  Shear localization occurs in various instances of material instability in
solid mechanics and is typically associated with Hadamard-instability for an
underlying model. While Hadamard instability indicates the catastrophic growth
of oscillations around a mean state, it does not by itself explain the
formation of coherent structures typically observed in localization. The latter
is a nonlinear effect and its analysis is the main objective of this article.
We consider a model that captures the main mechanisms observed in high
strain-rate deformation of metals, and describes shear motions of temperature
dependent non-Newtonian fluids. For a special dependence of the viscosity on
the temperature, we carry out a linearized stability analysis around a base
state of uniform shearing solutions, and quantitatively assess the effects of
the various mechanisms affecting the problem: thermal softening, momentum
diffusion and thermal diffusion. Then, we turn to the nonlinear model, and
construct localized states - in the form of similarity solutions - that emerge
as coherent structures in the localization process. This justifies a scenario
for localization that is proposed on the basis of asymptotic analysis in
\cite{KT}.
