Comments on loss of strong ellipticity in elastoplasticity

The loss of strong ellipticity is analyzed for a rate independent in®nitesimal elastoplastic model. This local stability condition corresponds to the loss of positive de®niteness of the symmetric part of the acoustic tensor. First, in the case of multisurface plasticity some expressions for the plastic hardening moduli are obtained for various bifurcation criteria. Next, explicit expressions for the critical plastic hardening modulus and the critical orientation are obtained in the case of single-surface plasticity (Hill type comparison solid). The analysis is based on a geometric method. Linear, isotropic elasticity, and a general nonassociative ¯ow rule are assumed. However, the principal axes of the second order tensors of the plastic potential and yield surfaces gradients are coaxial. It is shown that, similar to the loss of ellipticity, the direction of the critical orientation is identical to one of the principal directions, except in the particular case where the gradient of the plastic potential and yield surfaces each have a double eigenvalue. In particular, explicit expression for the plastic hardening modulus, using the same geometric method, is also presented for the Raniecki type comparison solid. As an illustrative example, the critical orientation for the loss of strong ellipticity and the classical shear band localization (loss of ellipticity) are compared for axially-symmetric compression and tension