# Minimum total potential energy principle

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).

## Structural mechanics

The total potential energy, ${\displaystyle {\boldsymbol {\Pi }}}$, 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]

${\displaystyle {\boldsymbol {\Pi }}=\mathbf {U} +\mathbf {V} \qquad \mathrm {(1)} }$

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

${\displaystyle \delta {\boldsymbol {\Pi }}=\delta (\mathbf {U} +\mathbf {V} )=0\qquad \mathrm {(2)} }$

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:

${\displaystyle \int _{S_{t}}\delta \ \mathbf {u} ^{T}\mathbf {T} dS+\int _{V}\delta \ \mathbf {u} ^{T}\mathbf {f} dV=\int _{V}\delta {\boldsymbol {\epsilon }}^{T}{\boldsymbol {\sigma }}dV\qquad \mathrm {(3)} }$

where

${\displaystyle \mathbf {u} }$ = vector of displacements
${\displaystyle \mathbf {T} }$ = vector of distributed forces acting on the part ${\displaystyle S_{t}}$ of the surface
${\displaystyle \mathbf {f} }$ = vector of body forces

In the special case of elastic bodies, the right-hand-side of (3) can be taken to be the change, ${\displaystyle \delta \mathbf {U} }$, 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]

${\displaystyle \mathbf {V} =-\int _{S_{t}}\mathbf {u} ^{T}\mathbf {T} dS-\int _{V}\mathbf {u} ^{T}\mathbf {f} dV}$

where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions, (3) becomes:

${\displaystyle -\delta \ \mathbf {V} =\delta \ \mathbf {U} }$

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.

## References

1. ^ a b 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