Projecting Low Dimensional Chaos from Spatio-temporal Dynamics in a Model for Plastic Instability

TL;DR

This study explores how low-dimensional chaos can be projected from the complex spatiotemporal dynamics of a model simulating plastic instability, revealing conditions under which chaos manifests and can be reduced to simpler models.

Contribution

It demonstrates the presence of low-dimensional chaos in a plasticity model and introduces a modified correlation dimension algorithm for analyzing such chaos in spatiotemporal signals.

Findings

Low strain rates show a one-to-one correspondence between dislocation bursts and stress drops.

Model equations are spatiotemporally chaotic with Lyapunov exponents scaling with system size.

Stress signals exhibit low-dimensional chaos at low strain rates, reducible to space-independent models.

Abstract

We investigate the possibility of projecting low dimensional chaos from spatiotemporal dynamics of a model for a kind of plastic instability observed under constant strain rate deformation conditions. We first discuss the relationship between the spatiotemporal patterns of the model reflected in the nature of dislocation bands and the nature of stress serrations. We show that at low applied strain rates, there is a one-to-one correspondence with the randomly nucleated isolated bursts of mobile dislocation density and the stress drops. We then show that the model equations are spatiotemporally chaotic by demonstrating the number of positive Lyapunov exponents and Lyapunov dimension scale with the system size at low and high strain rates. Using a modified algorithm for calculating correlation dimension density, we show that the stress-strain signals at low applied strain rates corresponding to spatially uncorrelated dislocation bands exhibit features of low dimensional chaos. This is made quantitative by demonstrating that the model equations can be approximately reduced to space independent model equations for the average dislocation densities, which is known to be low-dimensionally chaotic. However, the scaling regime for the correlation dimension shrinks with increasing applied strain rate due to increasing propensity for propagation of the dislocation bands.