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When a body is elastically deformed, work is done against restoring forces and stored as elastic potential energy. For a wire stretched by force F through extension Delta L:

Total energy: U = (1/2)FDelta L = F^2L/(2AY) = YA(Delta L)^2/(2L)

The factor 1/2 arises because force increases linearly from 0 to F (like a spring). The wire's spring constant is k = YA/L, so U = (1/2)k*(Delta L)^2.

Energy density (per unit volume): u = (1/2)sigmaepsilon = sigma^2/(2Y) = (1/2)Yepsilon^2

This equals the area under the stress-strain curve up to the working point. The three equivalent forms use Hooke's law (sigma = Y*epsilon) to express energy in terms of stress alone, strain alone, or both.

Important scaling: U is proportional to F^2 (doubling force quadruples energy), proportional to L (longer wire stores more for same force), inversely proportional to A and Y. For comparing two wires, identify which variable changes and apply the appropriate formula form.

For a wire hanging under its own weight: U = M^2g^2L/(6AY), where the 6 (instead of 2) comes from the linearly varying force requiring integration.