## 4 New Linear Contact Loading Law
### Verification  
Five sphere-half-space impact cases with different material combinations are selected for finite element calculations to further verify the general applicability of the linear contact loading law (10). The material properties, dimensions, constraints, impact velocities \(v_{0}\), and initial impact energy \(E_{0}\) of the spheres and half-spaces for the five impact cases are listed in Table 3. The results of the calculations are shown in Figure 9, which is chosen to be dimensionless in terms of the maximum impact force and the maximum indentation. From Figure 9, it can be seen that the predictions by the new linear contact loading law (10) match the finite element calculations. The predicted values of the maximum impact forces are calculated by using the linear contact loading law (10) based on the indentation \(\delta_{m}\) at the unloading moment of the finite element. The comparison between the predicted values and the finite element calculation results is shown in Table 4, with a maximum error of 15.2% and an average error of 4.8%. When the initial impact energy is less than 1.5 J, the maximum error is less than 5%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Num** & **1** & **2** & **3** & **4** & **5** \\ \hline \(\delta_{m}\) (mm) & 0.027 & 0.013 & 0.107 & 0.154 & 0.143 \\ FEM (N) & 1327.1 & 370.5 & 21,709.5 & 41,521.3 & 24,335.3 \\ prediction (N) & 1292.4 & 372.6 & 21,895.2 & 35,208.6 & 23,174.9 \\ error (\%) & 2.6 & 0.6 & 0.8 & 15.2 & 4.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the maximum impact force (N) for five sphere-half-space impact cases with different material combinations for lower impact velocities. Figure 9: \(F-\delta\) curves for the five impacts. To verify the linear contact loading law (10) for higher impact velocities, five sphere-half-space impact cases under different impact velocities are selected for finite element calculations to further verify the general applicability of the linear contact loading law (10). The material properties, dimensions, constraints, and impact velocities \(v_{0}\) of the spheres and half-spaces for the five impact cases are shown in Table 5. The results of the calculations are shown in Figure 10, which was chosen to be dimensionless in terms of the maximum impact force and the maximum indentation. From Figure 10, it can be seen that the predictions by the new linear contact loading law (10) match the finite element calculations. The predicted values of the maximum impact forces are calculated by using the linear contact loading law (10) based on the indentation \(\delta_{m}\) at the unloading moment of the finite element. The comparison between the predicted values and the finite element calculation results is shown in Table 6, with a maximum error of 11% and an average error of 4.3%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Num** & **1** & **2** & **3** & **4** & **5** \\ \hline \(v_{0}\) (m/s) & 3 & 5 & 10 & 15 & 20 \\ \(\delta_{m}\) (mm) & 0.078 & 0.200 & 0.405 & 0.314 & 0.420 \\ FEM (N) & 4139.2 & 4041.8 & 7724.3 & 8068.0 & 10,616.3 \\ prediction (N) & 4236.0 & 4302.0 & 8687.7 & 7886.6 & 10,523.7 \\ error (\%) & 2.3\% & 6\% & 11\% & 2.2\% & 0.09\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the maximum impact force (N) for five sphere-half-space impact cases with different material combinations for higher impact velocities. Figure 10: \(F-\delta\) curves for the five impact velocities.