A Geometrically Nonlinear Cosserat (Micropolar) Curvy Shell Model Via Gamma Convergence

Abstract

Using $\Gamma$-convergence arguments, we construct a nonlinear membrane-like Cosserat shell model on a curvy reference configuration starting from a geometrically nonlinear, physically linear three-dimensional isotropic Cosserat model. Even if the theory is of order O(h) in the shell thickness h, by comparison to the membrane shell models proposed in classical nonlinear elasticity, beside the change of metric, the membrane-like Cosserat shell model is still capable of capturing the transverse shear deformation and the Cosserat-curvature due to remaining Cosserat effects. We formulate the limit problem by scaling both unknowns, the deformation and the microrotation tensor, and by expressing the parental three-dimensional Cosserat energy with respect to a fictitious flat configuration. The model obtained via $\Gamma$-convergence is similar to the membrane (no $O(h^3)$ flexural terms, but still depending on the Cosserat-curvature) Cosserat shell model derived via a derivation approach, but these two models do not coincide. Comparisons to other shell models are also included.

Appendix

1.1 An 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 the first 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)

1.2 Calculations 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 $\langle (0|0|\overline{Q}_{e}^{\natural ,T}d^*-n_0)[(\nabla _x\Theta )^\natural ]^{-1},{{\mathbb {1}}}_3 \rangle _{\mathbb {R}^{3\times 3}}\,n_0 =\langle (\overline{Q}_{e}^{\natural ,T}d^*-n_0),n_0 \rangle _{\mathbb {R}^3}\,n_0=(\overline{Q}_{e}^{\natural ,T}d^*-n_0)\,n_0\otimes n_0$.

1.3 Calculations 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)

1.4 Homogenized 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 matrix X.