## 5 Phase Field Methods  
### Overview  
PFM emerged as an alternative to discrete fracture aiming to address some of the challenges of computational fracture mechanics, e.g., automatic crack initiation, robust resolution of branching and merging and also the treatment of curved crack paths. The PFM diffusive crack interface is represented by a scalar variable, i.e., the phase field. The latter evolves according to a set of governing equations arising from a robust variational structure. As a result, the method does not require numerical tracking of the evolving discrete crack topologies and complex problems as in the case of 3D crack paths (see, e.g., [227; 228; 229; 230]) and dynamic fragmentation are naturally resolved [231]. PFM emerged from the pioneering work of Francfort and Marigo [73] who proposed a variational theory of fracture based on energy minimization principles. Bourdin et al. [69] provided a regularised formulation by introducing a length scale parameter that rendered the approach more suitable for numerical approximations. The variational formulation was further modified and extended to multi-dimensional mixed-mode dynamic brittle fractures [228; 232; 233] also targeting the response of high performance composites [234; 235; 236]. The PFM for brittle fracture has been implemented in the commercial software Abaqus [237] via a User Element subroutine by Msekh et al. [64], which was later extended by Liu et al. [227]. Li et al. [238], see also [239], combined the variational phase field model of brittle fracture with an extended Cahn-Hilliard model [240; 241], and formulated a fourth-order  
Figure 5: Main steps in SBFEM crack propagation scheme. (**a**) Crack entering existing balanced quadtree region. ; (**b**) Refinement and balancing around crack tip; (**c**) Unifying cells into SBFEM macro element around crack tip; (**d**) New crack tip projected by gSIFs and \(\Delta a\). phase field model suitable resolving crack propagation in anisotropic materials. Rate-dependent PFM models for modelling fracture in visco-elastic solids [242] have also been established. The phase field representation of fracture has been extended to the ductile regime [79; 80; 243; 244] also within the context finite strains. The PFM has found application in the simulation of fractures in plates and shells [245; 246; 247; 245], which involve a 3-D degradation of induced stresses whereas the element kinematics and damage are defined at the mid-surface. Attempts to experimentally validate the method have also been provided (see, e.g., [79]). Verhoosel and de Borst [248] attempted to model cohesive fractures in composite materials using PFM by casting the cohesive zone approach in an energetic framework and introducing an auxiliary field in addition to the displacement and phase field which represents the jump in displacement across the cracked domain. The motivation to use an auxiliary field is to define the crack opening in cohesive fracture as a properly defined kinematic quantity, rather than an internal discontinuity as in the case of brittle fracture. Vignollet et al. [249] further extended the phase field-based cohesive fracture formulation for the case of propagating cracks. This approach succeeds in achieving convergence with lesser number of elements and in contrast to brittle fracture, confines the length scale parameter only to topological approximations hence rendering it uninfluential for the mechanical behaviour of the structure. Nguyen et al. [42] proposed a new phase field formulation which could model the interaction between interfacial damage and bulk brittle damage for complex topologies arising from voxel-based models of microtomography images. The formulation used a level-set method to describe the diffused jump in displacement field and used the phase field variable, instead of an additional internal variable as in [248], to model crack opening and reclosure during cohesive fractures.