Material instabilities in rate dependent solids

摘要

A continuum mechanics framework for the analysis of material instabilities in rate-dependent solids based upon a linear stability analysis of the partial differential equations of elastic-viscoplastic flow is presented. The derivation is general, three dimensional, with the effects of rate sensitivity, inertia, and elasticity included. Critical conditions for the formation of two qualitatively different types of material instability are discussed: shear band localization and Hopf bifurcation. The characteristic stability equation is given in explicit form for a general three dimensional state of stress. It is shown, that the elastic part of the constitutive law based on an additive decomposition of the rate of deformation tensor induces a relative rotation of the instability plane, i.e. the plane with the fastest incipient rate of instability growth with the respect to the principal directions of the stress deviator. Comparison with limiting cases of material behavior such as rigidly viscoplastic, and rate-independent plasticity in particular, is given also.