Clapeyron's theorem (elasticity)

In the linear theory of elasticity Clapeyron's theorem states that the potential energy of deformation of a body, which is in equilibrium under a given load, is equal to half the work done by the external forces computed assuming these forces had remained constant from the initial state to the final state.^[1]

It is named after the French scientist Benoît Clapeyron.

For example consider a linear spring with initial length L₀ and gradually pull on the string until it reaches equilibrium at a length L₁ when the pulling force is F. By the theorem, the potential energy of deformation in the spring is

$\frac{1}{2}F (L_1 - L_0).$

Note that the actual force increased from 0 to F during the deformation; the work done can be computed by integration in distance. Clapeyron's equation, which uses the final force only, may be puzzling at first, but is nevertheless true because it includes a corrective factor of one half.

Another theorem, the theorem of three moments used in bridge engineering is also sometimes called Clapeyron's theorem.

References

1. ^ Love, A.E.H., "A Treatise on the Mathematical Theory of Elasticity", 4th ed. Cambridge, 1927, p. 173