thin_shell.md

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 moment 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:

$$F_{z_2} - F_{z_1} - P . \delta s . r . \delta \theta . \cos \left< \alpha \right> = 0 \tag{linear z}$$

$$F_{r_2} - F_{r_1} - F_{\theta_r} + P . \delta s . r . \delta \theta . \sin \left< \alpha \right>= 0 \tag{linear r}$$

$$\left( F_{z_2} + F_{z_1} \right) . \frac{\delta r}{2} - \left( F_{r_2} + F_{r_1} \right) . \frac{\delta z}{2} + M_{\theta_2} - M_{\theta_1} - M_{s_{\theta}} = 0 \tag{angular 𝜃}$$

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 $\left. \alpha \left&lt; r \right&gt; \right|_{r=0} = 0$
• 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 $e_n$ 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:

• $\epsilon_{ss} = \frac{1}{E . h} . \left( \frac{F_s}{r . \delta \theta} - \nu . \frac{F_\theta}{\delta s} \right)$
• $\epsilon_{\theta \theta} = \frac{1}{E . h} . \left( \frac{F_\theta}{\delta s} - \nu . \frac{F_s}{r . \delta \theta} \right)$

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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.