Explain the Föppl–von Kármán equations 
I am a newbe to elasticity.
Could someone please explain to me briefly how the Föppl–von Kármán equations work?
What are we trying to solve for?
Is there some kind of intuition to the way they look?
Are the two terms being subtracted have any significance?
$$
\frac{Eh^3}{12(1 - \nu^2)} (\nabla^2)^2 \zeta - h \frac{\partial}{\partial x_\beta} \left(\sigma _{\alpha \beta}\frac{\partial \zeta}{\partial x_\alpha}\right)=P
^2 \\ \frac{\partial \sigma_{\alpha\beta}}{\partial x_\beta} =0
$$
Each letter is explained on Wikipedia.
 A: You are right, one of the variables you are solving for is $\zeta$, the vertical displacement or "how much the sheet rises on the y-axis", I would prefer to call it the "z-axis". There are other two variables which are the in-plane displacements $u_x$ and $u_y$, those are solved from the second equation (which in reality are two more equations because of the indexes $\alpha$ and $\beta$).
The second equation is just the equilibrium equation from linear elasticity with forces $f = 0$ and density $\rho = 0$, check Wikipedia.
Now, remember that the stresses $\sigma_{\alpha \beta}$ are related to the strains $\epsilon_{\alpha \beta}$ by a constitutive equation. In Foeppl-von karman a linear relation between stress and strain is assumed then Hooke's law is used as the constitutive equation. So we have $\sigma_{\alpha \beta} = E_{\alpha \beta \gamma \delta} \epsilon_{\gamma \delta}$ where the bad guy $E_{\alpha \beta \gamma \delta}$ is a  fourth-order rank tensor (do not get intimidated by that, given that we are working with Foeppl-von Karman remember that the stresses in the z-direction are zero, i.e. $\sigma_{zx} = \sigma_{zy} = \sigma_{zz} = 0$, so we can express the stresses as follow (remember that $\alpha$ and $\beta$ are just indexes):
$
\begin{pmatrix}
  \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy}
\end{pmatrix}
= \frac{E}{1-\nu^2}
\begin{pmatrix}
 1 & \nu & 0 \\
 \nu & 1 & 0 \\
 0 & 0 & \frac{1 - \nu}{2}
\end{pmatrix}
\begin{pmatrix}
 \epsilon_{xx} \\
 \epsilon_{yy} \\
 2 \epsilon_{xy}
\end{pmatrix}
$
where $E$ stands for Young's modulus and $\nu$ for Poisson ratio.
But you may be asking "OK, and what about the displacements $u_x$ and $u_y$ that you mention at the beginning?", well I am just going to explain that. The strain is related to the displacements by a kinematic equation, in general it is linear relation, however, in Foeppl-von Karman the relation is nonlinear, the equation is as follows:
$\epsilon_{\alpha \beta} = \frac{1}{2} (\frac{\partial u_\alpha}{\partial x_\beta} + \frac{\partial u_\beta}{\partial x_\alpha}) + \frac{1}{2} \frac{\partial \zeta}{x_\alpha} \frac{\partial \zeta}{x_\beta}$
In summary, the second equation that you asked for is the equilibrium equation for linear elasticity and the in-plane displacements $u_x$ and $u_y$ are the other two unknowns for which you are solving for.
Hope that helps at least for future references and other people who may be asking something similar.
