## 4 Numerical Experiments
The non-cracked symmetric line is subjected to the vertical displacement constraint of \(u_{y}=0\), and the right end point is enforced by the lateral displacement constrain of \(u_{x}=0\). A \(11\times 41\) uniform NEM grid is used, and all the NEM analyses, the stress recovery, and the modified interaction integrals \(\widetilde{M}^{(1,2)}\) were carried out with 13 Gauss points. The radius \(r_{int}\) for determining the domain of integration \(A\) was chosen using twice the side length of a square consisting of two Delaunay triangles [41]. Table 1 comparatively represents the normalized mode-I SIFs for uniform tensile loading and bending loading, respectively, where the values in parenthesis indicate the relative differences (%). It is confirmed that the proposed enriched PG-NEM provides the SIFs, which are consistent with the analytical solutions [40], for all the combinations of \(E_{2}/E_{1}\) and \(a/W\) of two types of loading, except for the relatively remarkable relative difference 6.807% in the tension case with \(E_{2}/E_{1}=10\) and \(a/W=0.3\). For the further comparison, the numerical results by [PERSON] and [PERSON] [15] and by [PERSON] and [PERSON] [34] are also given in Table 1. It is observed that the SIFs of proposed method are correlated satisfactorily with the FEM results by the \(I_{k}^{*}\)-integral and the EFGM results by the modified \(\widetilde{M}^{(1,2)}\)-interaction integral. Table 2 represents the comparison with the results obtained without using the enrichment. In the uniform tensile loading, the case without enrichment provides the SIFs, which are lower than those with enrichment, such that the range of relative differences is from 27.59% and 51.04%. Meanwhile, in the bending, it is observed that the difference between with and without enrichment is not significant. Furthermore, the case without using enrichment provides the SIFs which are higher than those obtained by the enrichment. It is inferred that the near-tip stress field produced by bending is totally different from that by uniform tensile loading. The relative difference becomes smaller as the relative crack length \(a/W\) increases. It is observed from the detailed numerical values that the relative difference ranges from 2.41% to 24.68%. Thus, it has been found that the proposed enrichment method is influenced by the loading type. Figure 7: A cracked rectangular functionally graded material (FGM) plate with a varying Young’s modulus: (**a**) Geometry dimensions, (**b**) tension loading, (**c**) bending. The third example is a slant edge crack in a 2-D FGM plate in plane stress condition with the height \(H=2\) units and the width \(W=1\) unit. Referring to Figure 8a, the crack angle \(\alpha\) is \(45^{\circ}\) and the relative crack length is \(a/W\) is \(0.4\sqrt{2}\), where the enrichment and the crack evaluation are performed within the inclined Cartesian co-ordinate system \(\{x^{\prime},y^{\prime}\}\) originated at the crack tip \(O^{\prime}\). The Poison's  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{5}{c}{**Relative Crack Length \(a/W\)**} \\ \hline  
ratio is kept uniform as \(\
u=0.3\), but the elastic modulus varies in an exponential function along the \(x-\)direction  
\[E(x)=\overline{E}exp[\eta(x-0.5)],\quad 0\leq x\leq W \tag{45}\]  
Here, \(\overline{E}=1\) unit and \(\eta\) is variable for the parametric study such as \(\eta a\ =0,0.1,0.25,0.5,\) 0.75, and 1.0. As external load, an exponentially varying distributed load is applied to the upper edge with \(\sigma_{yy}(x,1)=\overline{\epsilon}\overline{E}exp[\eta(x-0.5)]\), with \(\overline{\epsilon}\) being 1. The whole bottom edge is constrained in the vertical direction (i.e., \(u_{y}=0\)). At the same time, the right end is constrained in the horizontal direction. A uniform NEM grid, given in Figure 8b, is employed which is discretized by \(11\times 21\), where a darkened area indicates the domain A for the interaction integral.