Deformation Theory of Plasticity
Beyond the linear elastic range 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. Those questions were not resolved until Shanley’s landmark paper of 1947. Two principal approaches have been developed over the years for plastic analysis: (1) incremental theory and (2) deformation theory. Incremental theory, involving step-wise loading and unloading of stress increments, is acknowledged to be the more generally applicable approach, but bears the high price of complexity in application and in concepts of the theory. In contrast, deformation theory, in which only the final stress state is considered, is both simple in concepts and in practical application. However, deformation theory is limited in problems for which it is valid. The principal restriction of deformation theory is to proportional or near-proportional loading, i.e., the various stresses must all increase in approximately the same proportions throughout the loading process. In this book, we explore the concepts of deformation theory including yield criteria and loading concepts for isotropic metals in addition to some of the introductory fundamentals of incremental theory. Then, 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. Finally, a phenomenological or state-variable approach is developed to treat the nonlinear stress-strain behavior of particulate, laminated, and three-dimensional fiber-reinforced composite materials. Moreover, the nonlinear stress-strain behavior can be different under tensile loading than under compressive loading. The developed model is used to analyze buckling of laminated fiber-reinforced plates, uniaxial and biaxial deformation and the thermal-stress disk test for particulate graphite composites, off-axis behavior of laminated fiber-reinforced composite materials, and the off-axis loading of carbon-carbon composite materials (carbon fibers in a carbon matrix). Where possible, theoretical predictions are compared with experimental measurements to assess the validity of the theory.