# Unveiling the Strain Rate Sensitivity of G18 NiCrMo3-6 CAST Steel in Tension/Compression Asymmetry
## 3 Results
### Basic Formulation
Then, the accumulation of porosity level can directly be linked to the non-isochoric portion of strain tensor, as indicated in Equation (11) (this assumption omits the contribution of elastic strains to volume change, but it is still a proper strategy since elastic strains are extremely small compared to plastic counterparts, especially in moderate strain regime). \[\dot{f}=(1-f)\times\dot{\epsilon}_{kk} \tag{11}\]  
Again, the similar convention of Equation (9), \(f\) was defined at the end of the time increment as Equation (12):  
\[\sigma_{eq}=\sqrt{\frac{3}{2}S_{ii}S_{ij}}\,\ \ \sigma_{m}=\frac{\sigma_{kk}}{3} \tag{4}\]0  
By necessary mathematical manipulations, finally Equation (12) converts to Equation (13):  
\[\sigma_{eq}=\sqrt{\frac{3}{2}S_{ii}S_{ij}}\,\ \ \sigma_{m}=\frac{\sigma_{kk}}{3} \tag{4}\]1  
Combining Equation (12), Equations (8) and (13) finally yields Equation (14) for \(f_{n+1}\):  
\[\sigma_{eq}=\sqrt{\frac{3}{2}S_{ii}S_{ij}}\,\ \ \sigma_{m}=\frac{\sigma_{kk}}{3} \tag{4}\]2  
Equation (14) also means that at the end of time increment, when the stress and strain update is conducted, \(f\) can also be determined. Thus, the \(f\) parameter was processed like a state variable. Sticking to elastic predictor-plastic update methodology of computational plasticity, \(\bar{\sigma}_{n+1}\) can be written in an open form as (Equation (15)):  
\[\sigma_{eq}=\sqrt{\frac{3}{2}S_{ii}S_{ij}}\,\ \ \sigma_{m}=\frac{\sigma_{kk}}{3} \tag{4}\]3In Equation (15), G is the shear and \(\kappa\) is the bulk modulus; in addition, _dev_ and _tr_ represent the deviatoric part and trace of the tensors, respectively. Combining Equations (8), (9) and (15) yields (Equation (16)) for \(\sigma_{n+1}\):  
\[ \begin{array}{c}\sigma_{n+1}=2G\bigg{[}\;dev(\varepsilon_{n+1})-dev( \varepsilon_{n}^{p})-\Delta\lambda\;\frac{3}{2}\;\frac{dev(\sigma_{n+1})}{ \left(g_{1}\right)_{n}\bar{\sigma}_{n+1}}\;\bigg{]}+\kappa\;[\;tr(\varepsilon_ {n+1})-\ldots\\ \ldots tr(\varepsilon_{n}^{p})-\frac{1}{3}\;\frac{tr(\sigma_{n+1})\;\Delta \lambda}{\left(g_{2}\right)_{n}\bar{\sigma}_{n+1}}\big{]}\;\mathbf{I}\end{array}  \tag{16}\]  
By rearranging and conducting necessary mathematical manipulations, Equation (16), \(\sigma_{n+1}\) can be decomposed as Equations (17) and (18):  
\[dev(\sigma_{n+1})=\frac{2G\;[\;dev(\varepsilon_{n+1})-dev(\varepsilon_{n}^{p })]}{1+\frac{3G\Delta\lambda}{\left(g_{1}\right)_{n}\bar{\sigma}_{n+1}}} \tag{17}\]  
\[tr(\sigma_{n+1})=\frac{3\kappa\;[\;tr(\varepsilon_{n+1})-tr(\varepsilon_{n}^{p })]}{1+\frac{\kappa\Delta\lambda}{\left(g_{2}\right)_{n}\bar{\sigma}_{n+1}}} \tag{18}\]  
With the help of Equations (3) and (18), the yield locus definition for the plastic increment can be formulated as Equation (19). For the simplicity of formulation, the nominators of Equations (17) and (18) (expressions in blue color) were termed as P and Q, respectively. \[\phi_{n+1}=\sqrt{\frac{3}{2}\;\frac{[dev(\sigma_{n+1})\;:\;dev(\sigma_{n+1}) ]}{\left(g_{1}\right)_{n}}+\frac{1}{9}\;\frac{[tr^{2}(\sigma_{n+1})]}{\left(g _{2}\right)_{n}}} \tag{19}\]  
The basic motivation behind Equation (19) is to formulate the yield locus as a function of incremental plastic multiplier (\(\Delta\lambda\)). One can easily observe that in Equation (19) the first term was composed of all known quantities. In other words, the only unknown in Equation (19) is the incremental plastic multiplier (please note that \(\bar{\sigma}_{n+1}=f(\Delta\lambda)\)).