## 2 SPH Modeling of Cutting Model
### SPH Basic Principles
#### 2.1.2 Particle Approximation  
In the SPH algorithm, the computation domain can be discretized with a set of particles by using \(\Delta V_{j}\) instead of the \(dx_{i}\) in Equations (1) and (2):  
\[\Delta V_{j}=\frac{m_{j}}{\rho_{j}} \tag{4}\]  
where mj is the mass of particle j. \(\rho_{j}\) is the density of particle \(j\). \(N\) is the total number of particles which are involved in the computational field. Substituting Equation (4) into Equation (1) and Equation (2):  
\[\left\langle f(x_{i})\right\rangle=\sum_{j=1}^{N}\frac{m_{j}}{\rho_{j}}f\big{(} x_{i}\big{)}W\big{(}x_{i}-x_{j},h\big{)} \tag{5}\]  
\[\left\langle\
abla f(x_{i})\right\rangle=-\sum_{j=1}^{N}\frac{m_{j}}{\rho_{j} }f\big{(}x_{j}\big{)}\
abla W\big{(}x_{i}-x_{j},h\big{)} \tag{6}\]  
From above, it should be noticed that some macroscopic variables (temperature, pressure, density, etc.) are expressed as integral form by a set of disorderly point values in the SPH method. In this set of points, the interaction between various points are described and a kernel estimate of a point is obtained by an interpolation function. In the SPH equation, the governing equation is the Lagrangian form of Navier-Stokes(NS) equations [14], which is used for governing the materials. The mass conservation law and momentum conservation laws can be expressed as following:  
\[\frac{d\rho}{dt}=-\rho\frac{\partial v^{\alpha}}{\partial x^{\alpha}} \tag{7}\]  
\[\frac{dv^{\alpha}}{dt}=\frac{1}{\rho}\frac{\partial\sigma^{\alpha\beta}}{ \partial x^{\beta}}+f^{\alpha} \tag{8}\]  
where \(v\) is the velocity; \(t\) is the time; the coordinate direction can be denoted by \(\alpha\) and \(\beta\); \(f^{\alpha}\) is the component of acceleration; and \(\sigma^{\alpha\beta}\) is the total stress tensor which consists of two parts:  
\[\sigma^{\alpha\beta}=-P\delta^{\alpha\beta}+\tau^{\alpha\beta} \tag{9}\]  
where \(P\) is pressure; \(\delta^{\alpha\beta}\The shear stresses are generated and change with time when solids are in the dynamic velocity field with spatial gradients. The stress \(\tau\) can be expressed as following [14]:  
\[\frac{d\tau_{i}^{\alpha\beta}}{dt}=2G(\hat{\epsilon}_{i}^{\alpha\beta}-\frac{1}{ 3}\delta^{\alpha\beta}\hat{\epsilon}_{i}^{\gamma\gamma})+\tau_{i}^{\alpha \gamma}\cdot\dot{\dot{\dot{r}}}_{i}^{\beta\gamma}+\tau_{i}^{\gamma\beta}\cdot \dot{\dot{r}}_{i}^{\alpha\gamma} \tag{10}\]  
where \(\hat{\epsilon}^{\alpha\beta}\) is the strain rate tensor; \(G\) is the shear modulus which can be obtained by an experimental test; and \(\dot{r}^{\alpha\beta}\) is the rotation tensor. After elastic deformation, the material will enter the plastic deformation named plastic behavior and it can be identified by the critical point of plastic deformation [14]:  
\[f_{y}=\frac{\sigma_{y}}{\sqrt{3}J_{2}} \tag{11}\]  
where \(J_{2}\) is the second stress tensor invariant. \(\sigma_{y}\) is the flow stress and its initial value is the shear stress (\(\sigma_{y}=\sqrt{\tau^{\alpha\beta}\cdot\tau^{\alpha\beta}}\)). When the value of \(f_{y}\) is lower than 1 (\(f_{y}<1\)), the plastic deformation is said to occur. As mentioned above, the traditional SPH method has low accuracy [34]. Therefore, in this study, we employed an improved SPH method where density and kernel gradient are corrected by Moving Least Squares and a correction matrix. It has been proven to be efficient in improving the accuracy of the cutting model [15]. In the scheme of the improved SPH algorithm, we use the Moving Least Squares (MLS) method to correct the density field periodically; the linear variation of the density field can be exactly reproduced and a smoother pressure can be obtained [35]. The correction scheme is expressed as:  
\[\langle\rho_{i}\rangle=\sum_{j}^{N}\rho_{j}W_{ij}^{MLS}\frac{m_{j}}{\rho_{j}}= \sum_{j}^{N}m_{j}W_{ij}^{MLS} \tag{12}\]  
where \(W_{ij}^{MLS}\) is the MLS kernel; more details can be found in Reference [35]. The Kernel Gradient Correction (KGC) method is applied to improve the SPH approximation accuracy.