Dynamical approach to the spatiotemporal aspects of the Portevin-Le Chatelier effect:Chaos,turbulence,and band propagation

Abstract

The analysis of experimental time series, obtained from single and polycrystals subjected to constant strain rate tests, reports an intriguing dynamical crossover from a low-dimensional chaotic state at medium strain rates to an infinite-dimensional power-law state of stress drops at high strain rates. We present the results of an extensive study of all aspects of the Portevin-Le Chatelier (PLC) effect within the context of a recent model that reproduces this crossover. We characterize the dynamics of this crossover by studying the distribution of the Lyapunov exponents as a function of the strain rate, with special attention to system size effects. The distribution of the exponents changes from a small set of positive exponents in the chaotic regime to a dense set of null exponents in the scaling regime. As the latter is similar to the result in a shell model for turbulence,we compare the results of our model with that of the shell model. Interestingly, the null exponents in our model themselves obey a power law. The study is complimented by visualizing the configuration of dislocations through the slow manifold analysis. This shows that while a large proportion of dislocations are in the pinned state in the chaotic regime, most of them are pushed to the threshold of unpinning in the scaling regime, thus providing insight into the mechanism of crossover. We also show that this model qualitatively reproduces the different types of deformation bands seen in experiments. At high strain rates, where propagating bands are seen, the model equations can be reduced to the Fisher-Kolmogorov equation for propagative fronts, which in turn shows that the velocity of the propagation of the bands varies linearly with the strain rate and inversely with the dislocation density. These results are consistent with the known experimental results. We also discuss the connection between the nature of band types and the dynamics in the respective regimes. The analysis demonstrates that this simple dynamical model captures the complex spatiotemporal features of the PLC effect.