The concept of Representative Crack Elements (RCE) for phase‐field fracture ‐ Anisotropic elasticity and thermo‐elasticity (original) (raw)

International Journal for Numerical Methods in Engineering

The realistic representation of material degradation at a fully evolved crack is still one of the main challenges of the phase-field method for fracture. An approach with realistic degradation behavior is only available for isotropic elasticity in the small deformation framework. In this paper, a variational framework is presented for the standard phase-field formulation, which allows to derive the kinematically consistent material degradation. For this purpose, the concept of representative crack elements (RCE) is introduced to analyze the fully degraded material state. The realistic material degradation is further tested using the self-consistency condition, where the behavior of the phase-field model is compared to a discrete crack model. The framework is applied to isotropic elasticity, anisotropic elasticity and thermo-elasticity, but not restricted to these constitutive formulations. K E Y W O R D S consistent material degradation, finite element method, homogenization, phase-field fracture 1 INTRODUCTION Realistic modeling of load and direction depending material degradation, crack opening, and closure are of fundamental importance for reliable predictions of crack kinematics and crack evolution. Thus, describing these features is still under development for the phase-field method. May et al, 1 Strobl and Seelig, 2 Schlüter, 3 and Steinke and Kaliske 4 have shown, that the well-known volumetric-deviatoric split (V-D) 5 and spectral split 6 approaches with tension/compression decomposition lead to misleading predictions for the force transfer through the crack. They have proposed and developed a model for the crack kinematics in case of isotropic, linear elastic material. The directional decomposition 4,7 overcomes the observed discrepancies of the V-D and spectral split by performing the following steps: • local crack orientation is determined from a chosen criterion, • crack coordinate system is introduced and applied to the stresses and strains, • stress tensor is decomposed with respect to the kinematic considerations of an equivalent discrete crack, and • corresponding material tangent is derived from the decomposed stress tensor. Also in Teichtmeister et al, 8 Bryant and Sun, 9 and Levitas et al., 10 considerations on the crack kinematics are used with the aim to obtain a consistent material degradation for the phase-field method. In those approaches, a crack orientation is explicitly introduced into the phase-field method. This step requires a few reconsiderations of the interpretation of the phase-field concept in the context of crack irreversibility. 6,11,12 [The copyright line for this article was changed on 10 December 2020 after original online publication.] This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.