Minimal Integer Automaton behind Crystal Plasticity

Abstract

Power law fluctuations and scale free spatial patterns are known to characterize steady state plastic flow in crystalline materials. In this Letter we study the emergence of correlations in a simple Frenkel-Kontorova (FK) type model of 2D plasticity which is largely free of arbitrariness, amenable to analytical study and is capable of generating critical exponents matching experiments. Our main observation concerns the possibility to reduce continuum plasticity to an integer valued automaton revealing inherent discreteness of the plastic flow.At the macroscale one usually assumes that crystalline materials flow plastically when averaged stresses exceed yield thresholds. At the microscale plasticity evolves through a sequence of slow-fast events involving collective pinning and depinning of dislocational structures. Classical engineering theory has been very successful in reproducing the most important plasticity phenomenology such as yield, hardening and shakedown [1], however, a fully quantitative link between the phenomenological theory and the microscopic picture of plasticity remains elusive. The main reason is that the phenomenological approach implies spatial and temporal averaging in the system with poorly understood long range correlations.The presence of such correlations have been confirmed by numerous experiments revealing intermittent character of plastic activity with power law statistics of avalanches and self similar structure of dislocation cell structures [2]. The emergence of power laws suggests that in plasticity the relation between the microscopic and the macroscopic models is more akin to turbulence than to elasticity [3,4]. Similar critical features of stationary nonequilibrium states have been observed in a variety other driven systems with threshold nonlinearity and rate independent dissipation including, for instance, tectonic faults, magnets, and superconductors [5]. The origin of scale free attractors in such systems is a subject of active research, in particular, the problem of classifying the universality classes remains largely open [6]. In this situation finding the minimal representation of each class which is amenable to rigorous analysis is of significant general interest.The experimental evidence of plastic criticality has been corroborated by several numerical models [7]. The two main microscopic approaches are discrete dislocation dynamics, accounting for dislocation interactions on different slip planes [8][9][10], and a pinning-depinning model dealing with plasticity on a single slip plane [11,12]. Different meso-scopic continuum models implying partial averaging have also been shown to generate power law statistics of avalanches with realistic exponents [4,13]. Since scale free dislocation activity is expected to be independent of either microscopic or macroscopic details, one can try to maximally simplify the underlying physics while still capturing the observed exponents and even characteristic shape functions [14]. Presently the only analytically tractable models of pl...

From Mild to Wild Fluctuations in Crystal Plasticity

“…This identification, which we checked to be respected by the dislocation density fluctuations in the microscopic model [7], allows one to interpret observed behaviors in terms of the values of the parameters a, c, D…”

“…A different point of view emerged from the analysis of high resolution acoustic emission (AE) data in plastically deforming HCP crystals which showed that temporal fluctuations may be power-law distributed in size and energy [2] and may be clustered in both space [3] and time [4]. These observations, suggesting that averages do not represent typical behavior, were corroborated by the study of statistics of slip events in microand nano-pillars [5,6] for FCC and BCC metals and supported by numerical models [2,7,10,12].…”

“…In particular, the multiplicative noise describes the autocatalytic effect when dislocation clusters react to perturbations in a collective manner amplifying the effect of the noise proportionally to their size [14]. Such cooperative response implies the presence of long-range fields that are not explicitly resolved in our zero-dimensional model; we also neglect quenched disorder and diffusion whose account would allow one to model spatial intermittency [39] observed in microscopic models of crystal plasticity [7]. Multiplicative stochastic closure of the coarse grained models exemplified by (1) is rather common in the study of marginally stable driven systems [14,37] including turbulence [25], absorbing phase transitions [26] and depinning [27].…”

Dynamic Hysteresis in Cyclic Deformation of Crystalline Solids

“…However, our results suggest that this simple picture is incomplete, calling for novel theoretical ideas to properly describe the glassy, critical-like dynamics observed in the jammed phase of the system. Dislocation physics has been recently studied with many simplified models from discrete dislocation dynamics [17][18][19] to phase field [20] and automaton models [21]. Here, we consider the stress-controlled hysteretic dynamics within a two-dimensional discrete dislocations dynamics model [17].…”

The role of phase compatibility in martensite

“…45,[48][49][50][51]53 These systems evolve toward a critical state represented by the ensemble of metastable states through which the system passes via an avalanche-like dynamics whose amplitudes and durations follow power laws (see Ref. 52 for a review).…”