# Determination of the Equivalent Thickness of a Taiko Wafer Using ANSYS Finite Element Analysis
## 2 Warpage of an FSM Taiko Wafer According to the Theory of Elasticity in the Spherical Case
\begin{array}{l}M_{r,\;sub}(r)=\frac{E_{s}I_{sub}}{\left(1-v_{s}^{2} \right)}\left(\frac{\partial^{2}\zeta_{int}}{\partial r^{2}}+\frac{v_{s}}{r} \frac{\partial\zeta_{int}}{\partial r}\right)=F_{rsub}(z_{B}+h_{s})\\ \\ M_{\theta,\;sub}(r)=\frac{E_{s}I_{sub}}{\left(1-v_{s}^{2}\right)}\left(v_{s} \frac{\partial^{2}\zeta_{int}}{\partial r^{2}}+\frac{1}{r}\frac{\partial\zeta_{ int}}{\partial r}\right)=F_{\theta sub}(z_{B}+h_{s})\\ \\ M_{r,\;ring}(R_{ext})=\frac{E_{s}I_{ring}}{\left(1-v_{s}^{2}\right)}\left(\left. \frac{\partial^{2}\zeta_{ext}}{\partial r^{2}}\right|_{R_{ext}}+\frac{v_{s}}{ R_{ext}}\frac{\partial\zeta_{ext}}{\partial r}\right|_{R_{ext}}\right)=0\\ \\ M_{\theta,\;ring}(R_{ext})=\frac{E_{s}I_{ring}}{\left(1-v_{s}^{2}\right)}\left( v_{s}\frac{\partial^{2}\zeta_{ext}}{\partial r^{2}}\right|_{R_{ext}}+\frac{1}{R_{ext}} \frac{\partial\zeta_{ext}}{\partial r}\right|_{R_{ext}}\right)=0\\ \end{array} \right. \tag{7}\]  
where \(E_{s}I_{sub}\) and \(E_{s}I_{ring}\) are the flexural rigidity of the substrate and ring regions, respectively. Whereas \(I_{sub}\) and \(I_{ring}\) are the moments of inertia of a section of the taiko wafer, considered as a beam of width \(dw=rd\alpha\), with respect to the neutral axis [32], in the substrate and ring region, respectively. Being \(F_{r,\theta sub}\), the forces per unit length acting on the length \(dw=rd\alpha\), the moment equals \(F_{r,\theta sub}(z_{B}+h_{s})\). Since \(\frac{\partial^{2}\zeta_{int}}{\partial r^{2}}=2\kappa_{sub}\), and \(\frac{\partial\zeta_{int}}{\partial r}=2\kappa_{sub}r\), in the limit \(H\to h_{s}\), \(M_{r,sub}(r)=M_{\theta,sub}(r)=\frac{E_{s}I_{s}^{3}}{12(1-v_{s})}\frac{ \partial^{2}\zeta_{int}}{\partial r^{2}}\), which implies that \(F_{rsub}=F_{\theta sub}\). Moreover, in the limit \(H\to h_{s}\), \(z_{B}=-\frac{h_{s}}{2}\), and according to the Stoney formula \(\frac{E_{s}I_{s}^{3}}{12(1-v_{s})}\frac{\partial^{2}\zeta_{int}}{\partial r^{2}}= \sigma_{f}h_{f}\frac{h_{s}}{2}\), which implies that \(F_{rsub}=F_{\theta sub}=\sigma_{f}h_{f}\). Hence, the set of four equations for the moments can be written as:  
\[\left\{ \begin{array}{l}M_{r,\;sub}(r)=\frac{E_{s}I_{sub}}{(1-v_{s})}\frac{ \partial^{2}\zeta_{int}}{\partial r^{2}}=\sigma_{f}h_{f}(h_{s}+z_{B})\\ \\ M_{\theta,\;sub}(r)=\frac{E_{s}I_{sub}}{(1-v_{s})}\frac{\partial^{2}\zeta_{ int}}{\partial r^{2}}=\sigma_{f}h_{f}(h_{s}+z_{B})\\ \\ M_{r,\;ring}(R_{ext})=\frac{E_{s}I_{ring}}{\left(1-v_{s}^{2}\right)}\left(\left. \frac{\partial^{2}\zeta_{ext}}{\partial r^{2}}\right|_{R_{ext}}+\left.\frac{v _{s}}{R_{ext}}\frac{\partial\zeta_{ext}}{\partial r}\right|_{R_{ext}}\right)= 0\\ \\ M_{\theta,\;ring}(R_{ext})=\frac{E_{s}I_{ring}}{\left(1-v_{s}^{2}\right)}\left( v_{s}\frac{\partial^{2}\zeta_{ext}}{\partial r^{2}}\right|_{R_{ext}}+\left.\frac{1}{R_{ext}} \frac{\partial\zeta_{ext}}{\partial r}\right|_{R_{ext}}\right)=0\\ \end{array} \right. \tag{8}\]  
In general, \(I_{sub}\) and \(I_{ring}\) are functions of \(z_{B}\), such that in the limit \(H\to h_{s}\), \(I_{sub}=I_{ring}=\frac{h_{s}^{3}}{12}\). The equation  
\[\frac{E_{s}I_{sub}(z_{B})}{\left(1-v_{s}\right)}\frac{\partial^{2}\zeta_{int}}{ \partial r^{2}}=\sigma_{f}h_{f}(h_{s}+z_{B}) \tag{9}\]  
is the extension of the Stoney formula for the case of an FSM taiko wafer. It can be solved once that \(I_{sub}(z_{B})\) and \(z_{B}\) are known.