When you stretch a piece of material and allow it to return to its original shape multiple times, the molecules in the material start to deform and the bonds between them weaken. After a while, the material might break. This is elastic fatigue.

Example:

An elastic band. Most elastic bands snap because of elastic fatigue.

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fatigue is caused by repeated cycling of of the load.  It is a progressive localized damage due to fluctuating stresses and strains on the material.   fatigue cracks initiate and propagate in regions where the strain is most severe.   

The process of fatigue consists of three stages:

*  Initial crack initiation
*  Progressive crack growth across the part
*  Final sudden fracture of the remaining cross section       !!!!