Flexure hinges are susceptible to fatigue damage under cyclic loading, resulting in performance degradation. This paper investigates the stiffness degradation of the right circular flexure hinges (RCFHs) under cyclic loading. Fatigue damage experiments are conducted to obtain the stiffness degradation curves, which can be divided into several stages by feature points. A relationship between feature lives and alternating stress amplitudes is established. A fatigue damage stiffness degradation piecewise curve model for RCFHs is proposed. The effect of notch stress concentration on fatigue damage is analyzed. Fatigue damage experiments under non-zero mean stress are conducted, and an equivalent fatigue stress equation is obtained. Finally, a generalized fatigue damage stiffness degradation model for RCFHs is developed, which establishes a relationship between residual stiffness and cycle number. On this basis, a fatigue damage performance modeling method for flexure hinge mechanisms is proposed. The fatigue damage performance of a compliant bridge mechanism was modeled and tested. The experimental results of input stiffness degradation are generally in agreement with the predicted results, which verify the validity of the method.