A flow law with Arrhenius dependence on temperature is used to model shear localization and shear-band phenomena in thermoviscoplastic materials. Arrhenius dependence is suggested by microstructural arguments for some high-strength metals. Whereas this form has been used in numerical studies, this paper 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 combustion and chemical kinetics. 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 (or bifurcation diagram), which is new to the shear-band literature, The response curve is constructed asymptotically and verified numerically for a steady model in which stress is absent from the flow law but not from the problem. Stability analysis shows that both the lower and upper branches of the curve are 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. Finally, the effects of reinstating full stress dependence are analyzed.
ASJC Scopus subject areas
- Applied Mathematics