Phase-field approximation of a vectorial, geometrically nonlinear cohesive fracture energy

TL;DR

This paper develops a phase-field model for vectorial, nonlinear cohesive fracture energy, demonstrating it as a Gamma-limit of a phase-field approximation, capturing elastic, surface, and diffuse damage energies.

Contribution

It introduces a phase-field approximation for a complex vectorial cohesive fracture model, linking it to Gamma-convergence and providing explicit energy density relations.

Findings

The phase-field model converges to the cohesive fracture energy functional.

The energy densities can be expressed explicitly in terms of phase-field parameters.

The model incorporates invariance and diffuse damage in fracture analysis.

Abstract

We consider a family of vectorial models for cohesive fracture, which may incorporate $\mathrm{SO}(n)$-invariance. The deformation belongs to the space of generalized functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. We show that this type of functional can be naturally obtained as $\Gamma$-limit of an appropriate phase-field model. The energy densities entering the limiting functional can be expressed, in a partially implicit way, in terms of those appearing in the phase-field approximation.