Phase-field modelling of cohesive fracture. Part II: Reconstruction of the cohesive law

Abstract: This is the second paper of a three-part work the main aim of which is to provide a unified consistent framework for the phase-field modelling of cohesive fracture. Building on the theoretical foundations of the first paper, where {$\Gamma$-convergence} results have been derived, this second paper presents a systematic procedure for constructing phase-field models that reproduce prescribed cohesive laws. By either selecting the degradation function and determining the damage potential or vice versa, we enable the derivation of multiple phase-field models that exhibit the same cohesive fracture behavior but differ in their localized phase-field evolution. This methodology provides a flexible and rigorous strategy for tailoring phase-field models to specific cohesive responses, as shown by the several examples worked out. The mechanical responses associated with these examples, highlighting their features and validating the theoretical results, are investigated in the third paper from a more engineering-oriented and applied perspective.