2377:
1934:
2372:{\displaystyle {\begin{aligned}\sigma _{\text{v}}&={\sqrt {3J_{2}}}\\&={\sqrt {\frac {(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+\left(\sigma _{33}-\sigma _{11})^{2}+6(\sigma _{12}^{2}+\sigma _{23}^{2}+\sigma _{31}^{2}\right)}{2}}}\\&={\sqrt {\frac {(\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{3}-\sigma _{1})^{2}}{2}}}\\&={\sqrt {{\frac {3}{2}}s_{ij}s_{ij}}}\end{aligned}}\,\!}
2584:, because the stress tensor has six independent components. Therefore, it is difficult to tell which of the two specimens is closer to the yield point or has even reached it. However, by means of the von Mises yield criterion, which depends solely on the value of the scalar von Mises stress, i.e., one degree of freedom, this comparison is straightforward: A larger von Mises value implies that the material is closer to the yield point.
886:
2520:
4409:(1924) offered a physical interpretation of von Mises criterion suggesting that yielding begins when the elastic energy of distortion reaches a critical value. For this reason, the von Mises criterion is also known as the maximum distortion strain energy criterion. This comes from the relation between
2580:. As an example, the stress state of a steel beam in compression differs from the stress state of a steel axle under torsion, even if both specimens are of the same material. In view of the stress tensor, which fully describes the stress state, this difference manifests in six
3428:
4352:
4177:
2916:
4584:
reaches a critical value, i.e. the octahedral shear stress of the material at yield in simple tension. In this case, the von Mises yield criterion is also known as the maximum octahedral shear stress criterion in view of the direct proportionality that exists between
4770:
As shown in the equations above, the use of the von Mises criterion as a yield criterion is only exactly applicable when the following material properties are isotropic, and the ratio of the shear yield strength to the tensile yield strength has the following value:
3866:
3969:
4760:
Strain energy density consists of two components - volumetric or dialational and distortional. Volumetric component is responsible for change in volume without any change in shape. Distortional component is responsible for shear deformation or change in
4846:
Since no material will have this ratio precisely, in practice it is necessary to use engineering judgement to decide what failure theory is appropriate for a given material. Alternately, for use of the Tresca theory, the same ratio is defined as 1/2.
1636:
3753:
4061:
1012:
stress of the material in pure shear. As shown later in this article, at the onset of yielding, the magnitude of the shear yield stress in pure shear is â3 times lower than the tensile yield stress in the case of simple tension. Thus, we have:
3123:
2450:
1531:
4756:
4841:
4914:
4696:
2731:
3663:
4236:
1784:
3494:
3198:
3011:
1149:
4247:
4072:
2767:
4518:
2442:
4252:
4077:
3882:
3679:
1939:
3759:
1829:
1056:
4394:
2635:
1625:
926:
1219:
4570:
91:
3877:
3166:
1710:
1088:
is tensile yield strength of the material. If we set the von Mises stress equal to the yield strength and combine the above equations, the von Mises yield criterion is written as:
4637:
2578:
2551:
1926:
1891:
802:
767:
983:
2671:
3674:
2952:
1670:
4461:
3975:
1586:
877:
formulated the same criterion as von Mises independently in 1924. For the above reasons this criterion is also referred to as the "MaxwellâHuberâHenckyâvon Mises theory".
2759:
1860:
1086:
3019:
2410:
4610:
4434:
1249:
844:
701:
2515:{\displaystyle {\boldsymbol {\sigma }}^{\text{dev}}={\boldsymbol {\sigma }}-{\frac {\operatorname {tr} \left({\boldsymbol {\sigma }}\right)}{3}}\mathbf {I} \,\!}
1555:
1006:
640:
846:, it is applicable for the analysis of plastic deformation for ductile materials such as metals, as onset of yield for these materials does not depend on the
1261:
2761:
times lower than the yield stress in the case of simple tension. The von Mises yield criterion for pure shear stress, expressed in principal stresses, is
4707:
4777:
1928:
is used to predict yielding of materials under multiaxial loading conditions using results from simple uniaxial tensile tests. Thus, we define
4976:
4856:
4645:
2679:
873:
to some extent this criterion by properly relying on the distortion strain energy, not on the total strain energy as his predecessors.
