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

主要研究者将证明 的方法来研究剪切带的形成， 强调的材料，涉及到一个模型的建设， 纯剪切塑性阿耳忒弥斯定律和方法 高能量的渐近性。 以前的数学分析 剪切带的形成包括一个完整的数值 扰动增长到剪切带的研究， 剪切解的稳定性分析。 的 这项工作的结果表明，相当好的协议， 实验数据，但对这一潜在的理解 带形成的机制一直缺失。 校长 调查人员将设法通过以下方式提供这种理解 提出的建模方法。 许多物理过程在形成过程中起作用。 应力材料中的剪切带 到目前为止已应用 数学家和工程师主要依赖于两个 理解这些过程的方法。 第一种方法 包括在实际受压材料上进行实验， 而第二种方法涉及数值求解 偏微分方程组，控制 材料 每种方法都有优点和缺点，但是 没有一个人能够对这一问题作出令人满意的解释。 控制能带形成的基本物理机制。 的 主要研究者将采用不同的方法， 在用一组更简单的方程对材料进行建模时， 在某些帮助下提取更多有用的信息 渐近方法

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.