Shear localization with an Arrhenius flow law

A flow law with Arrhenius dependence on temperature is used to model shear localization and shear-band phenomena in thermoviscoplastic materials. Arrh enius dependence is suggested by microstructural arguments for some high-st rength metals. Whereas this form has been used in numerical studies, this p aper offers the first comprehensive analytical study. Results are presented for the one-dimensional problem governing the unidirectional shearing of a slab. A nonlinear analysis reveals the existence of multiple steady states whose stability is determined. The steady Arrhenius model is discussed and compared to similar models in c ombustion and chemical kinetics. Steady solutions are found to depend on a parameter related to both the stress applied at the boundary and to the com petition between diffusion and heat generation in the problem. Varying this parameter results in an S-shaped response curve (or bifurcation diagram), which is new to the shear-band literature. The response curve is constructed asymptotically and veri ed numerically fo r a steady model in which stress is absent from the ow law but not from the problem. Stability analysis shows that both the lower and upper branches o f the curve are stable. The lower branch corresponds to a low-temperature s teady state similar to those found in earlier studies. The upper branch, no t previously observed, is likely to represent a high-temperature state with fully formed shear bands. Finally, the effects of reinstating full stress dependence are analyzed.