## 3 Numerical Examples  
FGMs can generally be tailored through the power-law function, due to its easy control of properties and better implementation, so the following change function of [PERSON]'s modulus in the FG layer is used in numerical discussions [8]:  
\[E(r)=E_{p}\left(\frac{r}{r_{N}}\right)^{n}, \tag{26}\]  
where \(E_{p}\) is the Young's modulus in domain \(S_{p}\). The Poisson's ratio in the FG layer is assumed to be unchanged and the same as the plate \(v_{g}=v_{p}=0.3\). In order to check the accuracy of the present numerical analysis, the obtained solutions for several special cases were compared with Lubarda [41], in which the analytical solution of circumferential shear stress around circular and elliptical holes in a homogenous plate without FG layers is presented. The conformal mapping function Equation (7) for ellipses can be written as [31]  
\[z=\omega(\zeta)=R\left(\zeta+\frac{a-b}{a+b}\zeta^{-1}\right), \tag{27}\]  
where \(a\) and \(b\) are the semimajor and semiminor axes, respectively. The constant \(R\) in Equation (7) does not have an effect on the stress field for the infinite field, so we chose \(R=1\). The exponents of [PERSON]'s modulus in Equation (26) are chosen as \(n=0\), which corresponds to the case of the hole without an FG layer. In the case, the circumferential shear stresses around an elliptical hole for different values of the ellipse aspect ratio are compared with the analytical solution given by Equation (34) in Lubarda [41]. The results are shown in Figure 2. It was found that the present numerical solutions are in very close agreement with [PERSON] [41]. In the following, the effects of [PERSON]'s modulus on the shear stress distributions are discussed for some typically shaped holes. The conformal mapping functions for these typical shapes are summarized as follows [31]:  
\[\omega(\zeta)=\zeta\ (\text{circle}), \tag{28}\]  
Figure 2: The circumferential shear stresses around an elliptical hole without FG layer (comparison). \[\omega(\zeta)=\zeta+\frac{1}{3}\zeta^{-1}\ (\text{ellipse}), \tag{29}\]  
\[\omega(\zeta)=\zeta+\frac{1}{3}\zeta^{-2}\ (\text{triangle}), \tag{30}\]  
\[\omega(\zeta)=\zeta-\frac{1}{6}\zeta^{-3}\ (\text{square}), \tag{31}\]  
\[\omega(\zeta)=\zeta+\frac{1}{2}\zeta^{-1}-\frac{1}{8}\zeta^{-3}-\frac{1}{25} \zeta^{-5}\ (\text{rectangle}). \tag{32}\]  
The variations in shear stress \(\tau_{63}\) around different shaped holes under anti-plane shear loads \(\tau_{23}^{\infty}\) are shown in Figures 3-7, where the thickness of the layer in the image plane is assumed to be \(0.2\tau_{0}\). It can be easily observed that the variations in shear stress are noticeably different for different shaped holes. For circular and elliptical holes, the maximum shear stresses occur at \(\delta=0^{\circ}\), \(180^{\circ}\) of the hole surface, while they occur at \(\delta=0^{\circ}\) for a triangular hole, and at the corners for square and rectangular holes. Comparatively speaking, the FG coating around the triangular hole has the largest magnitude of concentrations of the five different shaped holes, and the maximum shear stress of an ellipse and rectangle is usually larger than that of a circle and square, respectively. On the other hand, according to the change of exponent \(n\), it is found the shear stresses of all shaped holes are largest for \(n=0\), which means the hole without an FG layer. For \(n\
eq 0\), i.e., the hole having an FG layer, the shear stresses get small, and they decrease further as \(n\) increases. Figure 4: The variation in shear stress \(\tau_{63}\) around the hole (ellipse). \[z=\omega(\zeta)=R\left(\zeta+\frac{a-b}{a+b}\zeta^{-1}\right), \tag{27}\]0 The variation in shear stress \(\tau_{63}\) around the hole (circle). \[z=\omega(\zeta)=R\left(\zeta+\frac{a-b}{a+b}\zeta^{-1}\right), \tag{27}\]1 The variation in shear stress \(\tau_{\theta 3}\) around the hole (triangle). \[z=\omega(\zeta)=R\left(\zeta+\frac{a-b}{a+b}\zeta^{-1}\right), \tag{27}\]2 The variation in shear stress \(\tau_{\theta 3}\) around the hole (square).