Equilibrium and constitutive partial and algebraic equations describing stresses and deformation of an axisymmetric elastic thin shell over a hole I want to design a vacuum table to clamp down a very thin plate and I want to know the stresses and deformations due to the atmospheric pressure. Consider the simplified model below:

               

a disk with radius of $r_2$ and an insignificant thickness of $h << r_2$ over a vacuum hole with a radius of $r_1$ with a $\rho_1$ filleted edge. I presume after deformation the disk should look like this:

               

where $P$ is uniform ambient pressure, $r_c$ is the radius of the contact point and $z_0$ is the vertical deformation in the center with respect to the contact point.
Free-body diagram:
Looking at a trapezoidal differential element of the deformed shell (the pressurized section inside the hole) from the $e_\theta$ direction:

               

where:

*

*$F_{z_1} \approx f_z \left< r - \frac{\delta r}{2} \right> . \left( r - \frac{\delta r}{2} \right) . \delta \theta$   and   $F_{z_2} \approx f_z \left< r + \frac{\delta r}{2} \right> . \left( r + \frac{\delta r}{2} \right) . \delta \theta$

*$F_{r_1} \approx f_r \left< r - \frac{\delta r}{2} \right> . \left( r - \frac{\delta r}{2} \right) . \delta \theta$   and   $F_{r_2} \approx f_r \left< r + \frac{\delta r}{2} \right> . \left( r + \frac{\delta r}{2} \right) . \delta \theta$

*$M_{\theta_1} \approx m_\theta \left< r - \frac{\delta r}{2} \right> . \left( r - \frac{\delta r}{2} \right) . \delta \theta$   and   $M_{\theta_2} \approx m_\theta \left< r + \frac{\delta r}{2} \right> . \left( r + \frac{\delta r}{2} \right) . \delta \theta$

*$F_{\theta_r} \approx 2 . f_\theta . \delta s . \sin \left< \frac{\delta \theta}{2} \right>$

*$M_{s_\theta} \approx 2 . m_s . \cos \left< \alpha \right> . \sin \left< \frac{\delta \theta}{2} \right> . \delta s$   assuming   $\cos \left< \alpha \right> \approx \frac{\delta z}{\delta s}$

*and $z$ is the vertical distance from the contact point in the negative direction

in the above equations:

*

*$f$   and   $m$ represent force and momentum per unit of length

*the   $* \left< * \right>$ notation has been misused to represent a function

*the   $.$   (dot) has been used to represent scalar multiplication

Equilibrium:
Now writing the equations of linear and angular equilibrium:
$$F_{z_2} - F_{z_1} - P . \delta s . r . \delta \theta . \cos \left< \alpha \right> = 0 \tag{linear z}$$
$$F_{r_2} - F_{r_1} - F_{\theta_r} + P . \delta s . r . \delta \theta . \sin \left< \alpha \right>= 0 \tag{linear r}$$
$$\left( F_{z_2} + F_{z_1} \right) . \frac{\delta r}{2} - \left( F_{r_2} + F_{r_1} \right) . \frac{\delta z}{2} + M_{\theta_2} - M_{\theta_1} - M_{s_{\theta}} = 0 \tag{angular }$$
Linear and angular equilibrium in other directions seem to be trivial. Also for the non-pressurized section outside the hole, equations are the same except the first one has no $P . \delta s . r . \delta \theta$ part.
Boundary:
Boundary conditions are:

*

*Due to axisymmetry $\left. \alpha \left< r \right> \right|_{r=0} = 0$

*curvature of the shell at contact point is differentiable

*no friction at contact point

Constitutive equations:
Now assuming the thin shell behaves as an Euler–Bernoulli beam, bending in $e_\theta$ and $e_s$ directions can be superpositioned, and $\delta \theta$ is infinitesimal:

*

*$\rho_s = \frac{E . I_s}{M_s}$   where   $I_s = \frac{\delta s . h^3}{12}$

*$\rho_\theta = \frac{E . I_\theta}{M_\theta}$   where   $I_\theta = \frac{r . \delta \theta . h^3}{12}$
and for in-plain deformations, the average strains are:

*

*$\tilde{\epsilon}_{ss} = \frac{1}{E . h} . \left( \frac{F_s}{r . \delta \theta} - \nu . \frac{F_\theta}{\delta s} \right)$

*$\tilde{\epsilon}_{\theta \theta} = \frac{1}{E . h} . \left( \frac{F_\theta}{\delta s} - \nu .  \frac{F_s}{r . \delta \theta} \right)$

This is s far as I have been able to go. It would be a great help if you could take look at my progress so far and tell me if I have done everything correctly? and what should I do next? have I missed anything? Thanks for your support in advance.
P.S. As a follow-up, here on Reddit I was reminded that I am naively trying to reinvent plate theory. Also here I am trying to do a numerical simulation using Elmer.
 A: I agree with the statement that you are trying to reinvent plate theory.
The simplest version is to consider Kirchhoff-Love plate theory. Assuming a linear behavior, the differential equation for deflections is
$$\nabla^2 \nabla^2 w = -\frac{q}{D} \,,$$
with $D=2h^3E/3(1 - \nu^2)$. This turns into
$$ \frac{1}{r}\cfrac{d }{d r}\left[r \cfrac{d }{d r}\left\{\frac{1}{r}\cfrac{d }{d r}\left(r \cfrac{d w}{d r}\right)\right\}\right] = -\frac{q}{D}\, ,$$
for axisymmetric problems. For your configuration, I think that you could consider simple-support boundary conditions, i.e., you have the following constraints
\begin{align}
w(a) &= 0\, ,\\
\left.\frac{\partial^2 w}{\partial r^2}\right|_{r=a} &= 0\, ,\\
\left.\frac{\partial w}{\partial r}\right|_{r=0} &= 0\, .
\end{align}
In that case,  you end up with the following solution (I would double-check it)
$$w(r) = -\frac{q}{64D}(r^2 - a^2)(r^2 - 5a^2)\, .$$
Keep that I am not considering the (possible) non-linear behavior that you might encounter in your problem, but it gives you a starting point to analyze the problem and a comparison expression for your simulations.
