Computing the effective crack energy of heterogeneous and anisotropic microstructures via anisotropic minimal surfaces

Abstract

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.

Investigations on the influence of the boundary conditions when computing the effective crack energy of random heterogeneous materials using fast marching methods

“…Classical fast marching algorithms are only applicable to isotropic crack resistances in the plane. To cover anisotropies in the crack resistance [46], anisotropic fast marching methods [89] may be explored.…”

“…From their work the effective crack energy is defined as the area of the crack resistance-weighted minimal surface cutting through the microstructure. Numerical approaches to computing the effective crack energy have been proposed by the authors [44][45][46] using FFT-based algorithms and periodic boundary conditions. Recently, Michel-Suquet addressed the approach based on Braides et al's [43] homogenization result using an alternative solution strategy by pointing out similarities with limit load analysis [47].…”

“…For the computations based on the minimum cut/maximum flow approach, we relied on an in-house FFT-based code [44][45][46]. The continuous equations were discretized with the CCMF discretization [68] and solved by a damped version of the alternating direction method of multipliers (ADMM) [71,72] with adaptive choice for the penalty parameter, see Ernesti-Schneider [45].…”

A sequential addition and migration method for generating microstructures of short fibers with prescribed length distribution

“…Even more striking, the size of the considered unit cells could be chosen to be extremely small, and still be representative [26][27][28]. Such a property is extremely desirable, in particular when it comes to nonlinear material properties of composites, and turned out to be critical when undertaking a number of scientific studies, including fatigue [29,30], fracture [31,32] and thermomechanically coupled problems [33,34].…”

Merits and limits of a variational definition of the effective toughness of heterogeneous materials

“…The topic of the effective toughness of a composite material considering the brittleness of its constituents as the only dissipative mechanism has been addressed in a number of studies over the last twenty years with an increasing interest in recent years. We will not review them in detail, referring the reader to the introductory sections in Ernesti and Schneider [17,18], limiting ourself to a reminder of the points which are essential for our purpose.…”