I am trying to figure out the shape that a disc of uniformly elastic material makes if you fix its circumference firmly to a flat table and slowly fill the interior with air.
So far, I've determined that the answer should be some kind of constrained optimality problem, where there's an equilibrium between forces due to air pressure and elasticity, and the boundary is fixed. And based on symmetry, the solution ought to be rotationally symmetric. But I'm unsure how to model elastic forces or how to set up the problem more formally or if there's a quick insight to see the solution. Maybe I should set it up as a variational calculus problem, where the candidate shape of the inflated material determines the distribution of elastic forces? Any advice is appreciated.
Edit: Maybe I can solve this equation in 2D instead of 3D, with a one-dimensional elastic string fixed at its endpoints and filled with air pressure? I imagine the solution to the 3D problem is something like a surface of revolution of the 2D solution. Is this problem equivalent to finding the shape of a meniscus between two fluids? Is the solution a catenary curve because it's like a hanging chain with air pressure replacing the force of gravity?