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This paper focuses on the numerical implementation of phase-field models of fracture using the Fast Fourier Transform (FFT) based numerical method. Recent studies on phase-field models focus on the discussions of the choice of the value of regularization length proposed to smear the discontinuity of the sharp crack. Some studies argue that it should be considered as a material property because it has a significant impact on the mechanical behavior of a material in some phase-field models, for instance, in the model proposed by Miehe. However, our results in this study for heterogeneous materials have shown that the choice of regularization length not only affects the macroscopic mechanical behavior but also the local crack propagation patterns. As a result, it can be challenging to select an appropriate value that produces both accurate macroscopic responses and local crack patterns for certain phase-field models, such as Miehe's model. Thus, the phase-field model proposed by Wu, which has been proven to reduce the length sensitivity for homogeneous material, has been successfully implemented in an FFT-based solver with the application of Newton–Krylov algorithm. The length sensitivity of Wu's phase-field model for heterogeneous materials is also investigated in this paper. Our tests show that Wu's phase-field model has partial length sensitivity for heterogeneous materials. However, the main reason for this sensitivity is that one phase enters the damage zone of other phase, which is different from Miehe's model. As a result, a set of criteria for accurately choosing the regularization value has been established for Wu's model in this work. Meanwhile, it has also been found that Wu's model may be more suitable than Miehe's model for brittle failure due to the introduction of an elastic stage. • Phase-field modeling of the fracture using the Fast Fourier Transform-based method. • Newton–Krylov algorithm to solve a phase-field problem in its complete formulation. • Investigation of the regularization length sensitivity of the complete formulation. • Investigation of regularization length and its effects on the local crack patterns. [ABSTRACT FROM AUTHOR]

Abstract The variational phase-field method is an attractive non-local approach of modeling fracture in heterogeneous materials. However, these materials usually require a fine mesh to resolve the fracture process zone. Consequently, the standard finite element solver becomes cumbersome due to the large number of elements in applications with highly heterogeneous materials. Motivated by this limitation, an algorithm based on FFT methods has been introduced in this paper to solve the phase-field model of brittle fracture. Relying on a staggered update scheme, the proposed algorithm solves the fracture problem and mechanical problem separately, both using the FFT technique. It inherits the advantages of classical FFT methods in terms of simplicity of mesh generation and parallel implementation. Introduced within a FFT-based code "AMITEX", it takes the advantage of massively parallel capabilities associated with a distributed memory implementation. The characteristics of the proposed method are analyzed in a single edge notched specimen benchmark. Representative numerical examples demonstrate that the proposed FFT solver is capable of predicting different crack modes and complex crack configuration, such as crack interaction, branching and coalescence. Finally, a model of an idealized continuous fiber composite with void involving over 32 million voxels is solved, illustrating the potential of the FFT solver in large-scale problems. [ABSTRACT FROM AUTHOR]

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. • Phase-field modeling of fracture using the Fast Fourier Transform based numerical method. • Development of a new complete formulation of the phase-field equation. • Comparison with a simplified formulation previously proposed in the literature. • Complete formulation shown to eliminate unphysical damage diffusion between phases. [ABSTRACT FROM AUTHOR]