A consistent multi-phase-field formulation for anisotropic brittle fracture

Anisotropic fracture modelling constitutes an important problem in fracture mechanics as many natural, composite and architected materials exhibit fracture anisotropy. The multi-phase-field technique offers a seemingly straightforward and easy-to-implement approach for handling a range of fracture anisotropies, some of which would otherwise necessitate the use of higher-order damage formulation. However, a closer examination reveals that the formulation contains a few inconsistencies, which hinder the establishment of precise definitions for crack through phase-fields and fracture toughness. In this article, we present modifications to the multi-phase-field approach aimed at rectifying these issues. Our model introduces a new degradation function, enabling an unambiguous definition of crack and leading to an analytical expression of the modelled fracture toughness. We conduct an in-depth analysis of several key components of the present formulation and discuss the nature of anisotropy it can handle in its current form, and also outline a roadmap for expanding its capability. A limited set of numerical simulations is also provided to illustrate the effectiveness of the current proposition. [Display omitted] •The shortcomings of the existing multi-phase-field method for anisotropic fracture are presented.•A modified multi-phase-field formulation addressing these limitations has been proposed.•Key components of the new method are analysed.•The effectiveness of the modified formulation is demonstrated using numerical experiments.•A roadmap for expanding the modelling capability has also been discussed.