I am developing a model for the respiratory system of a neonate (newborn child) for a BME project by using pressure/volume data of the lungs. To model these, I plan to use a rubber balloon. However. the calculations are more difficult than I thought (especially because rubber behaves non-linearly and with a hysteresis effect, but I will ignore these effects).
I will assume the balloon is perfectly spherical, that it behaves linearly and that the strain is dependent on the radius relative to the original radius ($R_0$).
Here are the equations for the basis of my model:
$$ V = \frac{4}{3} \pi R^3_1 \\ R_1 = \left( \frac{3V}{4\pi} \right)^\frac{1}{3} \\ \sigma = \frac{pR_1}{2t} \\ \epsilon = \frac{R_1 - R_0}{R_0} \\ E = \frac{\sigma}{\epsilon} = \frac{pR_0R_1}{2t(R_1-R_0)} = \frac{pR_0}{2t\left(1-\dfrac{R_0}{\left(\frac{3V}{4\pi}\right)^\frac{1}{3}}\right)} \\ $$
My conclusion is: $$E \propto \frac{p}{V^\frac{1}{3}}$$
Is this correct?