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• Biaxial fatigue crack growth is modeled using CFEM with the cyclic damage. • Effects of loading conditions on fatigue crack growth are explored systematically. • Both transverse stress and tensile overload reduce fatigue crack growth rate. • Fatigue crack growth is faster under low-high loading than high-low one. • The mechanisms of fatigue crack growth are elaborated by plastic dissipation. During the operation of proton exchange membrane in fuel cells, cyclic mechanical loads are introduced due to the humidity cycles. The resulting fatigue crack is known to be the main source for its mechanical degradation. In this paper, we use a relation between damage evolution of the cohesive elements and Paris law to simulate the biaxial fatigue crack growth. The effects of various loading conditions are investigated. The predicted crack growth is in a good agreement with experimental studies. Both the transverse stress and the tensile overload have retardation effects. It is also found that the fatigue crack growth rate associated with a shorter pre-crack depends more on the load waveform. Moreover, fatigue crack grows faster under the low-high loading history than the high-low one. These simulation results have provided significant insights into the fatigue failure behavior of membranes during operation conditions and can help improve their durability. Image, graphical abstract [ABSTRACT FROM AUTHOR]