## Appendix A
### Derivation Procedure of PD Material Parameters
\begin{array}{ll}W_{(k)}&=a\theta_{(k)}^{2}+b\sum\limits_{j=1}^{\infty} \delta|x_{(j)}-x_{(k)}|s_{(k)(j)}^{2}V_{(j)}+b_{\tilde{r}}\sum\limits_{j=1}^{ \tilde{l}}\delta|x_{(j)}-x_{(k)}|s_{(k)(j)}^{2}V_{(j)}+b_{\tilde{r}}\sum\limits _{j=1}^{\tilde{l}}\delta|x_{(j)}-x_{(k)}|s_{(k)(j)}^{2}V_{(j)}\\ &=a\xi^{2}+b\sum\limits_{\tilde{H}}\delta\xi(\zeta\cos^{2}\theta)^{2}dH+b_{ \tilde{r}}\sum\limits_{j=1}^{\tilde{l}}\delta|x_{(j)}-x_{(k)}|(\zeta\cos^{2} \theta)^{2}V_{(j)}+b_{\tilde{\rm Z}}(0)\\ &=a\xi^{2}+b\int_{0}^{\delta}\int_{0}^{2\pi}\delta\xi(\zeta\cos^{2}\theta)^{2} \xi d\theta d\xi+\xi^{2}b_{\tilde{r}}\sum\limits_{j=1}^{\tilde{l}}|x_{(j)}-x_ {(k)}|V_{(j)}\\ &=a\xi^{2}+\frac{\pi}{4}b\delta^{4}\xi^{2}+\xi^{2}b_{\tilde{r}}\sum\limits_{j= 1}^{\tilde{l}}|x_{(j)}-x_{(k)}|V_{(j)}=\frac{1}{2}Q_{11}\xi^{2}\end{array} .\] (A31)  
Comparing Equation (A28) with (A30) and (A31),  
\[d=\frac{2}{\pi\delta^{3}},\] (A32)  
\[a+\frac{\pi}{4}b\delta^{4}+b_{\tilde{r}}\delta\sum\limits_{j=1}^{\tilde{l}} \Big{|}x_{(j)}-x_{(k)}\Big{|}V_{(j)}=\frac{1}{2}Q_{11}.\] (A33)Substituting Equation (A27),  
\[a+b_{F}\delta\sum\limits_{j=1}^{J}|x_{(j)}-x_{(k)}|V_{(j)}=\frac{1}{2}Q_{11}- \frac{3}{2}Q_{55}.\] (A34)  
(3) Uniaxial stretch in thickness direction (\(\varepsilon_{33}\))  
Setting \(\varepsilon_{33}=\zeta\), similarly,  
\[\theta_{(k)}^{CM}=\zeta,\ W_{(k)}^{CM}=\frac{1}{2}Q_{33}\zeta^{2},\] (A35)  
\[s_{(k)(j)}=\zeta\sin^{2}\theta,\] (A36)  
\[\theta_{(k)} =d\sum\limits_{j=1}^{\infty}\delta s_{(k)(j)}\Lambda_{(k)(j)}V_{(j )}=d\sum\limits_{j=1}^{\infty}\delta\zeta\sin^{2}\theta\Lambda_{(k)(j)}V_{(j)} \quad,\] (A37) \[=d\int_{0}^{\delta}\int_{0}^{2\pi}\delta\zeta\sin^{2}\theta\xi d \theta d\zeta=\frac{\pi}{2}d\delta^{3}\zeta=\zeta\] \[W_{(k)} =a\theta_{(k)}^{2}+b\sum\limits_{j=1}^{\infty}\delta|x_{(j)}-x_{( k)}|s_{(k)(j)}^{2}V_{(j)}+b_{F}\sum\limits_{j=1}^{J}\delta|x_{(j)}-x_{(k)}|s_{(k)(j)}^ {2}V_{(j)}+b_{Z}\sum\limits_{j=1}^{J}\delta|x_{(j)}-x_{(k)}|s_{(k)(j)}^{2}V_{( j)}\] (A38) \[=a\zeta^{2}+b\int\limits_{H}\delta\xi(\zeta\sin^{2}\theta)^{2}dH+ b_{F}(0)+b_{Z}\sum\limits_{j=1}^{J}\delta|{\bf x}_{(j)}-{\bf x}_{(k)}|(\zeta \sin^{2}\theta)^{2}V_{(j)}\] \[=a\zeta^{2}+b\int_{0}^{\delta}\int_{0}^{2\pi}\delta\xi(\zeta\sin ^{2}\theta)^{2}\xi d\theta d\zeta+\zeta^{2}b_{Z}\delta\sum\limits_{j=1}^{J}|{ \bf x}_{(j)}-{\bf x}_{(k)}|V_{(j)}\] \[a+\frac{\pi}{4}b\delta^{4}+b_{\tilde{r}}\delta\sum\limits_{j=1}^{\tilde{l}} \Big{|}x_{(j)}-x_{(k)}\Big{|}V_{(j)}=\frac{1}{2}Q_{11}.\]0  
Comparing Equation (A35) with (A37) and (A38),  
\[a+\frac{\pi}{4}b\delta^{4}+b_{\tilde{r}}\delta\sum\limits_{j=1}^{\tilde{l}} \Big{|}x_{(j)}-x_{(k)}\Big{|}V_{(j)}=\frac{1}{2}Q_{11}.\]1 (A39)  
\[a+\frac{\pi}{4}b\delta^{4}+b_{\tilde{r}}\delta\sum\limits_{j=1}^{\tilde{l}} \Big{|}x_{(j)}-x_{(k)}\Big{|}V_{(j)}=\frac{1}{2}Q_{11}.\]2 (A40)  
Substituting Equation (A27),  
\[a+\frac{\pi}{4}b\delta^{4}+b_{\tilde{r}}\delta\sum\limits_{j=1}^{\tilde{l}} \Big{|}x_{(j)}-x_{(k)}\Big{|}V_{(j)}=\frac{1}{2}Q_{11}.\]3 (A41)  
(4) Biaxial tension on x-z plane (\(\varepsilon_{11},\ \varepsilon_{33}\))  
Setting \(\varepsilon_{11}=\zeta\), \(\varepsilon_{33}=\zeta\), similarly,  
\[a+\frac{\pi}{4}b\delta^{4}+b_{\tilde{r}}\delta\sum\limits_{j=1}^{\tilde{l}} \Big{|}x_{(j)}-x_{(k)}\Big{|}V_{(j)}=\frac{1}{2}Q_{11}.\]4 (A42)  
\[a+\frac{\pi}{4}b\delta^{4}+b_{\tilde{r}}\delta\sum\limits_{j=1}^{\tilde{l}} \Big{|}x_{(j)}-x_{(k)}\Big{|}V_{(j)}=\frac{1}{2}Q_{11}.\]5 (A43)  
\[a+\frac{\pi}{4}b\delta^{4}+b_{\tilde{r}}\delta\sum\limits_{j=1}^{\tilde{l}} \Big{|}x_{(j)}-x_{(k)}\Big{|}V_{(j)}=\frac{1}{2}Q_{11}.\]6 (A44)  
\[a+\frac{\pi}{4}b\delta^{4}+b_{\tilde{r}}\delta\sum\limits_{j=1}^{\tilde{l}} \Big{|}x_{(j)}-x_{(k)}\Big{|}V_{(j)}=\frac{1}{2}Q_{11}.\]7\[