# Analytical Study of a Circular Thin Plate Contacting with an Elastic Sphere
## 3 Results and Discussion  
In order to compare the results derived in the paper with the Hertz theory, we take a case as an example, and the parameters for calculation are listed in Table 1.  
Figure 7 shows the deflection of the whole circular thin plate. Figure 8 shows the deflection of the circular thin plate on the contact area in detail. As can be seen from these figures, the deflection of the circular thin plate presents in an axisymmetric distribution, and varies in a continuous and nonlinear manner, which initiates from zero at the boundary and reaches the peak value of 0.5377 mm at the center. Additionally, the radius of the contact area is 0.2616 mm.  
Furthermore, the normal stress on the circular thin plate also presents an axisymmetric distribution, which initiates from zero at the boundary of the contact area and reaches the peak value of 0.544 Gpa at the center (see Figure 9).  
\begin{table}
\begin{tabular}{c c c} \hline  & **Sphere** & **Circular Thin Plate** \\ \hline  & \(E_{b}=70\)Gpa & \(E=70\)Gpa \\ Parameter values & \(\mu_{b}=0.3\) & \(\mu=0.3\) \\  & \(r_{b}=2\)cm & \(a=0.05\)m \\  & \(P=78\)N & \(h=1\)mm \\ \hline \end{tabular}
\end{table}
Table 1: Parameters for calculation.  
Figure 8: Deflection of the circular thin plate on the contact area.  
Figure 7: Deflection of the circular thin plate.  
A comparison between the results based on the Hertz theory and ours is presented in Table 2. The relative penetration derived in the paper is less than 1% of that given by the Hertz theory in the case of identical force, while the radius of the contact area does not change significantly. Therefore, according to the data presented, we can conclude that the Hertz theory is not applicable to the elastic sphere in contact with the thin plate with large deformation.  
Next, the validity of the results derived in the paper were verified by the finite element method. The finite element software ANSYS was introduced to be the simulation tool. ANSYS is a general-purpose finite element program for static and dynamic, as well as multiphysics analysis and specializes in contact related problems. Figure 10 exhibits the finite element models of the sphere and the circular thin plate by ANSYS. All the model components were meshed with the element SOLID185, a 3D element of eight nodes and three degrees of freedom per node (displacements in nodal directions \(x,y,\) and \(z\)) [27]. This element had plasticity, large deflection, and large strain capabilities. The mesh at the contact locations was refined very well to accurately model the contact region, as also shown in Figure 10. The contact between the sphere and the circular thin plate was modeled using CONTA173 elements placed on the sphere and TARGET170 elements placed on the circular thin plate.  
\begin{table}
\begin{tabular}{c c c} \hline  & **This Paper** & **Hertz** \\ \hline Action of force & 78 N & 78 N \\ Relative penetration & 0.5406 mm & 0.004873 mm \\ Radius of contact area & 0.2616 mm & 0.3122 mm \\ \hline \end{tabular}
\end{table}
Table 2: Comparison results.  
Figure 9: Normal stress distribution on the circular thin plate.  
The deflection of the whole plate and the normal stress on the plate variation with respect to the radial coordinate \(r\) obtained theoretically and numerically can be seen in Figures 11 and 12. As these figures demonstrate, the theoretical and numerical results were in good agreement, and the relative deviation was in the acceptable range. Therefore, the theoretical model presented in this paper was fully verified.  
Figure 11: Deflection of the circular thin plate vs. \(r\).  
Figure 10: Finite element models.