Stresses and Strains

The theory of plasticity is the branch of mechanics that deals with the calculation of stresses and strains in a body, made of ductile material, permanently deformed by a set of applied forces. The theory is based on certain experimental observations on the macroscopic behavior of metals in uniform states of combined stresses. The observed results are then idealized into a mathematical formulation to describe the behavior of metals under complex stresses. Unlike elastic solids, in which the state of strain depends only on the final state of stress, the deformation that occurs in a plastic solid is determined by the complete history of the loading. The plasticity problem is, therefore, essentially incremental in nature, the final distortion of the solid being obtained as the sum total of the incremental distortions following the strain path. A metal may be regarded as macroscopically homogeneous and isotropic when the small crystal grains forming the aggregate are distributed with random orientations. As a result of plastic deformation, the crystallographic directions gradually rotate toward a common axis, producing a preferred orientation. An initially isotropic material thereby becomes anisotropic, and its mechanical properties vary with direction. The development of anisotropy with progressive cold work and the resulting strain-hardening are too complex to be successfully incorporated in the theoretical framework. In the mathematical theory of plasticity, it is generally assumed that the material remains isotropic throughout the deformation irrespective of the degree of cold work. Since the strain-hardening characteristic of a metal in a complex state of stress can be related to that in uniaxial tension or compression, it is necessary to examine the uniaxial stress–strain behavior in some detail before considering the general theory of plasticity.