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 ...