What are the criteria that govern the choice between titanium, composites, and more common materials, like aluminum?
What are the criteria that govern the choice between titanium, composites, and more common materials, like aluminum?


Three main criteria: cost, strength-to-weight ratio, and fatigue resistance.


Cost. Of the three, aluminium used to be the clear winner, with composites having made large advances due to improved manufacturing processes. Titanium is the most expensive and is difficult to machine. 
Strength-to-weight. 


Buckling. Titanium alloys have higher strength-to-weight than aluminium (this answer, more on temperature below), but aluminium is lighter and that gives it an advantage in structures loaded with compression stresses: buckling resistance is a function of cross sectional dimension as well. For the wing upper surface aluminium would be lighter than titanium, despite the lower specific strength. Composites have the highest specific strength of all.
Temperature. The graph in the linked answer also shows the influence of temperature on the specific strength of materials: aluminium is first to drop, and at higher temps titanium is the best next choice, as from the comment by @PeterKämpf.

Fatigue resistance. Titanium alloys, like steel allows, have a fatigue endurance limit. If stresses remain below this limit, the construction can endure an endless number of cycles. Aluminium does not have an endurance limit, and will eventually fail even from small stress amplitude cycles: aluminium constructions require careful monitoring and maintenance for prevention of fatigue failure.  



Composites do not have an endurance limit, but fibre orientation and material choice can improve fatigue life.


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