Abstract
Asymptotic results are presented for a simplified model for shear-band formation which neglects the effects of diffusion but still captures much of the important dynamics. Regular perturbation expansions fail, so a uniform expansion is constructed which tracks the divergent behavior of the simplified model. Severe computational difficulties exist in the form of finite-time blowup of the temperature and strain rate for the simplified model, but an adaptive numerical scheme tracks the severe blowup behavior well. This method, which is second-order accurate and has automatic mesh- and time-step refinement capabilities, also captures the severe band narrowing and strain rate growth in solutions to the full model with heat conduction. Comparison between the asymptotic and numerical methods shows good agreement, and remarks are made regarding asymptotic solutions of the full model.
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