Mathematical Sciences: Investigation of Shear-Band FormationUsing High Activation Energy Asymptotics

主要研究人员将演示一种研究应力材料中剪切带形成的方法的可行性，该方法包括使用塑性阿累尼乌斯定律和高能渐近方法建立纯剪切模型。以前对剪切带形成的数学分析包括对扰动发展到剪切带的完整的数值研究和对均匀剪切解的稳定性分析。这项工作的结果与实验数据显示出了相当好的一致性，但对这一带形成的潜在机制还缺乏了解。首席调查员将试图通过拟议的建模方法提供这种理解。在应力材料中剪切带的形成过程中，有许多物理过程在起作用。到目前为止，应用数学家和工程师主要依靠两种方法来理解这些过程。第一种方法涉及对实际受力材料进行实验，而第二种方法涉及用数值方法求解控制材料的偏微分方程组。每种方法都有优缺点，但都不能对支配带形成的基本物理机制给出令人满意的解释。首席研究人员将采用一种不同的方法，其基础是用一组更简单的方程对材料进行建模，并借助某些渐近方法提取更多有用的信息。

The principal investigator will demonstrate the plausibility of an approach to investigating the formation of shear bands in stressed materials that involves the construction of a model of pure shear using the Arrhenius Law for plasticity and the method of high energy asymptotics. Previous mathematical analysis of shear-band formation has included a complete numerical investigation of the growth of perturbations into shear bands and a stability analysis of the homogeneous-shear solution. The results of this work have shown reasonably good agreement with experimental data, but an understanding of this underlying mechanism of band formation has been missing. The principal investigator will seek to provide such an understanding by means of the proposed modelling approach. Many physical processes are at work during the formation of shear bands in stressed materials. Up to now applied mathematicians and engineers have relied largely on two approaches to understanding these processes. The first approach involves performing experiments on actual stressed materials, while the second approach involves solving numerically the systems of partial differential equations that govern the material. Each approach has advantages and disadvantages, but neither one is able to give a satisfactory explanation of the basic physical mechanisms that govern band formation. The principal investigator will employ a different approach based upon modelling the materials with a simpler set of equations and extracting more useful information with the aid of certain asymptotic methods.