resilience
Resilience is how much energy a material can soak up while flexing and then give straight back when released — like a good bow that stores your pull and hurls the arrow, or a trampoline that returns the bounce. The key word is 'give back': it is the elastic, recoverable energy, stored in stretched bonds, not the energy lost in permanent damage.
On a stress-strain curve, resilience is the area under the ELASTIC (straight) part, up to the yield point. The modulus of resilience, the energy stored per unit volume at yield, is U_r = (yield strength)^2 / (2 x E). Read that formula: to store a lot of springy energy you want a high yield strength but a modest modulus (not too stiff), so the material can bend far while resisting hard. Example: a spring steel with yield 1000 MPa and E = 200 GPa stores U_r = (1000e6)^2 / (2 x 200e9) = 2.5 x 10^6 J/m^3.
This is exactly why spring materials are chosen for high strength and moderate stiffness — a diving board, a watch spring, a pole-vault pole. Do not confuse resilience with toughness: resilience is energy stored and returned within the ELASTIC range, while toughness is the total energy absorbed all the way to fracture, most of which goes into permanent (plastic) work and is not returned. A material can be resilient (a spring) yet not especially tough, or tough yet not resilient.
A pole-vaulter's fibreglass pole bends deeply, stores the runner's energy elastically, then straightens and flings them over the bar — high resilience means little of that energy is wasted.
Resilience is the area under the elastic part of the curve — energy in and back out.
Resilience (elastic energy returned) is not toughness (total energy to fracture, mostly plastic and not returned). A spring is resilient; annealed copper is tough. They are different areas of the same curve.