Phase-field modelling of cohesive fracture. Part III: From mathematical results to engineering application

This paper concludes a three-part effort aimed at developing a consistent and unified framework for the phase-field modeling of cohesive fracture. Building on the theoretical foundations established in the first two parts, which included a $\Gamma$-convergence result for a broad class of phase-field energy functionals and the presentation of a rigorous analytical methodology for constructing models tailored to specific cohesive laws, this third paper explores the mechanical response of phase-field models, most of which are novel, associated with different cohesive fracture behaviors within a one-dimensional framework. Particular emphasis is placed on the possibility of formulating distinct phase-field models that, despite exhibiting different evolutions of their phase-field and displacement profiles, yield identical cohesive fracture responses. Thus, this work aims at providing a practical interpretation of the mathematical framework connecting the theoretical insights established in the previous parts for physical relevant applications.