Inflation of a balloon

We model the balloon as an isotropic spherical membrane with uniform radius and  thickness and that the spherical shape is preserved due to inflation. Uniform isotropic stretch of the membrane is described by the extension ratios; and the stretch in the thickness coordinate is .  

Then due to incompressibility, . Then the strain energy stored in the balloon is

(25.1)

Now let’s say, the balloon is inflated slightly, i.e. from radius  to so that, the work done against the outside atmospheric pressure is

Then the net work done is

(25.2)

This energy, which remains stored as the strain energy in the material of the balloon can be written as,

(25.3)