# What causes an inflated spherical cap to look more like a mountain?

I'm working with bulge tests of materials using biaxial loading (applying pressure from one side through a circular aperture). The materials are clamped on the edges.

The diameter of the aperture is >20 times the thickness of the material. Based on this, I expect membrane-like behavior, with almost uniform stress throughout the material, and the shape of a spherical cap. The experiments start with a flat sheet and proceed until displacements of the center are around >5-6 times the original thickness. Given this, I figured I could neglect bending and only focus on in-plane stresses.

However, I keep getting a shape that's more like a mountain (upside down version of the below shape):

instead of the actual spherical cap shape.

Why is this happening? I can't figure out why just looking at stress contours that are typically plotted -- usually the expected stress is largest at the center and less at the sides.

I have a theory that perhaps it's because in the center, there are "two directions" for stress to sustain the pressure, but at the clamped edges only one "direction" along the material experiences the stress due to the pressure. But I don't see anyone else having this issue online so I can't tell if my theory is total bogus/there's another reason this might be happening.

Any insight would be much appreciated. Thank you!

• So the image is exaggerated in z by a factor of ten or so? The actual z is only about 1/4 D? Commented Jul 9, 2020 at 1:02
• That may be the only issue here. Scale it properly and see what you have. Commented Jul 9, 2020 at 1:07
• An analytic solution for an assumed parameterization of the problem. This work was done by Alfonso Hermida of Goddard Space Flight Center. GSC-13783. techbriefs.com/component/content/article/tb/pub/techbriefs/… Commented Jul 9, 2020 at 1:56
• I think you're right -- I scaled it down and it doesn't look so bad! Commented Jul 14, 2020 at 18:22
• Strangely though, when the image has cubic voxels it does look like the above paraboloid Commented Jul 14, 2020 at 18:23

## 1 Answer

it can't be a sphere because the clamp limits the expansion to that diameter. I would expect that if the pressure resulted in a displacement equal to the clamp radius that you would have a sphere.

• This makes sense, but I've seen other studies where despite clamping the edges the inflated cap still looks like a sphere Commented Jul 14, 2020 at 18:23