3500:
5198:
5055:
4183:
633:
4951:
3423:{\displaystyle \sigma _{\text{v}}={\sqrt {{\frac {1}{2}}\left+3\left(\sigma _{12}^{2}+\sigma _{23}^{2}+\sigma _{31}^{2}\right)}}}
1724:
3439:
4347:{\displaystyle {\begin{aligned}\sigma _{2}&=\sigma _{3}=0\!\\\sigma _{12}&=\sigma _{31}=\sigma _{23}=0\!\end{aligned}}}
4172:{\displaystyle {\begin{aligned}\sigma _{1}&=\sigma _{2}=\sigma _{3}=0\!\\\sigma _{31}&=\sigma _{23}=0\!\end{aligned}}}
2957:
2911:{\displaystyle (\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{1}-\sigma _{3})^{2}=2\sigma _{y}^{2}\,\!}
1094:
2581:
606:
307:
144:
3861:{\displaystyle \sigma _{\text{v}}={\sqrt {\sigma _{11}^{2}-\sigma _{11}\sigma _{22}+\sigma _{22}^{2}+3\sigma _{12}^{2}}}\!}
5235:
4469:
2418:
1561:
as a circular cylinder (See Figure) whose yield curve, or intersection with the deviatoric plane, is a circle with radius
5245:
5159:
Hencky, H. (1924). "Zur
Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannngen".
1792:
1019:
626:
347:
233:
302:
4358:
1712:
within the red area. Because Tresca's criterion for yielding is within the red area, Von Mises' criterion is more lax.
211:
2594:
94:
5230:
4966:
3964:{\displaystyle {\begin{aligned}\sigma _{3}&=0\!\\\sigma _{12}&=\sigma _{31}=\sigma _{23}=0\!\end{aligned}}}
1591:
1160:
892:
4523:
218:
47:
513:
508:
123:
4946:
297:
290:
3131:
1675:
5240:
866:
576:
571:
240:
804:. The von Mises stress is used to predict yielding of materials under complex loading from the results of
128:
4615:
2556:
2529:
1904:
1869:
780:
773:. In this case, a material is said to start yielding when the von Mises stress reaches a value known as
745:
551:
169:
1639:
Von Mises yield criterion in 2D (planar) loading conditions: if stress in the third dimension is zero (
5022:
3748:{\displaystyle {\begin{aligned}\sigma _{3}&=0\!\\\sigma _{31}&=\sigma _{23}=0\!\end{aligned}}}
5168:
4581:
4056:{\displaystyle \sigma _{\text{v}}={\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}-\sigma _{1}\sigma _{2}}}\!}
2413:
1252:
946:
847:
816:
770:
720:
674:
389:
206:
186:
174:
118:
4577:
2640:
4956:
4931:
2924:
1642:
1009:
937:
870:
854:
712:
666:
654:
591:
439:
332:
38:
4439:
1564:
889:
The von Mises yield surfaces in principal stress coordinates circumscribes a cylinder with radius
738:, the von Mises yield criterion is also formulated in terms of the von Mises stress or equivalent
4941:
3118:{\displaystyle \sigma _{1}^{2}-\sigma _{1}\sigma _{2}+\sigma _{2}^{2}=3k^{2}=\sigma _{y}^{2}\,\!}
2740:
1838:
1064:
611:
245:
201:
196:
4996:
5194:
5051:
5045:
2737:
This means that, at the onset of yielding, the magnitude of the shear stress in pure shear is
862:
731:
716:
704:
228:
179:
5176:
4961:
2385:
566:
541:
454:
429:
424:
379:
4588:
4412:
1227:
822:
679:
4406:
874:
805:
724:
556:
480:
394:
325:
259:
161:
444:
314:
5172:
5104:
4971:
1540:
991:
774:
739:
561:
419:
384:
285:
191:
5224:
4926:
1558:
1526:{\displaystyle \sigma _{\text{v}}^{2}={\frac {1}{2}}\left={\frac {3}{2}}s_{ij}s_{ij}}
809:
601:
434:
5092:
Huber, M. T. (1904). "WĆaĆciwa praca odksztaĆcenia jako miara wytezenia materiaĆu".
4936:
2588:
929:
586:
581:
546:
278:
808:. The von Mises stress satisfies the property where two stress states with equal
1627:. This implies that the yield condition is independent of hydrostatic stresses.
858:
735:
596:
499:
17:
4751:{\displaystyle \tau _{\text{oct}}={\frac {\sqrt {2}}{3}}\sigma _{\text{y}}\,\!}
1893:, in agreement with the definition of tensile (or compressive) yield strength.
