[geeknerd99]

This all depends on its application. I'll use bike frames and components, as that's what I have the most experience with involving these materials.

Ti is extremely lightweight, but also pretty darn expensive. It is probably the strongest of the 3, but also has benefical vibration dampening features. It's been said that it has the ride quality of steel, the weight of carbon fiber, and the durability to the elements of aluminum. I wouldn't flex it though.

Carbon is really light, but often has no crush resistance. It might be a decent choice if flexing (within reason) is required.

Aluminum is light enough, strong, but does not fatigue well at all. When aluminum goes, it goes suddenly and often catastrophically.

I would generally pick Ti for the weight/strength ratio, but the addition of flex in your query makes it difficult. Could you elaborate more on what you need this data for?