On the nature of the response functions in rate-independent plasticity

The consequences of a work inequality used by Naghdi and Trapp [1, Q. J. Mech Appl. Math. 28, 25 (1975).] for rate independent elastic-plastic continua are examined. It is shown that the following two conditions are necessary and sufficient for the satisfaction of the work inequality: (i) the yield function ina certain space (which is referred to here as the Sigma space) is convex and further the yield,function in the stress and strain spaces can be derived from it; (ii) the rate of change of the variables which represent the inelastic response of the material is proportional to the gradient of this yield function. The introduction of the new Sigma space is a natural result of studying the rate of energy dissipation during the inelastic processes. The main impact of this work is the fact that it reduces the specification of the constitutive equations for rate independent elastic-plastic materials to two scalar valued functions-the strain energy function psi and a scalar valued function <(phi)over cap> from which the yield function g in the strain space, as well as the rate of change of the variables which represent the inelastic response of the material, can be determined; The results of this paper,which are valid for a very wide class of elastic-plastic materials, extends the work of Naghdi and Trapp, [2 J. Appl. Mech. 42, 61 (1975).] and Casey and Naghdi [3, Q. J. Mech. Appl. Math. 37, 231 (1984).] who showed that the work inequality implies the convexity of the yield surface in the stress space for a special class of constitutive equations (i.e. those for which the stress response depends only on the difference between the Green-St Venant strain and the ''plastic'' strain). Conditions under which the. elastic region in strain space is convex are also elucidated. Copyright (C) 1996 Elsevier Science Ltd.