## 2 Functionally Graded Porous GPLRC Cylindrical Panel
The size of the transient region is denoted by the length scale [44; 45], which is controlled by the grid density near the crack line when the phase field is approximated using the same grid constructed for the displacement field approximation. According to this concept, the total strain energy \(U\) and the total kinetic energy \(T\) of cracked structure are expressed as  
\[E_{\text{C}}=\frac{3}{8}\cdot\frac{1+\xi_{L}\eta_{L}f_{GPL}}{1-\eta_{L}f_{GPL} }E_{m}+\frac{5}{8}\cdot\frac{1+\xi_{T}\eta_{T}f_{GPL}}{1-\eta_{T}f_{GPL}}E_{m} \tag{4}\]4  
\[E_{\text{C}}=\frac{3}{8}\cdot\frac{1+\xi_{L}\eta_{L}f_{GPL}}{1-\eta_{L}f_{GPL} }E_{m}+\frac{5}{8}\cdot\frac{1+\xi_{T}\eta_{T}f_{GPL}}{1-\eta_{T}f_{GPL}}E_{m} \tag{4}\]5  
where \(\hat{U}\) and \(\hat{T}\) denote the total strain and total kinetic energies when the structure is completely undamaged. Letting \(\mathbf{d}=\left(u_{0},v_{0},w_{0},\beta_{x},\beta_{y}\right)^{T}\) be the displacement vector, the displacement field \(\mathbf{u}=\left\{u,v,w\right\}^{T}\) of FG-GPLRC cylindrical panel is expressed as  
\[\left\{ \begin{aligned} & u\\ & v\\ & w\end{aligned} \right\}_{(x,y,z)}=\left\{ \begin{aligned} & u_{0}\\ & v_{0}\\ & w_{0}\end{aligned} \right\}_{(x,y)}+z\cdot\left\{ \begin{aligned} & \beta_{x}\\ &\beta_{y}\\ & 0\end{aligned} \right\}_{(x,y)} \tag{17}\]  
according to the FSDT. The large defection of a cylindrical panel is represented by the von [PERSON] nonlinearity, which leads to the strain-displacement relations given by  
\[\varepsilon=\left\{ \begin{aligned} &\frac{w_{0}}{r}+\frac{\partial u_{0}}{ \partial x}+\frac{1}{2}\widetilde{w}_{x}\frac{\partial w}{\partial x}\\ &\frac{\partial w_{0}}{\partial x}+\frac{1}{2}\widetilde{w}_{y} \frac{\partial w}{\partial y}\\ &\frac{\partial w_{0}}{\partial x}+\frac{\partial u_{0}}{\partial y }+\frac{1}{2}\left(\widetilde{w}_{x}\frac{\partial w}{\partial x}+\widetilde{ w}_{y}\frac{\partial w}{\partial y}\right)\\ \end{aligned} \right\}+z\cdot\left\{ \begin{aligned} & \frac{\partial\beta_{x}}{\partial x}\\ &\frac{\partial\beta_{y}}{\partial y}\\ &\frac{\partial\beta_{x}}{\partial y}+\frac{\partial\beta_{y}}{ \partial x}\end{aligned} \right\}=(\mathbf{H}_{L}+\mathbf{H}_{NL})\mathbf{d} \tag{18}\]  
Figure 3: A central crack and phase field \(\phi(\mathbf{x})\) within a cylindrical panel. \[\gamma= \begin{cases}\quad\beta_{y}+\frac{\partial w_{0}}{\partial y}\\ \beta_{x}+\frac{\partial w_{0}}{\partial x}-\frac{w_{0}}{\tau}\end{cases} =\mathbf{H} _{s}\mathbf{d} \tag{19}\]  
with \(\varepsilon=\left\{\varepsilon_{xx},\varepsilon_{yy},2\varepsilon_{xy}\right\}^ {T}\), \(\gamma=\left\{\gamma_{yz},\gamma_{zx}\right\}^{T}\), and \(r=R+z\approx R\), where \(\mathbf{H}_{L}\), \(\mathbf{H}_{NL}\), and \(\mathbf{H}_{s}\) denote the \((3\times 5)\) and \((2\times 5)\) partial derivative matrices defined by  
\[\mathbf{H}_{L}= \begin{bmatrix}H_{x}&0&1/r&z\cdot H_{x}&0\\ 0&H_{y}&0&0&z\cdot H_{y}\\ H_{y}&H_{x}&0&z\cdot H_{y}&z\cdot H_{x}\end{bmatrix}  \tag{20}\]  
\[\mathbf{H}_{NL}= \begin{bmatrix}0&0&\widetilde{w}_{,x}H_{x}/2&0&0\\ 0&0&\widetilde{w}_{,y}H_{y}/2&0&0\\ 0&0&\left(\widetilde{w}_{,x}H_{y}+\widetilde{w}_{,y}H_{x}\right)/2&0&0\end{bmatrix}  \tag{21}\]  
\[\mathbf{H}_{s}= \begin{bmatrix}0&0&H_{y}&0&1\\ -1/r&0&H_{x}&1&0\end{bmatrix}  \tag{22}\]  
with \(H_{x}=\partial/\partial x\) and \(H_{y}=\partial/\partial y\). Here, \(\widetilde{w}_{,x}\) and \(\widetilde{w}_{,y}\) are the deflection derivatives of a panel, which are assumed to be known a priori, as described later. Then, the strain-stress constitutive relations become  
\[\sigma=\frac{E_{\mathrm{C}}}{1-\
u_{\mathrm{C}}^{2}} \begin{bmatrix}1&\
u_{ \mathrm{C}}&0\\ \
u_{\mathrm{C}}&1&0\\ 0&0&(1-\
u_{\mathrm{C}})/2\end{bmatrix}\begin{cases}\varepsilon_{xx}\\ \varepsilon_{yy}\\ 2\varepsilon_{xy}\end{cases} =\mathbf{D}(\mathbf{H}_{L}+\mathbf{H}_{NL})\mathbf{d} \tag{23}\]  
\[\tau= \begin{bmatrix}G_{\mathrm{C}}&0\\ 0&G_{\mathrm{C}}\end{bmatrix}\begin{cases}\gamma_{yz}\\ \gamma_{zx}\end{cases}