Extremum properties of finite-step solutions in elastoplasticity with nonlinear mixed hardening

Abstract

The class of elastic-plastic constitutive laws assumed herein can be described as follows. We envisage y ⩾ 1 yield criteria (or “modes”) which define the current yield surface; each yield function φα consists of two addends: an effective stress Φα and a yield limit Yα (α = 1,…, y). The former is an order-one, positively homogeneous function of the difference between the stresses σii and reference stresses αii, which are generally nonlinear functional of the plastic strain history, so that kinematic hardening is accounted for. The yield limit Yα is generally a nonlinear function of y nondecreasing internal variables λβ, so that another hardening mechanism (which may reduce to isotropic hardening) is accommodated in the model. The rates of these variables play the role of plastic multipliers in the flow rule.A finite step in the geometrically linear evolutive analysis of such solids is defined according to the backward-difference (or “stepwise holonomic”) strategy for approximate time integration. For the “finite-step” boundary value problem thus arising, various extremum characterizations of solutions are established and the underlying constitutive restrictions are pointed out and discussed.