## 1 Introduction
They simulated how performance degradation of concrete caused by fatigue loads led to stress redistribution along the cross-section by combining several actual cycles into one equivalent virtual cycle. Some scholars found that the stress-strain curve of concrete under fatigue loading followed a similar form as that under monotonic loading and derived the equivalent fatigue damage model by substituting parameters such as stiffness and strength after specific loading cycles into the monotonic stress-strain relationship of concrete [16; 17; 18; 10]. The above models discretize the fatigue analysis process and conduct concrete fatigue performance analysis in a jump-in-cycle manner. From an empirical and phenomenal perspective, this method is more convenient for engineering applications. However, these models have limitations in providing a physical explanation for fatigue damage evolution, and further research is crucial to understand the definition of fatigue damage variable and its evolution better. The progressive accumulation of internal damage causes concrete performance degradation at the macro level. Macroscopic mechanical properties of concrete evolve as fatigue loading progresses; the evolution of these concrete properties can then represent internal fatigue damage. Different parameters, such as maximum fatigue strain [19], residual strain [20; 21], secant modulus of elasticity [22], and residual strength [23], have been defined as concrete damage variables. Maximum fatigue strain (corresponding strain when the upper limit of fatigue loading is reached) reflects the deformation under fatigue loading and is convenient to measure in experiments [24]. Thus, defining concrete fatigue damage variable through maximum fatigue strain is valuable in revealing the damage deterioration mechanism [19]. However, the maximum fatigue strain measured during the first loading cycle is more significant than that generated in subsequent fatigue cycles due to initial defects such as pores and microcracks within the material. Initial damage induced by initial defects and preloading also shows strong dispersion affected by the manufacturing process and loading conditions. Therefore, defining fatigue damage variable with damage exclusively generated by fatigue loading is desirable. Fatigue strain evolution is generally accepted to exhibit a three-stage development pattern [24; 25], consisting of an initial rapid but gradually stabilizing stage, a secondary stabilized stage, and a final rapid growth stage. Multi-segment linear [26], polynomial [27], or exponential equations [28] have been applied to fit the fatigue damage evolution. When obtaining such equations, it is not easy to guarantee clear physical meaning and satisfactory fitting accuracy simultaneously. Because each fitted parameter's physical meaning and value range in these evolution equations have yet to be identified, the regression equation derived from a specific experiment might apply poorly to other experimental data with a large dispersion of fitted coefficients. The applicability of a regression equation to other experiments with similar loading conditions remains to be verified. To address a significant need in the field, we aim to develop a simplified, practical, and physically sound method to simulate the fatigue response of concrete based on Abaqus-integrated CDPM. This method should involve a discrete simulation of loading cycles and an equivalent fatigue stress-strain relationship that can reflect fatigue damage evolution. A damage variable is first constructed based on maximum fatigue strain, and parametrical analysis is conducted on the damage evolution equation. The physical meaning of each parameter is clarified, thus overcoming the limitation that fitted equations could only reflect the trend of specific test data. The proposed value range of each parameter and a family of the evolution curves make the fitted fatigue damage evolution equation under various stress levels valuable.