https://m.youtube.com/watch?v=8CLRTa_ocmo
In this video, the Mylar sheet is stretched into a plane and secured in a circle. Then, pressure is applied under the sheet and a bubble forms
In the video, he says that filling it with air causes it to create a paraboloid. Is this true and if so can it be derived mathematically? Because of rotational symmetry, I think problem this problem can be reduced to an ODE although I don't know the equation to set up.
To generalize the problem, if you were given a closed area on a plane and 'fill it with air', is it possible to determine the shape of the resulting volume?
Assume stress is proportionate to strain.
To try to clarify this problem, the pressure within the bubble is constant and the pressure outside is also constant, which means it has the same for force per surface area across the entire surface. Since force is equal to strain, it means that the surface will stretch the same amount all the way around. To accommodate all the stretching, what shape would be produced?
I think this equation is appropriate.
https://wikimedia.org/api/rest_v1/media/math/render/svg/77b3bbb9d1485775abe0dc7069df39c807ba6988