## 2 Equipartition Shear Zone Model of Orthogonal Cutting Based on [PERSON]'s Theory
### Shear Zone Analysis
Therefore, in order to consider the viscous behavior of this material, in the Oxley theoretical model, the S32760 two-phase constitutive model considering the viscous effect should be used to predict the flow stress in the shear zone. The MTS constitutive model is improved by also considering the viscous effect. The equation is divided into three parts as follows:  
\[\sigma=\sigma_{a}+f(\dot{\epsilon},T)\sigma_{th}+\sigma_{c} \tag{16}\]  
where \(\sigma_{a}\) is the non-thermal stress value; \(f(\dot{\epsilon},T)\) is the influence factor of the reaction strain rate and the temperature effect; \(\sigma_{th}\) is the thermal stress value; and \(\sigma_{c}\) is the stickiness effect. Due to the dual-phase property of the material, the three stresses of the ferrite and austenite phases need to be calculated separately, and the weighted sum is given by  
\[\sigma=l_{1}\sigma_{1}+l_{2}\sigma_{2} \tag{17}\]  
where \(l_{1}\) is the proportion of the ferrite phase; \(l_{2}\) is the proportion of the austenite phase; \(\sigma_{1}\) is the stress in the ferrite phase; and \(\sigma_{2}\) is the stress in the austenite phase. The non-thermal stress value is independent of the effect of strain rate and temperature change and is only affected by changes in strain. Since the influence of the strain rate and temperature on the ferrite's work-hardening behavior can be neglected, the flow stress increased by the work-hardening behavior of ferrite is included in the ferrite's non-thermal stress value. The non-thermal stress values in the constitutive equations of ferrite and austenite can be written as follows:  
\[\sigma_{a1}=\sigma_{i1}+m_{1}d_{1}^{-\frac{1}{2}}+K_{1}\epsilon^{n_{1}} \tag{18}\]  
\[\sigma_{a2}=\sigma_{i2}+m_{2}d_{2}^{-\frac{1}{2}} \tag{19}\]  
where \(\sigma_{i1}\) and \(\sigma_{i2}\) are the total stress of dislocation movement in ferrite and austenite, respectively; \(m_{1}\) and \(m_{2}\) are constants representing grain boundary strength of ferrite and austenite, respectively; \(d_{1}\) and \(d_{2}\) are the grain size of ferrite and austenite, respectively; \(K_{1}\) is the ferrite hardening coefficient; \(\epsilon\) is the true strain; \(n_{1}\) is the ferrite strain sensitivity index. The expression of the influence factor is given as follows [21]:  
\[f(\dot{\epsilon},T)=\left\{1-\left[-\frac{kT}{G_{0}}\ln(\frac{\dot{\epsilon}} {\dot{\epsilon}_{0}})\right]^{\frac{1}{6}}\right\}^{\frac{1}{6}} \tag{20}\]  
where \(k\) is the Boltzmann constant; \(T\) is the temperature; \(G_{0}\) is the reference thermal activation energy; \(\dot{\epsilon}_{0}\) is the reference strain rate; and \(p,q\) are the barrier constants. Since the thermal stress of the body-centered cubic structure is independent of the strain, the thermal stress value \(\sigma_{th1}\) of the ferrite is equal to the ferrite saturation stress value. Therefore, following [21], we have  
\[\sigma_{th1}=\hat{v}_{51}e^{\frac{kT}{b^{3}E_{40}}\ln\frac{\dot{\varepsilon}}{ \hat{v}_{40}}} \tag{21}\]  
where \(\hat{v}_{51}\) is the reference value of ferrite saturation threshold stress at \(T=0\) K; \(b\) is the Burgers vector of dislocation; \(E\) is the shear modulus of the material; and \(a_{0}\) is a constant. For austenite, the work-hardening rate is high. Therefore, the effect of strain rate and temperature needs to be considered in austenite strain hardening. Therefore, following [22],  
\[\sigma_{th2}=K_{2}e^{\kappa_{2}}\hat{v}_{32}e^{\frac{kT}{b^{3}E_{40}}\ln\frac{ \dot{\varepsilon}}{\hat{v}_{40}}} \tag{22}\]  
where \(K_{2}\) is the austenite strain-hardening coefficient; \(n_{2}\) is the austenite strain sensitivity index; and \(\hat{v}_{32}\) is the reference value of austenitic saturation threshold stress at \(T=0\) K.  
[PERSON] [23] studied the change in the flow stress of duplex stainless steel with the strain rate and found that the change in flow stress is similar to that of a Newtonian fluid after a certain strain rate value.