## 6 Numerical Examples
### Numerical Example 3: Notched Plate with Hole (NPwH)  
A notched plate containing a hole is considered with geometric description of the domain, boundary conditions and material parameters as defined in Figure 23. In [32; 243], a similar example has been analyzed previously. The specimen thickness is \(t=15\) mm and a state of plane-stress is treated. The Young's modulus, Poisson's ratio, length scale, fracture energy density and crack propagation length are chosen as \(E=5.98\) kN/mm\({}^{2}\), \(v=0.221\), \(l_{0}=0.35\) mm, \(\mathcal{G}_{c}=0.00228\) kN/mm and \(\Delta a=2\) mm, where applicable. For the PFM, the mesh-size is kept at a value of \(h_{\text{PFM}}\approx 0.34\) mm in the crack propagation region. A zero displacement boundary condition (\(u_{x}=0;u_{y}=0\)) is enforced on the bottom pin, whereas a vertical displacement \(u=u_{y}\geq 0\) is imposed on the top pin. The numerically predicted crack path is compared with the experimental results presented in [265]. Comparing PFM to discrete crack methods, the obtained peak load is similar (Figure 24), however, the crack paths differ significantly (Figures 25 and 26). Since the discrete methods do not possess an intrinsic method to nucleate cracks, once the crack tip has propagated into the hole, the algorithm terminates. This is apparent, since both XFEM and SBFEM report a final vertical displacement of approximately 0.33 mm. Due to this inherent limitation, expert judgment is required to interpret crack propagation results stemming from discrete crack methods as their termination is indistinguishable from crack arrest, when inspecting conventional results. The phase field methods circumvent these  
Figure 21: Shear test comparison of phase field diffusion widths with respect to decreasing \(l_{o}\), where (a) \(l_{o}=0.015\) mm (b) \(l_{o}=0.0075\) mm (c) \(l_{o}=0.00375\) mm. Figure 22: Shear test effect of length-scale parameter \(l_{o}\) variation on load displacement curvesissues resulting in a highly flexible and generalized method, at the cost of significantly increased computational effort. For the PF-FEM case, the effect of the number of staggered phase field iterations on the accuracy of the predicted peak fracture loads is examined. Four different cases with constant displacement increments \(\Delta u=10^{-2}\), \(\Delta u=5\times 10^{-3}\) mm, \(\Delta u=10^{-3}\) mm and \(\Delta u=5\times 10^{-4}\) mm are considered and the corresponding load-deflection paths are shown in Figure 27a. In all cases, the phase-field solution is predicted using a single staggered iteration step \(N_{staggs}=1\) and a tolerance of \(tol_{u}=10^{-5}\) is maintained. It can be seen that solution accuracy improves when the size of displacement increments \(\Delta u\) is sufficiently small, and convergence is achieved for \(\Delta u=1\times 10^{-3}\)mm. Further reduction of \(\Delta u\) marginally affects the results at the cost of increased number of calculations, with \(\Delta u=5\times 10^{-4}\) mm and \(\Delta u=1\times 10^{-3}\)mm yielding almost similar load-displacement curves. In Figure 27b the converged solution of Figure 27 is compared against the solution with \(\Delta u=5\times 10^{-3}\)mm when (i) only a single staggered iteration is performed and (ii) staggered iterations are performed until the phase field solution converges. It is evident that the peak fracture loads obtained in converged staggered iteration case is lower as compared to the \(N_{staggs}=1\) case, and are actually closer to the converged solution shown in Figure 27a. The evolution of phase field at successive monitored displacements is shown in Figure 28; results are obtained using single staggered iteration \(N_{staggs}=1\) and a constant displacement increment \(\Delta u=10^{-3}\) mm. The crack paths obtained from phase field calculations (Figure 29) show good agreement with the experimental fracture results presented from [265].