Second-order phase-field formulations for anisotropic brittle fracture

TL;DR

This paper introduces second-order variational phase-field models for anisotropic brittle fracture with two-fold and four-fold symmetry, simplifying numerical implementation and improving modeling of anisotropic fracture toughness.

Contribution

The paper develops two novel second-order phase-field models for anisotropic fracture, avoiding the complexity of fourth-order models and providing Gamma-convergence results for these formulations.

Findings

Models accurately simulate anisotropic fracture under shear loading.

Second-order models do not require C1 continuity basis functions.

The new degradation function improves four-fold symmetric fracture modeling.

Abstract

We address brittle fracture in anisotropic materials featuring two-fold and four-fold symmetric fracture toughness. For these two classes, we develop two variational phase-field models based on the family of regularizations proposed by Focardi (Focardi, M. On the variational approximation of free-discontinuity problems in the vectorial case. Math. Models Methods App. Sci., 11:663{684, 2001), for which Gamma-convergence results hold. Since both models are of second order, as opposed to the previously available fourth-order models for four-fold symmetric fracture toughness, they do not require basis functions of C1-continuity nor mixed variational principles for finite element discretization. For the four-fold symmetric formulation we show that the standard quadratic degradation function is unsuitable and devise a procedure to derive a suitable one. The performance of the new models is assessed via several numerical examples that simulate anisotropic fracture under anti-plane shear loading. For both formulations at fixed displacements (i.e. within an alternate minimization procedure), we also provide some existence and uniqueness results for the phase-field solution.