## 6 Numerical Examples
### Numerical Example 1: Single Edge-Notched Tension Test  
This example considers mode-I fracture behavior of a square panel, with geometric description of the domain, boundary conditions and material parameters as defined in Figure 11. A state of plane strain is assumed, the specimen thickness is \(t=1\) mm. The Young's modulus, Poisson's ratio, length scale, fracture energy density and crack propagation length are chosen as \(E=210\) kN/mm\({}^{2}\), \(v=0.30\), \(l_{0}=0.0075\) mm, \(\mathcal{G}_{c}=0.0027\) kN/mm, \(\sigma_{c}=2.5\) kN/mm\({}^{2}\) and \(\Delta a=0.02\) mm, where applicable. The bottom edge of the specimen is clamped in both \(x\) and \(y\) directions, such that \(u_{x}=0;u_{y}=0\). The loads and boundary conditions of the top edge by discrete and PFM are enforced differently, yet with equivalent outcome; for XFEM and SBFEM a prescribed displacement of \(u=u_{y}\geq 0\) is imposed on the top edge, while for PFM, a quasi-static displacement control analysis procedure is implemented considering a concentrated load applied at point C and kinematic coupling of the vertical displacement DOF along the top edge, such that \(u=u_{y}\geq 0\) is obtained. The analysis procedures for each approach as described in Section 6.1 apply. Two different solution procedures based on standard and cohesive phase field approaches, as described in Sections 5.5 and 5.6 respectively, are studied within this example. The resulting load deflection paths for all methods are shown in Figure 12; the standard SBFEM and XFEM implementations match the deflections and peak load, while the phase field method with \(G_{c}^{eff}\) approximates only the peak load closely. The nucleation and propagation of the crack at successive time-increments is shown in Figure 13. The nucleation of the crack automatically occurs at the notch-tip, and then this propagates linearly in the direction perpendicular to the applied load. It is known that the value of the length scale parameter \(l_{o}\) not only controls the width of the phase field diffusion zone, but also affects the peak fracture force values. This is illustrated in Figures 14 and 15, where a decreasing the value of \(l_{o}\) leads to sharper crack topologies and higher peak fracture forces, thus showcasing a more brittle-like fracture behaviour. It can be inferred from Figure 15 that if \(l_{o}\) is chosen sufficiently small, i.e., in the limit \(l_{o}\to 0\), the force-displacement curves converge towards the discrete solution, i.e., Griffith's description of brittle fracture; a property well-known as \(\Gamma\)-convergence of regularized phase field fractures. However, an important point to note is that a formal proof of \(\Gamma\)-convergence of anisotropic strain-energy splits (detailed in [232; 264]) towards Griffith's theory is not available yet, as also stated in [277]. It is evident from Figure 12 that both discrete crack methods, i.e., XFEM/SBFEM, predict similar fracture characteristics, whereas the critical fracture force obtained from phase-field method is slightly overestimated when the actual value of \(G_{c}^{actual}=0.0027\) kN/mm is used. Considering \(h_{PFM}=0.005\) mm and \(l_{0}=0.0075\) mm which have been used for the current analysis, an effective fracture energy \(G_{c}^{eff}=0.00231\) kN/mm can be calculated based on Equation (85). The critical fracture load thus obtained using \(G_{c}^{eff}\) shows very good agreement with those predicted by discrete methods XFEM/SBFEM. The difference in the elastic stiffness of the material between XFEM/SBFEM and PF-FEM cases is due the fact that in conventional PF-FEM formulations, as in [69], the phase-field variable evolution and consequently stress degradation start as soon as the material is loaded and hence, prevents recovery of a pure linear elastic limit. The crack paths, however, coalign as expected, although for the PF-FEM the resulting displacements are over-estimated as the fracture must initiate at the same critical load for a given value of \(\mathcal{G}_{c}\).