Question

What is the best approach or methods to investigate a fatigue failure of metal?

Answer 1

well it would depend on what you would like to investagate about fatigue failure? use a microscope to look at the fractured surface to tell if it has failed by fatigue. If it has failed you will normally see a number of squashed lines where the metal has plastically deformed under load and been squashed together when the material is unloaded, there will also be a region where fast fracture has occured.
for calculation of fatigue life of a material subjected to repeated loading of varying loads Miners Law is very useful, you should be able to find it by looking it up on google with reference to materials science