1635:
4836:{\displaystyle {\frac {F_{sy}}{F_{ty}}}={\frac {1}{\sqrt {3}}}\approx 0.577\!}
518:
414:
5180:
885:
490:
485:
319:
469:
374:
354:
340:
670:
223:
857:
in 1865, Maxwell only described the general conditions in a letter to
4909:{\displaystyle MS_{\text{yld}}={\frac {F_{y}}{\sigma _{\text{v}}}}-1}
364:
4691:{\displaystyle \tau _{\text{oct}}={\sqrt {{\frac {2}{3}}J_{2}}}\,\!}
2726:{\displaystyle \sigma _{12}=k={\frac {\sigma _{y}}{\sqrt {3}}}\,\!}
1634:
884:
708:
268:
769:. This is a scalar value of stress that can be computed from the
5027:
Nachrichten von der
Gesellschaft der Wissenschaften zu Göttingen
3658:{\displaystyle \sigma _{\text{v}}={\sqrt {{\frac {1}{2}}\left}}}
2553:, reaches the yield strength of the material in simple tension,
4231:{\displaystyle \sigma _{\text{v}}={\sqrt {3}}|\sigma _{12}|\!}
404:
5023:"Mechanik der festen Körper im plastisch-deformablen Zustand"
1901:
An equivalent tensile stress or equivalent von-Mises stress,
5193:
S. M. A. Kazimi. (1982). Solid
Mechanics. Tata McGraw-Hill.
4766:
Practical engineering usage of the von Mises yield criterion
1672:), no yielding is predicted to occur for stress coordinates
815:
Because the von Mises yield criterion is independent of the
4997:"Von Mises Criterion (Maximum Distortion Energy Criterion)"
1779:{\displaystyle \sigma _{1}\neq 0,\sigma _{3}=\sigma _{2}=0}
3489:{\displaystyle \sigma _{12}=\sigma _{31}=\sigma _{23}=0\!}
2526:
In this case, yielding occurs when the equivalent stress,
1631:
Reduced von Mises equation for different stress conditions
5105:"Specific Work of Strain as a Measure of Material Effort"
4402:
Physical interpretation of the von Mises yield criterion
3006:{\displaystyle \sigma _{12}=\sigma _{23}=\sigma _{31}=0}
1144:{\displaystyle \sigma _{v}=\sigma _{y}={\sqrt {3J_{2}}}}
5047:
Deformation Theory of
Plasticity, p. 151, Section 4.5.6
1557:
is called deviatoric stress. This equation defines the
707:
that mostly applies to ductile materials, such as some
1594:
895:
4859:
4780:
4710:
4648:
4618:
4591:
4526:
4472:
4442:
4415:
4361:
4250:
4186:
4075:
3978:
3880:
3762:
3677:
3503:
3442:
3201:
3134:
3022:
2960:
2927:
2770:
2743:
2682:
2643:
2597:
2559:
2532:
2453:
2421:
2388:
1937:
1907:
1872:
1841:
1795:
1727:
1678:
1645:
1567:
1543:
1264:
1230:
1163:
1097:
1067:
1022:
994:
949:
825:
783:
748:
682:
50:
5016:
5014:
4513:{\displaystyle W_{\text{D}}={\frac {J_{2}}{2G}}\,\!}
2437:{\displaystyle {\boldsymbol {\sigma }}^{\text{dev}}}
853:Although it has been believed it was formulated by
4908:
4835:
4750:
4690:
4631:
4604:
4564:
4512:
4455:
4428:
4388:
4346:
4230:
4171:
4055:
3963:
3860:
3747:
3657:
3488:
3422:
3160:
3117:
3005:
2946:
2910:
2753:
2725:
2665:
2629:
2572:
2545:
2514:
2436:
2404:
2371:
1920:
1885:
1854:
1824:{\displaystyle \sigma _{1}=\sigma _{\text{y}}\,\!}
1823:
1778:
1721:In the case of uniaxial stress or simple tension,
1704:
1664:
1619:
1580:
1549:
1525:
1243:
1213:
1143:
1080:
1051:{\displaystyle k={\frac {\sigma _{y}}{\sqrt {3}}}}
1050:
1000:
977:
920:
838:
796:
761:
695:
85:
4832:
4747:
4687:
4561:
4509:
4385:
4339:
4288:
4227:
4164:
4126:
4052:
3956:
3905:
3857:
3740:
3702:
3485:
