## 3 Governing Equation Based on State-Space Method  
For the benefit of direct derivations, a new variable \(z=r-R_{0}\) is introduced, such that \(-H/2\leq z\leq+H/2\). According to the theory of two-dimensional elasticity, for an arch, the constitutive equations for the state of plane stress read as follows:  
\[\sigma_{\theta}=c_{11}\varepsilon_{\theta}+c_{13}\varepsilon_{r},\ \sigma_{r}=c_{13} \varepsilon_{\theta}+c_{33}\varepsilon_{r},\ \tau_{r\theta}=c_{55}\gamma_{r\theta} \tag{7}\]  
where \(\sigma_{\theta}\), \(\sigma_{r}\) and \(\tau_{r\theta}\) are stresses, and \(\xi_{\theta}\), \(\xi_{r}\) and \(\gamma_{r\theta}\) are strains, respectively. For the GPL-reinforced nanocomposites, the material coefficients \(c_{ij}\) can be expressed as:  
\[c_{11}=c_{33}=\frac{(1-v)E}{(1+v)(1-2v)},\ c_{12}=c_{13}=\frac{vE}{(1+v)(1-2v )},\ c_{44}=c_{66}=\frac{E}{2(1+v)}\]  
Figure 2: GPL distribution patterns along the thickness of the FG-GPLRC circular arch. where \(E=E(z)\) and \(\
u=\
u(z)\) are the \(z\)-dependent elastic constants, and can be determined by Equations (3) and (6), respectively. It should be pointed all the formulations in the present can be extend to the case of orthotropic composites easily. The kinematic conditions give:  
\[\varepsilon_{\theta}=\frac{1}{R_{0}+z}\frac{\partial u_{\theta}}{\partial\theta }+\frac{u_{r}}{R_{0}+z},\ \varepsilon_{r}=\frac{\partial u_{r}}{\partial z},\ \gamma_{r \theta}=\frac{\partial u_{\theta}}{\partial z}-\frac{u_{\theta}}{R_{0}+z}+ \frac{1}{R_{0}+z}\frac{\partial u_{r}}{\partial\theta} \tag{8}\]  
where \(u_{\theta}\) and \(u_{r}\) are the displacement components in the \(\theta\)- and \(r\)-directions, respectively. The differential equations of motion of the elastic circular arch can be written as:  
\[ \begin{array}{l}\frac{1}{R_{0}+z}\frac{\partial\sigma_{\theta}}{\partial \theta}+\frac{\partial\tau_{r\theta}}{\partial z}+\frac{2\tau_{r\theta}}{R_{0 }+z}=\rho\frac{\partial^{2}u_{\theta}}{\partial t^{2}}\\ \frac{1}{R_{0}+z}\frac{\partial\tau_{r\theta}}{\partial\theta}+\frac{\partial \sigma_{r}}{\partial z}+\frac{\sigma_{r}-\sigma_{\theta}}{R_{0}+z}=\rho \frac{\partial^{2}u_{r}}{\partial t^{2}}\end{array}  \tag{9}\]  
where \(\rho=\rho(z)\) is the \(z\)-dependent mass density of the nanocomposites, and \(t\) is the time. With the help of the state space method [32; 33], the following equations can be derived from Equations (7)-(9):  
\[ \begin{array}{l}\frac{\partial u_{\theta}}{\partial z}=\frac{u_{\theta}}{R_{ 0}+z}-\frac{1}{R_{0}+z}\frac{\partial u_{r}}{\partial\theta}+\frac{1}{c_{55}} \tau_{r\theta}\\ \frac{\partial\sigma_{r}}{\partial z}=\rho\frac{\partial^{2}u_{r}}{\partial t ^{2}}+\frac{k_{11}}{\left(R_{0}+z\right)^{2}}\Bigg{(}\frac{\partial u_{\theta }}{\partial\theta}+u_{r}\Bigg{)}+\frac{1}{R_{0}+z}\Bigg{(}\frac{c_{13}}{c_{33} }-1\Bigg{)}\sigma_{r}-\frac{1}{R_{0}+z}\frac{\partial\tau_{r\theta}}{\partial \theta}\\ \frac{\partial u_{r}}{\partial z}=-\frac{c_{13}}{c_{33}}\frac{1}{R_{0}+z}\frac{ \partial u_{\theta}}{\partial\theta}-\frac{c_{13}}{c_{33}}\frac{1}{R_{0}+z}u_ {r}+\frac{1}{c_{33}}\sigma_{r}\\ \frac{\partial\tau_{r\theta}}{\partial z}=\rho\frac{\partial^{2}u_{\theta}}{ \partial t^{2}}-\frac{k_{11}}{\left(R_{0}+z\right)^{2}}\Bigg{(}\frac{\partial ^{2}u_{\theta}}{\partial\theta^{2}}+\frac{\partial u_{r}}{\partial\theta} \Bigg{)}-\frac{c_{13}}{c_{33}}\frac{1}{R_{0}+z}\frac{\partial\sigma_{r}}{ \partial\theta}-\frac{2}{R_{0}+z}\tau_{r\theta}\end{array}  \tag{10}\]  
where \(k_{11}=c_{11}-c_{13}^{2}/c_{33}\), \(u_{\theta}\), \(\sigma_{r}\), \(u_{r}\) and \(\tau_{r\theta}\) are termed as the state variables, and:  
\[\sigma_{\theta}=\frac{k_{11}}{R_{0}+z}\Bigg{(}\frac{\partial u_{\theta}}{ \partial\theta}+u_{r}\Bigg{)}+\frac{c_{13}}{c_{33}}\sigma_{r} \tag{11}\]  
is obtained as the induced variable. The end boundary conditions for the FG-GPLRC circular arches can be expressed in terms of the state variables and/or induced variable.