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

Abstract

This work is a continuation of our efforts to quantify the irregular scalar stress signals from the Ananthakrishna model for the Portevin-Le Chatelier instability observed under constant strain rate deformation conditions. Stress related to the spatial average of the dislocation activity is a dynamical variable that also determines the time evolution of dislocation densities. We carry out detailed investigations on the nature of spatiotemporal patterns of the model realized in the form of different types of dislocation bands seen in the entire instability domain and establish their connection to the nature of stress serrations. We then characterize the spatiotemporal dynamics of the model equations by computing the Lyapunov dimension as a function of the drive parameter. The latter scales with the system size only for low strain rates, where isolated dislocation bands are seen, and at high strain rates, where fully propagating bands are seen. At intermediate applied strain rates corresponding to the partially propagating bands, the Lyapunov dimension exhibits two distinct slopes, one for small system sizes and another for large. This feature is rationalized by demonstrating that the spatiotemporal patterns for small system sizes are altered from the partially propagating band types to isolated burst type. This in turn allows us to reconfirm that low-dimensional chaos is projected from the stress signals as long as there is a one-to-one correspondence between the bursts of dislocation bands and the stress drops. We then show that the stress signals in the regime of partially to fully propagative bands have features of extensive chaos by calculating the correlation dimension density. We also show that the correlation dimension density also depends on the system size. A number of issues related to the system size dependence of the Lyapunov dimension density and the correlation dimension density are discussed.

Spatio-temporal dynamics of jerky flow in high-entropy alloy at extremely low temperature

“…Here, we use a method of compensating the errors arising from the finite size of the attractor by normalising the local slope D 2 (r, d E ) by the slope of C 2 (r, 2) 1/2 . This normalisation extends the scaling regime considerably [62,63]. Then, we can define…”

“…Here, we emphasise that the large amplitude nature of the stress drops in our case is a general feature of low dimensional chaos in several models of chaos [60] and also types C and B serrations of the PLC effect. In contrast, amplitudes of varying magnitudes are a typical high dimensional attractor such as the Mackey-Glass equations [65] or the PLC time series at high strain rates (see for example [63]).…”

Macroscopic criteria for the deformation and fracture of iron based alloys

“…Thus, inefficient dislocation models in the physics of strength and plasticity were replaced with the complex concept of a multilevel approach. A multiscale approach to the problems of plastic deformation and fracture was developed due to the works of domestic and foreign researchers [11][12][13][14][15][16][17][18][19] and currently is used by the scientific community. Thus, at present a deformable material is considered to be a nonlinear medium, an appropriate description of which is impossible without the approach based on the concepts for the mechanics of nonlinear media, synergetics and nonlinear acoustics.…”