## 4 Numerical Examples
### The Case for an FGM Ring Reinforced in a Homogeneous Plate  
There is an important theoretical significance of the above conclusions on the bending stress concentration in a whole FGM plate. However, considering the feasibility of material preparation and engineering application, it would be better to reinforce a thin FGM ring around the hole to reduce the stress concentration, in comparison with manufacturing a whole FGM perforated plate [23]. Hence, the bending moments in a homogeneous plate with a circular hole reinforced with an FGM ring are further analyzed here. The schematic diagram of the structure is displayed in Figure 5a. The solution of the bending moments for this case can be easily derived by taking \(r_{s}\) in Section 3 as the outer radius of the reinforcing ring. Since the varying Poisson's ratio has little effect on the bending moments, Poisson's ratio is assumed as constant \(v(r)=v_{0}=0.3\) in the next discussions. Two Young's moduli, \(E_{h}\) at the edge of the hole and \(E_{p}\) in the plate, are taken as the main parameters. Meanwhile, four types of the reinforcing ring listed in Table 2 are considered. The varying curves of Young's modulus in four different reinforcing rings are presented in Figure 5b. The solid line in the figure represents a special case of a homogeneous plate with a circular hole with no ring. The dash-dotted line, dotted line, and dashed line mean the edge of the hole reinforced by, respectively, a homogeneous ring with unchanged Young's modulus, FGM ring with linearly varying Young's modulus, and FGM ring with exponentially varying Young's modulus. It is obvious that [PERSON]'s modulus is always decreasing as \(E_{h}>E_{p}\), but increasing as \(E_{h}<E_{p}\) in Figure 5.  
The distribution of bending moments \(M_{\theta}/M_{0}\) for different types of the ring is presented in Figure 6 for the cases of \(E_{h}\geq E_{p}\) and \(E_{h}\leq E_{p}\). The radius ratio of the ring is taken as \(r_{N}/r_{0}=1.5\). It is observed from Figure 6a that the bending moments at the hole for three cases of \"with-ring\" are much higher than that of non-ring as [PERSON]'s modulus at the edge of the hole \(E_{h}\) larger than that in the plate \(E_{p}\). On the contrary, the moments near the hole for \"with-ring\" are less than that of non-ring as \(E_{h}<E_{p}\) in Figure 6b. It is obvious that the moment concentration near the hole can be \begin{table}
\begin{tabular}{c c} \hline
**The Types of Ring** & **The Functions of Young’s Modulus** \\ \hline Non-ring & \(E(r)=E_{h}=E_{p}\) \\ Homo-ring & \(E(r)=E_{h}\
eq E_{p}\) \\ FGM-ring (linear) & \(E(r)=\left(E_{p}-E_{h}\right)\left(r-r_{0}\right)/\left(r_{s}-r_{0}\right)+E_{h}\) \\ FGM-ring (exponential) & \(E(r)=\left(E_{p}-E_{h}\right)\left(1-e^{5}\right)*e^{5(1-\left(r-r_{0}\right)/ \left(r_{s}-r_{0}\right))}-\left(e^{5}E_{p}-E_{h}\right)\left/\left(1-e^{5} \right)\right.\) \\ \hline \end{tabular}
\end{table} 
Table 2: Four types of the reinforcing ring with different functions of Young’s modulus. Figure 5: **(a)** A homogeneous thin plate with a circular hole reinforced with a ring; **(b)** variation of Young’s modulus along the radial direction for four types of the ring. reduced in this case. However, it is also worth noting that the moments in the plate for \"with-ring\" are slightly higher than that of non-ring. For this reason, it is found that a sudden change of the moments occurs at the interface between the homogeneous ring and the plate. The sudden change of the moments will lead to a seriously interfacial stress concentration, which may promote the crack initiation and propagation along the interface and ultimately result in the failure of the whole structure. Therefore, the homogeneous ring can relieve the stress concentration near the hole, but the mismatch of material properties at the interface between the ring and the plate will cause a new interfacial stress concentration. The FGM ring can effectively avoid this problem due to continuous variations of elastic properties from the edge of the hole to the plate.