In the linear theory of
elasticity Clapeyron's theorem states that the
potential energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
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.
[Love, A.E.H., "A Treatise on the Mathematical Theory of Elasticity", 4th ed. Cambridge, 1927, p. 173]
It is named after the French scientist
Émile Clapeyron
Benoît Paul Émile Clapeyron (; 26 January 1799 – 28 January 1864) was a French engineer and physicist, one of the founders of thermodynamics.
Life
Born in Paris, Clapeyron studied at the École polytechnique, graduating in 1818. He also studi ...
.
For example, consider a linear spring with initial length ''L''
0 and gradually pull on the spring until it reaches equilibrium at a length ''L''
1 when the pulling force is ''F''. By the theorem, the potential energy of deformation in the spring is given by:
:
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
In civil engineering and structural analysis Clapeyron's theorem of three moments (by Émile Clapeyron) is a relationship among the bending moments at three consecutive supports of a horizontal beam.
Let ''A,B,C-D be the three consecutive points o ...
used in bridge engineering is also sometimes called Clapeyron's theorem.
References
* Roger Fosdick & Lev Truskinovsky (2003
About Clapeyron's Theorem in Linear Elasticity Journal of Elasticity
''Journal of Elasticity: The Physical and Mathematical Science of Solids'' is a peer-reviewed scientific journal covering all aspects of elasticity. It is published seven times a year by Springer Science+Business Media. The editor-in-chief is Roge ...
72(1–3): 145–72, Springer.
Structural analysis
Continuum mechanics
Eponymous theorems of physics
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