# Unveiling the Strain Rate Sensitivity of G18 NiCrMo3-6 CAST Steel in Tension/Compression Asymmetry
## 3 Results  
### Basic Formulation  
The yield locus was defined in the Cocks'89 material model as in Equation (3):  
\[\phi=\underbrace{\sqrt{\frac{\sigma_{eq}^{2}}{\text{g}_{1}(f)}+\frac{\sigma_{m}^{ 2}}{\text{g}_{2}(f)}}}_{\widetilde{\sigma}}-\sigma_{y}=0 \tag{3}\]  
In Equation (3), \(\sigma_{eq}\) is the equivalent von Mises stress and \(\sigma_{m}\) is the mean stress which were explicitly given in Equation (4); moreover, \(\text{g}_{1}\) and \(\text{g}_{2}\) are functions of \(f\) which is nothing but the void volume fraction; and finally \(\phi\) is the yield potential. In particular, \(\widetilde{\sigma}\) is the equivalent stress definition of the Cocks'89 plasticity model:  
\[\sigma_{eq}=\sqrt{\frac{3}{2}S_{ii}S_{ij}}\,\ \ \sigma_{m}=\frac{\sigma_{kk}}{3} \tag{4}\]  
In Equation (4) \(\sigma\) is the (Cauchy) stress tensor and \(S_{ij}\) is the deviatoric part of it (please note that [PERSON]'s summation convention is in place for the formulation). Therefore, \(\sigma_{kk}\) is actually a scalar value and equals to \(I_{1}\) which is the first invariant of the stress tensor (aiming to eliminate any possible confusions, the tensorial expressions were designated by bold character to be easily distinguished from scalar quantities). To clarify the yield locus definition in Cocks'89, the two separate functions of \(f\) were also provided in Equation (5). As expected, when \(f\to 0\), i.e., void volume fraction is zero (conventional metal plasticity approach), the yield locus definition turns out to the von Mises yield criteria \(\left[\sqrt{3}\right]_{2}-\sigma_{y}=0\)]:  
\[\text{g}_{1}=\frac{(1-f)^{2}}{1+\frac{2}{3}f},\ \text{g}_{2}=\frac{2}{9}\frac{ (1+f)(1-f)^{2}}{f} \tag{5}\]  
By assuming the associative flow rule, the plastic strain rate equation can be written as Equation (6). (Similar to the conventional computational plasticity approach, Equation (6) was constructed as a time rate of change form since the analyses are conducted via a time integration scheme). \[\dot{\epsilon_{p}}=\lambda\ \underbrace{\frac{\partial\phi}{\partial\sigma}}_{N_{p}} \tag{6}\]  
In Equation (6), \(\epsilon_{p}\) is the plastic strain tensor, \(\lambda\) is the plastic multiplier and \(N_{p}\) is the unit normal vector of the plastic flow, while the dot operator denotes the time derivative of  
Figure 5: SRS parameter for tension and compression modes. the tensors (unit normal vector was shown via a capital letter to eliminate the confusion with \(n\) and \(n+1\) subscripts of time integration). By taking the partial derivative of Equation (4), reads as (Equation (7)):  
\[\mathbf{N}^{p}=\left(\frac{3}{2}\frac{\mathbf{S_{ij}}}{\left(\mathsf{g_{1}}\,\bar{ \sigma}\right)}+\frac{1}{9}\frac{\sigma_{kk}}{\left(\mathsf{g_{2}}\,\bar{ \sigma}\right)}\mathbf{\mathrm{I}}\right) \tag{7}\]  
In this study, a [PERSON] backward time integration scheme was used, then \(\mathbf{N}^{p}\) calculation (Equation (8)) dictates under Equation (7), where \(\mathbf{\mathrm{I}}\) is the second order identity tensor:  
\[\mathbf{N}^{p}_{n+1}=\left(\frac{3}{2}\,\frac{\left(\mathsf{S_{ij}}\right)_{n+1}}{ \left(\mathsf{g_{1}}\,\bar{\sigma}_{n+1}\right)}+\frac{1}{9}\,\frac{\left( \sigma_{kk}\right)_{n+1}}{\left(\mathsf{g_{2}}\,\bar{\sigma}_{n+1}\right)}\mathbf{ \mathrm{I}}\right) \tag{8}\]  
By combining Equations (6) and (8), one can obtain the plastic strain tensor at the end of time increment as (Equation (9)):  
\[\varepsilon^{p}_{n+1}=\underbrace{\Delta\lambda}_{\lambda dt}\mathbf{N}^{p}_{n+1} \tag{9}\]  
Equation (9) can also be written in a scalar form as Equation (10), where \(\bar{\varepsilon}^{\ p}\) denotes equivalent plastic strain:  
\[\bar{\varepsilon}^{\ p}_{n+1}=\Delta\lambda\ \mathbf{N}^{p}_{n+1}+\bar{\varepsilon}^{ \ p}_{n} \tag{10}\]  
On the other hand, with the assumption of incompressible plastic flow of the defect-free matrix material (without any porosity), the only contribution to volume change is the change in porosity level.