Robot tendons aren't much different from any other load-bearing cable. The same sources of stress and wear apply: the load tension, bending around pulleys, friction with other objects and friction between its own strands.
Fatigue is much of what limits the life of a cable/wire rope, and both bending stress and abrasion play a role. Each time a section of the cable passes through a pulley or is wound around a drum, it must bend and straighten again. This creates a repeated reversing stress that causes cracks to grow through the material until it breaks (fatigue). Abrasion has a synergistic effect, by roughening the outside of the wires and making it easier for cracks to form. Both the abrasion and bending stresses are greatest at the outside of the cable, so those strands tend to fail first.
It might not even be the cable itself that breaks. Consider an everyday cable actuated mechanism: the window regulator that raises and lowers side windows in an automobile. It's not usually the cable or motor that fails, but rather the pulleys break. The pulleys are usually plastic and only supported on one side, so they're not as strong as the steel cable they're supporting. Sometimes the manufacturers don't even use pulleys, and instead just use a U-shaped plastic channel. Then, the cable slowly saws through the plastic piece. I suppose it only has to outlast the warranty.
Some guidelines for maximum life are:
- Keep the diameters of pulleys and winding drums large relative to the cable. 10 times the diameter of the cable is barely enough, and 40 times the diameter of the cable is not too much.
- Lubricating the cable helps extend life both by reducing abrasion and by letting the strands slide against each other, making the cable more flexible and reducing bending stress.
- Don't neglect the other parts of the system, such as the pulleys the cable wraps around, guides it runs through, and its attachment points.
- Don't neglect the environment. Make sure any materials can handle the temperatures and chemicals they may be exposed to. Keep abrasive dust away from anything that moves.
As far as fatigue testing materials goes, there are commercial testing machines for fatigue testing wire and cable, and testing houses that can perform the tests as a service. Actually getting a sample to fail in fatigue can take days at low stress levels, and accelerated (high stress) tests don't map easily across different materials. Narrowing options down based on calculations before testing is a good idea.
Shigley et al. Mechanical Engineering Design, 7th ed. McGraw Hill, 2004 (specifically, chapter 17-6 on Wire Rope)
Wirerope Works Technical Bulletin on Fatigue