A variational approach to fracture incorporating any convex strength criterion

TL;DR

This paper introduces a variational phase-field model for fracture that incorporates arbitrary convex strength criteria, unifying various failure concepts within a single theoretical framework and providing explicit cohesive limits.

Contribution

It develops a novel phase-field approach that accounts for any convex strength domain without altering material stiffness, and derives cohesive laws from the model.

Findings

Solution characterized as a transverse cohesive crack under certain conditions

Hierarchy of strength criteria determining fracture possibility

Explicit derivation of cohesive limits from phase-field models

Abstract

We propose a variational phase-field model of fracture capable of accounting for arbitrary closed convex strength domains. Unlike traditional models based on Ambrosio and Tortorelli regularization, the phase-field variable does not affect the material stiffness. Instead, our elastic energy exhibits linear growth outside a strength domain, which shrinks to 0 as the phase-field variable goes to 1. We characterize this model through a fundamental problem on a cube subject to boundary loads. We show that the solution of this problem is a transverse cohesive crack, provided that the applied load and the direction of the displacement jumps satisfy a compatibility criterion, which we formulate in terms of Mohr's circles for isotropic strength domains. This allows us to derive a hierarchy of strength criteria for which fracture is never possible, sometimes possible or always possible, depending on the direction of the stress tensor. We discuss the properties of the model and postulate a ``sharp-interface'' limit in the form of a cohesive law that can be explicitly derived from the form of the phase-field model. We give several examples of phase-field models and their cohesive limits. The proposed framework unifies within a single consistent variational theory key concepts developed over the centuries to predict or prevent material failure: Griffith and cohesive crack models, damage models, plasticity, strength criteria, and limit analysis.