Localization emerging from competition of rate dependence and Hadamard instability

Abstract eng:
Materials deformed under high strain rates can exhibit failure through formation of shear bands. Shear band formation is often associated with Hadamard instability and strain localization into an emerging coherent structure. We assess the contributions of various effects leading to localization in the simplest model for which the stress exhibits strain softening and strain rate sensitivity. A linearized analysis indicates that the combined effect leads to Turing instability. Then we consider the nonlinear problem and introduce a set of selfsimilar variables. A class of special solutions is obtained which pinpoints the emergence of a coherent localized structure and the formation of a shear band. This solution is associated to a heteroclinic orbit of a certain dynamical system. The orbit is constructed numerically and yields almost explicit shear localizing solutions.

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