Models based on plastic rate equations

The range of material behavior considered next is broadened significantly by appeal to the notion of a plastic rate equation as a model for any possible physical mechanism of deformation that may be operative. The ideas will be developed for general states of stress, but will be applied primarily for the case of thin films in equi-biaxial tension. Constitutive relationships that serve as models for inelastic response of materials for a wide variety of physical mechanisms of deformation have been compiled by Frost and Ashby (1982). These constitutive equations are represented as scalar equations expressing the inelastic equivalent strain rate /3e in terms of the effective stress (Tm/ /3 and temperature T. These strain rate and stress measures are denoted by 7 and as by Frost and Ashby (1982), and the rate equations representing models of material behavior all take the form 

For a general state of stress and deformation at a material point, how are individual components of plastic strain rate related to stress components in this framework An answer is provided through the work of Rice (1970) on the general structure of stress-strain relations for time-dependent plastic deformation. In the present setting, it is most conveniently expressed in terms of deviatoric stress components Sij defined in terms of stress in 

given the state of stress and the temperature, the current rate of plastic straining is specified through (7.79) and (7.80). 

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