No abstract

This work is concerned with computing the effective crack energy of periodic and random media which arises in mathematical homogenization results for the Francfort-Marigo model of brittle fracture. A previous solver based on the fast Fourier transform (FFT) led to solution fields with ringing or checkerboard artifacts and was limited in terms of the achievable accuracy. As computing the effective crack energy may be recast as a continuous maximum flow problem, we suggest using the combinatorial continuous maximum flow discretization introduced by Couprie et al. The latter is devoid of artifacts, but lacks an efficient large-scale solution method. We fill this gap and introduce a novel solver which relies upon the FFT and a doubling of the local degrees of freedom which is resolved by the alternating direction method of multipliers (ADMM). Last but not least we provide an adaptive strategy for choosing the ADMM penalty parameter, further speeding up the solution procedure. We demonstrate the salient features of the proposed approach on problems of industrial scale.

A variety of materials, such as polycrystalline ceramics or carbon fiber reinforced polymers, show a pronounced anisotropy in their local crack resistance. We introduce an FFT-based method to compute the effective crack energy of heterogeneous, locally anisotropic materials. Recent theoretical works ensure the existence of representative volume elements for fracture mechanics described by the Francfort–Marigo model. Based on these formulae, FFT-based algorithms for computing the effective crack energy of random heterogeneous media were proposed, and subsequently improved in terms of discretization and solution methods. In this work, we propose a maximum-flow solver for computing the effective crack energy of heterogeneous materials with local anisotropy in the material parameters. We apply this method to polycrystalline ceramics with an intergranular weak plane and fiber structures with transversely isotropic crack resistance.

The description of material failure as an energy minimization problem, i.e., the Francfort-Marigo model, has been studied widely in recent years. The approximation of the crack surface as a phase field, i.e., smeared interface, enjoys great popularity, as it allows describing fracture as a set of partial differential equations. In numerical homogenization, FFT-based solution methods have been established over the past two decades. Their purpose is to compute the overall response of a heterogeneous microstruture w.r.t. a macroscopic loading and can be applied to a variety of nonlinear materials. The benefits lie in a fast implementation and the possibility to use image data like CT-scans as input without further need for meshing. Based on the results of the master thesis of the first author, we investigate phase field crack propagation on heterogeneous microstructures using FFT-based solvers.