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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 = $\frac{1}{2}$FDelta L = $F^2$$\frac{L}{2AY}$ = YA(Delta L)^$\frac{2}{2L}$

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

Energy density (per unit volume): u = $\frac{1}{2}$sigmaepsilon = sigma^$\frac{2}{2Y}$ = $\frac{1}{2}$Y$epsilon^2$

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^2$$g^2$$\frac{L}{6AY}$, where the 6 (instead of 2) comes from the linearly varying force requiring integration.