Minimum total potential energy principle explained

Minimum total potential energy principle should not be confused with principle of minimum energy.

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).

Some examples

Structural mechanics

The total potential energy,

\boldsymbol{\Pi}

, is the sum of the elastic strain energy,, stored in the deformed body and the potential energy,, 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:

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

u

= vector of displacements

T

= vector of distributed forces acting on the part

St

of the surface

f

= vector of body forces

In the special case of elastic bodies, the right-hand-side of can be taken to be the change,

\deltaU

, of elastic strain energy due to infinitesimal variations of real displacements.In addition, when the external forces are conservative forces, the left-hand-side of can be seen as the change in the potential energy function of the forces. The function is defined as:[2] \mathbf = -\int_ \mathbf^T \mathbf dS - \int_ \mathbf^T \mathbf dV where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions, becomes: -\delta\ \mathbf = \delta\ \mathbf This leads to as desired. The variational form of is often used as the basis for developing the finite element method in structural mechanics.

Notes and References

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