## 3 General Solutions  
For convenience, we introduce a cylindrical coordinate \((r,\theta)\) in the plane \(xoy\) with the application of the hole center as the pole, the \(x\)-axis and \(\theta\) as the polar axis and polar angle, respectively. The moments and transverse shear forces in the cylindrical coordinate can be expressed with two complex potential functions, \(\varphi(z)\) and \(\psi(z)\), by formulas [32] (1) and (2):  
\[M_{\theta}-M_{r}+2 iH_{r\theta}=2D(1-v)e^{2i\theta}[\overline{z}\varphi^{\prime \prime}(z)+\psi^{\prime\prime}(z)], \tag{1}\]  
\[M_{\theta}+M_{r}=-4D(1+v)\text{Re}[\varphi^{\prime}(z)], \tag{2}\]  
where \(M_{\theta}\), \(M_{r}\), and \(H_{r\theta}\) are the bending moments and twisting moments, respectively, and the bending rigidity of the plate \(D=El^{2}/\left[12\big{(}1-v^{2}\big{)}\right]\). The complex potential functions \(\varphi(z)\) and \(\psi(z)\) can be determined by the following boundary conditions when the components of force \(f_{1}\), \(f_{2}\), components of displacement \(u\), \(v\) and deflection \(w\) are given  
\[D\Big{[}(1-v)\Big{[}\overline{z}\varphi^{\prime}(z)+\psi(z)\Big{]}-(3+v) \overline{\varphi(z)}\Big{]}=f_{1}-if_{2}, \tag{3}\]  
\[\psi(z)+\overline{z}\varphi^{\prime}(z)+\overline{\varphi(z)}=u-iv, \tag{4}\]  
\[\text{Re}[\chi(z)+\overline{z}\varphi(z)]=w(x,y). \tag{5}\]  
Figure 1: A functionally graded thin plate with a circular hole subjected to remote bending moment. Introducing Equations (9) and (10) with \(j=1,2,\cdots,s-1\) and \(z=r_{j}e^{i\theta}\) into Equations (13)-(15), and instead of \(n_{j}=r_{j}/R\), we obtain the following lengthy equations after a series of somewhat arduous simplifications  
\[D_{j}\cdot\left\{\left(1-v_{j}\right)\!\!\left[a^{(j)}e^{-i\theta}n_{j}^{-1}+ \sum_{k=2}^{+\infty}a^{(j)}_{-(k-1)}(1-k)e^{-ik\theta}n_{j}^{-k}+\sum_{k=0}^{+ \infty}a^{(j)}_{k+1}(k+1)e^{ik\theta}n_{j}^{k}\right.\right.\]  
\[\left.\left.+\sum_{k=3}^{+\infty}b^{(j)}_{-(k-2)}(2-k)e^{-ik\theta}n_{j}^{-(k-2 )}+\sum_{k=-1}^{+\infty}b^{(j)}_{k+2}(k+2)e^{ik\theta}n_{j}^{k+2}\right]\right.\]  
\[\left.-\left(3+v_{j}\right)\!\!\left[\sum_{k=1}^{+\infty}\overline{b^{(j)}_{- k}}e^{ik\theta}n_{j}^{-k}+\sum_{k=0}^{+\infty}\overline{b^{(j)}_{k}}e^{-ik \theta}n_{j}^{k}\right]\!\!\right\}\]  
\[=D_{j+1}\cdot\left\{\!\!\left(1-v_{j+1}\right)\!\!\left[a^{(j+1)}e^{-i\theta} n_{j}^{-1}+\sum_{k=2}^{+\infty}a^{(j+1)}_{-(k-1)}(1-k)e^{-ik\theta}n_{j}^{-k} \!+\!\sum_{k=0}^{+\infty}a^{(j+1)}_{k+1}(k+1)e^{ik\theta}n_{j}^{k}\right.\right.\]  
\[M_{\theta}+M_{r}=-4D(1+v)\text{Re}[\varphi^{\prime}(z)], \tag{2}\]0  
\[M_{\theta}+M_{r}=-4D(1+v)\text{Re}[\varphi^{\prime}(z)], \tag{2}\]1  
\[M_{\theta}+M_{r}=-4D(1+v)\text{Re}[\varphi^{\prime}(z)], \tag{2}\]2  
\[M_{\theta}+M_{r}=-4D(1+v)\text{Re}[\varphi^{\prime}(z)], \tag{2}\]3  
\[M_{\theta}+M_{r}=-4D(1+v)\text{Re}[\varphi^{\prime}(z)], \tag{2}\]4  
\[M_{\theta}+M_{r}=-4D(1+v)\text{Re}[\varphi^{\prime}(z)], \tag{2}\]5  
\[M_{\theta}+M_{r}=-4D(1+v)\text{Re}[\varphi^{\prime}(z)], \tag{2}\]6  
Similar to the calculations at the edge of the hole, we can get a set of \(4m(s-1)+7(s-1)\) linear equations listed as Equations (A6)-(A16) in Appendix A by comparing the coefficients of the same power of \(-2\), \(-1\), \(0\)\(k\) and \(-k\) of \(e^{i\theta}\) in Equations (16)-(18). Finally, there are also the same continuous conditions of Equations (13)-(15) at the interface \(r=r_{s}\) between the ring \(K_{s}\) and the outer plate \(K_{s+1}\). Substituting Equations (7)-(10) with \(z=r_{s}e^{i\theta}\) into Equations (13)-(15), and using \(n_{s}=r_{s}/R\), we can get another set of lengthy equations.