Minimum total potential energy principle

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The minimum total potential energy principle is a fundamental concept used in physics and engineering. It dictates that at low temperatures a structure or body shall deform or displace to a position that (locally) minimizes the total potential energy, with the lost potential energy being converted into kinetic energy (specifically heat).

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Some examples

Structural mechanics

The total potential energy, , is the sum of the elastic strain energy, U, stored in the deformed body and the potential energy, V, associated to the applied forces: [1]

This energy is at a stationary position when an infinitesimal variation from such position involves no change in energy: [1]

The principle of minimum total potential energy may be derived as a special case of the virtual work principle for elastic systems subject to conservative forces.

The equality between external and internal virtual work (due to virtual displacements) is:

where

In the special case of elastic bodies, the right-hand-side of ( 3 ) can be taken to be the change, , of elastic strain energy U due to infinitesimal variations of real displacements. In addition, when the external forces are conservative forces, the left-hand-side of ( 3 ) can be seen as the change in the potential energy function V of the forces. The function V is defined as: [2] where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions, Equation 3 becomes: This leads to ( 2 ) as desired. The variational form of ( 2 ) is often used as the basis for developing the finite element method in structural mechanics.

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References

  1. 1 2 Reddy, J. N. (2006). Theory and Analysis of Elastic Plates and Shells (2nd illustrated revised ed.). CRC Press. p. 59. ISBN   978-0-8493-8415-8. Extract of page 59
  2. Reddy, J. N. (2007). An Introduction to Continuum Mechanics. Cambridge University Press. p. 244. ISBN   978-1-139-46640-0. Extract of page 244