3128:This equation represents an ellipse in the plane
3114:
2907:
2722:
2511:
2368:
1820:
974:
4389:{\displaystyle \sigma _{\text{v}}=\sigma _{1}\!}
5124:
5122:
2630:{\displaystyle \sigma _{12}=\sigma _{21}\neq 0}
1835:which means the material starts to yield when
1620:{\textstyle {\sqrt {\frac {2}{3}}}\sigma _{y}}
921:{\textstyle {\sqrt {\frac {2}{3}}}\sigma _{y}}
715:, material response can be assumed to be of a
1214:{\displaystyle \sigma _{v}^{2}=3J_{2}=3k^{2}}
634:
8:
4565:{\displaystyle G={\frac {E}{2(1+\nu )}}\,\!}
4436:and the elastic strain energy of distortion
1786:, the von Mises criterion simply reduces to
928:around the hydrostatic axis. Also shown is
86:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
848:hydrostatic component of the stress tensor
703:reaches a critical value. It is a part of
641:
627:
474:
264:
107:
29:
4892:
4882:
4876:
4867:
4858:
4850:The yield margin of safety is written as
4814:
4800:
4787:
4781:
4779:
4746:
4740:
4724:
4715:
4709:
4686:
4678:
4664:
4662:
4653:
4647:
4623:
4617:
4596:
4590:
4560:
4533:
4525:
4508:
4492:
4486:
4477:
4471:
4447:
4441:
4420:
4414:
4379:
4366:
4360:
4327:
4314:
4297:
4276:
4259:
4251:
4249:
4222:
4216:
4207:
4200:
4191:
4185:
4152:
4135:
4114:
4101:
4084:
4076:
4074:
4044:
4034:
4021:
4016:
4003:
3998:
3992:
3983:
3977:
3944:
3931:
3914:
3889:
3881:
3879:
3849:
3844:
3828:
3823:
3810:
3800:
3787:
3782:
3776:
3767:
3761:
3728:
3711:
3686:
3678:
3676:
3642:
3632:
3619:
3603:
3593:
3580:
3564:
3554:
3541:
3519:
3517:
3508:
3502:
3473:
3460:
3447:
3441:
3407:
3402:
3389:
3384:
3371:
3366:
3340:
3330:
3317:
3301:
3291:
3278:
3262:
3252:
3239:
3217:
3215:
3206:
3200:
3152:
3139:
3133:
3113:
3107:
3102:
3089:
3073:
3068:
3055:
3045:
3032:
3027:
3021:
2991:
2978:
2965:
2959:
2932:
2926:
2906:
2900:
2895:
2879:
2869:
2856:
2840:
2830:
2817:
2801:
2791:
2778:
2769:
2744:
2742:
2721:
2708:
2702:
2687:
2681:
2648:
2642:
2615:
2602:
2596:
2564:
2558:
2537:
2531:
2510:
2505:
2490:
2477:
2469:
2460:
2455:
2452:
2428:
2423:
2420:
2393:
2387:
2367:
2352:
2339:
2325:
2323:
2300:
2290:
2277:
2261:
2251:
2238:
2222:
2212:
2199:
2188:
2160:
2155:
2142:
2137:
2124:
2119:
2100:
2090:
2077:
2059:
2049:
2036:
2020:
2010:
1997:
1986:
1968:
1959:
1946:
1938:
1936:
1912:
1906:
1877:
1871:
1846:
1840:
1819:
1813:
1800:
1794:
1764:
1751:
1732:
1726:
1696:
1683:
1677:
1650:
1644:
1611:
1595:
1593:
1568:
1566:
1542:
1514:
1501:
1487:
1468:
1463:
1450:
1445:
1432:
1427:
1406:
1396:
1383:
1367:
1357:
1344:
1328:
1318:
1305:
1283:
1274:
1269:
1263:
1235:
1229:
1205:
1189:
1173:
1168:
1162:
1133:
1124:
1115:
1102:
1096:
1072:
1066:
1035:
1029:
1021:
993:
973:
967:
954:
948:
912:
896:
894:
830:
824:
788:
782:
753:
747:
687:
681:
63:
49:
4580:suggested that yielding begins when the
3175:
4988:
3161:{\displaystyle \sigma _{1}-\sigma _{2}}
2921:In the case of principal plane stress,
2491:
2470:
2456:
2424:
1705:{\displaystyle \sigma _{1},\sigma _{2}}
498:
453:
403:
363:
267:
136:
110:
37:
869:(1904), in a paper written in Polish,
5212:Theory of Flow and Fracture of Solids
5131:The Mathematical Theory of Plasticity
5029:. Mathematisch-Physikalische Klasse.
