A constitutive inequality in plasticity involving left-hand and right-hand rates

An incremental inequality for elastic–plastic materials involving right-hand (forward) and left-hand (backward) stress and strain rates is presented. This inequality is demonstrated as a theorem for standard elastic–plastic materials having a symmetric hardening matrix and also for generalized standard materials [J. Mec. 14 (1975) 39]. It imposes some restrictions on the stress and strain rate discontinuities of these materials. Some applications of this inequality to deformation models of crystals and to micro-macro models are given. It is shown that this inequality generalizes some constitutive inequalities in plasticity and in particular the bi-incremental form of the normality flow rule [J. Mech. Phys. Solids 15 (1967) 79; J. Mech. Phys. Solids 16 (1968) 315]. It includes and generalizes also some results of the [J. Mech. Phys. Solids 37 (1989) 265] inequality. Finally, it is shown that this inequality can be adopted as a postulate for elastic-plastic materials and that, in this case, it synthesizes the postulates of symmetry of the elastic compliant tensor, normality flow rule and, for models containing a hardening matrix and normal (independent) internal variables, the local symmetry of this matrix.