## 4 New Linear Contact Loading Law
### Piecewise Linear Feature  
Figure 5 shows the results of the equivalent stiffness calculation. In Figure 5, the horizontal coordinate is the logarithm of \(\sigma_{Yh}/\sigma_{Y\mathrm{s}}\) with base 10, and the vertical coordinate is \(\sqrt{K_{eq}/K_{h}}\). From Figure 5, it can be seen that \(K_{eq}/K_{h}=1\) is not universal and is usually less than or equal to 1 in most cases. As the yield strength of the sphere increases, \(K_{eq}\) also experiences a corresponding increase. Similarly, a higher elastic modulus of the sphere results in a greater increase in \(K_{eq}\). The smaller the effective yield strain \(\sigma_{Yh}/E_{h}^{*}\) of the half-space material, the larger \(K_{eq}\). The results in Figure 5 show that, for a given half-space material and \(E_{h}/E_{\mathrm{s}}\), the variation of \(\sqrt{K_{eq}/K_{h}}\) exhibits segmented linear characteristics. Before a certain value of \(\sigma_{Yh}/\sigma_{Y\mathrm{s}}=k\leq 1\), it is a horizontal line segment, after that, it is a downward sloping line. Its slope decreases as the effective yield strain \(\sigma_{Yh}/E_{h}^{*}\) of the half-space material increases. To analyze the segmental linear characteristics, in Figure 5f, we can select three impact cases corresponding to point 1, point 2, and point 3 when \(E_{h}/E_{\mathrm{s}}=0.25\). The calculated Mises stress distributions are depicted in Figure 6, enabling the observation of contact elastic-plastic deformation behaviors both before and after the intersection \(k\). Among them, \(\sigma_{Yh}=275\) MPa for the half-space and \(\sigma_{Y\mathrm{s}}\) for the spheres at points 1, 2, and 3 are 550, 275, and 137.5 MPa, respectively. Figure 6a shows that the Mises stress of half-space 6 reaches the yield strength and exhibits elastic-plastic deformation. Meanwhile, the Mises stress of the sphere does not reach the yield strength and exhibits elastic deformation. This type of contact behavior can be viewed as an indentation type [31]. From Figure 6b, it can be seen that the Mises stresses of both the half-space 6 and the sphere reach the yield strength, and they exhibit elastic-plastic deformation at the same time. This contact behavior can be regarded as the combined type. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Half-Space** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(E_{h}(\mathrm{GPa})\) & 60 & 123 & 210 & 207 & 206 & 206 \\ \(\sigma_{Yh}(\mathrm{MPa})\) & 800 & 1070 & 1079 & 540 & 345 & 275 \\ \(v_{h}\) & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 \\ \(v_{s}\) & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 \\ \(\rho_{h}(\mathrm{kg}/\mathrm{m}^{3})\) & 7800 & 7800 & 7800 & 7800 & 7800 & 7800 \\ \(\rho_{s}(\mathrm{kg}/\mathrm{m}^{3})\) & 7800 & 7800 & 7800 & 7800 & 7800 & 7800 \\ \(E_{h}^{*}/\sigma_{Yh}\) & 82.42 & 126.32 & 213.87 & 421.25 & 656.16 & 823.18 \\ \(E_{h}/E_{\mathrm{s}}\) & 0.25–4 & 0.25–4 & 0.25–4 & 0.25–4 & 0.25–4 & 0.25–4 \\ \(\sigma_{Yh}/\sigma_{Y\mathrm{s}}\) & 0.25–4 & 0.25–4 & 0.25–4 & 0.25–4 & 0.25–4 & 0.25–4 \\ \hline \end{tabular}
\end{table} 
Table 2: Material properties of half-space and spheres for different impact scenarios. Figure 5: Piecewise linear features for six half-space materials for \(K_{eq}/K_{h}\) as a function of \(\sigma_{Yh}/\sigma_{Ys}\), varies with \(E_{h}/E_{s}\). (**a–f**) represents half-space 1, 2, 3, 4, 5 and 6, respectively. Figure 6c shows that the Mises stress of half-space 6 does not reach the yield strength, and the half-space exhibits elastic deformation; the Mises stress of the sphere reaches the yield strength and exhibits elastic-plastic deformation. This contact behavior can be viewed as a flattening type [31]. A similar careful analysis of all the impacts in Figure 5 reveal that \(\sigma_{Yh}/\sigma_{Ys}<k\) is of the indentation type and \(\sigma_{Yh}/\sigma_{Ys}>k\) is mostly of the indentation type. Sometimes \(\sigma_{Yh}/\sigma_{Ys}>k\) also appears as the combined type. In summary, it can be concluded that \(\sqrt{K_{eq}/K_{h}}\) for the indentation type is only related to \(E_{h}/E_{s}\) and \(E_{h}^{*}/\sigma_{Yh}\).