I have derived a Paris Law type relationship with an extra term in it, which is the ratio of the maximum stress in the cycle divided by the yield stress, raised to an exponent. I did this by deriving the equation of motion of the crack using Euler-Lagrange, see Analysis of Fracture and Fatigue using Lagrangian Mechanics arXiv 1212. 6704. This answers many questions about the Paris Law. However, I cannot find much data where the relevant parameters are quoted.

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