The phase-field paradigm provides a robust damage modelling approach which is equipped with capabilities of automatically predicting initiation, propagation, branching and merging of complex curvilinear crack topologies. To this point, phase-field modelling has been widely applied to study brittle fractures based on Griffith’s theory, with some extensions also to ductile and cohesive fractures. Despite its immense popularity, its application to complex high-performance materials, for e.g. composites, has been limited. This is because most composites are not brittle in the Griffith’s sense. Rather, they display quasi-brittle fracture characteristics wherein a crack is driven by cohesive forces present within the fracture process zone, which ...