We set up our problem like so: We assume that we want to launch a 1kg bag of sugar with our elastic band. We assume that our elastic band can be approximated to a very long hyper-elastic cylinder. One end is fixed at the surface of the Earth while the other end dangles into a purpose made hole in the radial direction towards the core. The bag can be attached to the free end of the cylinder which is then stretched along its axis before it can be released. This will require a large deformation so we make use of Finite Strain Theory. We'll also assume that the deformation is volume preserving to make things easier.

So, the cylinder will increase in length along its axis and will compress radially to preserve volume into a new deformed cylinder. The Deformation gradient is easy to get from that, it's just the principal stretches along the diagonal. Using just this I can find the Neo-Hookean Strain Energy Density and can Integrate that over some volume but I' left with several problems:

1. How can I even start with finding equations of motion?

2. How can I determine necessary radius to avoid exceeding yield strength?

3. How can I determine the maximum velocity of the free end of the band?