Integration by parts with cross derivatives to obtain the weak form [duplicate]

I’m trying to write the weak form of the Navier-Cauchy equation in the component form, where $$u_1$$ and $$u_2$$ are the displacement components:

$$-(2 \mu +\lambda) \frac{\partial ^2 u_1}{\partial x_1 ^2} - \mu \frac{\partial ^2 u_1}{\partial x_2 ^2} - (\mu + \lambda) \frac{\partial ^2 u_2}{\partial x_1 \partial x_2} - \rho \omega ^2 u_1 = 0$$

$$-(2 \mu +\lambda) \frac{\partial ^2 u_2}{\partial x_1 ^2} - \mu \frac{\partial ^2 u_2}{\partial x_2 ^2} - (\mu + \lambda) \frac{\partial ^2 u_1}{\partial x_1 \partial x_2} - \rho \omega ^2 u_2 = 0$$

The genral procedure is to multiply these equations by a test function $$q$$ and integrate them over the domain $$\Omega$$. Using Integration by parts and Green's Theorem I believe I can do this. However, I'm getting stuck in the term with the cross derivatives. Can someone please explain how to proceed?

• Your second equation is incorrect, it should read $\iint(v \frac{\partial ^2 u}{\partial x ^2})dxdy = \int (v \frac{\partial u}{\partial x} \hat{n}_x)ds - \iint (\frac{\partial v}{\partial x} \frac{\partial u}{\partial x})dxdy$. Your third equation is also incorrect. Jul 1, 2020 at 4:25
• Yes, I've noticed that afterwards, thank you. Jul 1, 2020 at 22:53
• I have added an answer in your other question. You can delete this question since it is a duplicate. Jul 4, 2020 at 4:03
• I'd like to, but I'm not allowed to do so since there are answers to the question. Could you first delete your answer and then I'll delete my post? Jul 4, 2020 at 18:00

The given $$\int_\Omega v \frac{\partial}{\partial x}\left( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}\right) dx dy = 0$$ becomes $$\int_\Omega \frac{\partial}{\partial x}\left[v\left( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}\right) \right] dx dy - \int_\Omega \frac{\partial v}{\partial x}\left( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}\right) dx dy = 0$$ This follows from basic differential calculus $$f = \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}, \qquad v \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(vf) - \frac{\partial v}{\partial x} f$$ Converting first integral to a surface integral, we get $$\int_{\partial \Omega} v \left( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}\right) n_x ds - \int_\Omega \frac{\partial v}{\partial x}\left( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}\right) dx dy = 0$$ But where did you get such an equation $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial x \partial y} = 0$$ which is not elliptic. It can be written as $$\nabla \cdot (A \nabla u) = 0, \qquad A = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$$ but the matrix $$A$$ is not positive definite.

Under some conditions you may be able to integrate it once $$\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = f(y)$$ which is a hyperbolic equation. You can solve this using method of characteristics.

• Your first equation is correct. In your second equation, v appears on the right side of ∂/∂x, which does not match the question posed by the OP. Jun 30, 2020 at 8:44
• It follows from basic differential calculus, see my updated answer. Jul 1, 2020 at 4:07
• Thank you @cfdlab. I added more details to my question in the hope it will clarify where I got this equation from. Jul 1, 2020 at 23:11
• Have you deleted your answer @CharlieS ? There was something I wanted to check out one more time. Jul 1, 2020 at 23:25

I'm trying to obtain the weak form of the Navier-Cauchy equation in the component form

$$-(2 \mu +\lambda) \frac{\partial ^2 u_1}{\partial x_1 ^2} - \mu \frac{\partial ^2 u_1}{\partial x_2 ^2} - (\mu + \lambda) \frac{\partial ^2 u_2}{\partial x_1 \partial x_2} - \rho \omega ^2 u_1 = 0$$

$$-(2 \mu +\lambda) \frac{\partial ^2 u_2}{\partial x_1 ^2} - \mu \frac{\partial ^2 u_2}{\partial x_2 ^2} - (\mu + \lambda) \frac{\partial ^2 u_1}{\partial x_1 \partial x_2} - \rho \omega ^2 u_2 = 0$$

What I did was multiply these equations by a test function $$q$$ and integrated them over the domain, but I got stuck in the $$\frac{\partial ^2 u}{\partial x \partial y}$$ part. Now I believe I can continue with it. I'm not sure where the non elliptic equation issue enters to it though.

• Can I suggest that you delete the contents of your first question and post this instead. Jul 2, 2020 at 10:33
• Lucas, I previously assumed $u$ was a vector quantity which created some commotion. Admittedly, if $u$ wasn't a vector, then my answer was totally invalid. Here is the wiki link I used previously: en.wikipedia.org/wiki/… Jul 2, 2020 at 14:16
• Yes of course @cfdlab, it's done. I'd be very thankful if you could give an answer to the problem in the referred context. Jul 2, 2020 at 20:34