Pattern formation in a minimal model of continuum dislocation plasticity

Abstract

The spontaneous emergence of heterogeneous dislocation patterns is a conspicuous feature of plastic deformation and strain hardening of crystalline solids. Despite long-standing efforts in the materials science and physics of defect communities, there is no general consensus regarding the physical mechanism which leads to the formation of dislocation patterns. In order to establish the fundamental mechanism, we formulate an extremely simplified, minimal model to investigate the formation of patterns based on the continuum theory of fluxes of curved dislocations. We demonstrate that strain hardening as embodied in a Taylor-type dislocation density dependence of the flow stress, in conjunction with the structure of the kinematic equations that govern dislocation motion under the action of external stresses, is already sufficient for the formation of dislocation patterns that are consistent with the principle of similitude.

Dislocation patterning in a two-dimensional continuum theory of dislocations

“…As it is shown above there are two alternative ways (the diffusionlike term associated with the stressτ b , or the ∂ xx type term in the flow stress) leading to characteristic length scale of the dislocation patten. Irrespective of the pattern selection mechanism and in line with previous work [47], we find that there are two requirements for patterning: First, the system must be in the plastically deforming phase, and second, the rate of shear must not be too high (γ α ). This condition indicates that patterning as studied here cannot be understood as a energy minimization process, despite the fact that the dynamics which we investigate minimizes an energy functional.…”

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

“…However, it has soon become clear that in CDD models the emergence of heterogeneous dislocation patterns turns out to be an almost inevitable feature of the collective dynamics. Simulations of CDD models in 3D demonstrate an intrinsic tendency towards dislocation patterning [17,18] as they relate the emergence of patterning to the same dislocation interactions that govern strain hardening, in line with the 'principle of similitude' [19]. This principle has been related to fundamental invariance properties of the equations that govern the properties of discrete dislocation systems, and indeed all physically based models of dislocation patterning published in recent years are consistent with these invariance properties [20].…”

Annihilation and sources in continuum dislocation dynamics

“…Therefore CDD (0) cannot capture these stages. To capture these stages, one might need to use higher order non local models such as CDD (1) and CDD (2) which are cable of accounting for dislocation transport and capture cell formation (Sandfeld and Zaiser, 2015). In our treatment we have neglected the slip contribution of segments that move on the cross slip plane during cross-slip induced recombination processes.…”

“…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).…”