I did a quick read and a search and didnt find any discussion of ribbon fatigue. The issue is that the various stressings on the ribbon thermal, bending, tension, and curving through the 28 km loop over many cycles. In general the mechanical analysis of ribbon strength was unconvincing to me.
The reason inertial schemes like the space fountain and the launch loop were created is to overcome those material limitations. Aside from the scale, the engineering and material science of a launch loop is relatively straight forward.
The seminal paper, which discusses rotor fatigue and much more: http://launchloop.com/LaunchLoop%3Faction=AttachFile%26do=ge...