词条 | Minimum total potential energy principle |
释义 |
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 mechanicsThe 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 = 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 (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, (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. References1. ^1 {{cite book |title=Theory and Analysis of Elastic Plates and Shells |edition=2nd illustrated revised |first1=J. N. |last1=Reddy |publisher=CRC Press |year=2006 |isbn=978-0-8493-8415-8 |page=59 |url=https://books.google.com/books?id=qA9A54UD0gkC}} [https://books.google.com/books?id=qA9A54UD0gkC&pg=PA59 Extract of page 59] 2. ^{{cite book |title=An Introduction to Continuum Mechanics |first1=J. N. |last1=Reddy |publisher=Cambridge University Press |year=2007 |isbn=978-1-139-46640-0 |page=244 |url=https://books.google.com/books?id=7nRGJMEmvJMC}} [https://books.google.com/books?id=7nRGJMEmvJMC&pg=PA244 Extract of page 244] 2 : Thermodynamics|Solid mechanics |
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