I am trying to evaluate the variation of strain energy for a thin plate to obtain the correct form of the boundary conditions associated with the problem. The strain energy, $U$ is given in terms of displacement, $w$, Poisson's ratio, $\nu$ and plate rigidity, $D$ as follows:
$$ U = \iint_A \frac{1}{2}D \left[ \left( \frac{\partial^2 w}{\partial x^2} \right)^2 + \left( \frac{\partial^2 w}{\partial y^2} \right)^2 + 2\nu \left( \frac{\partial^2 w}{\partial x^2} \right)\left( \frac{\partial^2 w}{\partial y^2} \right) + 2(1-\nu)\left( \frac{\partial^2 w}{\partial x \partial y} \right)^2\right] dA$$
Now, I take the variation of $U$ and integrate by parts using two dimensional integral theorems such as:
$$ \iint_A \frac{\partial f}{\partial x} dA = \oint_C f n_x dC $$ $$ \iint_A \frac{\partial f}{\partial y} dA = \oint_C f n_y dC $$
I only care for the line integral part of the variation, so the remaining parts are not written for the sake of clarity. I end up with the following form:
$$ D\oint_C \frac{\partial^2 w}{\partial x^2} \left( \frac{\partial \delta w}{\partial x} n_x \right) + \frac{\partial^2 w}{\partial y^2} \left( \frac{\partial \delta w}{\partial y} n_y \right) + \nu \frac{\partial^2 w}{\partial x^2} \left( \frac{\partial \delta w}{\partial y} n_y \right) + \nu\frac{\partial^2 w}{\partial y^2} \left( \frac{\partial \delta w}{\partial x} n_x \right) + (1-\nu) \frac{\partial^2 w}{\partial x \partial y} \left( \frac{\partial \delta w}{\partial y} n_x \right) + (1-\nu) \frac{\partial^2 w}{\partial x \partial y} \left( \frac{\partial \delta w}{\partial x} n_y \right) dC $$
By making some manipulations, I am able to reduce this equation to this form:
$$ D \oint_C \nabla^2w \left( \frac{\partial \delta w}{\partial n} \right) + (1-\nu) \left( \frac{\partial^2 w}{\partial y^2} n_x + \frac{\partial^2 w}{\partial x \partial y} n_y \right) \left( \frac{\partial \delta w}{\partial x} \right) + (1-\nu) \left( \frac{\partial^2 w}{\partial x^2} n_y + \frac{\partial^2 w}{\partial x \partial y} n_x \right) \left( \frac{\partial \delta w}{\partial y} \right) dC$$
The first part of this integral is in the form that I seek but I cannot say so for the second part. In the end, I must obtain the following form:
$$ D \oint_C \left[ \nabla^2w - (1-\nu) \left( \frac{\partial^2 w}{\partial x^2} n_y^2 - 2 \frac{\partial^2 w}{\partial x \partial y} n_x n_y + \frac{\partial^2 w}{\partial y^2} n_x^2 \right) \right] \left( \frac{\partial \delta w}{\partial n} \right) dC $$
Can you help me to obtain the final form of the integral? Thank you.
I have found the way to correctly have the form of the integral that I seek. Since I am integrating over a curve, the partial derivatives can be expanded as follows:
$$ \frac{\partial \delta w}{\partial x} = -\frac{\partial \delta w}{\partial s} n_y + \frac{\partial \delta w}{\partial n} n_x $$
$$ \frac{\partial \delta w}{\partial y} = \frac{\partial \delta w}{\partial s} n_x + \frac{\partial \delta w}{\partial n} n_y $$
When these expressions are substituted into the integral, one obtains the required form:
$$ D \oint_C \left[ \nabla^2w - (1-\nu) \left( \frac{\partial^2 w}{\partial x^2} n_y^2 - 2 \frac{\partial^2 w}{\partial x \partial y} n_x n_y + \frac{\partial^2 w}{\partial y^2} n_x^2 \right) \right] \left( \frac{\partial \delta w}{\partial n} \right) dC $$