675:second invariant of deviatoric stress
27:Failure Theory in continuum mechanics
7:
3013:, the von Mises criterion becomes:
659:maximum distortion energy criterion
4632:{\displaystyle \tau _{\text{oct}}}
2573:{\displaystyle \sigma _{\text{y}}}
2546:{\displaystyle \sigma _{\text{v}}}
1921:{\displaystyle \sigma _{\text{v}}}
1886:{\displaystyle \sigma _{\text{y}}}
865:rigorously formulated it in 1913.
797:{\displaystyle \sigma _{\text{y}}}
762:{\displaystyle \sigma _{\text{v}}}
25:
4977:BigoniâPiccolroaz yield criterion
4612:and the octahedral shear stress,
5146:History of strength of materials
2506:
812:have an equal von Mises stress.
5079:Advanced Mechanics of Materials
4520:with the elastic shear modulus
2673:, von Mises criterion becomes:
978:{\displaystyle J_{2}=k^{2}\,\!}
5044:Jones, Robert Millard (2009).
4554:
4542:
4223:
4208:
3639:
3612:
3600:
3573:
3561:
3534:
3337:
3310:
3298:
3271:
3259:
3232:
2876:
2849:
2837:
2810:
2798:
2771:
2666:{\displaystyle \sigma _{ij}=0}
2297:
2270:
2258:
2231:
2219:
2192:
2112:
2097:
2056:
2029:
2017:
1990:
1403:
1376:
1364:
1337:
1325:
1298:
1:
2947:{\displaystyle \sigma _{3}=0}
1897:Multi-axial (2D or 3D) stress
1665:{\displaystyle \sigma _{3}=0}
936:Mathematically the von Mises
4952:HoekâBrown failure criterion
4456:{\displaystyle W_{\text{D}}}
1581:{\displaystyle {\sqrt {2}}k}
2754:{\displaystyle {\sqrt {3}}}
1855:{\displaystyle \sigma _{1}}
1081:{\displaystyle \sigma _{y}}
940:criterion is expressed as:
932:'s hexagonal yield surface.
5262:
5133:. Oxford: Clarendon Press.
5050:. Bull Ridge Corporation.
4639:, which by definition is
673:material begins when the
663:von Mises yield criterion
5214:. New York: McGraw-Hill.
5181:10.1002/zamm.19240040405
5148:. New York: McGraw-Hill.
881:Mathematical formulation
145:ClausiusâDuhem (entropy)
95:Fick's laws of diffusion
5144:Timoshenko, S. (1953).
4582:octahedral shear stress
3872:Principal plane stress
867:Tytus Maksymilian Huber
863:Richard Edler von Mises
303:NavierâStokes equations
241:Material failure theory
5021:von Mises, R. (1913).
4910:
4837:
4752:
4692:
4633:
4606:
4566:
4514:
4457:
4430:
4390:
4348:
4232:
4173:
4057:
3965:
3862:
3749:
3659:
3490:
3424:
3162:
3119:
3007:
2948:
2912:
2755:
2727:
2667:
2631:
2574:
2547:
2516:
2438:
2414:stress deviator tensor
2406:
2405:{\displaystyle s_{ij}}
2373:
1922:
1887:
1856:
1825:
1780:
1713:
1706:
1666:
1621:
1582:
1551:
1527:
1245:
1215:
1145:
1082:
1052:
1002:
979:
933:
922:
840:
817:first stress invariant
806:uniaxial tensile tests
798:
763:
697:
87:
5109:Archives of Mechanics
5094:Czasopismo Techniczne
4911:
4838:
4753:
4693:
4634:
4607:
4605:{\displaystyle J_{2}}
4567:
4515:
4458:
4431:
4429:{\displaystyle J_{2}}
4391:
4349:
4233:
4174:
4058:
3966:
3863:
3750:
3669:General plane stress
3660:
3491:
3425:
3163:
3120:
3008:
2949:
2913:
2756:
2728:
2668:
2632:
2575:
2548:
2517:
2439:
2407:
2374:
1923:
1888:
1857:
1826:
1781:
1707:
1667:
1638:
1622:
1583:
1552:
1528:
1246:
1244:{\displaystyle J_{2}}
1216:
1146:
1083:
1053:
1003:
980:
923:
888:
841:
839:{\displaystyle I_{1}}
799:
764:
698:
696:{\displaystyle J_{2}}
298:Bernoulli's principle
291:Archimedes' principle
88:
5236:Plasticity (physics)
5161:Z. Angew. Math. Mech
4857:
4778:
4708:
4646:
4616:
4589:
4524:
4470:
4440:
4413:
4359:
4248:
4184:
4073:
3976:
3878:
3760:
3675:
3501:
3440:
3199:
3185:von Mises equations
3182:Boundary conditions
3132:
3020:
2958:
2925:
2768:
2741:
2680:
2641:
2595:
2587:In the case of pure
2557:
2530:
2451:
2419:
2386:
1935:
1905:
1870:
1839:
1793:
1725:
1717:Uniaxial (1D) stress
1676:
1643:
1592:
1565:
1541:
1262:
1253:Cauchy stress tensor
1228:
1161:
1095:
1065:
1020:
992:
947:
893:
823:
781:
771:Cauchy stress tensor
746:
680:
390:Cohesion (chemistry)
212:Infinitesimal strain
48:
5246:Structural analysis
5173:1924ZaMM....4..323H
5081:. London: Longmans.
