# Analytical Study of a Circular Thin Plate Contacting with an Elastic Sphere
## 2 Formulation and Governing Equations
### Contact Deformation of the Circular Thin Plate  
We now proceed to a discussion on the large deformation of the circular thin plate under the axisymmetric transverse distributed load \(q(r)\). As shown in Figure 6, point \(o\) is the center of the middle plane of the circular thin plate, and \(r\) denotes the radial coordinate. The governing equations of axisymmetric large deflection bending of the circular thin plate with radius \(a\) and thickness \(h\) take the forms of [22]  
\[\left\{ \begin{array}{l}r^{2}\frac{\text{d}^{2}N_{r}}{\text{d}r^{2}}+3\tau \frac{\text{d}N_{r}}{\text{d}r}+\frac{E_{b}}{2}\Big{(}\frac{\text{d}w}{\text{ d}r}\Big{)}^{2}=0\\ D\Big{(}\frac{\text{d}^{4}w}{\text{d}r^{4}}+\frac{2}{r}\frac{\text{d}^{3}w}{ \text{d}r^{3}}-\frac{1}{r^{2}}\frac{\text{d}^{2}w}{\text{d}r^{2}}+\frac{1}{r^ {3}}\frac{\text{d}w}{\text{d}r}\Big{)}-N_{r}\frac{\text{d}^{2}w}{\text{d}r^{ 2}}-\frac{1}{r}\frac{\text{d}w}{\text{d}r}\Big{(}N_{r}+r\frac{\text{d}N_{r}}{ \text{d}r}\Big{)}=q(r)\end{array} \right., \tag{25}\]  
Figure 5: Analysis of the vertical displacement of point M2. in which \(D=Eh^{3}/12\big{(}1-\mu^{2}\big{)}\) is the flexural rigidity of the plate; \(w\) is the deflection of the plate; \(N_{r}\) is the radial stress per unit length; and \(E\) and \(\mu\) are the Young's modulus and Poisson ratio of the plate, respectively. The boundary conditions of the plate can be written as  
\[\left\{ \begin{array}{ll}r=0:&\frac{\mathrm{d}w}{\mathrm{d}r}=0&N_{r}=\text{ finite}\\ r=a:&w=\frac{\mathrm{d}w}{\mathrm{d}r}=0&\frac{hEu}{r}=r\frac{\mathrm{d}N_{r}}{ \mathrm{d}r}+(1-\mu)N_{r}=0\end{array} \right., \tag{26}\]  
where \(u\) is the radial displacement of the plate. With the aid of the following dimensionless parameters  
\[\left\{ \begin{array}{ll}y=\frac{r^{2}}{\mu^{2}}&W=\sqrt{3(1-\mu^{2})}\frac{w }{h}\\ \varphi(y)=y\frac{dW}{dy}&S(y)=3\big{(}1-\mu^{2}\big{)}\frac{a^{2}N_{r}}{Eh^{3 }}y\\ \psi(y)=\frac{3a^{2}\big{(}1-\mu^{2}\big{)}\sqrt{3(1-\mu^{2})}}{4\pi E\mathrm{ d}^{3}y}\int_{0}^{r}q(r)2\pi r\mathrm{d}r\end{array} \right., \tag{27}\]  
Equations (25) and (26) can be reduced into  
\[\left\{ \begin{array}{ll}y^{2}\varphi^{\prime\prime}(y)=\varphi(y)S(y)+y^{2} \psi(y)\\ y^{2}S^{\prime\prime}(y)=-\frac{1}{2}\varphi^{2}(y)\end{array} \right. \tag{28}\]  
and  
\[\left\{ \begin{array}{ll}y=0:&\varphi(y)=0&S(y)=0\\ y=1:&\varphi(y)=0&S(y)=\frac{2}{1+\mu}S^{\prime}(y)\end{array} \right.. \tag{29}\]  
When \(0\leq r\leq b\), we can obtain the expression of \(\psi(y)\) by substituting Equation (9) into Equation (27)  
\[ \begin{array}{ll}\psi(y)&=\frac{3a^{2}\big{(}1-\mu^{2}\big{)}\sqrt{3(1-\mu^{ 2})}}{4\pi E\mathrm{d}^{3}y}\frac{p}{b^{3}}\left[b^{3}-\big{(}b^{2}-r^{2}\big{)} ^{\frac{3}{2}}\right]\\ &=\frac{3a^{2}\big{(}1-\mu^{2}\big{)}\sqrt{3(1-\mu^{2})}}{4\pi E\mathrm{d}^{ 3}}P\sum\limits_{k=1}^{\infty}\frac{(-1)^{k+1}\frac{3}{2}\cdot\big{(}\frac{3}{ 2}-1\big{)}\cdots\big{(}\frac{3}{2}-k+1\big{)}}{k!}y^{k-1}\Big{(}\frac{a^{2}}{ b^{2}}\Big{)}^{k}\end{array}  \tag{30}\]  
In the above equation, it should be noted that \(\psi(y)\) is a polynomial with respect to \(y\). Hence, the exact solution to Equation (28) for this case can be given by [23; 24]  
\[\left\{ \begin{array}{ll}\varphi(y)=\sum\limits_{i=1}^{\infty}a_{i}y^{i}\\ S(y)=\sum\limits_{i=1}^{\infty}b_{i}y^{i}\end{array} \right., \tag{31}\]  
Figure 6: Contact deformation analysis of the circular thin plate. where \(a_{i}\) and \(b_{i}\) are the constant coefficients to be determined. Then, the deflection of the circular thin plate on the contact area can be obtained from Equations (27) and (31)  
\[w_{1}(r)=\frac{h}{\sqrt{3(1-\mu^{2})}}\sum\limits_{i=1}^{\infty}\frac{a_{i}}{i} \left(\frac{r}{a}\right)^{2i}+C_{1}, \tag{32}\]  
where \(C_{1}\) is also an arbitrary constant to be determined.