View source for Clapeyron's theorem (elasticity)

In the linear theory of [[Physics:Elasticity|elasticity]] '''Clapeyron's theorem''' states that the [[Physics:Potential energy|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.<ref>Love, A.E.H., "A Treatise on the Mathematical Theory of Elasticity", 4th ed. Cambridge, 1927, p. 173</ref>

It is named after the French scientist [[Biography:Benoît Paul Émile Clapeyron|Benoît Clapeyron]].

For example, consider a linear spring with initial length ''L''<sub>0</sub> and gradually pull on the spring until it reaches equilibrium at a length ''L''<sub>1</sub> when the pulling force is&nbsp;''F''.  By the theorem, the potential energy of deformation in the spring is given by:

:<math>\frac{1}{2}F (L_1 - L_0).</math>

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|theorem of three moments]] used in bridge engineering is also sometimes called Clapeyron's theorem.

==References==
{{reflist}}
* Roger Fosdick & Lev Truskinovsky (2003) [https://link.springer.com/article/10.1023/B:ELAS.0000018757.56138.f3 About Clapeyron's Theorem in Linear Elasticity], [[Physics:Journal of Elasticity|Journal of Elasticity]] 72(1–3): 145–72, Springer.

[[Category:Structural analysis]]
[[Category:Continuum mechanics]]



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