## 6 Numerical Examples
### Numerical Example 4: L-Shaped Panel (LSP) Test with Crack at Re-Entrant Corner  
Figure 31b depicts the geometric description of the domain, boundary conditions and material parameters for an L-shaped panel. A state of plane stress is considered with specimen thickness \(t=100\) mm. The Young's modulus, Poisson's ratio, length scale, fracture energy density and crack propagation length are chosen as \(E=5.98\) kN/mm\({}^{2}\), \(v=0.2\), \(l_{0}=2.5\) mm, \(\mathcal{G}_{c}=0.0089\) kN/mm, \(h_{PFM}\approx 1.4\) mm and \(\Delta a=10\) mm, where applicable. A zero displacement boundary condition (\(u_{x}=0;u_{y}=0\)) is enforced on the bottom side, while a cyclic imposed displacement envelope is considered at a distance \(d_{l}=30\) mm from the rightmost edge of the panel with a constant displacement increment \(\Delta u=10^{-3}\) mm and the load history as shown in Figure 31a. The analysis procedures described in Section 6.1 for each method apply. Through this application, we simulate the experimental program undertaken in [278] which has also been investigated in previous publications pertinent to computational fracture mechanics [265]. Since the discrete crack methods do not intrinsically posses the capability to avoid crack over-closure and interpenetration, without introducing contact, the numerical  
Figure 29: NPwH comparison of crack topologies depicting experiments from (**a**) [243] vs. (**b**) phase field simulations.  
Figure 28: NPwH phase field at monitored displacement (**a**) \(u=0.35\) mm (**b**) \(u=0.96\) mm (**c**) \(u=1.20\) mm, with a displacement increment \(\Delta u=1e^{-3}\) mm and a single stagger iteration.  
simulations employing XFEM and SBFEM follow a modified loading path (Figure 31 left) starting from time step 1000.  
The load-displacement curve and the peak fracture force (Figure 32) are in accordance with existing literature [265] and a good agreement is observed between all methods. The crack paths obtained from all methods remain within the envelope of the experimental results (Figures 33 and 34). Furthermore, the crack path obtained in Figure 35 coincides with the experimentally observed crack in [278]. For the case of SBFEM, the crack tip does not coincide with the re-entrant corner, since the implementation requires the crack tip to reside within the domain and not on the boundary. Hence, the crack tip was perturbed by a small value and thus the peak load is slightly overestimated.  
Furthermore, a comparison is drawn between the load-displacement curves obtained using the spectral strain decomposition [232] and the constrained hybrid phase field model proposed in [265].  
Figure 30: NPwH comparison between anisotropic phase field models with strain energy splits proposed in [232; 264]. **(a)** Crack path from analysis implementing the anisotropic split proposed in [232]; **(b)** Crack path from analysis implementing the anisotropic split proposed [264]; **(c)** Force-displacement response comparison between the anisotropic phase field models.  
The resulting load-deflection curves are shown in Figure 36. It is of interest to note that the anisotropic spectral split [232] naturally avoids crack face overlapping during crack closure when cyclic loads are considered. On the other hand, the hybrid phase-field model in [265] requires an additional constraint to prohibit interpenetration of crack faces during compression phase.  
Figure 31: (a) LSP geometry, material parameters, loading and boundary conditions (b) Cyclic envelope of monitored displacement.  
Figure 32: LSP load-deflection curves.  
Figure 34: LSP crack paths for SBFEM, XFEM and PFM.  
Figure 35: (**a**) LSP phase field (**b**) load-deflection response under the cyclic loading defined in Figure 31**b**.  
Figure 33: LSP meshes for SBFEM (**top**) and XFEM (**bottom**), with focus on crack path region.