RESUMEN
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 lowerbranch
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.