Plasticity Theory

Plasticity theory provides the framework for analyzing materials stressed beyond their elastic limit. The transition from elastic to plastic behavior is governed by a yield criterion, which defines a surface in stress space (the yield surface). For stress states inside this surface, the material behaves elastically. When the stress state reaches the surface, plastic deformation begins. Two of the most common yield criteria for metals are the Tresca (maximum shear stress) and von Mises (maximum distortion energy) criteria.

Once yielding occurs, the evolution of plastic strain is described by a flow rule. The most common is the associated flow rule, which states that the increment of plastic strain occurs in a direction normal to the yield surface. This determines the relative proportions of plastic strain components.

Finally, a hardening rule describes how the yield surface changes as plastic deformation accumulates. Isotropic hardening assumes the yield surface expands uniformly in all directions, meaning the material’s yield strength increases equally regardless of the loading direction. Kinematic hardening, on the other hand, assumes the yield surface translates in stress space without changing its size, which is useful for modeling the Bauschinger effect observed in cyclic loading. More complex models combine both isotropic and kinematic hardening to accurately capture real material behavior under complex loading paths. These three components—yield criterion, flow rule, and hardening rule—form the core of classical plasticity theory.