Strain energy density decompositions in phase-field fracture theories for orthotropy and anisotropy

Abstract

In phase-field theories of fracture, decompositions of the strain energy density into tensile and compressive parts are often necessary to avoid interpenetration of cracked surfaces and to select physically trustworthy crack paths. General formulations accounting for orthotropy of the well-known spectral and hydrostatic-deviatoric decompositions of the strain tensor (often referred to as Miehe and Amor decompositions) are presented in this study. Additionally, a new principal energy decomposition based on spectral decomposition of the stiffness tensor is proposed for general anisotropic materials. The decompositions are evaluated numerically in a quadratic specimen with an initial stationary edge crack subject to both tensile and shear global remote loading. It is shown that when an isotropic case is considered, solutions agree well with results reported elsewhere for both the spectral and the hydrostatic-deviatoric approaches. The principal energy decomposition results in similar crack paths as the other approaches, with only subtle differences. When orthotropy is considered, however, significant differences in the resulting crack paths as well as global force-displacement behavior are obtained, especially when the crack is subjected to shear loading. For global tensile loading, the decompositions result in similar crack paths and force-displacement relations. The results provide a step forward when developing phase-field fracture theories for brittle materials with an orthotropic nature and highlight the importance of a proper decomposition of the strain energy density.