# Clapeyron's theorem (elasticity)

"Clapeyron's theorem" redirects here. For Clapeyron's theorem used in bridge engineering, see Theorem of three moments.

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 spring 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 given by:

${\displaystyle {\frac {1}{2}}F(L_{1}-L_{0}).}$

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