What is the relationship between the pressure inside of a balloon and the decibels at which it pops? 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?
 A: 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.

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


*

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

