473:
334:
760:
297:
212:
811:
661:
468:{\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}
157:
624:
589:
538:
514:
246:
105:
A rolling ball will end up stationary at the bottom of a hill, the point of minimum potential energy. The reason is that as it rolls downward under the influence of
565:
179:
771:
826:
762:
where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions,
38:. It dictates that at low temperatures a structure or body shall deform or displace to a position that (locally) minimizes the total
890:
860:
755:{\displaystyle \mathbf {V} =-\int _{S_{t}}\mathbf {u} ^{T}\mathbf {T} dS-\int _{V}\mathbf {u} ^{T}\mathbf {f} dV}
114:
140:
604:
125:
20:
128:. In this case, the dissipation takes the form of vibration of atoms within or adjacent to the protein.
95:
920:
572:
521:
497:
915:
632:
due to infinitesimal variations of real displacements. In addition, when the external forces are
72:
897:
867:
543:
326:
The equality between external and internal virtual work (due to virtual displacements) is:
91:
8:
80:
315:
The principle of minimum total potential energy may be derived as a special case of the
633:
320:
886:
856:
643:
231:
39:
880:
850:
292:{\displaystyle \delta {\boldsymbol {\Pi }}=\delta (\mathbf {U} +\mathbf {V} )=0}
43:
909:
235:
68:
316:
76:
64:
35:
113:
produced by its motion transfers energy in the form of heat of the
110:
87:
84:
60:
121:
106:
99:
31:
16:
Principle in physics regarding total potential energy of a system
595:
In the special case of elastic bodies, the right-hand-side of (
56:
207:{\displaystyle {\boldsymbol {\Pi }}=\mathbf {U} +\mathbf {V} }
806:{\displaystyle -\delta \ \mathbf {V} =\delta \ \mathbf {U} }
855:(2nd illustrated revised ed.). CRC Press. p. 59.
238:
variation from such position involves no change in energy:
63:
will tend to combine to form the lowest energy state (the
165:, stored in the deformed body and the potential energy,
42:, with the lost potential energy being converted into
774:
664:
607:
575:
546:
524:
500:
337:
249:
182:
143:
90:. The dissipation in this system takes the form of
805:
754:
618:
583:
559:
540:= vector of distributed forces acting on the part
532:
508:
467:
291:
206:
151:
907:
852:Theory and Analysis of Elastic Plates and Shells
825:) is often used as the basis for developing the
75:. This is because that state's energy is 13.6
827:finite element method in structural mechanics
885:. Cambridge University Press. p. 244.
159:, is the sum of the elastic strain energy,
319:principle for elastic systems subject to
844:
842:
28:minimum total potential energy principle
819:) as desired. The variational form of (
455:
444:
132:
908:
882:An Introduction to Continuum Mechanics
117:with an attendant increase in entropy.
878:
872:
848:
839:
328:
240:
173:
171:, associated to the applied forces:
152:{\displaystyle {\boldsymbol {\Pi }}}
642:) can be seen as the change in the
619:{\displaystyle \delta \mathbf {U} }
13:
14:
932:
601:) can be taken to be the change,
30:is a fundamental concept used in
799:
785:
742:
731:
706:
695:
666:
612:
577:
526:
502:
416:
405:
374:
363:
276:
268:
254:
200:
192:
184:
145:
49:
124:folds into the state of lowest
280:
264:
1:
832:
652:of the forces. The function
584:{\displaystyle \mathbf {f} }
533:{\displaystyle \mathbf {T} }
509:{\displaystyle \mathbf {u} }
137:The total potential energy,
7:
821:
815:
764:
638:
626:, of elastic strain energy
597:
481:
305:
220:
21:principle of minimum energy
10:
937:
18:
636:, the left-hand-side of (
516:= vector of displacements
96:electromagnetic radiation
79:lower than when the two
19:Not to be confused with
591:= vector of body forces
807:
756:
620:
585:
561:
534:
510:
469:
293:
208:
153:
98:, which increases the
879:Reddy, J. N. (2007).
849:Reddy, J. N. (2006).
808:
757:
621:
586:
562:
560:{\displaystyle S_{t}}
535:
511:
470:
294:
209:
154:
46:(specifically heat).
772:
662:
605:
573:
544:
522:
498:
335:
247:
230:This energy is at a
180:
141:
133:Structural mechanics
102:of the surroundings.
