## 1 Introduction  
The continuum-level thermo-mechanical problem of predicting the behavior of ductile metallic components subjected to intensive dynamic loading and undergoing plastic flow is of central importance in a number of applications. Consequently, this is a field of study that has received significant attention over at least the last 170 years. However, even with such sustained attention, the problem remains one that is not fully resolved. One reason for this is that closed form solutions of the initial boundary value problem (IBVP) are almost always untenable, and numerical approaches are inherently limited by approximations that the modeler is forced make. Numerical solution of the IBVP requires the simultaneous solution of a coupled set of governing field equations (typically: conservation of mass, conservation of linear momentum, and conservation of energy), along with an appropriate set of initial and boundary conditions. Note that within the context of plasticity, resultant from intensive dynamic loading, the field equations are typically formulated in rate form. That being the case, they must be solved in incremental fashion as time goes on. Together, the governing field equations form a hyperbolic set, for which a numerical solution may be attempted using Lagrangian, Eulerian, or Augmented Lagrangian-Eulerian (ALE) approaches employing either explicit or implicit numerical time integration schemes depending on the specifics of the IBVP and preferences of the modeler. Numerical approaches include the finite element method and the finite volume method, among others. If all one has to work with are the governing field equations plus initial and boundary conditions, the IBVP is inherently underconstrained and as such is ill-posed. This is because there are more unknown variables than equations to constrain them. It is through certain constitutive equations that the IBVP attains closure and is well-posed, and the solution of the same becomes tenable. There are a number of constitutive relations that may be critical in this regard, but the focus of this work is on only one--namely that related to the deviatoric Cauchy stress, and it is within the context of classical plasticity that it is herein addressed. By way of review, classical plasticity consists of the following fundamental parts: (1) a stress-strain relationship (the constitutive equations that are required for closure of the IBVP), and (2) several component parts that feed into the constitutive equations. Those component parts include a yield criterion, a flow rule, and a hardening rule. Note that the hardening rule includes, or should include, mechanisms for softening as well as hardening, so it should perhaps be referred to as a hardening/softening rule, but we shall, out of adherence to convention, refer to it herein as a hardening rule with the understanding that mechanisms for softening are included. The stress-strain equations can be formulated in several different ways. One possibility is an algebraic relationship between the Cauchy stress and Eulerian strain:  
\[\sigma_{ij}=C_{ijkl}\left(\epsilon_{kl}-\epsilon_{kl}^{p}\right) \tag{1}\]  
where \(\sigma_{ij}\) is the Cauchy stress, \(C_{ijkl}\) is a fourth order tensor of material properties, \(\epsilon_{kl}\) is the Eulerian strain, and the superscript \(p\) is used to denote the plastic part of the strain tensor. Another possibility is to separate the Cauchy tensor into deviatoric and spherical parts and to focus our plasticity-related attention exclusively on the deviator since it is within the deviator that plastic flow characteristics normally manifest. In this case, the spherical part of the Cauchy tensor is determined separately--for example, through something like an equation of state (constitutive relationship for the spherical part of the Cauchy tensor).