In this case, the strap's polymer molecules have engaged in a process called "cross linking". That is, rather than just being sort of wrapped around each other, the long chain molecules in it have actually formed chemical bonds along their lenghts with each other. When this happens, the material will loose its elasticity and stiffen up. When this process reaches a certain point, if the material is flexed suddenly, it can actually crack in two!
If you can figure out a way to break the cross linkages within strap's material, then if MIGHT restore its elasticity. You might try soaking the strap in very hot water for awhile. That will get its molecules vibrating rapidly and that might help break the cross linkages.
You must be careful about soaking the strap in any type of plasticizer solutions because they might make it swell up or even dissolve!