There are a number of different failure mechanisms. I'll keep this simple in layman's terms: -
First some definitions:
1) Elastic - structure will return to original "position" on removal of load. i.e. deformation proportional to load - no load - no deformation. Plotting out a graph of deformation vs load will be a straight line.
2) Plastic - when the elastic limit is reached, the structure continues to deform. Howver, the realtionship is no longer linear (or elastic) and when load is removed there will be some permanent deformation of the structure.
3) Plastic hinge - structural term for the location where the structure is likely to / will fail. To form a hinge, the structure reaches its plastic limit. From then on there will be no elastic recovery at all as a hinge has formed.
4) Fatigue - failure due to repeated loading and unloading of a structure. (This is why aircraft have reference cycles - it is a connection back to the certifcation process and how many fatigue cycles the aircraft has been designed for... think Comet disasters....)
5) Crack propagation - Cracks can occur as a result of either fatigue or overload (Brittle fracture - where deformation by load occurs at a rate quicker then the material can deformm which itself is temperature dependent) However, once a crack occurs, cyclic loading is usually the main driver for propagation (lengthening) of the crack.
So to answer the question (I hope) then there are two possible failure mechanisms....
1) Overload. This is the classic bend and break scenario. i.e. if an elevator or other control surace is overloaded, the structure will remain elastic then go plastic before a hinge forms and the offending bit breaks off....
2) Crack failure - A crack has formed and then the repeated cyclic loading eventually causes the crack to propage until eventually overload occurs and the offending bit breaks off. The relative lengths of the crack to the failure surface are a big clue as to what actually caused the failure but would require a bit more typing to explain.
The examples posted by barit are typical of plastic deformation.
Regards,
Shuttlebus