Assume I have two pieces of metal. I want to test the material fatigue (e.g. how many cycles can the material stand before it will break). Onto one of them I apply a tension force every second. Onto the other one I apply a tension force every hour. Should the results have any difference?
Yes, depending on the material there might be a difference. I assume you mean a short interval of force applied followed by an hour of rest, not a force applied constantly for an hour.
One hour is a long time and allows the material to return to a relaxed state. A shorter time interval will give the material less time to "repair the damage" done by the force you applied (atoms can not move to fill lattice errors, e.g.).
Basically, the shorter the time interval the faster it will break (maybe except for some special materials). On the extreme you test the material for its ability to withstand vibration.
No, the sample should fail after the same number of load cycles. For a metal piece tested under room conditions, the stress must exceed the elastic limit (somewhere in the specimen) to cause any damage to the sample. The length of time between the time when the stress is greater than this is generally irrelevant. This is the premise for conducting fatigue experiments to determine the 'lifetime' of a piece on an accelerated schedule.
The differences between one cycle per hour and one cycle per second, at 25 degrees C, is going to be very small for metal. Thermally activated 'recovery' processes just won't proceed fast enough to have any effect at this temperature.
ASTM defines fatigue life, as the number of stress cycles of a specified character that a specimen sustains before failure of a specified nature occurs.
However, if the two experiments give significantly different results this could be an indication that there are time-dependent processes contributing to failure. But this is not the behavior of metals. Silica glass is an example of a material that has a strength strongly affected by aging.
In theory: Provided that there is no significant increase in temperature between a 1 cycle/s and 1 cycle/hr, the failure mode of the 2 tests will remain the same and hence results should be identical...
In real life: Fatigue problem is also a statistical problem, you have to test batches of samples at the two speeds in order to tell if there is difference.
Metal Fatigue will occur even below elastic limit [or yield strength]. The 'Fatigue limit' maybe a more appropriate term to consider in engineering design, depends in your desire lifetime.