# Finitely Deforming an Elastic Band to Launch a Payload

Hi. I've set myself a project with a couple of my classmates and despite the impracticability of it we want to do it as well as possible. Our aim is to look at using a massive elastic band in order to launch a payload at massive velocities, possibly even at escape velocity. To this end I've self-studied a lot of the linear theory of elasticity, Landau and Lifschitz was a pain to slog through but I'm familiar with the rest so it could've been worse, and I've looked extensively at Finite Strain Theory. It seems however that the kind of problems we solve with springs for introductory mechanics are never touched upon in elasticity so I'm at a loss as to where I should go and whether what I've been doing is correct.

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: