Mechanical hysteresis can someone explain what hysteresis is in a mechanical system. I briefly think I understand, say we have a rubber cylinder and compress it with 15N then to 30N when we reduce the force back to 15N again there will be a difference in the overall length of the cylinder? is this correct?
 A: Take the old Galilei experient with the inclined plane. The plane is horizontal at first, you put an object on it, and then you start to incline the plane.
When the angle reaches a certain value $\alpha_1$, the object starts to slide down the slope and it won't stop until it reaches the bottom.
If, however, you start to lower the plane, you'll notice that the angle $\alpha_2$ at which the objet slows down and stops is lower than $\alpha_1$.
In other words, when the inclination of the plane is between $\alpha_1$ and $\alpha_2$, the angle isn't enough to know what happens: if the object is at rest, it'll remain at rest, if it moves, it'll keep moving.
That's hysteresis in a mechanical setting. The state of an object doesn't depend only on its mechanical parameters at the given time $t$, but also on those parameters at a previous time.
A: Hysteresis is when you have a repeted cycling loading, the force needed to go one way is different from the force needed to return back. Very roughly speaking if you plot Force vs. Displacement you would see the following

The area inside the curve represents the total work done every cycle lost to heat.
See page 2 of this paper for a more analytical approach to structural damping.

