## 4 Numerical Examples
### The Case for a Whole FGM Plate with a Circular Hole  
It is assumed that the Young's modulus and Poisson's ratio in the FGM plate have three varying forms as decreasing, unchanged, and increasing, respectively, along the radial direction. The varying functions of Young's modulus and Poisson's ratio are listed in Table 1, and their varying characteristics can be clearly seen in Figure 2. The unchanged elastic parameters correspond to the case of a homogeneous plate with a circular hole. For the decreasing and increasing Young's modulus and Poisson's ratio, it can be seen they vary obviously near the hole but change slowly and ultimately tend to constants after a radius larger than about \(5r_{0}\). In the following, we will discuss the distribution of bending moments in the FGM plate under uniaxial bending and balanced biaxial bending, for the above three functions of Young's modulus and Poisson's ratio in Table 1.  
Firstly, the effects of different varying Young's modulus on the distribution of normalized bending moments \(M_{\theta}/M_{0}\) in the plate are discussed, and the results are shown in Figure 3, where the Poisson's ratio is assumed to remain unchanged \(v(r)=v_{0}=0.3\). The solid lines in the figure represent the results of the homogeneous perforated plate. For the case of uniaxial bending in Figure 3a,b, it is found the maximum values of the moments are at the points \(\theta=\pm 90^{0}\) at the edge of the hole and equal to \(1.79M_{0}\). For the balanced biaxial bending in Figure 3c,d, the value of moments around the hole is a constant, and identically equal to \(2M_{0}\). As shown in the figure, these results agree with those of [PERSON] [5] and [PERSON] [6]. In addition, it is most important to find from Figure 3 that the bending stress concentration near the hole significantly increases as the [PERSON]'s modulus decreases in radial direction, while it reduces with the Young's modulus increasing under both uniaxial bending and biaxial bending. The normalized bending moment \(M_{\theta}/M_{0}\) is close to 1 in the whole plate for \(E_{+}(r)\). That means the phenomenon of bending stress concentration around the hole is almost eliminated in this case. Therefore, it can be concluded that the problem of bending stress concentration, caused by the geometric discontinuity of the hole in the traditional homogeneous plate, can be well relieved or even eliminated by properly choosing the radially increasing [PERSON]'s modulus in the plate. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Varying Forms** & **The Functions of Young’s Modulus** & **The Functions of Poisson’s Ratio** \\ \hline Decreasing & \(E_{-}(r)=E_{0}\big{(}1+0.5e^{1-r/r_{0}}\big{)}\)\({}^{1}\) & \(v_{-}(r)=v_{0}\big{(}1+0.5e^{1-r/r_{0}}\big{)}\)\({}^{1}\) \\ Unchanged & \(E_{0}(r)=E_{0}\) & \(v_{0}(r)=v_{0}\) \\ Increasing & \(E_{+}(r)=E_{0}\big{(}1-0.5e^{1-r/r_{0}}\big{)}\) & \(v_{+}(r)=v_{0}\big{(}1-0.5e^{1-r/r_{0}}\big{)}\) \\ \hline \hline \end{tabular} \({}^{1}\)\(E_{0}\) and \(v_{0}\) are constants.  
\end{table} 
Table 1: Three different functions of Young’s modulus and Poisson’s ratio in the functionally graded material (FGM) plate. Figure 2: Variation of Young’s modulus and Poisson’s ratio along the radial direction. The distribution of circumferential bending moments \(M_{\theta}/M_{0}\) for different radially varying [PERSON]'s ratios are shown in Figure 4, where the [PERSON]'s modulus is taken as constant. It is clear to see that the effect of a varying Poisson's ratio on the bending moments is very slight in comparison with that of [PERSON]'s modulus. Three curves of the moments for different varying Poisson's ratios almost overlap under uniaxial bending in Figure 4a. The influence of the radial variations of [PERSON]'s ratio on the bending moments is so small that it can be ignored. Figure 4: Distribution of moments \(M_{\theta}/M_{0}\) around the hole for different varying Poisson’s ratio: (**a**) uniaxial bending; (**b**) balanced biaxial bending.