Nonlinear differential equation for fatigue damage evolution, using a micromechanical model

Abstract

A 1-D discrete (cycle by cycle) micromechanical fatigue model, which represents a material made from an ensemble of elements having statistical (Weibull) strength distribution, has been developed in previous works. In the present study, the recursive type equations have been transformed into a continuous form, leading to a nonlinear second order differential equation of damage evolution (stiffness reduction), for which an exact solution is possible in some cases. By an analytical treatment, explicit relations between microscale parameters and a well-known macro-S–N material response were found. Two major results have been obtained analytically: (a) A relation between the ensemble statistical strength distribution and a commonly observed fatigue power law, and (b) a relation between the scale of microfailure probability of neighbor elements and the macroendurance limit phenomenon, without any a priori condition. In addition, the model predicts three distinct types of fatigue damage evolution characteristics, commonly seen in fatigue tests. At high stresses, damage rate increases monotonically until failure. At medium level stresses, three regions (primary, secondary and tertiary) are observed, resembling strain-time creep response. Finally, at stresses lower than the endurance limit, an exponentially decaying behavior is observed.