Energy-norm positive/negative projection scheme for phase-field fracture with anisotropic elasticity

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Abstract
Phase-field theory has proven to be an efficient approach for predicting the damage and fracture of engineering materials. It can simulate complex crack patterns without previously defined crack paths. In this work, a novel framework was developed for quantifying the fracture behavior of anisotropic brittle materials. This novel approach includes a phase-field formulation for fracture with (i) distinct contributions arising from anisotropy in the elastic moduli and (ii) a positive/negative projection (PNP) scheme in the complementary-energy norm for anisotropic elasticity, so that the crack-driving energy is consistently associated with tensile deformation while suppressing unphysical damage growth under compression. The present work considers anisotropy in the elastic constitutive response, while the fracture surface energy is assumed isotropic. The resulting coupled displacement–damage problem is implemented in Abaqus/Standard through a user element subroutine (UEL), while a user material subroutine (UMAT) is used to transfer stresses, strains, and damage variables for post-processing. A monolithic solution strategy is adopted to solve the displacement and phase-field equations simultaneously. The implementation is verified using one-element tests under tensile, compressive, and shear loading, patch tests with distorted meshes, and several quasi-static fracture benchmarks, including single-edge-notched tension, compression, shear, and three-point bending problems. Parametric studies are also performed to examine the effects of mesh size, regularization length scale, and critical energy release rate on the predicted fracture response. For anisotropic trigonal materials, the material-point simulations capture the orientation-dependent elastic and damage responses, while the notched fracture simulations demonstrate orientation-dependent crack trajectories arising from elastic anisotropy.

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