Beyond the linear elastic range of stress-strain behavior lies nonlinear stress-strain behavior otherwise known as plastic behavior or plasticity. Concern for nonlinear stress-strain behavior apparently surfaced in the mid-1800s with questions about how the bar buckling load is properly analyzed for materials that are not linear elastic. Those questions were not resolved until Shanley’s landmark paper of 1947.

Two principal approaches were developed over the years for plastic deformation analysis: (1) incremental theory and (2) deformation theory. Incremental theory, involving stepwise loading and unloading of stress increments, is acknowledged to be the more generally applicable approach, but bears the high price of complexity both in concepts of the theory and in application. In contrast, deformation theory, in which only the final stress state is considered and not the loading path to that final state, is simple in practical applications and in theoretical concepts. However, deformation theory is limited in problems for which it is valid. The principal limitation of deformation theory is to proportional or near-proportional loading, i.e., the various stresses must all increase in approximately the same proportions dur-ing the loading process.

Concepts of deformation theory are explored including yield criteria and loading concepts for isotropic metals in addition to some of the introductory fundamentals of incremental theory to give some additional contrast between the two theo-ries. Deformation theory is applied to problems of beam bending; a hollow sphere under external pressure; plastic buckling of bars, plates, and shells; and thermal plastic buckling of bars and plates with temperature-dependent, nonlinear material properties. A phenomenological or state-variable approach is developed to treat nonlinear stress-strain behavior of particulate, laminated, and three-dimensional fiber-reinforced composite materials. Moreover, the nonlinear stress-strain behav-ior can be different under tensile loading than under compressive loading. The model is used to analyze buckling of laminated fiber-reinforced plates; uniaxial and biaxial deformation and the thermal stress disk test for particulate graphite com-posite materials; off-axis behavior of laminated fiber-reinforced composite mate-rials; and the off-axis loading of carbon-carbon composite materials (carbon fibers in a carbon matrix). Where possible, theoretical predictions are compared to expe-rimental measurements to assess the validity of the theory. The comparisons are quite favorable.