Current theoretical approaches to collective behavior of dislocations

A model for dynamic strain aging is proposed. It is written as a local constitutive formulation but can easily be extended to account for spatial interactions. The model is obtained by embedding accepted ‘slow’ time scale formulations for dislocation dynamics and aging kinetics in a multiple time scales dynamic system. Dislocation and solute dynamics are coupled at the ‘fast’ time scale pertaining to dynamic aging. Features of jerky flow described by standard models such as the negative strain rate sensitivity of the flow stress and critical strains for the occurrence of the Portevin-Le Chatelier effect are retrieved. In addition, negative strain rate sensitivity of the threshold stress for instability is predicted.

Dynamic strain aging: A coupled dislocation—Solute dynamic model

http://inaesp.org/PublicJG/fg_stern-brocot_mmo_organization_excitable_sys_PLA_375_1097_2011.pdf

Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems

Deformation behavior of an ultrafine-grained Al¿Mg alloy at different strain rates

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\char21{}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\penalty1000-\hskip0ptKolmogorov 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.

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

A characteristic feature of the Portevin-Le Chatelier effect, or the jerky flow, is the stick-slip nature of stress-strain curves which is believed to result from the negative strain rate dependence of the flow stress. The latter is assumed to result from the competition of a few relevant time scales controlling the dynamics of jerky flow. We address the issue of time scales and its connection to the negative strain rate sensitivity of the flow stress within the framework of a model for the jerky flow, which is known to reproduce several experimentally observed features, including the negative strain rate sensitivity of the flow stress. We attempt to understand the above issues by analyzing the geometry of the slow manifold underlying the relaxational oscillations in the model. We show that the nature of the relaxational oscillations is a result of the atypical bent geometry of the slow manifold. The analysis of the slow manifold structure helps us to understand the time scales operating in different regions of the slow manifold. Using this information we are able to establish connection with the strain rate sensitivity of the flow stress. The analysis also helps us to provide a proper dynamical interpretation for the negative branch of the strain rate sensitivity.

Relaxation oscillations and negative strain rate sensitivity in the portevin-Le chatelier effect

Incomplete approach to homoclinicity in a model with bent-slow manifold geometry

Relaxation oscillations or stick-slip dynamics exhibited by a model, originally proposed for a form of plastic instability namely, Portevin Le-Chatelier effect, has been analysed. The model exhibits atypical slow manifold which has a bent structure. It is this geometry that gives rise to a new mechanism of relaxation oscillations. A partial representation of the slow manifold in the form of the next maximal amplitude (NMA) maps has also been analysed. Minimal information of principal periodic orbit embedded in four dimensions and the slow manifold structure is shown to be sufficient to reproduce the qualitative features of the NMA maps.

Effect of slow manifold structure on relaxation oscillations and one-dimensional map in a model for plastic instability

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Papers

Relaxation oscillations and negative strain rate sensitivity in the portevin-Le chatelier effect

Incomplete approach to homoclinicity in a model with bent-slow manifold geometry

Effect of slow manifold structure on relaxation oscillations and one-dimensional map in a model for plastic instability