Hello! I am trying to obtain the weak form of the Navier-Cauchy equation, which is
- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}
and can be written in the component form
-(2 \mu +\lambda) \frac{\partial ^2 U_1}{\partial x_1 ^2} - \mu \frac{\partial ^2 U_1}{\partial x_2 ^2} - (\mu + \lambda) \frac{\partial ^2 U_2}{\partial x_1 \partial x_2} - \rho \omega ^2 U_1 = F_1
-(2 \mu +\lambda) \frac{\partial ^2 U_2}{\partial x_2 ^2} - \mu \frac{\partial ^2 U_2}{\partial x_1 ^2} - (\mu + \lambda) \frac{\partial ^2 U_1}{\partial x_1 \partial x_2} - \rho \omega ^2 U_2 = F_2
The general procedure is to multiply the PDE by a test function \textbf{v} in the space \textbf{V}, or v in the space V, and integrate it over the domain \Omega. I will proceed with the component form, for I believe it is easier for me to understand. Setting \textbf{F} = 0 and rearranging the terms
-(2 \mu +\lambda) \int_\Omega v \left[ \frac{\partial ^2 U_1}{\partial x_1 ^2} + \frac{\partial ^2 U_2}{\partial x_2 ^2} \right]dxdy - \mu \int_\Omega v \left[ \frac{\partial ^2 U_1}{\partial x_2 ^2} + \frac{\partial ^2 U_2}{\partial x_1 ^2} \right]dxdy -(\mu + \lambda)\int_\Omega v \left[ \frac{\partial ^2 U_2}{\partial x_1 \partial x_2} + \frac{\partial ^2 U_1}{\partial x_1 \partial x_2} \right]dxdy - \rho \omega ^2 \int_\Omega v \left[ U_1+U_2 \right]dxdy = 0
From Greenās theorem I know that
\int_{\Omega} \left(v \frac{\partial ^2 u}{\partial x ^2} \right)dxdy = \int_\Gamma \left(vu\hat{n}_x \right)ds - \int_{\Omega} \left( \frac{\partial v}{\partial x} \frac{\partial u}{\partial x} \right)dxdy
Which is sufficient to deal with the first and second integrals. However, I do not know how to proceed with the cross derivatives \partial ^2 / \partial x_1 \partial x_2 of the third integral. Can someone help me with this?