Mechanical modeling of inflating a elastomeric balloon

[spiri]

Hello PF,

Can anyone recommend a set of governing equations for the design of a simple elastomeric balloon that will expand to a "hotdog" shape when inflated to a certain pressure? I have the stress-strain curve for the material, I know what pressure I need to inflate it to, I just can't figure out how to determine what the "empty" geometry needs to be to get me to the inflated shape and wall thickness with a polyisoprene. I'd like to stay within the elastic range of the material so that it deflates with a constant pressure for a specific length of time. Any thoughts?

John

 

[Chestermiller]

This is not a simple problem. Setting it up properly requires knowledge of differential geometry and elastic material behavior. Just having a stress strain curve is not enough. You need to have the stress strain behavior in large bi-axial deformations of the sheet , with unequal stretches in the two principal directions. You also need a differential force balance on each arbitrary patch of surface.

Chet

 

[spiri]

Thanks Chet. It's a very complex problem. I'm hoping to get an approximation so that I can build some prototypes and then test it and optimize. Any ideas?

John

 

[Chestermiller]

Start out with analyzing a spherical balloon. Why? If you can't do that, you won't be able to do a hot dog. The sphere is a 1D problem, and the deformation is equal biaxial.

Chet

 

[spiri]

Thanks so much Chet! That's what I was thinking. I was thinking of treating the system as a thin-walled pressure vessel to calculate the stress on the wall at the inflated pressure. Knowing the internal pressure and assuming a thickness, that will give me a stress that I can look up to see what region of the S/S curve I am in. The problem I have is, since it's an elastomer, there is a region where the S/S curve is fairly flat (and that's where I want to be) so I'd like to design the empty so that I end up with a balloon that will inflate to a certain sized sphere and then possibly elongate into a "hotdog" shape and hopefully continue to retain the same internal pressure (+/- some tolerance) while increasing in volume. Any further thoughts?

 

[Chestermiller]

Yes. I assume you are talking about a uni axial stress strain curve. For a sphere, you need to stretch biaxiallly, with equal stretches in both directions. For a hot dog, you need to stretch unequally in both directions, covering a broad range of ratios. These are the kind of measurements you will need.

Chet

 

[Chestermiller]

In the previous reply, by measurements, I meant two principal stresses and two principal stretches.