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If I've got a circle of elastic material that's fixed at the circumference, and I drop a point weight on it, what will the cross section look like?

For example, if I took some material from a balloon, clamped it between two metal sheets that had a circle cut out of them, and then dropped an infinitely small bowling ball in the middle, what would that shape look like? I'm assuming it would be symmetrical and thus asking for the cross section. Note that a 1D simplification wouldn't be valid because of Poisson's ratio.


Edit: Some clarification was requested by the comments:

The scenario I've got in mind is that there is a circle of elastic material, which is fixed at its (circular) boundary so that no matter how much you pull on it, the circumference of the circle will not move nor deform. Kind of like a trampoline, but instead of springs along the edge, the material itself can deform when a force is applied.

Onto this circular elastic material, a force is applied. In this example, a point mass is placed above the center of the circle and allowed to fall until it hits the circle. When it hits the circle, the elastic material will deform due to the weight of the point mass.

My question: What is the new (deformed) shape in 3D space of the elastic circle (or "trampoline") when this point mass is placed onto it? I expect that the point mass might move around a bit when it's placed, so I'm only interested in what shape the elastic circle will take when the point mass stops moving.

When I spoke about a "cross section", I was asking what shape you would see if you took the deformed elastic material and sliced it vertically through its upright axis of symmetry. I know it's not going to be a "V" shape, but I don't know what it is going to be.

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  • $\begingroup$ your question isn't very clear. what exactly do you mean by cross section? Your example isn't very clear either. I would guess that you have a particular scenario in your mind. please try explaining it better. $\endgroup$ Commented Jan 14 at 14:04
  • $\begingroup$ Some edits made, I hope they help! (let me know if they do not) $\endgroup$
    – beyarkay
    Commented Jan 14 at 14:38
  • $\begingroup$ the question is wayyy better now. $\endgroup$ Commented Jan 14 at 14:48
  • $\begingroup$ <3 thanks for your help $\endgroup$
    – beyarkay
    Commented Jan 14 at 14:52
  • $\begingroup$ If the problem you're describing is the one depicted in the answer below, how thick is the elastic membrane? Depending on the thickness and the bending stiffness, and the constraints on the edge of the circular membrane (providing boundary conditions of the mathematical problem), you could get quite a different shape of the deformed membrane. If it's not so, a sketch representing the system could really improve your question $\endgroup$
    – basics
    Commented Jan 14 at 15:52

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Since the shape of the deformed material is asked, I am assuming the point mass($m$) to to be placed at the center of the circle and slowly otherwise the point mass would start oscillating.

Now the center is being pulled down with a force $mg$.

The shape formed by the sheet would be a curved v shape.

Side 1

The curved shape can be explained when looking at any random horizontal level from the top. Initially, it would form a v shape and then the v shape would be pulled in at every point, making it curved.

Top 1

The mass distribution of the sheet would be more towards the middle when we look at it from the top

Cross section from top

The circular sheet can be split into infinite rubber bands(circles) and each circle is being expanded. this would cause an uniform force inward from every point on that circle just like when we expand a rubber band.

and finally, the point where the point mass resides would be vertical or in case of a bigger object, almost vertical and this part would provide the force required to keep the point mass at rest.

FBD

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  • $\begingroup$ if you still can't visualize the shape, lmk and I would make and add more pics from different angles to help you understand it better :) $\endgroup$ Commented Jan 14 at 15:34

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