Phase-field modelling of cohesive fracture. Part III: From mathematical results towards engineering applications

This paper concludes a three-part research program devoted to the development of a consistent and unified variational framework for the phase-field modeling of cohesive fracture. Building on the theoretical foundations established in the first two parts, namely a Gamma-convergence result for a broad class of phase-field energy functionals and a rigorous analytical methodology for constructing models tailored to prescribed cohesive laws, this third contribution investigates the mechanical response of a wide class of phase-field models, most of which are novel. Particular emphasis is placed on the non-uniqueness of the phase-field representation of cohesive fracture, demonstrating that distinct models, characterized by different phase-field and displacement evolutions, may reproduce the same macroscopic traction–separation response. To bridge theory and applications, the key modeling aspects are further assessed in a higher-dimensional context through numerical benchmark examples addressing crack nucleation and propagation. While these simulations provide a first validation of the proposed framework beyond one dimension, they also highlight the challenges and perspectives associated with its extension to more general settings. Overall, this work offers a practical interpretation of the underlying mathematical structure and establishes a link between rigorous analysis and physically relevant applications, paving the way for the systematic development of more advanced higher-dimensional cohesive phase-field models.