Bridging perturbation and variational approaches in brittle fracture

TL;DR

This paper introduces a variational reduced-order model for 3D crack propagation in brittle solids, combining fracture theory and perturbation methods, validated through analytical solutions and extensive simulations.

Contribution

It develops a novel computational approach that efficiently links variational fracture models with perturbation theory, enabling large-scale simulations of crack dynamics in heterogeneous materials.

Findings

Reproduces the transition from smooth to intermittent crack growth.

Shows mode mixity influences crack front shape but not the onset of intermittency.

Identifies a size-dependent crossover from weakening to toughening due to depinning.

Abstract

We present a variational reduced-order model for three-dimensional coplanar propagation of sharp cracks in heterogeneous perfectly brittle solids under mixed-mode I+II+III loading. The approach connects the variational fracture formulation of Francfort and Marigo (1998) and the perturbation theory of Rice (1985) by computing equilibrium crack-front configurations through minimization of the total energy defined as the sum of (i) the elastic potential energy, evaluated asymptotically from front deformations, and (ii) the dissipated energy, set by the fracture energy field. The potential energy and its derivatives are evaluated efficiently using the Fast Fourier Transform. The resulting nonconvex box-constrained minimization problem is solved with a matrix-free Newton conjugate gradient algorithm with a trust region and physics-based preconditioning, enforcing irreversibility while resolving energy barriers and long-range elastic interactions. We validate our implementation against newly derived analytical solutions. We then perform 116,000 large-scale simulations of tensile and shear crack propagation in disordered media to quantify the impact of finite-size effects, disorder intensity, and mode mixity. The simulations reproduce the transition from smooth to intermittent crack growth, and show that mode mixity has limited influence on the onset of intermittency but induces quasi-elliptic fronts in mixed II+III loading. They reveal a size-dependent crossover from disorder-induced weakening to toughening controlled by the emergence of depinning instabilities.