For those who are not too familiar with fatigue it seems always be attractive to use equivalent stresses like Von Mises for fatigue life analysis. I am sorry, but I have to disappoint you. An equivalent stress has nothing to do with fatigue, it is merely a calculation number to estimate the onset of yielding on macro scale in a multiaxial stress situation. Often for shear a factor of (1/3)√3 is applied on the fatigue strength for tensional fatigue. This factor gives a nice estimation, but that is pure coincidence.
Fatigue damage is propagating perpendicular to the largest principal stress range, therefore this stress range determines the fatigue behaviour. Principal stresses in tension in the other directions have hardly any influence on the crack growth, these stresses do not affect the shear stress in the activated slip planes. In case of shear (2D stress state with bi-axiality ratio of, or close to, -1), fatigue data for shear should be used.
3D stress states are hardly ever relevant for fatigue, cracks always start at a free surface. Even with sub-surface initiation it can be argued that the stress state is 2D, cracks start at inclusions or voids.
If there is a 2D or 3D stress state with varying largest principal stress direction, it is sometimes thought that using equivalent stresses is at least in this case attractive. This is still not true, equivalent stresses have no direction and certainly not a varying one. The best approach would be the “Critical Plane Approach”, i.e. analysing different crack growth directions (planes) and using for each plane the stress components (determined using the Mohr circles) perpendicular to that plane. Note that such an approach must be performed twice; viz. also for shear stresses.