## 4 New Linear Contact Loading Law
### Combined New Linear Contact Loading Law  
Since the linear loading law (1) cannot be applied to more general impact situations, a new linear contact loading law is proposed:  
\[F=K_{eq}\delta, \tag{6}\]  
Due to the significant segmental linear character of \(K_{eq}\), the analytical expression of \(K_{eq}\) can be obtained by fitting the finite element calculation results of 150 impacts in Section 3.1 as follows:  
\[\sqrt{K_{eq}/K_{h}}=\zeta, \tag{7}\]  
\[\xi=\left\{\begin{matrix}a-b(E_{h}/E_{s})+c(E_{h}/E_{s})^{2}&0\leq\sigma_{Yh} /\sigma_{Ys}<k\\ d-e(\sigma_{Yh}/\sigma_{Ys})&k\leq\sigma_{Yh}/\sigma_{Ys}\end{matrix},\right. \tag{8}\]  
where:  
\[a=0.44911+0.0014(E_{h}^{*}/\sigma_{Yh})-\left(8.71326\times 10^{-7}\right)(E_{ h}^{*}/\sigma_{Yh})^{2},\]  
\[b=0.11699-\left(1.21254\times 10^{-2}\right)(E_{h}^{*}/\sigma_{Yh})^{3.62294},\]  
\[c=0.01557-\left(1.27765\times 10^{-5}\right)(E_{h}^{*}/\sigma_{Yh}),\]  
where \(a\), \(b\), and \(c\) are the coefficients when \(0\leq\sigma_{Yh}/\sigma_{Ys}<k\), while \(d\) and \(e\) are the coefficients when \(k\leq\sigma_{Yh}/\sigma_{Ys}\):  
\[d=\lambda-\beta(E_{h}/E_{s})+\gamma(E_{h}/E_{s})^{2},\]  
\[e=\omega-\eta(E_{h}/E_{s})+\psi(E_{h}/E_{s})^{2},\]  
\[\lambda=0.46508+0.00161(E_{h}^{*}/\sigma_{Yh})-(9.92809\times 10^{-7})(E_{h}^{*}/ \sigma_{Yh})^{2},\]  
\[\beta=0.10704+(3.14676\times 10^{-4})(E_{h}^{*}/\sigma_{Yh})-(6.92422\times 10^{- 7})(E_{h}^{*}/\sigma_{Yh})^{2}\]  
\[+(4.20916\times 10^{-10})(E_{h}^{*}/\sigma_{Yh})^{3},\]  
\[\gamma=0.01387+(2.8986\times 10^{-5})(E_{h}^{*}/\sigma_{Yh})-(7.95567\times 10^{- 8})(E_{h}^{*}/\sigma_{Yh})^{2}\]  
\[+(5.35956\times 10^{-11})(E_{h}^{*}/\sigma_{Yh})^{3},\]  
\[\omega=0.00439+(2.87402\times 10^{-4})(E_{h}^{*}/\sigma_{Yh})-(1.5494\times 10^{- 7})(E_{h}^{*}/\sigma_{Yh})^{2},\]  
\[\eta=0.00714+(6.1706\times 10^{-5})(E_{h}^{*}/\sigma_{Yh})-(4.51026\times 10^{- 8})(E_{h}^{*}/\sigma_{Yh})^{2},\]  
\[\psi=(9.89625\times 10^{-4})+(6.36574\times 10^{-6})(E_{h}^{*}/\sigma_{Yh})\]  
\[-(4.51709\times 10^{-9})(E_{h}^{*}/\sigma_{Yh})^{2},\]  
the intersection point \(k\) of two lines is as follows:  
\[k=\frac{1}{e}\Big{[}d-a+b(E_{h}/E_{s})-c(E_{h}/E_{s})^{2}\Big{]}, \tag{9}\]it follows that the following is true:  
\[K_{eq}=2\pi R\xi^{2}p_{0}, \tag{10}\]  
the fitting errors of the \(K_{eq}\) for all 150 impacts were less than 5.9%.  
To clearly illustrate the variations of \(\xi\), Equation (8) is graphed as a response surface. For \(0\leq\sigma_{Yh}/\sigma_{Ys}<k\), the response surface is shown in Figure 7. It can be seen that \(\xi\) monotonelydecreases as \(E_{h}^{*}/\sigma_{Yh}\) decreases or \(E_{h}/E_{s}\) increases. For large \(E_{h}^{*}/\sigma_{Yh}\), \(\xi\) almost linearly varies with \(E_{h}/E_{s}\). For small \(E_{h}^{*}/\sigma_{Yh}\), \(\xi\) varies parabolically with \(E_{h}/E_{s}\) as a parabola. For a given \(E_{h}/E_{s}\), the variation of \(\xi\) is dominated by the parabolic forms.  
For \(k\leq\sigma_{Yh}/\sigma_{Ys}\), the response surfaces for \(\sigma_{Yh}/\sigma_{Ys}=1\), 2, and 4 are shown in Figure 8. It can be seen that \(\xi\) monotonely decreases as \(E_{h}^{*}/\sigma_{Yh}\) decreases or \(E_{h}/E_{s}\) increases. \(\xi\) also monotonely decreases as \(\sigma_{Yh}/\sigma_{Ys}\) increases. For a given \(\sigma_{Yh}/\sigma_{Ys}\) and a given \(E_{h}^{*}/\sigma_{Yh}\), \(\xi\) varies parabolically with \(E_{h}/E_{s}\). For a given \(\sigma_{Yh}/\sigma_{Ys}\) and a given \(E_{h}/E_{s}\), the variation of \(\xi\) is dominated by the parabolic forms.