Modeling multiaxial fatigue damage using polar equations

It is estimated that more than 70% of failures in engineering components are associated with fatigue loading. Therefore, fatigue is a major design tool for mechanical components. These components are usually subjected to multiaxial cyclic loading. In fact, multiaxial state is very common as tension specimen is under triaxial strain state even though its stress state is uniaxial. There are three approaches to modeling fatigue damage: stress, strain and energy. Critical plane concept is established based on the fact that fatigue cracks initiate at specific plane(s), therefore, multiaxial fatigue damage parameter is evaluated at these plane(s). Critical plane fatigue models such as Fatemi-Socie is among the popular strain-based models. Because it was shown to provide estimation mostly within two factors of life for different materials and different multiaxial loading conditions. This paper presents a new method for analyzing critical plane damage parameters. Using plane stress-strain transformation, maximum values of normal and shear stresses and strains from hysteresis loops are obtained at 360 planes. Plotting these values on polar diagrams shows that multiaxial cyclic responses represent polar curves that can successfully be fitted with definitive known polar equations. In principle, this means that both critical plane and fatigue damage can be determined analytically for a given loading path. However, fitting constants must first be determined. A systematic analysis is performed on different experimental data that were obtained by testing two extruded magnesium alloys at proportional and 90° out of phase loading paths. A closed-form solution for Fatemi-Socie damage parameter is presented for these two loading paths.