Mechanical modeling of inflating a elastomeric balloon

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Discussion Overview

The discussion revolves around the mechanical modeling of an elastomeric balloon, specifically focusing on how to design its geometry to achieve a desired "hotdog" shape upon inflation. Participants explore the governing equations and material behavior necessary for this design, considering factors such as stress-strain relationships and deformation characteristics.

Discussion Character

  • Technical explanation
  • Exploratory
  • Debate/contested

Main Points Raised

  • John seeks recommendations for governing equations to design an elastomeric balloon that inflates to a specific shape and pressure, emphasizing the need to remain within the elastic range of the material.
  • Chet highlights the complexity of the problem, noting that knowledge of differential geometry and elastic material behavior is essential, and that a simple stress-strain curve is insufficient.
  • John expresses a desire for an approximation to facilitate prototyping and testing, indicating a willingness to optimize based on experimental results.
  • Chet suggests starting with a spherical balloon analysis, arguing that understanding this simpler shape is crucial before tackling the more complex "hotdog" shape.
  • John considers treating the balloon as a thin-walled pressure vessel to calculate wall stress at the inflated pressure, aiming to design the empty geometry to achieve the desired inflated shape while maintaining internal pressure.
  • Chet clarifies that for a spherical balloon, equal biaxial stretching is required, whereas for the "hotdog" shape, unequal stretching in both directions is necessary, indicating the need for specific measurements of principal stresses and stretches.

Areas of Agreement / Disagreement

Participants generally agree on the complexity of the problem and the need for a thorough understanding of material behavior and geometry. However, there is no consensus on a specific approach or solution, as various methods and considerations are proposed without resolution.

Contextual Notes

The discussion reflects limitations in the assumptions about material behavior, the need for precise measurements of stresses and strains, and the challenges associated with transitioning from a spherical to a more complex shape.

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

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