Instability of dislocation fluxes in a single slip: Deterministic and stochastic models of dislocation patterning

Abstract

We study a continuum model of dislocation transport in order to investigate the formation of heterogeneous dislocation patterns. We propose a physical mechanism which relates the formation of heterogeneous patterns to the dynamics of a driven system which tries to minimize an internal energy functional while subject to dynamic constraints and state dependent friction. This leads us to a novel interpretation which resolves the old 'energetic vs. dynamic' controversy regarding the physical origin of dislocation patterns. We demonstrate the robustness of the developed patterning scenario by considering the simplest possible case (plane strain, single slip) yet implementing the dynamics of the dislocation density evolution in two very different manners, namely (i) a hydrodynamic formulation which considers transport equations that are continuous in space and time while assuming a linear stress dependency of dislocation motion, and (ii) a stochastic cellular automaton implementation which assumes spatially and temporally discrete transport of discrete 'packets' of dislocation density which move according to an extremal dynamics. Despite the huge differences between both kinds of models, we find that the emergent patterns are mutually consistent and in agreement with the prediction of a linear stability analysis of the continuum model. We also show how different types of initial conditions lead to different intermediate evolution scenarios which, however, do not affect the properties of the fully developed patterns.

Cell structure formation in a two-dimensional density-based dislocation dynamics model

“…A closed mathematical model is then specified by relating the effective shear stresses to the dislocation densities. In line with the single-slip model of Groma and co-workers (Groma et al 2016;Wu et al 2018), we consider the effective driving stresses T s i to result from the combination of sign-dependent local driving stresses τ s,dr i and friction stresses τ s,f i :…”

“…The considered slip geometry has the peculiarity that this stress is the same in both slip systems and equals the xy component of the stress tensor, τ 1 = τ 2 = σ xy . In our calculations, we consider a bulk system with periodic boundary conditions and calculate this stress from the plastic strain γ using a Green's function formalism (Zaiser and Moretti 2005;Wu et al 2018):…”

“…To this end we rely on a most basic version of den-sity based dislocation dynamics in multiple-slip conditions. We start from the model used by Zaiser, Groma and co-workers (Groma et al 2016;Wu et al 2018) for analysing the conditions for pattern formation in single slip, and generalize this to symmetrical double slip along lines proposed in earlier work of Groma and co-workers (Yefimov and Van der Giessen 2005;Limkumnerd and der Giessen 2008). This framework not only provides us with some degree of analytical tractability but also with a solid theoretical foundation: The equations we use have been rigorously derived from statistical averaging of the underlying discrete dynamics (Groma et al 2003;Valdenaire et al 2016) and can be related via variational calculus to the statistically averaged energy functional of the dislocation system (Zaiser 2015;Groma et al 2016).…”

Implementation of annihilation and junction reactions in vector density-based continuum dislocation dynamics

“…So the energy criterion can be written as a function of reacting dislocation density vectors, (1)  and (2)  , the corresponding Burgers vectors, (1) b and , together with the line direction of the junction e , which is a constant vector for each type of junction, by substituting equations (19) and (20) into equation (18). When using the criterion (18), the dislocation density vectors () k  should be consistent with the direction of the intersection vector, which means () 0 k  e  . If not, we can change the sign of both the density () k  and its Burgers vector () k b to make it satisfied, since the physical dislocation does not change by changing the sign of () k  and () k b simultaneously .…”

Annihilation and sources in continuum dislocation dynamics

“…The components of the dislocation density alignment tensors can be envisaged as density-like fields which contain more and more detailed information about the orientation distribution of dislocations. CDD has been used to simulate various phenomena including dislocation patterning (Sandfeld and Zaiser, 2015;Wu et al, 2017b) and co-evolution of phase and dislocation microstructure (Wu et al, 2017a). The formulation in terms of alignment tensors has proven particularly versatile since one can formulate the elastic energy functional of the dislocation system in terms of dislocation density alignment tensors (Zaiser, 2015) and then use this functional to derive the dislocation velocity in a thermodynamically consistent manner (Hochrainer, 2016).…”