Elastic Strain Energy

Work done = force × distance travelled in the direction of the force.To calculate the work done we use the average force ½F.ΔE_el = ½ F Δ xWhen a spring or wire is stretched, the work done on it is stored as elastic strain energy. If it obeys Hooke's law, the force rises steadily from 0 to F, so the average force is ½F and the work done is ½F multiplied by the extension Δx.This is the same as the area under the force-extension graph, which for the straight-line region is a triangle of base Δx and height F.Since F = kΔx, the energy can also be writtenΔE_el = ½ k (Δx)²so doubling the extension stores four times as much energy.Worked exampleA spring of stiffness 250 N m^-1 is stretched by 0.080 m.Force: F = kΔx = 250 × 0.080 = 20 NEnergy stored: ΔE_el = ½ × 20 × 0.080 = 0.80 JCheck: ½ × 250 × 0.080² = 0.80 JIf a material is stretched beyond its limit of proportionality the graph is curved, so ½FΔx no longer applies and the energy is found by estimating the area under the curve, for example by counting squares. The stored energy is only fully recovered when the load is removed if the deformation was elastic.