Assessment of four strain energy decomposition methods for phase field fracture models using quasi-static and dynamic benchmark cases

Abstract

Strain energy decomposition methods in phase field fracture models separate strain energy that contributes to fracture from that which does not. However, various decomposition methods have been proposed in the literature, and it can be difficult to determine an appropriate method for a given problem. The goal of this work is to facilitate the choice of strain decomposition method by assessing the performance of three existing methods (spectral decomposition of the stress or the strain and deviatoric decomposition of the strain) and one new method (deviatoric decomposition of the stress) with several benchmark problems. In each benchmark problem, we compare the performance of the four methods using both qualitative and quantitative metrics. In the first benchmark, we compare the predicted mechanical behavior of cracked material. We then use four quasi-static benchmark cases: a single edge notched tension test, a single edge notched shear test, a three-point bending test, and a L-shaped panel test. Finally, we use two dynamic benchmark cases: a dynamic tensile fracture test and a dynamic shear fracture test. All four methods perform well in tension, the two spectral methods perform better in compression and with mixed mode (though the stress spectral method performs the best), and all the methods show minor issues in at least one of the shear cases. In general, whether the strain or the stress is decomposed does not have a significant impact on the predicted behavior.

Crack-path selection in phase-field models for brittle fracture

“…These are two of the most widely studied tension/compression splits (see, e.g., Refs. [2,3,60,44,16,17]). The strain-spectral split has terms ψ + 0 and ψ − 0 of the form…”

“…[65] for a form of F f incorporating the Laplacian of φ, and Refs. [60,16,17] for alternative decompositions of the elastic energy density ψ(φ, ε). Instead of F itself, evolution methods use F φ and F u , respectively the variational derivatives of F with respect to φ and u.…”

“…The different forms of the stress affect the mechanical response of the crack and other regions with non-zero φ. For this reason, we refer to forms of ∂ψ/∂ε as contact models, even if they fail to reproduce realistic contact physics [42,44,17]. The earliest example of a non-variational phase field fracture model may be the use of a history function in place of ψ + 0 in the crack driving force in order to satisfy a damage-type irreversibility condition [53,46].…”

“…For researchers interested in using phase field fracture, systematic comparisons of these models are valuable in determining what is physically appropriate for their system. However, such comparisons [2,13,14,15,16,17] have so far focused on homogeneous systems, despite the growing number of phase field fracture studies that are explicitly interested in the effects of material heterogeneity [18,19,20,21,22]. The present study seeks to address this gap by focusing on how different phase field fracture formulations affect crack paths in a set of randomly generated, elastically heterogeneous two-dimensional (2-D) microstructures.…”

“…One example is the possibility for interpenetration of crack faces and crack nucleation in compression in the initial model of Bourdin et al [1]. Multiple formulations were subsequently proposed to restrict the driving force for fracture to tensile or shear conditions and to enforce some form of elastic contact between crack faces [42,43,44,45,17] (see also Ref. [3] for a review).…”