## 2 Theoretical Analysis
\begin{array}{l}D_{11}\frac{dw}{dy}\right|_{-\frac{s}{2}}^{\frac{s}{2} }=D_{11}\left[\begin{array}{cc}2\alpha\;c_{1}\;\cos\left(\frac{\kappa s}{2} \right)\;\sinh\left(\frac{\kappa s}{2}\right)-2\alpha\;c_{1}\;\cos\left(\frac{ \kappa s}{2}\right)\;\sin\left(\frac{\kappa s}{2}\right)+\\ 2\alpha\;c_{2}\;\cos\left(\frac{\kappa s}{2}\right)\;\sinh\left(\frac{\kappa s}{ 2}\right)+2\alpha\;c_{2}\;\cos\left(\frac{\kappa s}{2}\right)\;\sin\left(\frac{ \kappa s}{2}\right)\end{array}\right]\\ \int_{-\frac{s}{2}}^{\frac{s}{2}}\left[\frac{w^{2}}{r}\left(\frac{A_{12}{}^{2} }{A_{22}}-A_{11}\right)\right]dy=\frac{\left(\frac{4s_{12}{}^{2}}{A_{22}}-A_{1 1}\right)}{r}\left\{\begin{array}{l}\frac{\left(c_{12}-c_{2}^{2}\right)}{16 \alpha}\left[\left.\left(\kappa s\sin\alpha+\cos\alpha s\right)+e^{-\alpha s}( \sin\alpha s-\cos\alpha s)\right]+\\ \frac{c_{12}\;\left(\kappa s\right)}{8\alpha}\left[\left.\left(\kappa s\sin \alpha-\cos\alpha s\right)+e^{-\alpha s}(\sin\alpha s+\cos\alpha s)\right]\\ +\frac{e^{-\alpha s}-e^{-\alpha s}}{8\alpha}\left(c_{1}\;^{2}+c_{2}^{2}\right) +\frac{\sin\left(\kappa s\right)}{4}\left(c_{1}^{2}+c_{2}^{2}\right)+\frac{s}{4 }\left(c_{1}^{2}+c_{2}^{2}\right)+\frac{s}{4}\left(c_{1}^{2}-c_{2}^{2}\right. \end{array}\right)\right\}\end{array} \right\} \tag{21}\]  
For isotropic materials:  
\[ \begin{array}{l}E_{1}=E_{2}=E,\
u_{21}=\
u_{12}=\
u\\ A_{11}=\frac{El}{1-\
u^{2}}=A=A_{22},\;A_{12}=\
u A,A_{66}=\frac{El}{2(1+\
u)}= \frac{1-\
u}{2}A,\;A_{16}=A_{26}=0,B_{ij}=0,\\ D_{11}=\frac{El^{3}}{12(1-\
u^{2})}=D=D_{22},\;D_{12}=\
u D,D_{66}=\frac{El^{3}} {2(1+\
u)}=\frac{1-\
u}{2}D,D_{16}=D_{26}=0\end{array}  \tag{22}\]Then  
\[\begin{array}{l}M=\frac{vD}{r}\left\{ \begin{array}{l}\frac{\frac{a^{2}c_{1}c_{2}} {4}}{4}[\mathrm{e}^{\alpha s}(\sin\alpha s+\cos\alpha s)+\mathrm{e}^{-\alpha s}( \sin\alpha s-\cos\alpha s)]+\\ \frac{a^{2}\left(c_{2}^{2}-c_{1}^{2}\right)}{8}[\mathrm{e}^{\alpha s}(\sin \alpha s-\cos\alpha s)+\mathrm{e}^{-\alpha s}(\sin\alpha s+\cos\alpha s)]+a^{2} c_{1}c_{2}s\end{array} \right\}-\\ Dv\left[ \begin{array}{l}2\alpha c_{1}\cos\left(\frac{\alpha s}{2}s\right) \sinh\left(\frac{\alpha s}{2}s\right)-2\alpha\ c_{1}\ \cosh\left(\frac{\alpha s}{2}\right) \sin\left(\frac{\alpha s}{2}s\right)+\\ 2\alpha c_{2}\ \cos\left(\frac{\alpha s}{2}s\right)\sinh\left(\frac{\alpha s}{2}s \right)+2\alpha\ c_{2}\ \cosh\left(\frac{\alpha s}{2}s\right)\ \sin\left(\frac{\alpha s}{2}s\right) \end{array} \right]+\\ \left(\frac{T}{r}+\frac{D}{r^{3}}-\frac{T^{2}}{r}\right)\left\{ \begin{array}[ ]{l}\frac{\left(c_{1}^{2}-c_{2}^{2}\right)}{16\alpha s}\big{[}\mathrm{e}^{ \alpha s}(\sin\alpha s+\cos\alpha s)+\mathrm{e}^{-\alpha s}(\sin\alpha s-\cos \alpha s)]+\\ \frac{\mathrm{e}^{\alpha s}-c\alpha s}{8\alpha}\big{[}\mathrm{e}^{\alpha s}( \sin\alpha s-\cos\alpha s)+\mathrm{e}^{-\alpha s}(\sin\alpha s+\cos\alpha s)]+ \\ \frac{\mathrm{e}^{\alpha s}-c\alpha s}{8\alpha}\big{(}c_{1}^{2}+c_{2}^{2}\big{)} +\frac{\sin\left(\alpha s\right)}{4\alpha}\big{(}c_{1}^{2}+c_{2}^{2}\big{)} +\frac{\mathrm{e}^{\alpha}}{4}\big{(}c_{1}^{2}-c_{2}^{2}\big{)}\end{array}  \right\}-\\ \left(\frac{vD}{r}\frac{1}{R}+\frac{D}{r}\right)\left[ \begin{array}{l}\left( \frac{c_{1}}{2}\frac{1}{4}\mathrm{e}^{\alpha\frac{\alpha}{2}}+\frac{c_{2}}{2} \frac{1}{4}\mathrm{e}^{-\frac{\alpha}{2}}\right)(\sin\frac{\alpha s}{2}+\cos \frac{\alpha s}{2})+\\ \left(\frac{\mathrm{e}^{\frac{1}{2}}\mathrm{e}^{-\frac{\alpha}{2}}+\frac{c_{2} }{2}\frac{1}{4}\mathrm{e}^{\frac{\alpha}{2}}}\right)(\sin\frac{\alpha s}{2}- \cos\frac{\alpha s}{2})\end{array} \right]+\frac{Dvs}{R} \tag{23}\]  
For the negative [PERSON]'s ratio honeycomb structures, \(A_{ij}=Q_{ij}t,B_{ij}\equiv 0,D_{ij}=Q_{ij}\frac{t^{3}}{\Gamma 2}\), then  
\[\begin{array}{l}M=\left(D_{11}\frac{1}{r}+D_{22}\frac{1}{R}\right)\Big{|}_{ -\frac{5}{2}}^{\frac{\delta}{2}}-D_{22}\frac{dw}{dx}\Big{|}_{-\frac{5}{2}}^{ \frac{\delta}{2}}-\frac{\delta}{2}\Big{[}\frac{w^{2}}{r}\Big{(}\frac{A_{12} ^{2}}{A_{11}}-A_{22}\Big{)}\Big{]}dx\\ =\left(D_{11}\frac{1}{r}+D_{22}\frac{1}{R}\right)s-D_{22}\Bigg{[}