Summary: The methods of convex analysis are used to explore in greater depth the nature of the evolution equation in internal variable formulations of elastoplasticity. The evolution equation is considered in a form in which the thermodynamic force belongs to a set defined by a multi-valued map G. It is shown that the maximum plastic work inequality together with the assumption that G is maximal responsive, is necessary and sufficient to give a theory equivalent to that proposed by J. J. Moreau [C. R. Acad. Sci. Paris, Ser. II, 271, 608-611 (1970)]. Further consequences are investigated or elucidated, including the relationship between the yield function and the dissipation function; these functions are polars of each other. Examples are given to illustrate the theory.