## 1 Introduction
Taking this approach, as is most common in the field of plasticity resultant from intensive dynamic loading, the constitutive equations are typically formulated, not in terms of the deviator directly, but in terms of a deviatoric stress rate, \(\dot{s}_{ij}\), and may be expressed as follows:  
\[\dot{s}_{ij}=C_{ijkl}\left(\eta_{kl}-\eta_{kl}^{p}\right) \tag{2}\]  
where \(s_{ij}\) is the deviatoric stress, \(C_{ijkl}\) is, as above, a fourth order tensor of material properties, \(\eta_{kl}\) denotes the deviatoric part of the rate of deformation tensor (deviatoric natural strain rate) [1], and the superscript \(p\) denotes the plastic part of the rate of deformation tensor. Note that, for isotropic systems, \(C_{ijkl}\) simplifies to \(2G\), where \(G\) is the shear modulus. The deviatoric stress rate is typically cast in terms of a Jaumann stress rate or, alternatively, a Green-Nagdi stress rate [2]. Note that the choice of formulating the constitutive equations in rate form is natural since the governing field equations, as mentioned earlier, are most often expressed in rate form, and must, consequently, be solved in an incremental fashion over time with an update of the the stress-strain relationship occurring on each time step. Whatever form the constitutive equations take, they should be developed in accordance with four fundamental principles: (1) the principle of local action (if appropriate), (2) the principle of equipresence, (3) the principle of determinism (assures uniqueness), and (4) the principle of frame indifference [3]. Please note that, although not shown explicitly in Equations (1) and (2), \(\sigma_{ij}\) and \(s_{ij}\) are functions of temperature as well as whatever measure of strain is chosen as pertinent. Let us now briefly address the component parts that feed into the constitutive equations discussed above. The yield criterion is typically formulated in terms of strain, temperature, and possibly some number of internal state variables that evolve over time (i.e., \(F=F(\epsilon_{kl},T,\alpha_{kl}^{u})\)). Options for the precise form of the yield criterion are several. One possibility is to express yield in a stress formulation involving the deviator, for example, as follows:  
\[F=\frac{1}{2}\alpha_{ijkl}\,s_{ij}\,s_{ij}-\sigma^{2}=0 \tag{3}\]  
where \(F\) represents the yield surface, \(\alpha_{ijkl}\) is a fourth order tensor of coefficients that define the shape of the yield surface, and \(\sigma\) is the flow stress, which incorporates hardening and softening effects. Note that the yield surface, as expressed here, is a function of deviatoric stress only (as opposed to the full Cauchy tensor) and since \(s_{kk}=0\), this is a five dimensional space. Note also that von Mises \(J_{2}\) plasticity [2] exists as a simplified subset of Equation (3) and is expressed as follows:  
\[F=\frac{1}{2}\,s_{ij}\,s_{ij}-\sigma^{2}=0 \tag{4}\]  
where \(J_{2}\equiv\frac{1}{2}\,s_{ij}\,s_{ij}\). As was the case with the yield criterion, the flow rule, which is a rule for the evolution of plastic strain and can be thought of as an internal state growth law, can be formulated in terms of strain, temperature, and perhaps some number of internal state variables (i.e., \(\dot{e}_{ij}^{p}=\epsilon_{ij}^{p}(\epsilon_{kl},T,\alpha_{kl}^{\mu})\)). However, it is most often given in a stress formulation, as in the Levy-Mises flow rule (which is in fact a special case of the more general Prandtl-Reuss flow rule) and is given as:  
\[de_{ij}^{p}=d\lambda\,s_{ij} \tag{5}\]  
where \(\lambda\) is a history-dependent scalar and all other variables are as already defined.