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Looking for this on the web I was provided with little information, only that as the pressure increases so does tension, and if there is a higher tension it results in a louder pop, how may that be expressed in the form of an equation?

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  • $\begingroup$ Check out thin shell membrane sress. $\endgroup$ – user207455 Jun 17 at 8:05
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It is true that tension increases when you inflate a balloon, but the pressure has a peak value. Once the pressure reaches the peak value there is a sudden expansion of the balloon with a decrease of the internal pressure (see reference 1), this is due to the nonlinear behavior of the material.

To obtain a mathematical expression let us assume the following:

  • The material behaves as a neo-Hookean hyperelastic material and it behaves nearly incompressible.
  • The shape of the balloon is spherical for the whole inflation process. This is not true but it simplifies things, so we can do it by hand.
  • It is inflated slowly, so we can neglect inertial effects.

The radius ($r$) and thickness ($t$) of the balloon at any instant are given by

$$r = \lambda r_0\, ,\quad t = \frac{t_0}{\lambda^2}\, ,$$

where $r_0$ is the initial radius, and $t_0$ the initial thickness. $\lambda$ is termed the stretch, and the thickness should decrease quadratically because of the incompressibility.

Since we have a neo-Hookean material we can say that the tension is given by

$$\sigma_{11} = \sigma_{22} \equiv T = 2\alpha\left(\lambda^2 - \frac{1}{\lambda^4}\right)\, ,$$

and $\alpha$ is a material constant measuring how difficult is to inflate the balloon.

Equilibrium between the tension (membrane stress) and the pressure gives

$$P = \frac{2 T t}{r}\, ,$$

and, putting it all together,

$$P = 2\alpha\left(\frac{1}{\lambda} - \frac{1}{\lambda^7}\right) \frac{2t_0}{r_0}\, .$$

And you end up with the behavior shown in the following figure.

enter image description here

Now, we need to know the energy stored to estimate how loud it will sound when popped

$$E = PV = P \frac{4}{3}\pi r^3 = 4\alpha\left(\lambda^2 - \frac{1}{\lambda^4}\right) t_0 r_0^2\, .$$

This expression can be rewritten in terms of the mass of the balloon and density of the material, as well.

References

  1. Jame F. Doyle and C.T. Sun (2007). Theory of Elasticity, Purdue University.
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