## 2 Hardening Models
### Ptw  
As with the model of Johnson and Cook, the PTW model is a simple algebraic history independent model within which the flow stress depends upon the current state of the equivalent plastic strain, equivalent plastic strain rate, and temperature: \(\sigma=\sigma(\epsilon,\dot{\epsilon},T)\). That being said, the PTW model represents a significant advancement in capability in that it is applicable over a much wider range of strain rate. This expansion in strain rate applicability is accomplished by \"pasting together\" what is essentially multiple models, each of which is applicable to a particular strain rate regime. For strain rates up to about \(10^{4}\) s\({}^{-1}\), the primary mechanism for plastic flow is thermally activated dislocation glide. For strain rates on the order of \(10^{9}\) to \(10^{12}\), occurring in what is sometimes referred to as the overdriven shock regime, the predominant mechanism for plastic flow is no longer thermally activated dislocation glide, but dislocation drag. At intermediate strain rates (e.g., \(10^{5}\) to \(10^{9}\)) phonon friction may also be decisive. Sub-models within PTW are developed for each of these two strain rate regimes. The gap between strain rates of \(10^{4}\) and \(10^{9}\) is bridged through simple extrapolation. The PTW model incorporates a model for the shear modulus that is potentially temperature and density dependent. It also incorporates an explicit model for the saturation stress (maxima on a stress-strain curve). The model is structured in such a way that many of its intermediate steps (and many of its input parameters) are dimensionless. The model shall be presented here in building block fashion involving several sub-models that feed into the flow stress. The first building block is the sub-model for the shear modulus, given as:  
\[G=G_{0}\left(1-\alpha\widehat{T}\right) \tag{14}\]  
where \(G\) is the shear modulus, \(\widehat{T}\) is a dimensionless temperature variable defined as: \(\widehat{T}=T/T_{m}\) (where \(T_{m}\) is the melt temperature), \(G_{0}\) is the zero temperature shear modulus, and \(\alpha\) is an input parameter. The second building block is \(\dot{\xi}\), defined as:  
\[\dot{\xi}=\frac{1}{2}\left(\frac{4\pi\rho}{3M}\right)^{1/3}\sqrt{\frac{G}{\rho}} \tag{15}\]  
where \(\dot{\xi}\) is the reciprocal of the time required for a shear wave to traverse an atom (units are s\({}^{-1}\)), \(\rho\) is the density, and \(M\) is the atomic mass (mass of a single atom). The third building block is the sub-model for \(\widehat{\tau}_{s}\), given as:  
\[\widehat{\tau}_{s}=\max\left\{s_{0}-(s_{0}-s_{\infty})\;\text{erf}\left[\kappa \widehat{T}\,\ln\left(\frac{\gamma\dot{\xi}}{\dot{\epsilon}}\right)\right],\,s _{0}\left(\frac{\dot{\epsilon}}{\gamma\dot{\xi}}\right)^{\beta}\right\} \tag{16}\]  
where \(\widehat{\tau}_{s}\) is a dimensionless saturation stress, and \(s_{0}\), \(s_{\infty}\), \(\kappa\), \(\gamma\), and \(\beta\) are dimensionless parameters. The first segment of Equation (16) (\(s_{0}-(s_{0}-s_{\infty})\) erf\(\left[\kappa\widehat{T}\,\ln\left(\frac{\gamma\dot{\xi}}{\dot{\epsilon}} \right)\right]\)) relates to plastic flow resultant from thermally activated dislocation glide. The second segment of Equation (16) (\(s_{0}\left(\frac{\dot{\epsilon}}{\gamma\dot{\xi}}\right)^{\beta}\)) relates to plastic flow occurring in an overdriven shock regime where dislocation drag is the pre-eminate mechanism for plastic flow. The fourth building block is the sub-model for \(\widehat{\tau}_{y}\), given as:  
\[\widehat{\tau}_{y}=\max\left\{y_{0}-(y_{0}-y_{\infty})\;\text{erf}\left[\kappa \widehat{T}\,\ln\left(\frac{\gamma\dot{\xi}}{\dot{\epsilon}}\right)\right],\, \min\left[y_{1}\left(\frac{\dot{\epsilon}}{\gamma\dot{\xi}}\right)^{y_{2}}\,s _{0}\left(\frac{\dot{\epsilon}}{\gamma\dot{\xi}}\right)^{\beta}\right]\right\} \tag{17}\]  
where \(\widehat{\tau}_{y}\) is a dimensionless dynamic yield stress, and \(y_{0}\), and \(y_{\infty}\) are dimensionless parameters.