## 2 SPH Modeling of Cutting Model
### Material Constitutive Model  
Johnson-Cook (JC) is the most frequently adapted constitutive law to describe the dynamic mechanism properties (strain hardening, strain hardening rate and thermal softening) of mental material when the elastic deformation transforms to the plastic deformation stage. The JC constitutive law, however, does not take into account the strain-softening of the material under high strain conditions. Cutting Ti-6 AL-4V, however, is a high stress and high stress rate process, and JC law cannot describe this process accurately. To solve the problems of JC law, [PERSON] proposed a TANH constitutive law which modifies the stress, strain and temperature components of JC law by adding \(a\), \(b\), \(c\), \(d\) parameters and a tanh function [20]. Later, [PERSON] and [PERSON] introduced exponent \(r\) into the TANH constitutive model to further control the tanh function for thermal softening [21]. The TANH constitutive model not only considers the stress softening of the material as shown in Figure 1, but also considers the recrystallization phenomenon. Figure 1 is the flow stress-strain curve with the TANH law. \[\sigma_{y}=\left[A+B(\frac{1}{\exp(\epsilon_{eff}^{2})})^{N}\right]\left[1+Cln \left(\frac{i_{eff}}{\dot{\epsilon}_{0}}\right)\right]\left[1-\left(\frac{T-T_ {nom}}{T_{melt}-T_{nom}}\right)^{M}\right]\left(D+(1-D)(\tanh(\frac{1}{( \epsilon+S)^{2}}))^{r}\right) \tag{17}\]  
\[D=1-\left(\frac{T}{T_{m}}\right)^{d} \tag{18}\]  
\[S=\left(\frac{T}{T_{m}}\right)^{b} \tag{19}\]  
\[T=e/C_{p}+T_{room} \tag{20}\]  
where \(\epsilon_{eff}\) is the equivalent plastic strain; \(\dot{\epsilon}_{eff}\) is the equivalent plastic strain rate; \(\dot{\epsilon}_{0}\) is the reference plastic strain rate; \(A\), \(B\), \(C\), \(N\), and \(M\) are material-dependent constants which are the same parameters as in the JC equation and are obtained by SHPB tests [36]. \(T_{room}\) is the room temperature. \(T_{melt}\) is the melting temperature of Ti-6 AL-4V. \(T\) is the current temperature which is computed by Equation (20). \(a\), \(b\), \(c\), \(d\) and \(r\) are corrected parameters which are obtained by the control variable method. We vary one parameter while keeping the others constant to investigate the effects of the parameters on strain-stress curves and make the strain-curve of the TANH model be in good agreement with that of the JC model under the low strain (the strain can be considered low strain when the strain is lower than 0.5). As shown in Figure 2a, parameter \(a\) controls the strain-hardening part by decreasing flow stress after a critical strain value, and a low value of the parameter will lead to a big difference between TANH and the SHPB test data. As \(a\) increases, the maximum stress will increase. Parameter \(b\) in (Equation (19)) controls the temperature-dependent flow-softening effect and points out where the maximum flow stress is attained. The lower the value of parameter \(b\) is, the lower the strain value of the peak flow stress that occurs, as shown in Figure 2b. Parameter \(d\) is an exponent for the temperature that controls the degree of temperature dependency of parameter \(D\) as given in (Equation (18)). As shown in Figure 2c, parameter \(d\) has a strong impact on the value of flow softening and it determines the minimum flow stress value. As shown in Figure 2d, parameter \(r\) controls the softening trend. The higher the parameter \(r\) is, the faster the flow stress-strain curve enters the softening region. With the strain increasing, the slope of strain softening curves will decrease and the maximum flow stress remains unchanged. The \(c\) has similar effects to \(r\) on the flow stress [16]. After observing the trend of the curve with different parameters, the best fit parameters and best stress-strain curve are given in Table 1 and Figure 1.  
Figure 1: The stress-strain curve.