## 2 Linear Contact Law and Finite Element Model
### Finite Element Model  
For an in-depth exploration of the low-velocity impact problems between an elastic-plastic sphere and an elastic-plastic half-space, the LS-DYNA program conducts numerical simulations to analyze the response of the elastic-plastic half-space to a center impact from the elastic-plastic sphere. The half-space is an isotropic material with radius \(r\), height \(h\), elastic modulus \(E_{h}\), yield limit \(\sigma_{Yh}\), and Poisson's ratio \(v_{h}\). The mass of the sphere is \(m_{s}\), modulus of elasticity is \(E_{s}\), yield strength \(\sigma_{Ys}\), and Poisson's ratio is \(v_{s}\). The dimensions and material properties of the sphere and the half-space are listed in Table 1. The initial impact velocity of the sphere is 1 m/s.  
A 2D axisymmetric finite element model, as depicted in Figure 1, is developed for the analysis according to the axisymmetric nature. For the contact region, a regular dense mesh is divided. The number of meshes in the contact area in this model is 80 elements. In the outer region away from the contact area, in order to avoid the sudden change of mesh size, a regular mesh with gradually increasing size from fine to coarse is divided. According to the convergence test, the element contact penalty stiffness and tolerance are set to be 1 and 0.1, respectively, and the contact area needs to be guaranteed to be more than 40 elements discrete. Both the sphere and the half-space use PLANE 162 axisymmetric elements. The sphere comprises 7600 elements with 7811 nodes, while the half-space consists of 6800 elements with 7001 nodes. To simulate the half-space constraints, full displacement constraints are applied to the bottom edge mesh nodes of the half-space. According to the convergence test, both the height and radius of the half-space are selected to be larger than \(20R\). The impact initial velocity is set for all nodes of the sphere. The contact between the sphere and the half-space is chosen as surface-to-surface contact without considering the friction between the contact surfaces. It is assumed that both the half-space and the sphere are perfectly elastic-plastic bodies and the Mises yield criterion determines the plastic yield. Due to the short impact time and the high acceleration produced by impacts, the explicit analysis strategy for general transient analysis is employed in the present study of low impacts. The forces resulting from gravity and damping remain negligibly small and are not considered in the finite element model.  
In order to verify the accuracy of the finite element model, the impact force-indentation curve for an elastic impact is simulated in Figure 2 and compared with the Hertz theoretical solution [27]. The Hertz contact law is as follows:  
\[F=\frac{4}{3}E^{*}\sqrt{R^{*}}\delta^{1.5}, \tag{4}\]  
where \(E^{*}\) is the equivalent contact modulus, \(1/E^{*}=1/E_{h}+1/E_{s}\), and \(R^{*}\) is the effective radius equal to the sphere radius for sphere-half-space impacts. The validation results show that the finite element simulation results are consistent with the Hertz solution. The maximum impact force and maximum indentation errors are 1.9% and 2%, respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Half-Space** & \multicolumn{2}{c}{**Sphere**} \\ \hline \(r\times h\) & 800 \(\times\) 800 (mm) & \(R\) & 35 (mm) \\  & & \(m_{s}\) & 1.4 (kg) \\ \(E_{h}\) & 206 (GPa) & \(E_{s}\) & 206 (GPa) \\ \(\sigma_{Yh}\) & 275 (MPa) & \(\sigma_{Ys}\) & 275 (MPa) \\ \(v_{h}\) & 0.3 & \(v_{s}\) & 0.3 \\ \(\rho_{h}\) & 7800 (kg/m\({}^{3}\)) & \(\rho_{s}\) & 7800 (kg/m\({}^{3}\)) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensions and material properties.