## 2 Theoretical Formulation
### 3D-GrF Circular Cylindrical Microshell
The displacement fields, based on [PERSON]'s thin shell theory, are expressed as [47]  
\[u_{1}(x,\theta,z,t)=u(x,\theta,t)-z\frac{\partial w(x,\theta,t)}{\partial x} \tag{25}\]  
\[u_{2}(x,\theta,z,t)=v(x,\theta,t)-\frac{z}{R}\bigg{[}\frac{\partial w(x,\theta, t)}{\partial\theta}-v(x,\theta,t)\bigg{]} \tag{26}\]  
\[u_{3}(x,\theta,z,t)=w(x,\theta,t) \tag{27}\]  
where \(t\) is time, and \(u_{1}(x,\theta,z,t)\), \(u_{2}(x,\theta,z,t)\), and \(u_{3}(x,\theta,z,t)\) are displacements of an arbitrary point of the microshell along \(x\)-, \(\theta\)-, and \(z\)-axes, respectively. By substituting Equations (25)-(27) and Equation (10) into Equation (7), strain-displacement relations are expressed as:  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}} \tag{28}\]  
\[\varepsilon_{\theta\theta}=\frac{1}{R}\frac{\partial v}{\partial\theta}+ \frac{w}{R}-\frac{z}{R^{2}}\frac{\partial^{2}w}{\partial\theta^{2}} \tag{29}\]  
\[\gamma_{x\theta}=\frac{\partial v}{\partial x}+\frac{1}{R}\frac{\partial u}{ \partial\theta}-\frac{2z}{R}\frac{\partial^{2}w}{\partial\theta\partial x}. \tag{30}\]  
Nonzero components of \(\chi\) can be obtained by substituting Equations (25)-(27) into Equation (11) and using Equation (5):  
\[\chi_{xx}=-\frac{1}{R}\bigg{(}\frac{\partial v}{\partial x}-\frac{\partial^{2 }w}{\partial x\partial\theta}\bigg{)} \tag{31}\]  
\[\chi_{zz}=\frac{1}{2R^{2}}\bigg{(}\frac{\partial u}{\partial\theta}-2z\frac{ \partial^{2}w}{\partial x\partial\theta}+R\frac{\partial v}{\partial x}\bigg{)} \tag{32}\]  
Figure 2:0  
Figure 2:1  
Figure 2:2  
Figure 2:3 \begin{table}
\begin{tabular}{c c c} \hline \hline \(\kappa_{\theta}\) & \(\kappa_{\theta}^{*}\) & \(\theta\) \\ \hline
0.1 & 0.1734 & 0.9360 \\
0.2 & 0.3426 & 0.8713 \\
0.3 & 0.5065 & 0.8058 \\
0.4 & 0.6637 & 0.7391 \\
0.5 & 0.8112 & 0.6711 \\
0.6 & 0.9432 & 0.6012 \\
0.65 & 0.9976 & 0.5653 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Foam coefficients for different distributions. According to Equation (1), the strain energy can be written as  
\[ \begin{array}{ll}U_{\rm s}=&\frac{1}{2}\int_{0}^{L}\int_{0}^{2\pi}\Bigl{[}N_{xx} \frac{\partial u}{\partial x}+\Bigl{(}\frac{N_{\theta\theta}}{R}-\frac{Y_{ \theta\theta}}{2R^{2}}+\frac{Y_{xx}}{2R^{2}}\Bigr{)}\frac{\partial u}{\partial \theta}-\frac{Y_{\theta\theta}}{2R^{2}}\frac{\partial^{2}u}{\partial\theta^{2 }}\\ &-\frac{Y_{xx}}{2R}\frac{\partial^{2}u}{\partial\theta^{2}}+\Bigl{(}N_{x\theta }-\frac{Y_{xx}}{R}+\frac{Y_{xx}}{2R}+\frac{Y_{xx}}{2R}+\frac{Y_{xx}}{2R}\Bigr{)} \frac{\partial v}{\partial x}+\frac{Y_{xx}}{2}\frac{\partial^{2}v}{\partial x^ {2}}+\frac{Y_{xx}}{R}\frac{\partial w}{\partial x}\\ &+\Bigl{(}\frac{N_{\theta\theta}}{R}-\frac{Y_{\theta\theta}}{R^{2}}\Bigr{)} \frac{\partial v}{\partial\theta}+\frac{Y_{\theta\theta}}{2R}\frac{\partial^{ 2}v}{\partial x\partial\theta}+\Bigl{(}Y_{xx}-M_{xx}-Y_{x\theta}\Bigr{)} \frac{\partial^{2}v}{\partial x^{2}}\\ &+\Bigl{(}Y_{\theta\theta}-\frac{M_{\theta\theta}}{R^{2}}+\frac{Y_{xx}}{R^{2}} \Bigr{)}\frac{\partial^{2}w}{\partial\theta^{2}}+\Bigl{(}Y_{x\theta}+Y_{z \theta}-\frac{2M_{x\theta}}{R}+\frac{Y_{xx}}{R}-\frac{Y_{xx}}{R}-\frac{Y_{xx} }{R^{2}}\Bigr{)}\frac{\partial^{2}w}{\partial x\partial\theta}\\ &+\frac{N_{\theta\theta}}{R}w\Bigr{]}Rd\theta dx\end{array}  \tag{37}\]  
The non-classical and classical moments and forces are given by  
\[ \begin{array}{ll}T_{ij}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}m_{ij}z{\rm d}z,&Y_{ij}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}m_{ij}{\rm d}z,\\ M_{ij}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sigma_{ij}z{\rm d}z,&N_{ij}=\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\sigma_{ij}{\rm d}z.\end{array}