Simplified and complete phase-field fracture formulations for heterogeneous materials and their solution using a Fast Fourier Transform based numerical method

This paper focuses on the numerical implementation of phase-field models of fracture using the Fast Fourier Transform based numerical method. Recent developments in that field rely on the separate solution of a coupled problem where the mechanical equilibrium problem is solved first, and then the phase-field evolution equation. The latter involves a diffusion term which has been simplified in previous works relying on the Fast Fourier Transform based numerical method. This simplification has theoretically no effect for homogeneous materials, but might influence predictions significantly for heterogeneous materials where fracture properties vary between the different components.In this paper, the influence of this simplification is assessed and a complete formulation is proposed as well as a novel implementation of this formulation using the Fast Fourier Transform based numerical method. The assessment relies on simulations with a material containing two components, one of them being defined as unbreakable by using higher fracture properties. Using the simplified formulation, the presence of an artificial diffusion of damage between the two components is evidenced, and non-zero damage values are observed in the unbreakable component. Although the complete formulation leads to an increase of the number of iterations to solve the phase-field evolution equation, it suppresses completely the diffusion of damage towards the unbreakable component. The two formulations, in fact, lead to identical results when the fracture properties are homogeneous, but the results diverge both in terms of local fracture patterns and global stress-strain relations when the fracture properties contrast increases. This difference is also more pronounced when the regularization length introduced by the phase-field model increases.