Gamma convergence for a phase-field cohesive energy

TL;DR

This paper introduces a new variational phase-field model for fracture that accurately captures complex strength surfaces and demonstrates Gamma-convergence to a sharp fracture energy, with numerical validation.

Contribution

It develops a novel cohesive phase-field energy functional with a decoupled strength surface control and proves its Gamma-convergence in 1D and 2D settings.

Findings

The model captures complex strength surfaces more accurately.

Gamma-convergence to sharp fracture energy is established.

Numerical simulations show robustness to mesh anisotropy.

Abstract

Reproducing the key features of fracture behavior under multiaxial stress states is essential for accurate modeling. Experimental evidence indicates that three intrinsic material properties govern fracture nucleation in elastic materials: elasticity, strength, and fracture toughness. Among these, strength remains the most often misunderstood, as it is not a single scalar quantity but rather a full surface in stress space. The flexibility in defining this strength envelope in phase-field models poses significant challenges, especially under complex loading conditions. Existing models in the literature often fail to capture both the qualitative shape and the quantitative fit of experimentally observed strength surfaces. To address this limitation, recent work introduces a new energy functional within a cohesive phase-field framework, specifically designed to control the shape of elastic domains. This model introduces an internal variable to describe the inelastic response. Notably, the strength is decoupled from the internal length, that is not interpreted as a material length scale, as often done in literature, but rather as a purely variational tool. The proposed functional allows for a rigorous variational framework, enabling the use of tools from the calculus of variations. We investigate the Gamma-convergence of the model to a sharp cohesive fracture energy in the one- and two-dimensional (anti-plane) setting, using a finite element discrete formulation and exploiting the strong localization of the damage variable. Notably, unlike classical models where the elastic and fracture energies converge independently, this model exhibits a coupling of all energy terms. We also present numerical simulations exploring the sensitivity of the model to mesh anisotropy, offering insight into both its theoretical robustness and its practical implementation.