# Analytical Study of a Circular Thin Plate Contacting with an Elastic Sphere
## 2 Formulation and Governing Equations
### Contact Deformation of the Sphere  
Consider the force element \(q\)ds acting normal to the circular contact area of the sphere, as shown in Figure 3. A rectangular coordinate system (\(o_{p}xy\)) is established with the center of the area element ds as origin. Because the problem is axisymmetric, it is necessary to use the cylindrical coordinate (\(\rho\), \(\varphi\), and \(z\)), and all the field quantities are independent of the angular coordinate. For such a system, the stress components are defined on the differential element by dropping all \(\varphi\)-terms shown in Figure 4.  
Figure 2: Contact area. By applying the equilibrium principle to the differential element, the equilibrium equations with zero body forces can be expressed as follows [20]  
\[\rho-\text{direction}\ \frac{\partial\sigma_{\rho}}{\partial\rho}+\frac{\partial \tau_{z\rho}}{\partial z}+\frac{\sigma_{\rho}-\sigma_{\varphi}}{\rho}=0 \tag{11}\]  
\[z-\text{direction}\ \ \ \ \ \ \ \ \frac{\partial\sigma_{z}}{\partial z}+\frac{ \partial\tau_{\varphi z}}{\partial\rho}+\frac{\tau_{\varphi z}}{\rho}=0 \tag{12}\]  
Thus, all elasticity stress fields must satisfy these relations in order to be in static equilibrium. For the axisymmetric case, the displacement field is of the form \(u_{\rho}=u_{\rho}(\rho,z)\), \(u_{z}=u_{z}(\rho,z)\), and \(u_{\varphi}=0\). Therefore, the strain field is given by  
\[\left\{ \begin{array}{l}\varepsilon_{\rho}=\frac{\partial u_{\rho}}{\partial \rho}\\ \varepsilon_{\varphi}=\frac{u_{\rho}}{\rho}\\ \varepsilon_{z}=\frac{\partial u_{z}}{\partial z}\\ \gamma_{z\rho}=\gamma_{\rho z}=\frac{\partial u_{\rho}}{\partial z}+\frac{ \partial u_{z}}{\partial\rho}\end{array} \right.. \tag{13}\]  
Figure 4: Stress components in cylindrical coordinates. Figure 3: Force element qds acting normal to the circular contact area of the sphere. According to the generalized Hooke's law for linear isotropic elastic solids, the stress-strain relations can be written as  
\[\left\{ \begin{array}{l}\sigma_{\rho}=\frac{E_{b}}{1+\mu_{b}}\left(\frac{\mu_{b}} {1-2\mu_{b}}\theta+\varepsilon_{\rho}\right)\\ \sigma_{\varphi}=\frac{E_{b}}{1+\mu_{b}}\left(\frac{\mu_{b}}{1-2\mu_{b}}\theta +\varepsilon_{\varphi}\right)\\ \sigma_{z}=\frac{E_{b}}{1+\mu_{b}}\left(\frac{\mu_{b}}{1-2\mu_{b}}\theta+ \varepsilon_{z}\right)\\ \tau_{z\rho}=\tau_{\rho z}=\frac{E_{b}}{2(1+\mu_{b})}\gamma_{z\rho}=G_{b}\gamma _{z\rho}\end{array} \right., \tag{14}\]  
where \(\theta=\varepsilon_{\rho}+\varepsilon_{\varphi}+\varepsilon_{z}\), and \(E_{b}\), \(\mu_{b}\), and \(G_{b}\) are the [PERSON]'s modulus, Poisson ratio, and shear modulus of the sphere, respectively. Combining Relations (11)-(14), we can obtain the equilibrium equations solely in terms of displacements denoted as  
\[\left\{ \begin{array}{l}\frac{1}{1-2\mu_{b}}\frac{\partial\theta}{\partial \rho}+\
abla^{2}u_{\rho}-\frac{u_{\rho}}{\rho^{2}}=0\\ \frac{1}{1-2\mu_{b}}\frac{\partial\theta}{\partial z}+\
abla^{2}u_{z}=0\end{array}  \right., \tag{15}\]  
where the Laplacian is given by \(\
abla^{2}=\frac{\partial^{2}}{\partial\rho^{2}}+\frac{1}{\rho}\frac{\partial} {\partial\rho}+\frac{\partial^{2}}{\partial z^{2}}\). This system is difficult to solve, and additional mathematical techniques have been developed to further simplify these equations for problem solution. Common methods normally employ the use of displacement potential functions [21]. Here, we use the Love-Boussinesq potential defined by  
\[\zeta(\rho,z)=A_{1}R+A_{2}\ln(R+z), \tag{16}\]  
where \(R=\sqrt{\rho^{2}+z^{2}}\), \(A_{1}\), and \(A_{2}\) are constants to be determined. The displacement fields corresponding to the potential can be written in the form  
\[\left\{ \begin{array}{l}u_{\rho}=-\frac{1}{2G_{b}}\frac{\partial^{2}\zeta}{ \partial\rho\partial z}\\ u_{z}=\frac{1}{2G_{b}}\Big{[}2(1-\mu_{b})\
abla^{2}-\frac{\partial^{2}}{ \partial z^{2}}\Big{]}\zeta\end{array} \right.. \tag{17}\]  
By combining Equations (14), (16) and (17), we can obtain  
\[\left\{