Viscoplasticity for instabilities due to strain softening and strain-rate softening

Three viscoplastic approaches are examined in this paper. First, the overstress viscoplastic models (i.e. the Perzyna model and the DuvautÐLions model) are outlined. Next, a consistency viscoplastic approach is presented. In the consistency model a rate-dependent yield surface is employed while the standard KuhnÐTucker conditions for loading and unloading remain valid. For this reason, the yield surface can expand and shrink not only by softening or hardening e¤ects, but also by softening/hardening rate e¤ects. A full algorithmic treatment is presented for each of the three models including the derivation of a consistent tangential sti¤ness matrix. Based on a limited numerical experience it seems that the consistency model shows a faster global convergence than the overstress approaches. For softening problems all three approaches have a regularising e¤ect in the sense that the initial-value problem remains well-posed. The width of the shear band is determined by the material parameters and, if present, by the size of an imperfection. A relation between the length scales of the three models is given. Furthermore, it is shown that the consistency model can properly simulate the so-called S-type instabilities, which are associated with the occurrence of travelling Portevin-Le Chatelier bands