I have run a few simulations of a space elevator breaking. This page summarizes some results, and gives you access to animations.
The elevator that is simulated is an equatorial uniform stress elevator with Brad Edwards' standard parameters. Length is 91000 km, density is 1300 kg/m^3, strength is 130 GPa with a factor of safety of 2, Young's modulus is 1 TPa.
The elevator is broken up into 200 pieces and simulated by a springs and masses model. Heavy damping is added by placing dampers in parallel with the springs, this eliminates spurious high frequency noise caused by the discrete pieces of elevator hitting the Earth. Moreover, it is plausible that there will be a certain amount of longitudinal damping in the real material (it could even be engineered in). A time step of 0.5 seconds was used. A simple Euler algorithm is used for solving the differential equation. The simulation is done in a geocentric reference frame rotating with the Earth. Only gravity, centrifugal force and coriolis forces are taken into account. In addition, the Earth is modeled as an impenetrable body with friction on its surface.
The code is in a horrible state of disarray. It uses glut/openGL for the graphics and the giflib for saving the animation. Use at your own risk. I intend to rewrite it nearly from scratch. Get it here.
The animations do 800 time steps per frame (6 minutes and 40 seconds), the first frame where the elevator is intact is missing (sorry). The color of the tether indicates the amount of stress. It goes from blue to red to yellow to white to snap! The Earth is in blue, and the red sphere is at the geosynchronous altitude.
- An elevator that breaks at the anchor.
- An elevator that breaks a quarter of the way up.
- An elevator that breaks half way up.
- An elevator that breaks three quarters of the way up.
- An elevator that breaks at the counterweight.
Here are some comments I can make about the animations and some further playing I have done. (Note: Since writing this, I have considered that the simulations I did do not actually prove that the top piece always leave the Earth, I simply know that they are thrown very far away. More careful simulation and analysis are needed before I can distinguish between a very elongated ellical orbit and one that truly leaves the Earth's influence. In any case, I can say with confidence that the upper fragment does get past the moon, at which point the Earth-centric assumptions of this simulation can be considered crude at best.)
- The piece that falls to Earth ends up wrapping faster and faster, this causes centrifugal force on the tip, increasing the tension in the ribbon. Often the ribbon breaks on its way down and some fragments go flying out of Earth's gravity well. I didn't expect this at all.
- The top piece goes up and away, rotating end over end, escaping Earth's gravity well. Within a longitude of less than 90 degrees, the bottom of the elevator has cleared the original counterweight altitude. So an elevator that is more than 90 degrees away is not at risk from the top piece in the event of a break.
- For a non-equatorial elevator (5 degree latitude), the top fragment falls towards the equatorial plane with a >24h period. The bottom fragment quickly falls towards the equatorial plane as soon as the tension drops at the anchor, then there are various <24h oscillations as it falls in. Bob, for non-equatorial elevators, the angle between a deployed and a broken elevator is REALLY not zero. This animation has not been included.
- The exact sequence of events, in particular, the secondary breaks, is very sensitive to the exact position of the break.
- So far no atmospheric effects are considered. The elevator will probably start burning up on re-entry at some point. That may cause a tether fragment to end up in a long duration orbit.