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,
, 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
= vector of displacements
= vector of distributed forces acting on the part
of the surface
= vector of body forces
In the special case of elastic bodies, the right-hand-side of can be taken to be the change,
, 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] where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions, becomes:
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
- 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
- Book: An Introduction to Continuum Mechanics . J. N. . Reddy . Cambridge University Press . 2007 . 978-1-139-46640-0 . 244 . Extract of page 244