## 2 SPH Modeling of Cutting Model
### Contact Algorithms  
Figure 3a provides a conceptual explanation of the particle distribution and interaction and how the contact algorithm is used to calculate the contact force. In this work, the total number of particles is 25,000 and the time step is 3.0\({}^{\text{-9}}\) s. (When the number of particles is greater than 25,000 or the time step is less than 3.0\({}^{\text{-9}}\) s. The accuracy of the model does not change obviously, while the running time of the program will increase with the increase of particles and decrease of time step). The cutting tool and material are regarded as groups of particles in our improved SPH cutting model. We treat the cutting tool as a rigid body without carrying any attributes as the workpiece material, such as density and quality. The location information \((x_{k},y_{k})\) and velocity information \((u_{k},v_{k})\) are only needed to set up for the particles of the cutting tool, which is computed by the initial location and velocity of the cutting tool respectively. Then, this information is used to calculate surface vectors (unit tangent vector and unit normal vector). We can use them to determine the direction of the cutting force and contact location between the cutter and workpiece respectively. When the cutter contacts the workpiece, two different kinds of particles (i.e. particles of the workpiece and particles of the cutter) occur contact force. To acquire the contact force, we need to 1) set a threshold \(d_{0}\) to decide when the contact is considered to occur, and 2) calculate the contact force. Figure 3b illustrates how the cutting tool and workpiece are regarded as the contact. \(P\) is the vertical point from particle i to the surface of the cutting tool. \(d_{p}\) is the actual vertical distance between particle i and the cutting tool. \(d_{0}\) is the contact threshold for these two kinds of particles. When \(d_{p}\) is smaller than \(d_{0}\), these two particles contact. The tangent vector \(\mathbf{\tau_{p}}\) and normal vector \(\mathbf{n_{p}}\) at point \(\mathbf{p}\) are calculated by the following Equation:  
\[\left\{ \begin{array}{l}\mathbf{\tau_{p}}=\left(t_{x},t_{y}\right)=\left(\frac{x_ {k+1}-x_{k}}{|x_{k+1}-x_{k}|},\frac{y_{k+1}-y_{k}}{|x_{k+1}-x_{k}|}\right)\\ \mathbf{n_{p}}=\left(-t_{y},t_{x}\right)=\left(-\frac{y_{k+1}-y_{k}}{|x_{k+1}-x_{k }|},\frac{x_{k+1}-x_{k}}{|x_{k+1}-x_{k}|}\right)\end{array} \right. \tag{21}\]  
where \(\mathbf{x}_{k+1}\) and \(\mathbf{x}_{k}\) are the coordinate values of cutting tool particle \(\mathbf{k}\) and \(\mathbf{k}+1\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Physical Parameter** & **Constant** & **Work Material (Ti–6 AL–4V)** \\ \hline Density, \(\rho_{0}\) (Kg/m\({}^{3}\)) & & 4430 \\ Possion’s ratio \(v\) & & 0.342 \\ Shear modulus (G) & & 110 Gpa \\ Specific heat, \(C_{p}\) J/(Kg\(\cdot\)K) & & 580 \\ Thermal conductivity (W/mk) & & 7.3 \\ \(T_{melt}\) (\({}^{\circ}\)C) & & 1878 \\ \(T_{room}\) (\({}^{\circ}\)C) & & 25 \\ \(A\) & & 968 Mpa \\ \(B\) & & 380 Mpa \\ C & & 0.02 \\ \(N\) & & 0.577 \\ TANH model (Equation (17)) & \(M\) & 0.421 \\ \(a\) & & 0.05 \\ \(b\) & & 2 \\ \(c\) & & 1 \\ \(d\) & & 1 \\ \(r\) & & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of Ti–6 AL–4V for workpiece [36; 37]. The next step is to calculate the contact force between the particles of the workpiece and cutting tool. The forces on point \(i\) have two parts, the force along the normal direction and the force along the tangential direction. The forces are calculated as the following Equation:  
\[\left\{ \begin{array}{l}\textbf{{F}}_{ni}=\left[\frac{{{m}_{i}}}{{\left({ \Delta t}\right)^{2}}}\left({{d}_{0}-{d}_{p}}\right)\right]\cdot\textbf{{n}}_{ \textbf{p}}\\ \textbf{{F}}_{\textbf{r}i}=min\Big{\{}\Big{(}\frac{{\textbf{{n}}_{\textbf{r} }}}{{{|{\textbf{F}}_{\textbf{r}i}|}}}{{|{\textbf{F}}_{ni}|}}\Big{)},\frac{{{m} _{i}}}{{\Delta t}}\left({{\textbf{{v}}_{\textbf{p}i}}\cdot{\textbf{{v}}_{ \textbf{p}}}}\right)\cdot{\textbf{{v}}_{\textbf{p}}}\Big{\}}\end{array} \right.