Phase-Field Model of Mode III Dynamic Fracture

Abstract

We introduce a phenomenological continuum model for mode III dynamic fracture that is based on the phase-field methodology used extensively to model interfacial pattern formation. We couple a scalar field, which distinguishes between "broken" and "unbroken" states of the system, to the displacement field in a way that consistently includes both macroscopic elasticity and a simple rotationally invariant short scale description of breaking. We report two-dimensional simulations that yield steady-state crack motion in a strip geometry above the Griffith threshold.

Thermal fracture as a framework for quasi-static crack propagation

“…This function is such that in regions where the material is broken (φ = 0), the contribution to the elastic energy is zero, while in regions where the material is intact, the contribution to the elastic energy recovers the one prescribed by linear elasticity. In (Karma et al, 2001), it is shown that the particular choice of g does not affect the results as long as g(0) = g ′ (0) = g ′ (1) = 0 and lim φ=0 g(φ) ∼ φ α , with α > 2. The additional term T (x) corresponds to the thermal expansion.…”

“…In this context, the phase field approach to crack propagation (Aranson et al, 2000;Karma et al, 2001;Eastgate et al, 2002;Marconi and Jagla, 2005) is very useful as it allows to study problems in arbitrary geometries and go beyond simple crack paths. It has succeeded in reproducing qualitative behavior of cracks such as branching (Karma and Lobkovsky, 2004) and oscillations under biaxial strain (Henry and Levine, 2004).…”

“…The size of the domain [L] × [2] was typically 1500 × 300 grid points, and numerical checks showed that doubling L did not affect the results. Finally, the fracture energy Γ = K 2 Ic /(2µ) was computed using the expression (Karma et al, 2001)…”

“…Extensions of this work allowed quantitative modeling of dendritic growth (Karma and Rappel, 1998) and since then, phase field methods have been applied to many solidification problems such as growth of binary alloys (Etchebarria et al, 2004) and formation of polycristalline solids (Kobayashia and Warren, 2005). Furthermore, they have been extended to other free boundary problems including viscous fingering (Folch et al, 2000) and crack propagation (Karma et al, 2001;Karma and Lobkovsky, 2004;Henry and Levine, 2004;Karma, 2005, 2009). In the latter case, the phase field approach turns out to be very similar to the variational approach of fracture which uses the description of free discontinuity problems (Braides, 1998;Acerbi and Braides, 1999) using the Γ-convergence Tortorelli, 1990, 1992).…”

A Higher Order Phase-Field Approach to Fracture for Finite-Deformation Contact Problems

“…It is further possible to give a variational formulation for the crack propagation problem (see e.g. [23,13,12]). The phase-fracture method has been developed in the small strain regime (see e.g.…”