Recent Developments in the Mathematical Theory of Plasticity

Abstract

The first part of the paper (Sections I to V) is concerned with the general technique used in the discussion of stress‐strain laws for inviscid elastic‐plastic materials with work hardening. It is postulated that when the ``mechanical state'' of such a material is known a given infinitesimal change of stress produces a uniquely defined infinitesimal change of strain. Once the assumption is made as to which variables determine the mechanical state, the conditions of continuity, uniqueness, irreversibility, and consistency can be used to extract rather far‐reaching information regarding the structure of the stress‐strain law. The procedure is applied to the case where the mechanical state is determined by the components of stress and permanent strain. The second part of the paper (Sections VI to VIII) is concerned with various problems of plastic equilibrium. Structural stability in the plastic range is discussed and the difficulties arising in the formulation of stability problems for nonconservative systems are pointed out. Statically determinate problems in the Saint‐Venant‐Mises theory of plasticity are reviewed with special reference to discontinuous solutions. Finally, the ``shake‐down'' problem is discussed for an elastic‐plastic structure under the action of loads, each of which varies in a random manner between given extreme values.