symmetric, we obtain
$$\begin{aligned}&\Vert \mathop {{\textrm{skew}}}\nolimits ((0|0|\overline{Q}_{e}^{\natural ,T}d^*)[(\nabla _x\Theta )^\natural ]^{-1})\Vert ^2\\&\quad =\Vert \mathop {{\textrm{skew}}}\nolimits (n_0\otimes n_0+\frac{\mu _c\,-\mu \,}{\mu _c\,+\mu \,} \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T \, n_0\otimes n_0-\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T )\, n_0\otimes n_0)\Vert ^2\\&\quad =\frac{(\mu _c\,-\mu \,)^2}{(\mu _c\,+\mu \,)^2}\Vert \mathop {{\textrm{skew}}}\nolimits ( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T\, n_0\otimes n_0)\Vert ^2. \end{aligned}$$
But, we have
$$\begin{aligned} \Vert \mathop {{\textrm{skew}}}\nolimits ( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ^Tn_0\otimes n_0)\Vert ^2&=\frac{1}{4}\left\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T\, n_0\otimes n_0 \right\rangle \nonumber \\&\quad -\frac{1}{4}\left\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0,n_0\otimes n_0 \, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \right\rangle \nonumber \\&\quad -\frac{1}{4}\left\langle n_0\otimes n_0 \, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } , \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T\, n_0\otimes n_0 \right\rangle \nonumber \\&\quad +\frac{1}{4}\left\langle n_0\otimes n_0\, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,n_0\otimes n_0 \, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \right\rangle =\frac{1}{2}\Vert \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\Vert ^2\,, \end{aligned}$$
(A.38)
where we used the fact that\((n_0\otimes n_0)^2=(n_0\otimes n_0)\). The difficulty in the skew symmetric part of (A.37) is solved in the following calculation
$$\begin{aligned}&2\bigl \langle \mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}), \mathop {{\textrm{skew}}}\nolimits ((0|0|\overline{Q}_{e}^{\natural ,T}d^*)[(\nabla _x\Theta )^\natural ]^{-1})\bigr \rangle \nonumber \\&\quad = 2\,\frac{(\mu _c\,-\mu \,)}{(\mu _c\,+\mu \,)}\left\langle \mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}),\mathop {{\textrm{skew}}}\nolimits ( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0) \right\rangle \nonumber \\&\quad =\frac{(\mu _c\,-\mu \,)}{2(\mu _c\,+\mu \,)}\langle \overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0 \rangle \nonumber \\&\qquad -\frac{(\mu _c\,-\mu \,)}{2(\mu _c\,+\mu \,)}\langle \overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1},n_0\otimes n_0\, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \rangle \nonumber \\&\qquad -\frac{(\mu _c\,-\mu \,)}{2(\mu _c\,+\mu \,)}\langle (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1})^T, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0 \rangle \nonumber \\&\qquad +\frac{(\mu _c\,-\mu \,)}{2(\mu _c\,+\mu \,)}\langle (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1})^T,n_0\otimes n_0 \, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } \rangle \nonumber \\&\quad =-\frac{(\mu _c\,-\mu \,)}{(\mu _c\,+\mu \,)}\Vert \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\Vert ^2. \end{aligned}$$
(A.39)
Therefore,
$$\begin{aligned}&2\bigl \langle \mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1}), \mathop {{\textrm{skew}}}\nolimits ((0|0|\overline{Q}_{e}^{\natural ,T}d^*)[(\nabla _x\Theta )^\natural ]^{-1})\bigr \rangle \nonumber \\&\quad =-\frac{(\mu _c\,-\mu \,)}{(\mu _c\,+\mu \,)}\Vert \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0\Vert ^2\,, \end{aligned}$$
(A.40)
and we obtain
$$\begin{aligned}&\Vert \mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1})\Vert ^2\nonumber \\&\quad =\Vert \mathop {{\textrm{skew}}}\nolimits (\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1})\Vert ^2 +\frac{(\mu _c\,-\mu \,)^2}{2(\mu _c\,+\mu \,)^2}\Vert \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0\Vert ^2\nonumber \\&\qquad -\frac{(\mu _c\,-\mu \,)}{(\mu _c\,+\mu \,)}\Vert \mathcal {E}^T_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } n_0\Vert ^2. \end{aligned}$$
