Composites do not 'fatigue' in the same way that metals do. Metals such as steel have a nearly infinite fatigue life, so long as you stay away from the plastic deformation area - aluminum is somewhat unique it that it does eventually fatigue even if you are operating well away from the plastic deformation area. Corrosion is a different matter - many fatigue failures in metal structures are really due to corrosion. In contrast, composites are pretty immune to corrosion, what eventually happens is the binding resin start to break down (which can sometimes be more of an 'age' issue than 'stress' issue - the resin gets old and brittle)
The one big area of concern with composite structure is that it basically does not plastically deform. Over the years, several aircraft have been able to land safely after exceeding structural limits because the metal structure bent but didn't break. Composites will simply break when the structural limit is exceeded, so composite structure need to be designed with that in mind. This was demonstrated rather dramatically when Formula 1 first started using composite constructions - in a big wreck the chassis would pretty much shatter. Now they design for that and the result is the safest race cars ever built.