# Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow
## 5 Realization of the Discrete Nonlinear Elasticity Problem
### Derivatives in the Case of the St. Venant-Kirchhoff Material  
Let \(\mathbf{P}=\mathbf{P}(\
abla\mathbf{u}_{h}(\mathbf{\alpha}))=(P_{ij})_{ij=1}^{2}\) be the first Piola-Kirchhoff tensor of the St. Venant-Kirchhoff material as defined in (34). Let \(\mathbf{u}_{h}(\mathbf{\alpha})=(u_{1},u_{2})\). Then, we have  
\[ \begin{array}{ll}P_{11}&=\mu\frac{\partial u_{1}}{\partial x_{2}}\frac{ \partial u_{2}}{\partial x_{1}}\Big{(}\frac{\partial u_{2}}{\partial x_{2}}+1 \Big{)}+\frac{\lambda}{2}\Big{(}\frac{\partial u_{1}}{\partial x_{1}}+1 \Big{)}\bigg{(}\Big{(}\frac{\partial u_{2}}{\partial x_{2}}+1\Big{)}^{2}-1 \bigg{)}\\ &\quad+\Big{(}\mu+\frac{\lambda}{2}\Big{)}\Big{(}\frac{\partial u_{1}}{ \partial x_{1}}+1\Big{)}\bigg{(}\frac{\partial u_{1}}{\partial x_{2}}^{2}+ \frac{\partial u_{2}}{\partial x_{1}}^{2}+\Big{(}\frac{\partial u_{1}}{ \partial x_{1}}+1\Big{)}^{2}-1\bigg{)},\end{array}  \tag{81}\]  
\[ \begin{array}{ll}P_{12}&=\mu\Big{(}\frac{\partial u_{1}}{\partial x_{1}}+1 \Big{)}\frac{\partial u_{2}}{\partial x_{1}}\Big{(}\frac{\partial u_{2}}{ \partial x_{2}}+1\Big{)}+\frac{\lambda}{2}\frac{\partial u_{1}}{\partial x_{2} }\Big{(}\frac{\partial u_{2}}{\partial x_{1}}^{2}-1\Big{)}\\ &\quad+\Big{(}\mu+\frac{\lambda}{2}\Big{)}\frac{\partial u_{1}}{\partial x_{2} }\Big{(}\frac{\partial u_{1}}{\partial x_{2}}^{2}+\Big{(}\frac{\partial u_{1} }{\partial x_{1}}+1\Big{)}^{2}+\Big{(}\frac{\partial u_{2}}{\partial x_{2}}+1 \Big{)}^{2}-1\bigg{)},\end{array}  \tag{82}\]  
\[ \begin{array}{ll}P_{21}&=\mu\Big{(}\frac{\partial u_{2}}{\partial x_{2}}+1 \Big{)}\frac{\partial u_{1}}{\partial x_{2}}\Big{(}\frac{\partial u_{1}}{ \partial x_{1}}+1\Big{)}+\frac{\lambda}{2}\frac{\partial u_{2}}{\partial x_{ 1}}\bigg{(}\frac{\partial u_{1}}{\partial x_{2}}^{2}-1\bigg{)}\\ &\quad+\Big{(}\mu+\frac{\lambda}{2}\Big{)}\frac{\partial u_{2}}{\partial x_{1} }\Big{(}\frac{\partial u_{2}}{\partial x_{1}}^{2}+\Big{(}\frac{\partial u_{1} }{\partial x_{1}}+1\Big{)}^{2}+\Big{(}\frac{\partial u_{2}}{\partial x_{2}}+1 \Big{)}^{2}-1\bigg{)},\end{array}  \tag{83}\]  
\[ \begin{array}{ll}P_{22}&=\mu\frac{\partial u_{2}}{\partial x_{1}}\frac{ \partial u_{1}}{\partial x_{2}}\Big{(}\frac{\partial u_{1}}{\partial x_{1}}+1 \Big{)}+\frac{\lambda}{2}\frac{\partial u_{2}}{\partial x_{2}}+1\Big{)}\bigg{(} \Big{(}\frac{\partial u_{1}}{\partial x_{1}}+1\Big{)}^{2}-1\bigg{)}\\ &\quad+\Big{(}\mu+\frac{\lambda}{2}\Big{)}\Big{(}\frac{\partial u_{2}}{ \partial x_{2}}+1\Big{)}\bigg{(}\frac{\partial u_{1}}{\partial x_{2}}^{2}+ \frac{\partial u_{2}}{\partial x_{1}}^{2}+\Big{(}\frac{\partial u_{2}}{\partial x _{2}}+1\Big{)}^{2}-1\bigg{)}.\end{array}  \tag{84}\]  
Now let \(\mathbf{u}_{h}(\mathbf{\alpha})=(u_{1},u_{2})=\sum_{k=1}^{2N}\alpha_{k}\mathbf{\phi}_{k}\), where \(\mathbf{\phi}_{k}=(\psi_{k},0)\) for \(1\leq k\leq N\) and \(\mathbf{\phi}_{k}=(0,\psi_{k-N})\) for \(N<k\leq 2N\). The derivatives of \(\mathbf{P}(\
abla_{\mathbf{\chi}}\mathbf{u}_{h}(\mathbf{\alpha}))\) with respect to the coefficient \(\alpha_{k}\) are given as follows: for \(1\leq k\leq N\), \(i=k\):  
\[ \begin{array}{ll}\frac{\partial}{\partial a_{k}}P_{11}&=\mu\frac{ \partial\psi_{i}}{\partial x_{2}}\frac{\partial u_{2}}{\partial x_{1}}\Big{(} \frac{\partial u_{2}}{\partial x_{2}}+1\Big{)}+\frac{\lambda}{2}\frac{\partial \psi_{i}}{\partial x_{1}}\bigg{(}\Big{(}\frac{\partial u_{2}}{\partial x_{2}}+1 \Big{)}^{2}-1\bigg{)}\\ &\quad+\Big{(}\mu+\frac{\lambda}{2}\Big{)}\frac{\partial\psi_{i}}{\partial x_{1} }\bigg{(}\frac{\partial u_{1}}{\partial x_{2}}^{2}+\frac{\partial u_{2}}{ \partial x_{1}}^{2}+\Big{(}\frac{\partial u_{1}}{\partial x_{1}}+1\Big{)}^{2}-1 \bigg{)}\\ &\quad+2\Big{(}\mu+\frac{\lambda}{2}\Big{)}\Big{(}\frac{\partial u_{1}}{ \partial x_{1}}+1\Big{)}\Big{(}\frac{\partial u_{1}}{\partial x_{2}}\frac{ \partial\psi_{i}}{\partial x_{2}}+\Big{(}\frac{\partial u_{1}}{\partial x_{1}}+1 \Big{)}\frac{\partial\psi_{i}}{\partial x_{1}}\Big{)},\end{array}  \tag{85}\]  
\(C_{W}^{b}=6\cdot 10^{6}\).