## 4 Solution Methods  
### Analytical Solution of a Simply Supported FG-GPLRC Circular Arch  
For the S-S FG-GPLRC circular arch, an exact solution can be assumed as:  
\[ \begin{split} u_{\theta}&=H\sum\limits_{n=1}^{\infty}U( \zeta)\cos(n\pi\theta/\theta_{0})\mathrm{e}^{\mathrm{i}\omega t}\\ \sigma_{r}&=c_{0}\sum\limits_{n=1}^{\infty}Y( \zeta)\sin(n\pi\theta/\theta_{0})\mathrm{e}^{\mathrm{i}\omega t}\\ u_{r}&=H\sum\limits_{n=1}^{\infty}W(\zeta)\sin(n \pi\theta/\theta_{0})\mathrm{e}^{\mathrm{i}\omega t}\\ \tau_{r\theta}&=c_{0}\sum\limits_{n=1}^{\infty} \Gamma(\zeta)\cos(n\pi\theta/\theta_{0})\mathrm{e}^{\mathrm{i}\omega t}\end{split}  \tag{15}\]  
where \(\zeta=z/H\), \(n\) is the half-wave number along the \(\theta\)-direction, \(\omega\) is the circular frequency, \(\mathrm{i}=\sqrt{-1}\) is the imaginary unit, and \(c_{0}\) denotes the elastic constant of the matrix of the GPL-reinforced nanocomposites. Substitution of Equation (15) into Equation (10) yields the following nondimensional state ordinary differential equation:  
\[\frac{\mathrm{d}}{\mathrm{d}\zeta}\boldsymbol{\delta}(\zeta)=\mathbf{M}( \Omega,\zeta)\boldsymbol{\delta}(\zeta) \tag{16}\]  
where \(\boldsymbol{\delta}(\zeta)=[U\,Y\,W\,\Gamma]^{\mathrm{T}}\) is called the dimensionless state vector, and \(\mathbf{M}\) is the coefficient matrix of the state space equation, which is:  
\[\mathbf{M}=\left[ \begin{array}{ccc}\dfrac{\lambda}{\eta}&0&-\dfrac{k_{n}}{ \eta}&\dfrac{c_{0}}{c_{55}}\\ -\dfrac{k_{11}}{c_{0}}\dfrac{k_{n}\lambda}{\eta^{2}}&\dfrac{\lambda}{\eta} \bigg{(}\dfrac{c_{13}}{c_{33}}-1\bigg{)}&\dfrac{k_{11}}{c_{0}}\bigg{(}\dfrac{ \lambda}{\eta}\bigg{)}^{2}-\dfrac{\rho}{\rho_{0}}\Omega^{2}&\dfrac{k_{n}}{ \eta}\\ \dfrac{c_{13}}{c_{33}}\dfrac{k_{n}}{\eta}&\dfrac{c_{0}}{c_{33}}&-\dfrac{c_{13}} {c_{33}}\dfrac{\lambda}{\eta}&0\\ \dfrac{k_{11}}{c_{0}}\bigg{(}\dfrac{k_{n}}{\eta}\bigg{)}^{2}-\dfrac{\rho}{\rho _{0}}\Omega^{2}&-\dfrac{c_{13}}{c_{33}}\dfrac{k_{n}}{\eta}&-\dfrac{k_{11}}{c_ {0}}\dfrac{k_{n}\lambda}{\eta^{2}}&-\dfrac{2\lambda}{\eta}\\ \end{array} \right] \tag{17}\]  
where \(\lambda=H/R_{0}\), \(\eta=1+\lambda\zeta\), \(k_{n}=n\pi H/L_{0}\), \(\rho_{0}\) denotes the density of the matrix, and \(\Omega=\omega H\sqrt{\rho_{0}/c_{0}}\) is the nondimensional frequency. However, it is seen that Equation (16) is a variable coefficient differential equation with respective to the state vector \(\boldsymbol{\delta}(\zeta)\), and it is quite difficult to solve analytically. Herein, the approximate laminate model (ALM) is adopted to break this obstacle [35]. The arch is divided into \(p\) layers with the identical thickness \(h=H/p\) through the thickness direction. For each sufficiently thin layer, the coordinate \(\zeta\) can be regarded as constant. With these approximations, Equation (16) is then reduced to a differential equation with constant coefficient matrix for an individual layer, say the \(j\)-th layer, by setting:  
\[\zeta=\zeta_{j_{m}}=\big{(}\zeta_{j_{0}}+\zeta_{j_{1}}\big{)}/2,\ (j=1,2,\cdots,p) \tag{18}\]  
the \(\zeta\)-dependent variable \(\eta\) in Equation (16) turns out to be constant, where \(\zeta_{j_{0}}=-0.5+\frac{j-1}{p}\), \(\zeta_{j_{1}}=-0.5+\frac{j}{p}\) are the positions of the surfaces of the \(j\)-th suppositional layer, respectively.