## 6 Numerical Examples
### Numerical Example 2: Single Edge-Notched Shear Test  
In the present example, the mode-II fracture behavior of a square panel is examined, with geometric description of the domain and boundary conditions as shown in Figure 17. This is a standard benchmark test to evaluate damage characteristics under shear loads, and has been analyzed extensively in the literature, see for e.g., [229; 232]. The specimen thickness is \(t=1\) mm and a state of plane-strain is assumed. The material parameters are chosen as \(E=210\) kN/mm\({}^{2}\), \(v=0.30\), \(l_{0}=0.0075\) mm, \(\mathcal{G}_{c}=0.0027\) kN/mm with crack propagation increment \(\Delta a=0.02\) mm, in accordance with [232]. For the phase-field analysis, the mesh is refined with \(h_{PFM}=0.005\) mm in the regions where the crack is expected to propagate. Zero y-displacement boundary conditions are enforced (\(u_{y}=0\), Figure 17) on all outer edges of the plate. Furthermore, the bottom edge of the specimen is retrained in the horizontal direction (\(u_{x}=0\)). For the discrete crack methods, a horizontal displacement \(u=u_{x}\geq 0\) is imposed on the top edge of the specimen, while the PFM applies a concentrated load P at point C, kinematically couples the horizontal DOF on the top edge and solves enforcing quasi static  
Figure 16: Tension test with cohesive phase field formulation studying effect of shape parameter \(p\) on the peak fracture loads  
Figure 15: Tension test effect of length-scale parameter variation on load displacement curves. displacement control. The second order quadratic phase field formulation described in Section 5.2.1 is employed in this example. The analysis procedures for each approach are as described in Section 6.1. The corresponding load-deflection paths are shown in Figure 18.  
The shear test results in a biaxial stress state developed at the notch-tip which leads to an inclined crack propagation at an angle \(45^{\circ}\) to the horizontal. The crack paths are closely aligned (Figure 19), however, the origin of the discontinuity differs slightly between the PFM and the discrete crack methods, resulting in a slight differentiation of the crack paths upon crack propagation. Such behaviour is a consequence of the discrete crack methods mandating the crack propagate starting from the proceeding crack tip, whereas the PFM permits the evolution along the notch. Various stages of the phase field evolution are shown in Figure 20.  
Further discrepancy is also observed in the significantly differentiated behaviour of the associated load-deflection curve. The higher peak load obtained by discrete crack methods and the snap back behaviour is not mirrored in the PFM result. The difference in snap back behaviour between the SBFEM and XFEM is attributed to the adaptivity of the SBFEM mesh about the crack tip, while the XFEM relies on the initial mesh topology. After this oscillatory step, the respective load deflection curves coincide closely. Contrary to the discrete methods where the equilibrium path is derived from sequential linear solutions, PFM relies on incremental iterative solvers; hence the snap back response would not be captured with a displacement control nonlinear analysis procedure; rather, a generalized, e.g., arc-length, analysis is required. Eventhough the PFM results shown in Figure 19 are identical to the results provided in the literature (see, e.g., [232; 265]), the 8% difference in the peak load compared to discrete methods highlights the importance of the length scale parameter on the solution. The effect of the length scale \(l_{0}\) on the crack topology and the peak fracture loads is shown in Figures 21 and 22, respectively. It can be noted that the shear crack paths and load-displacement curves show a similar trend as already seen in Section 6.2, wherein decreasing \(l_{0}\) leads to sharper and more brittle cracks with higher peak fracture forces which converge to the discrete fracture solution. Figure 17: Shear test geometry, material parameters, loading and boundary conditions.