Phase Field Modeling of Brittle and Ductile Fracture

Abstract

The phase field modeling of brittle fracture was a topic of intense research in the last few years and is now well-established. We refer to the work [1][2][3], where a thermodynamically consistent framework was developed. The main advantage is that the phase-field-type diffusive crack approach is a smooth continuum formulation which avoids the modeling of discontinuities and can be implemented in a straightforward manner by multi-field finite element methods. Therefore complex crack patterns including branching can be resolved easily. In this paper, we extend the recently outlined phase field model of brittle crack propagation [1-3] towards the analysis of ductile fracture in elastic-plastic solids. In particular, we propose a formulation that is able to predict the brittle-to-ductile failure mode transition under dynamic loading that was first observed in experiments by Kalthoff and Winkler [4] . To this end, we outline a new thermodynamically consistent framework for phase field models of crack propagation in ductile elastic-plastic solids under dynamic loading, develop an incremental variational principle and consider its robust numerical implementation by a multi-field finite element method. The performance of the proposed phase field formulation of fracture is demonstrated by means of the numerical simulation of the classical Kalthoff-Winkler experiment that shows the dynamic failure mode transition. subject to the Dirichlet constraints W Γ(t) = {d | d(x, t) = 1 at x ∈ Γ(t)}. The crack functional Γ l in terms of the crack surface density function γ(d, ∇d) governs the regularization by the length scale parameter l giving for l → 0 the sharp crack topology. The reader is referred to the recent work [1] for a more detailed derivation of the diffusive crack topology.

Numerical simulations of crack propagation in screws with phase-field modeling

“…The energy release rate G c is kept fixed. For other modeling choices of G c , we refer the reader to [17,19]. In elasto-plastic settings we expect moderate crack propagation in comparison to brittle materials.…”

“…We finally recall that the energy release rate should be higher in plasticity due to plastic dissipation. In our work, we just choose a higher number for the energy release rate and refer to the literature, e.g., [17,19] and similar references for other modeling choices. In our work, we compare the elastic G c to a number that is twice higher.…”

“…While it is fairly well understood how phase-field fracture models act in brittle materials with elasticity, only a few references exist to date on ductile crack propagation (e.g., [16][17][18][19]). Here, the last reference focuses on numerical tests of I-shaped specimen that have some similarities to screw simulations as we have them in mind.…”

Phase‐field modelling of fracture in single crystal plasticity

“…A few very recent contributions deal with phase-field modelling of fracture in ductile materials (see e.g. Ulmer et al, 2013;Borden, 2012;Duda et al, 2015;Ambati et al, 2015;Alessi et al, 2015;Wick et al, 2015). Most relevant for the present investigation is the phase-field model for quasi-static ductile fracture proposed by Ambati et al (2015) for classical J 2 -plasticity in a kinematically linear framework and later extended by Ambati et al (2016) to large deformations and by Ambati and De Lorenzis (2016) to modelling of fracture in shells.…”

Small strain elasto-plastic multiphase-field model

“…We remark that a reduction of crack resistance due to the generation of microvoids in the plastically highly stressed areas is thereby neglected. Despite the choice of the hardening parameter, this effect can be accounted for by introducing an additional indicator function dependent on a critical value of the accumulated plastic strain, as done by Ulmer et al [17]. Figure 5a, b show the resulting stress field profiles for brittle and ductile material, respectively.…”