## 1 Introduction
An often-used alternative to the Levy-Mises formulation is what is commonly referred to as an associative flow rule (since it is associated with a particular yield criterion), or the normality condition, and is given as:  
\[d\epsilon_{ij}^{p}=d\lambda\,\frac{\partial F}{\partial\sigma_{ij}} \tag{6}\]  
However, if we assume that the yield surface is a function of the Cauchy deviator only (as suggested in Equations (3) and (4)), then the normality condition is perhaps more appropriately depicted in terms of the deviator as opposed to the full Cauchy tensor:  
\[de_{ij}^{p}=d\lambda\,\frac{\partial F}{\partial s_{ij}} \tag{7}\]  
where all variables are as already defined. The final component that feeds into the constitutive equations is the hardening rule. It is a rule for the evolution of the yield surface. This evolution is brought about through the effect of certain internal state varibles, such as the drag stress, \(\alpha_{2}\), in the case of isotropic hardening, or the back stress, \(\alpha_{3_{ij}}\), in the case of kinematic hardening, along with whatever internal state variable growth laws (i.e., \(\dot{\alpha}_{ij}^{\eta}=\alpha_{ij}^{\eta}(\epsilon_{kl},T,\alpha_{kl}^{\mu})\)) may be appropriate. In a rather generic form, isotropic and kinematic hardening can be expressed, respectively, by what is given in the two equations that follow:  
\[F=J_{2}-\frac{\alpha_{2}^{2}}{3}=0 \tag{8}\]  
\[F=\frac{1}{2}\left(s_{ij}-\alpha_{3_{ij}}^{\prime}\right)\left(s_{ij}-\alpha_{ 3_{ij}}^{\prime}\right)-\frac{\sigma^{2}}{3}=0 \tag{9}\]  
where the superscripted \(\prime\) denotes deviatoric part. This completes our review of classical plasticity. For a more complex treatment of plasticity--for example, a treatment that includes anisotropy and complications related to material frame indifference--the interested reader may find the publication of [PERSON] et al. [4] to be of benefit. To keep things simple, going forward we shall limit our consideration to isotropic materials undergoing \(J_{2}\) plastic flow in accordance with an associative flow rule and isotropic hardening. Thus, the most pertinent framework for what follows is that represented by Equations (2), (4), (7) and (8). Within that framework, the flow stress relates to the drag stress and to \(J_{2}\), as follows:  
\[\sigma=\frac{\alpha_{2}}{\sqrt{3}}=\sqrt{J_{2}} \tag{10}\]  
In what follows, we shall focus our attention on the hardening model only (i.e., Equation (10)), or more precisely, on some specific forms that the generic Equation (10) may take. Those specific forms include the models of [PERSON] and Cook, [5] (which we shall refer to in \"shorthand\" fashion as the JC model, though the model is not universally referred to in this manner), [PERSON], [PERSON], and Guinan, [6] (which we shall refer to in \"shorthand\" fashion as the SCG model, though the model is not universally referred to in this manner), [PERSON] [7] (the MTS model), and [PERSON] [8] (the PTW model). Our ultimate objective will be the development of a hardening model for \(\gamma\)-phase cerium that is in the style of the SCG model. Cerium has a very complex phase diagram. There are four solid phases at zero pressure and at least three more high pressure phases. Gamma phase, the stable room temperature zero pressure phase, exhibits unusual hydrostatic constitutive behavior, wherein the bulk modulus decreases with increasing pressure. Cerium is unique in that it is the only pure element possessing a solid-solid critical point. One of the more interesting features of cerium is the isomorphic \(\gamma\)-\(\alpha\) solid-solid phase transition (fcc to fcc), which occurs at about 0.75 GPa (precipitated by localization/delocalization mechanisms involving 4f electrons). This transition is accompanied by a substantial volume change of almost 15% and has a profound impact upon the dynamic response of cerium.