In the linear theory of
elasticity Clapeyron's theorem states that the
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
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
Benoît Clapeyron.
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 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
Physics theorems
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