(A.41)
The last requirement for our calculations, is
$$\begin{aligned}&\Big [{\textrm{tr}}\Big ({\textrm{sym}}(\overline{Q}_{e}^{\natural ,T}(\nabla _{(\eta _1,\eta _2)} \varphi ^\natural |d^*)[(\nabla _x\Theta )^\natural ]^{-1}-{{\mathbb {1}}}_3)\Big )\Big ]^2\nonumber \\&\quad =\Big ({\textrm{tr}}\big ({\textrm{sym}}((\overline{Q}_{e}^{\natural ,T}\nabla _{(\eta _1,\eta _2)} \varphi ^\natural -[\nabla y_0]^\natural |0)[(\nabla _x\Theta )^\natural ]^{-1})\big )\nonumber \\&\qquad +{\textrm{tr}}\big ({\textrm{sym}}((0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1})\big )\Big )^2\nonumber \\&\quad =\Big ({\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )+\frac{(\mu _c\,-\mu \,)}{2(\mu _c\,+\mu \,)}(\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T n_0\otimes n_0,{{\mathbb {1}}}_3 \rangle +\langle n_0\otimes n_0\, \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,{{\mathbb {1}}}_3 \rangle )\nonumber \\&\qquad -\frac{\lambda }{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )\underbrace{\langle n_0\otimes n_0,{{\mathbb {1}}}_3 \rangle }_{\langle n_0,n_0 \rangle =1}\Big )^2\nonumber \\&\quad =\Big (\frac{2\,\mu \,}{2\,\mu \,+\lambda }{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )+\frac{(\mu _c\,-\mu \,)}{2(\mu _c\,+\mu \,)}(\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } }^T ,n_0\otimes n_0 \rangle +\langle \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } ,n_0\otimes n_0 \rangle )\Big )^2\nonumber \\&\quad =\frac{4\mu \,^2}{(2\,\mu \,+\lambda )^2}{\textrm{tr}}( \mathcal {E}_{\varphi ^\natural ,\overline{Q}_{e}^{\natural } } )^2. \end{aligned}$$
(A.42)
1.4Homogenized Quadratic Curvature Energy
The explicit expression of\(\widetilde{W}^{\textrm{hom}}_{\textrm{curv}}(\mathcal {K}_{e,s})\) is announced in this appendix, but its explicit calculation will be provided in a forthcoming paper in which the authors obtained the homogenized curvature energy for the following curvature energy
$$\begin{aligned} W_{\text {curv}}(\Gamma ^\natural )&=\mu L_c^2\Big (b_1\Vert {\textrm{sym}}\Gamma ^\natural \Vert ^2+b_2\,\Vert \mathop {{\textrm{skew}}}\nolimits \Gamma ^\natural \Vert ^2+b_3{\textrm{tr}}(\Gamma ^\natural )^2\Big )\,, \end{aligned}$$
(A.43)
as
$$\begin{aligned} W_{\text {curv}}^{\text {hom}}(\mathcal {K}_{e,s})&=\mu L_c^2\Big (b_1\Vert {\textrm{sym}}\mathcal {K}_{e,s}\Vert ^2+b_2\Vert \mathop {{\textrm{skew}}}\nolimits \mathcal {K}_{e,s}\Vert ^2\nonumber \\&\quad -\frac{(b_1-b_2)^2}{2(b_1+b_2)}\Vert \mathcal {K}_{e,s}^Tn_0\Vert ^2+\frac{b_1b_3}{(b_1+b_3)}{\textrm{tr}}(\mathcal {K}_{e,s})^2\Big )\nonumber \\&=\mu L_c^2\Big (b_1\Vert {\textrm{sym}}\mathcal {K}_{e,s}^\parallel \Vert ^2+b_2\Vert \mathop {{\textrm{skew}}}\nolimits \mathcal {K}_{e,s}^\parallel \Vert ^2-\frac{(b_1-b_2)^2}{2(b_1+b_2)}\Vert \mathcal {K}_{e,s}^Tn_0\Vert ^2\nonumber \\&\quad +\frac{b_1b_3}{(b_1+b_3)}{\textrm{tr}}(\mathcal {K}_{e,s}^\parallel )^2+\frac{b_1+b_2}{2}\Vert \mathcal {K}_{e,s}^Tn_0\Vert \Big )\nonumber \\&=\mu L_c^2\Big (b_1\Vert {\textrm{sym}}\mathcal {K}_{e,s}^\parallel \Vert ^2+b_2\Vert \mathop {{\textrm{skew}}}\nolimits \mathcal {K}_{e,s}^\parallel \Vert ^2+\frac{b_1b_3}{(b_1+b_3)}{\textrm{tr}}(\mathcal {K}_{e,s}^\parallel )^2\nonumber \\&\quad +\frac{2b_1b_2}{b_1+b_2}\Vert \mathcal {K}_{e,s}^\perp \Vert \Big )\,, \end{aligned}$$
(A.44)
where\(\mathcal {K}_{e,s}=(\Gamma _1|\Gamma _2|0)[(\nabla _x\Theta )^\natural ]^{-1}\) with the decomposition
$$\begin{aligned} X=X^\parallel +X^\perp , \qquad \qquad \qquad X^\parallel :=\textrm{A}_{y_0} \,X, \qquad \qquad \qquad X^\perp :=({{\mathbb {1}}}_3-\textrm{A}_{y_0}) \,X, \end{aligned}$$
(A.45)
for every matrixX.
