| A flow law with Arrhenius dependence on temperature is used to realistically model shear localization and possible shear-band phenomena. Although Arrhenius dependence is suggested by microstructural arguments and used in numerical studies, this dissertation offers the first comprehensive analytical study. Results are presented for the one-dimensional problem governing the unidirectional shearing of a slab. A nonlinear analysis shows that multiple stable steady states are obtained with Arrhenius flow.;The full Arrhenius model is first discussed and compared to similar models in combustion and chemical kinetics. A steady problem, formulated as both a boundary-value problem and as an integral equation, then becomes the focus of the study. Steady solutions are found to depend on a parameter related to both the stress applied at the boundary and to the competition between diffusion and heat generation in the problem. Varying this parameter results in an S-shaped response curve. This curve can be interpreted as a bifurcation diagram in which each branch represents a steady state for the slab.;The response curve is constructed asymptotically and verified numerically for a model problem in which stress is absent from the flow law. The effects of reinstating stress dependence are then analyzed. Finally, stability is determined asymptotically and both the lower and upper branches of the curve are found to be stable. The lower branch corresponds to a low-temperature steady state similar to those found in earlier studies. The upper branch, not previously observed, is likely to represent a high-temperature state with fully formed shear-bands. |
