## 4 Model Verification
### Cutting Force of a Single Tool
The first phase (\(OA\)) is linear elastic and expressed as follows: \[p=K\mu\ ;\ p\leq p_{c},\ \mu\leq\mu_{c}\] (8) where \(K=p_{c}/\mu_{c}\) is the elastic bulk modulus; \(p_{c}\) elastic limit pressure; \(\mu_{c}\) is the elastic limit volumetric strain. In the second phase (AB), the void inside the material is gradually compressed and the plastic volumetric strain appears. \[p=p_{c}+K_{1}(\mu-\mu_{c})\ ;\ p\leq p_{1},\ \mu\leq\mu_{1}\] (9) where \(K_{1}=(p_{1}-p_{c})/(\mu_{1}-\mu_{c})\) is the slope. In the third phase (BC), the void inside the material disappears and the material is completely compacted. \[p=K_{1}\overline{\mu}+K_{2}\overline{\mu}^{2}+K_{3}\overline{\mu}^{3}\] (10) where \(K_{1}\), \(K_{2}\) and \(K_{3}\) are the material constants; the modified volumetric strain is \(\overline{\mu}=(\mu-\mu_{\rm L})/(1+\mu_{\rm L})\); \(\mu_{\rm L}\) is the grain density of the material. Figure 5: HJC model: (**a**) Strength model; (**b**) Damage of HJC model; (**c**) Equation of state. Appropriate model parameters are critical to the accuracy of cutting force calculations. Many scholars have fitted the parameters of the HJC model and given the results. The parameters of the HJC model below are determined based on the available static and modified according to laboratory dynamic test data [40]. The tool parameters are based on the special tools used in the field test and engineering practice case below. The blade width of the special tools in the model is 4 mm, the edge angle is 90 degrees, the rake angle is \(-45\) degrees and the relief angle is 0 degrees. These are common parameter values for special tools (shell-shaped tools) used for cutting reinforced concrete piles. In consideration of computational efficiency and accuracy, the unit size ratio between the cutting area and other areas should be reasonably adjusted. The size of the concrete unit inside the cutting area is about 0.1 mm. The cutter model is set as a rigid body. The parameters of the HJC model are shown in Table 2. The diagram of model and an example of cutting process are shown in Figure 6.  
In order to facilitate the calculation of the following engineering cases in this paper, the cutting force with different penetration was obtained through this model. The results are shown in Figure 7. It can be seen that the relation between cutting force and penetration is nearly linear and the cutting force is also linearly related to the penetration. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \multicolumn{3}{c}{**Strength Parameters**} & \multicolumn{3}{c}{**Damage Parameters**} \\ \hline \(A\) & \(B\) & \(N\) & \(S_{\text{FMAX}}\) & \(D_{1}\) & \(D_{2}\) & \(E_{\text{FMIN}}\) \\ \hline
0.27 & 1.5 & 0.87 & 20 & 0.04 & 1.0 & 0.01 \\ \hline \multicolumn{6}{c}{**Equation of State Parameters**} \\ \hline \(p_{1}\) (GPa) & \(p_{c}\) (GPa) & \(\mu_{1}\) & \(\mu_{c}\) & \(K_{1}\) (GPa) & \(K_{2}\) (GPa) & \(K_{3}\) (GPa) \\ \hline
1.780 & 0.008 & 0.16 & \(7\times 10^{-4}\) & 9.23 & 141.24 & 136.50 \\ \hline \multicolumn{6}{c}{**Basic Parameters**} & \multicolumn{3}{c}{**Strain Rate**} \\ \hline \(\rho\) (kg/m\({}^{3}\)) & \(G\) (MPa) & \(T\) (MPa) & \(f_{c}\) (MPa) & \(C\) & \\ \hline
2110 & 8750 & 1.62 & 24.45 & 0.012 & \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters for the HJC model. Figure 6: The 3D numerical model of cutting concrete: (**a**) Diagram of model; (**b**) Equivalent stress diagram of the concrete.