  We study localization occurring during high speed shear deformations of
metals leading to the formation of shear bands. The localization instability
results from the competition among Hadamard instability (caused by softening
response) and the stabilizing effects of strain-rate hardening. We consider a
hyperbolic-parabolic system that expresses the above mechanism and construct
self-similar solutions of localizing type that arise as the outcome of the
above competition. The existence of self-similar solutions is turned, via a
series of transformations, into a problem of constructing a heteroclinic orbit
for an induced dynamical system. The dynamical system is four dimensional but
has a fast-slow structure with respect to a small parameter capturing the
strength of strain-rate hardening. Geometric singular perturbation theory is
applied to construct the heteroclinic orbit as a transversal intersection of
two invariant manifolds in the phase space.
