## 7 Discussion and Conclusions
### Crack Propagation by PFM  
In PFM fracture is not introduced as an explicit or implicit discontinuity in the displacement field. Rather, it is associated with the evolution of a continuous field, i.e., the phase field. The governing equations of the crack propagation problem emerge through the minimization of the total potential energy established in Equation (70), see, e.g., [69]. This gives rise to the coupled system of equilibrium and phase field governing equations established in Equations (71) and (72). The crack is not explicitly represented but derived from the solution of the coupled system as the region where \(c=0\) (typically values of \(c<10e-3\)). Within the setting of an incremental solution procedure, the phase field is updated at each time step and with it the crack topology. Nucleation, merging, branching and arrest of cracks as well as the associated crack propagation increment is a natural byproduct of the phase field evolution. The mechanical/phase field coupling is enforced by introducing a material degradation function that is dependent on the phase field. The evolution of fracture follows through the solution of this coupled strong form. Existing discontinuities may be introduced into the domain by providing initial values to the phase field. Mesh density is contingent on sufficient resolution of the fracture process zone, mandating a highly refined mesh in its vicinity. The combination of length scale and level of mesh refinement interact and affect the estimation of the fracture energy hence necessitating the scaling of the critical energy release rate. The numerical solution of the PFM-coupled governing equations is performed using either monolithic or staggered solvers. Monolithic solvers are typically based on the Newton-Raphson solution procedure and have been proven to provide accurate fracture paths. However, they have been shown to suffer from poor convergence due to the non-convex nature of the underlying energy functional [277]. Yet, the accuracy provided by monolithic solvers renders them a favourable solution, especially in the case of dynamic fracture problems and several attempts have been suggested in the literature to improve the robustness of monolithic procedures (see, e.g., [279; 280; 281]). In staggered methods the displacement and phase field equations are decoupled and solved separately within each load increment. In principle, a staggered algorithm for coupled field problems is based on freezing one field variable at a constant value, solving for the other until convergence is achieved. The staggered approach (also known as alternate minimization approach) provides better convergence rates than the monolithic due to the convexity of the energy functional (Equation (70)) with respect to the two unknown fields \(\{\mathbf{u},\phi\}\) separately. However, its accuracy is dependent on the incremental step unless stagger iterations are performed; these however increase the computational burden of the analysis. Very recent developments aim towards providing more robust staggered solvers, (see, e.g., [282]). The steps to a typical PFM solution procedure with a staggered solution scheme are summarized in Figure 43.