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If I have a circular membrane that is fixed by the edges and then I apply a uniform pressure to the membrane (let's say the membrane is the top of a circular chamber that has a rigid wall and flat bottom) by injecting compressed air, what shape will the membrane take? Parabolic? Hyperbolic? Is the shape very dependent if the membrane is parallel to the ground or perpendicular? Will it be different at the edges?

Mike Wong
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1 Answers1

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Without pressure, as a result of gravity it is a catenary in any cross section, https://en.wikipedia.org/wiki/Catenary

It is assumed that the density of the membrane (and therefore its tension) is uniform. The catenary deviates according to the direction of gravity.

The force of pressure is perpendicular to the membrane at all points and would form a perfectly spherical shape if acting alone; depending upon the pressure magnitude and the weight of the membrane this may be an effect that dominates that of gravity.

JMLCarter
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  • Is there a reference that explains why the membrane would have a spherical shape (assuming gravity was negligible)? – Mike Wong Sep 12 '17 at 21:26
  • https://physics.stackexchange.com/questions/137980/why-does-a-bubble-take-a-spherical-shape – JMLCarter Sep 12 '17 at 21:53
  • Interesting, but does that justification hold in general or is it specific for cases with surface tension like a soap film (in my particular case, the membrane is solid). – Mike Wong Sep 12 '17 at 23:53
  • A (thin, idealized) membrane has no bending stiffness. Therefore, the pressure can only be resisted by tension in the membrane, since by symmetry the shear is zero (at least in your case of a circular boundary). It follows that the Gaussian curvature of the membrane shape is determined only by the pressure. For any constant pressure source, this means the shape will be a section of a sphere. If the bending stiffness for a (thick, solid) "membrane" is not negligible, you don't have a membrane but a thin or thick shell structure, and the general solution is much more complicated. – alephzero Sep 13 '17 at 03:11
  • Without pressure, as a result of gravity it is a catenary in any cross section. This does not seem correct to me. Do you have a proof? A rotated catenary would be denser at the bottom than at the top, whereas the chain has uniform density. So I think a membrane of uniform density would hang shallower than a catenary. ... Also, you are assuming that (like a chain) there is no stretching. – sammy gerbil Sep 13 '17 at 12:27
  • I assume uniform density/tension/stretching. – JMLCarter Sep 13 '17 at 17:58
  • I think I see the question. Here is a verbal proof. Imagine a series of parallel cateniary infintiesimal strips,hung from one point on a circle to another. Now arrange a set of similar catenniaries perpendicular to them? If these were physically attached to each other to form a circular membrane of double the density, would their shape change? – JMLCarter Sep 13 '17 at 18:23