Nonlinear analysis techniques for shear band formation at high strain-rates

One of the most striking manifestations of instability in solid mechanics is the localization of shear strain into narrow bands during high speed, plastic deformations of metals. According to one theory, the formation of shear bands is attributed to effective strainsoftening response, which results at high strain rates as the net outcome of the influence of thermal softening on the, normally, strain-hardening response of metals. Our objective is to review some of the insight obtained by applying nonlinear analysis techniques on simple models of nonlinear partial differential equations simulating this scenario for instability. First, we take up a simple system, intended as a paradigm, that describes isothermal shear deformations of a material exhibiting strain softening and strain-rate sensitivity. As it turns out, for moderate amounts of strain softening strain-rate sensitivity exerts a dissipative effect and stabilizes the motion. However, once a threshold is exceeded, the response becomes unstable and shear strain localization occurs. Next, we present extensions of these results to situations where explicit thermal effects are taken into account.