Sarmah Ritupan, Ananthakrishna G
Materials Research Centre, Indian Institute of Science, Bangalore 560012, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 May;87(5):052907. doi: 10.1103/PhysRevE.87.052907. Epub 2013 May 10.
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.
这项工作是我们为量化在恒应变率变形条件下观察到的Portevin-Le Chatelier不稳定性的Ananthakrishna模型的不规则标量应力信号所做努力的延续。与位错活动的空间平均值相关的应力是一个动态变量，它也决定了位错密度的时间演化。我们对以在整个不稳定性域中看到的不同类型位错带形式实现的模型的时空模式的性质进行了详细研究，并建立了它们与应力锯齿性质的联系。然后，我们通过计算作为驱动参数函数的Lyapunov维数来表征模型方程的时空动力学。后者仅在低应变率下与系统大小成比例，此时可以看到孤立的位错带，而在高应变率下，可以看到完全传播的带。在对应于部分传播带的中间施加应变率下，Lyapunov维数呈现出两个不同的斜率，一个对应于小系统大小，另一个对应于大系统大小。通过证明小系统大小的时空模式从部分传播带类型转变为孤立爆发类型，这一特征得到了合理的解释。这反过来又使我们能够再次确认，只要位错带的爆发与应力下降之间存在一一对应关系，低维混沌就会从应力信号中投射出来。然后，我们通过计算关联维数密度表明，在部分到完全传播带的区域中的应力信号具有广泛混沌的特征。我们还表明，关联维数密度也取决于系统大小。讨论了一些与Lyapunov维数密度和关联维数密度的系统大小依赖性相关的问题。
