Research On Mathematical Problems In Material During Compressive Deformation

In this paper, we investigate the plastic dynamics of bulk metal glass during compression by several mathematical methods.We analysis the plastic serrated flow dynamics of bulk metal glass at di?erent strain rate. Using chaotic time analysis of the stress-time curves during serrated flow, the positive Lyapvnov exponent suggests chaos, while the negative Lyapunov exponent means stable. In addition, the statistical analysis suggests self-organized critical behavior at larger strain rate. We characterize the distinct spatiotemporal dynamical regimes and find that the plastic dynamic behavior of bulk metal glass changes from chaotic to selforganized critical behavior with increasing strain rate.Then we analysis the plastic dynamics at di?erent temperature. It find that as temperature decreasing, the plastic serrated flow transits from chaotic to self-organized critical state, which is the same with the trend as the strain rate increasing. To explore the dynamics of the transition zone, we introduce the multifractal analysis, and find at the transition zone, there is larger multifractal spectrum, which proof that at this condition,there are obvious multifractal in the transition zone.Based on the above data analysis, we establish a spatiotemporal dynamic model considering the interaction between multiple shear bands in the plastic deformation of metallic glasses during compression. Microscopic creep events and delocalized sliding events are analyzed based on the established model. We discuss the spatially uniform solutions and traveling wave solution as well as the approximate solution based on multi-scale analysis.The phase space of the spatially uniform system applied in this study reflects the unstable state of the system at a lower strain rate. The system is unstable at smaller parameter perturbation, while the system is stable at larger parameter perturbation based on the multi-scale analysis, which is consistent with the above dynamical analysis. Moreover,numerical simulation shows that the microscopic creep events were manifested at a lower strain rate, whereas the delocalized sliding events are observed at a higher strain rate. At larger strain rate, the statistics based on the numerical simulation manifests power law distribution, which is consistent with the experimental results.At last, we investigate the global characteristics of the serrated flow signal in time scale at low temperature, including the scale free self-similarity and the long-range correlation. Fractal analysis suggests there is larger range of self-similarity at lower temperature,and the fractal dimension increases as temperature decreasing. The detrended fluctuation analysis suggests that there is a stronger negative correlation for the stress-rate signal at lower or higher temperature, while there is weaker negative correlation at intermediate temperature, reflecting the homogeneous shear-branching process, and a degree of heterogeneity at other temperatures. In addition, a stochastic model is given to describe the self-similar shear-branching process during plastic deformation, and in which the Hurst exponent and fractal dimension are independent with each other.