We provide a minimal continuum model for mesoscale plasticity, explaining the cellular dislocation structures observed in deformed crystals. Our dislocation density tensor evolves from random, smooth initial conditions to form self-similar structures strikingly similar to those seen experimentally-reproducing both the fractal morphologies and some features of the scaling of cell sizes and misorientations analyzed experimentally. Our model provides a framework for understanding emergent dislocation structures on the mesoscale, a bridge across a computationally demanding mesoscale gap in the multiscale modeling program, and a new example of self-similar structure formation in nonequilibrium systems.

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 ...

Intriguing analogies were found between a model of plastic deformation in crystals and turbulence in fluids. A study of this model provides remarkable explanations of known experiments and predicts fractal dislocation pattern formation. Further, the challenges encountered resemble those in turbulence, which is exemplified in a comparison with the Rayleigh-Taylor instability.From horseshoes and knives to bridges and aircrafts, mankind has spent five millennia studying how the structural properties of metals depend not only on their constituents, but also how the atoms are arranged and rearranged as metals are cast, hammered, rolled, and bent into place. A key part of the physics of this plastic distortion is played by the motion of intrinsic line defects called dislocations, and how they move and rearrange to allow the crystal to change shape.Here, we describe the intriguing analogies we found between our model of plastic deformation in crystals and turbulence in fluids. Studying this model led us to remarkable explanations of existing experiments and let us predict fractal dislocation pattern formation. The challenges we encountered resemble those in turbulence, which we describe here with a comparison to the Rayleigh-Taylor instability.For brevity, we offer a minimal problem description (a) Continuum Dislocation Dynamics (b) Turbulence FIG. 1: Comparison of our continuum dislocation dynamics (CDD) with turbulence. (a) Dislocation density profile as it evolves from a smooth random initial condition. The structures form fractal cell wall patterns. Dark regions represent high dislocation density. (b) Rayleigh-Taylor instability at a late time. The fluid (air) with two layers of different densities mix under the effect of gravity. The emerging flows exhibit complex swirling turbulent patterns. The color represents density (red for high, blue for low).that ignores many important features of plastic deformation of crystals, including yield stress, work hardening, dislocation entanglement, and dependence on material properties [1]. We focus on the complex cellular structures that develop in deformed crystals, which appear to be fractal in some experiments [2]. These fractal structures are reproduced by our continuum dislocation dynamics (CDD) [3] theory (see Figure 1a). Not only do the resulting patterns match the experimental ones, but the theory also has rich dynamics, akin to turbulence. This raises a question: Is the dislocation flow turbulent? Here, we focus on exploring this question by building analogies to an explicit turbulence example: the Rayleigh-Taylor instability. As we describe, our theory displays similar conceptual and computational challenges as does this example, which reassures us that we're on firm ground. This CDD model [3][4][5] provides a deterministic explanation for the emergence of fractal wall patterns [3,4] in mesoscale plasticity. The crystal's state is described by the deformation-mediating dislocation density ̺ ijwhere i denotes the direction of the dislocation lines and j their Burgers ...

We explore new ways of regulating defect behavior in systems of conservation laws. Contrary to usual regularization schemes (such as a vanishing viscosity limit), which attempt to control defects by making them smoother, our schemes result in defects which are more singular, and we thus refer to such schemes as "irregularizations". In particular, we seek to produce delta shock defects which satisfy a condition of stationarity. We are motivated to pursue such exotic defects by a physical example arising from dislocation dynamics in materials physics, which we describe.

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