Second-order phase-field formulations for anisotropic brittle fracture

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 (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. (c) 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

In this study, we address brittle fracture in anisotropic materials with two-fold and four-fold symmetric fracture toughness, developing two variational phase-field models based on regularizations by Focardi. The new second-order models do not require C1-continuity basis functions or mixed variational principles for finite element discretization, and outperform previously available fourth-order models for four-fold symmetric fracture toughness. The performance of the new models is evaluated through numerical examples simulating anisotropic fracture, and existence and uniqueness results for the phase-field solution under fixed displacements are provided.