Question

find an expression for the energy density in terms of sterr and strain

Answer 1

Elastic Potential energy

Without load, restoring force = 0

after removal of load , Restoring force = F

average restoring force = F + 0/2 = F/2

Work Done to stetch = avarage force x displacement

Work = F/2 x delta L x AL / AL

= 1/2 x F/A x delta L / L x AL

P.E = 1/2 x Stress x Strain x AL

P.E/AL = 1/2 x Stess x Strain.

Answer 2

Elastic potential energy density per volume =U/volume
u = U/volume
= 1/2*stress*strain*volume/volume
(volume/volume ,cancel out)
= 1/2*young modulus *strain
(because Y=stress/strain)
= 1/2* y* (strain)square
u= 1/2*young's modulus * strain square

hope it will helps u.