Estimating balloon internal pressure

[kiwi_zt]

Hi guys, firstly this is not a homework question despite looking like one.

I'm curious about how one would go about estimating the internal pressure in a compliant balloon.

I am assuming that I would need the following:

1. Stress-strain curve of the material
2. Diameter of the uninflated balloon
3. Thickness of the uninflated balloon
4. Final size of the inflated balloon
5. Assume that the balloon is spherical and expands so

How would I use this data to help me get a reliable estimation?

Thanks in advance!

 

[Chestermiller]

If it's made out of rubber, a linear stress-strain curve would not be sufficient. If the balloon remains spherical, the deformation of the balloon membrane can be characterized as a transversely isotropic equal biaxial stretching, and the principal in-plane stretch ratio will be $\lambda=r/r_0$, where $r_0$ is the original radius. The stress $\sigma$ within the membrane, for a stretch ratio of $\lambda$ can be expressed as $\sigma=\sigma(\lambda)$. If one does a force balance on the upper hemisphere, one obtains:$$\pi r^2\Delta p=2\pi r h\sigma$$or $$\Delta p=\frac{2\sigma h}{r}$$where h is the current thickness of the membrane and $\Delta p$ is the pressure difference. Since rubber is nearly incompressible, we can write $h=h_0/\lambda^2$, where $h_0$ is the initial thickness. Substituting this gives:
$$\frac{(\Delta p) r_0}{2h_0}=\frac{\sigma(\lambda)}{\lambda}=\sigma_E(\lambda)$$
where $\sigma_E$ is called the "engineering stress", and represents the stress in the membrane per unit initial (undeformed) of cross sectional area of the membrane. To apply this equation, one would have to perform laboratory measurements in a equal biaxial stretching device to measure the relationship between the engineering stress $\sigma_E$ and the stretch ratio $\lambda$. In the laboratory stretcher, $\lambda$ would be the ratio of the final length to the initial length of a square sample. Alternately, one could use the balloon itself (at various imposed pressure differences) to measure this material property of the rubber.

 

[kiwi_zt]

Hi Chestermiller, thanks for taking the time to answer my question.

πr2Δp=2πrhσ

I am a little confused with how you arrived at that equation. I know you mentioned doing a force balance on the upper hemisphere, but I don't really understand. I apologise if the answer is obvious!

 

[Chestermiller]

Hi Chestermiller, thanks for taking the time to answer my question.
I am a little confused with how you arrived at that equation. I know you mentioned doing a force balance on the upper hemisphere, but I don't really understand. I apologise if the answer is obvious!

You conceptually cut the balloon in half, and do a force balance on half the balloon. The cross sectional area of rubber exposed by the cut is $2\pi r h$ and the stress on this exposed surface is $\sigma$. This is balanced by the net pressure force on the open area $\pi r^2$.

 

[russ_watters]

Is this a balloon you blow-up yourself...?

 

[kiwi_zt]

You conceptually cut the balloon in half, and do a force balance on half the balloon. The cross sectional area of rubber exposed by the cut is $2\pi r h$ and the stress on this exposed surface is $\sigma$. This is balanced by the net pressure force on the open area $\pi r^2$.

I see! Just to check - this is assuming that the thickness of the stretched rubber is significantly smaller than the stretched radius, right? So you're taking $2\pi r$ (circumference) multiplied by the thickness, $h$ to get the cross sectional area?

Is this a balloon you blow-up yourself...?

Technically I wouldn't be able to blow it up myself.

To be more specific, I am trying to guess the internal balloon pressure of balloons made of different materials. I wouldn't have the capability to make the balloons and blow them up myself!

 

[Chestermiller]

I see! Just to check - this is assuming that the thickness of the stretched rubber is significantly smaller than the stretched radius, right? So you're taking $2\pi r$ (circumference) multiplied by the thickness, $h$ to get the cross sectional area?

Sure. Even for the unstretched rubber, it's small.

 

[kiwi_zt]

Sure. Even for the unstretched rubber, it's small.

Got it. Thanks a lot!

Regarding testing of the material properties, could I get away with using a uniaxial testing machine?

 

[Chestermiller]

Got it. Thanks a lot!

Regarding testing of the material properties, could I get away with using a uniaxial testing machine?

This is a matter of judgment. In my judgment, no. But, as I said, a balloon could be used to measure the key material property.