Elastic strain energy density via Hooke's law

Steps:

• Hooke's law gives strain ε = S / E, where S is tensile stress and E is Young's modulus.
• Elastic energy per unit volume equals work done per unit volume, (1/2) × stress × strain for linear response.
• Substitute ε into formula: u = (1/2) S × (S / E).
• Simplify: u = S² / (2E).

Why the correct expression is S² / (2E):

• It derives from integrating stress over strain, (1/2) σ ε, per Hooke's law definition.

Why the others are wrong:

• A. \frac{E}{2S}: Reverses variables, yields incorrect dimensions for energy/volume.
• B. \frac{E^{2}}{S}: Wrong functional form, unrelated to strain energy derivation.
• C. \frac{S^{2}}{E}: Omits 1/2 factor, overestimates energy by double.
• D. \frac{SE}{2}: Treats E as strain instead of modulus, invalid substitution.

Final answer: \frac{S^{2}}{2E}