Modelling and simulation of fracture in anisotropic brittle materials by the phase-field method with novel strain decompositions

When brittle and quasi-brittle materials, are concerned, it is essential to account for the fundamental fact that they behave differently according as they undergo tensile or compressive loadings. Due to this fact, in modelling and simulating the fracture of brittle and quasi-brittle materials by the phase-field method (PFM), the strain tensor is usually decomposed into a tensile part and a compressive part, and it is argued that only the strain energy related to the tensile part controls the nucleation and propagation of cracks. The present work improves the phase-field method by incorporating into it a novel strain decomposition strategy giving rise to strain decompositions being orthogonal in the sense of the natural inner product with the elastic stiffness tensor acting as a metric. The improved phase-field method involving only a scalar phase-field variable is applied to model and simulate the fracture of brittle anisotropic materials. The results thus obtained are compared with the relevant analytical and numerical ones reported in the literature. It turns out from these comparisons that the improved phase-field method is accurate and efficient.