On the role of dislocation conservation in single-slip crystal plasticity

Abstract

The objective of this paper is to evaluate the relevance of dislocation conservation within the context of dislocation-based crystal plasticity. In advanced crystal plasticity approaches, dislocations play a prominent role. Their presence, nucleation, motion, and interactions enable the explanation and description of physical phenomena such as plastic slip, hardening and size effects. While the conceptual aspects of the evolution and mechanical consequences of dislocations are treated analogously in a wide range of advanced crystal plasticity formulations, these formulations differ significantly with respect to the underlying dislocation conservation properties. This paper identifies and compares two essentially different approaches to model plastic deformations in a single crystal. Both approaches have in common that they rely on the geometrical relation between the plastic slip and the densities of geometrically necessary dislocations. In the first approach, the geometrical relation serves as a balance and is supplemented with an evolution law for the statistically stored dislocations. In the second approach the local conservation of the total number of dislocations is enforced in addition to the first balance instead of the evolution. Considering a single-slip model pileup problem, the two representative frameworks are elaborated and confronted theoretically and numerically on the basis of a dimensionless finite-element analysis to evaluate the intrinsic role of dislocation conservation for model predictions.

Coupling continuum dislocation transport with crystal plasticity for application to shock loading conditions

“…Several researchers pursuing nonlocal continuum theories of crystal plasticity use a plane strain solution assuming isotropic elasticity for the stress field surrounding the Volterra dislocation to compute either the internal stress field or a slip-system associated back stress (Evers et al, 2004a,b;Yefimov and van der Giessen, 2005a;Bayley et al, 2006;Geers et al, 2007;Gerken and Dawson, 2008;Hirschberger et al, 2011;Liu et al, 2011;Leung et al, 2015;Schulz et al, 2014). Leung et al (2015), incorporate a physical representation of the long-range internal stress by generalizing relations for the elastic interaction force between two dislocations from Hirth and Lothe (1982).…”

“…These include not only the theory and implementation presented here, but also models in the recent literature, e.g. (Yefimov and van der Giessen, 2005a;Gerken and Dawson, 2008;Hirschberger et al, 2011;Reuber et al, 2014), which incorporate a subset of -or similar approaches to -the features depicted in Figure 1. The novelty of our approach is built upon explicit representation of dislocation transport (CDT), detailed accounting for elastic interactions involving geometrically necessary dislocations (DDC), classical momentum balance (DMB), and the tight coupling between all three of these sub-problems, in balance with constitutive theories that are germane to the high strain-rate and large elastic and plastic deformation regimes.…”

A new dislocation-density-function dynamics scheme for computational crystal plasticity by explicit consideration of dislocation elastic interactions

“…In stark contrast with these approaches which emphasize on individual atoms or dislocation segments, a number of researchers have advocated the use of a modeling strategy that focuses on dislocation density (Walgraef and Aifantis, 1985;Groma, 1997;Acharya, 2001;Arsenlis et al, 2004;Zhou and Sun, 2004;Evers et al, 2004;Yefimov and Van der Giessen, 2005;Pontes et al, 2006;Ma et al, 2006;Hochrainer et al, 2007;Lee et al, 2010;Watanabe et al, 2010;Alankar et al, 2011;Hirschberger et al, 2011;Bargmann et al, 2011;Liu et al, 2011;Shanthraj and Zikry, 2012;Engels et al, 2012;Aghababaei and Joshi, 2013;Li et al, 2014). Unlike DDD which becomes handicapped at high strains, such a strategy would be well suited for large-strain problems with high quantities of dislocations, since any amount of dislocations can still be represented a dislocation density.…”

“…Typically, the shear of slip systems against a critical resolved shear stress governed by certain basic dislocation-level physics, such as Taylor's forest hardening, is considered (Busso et al, 2000;Dunne et al, 2007Dunne et al, , 2012Alankar et al, 2009;Cordero et al, 2012). Other models have focused on dislocation densities as continuous functions of space, with conservation including generation and annihilation duly taken into account (Acharya, 2001;Arsenlis et al, 2004;Zhou and Sun, 2004;Evers et al, 2004;Yefimov and Van der Giessen, 2005;Pontes et al, 2006;Hochrainer et al, 2007;Hirschberger et al, 2011;Bargmann et al, 2011;Puri et al, 2011). These models are based on crystal kinematics rules which govern the relationship between the evolution of geometrically necessary dislocations (GNDs) and the rate of change of the crystal shape (Asaro and Rice, 1977).…”

“…The evolution of the GNDs is coupled to that of SSDs, the motion of which is modeled in terms of a phenomenological back-stress interaction resistance which involves empirical hardening and recovery coefficients (Puri et al, 2011). The models by Yefimov and Van der Giessen (2005) and Hirschberger et al (2011) also involve a similar empirical back stress resistance, which lacks details of the long-range elastic interactions between dislocations. In the model by Arsenlis et al (2004), a configurational resistance is also involved which is an empirical back stress pertinent only to elastic interactions in 2-D dislocation arrays (Groma et al, 2003).…”