## 2 Theoretical Formulation
### 3D-GrF Circular Cylindrical Microshell  
In Figure 1, a 3D-GrF microshell with the thickness \(h\), the middle-plane radius \(R\), and the length \(L\) is shown. \(u(x,\,\theta,\,t),\,v(x,\,\theta,\,t)\), and \(w(x,\,\theta,\,t)\) represent the in-plane and transverse displacements of points at the middle plane; \(N^{0}_{xx}\) represents the axial load applied to the microshell. Figure 1: Schematic of a three-dimensional graphene foam (3D-GrF) circular cylindrical microshell. Three types of foam distributions in the thickness direction were considered, as shown in Figure 2. Herein, Figure 2a,b depict non-uniform foam distribution while Figure 2c shows uniform foam distribution. They are denoted by 3D-GrF-I, 3D-GrF-II, and 3D-GrF-U, respectively. As shown in Figure 2, the largest foams are located on the mid-plane for 3D-GrF-I while on the top and bottom surfaces for 3D-GrF-II, leading to the variations of material properties given in Equations (12)-(14) for 3D-GrF-I, and Equations (15)-(17) for 3D-GrF-II. Material properties of 3D-GrF-U are described in Equations (18)-(20). They are given by [44; 45; 46]:  
\[E(z)=E_{1}[1-\kappa_{0}\cos(\pi z/h)] \tag{12}\]  
\[\rho(z)=\rho_{1}[1-\kappa_{m}\cos(\pi z/h)] \tag{13}\]  
\[G(z)=G_{1}[1-\kappa_{0}\cos(\pi z/h)] \tag{14}\]  
\[E(z)=E_{1}\{1-\kappa_{0}^{*}[1-\cos(\pi z/h)]\} \tag{15}\]  
\[\rho(z)=\rho_{1}\{1-\kappa_{m}^{*}[1-\cos(\pi z/h)]\} \tag{16}\]  
\[G(z)=G_{1}\{1-\kappa_{0}^{*}[1-\cos(\pi z/h)]\} \tag{17}\]  
\[E(z)=E_{1}\theta \tag{18}\]  
\[\rho(z)=\rho_{1}\theta^{\prime} \tag{19}\]  
Figure 2.0  
where \(E(z)\), \(\rho(z)\), and \(G(z)\) are general Young's modules, mass density, and shear modules of the 3D-GrF microshell, respectively; \(E_{1}\), \(G_{1}\), and \(\rho_{1}\) represent corresponding properties of solid graphenes without internal foams; \(\kappa_{0}\) and \(\kappa_{m}\) represent coefficients of foams and mass density for 3D-GrF-I, respectively; \(\kappa_{0}^{*}\) and \(\kappa_{m}^{*}\) are corresponding coefficients for 3D-GrF-II; \(\theta\) and \(\theta^{\prime}\) are corresponding coefficients for 3D-GrF-U. Thereinto, shear modulus \(G_{1}\) is calculated by  
Figure 2.1  
where \(\
u\) represents [PERSON]'s ratio. Figure 2.2 _Cont._  
The typical mechanical property of 3D-GrFs [3; 20; 21], shown in Equation (22), is employed to establish the relationships in Equation (23) between mass density coefficients and foam coefficients for different foam distributions:  
Figure 2.3  
\[\left\{ \begin{array}{rl}1-\kappa_{m}\cos(\pi z/h)=&\
icefrac{{2.73}}{{\sqrt{1- \kappa_{0}\cos(\pi z/h)}}}&\mathrm{3D-GrF-I}\\ 1-\kappa_{m}^{*}[1-\cos(\pi z/h)]=&\
icefrac{{2.73}}{{\sqrt{1-\kappa_{0}^{*}[1 -\cos(\pi z/h)]}}}&\mathrm{3D-GrF-II}\\ \theta^{\prime}=&\
icefrac{{2.73}}{{\sqrt{\theta}}}&\mathrm{3D-GrF-U}\end{array} \right. \tag{23}\]  
The masses of all 3D-GrF microshells with varying foams are set to be equivalent, namely  
\[\left\{ \begin{array}{rl}\int_{0}^{h/2}\
icefrac{{2.73}}{{\sqrt{1-\kappa_{0} ^{*}[1-\cos(\pi z/h)]}}}\mathrm{d}z=&\int_{0}^{h/2}\
icefrac{{2.73}}{{\sqrt{1- \kappa_{0}\cos(\pi z/h)}}}\mathrm{d}z\\ \int_{0}^{h/2}\
icefrac{{2.73}}{{\sqrt{\theta}}}\mathrm{d}z=&\int_{0}^{h/2} \
icefrac{{2.73}}{{\sqrt{1-\kappa_{0}\cos(\pi z/h)}}}\mathrm{d}z\end{array} \right. \tag{24}\]  
which can be used to determine \(\kappa_{0}^{*}\) and \(\vartheta\) with a given value of \(\kappa_{0}\), as tabulated in Table 1. It is seen that \(\kappa_{0}^{*}\) rises dramatically with the increase of \(\kappa_{0}\). When \(\kappa_{0}\) reaches 0.65, \(\kappa_{0}^{*}\) is close to the upper limit (\(\kappa_{0}^{*}=0.9976\)). Therefore, the selected range of \(\kappa_{0}\in[0,\,0.65]\) is applied in the following numerical calculations. Figure 2: 3D-GrF distributions in the thickness direction: (**a**) 3D-GrF-I; (**b**) 3D-GrF-II; (**c**) 3D-GrF-U.