Chapter 3: Theory of Plasticity

Unlike elastic behavior where deformation is reversible and governed by Hooke’s Law, plastic deformation is a non-reversible process where the final state of a material depends heavily on its specific loading history rather than just the initial and final states. Central to this study is the flow curve, which tracks the relationship between true stress and true strain, typically represented by a power law where stress equals a strength coefficient multiplied by strain raised to a strain-hardening exponent. The text emphasizes the necessity of using true stress and strain based on instantaneous dimensions—rather than engineering stress and strain based on original dimensions—to accurately reflect the large deformations encountered in metalworking. A fundamental principle discussed is the constancy of volume during plastic flow, which implies that the sum of the three principal true strains is zero, leading to significant mathematical simplifications in plasticity problems. To predict when a material will transition from elastic to plastic behavior under complex, multi-axial loading, the chapter introduces critical yielding criteria, primarily the von Mises (distortion-energy) criterion and the Tresca (maximum-shear-stress) criterion. While the von Mises criterion suggests yielding occurs when the energy involved in changing a material's shape reaches a critical value, the Tresca criterion focuses on the maximum shear stress reaching a limit equal to half the tensile yield stress. These concepts are visualized geometrically through yield surfaces, such as the von Mises cylinder, and the normality rule, which dictates that the plastic strain increment vector must be normal to the yield surface. The chapter also addresses material anisotropy, where properties vary by direction due to manufacturing processes like rolling, and introduces the concept of texture hardening in highly oriented materials. Advanced sections detail the invariants of stress and strain, such as effective stress and effective strain, which provide a way to simplify complex stress states into a single value that can be directly compared to a simple tensile test. Finally, the chapter covers plastic stress-strain relations through the Levy-Mises and Prandtl-Reuss equations and introduces slip-line field theory for solving two-dimensional plane-strain problems, such as the indentation of a thick block, by using Hencky equations to calculate stress variations along geometric slip lines.