Strain Energy
When external forces are applied to a body the resisting force starts developing gradually till the deformation is fully developed. Work is done by the resisting force due to the deformation. This work is stored in the form of energy. We call this Strain Energy.
Let's consider a bar of length L with cross-sectional area A under axial load P. Let Resistance developed be R.
When deformation = 0, Resistance R = 0
When deformation = Δ, Resistance R = P
Deformation = Δ = strain * original length = ε * L
Work done by resisting force
= average resistance * deformation
= 0.5* (0+P) * Δ
= P * ε * L/2
= p * A * ε * L/2 [Because, P = Force = Stress * Area = p * A ]
= p * ε * A * L/2
= p * ε * V/2
= p * (p/E) * (V/2) [Because, ε = Strain = p/E ]
= (p^2 * V) / 2E
Resilience
Strain Energy per unit Volume is defined as Resilience.
Substituting V with 1,
Work done by resisting force per unit volume = Resilience = p^2 / 2E
The same derivation can be done as well using the graphical method as shown below:
Proof Resilience
Maximum strain energy which may be stored in a body, i.e. when corresponding stress of the strain reaches yield stress, is termed as Proof Resilience.
Proof Resilience = py^2 / 2E
where, py = stress at elastic limit