SDE models with exponential drift and diffusion for approximating fatigue crack growth dynamics

A scalar stochastic differential equation (SDE) is studied for modeling fatigue crack growth dynamics. The drift and diffusion coefficients in the SDE model are exponential functions of the number of fatigue cycles. Exponential coefficients are shown to be reasonable based on experimental, physical, and theoretical arguments. The SDE model is examined for two fatigue crack growth datasets. For each dataset, the four parameters in the SDE model are fit using observed crack lengths for two different low values for the number of cycles. Then, using the SDE model with the fitted parameters, probability distributions for several higher values of the number of cycles and several first-passage probability distributions are calculated and compared to the observed distributions. The comparisons indicate that the proposed SDE model provides a good approximation for fatigue crack growth.