# Balloon change of thickness, diameter and pressure

1. Nov 1, 2013

### dislect

Hi guys,

Say we have a balloon, not completely inflated like in the sketch picture below.

I would like to come up with a set of equations connecting between the increase in pressure inside the balloon and the change in the following dimensions as a result:
1. "neck" length (shown as 40mm )
2. overall thickness (shown as 15mm)
3. balloon radius (shown as 100mm)
4. balloon material properties (young's modulus, yield strength, density if relevant)

So in theory I would be able to input the pressure inside the balloon and receive the "neck"'s new length (shortening) + new thickness + new radius as a result of the balloon inflating and 'taking' away from their dimensions

I would love someone to guide me and send me the right way.
I can't really use laplace's/pascal law because the thickness is not negligible.

Thanks a lot!

2. Nov 1, 2013

### Simon Bridge

It's non-trivial.
You have to work on how you will model the balloon - work out an action and minimize it.
Probably model it as a lot of masses joined together by springs?

The neck must be pinched off in some way - so I guess that "how" will be the linchpin for your question.

3. Nov 3, 2013

### Stargazer19385

Is this balloon in the atmosphere or in a vacuum? The vast majority of the balloons resistance to expansion comes from atmospheric pressure.

The thickness of the balloon skin can probably be estimated by assuming conservation of volume, though it really depends on the, I think, Poison's ratio of the material.

I can't help you with the neck, though in your picture it is pinched off, so I don't see any pressure reaching it.

The balloon skin density is not that relevant except in calculating its total mass to determine the volume you need to displace to float. Is that your goal?

As for modulus of elasticity and yield strength, those matter, but again the atmospheric pressure is the biggest factor. Some materials are elastomers, and others will rip if you try to expand them much. And you can tell which it is by its modulus and yield strength just by calculating two ways and comparing.

What exactly are you trying to do with the balloon? The same mass of hydrogen needed to lift a person is just as good at any altitude, because the gas expands. And the greater the excess lift, the faster it climbs. It pops when it stretches too far. You can avoid ripping by getting the lift and weight close, but that slows the climb, unless you vent gas when you get high. The main strength the balloon needs is the cross sectional tensile strength to carry the cargo. The atmosphere gives the rest of the strength.

If the balloon is in a vacuum, here is what you must do:
Look at the surface area of its shadow, and multiply that by the pressure inside it. That is how much force is pulling it in two direction. Then slice it in half at the circumference. Look at the cross sectional area of the ring. That ring cross sectional area must handle the tension force previously calculated. If you have 14 psi on a big balloon in space, then its walls will need to be very thick and strong. And if there is a flaw and one part starts to rip, it can spread explosively.

4. Nov 3, 2013

### dislect

Hi Stargazer, thanks for the detailed reply but I'm kind of going another way with the balloon :)
I'm using it as an approximation to how a human female uterus develops over weeks (from around week 20) -
The uterus expands outwards because of the growing baby which puts radial pressure of the muscle of the uterine -> becomes less thick and more dense while expanding radius R -> gradually "takes" material from the "neck" (cervix) to expand, so size of "neck" (40mm) becomes shorter with time untill there's nothing to take from and that's when child delivery begins

5. Nov 4, 2013

### Stargazer19385

I read a book on the properties of biological materials. Their stress strain curves have a highly variable modulus of elasticity that is even unlike rubber. The author mentioned J curves vs S curves, resulting from the difference in microscopic structure. None of what you google about yield strength and modulus of elasticity will apply here unless you get highly material specific from a paper you find somewhere.

The tension is dependent on the stretch, which has little to do with air pressure. Skin bends and stretches quite differently than other non-biological materials. There are fats, muscles, tendons. Even the baby is partially surrounded with fluid. And the uterus wall gets thicker or smaller during a woman's cycle. Weeks is plenty of time for wall thickness to grow or shrink by cell division rather than by force.

Good luck modeling this.

6. Nov 4, 2013

### dislect

Hi again,
Could you refer me to that article?

7. Nov 5, 2013

### Stargazer19385

It was a book, not an article. I searched through 30 pages on Amazon and could not find the book. Many solid biological materials follow standard materials equations. It is just many of the soft ones that don't follow it because their microscopic structure is better modeled by a string unwinding rather than as a spring stretching, which produces a different shaped stress strain curve. I did a Google search on biological materials, but all the articles talked about solid stuff like bone and feathers.