Projecting low-dimensional chaos from spatiotemporal dynamics in a model for plastic instability

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.

Influence of system size on spatiotemporal dynamics of a model for plastic instability: Projecting low-dimensional and extensive chaos

“…As shown in I [29], the model reproduces all three types of bands (C, B, and A) observed in experiments with increasing strain rate. While the metallurgical literature identifies the three band types and the associated stress-strain curves, there is no clear understanding of the evolution of bands with increasing˙ a .…”

“…Specifically, well separated bands are seen for small system sizes in the region of strain rates where partially propagative bands are seen for large system sizes (say N = 100). This property is used to reconfirm that low-d chaos is projected as long as the one-to-one correspondence between the isolated bursts of mobile dislocation density and the stress drops remains valid, as is the case for the randomly nucleated isolated bands [29]. In this regime of strain rates corresponding to the partially to fully propagating bands, we show the existence of the correlation dimension density.…”

“…The well separated time scales subsumed in the dynamic interaction of dislocations with solute atoms are captured by the three types of dislocations, namely, the fast mobile ρ m , immobile ρ im , and the "decorated" Cottrell-type dislocations ρ c . Here again we ignore the strain hardening term as was done in I [29] (due its irrelevance to dynamical properties); the scaled form of the evolution equations are…”

“…Recently, using the Ananthakrishna (AK) model for the PLC effect, we demonstrated that the stress signal, which is directly related to the spatial average of dislocation activity in the entire sample, could be unambiguously identified as having all features of low-d chaos while the full set of equations was spatiotemporally chaotic [29]. We further showed that this was entirely due to the one-to-one correspondence between the randomly nucleated isolated bursts of mobile dislocation density (or, equivalently, dislocation bands) and the stress drops.…”

“…As in the previous investigation, hereafter referred to as I [29], we use the AK model for the PLC effect. We first elucidate the relationship between the spatiotemporal patterns (dislocation band types) of the model and the nature of stress serrations.…”