Continuing equilibrium assumption over-restricts bifurcation condition in the classical localization theory

The assumption of continuing equilibrium plays a fundamental role in the traditional bifurcation formulation. Within the scope of time-independent plasticity, continuing equilibrium requires that equilibrium be satisfied beyond the bifurcation point as well as at the bifurcation point for all permissible paths. It automatically rules out the possibility that a non-equilibrium path may be followed immediately after the bifurcation. Continuing equilibrium condition leads to often unrealistic predictions by the classical shear band theory. As a natural outcome of relaxation of the continuing equilibrium condition, this paper introduces a stress-gradient into the basic formulation of localization problem, conceptually different from the commonly employed strain gradient approach which stays within the classical localization framework but introduces spatial scales through the strain gradient in the constitutive relations (non-local plasticity theory). Without assuming vertices on the current yield surface or unrealistically large material imperfections, the new theory shows that it is at least kinematically and kinetically permissible for localization to occur in the hardening regime.