Scaling theory of continuum dislocation dynamics in three dimensions: Self-organized fractal pattern formation

Abstract

We focus on mesoscopic dislocation patterning via a continuum dislocation dynamics theory (CDD) in three dimensions (3D). We study three distinct physically motivated dynamics which consistently lead to fractal formation in 3D with rather similar morphologies, and therefore we suggest that this is a general feature of the 3D collective behavior of geometrically necessary dislocation (GND) ensembles. The striking self-similar features are measured in terms of correlation functions of physical observables, such as the GND density, the plastic distortion, and the crystalline orientation. Remarkably, all these correlation functions exhibit spatial power-law behaviors, sharing a single underlying universal critical exponent for each type of dynamics. PACS numbers: 61.72.Bb, 61.72.Lk, 05.45.Df, 05.45.Pq I. INTRODUCTIONDislocations in plastically deformed crystals, driven by their long-range interactions, collectively evolve into complex heterogeneous structures where dislocation-rich cell walls or boundaries surround dislocation-depleted cell interiors. These have been observed both in single crystals 1-3 and polycrystals 4 using transmission electron microscopy (TEM). The mesoscopic cellular structures have been recognized as scale-free patterns through fractal analysis of TEM micrographs 5-8 . The complex collective behavior of dislocations has been a challenge for understanding the underlying physical mechanisms responsible for the development of emergent dislocation morphologies.Complex dislocation microstructures, as an emergent mesoscale phenomenon, have been previously modeled using various theoretical and numerical approaches 9 . Discrete dislocation dynamics (DDD) models have provided insights into the dislocation pattern formations: parallel edge dislocations in a two-dimensional system evolve into 'matrix structures' during single slip 10 , and 'fractal and cell structures' during multiple slip 11,12 ; random dislocations in a threedimensional system self-organize themselves into microstructures through junction formation, cross-slip, and shortrange interactions 13,14 . However, DDD simulations are limited by the computational challenges on the relevant scales of length and strain. Beyond these micro-scale descriptions, CDD has also been used to study complex dislocation structures. Simplified reaction-diffusion models have described persistent slip bands 15 , dislocation cellular structures during multiple slip 16 , and dislocation vein structures 17 . Stochasticity in CDD models 7,10,18 or in the splittings and rotations of the macroscopic cells 19-21 have been suggested as an explanation for the formation of organized dislocation structures. The source of the noise in these stochastic theories is derived from either extrinsic disorder or short-length-scale fluctuations.In a recent manuscript 22 , we analyzed the behavior of a grossly simplified continuum dislocation model for plasticity 22-26 -a physicist's 'spherical cow' approximation designed to explore the minimal ingredients necessary to explain key ...

Dislocation patterning in a two-dimensional continuum theory of dislocations

“…For example Acharya [28] and later on Chen et al [29] proposed models in which the coarsegrained dislocation density (Nye's) tensor plays a central role. This tensor provides information about the distribution of "geometrically necessary" dislocations with excess Burgers vector.…”

Multiple scattering of elastic waves by pinned dislocation segments in a continuum

“…Finally, the Portevin-Le Chatelier effect 11 -in which plastic instabilities generate localized bands-seriously hampers the use of some alloys 12 and remains, to a large extent, a mystery. Recent attempts at developing a conceptual framewok to understand the formation of patterns by dislocations include those of Sethna and collaborators 13 , Limkumnerd and van der Giessen 14 , and Rickman, Haataja and Le Sar 15 .…”

Microstructure in plasticity without nonconvexity

“…We remark that in problems where a system with more than one plastic strain component is involved, it is known that, at least in 1 (Case IV) spatial dimension and time, the system is weakly hyperbolic without core regularization even without any involvement from hardening (Acharya and Tartar, 2011), and again the development of microstructure is to be expected (cf. Chen et al, 2013 Fig. 7 (a) is a plot of the evolution of  when the wiggly back-stress term and 0 k is set to be equal to zero.…”

“…PMFDM Acharya, 2010, building on work from the same group from 2001 onwards) produces spatial patterning with similarities to 'cell-wall' fatigue microstructures without any non-monotone equations of state but due to wave-propagative dislocation transport. An important motivation for our study is that (P)MFDM in its full-blown multidimensional system form, as well as very closely related, if not identical, models have been shown to produce the emergence of microstructural features without any non-convexity in extensive numerical computations Limkumnerd and Sethna, 2006;Chen et al, 2010;Choi et al, 2012;Chen et al 2013). Another recent work with many mathematically similar features as PMFDM but more complicated constitutive structure related to multiple-slip behavior (Xia and El-Azab, 2015) shows the development of realistic microstructure related to plastic response.…”