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

首席研究员将展示一种研究应力材料中剪切带形成方法的可行性，该方法涉及使用塑性的Arrhenius定律和高能渐近方法构建纯剪切模型。以前对剪切带形成的数学分析包括对扰动进入剪切带的完整数值研究和均匀剪切解的稳定性分析。这项工作的结果已经显示出与实验数据相当好的一致性，但是对这种潜在的能带形成机制的理解一直缺失。首席研究员将通过建议的建模方法寻求提供这样的理解。在受压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。第一种方法涉及在实际受力材料上进行实验，而第二种方法涉及对控制材料的偏微分方程系统进行数值求解。每种方法都有优点和缺点，但没有一种方法能够对控制能带形成的基本物理机制给出令人满意的解释。首席研究员将采用一种不同的方法，基于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.