## 4 Numerical Examples
### A Slanted Single-Edge Crack Plate  
The rectangular plate under end shearing with oblique crack is shown in Figure 4. The bottom of the model is fixed, and the top is imposed to mixed-mode shear load \(\tau=1\)MPa, where \(W=L=10\)m, the initial crack length \(a=6\)m, and the initial crack direction \(\beta=\)30\({}^{\circ}\). The material parameters of the model are elastic modulus, \(E=210\) GPa, [PERSON]'s ratio \(\
u=0.25\). The model is computed under plane stress condition. Figure 3: The stress nephogram at the left crack tip: (**a**) \(x\)-axis direction; (**b**) \(y\)-axis direction. The present computation results are shown in Table 3, compared with results in the literature [6; 38], which are in good agreement and verify effectiveness of the present method. The results \(K_{I}=20.485\,\text{MPa}\cdot\text{m}^{1/2}\), \(K_{II}=9.8188\,\text{MPa}\cdot\text{m}^{1/2}\), \(T=5.7802\,\text{MPa}\) in [6] are referred as the reference solutions, the present computation results are \(K_{I}=20.490\,\text{MPa}\cdot\text{m}^{1/2}\), \(K_{II}=9.821\,\text{MPa}\cdot\text{m}^{1/2}\), \(T=5.781\,\text{MPa}\). The relative errors of \(K_{I}\), \(K_{II}\), and \(T\) are 0.024%, 0.022% and 0.0138% respectively. The errors of energy norm of the P-FEM solutions and relative errors of normalized SIFs comparison with Abaqus solutions [6] for the example are listed in Table 4.  
Figure 4: (**a**) A slanted single-edge cracked plate under end shearing; (**b**) Global and local meshes (the total number of elements is 39). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**SIFs** & \(K_{I}\)/\(\tau\sqrt{\pi a}\) & \(\mathbf{T}\)/\(\tau\) & \(\mathbf{K_{II}}\)/\(\tau\sqrt{\pi a}\) \\ \hline Present & & 4.7195 & 5.7810 & 2.2621 \\ \hline  & \(r/a=1/12\) & 4.7290 & 5.7730 & 2.2613 \\  & \(r/a=1/6\) & 4.7255 & 5.7737 & 2.2611 \\ EMM [6] & \(r/a=1/3\) & 4.7236 & 5.7745 & 2.2612 \\  & \(r/a=1/2\) & 4.7229 & 5.7752 & 2.2614 \\  & \(r/a=2/3\) & 4.7225 & 5.7758 & 2.2617 \\ \hline Abaqus solution [6] & & 4.7183 & 5.7802 & 2.2615 \\ \hline FEOD [38] & & 4.6883 & 5.7376 & 2.2400 \\ BCM [38] & & 4.1970 & 5.1776 & 2.0184 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Normalized SIF for a slanted-edge crack plate under end shearing. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(c\)/\(L\) & **0.1** & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** \\ \hline \multirow{2}{*}{Present} & \(K_{l}(\text{MPa}\cdot\text{m}^{1/2})\) & 8.546 & 7.120 & 7.002 & 7.108 & 7.287 & 7.646 \\  & \(K_{l}/\sigma\sqrt{\pi a}\) & 3.409 & 2.840 & 2.793 & 2.836 & 2.907 & 3.050 \\ \hline \multirow{2}{*}{[PERSON] et al. [7]} & \(K_{l}(\text{MPa}\cdot\text{m}^{1/2})\) & 8.372 & 7.096 & 6.981 & 7.086 & 7.264 & 7.620 \\  & \(K_{l}/\sigma\sqrt{\pi a}\) & 3.340 & 2.831 & 2.785 & 2.827 & 2.898 & 3.040 \\ \hline \multirow{2}{*}{[PERSON] et al. [8]} & \(K_{l}(\text{MPa}\cdot\text{m}^{1/2})\) & 8.528 & 7.119 & 7.004 & 7.109 & 7.289 & 7.648 \\  & \(K_{l}/\sigma\sqrt{\pi a}\) & 3.402 & 2.840 & 2.794 & 2.836 & 2.908 & 3.051 \\ \hline \multirow{2}{*}{Present} & \(K_{ll}(\text{MPa}\cdot\text{m}^{1/2})\) & \(-0.313\) & \(-0.458\) & \(-0.403\) & \(-0.333\) & \(-0.407\) & \(-0.736\) \\  & \(K_{ll}/\sigma\sqrt{\pi a}\) & \(-0.125\) & \(-0.183\) & \(-0.161\) & \(-0.133\) & \(-0.162\) & \(-0.294\) \\ \hline \multirow{2}{*}{[PERSON] et al. [7]} & \(K_{ll}(\text{MPa}\cdot\text{m}^{1/2})\) & \(-0.318\) & \(-0.456\) & \(-0.401\) & \(-0.333\) & \(-0.406\) & \(-0.732\) \\  & \(K_{ll}/\sigma\sqrt{\pi a}\) & \(-0.127\) & \(-0.182\) & \(-0.160\) & \(-0.133\) & \(-0.162\) & \(-0.292\) \\ \hline \multirow{2}{*}{[PERSON] et al. [8]} & \(K_{ll}(\text{MPa}\cdot\text{m}^{1/2})\) & \(-0.326\) & \(-0.461\) & \(-0.406\) & \(-0.338\) & \(-0.411\) & \(-0.742\) \\  & \(K_{ll}/\sigma\sqrt{\pi a}\) & \(-0.130\) & \(-0.184\) & \(-0.162\) & \(-0.135\) & \(-0.164\) & \(-0.296\) \\ \hline \end{tabular}
\end{table} 
Table 6: SIFs with different crack spacing (\(L/W=2\)).