The Sky Road
It's a SF novel by Ken MacLeod where this event plays a key role in human affairs for about 2 centuries. He calls it an "ablation cascade". Not a bad little read if you're in to that sort of thing.
This is something I'd worry about for a space elevator. Early single tether versions, not the solid diamond tower or equally impenetrable version. Any pissed off nation with sub-orbital rocketc could launch a bucket of chains at it.