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.
