323 ON THE FORMULATION AND ITERATIVE SOLUTION OF SMALL STRAIN PLASTICITY PROBLEMS* BY KERRY S. HAVNER Douglas Aircraft Co., Inc., Santa Monica, Calif. Abstract. This paper is concerned with a general method of formulation and iterative solution of small displacement plasticity problems, using the Hencky-Nadai hardening law as mathematical model for the material behavior. Beginning with a mini- mum energy principle for small thermal-mechanical strains under simple external loading, quasi-linear partial differential equations are formulated and a method of iteration by successive solutions is proposed. A finite-difference discretization of the equations (in two dimensions) is obtained through minimization of the total potential energy function, leading to positive definite symmetric matrices for general boundary configurations. 1. Introduction. The general stress-strain law of the linear incremental theory of small plastic deformations has been shown by Drucker to rest solely upon the funda- mental postulate of material stability [1, 2], the extended stability postulate [3, 4], and the assumption of a smooth (regular) loading surface / in stress space. If the alternate assumption is made that a corner forms on the yield surface at the point of loading, a non-linear incremental theory results. A significant body of experimental evidence has been reported since 1953 indicating the regular appearance of corners (see, for example, the discussions in [5] and [6]), and although there is also contrary evidence which tends to refute the concept of a corner carried with the point of loading, the matter does not appear to be sufficiently resolved to justify the acceptance of linearity and exclusion of non-linearity solely on experi- mental grounds. For the purpose of mathematical stress analysis it would seem partic- ularly appropriate to utilize a non-linear incremental theory should that theory lead to a more feasible method of solution of complex problems. To this end a number of in- vestigators have succeeded in rigorously justifying the relatively simple total stress- strain laws (deformation theory) within a theoretical framework of the fundamental postulate and singular loading surfaces [7-10]. In particular, Budiansky [8] has used the lack of uniqueness in direction of the incremental plastic strain vector at a corner (as given by Drucker's postulate [2]) to establish that the Hencky-Nadai deformation theory can be viewed as an integrable non-linear incremental theory that is mathematically and physically consistent for a range of loading paths including but not limited to propor- tional loading. In differential form the Hencky-Nadai law can be written deft = dAi(J2) Su + A^JY)dsn (1) where Sa = <ru — lakkSii is the deviatoric stress tensor and A,(./2) is an experimentally specified function of the second deviatoric stress invariant. Budiansky's theory of the validity of this law has served as a theoretical basis for several papers [11-13] devoted to the solution of particular boundary value problems. In each of the problems considered *Received October 9, 1964; revised manuscript received April 19, 1965.