92:spontaneous emission
898:Extract of page 244
634:conservative forces
321:conservative forces
232:stationary position
77:electron volts (eV)
868:Extract of page 59
803:
752:
616:
581:
557:
530:
506:
465:
289:
204:
149:
71:, the most stable
892:978-1-139-46640-0
862:978-0-8493-8415-8
797:
783:
489:
488:
402:
360:
313:
312:
228:
227:
928:
900:
896:
876:
870:
866:
846:
812:
810:
809:
804:
802:
795:
788:
781:
761:
759:
758:
753:
745:
740:
739:
734:
728:
727:
709:
704:
703:
698:
692:
691:
690:
689:
669:
657:
651:
644:potential energy
631:
625:
623:
622:
617:
615:
590:
588:
587:
582:
580:
566:
564:
563:
558:
556:
555:
539:
537:
536:
531:
529:
515:
513:
512:
507:
505:
483:
474:
472:
471:
466:
458:
453:
452:
447:
438:
437:
419:
414:
413:
408:
400:
396:
395:
377:
372:
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366:
358:
354:
353:
352:
351:
329:
307:
298:
296:
295:
290:
279:
271:
257:
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222:
213:
211:
210:
205:
203:
195:
187:
174:
170:
164:
158:
156:
155:
150:
148:
126:potential energy
83:separated by an
40:potential energy
936:
935:
931:
930:
929:
927:
926:
925:
921:Solid mechanics
906:
905:
904:
903:
893:
877:
873:
863:
847:
840:
835:
813:This leads to (
798:
784:
773:
770:
769:
741:
735:
730:
729:
723:
719:
705:
699:
694:
693:
685:
681:
680:
676:
665:
663:
660:
659:
658:is defined as:
653:
647:
627:
611:
606:
603:
602:
576:
574:
571:
570:
551:
547:
545:
542:
541:
525:
523:
520:
519:
501:
499:
496:
495:
454:
448:
443:
442:
433:
429:
415:
409:
404:
403:
391:
387:
373:
367:
362:
361:
347:
343:
342:
338:
336:
333:
332:
275:
267:
253:
248:
245:
244:
199:
191:
183:
181:
178:
177:
166:
160:
144:
142:
139:
138:
135:
52:
24:
17:
12:
11:
5:
934:
924:
923:
918:
916:Thermodynamics
902:
901:
891:
871:
861:
837:
836:
834:
831:
801:
794:
791:
787:
780:
777:
751:
748:
744:
738:
733:
726:
722:
718:
715:
712:
708:
702:
697:
688:
684:
679:
675:
672:
668:
614:
610:
593:
592:
579:
568:
567:of the surface
554:
550:
528:
517:
504:
487:
486:
477:
475:
464:
461:
457:
451:
446:
441:
436:
432:
428:
425:
422:
418:
412:
407:
399:
394:
390:
386:
383:
380:
376:
370:
365:
357:
350:
346:
341:
311:
310:
301:
299:
288:
285:
282:
278:
274:
270:
266:
263:
260:
256:
252:
226:
225:
216:
214:
202:
198:
194:
190:
186:
147:
134:
131:
130:
129:
118:
103:
51:
48:
44:kinetic energy
15:
9:
6:
4:
3:
2:
933:
922:
919:
917:
914:
913:
911:
899:
894:
888:
884:
883:
875:
869:
864:
858:
854:
853:
845:
843:
838:
830:
828:
824:
823:
818:
817:
792:
789:
778:
775:
767:
766:
749:
746:
736:
724:
720:
716:
713:
710:
700:
686:
682:
677:
673:
670:
656:
650:
645:
641:
640:
635:
630:
608:
600:
599:
569:
552:
548:
518:
494:
493:
492:
485:
478:
476:
462:
459:
449:
439:
434:
430:
426:
423:
420:
410:
397:
392:
388:
384:
381:
378:
368:
355:
348:
344:
339:
331:
330:
327:
324:
322:
318:
309:
302:
300:
286:
283:
272:
261:
258:
250:
243:
242:
239:
237:
236:infinitesimal
233:
224:
217:
215:
196:
188:
176:
175:
172:
169:
163:
127:
123:
119:
116:
112:
108:
104:
101:
97:
93:
89:
86:
82:
78:
74:
73:configuration
70:
69:hydrogen atom
66:
62:
58:
54:
53:
50:Some examples
47:
45:
41:
37:
33:
29:
22:
881:
874:
851:
820:
814:
763:
654:
648:
637:
628:
596:
594:
490:
479:
325:
317:virtual work
314:
303:
229:
218:
167:
161:
136:
115:surroundings
65:ground state
27:
25:
36:engineering
910:Categories
833:References
765:Equation 3
793:δ
779:δ
776:−
768:becomes:
721:∫
717:−
678:∫
674:−
646:function
609:δ
456:σ
445:ϵ
440:δ
431:∫
398:δ
389:∫
356:δ
340:∫
262:δ
255:Π
251:δ
185:Π
146:Π
81:particles
59:and free
234:when an
111:friction
88:distance
85:infinite
61:electron
122:protein
107:gravity
100:entropy
67:) of a
55:A free
32:physics
889:
859:
796:
782:
491:where
401:
359:
57:proton
887:ISBN
857:ISBN
34:and
26:The
94:of
912::
841:^
829:.
323:.
120:A
109:,
895:.
865:.
822:2
816:2
800:U
790:=
786:V
750:V
747:d
743:f
737:T
732:u
725:V
714:S
711:d
707:T
701:T
696:u
687:t
683:S
671:=
667:V
655:V
649:V
639:3
629:U
613:U
598:3
578:f
553:t
549:S
527:T
503:u
484:)
482:3
480:(
463:V
460:d
450:T
435:V
427:=
424:V
421:d
417:f
411:T
406:u
393:V
385:+
382:S
379:d
375:T
369:T
364:u
349:t
345:S
308:)
306:2
304:(
287:0
284:=
281:)
277:V
273:+
269:U
265:(
259:=
223:)
221:1
219:(
201:V
197:+
193:U
189:=
168:V
162:U
23:.
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