4957:Yield (engineering)
4947:MohrâCoulomb theory
4026:
4008:
3854:
3833:
3792:
3434:Principal stresses
3412:
3394:
3376:
3112:
3078:
3037:
2905:
2165:
2147:
2129:
1473:
1455:
1437:
1279:
1255:components, we get
1178:
855:James Clerk Maxwell
655:continuum mechanics
308:Poiseuille equation
39:Continuum mechanics
33:Part of a series on
5210:Nadai, A. (1950).
4942:Stephen Timoshenko
4906:
4833:
4748:
4688:
4629:
4602:
4562:
4510:
4453:
4426:
4386:
4344:
4342:
4228:
4169:
4167:
4053:
4012:
3994:
3961:
3959:
3858:
3840:
3819:
3778:
3745:
3743:
3655:
3486:
3420:
3398:
3380:
3362:
3158:
3115:
3098:
3064:
3023:
3003:
2944:
2908:
2891:
2751:
2723:
2663:
2637:, while all other
2627:
2582:degrees of freedom
2570:
2543:
2512:
2434:
2412:are components of
2402:
2369:
2365:
2151:
2133:
2115:
1918:
1883:
1852:
1821:
1776:
1714:
1702:
1662:
1617:
1578:
1547:
1523:
1459:
1441:
1423:
1265:
1241:
1211:
1164:
1141:
1078:
1048:
998:
975:
934:
918:
836:
794:
759:
693:
514:Magnetorheological
509:Electrorheological
246:Fracture mechanics
83:
5231:Materials science
5129:Hill, R. (1950).
4999:. Engineer's edge
4898:
4895:
4870:
4824:
4823:
4809:
4743:
4734:
4730:
4718:
4684:
4672:
4656:
4626:
4558:
4506:
4480:
4450:
4399:
4398:
4369:
4205:
4194:
4050:
3986:
3855:
3770:
3653:
3527:
3511:
3418:
3225:
3209:
2749:
2719:
2718:
2567:
2540:
2503:
2463:
2431:
2361:
2333:
2311:
2310:
2176:
2175:
1974:
1949:
1915:
1880:
1816:
1605:
1604:
1573:
1550:{\displaystyle s}
1495:
1291:
1272:
1139:
1046:
1045:
1001:{\displaystyle k}
906:
905:
810:distortion energy
791:
756:
732:materials science
721:nonlinear elastic
705:plasticity theory
651:
650:
526:
525:
460:
459:
229:Contact mechanics
152:
151:
81:
16:(Redirected from
5253:
5216:
5215:
5207:
5201:
5191:
5185:
5184:
5156:
5150:
5149:
5141:
5135:
5134:
5126:
5117:
5116:
5115:: 173â190. 2004.
5101:
5089:
5083:
5082:
5074:
5068:
5067:
5065:
5064:
5041:
5035:
5034:
5018:
5009:
5008:
5006:
5004:
4993:
4932:Huber's equation
4915:
4913:
4912:
4907:
4899:
4897:
4896:
4893:
4887:
4886:
4877:
4872:
4871:
4868:
4842:
4840:
4839:
4834:
4825:
4819:
4815:
4810:
4808:
4807:
4795:
4794:
4782:
4757:
4755:
4754:
4749:
4745:
4744:
4741:
4735:
4726:
4725:
4720:
4719:
4716:
4697:
4695:
4694:
4689:
4685:
4683:
4682:
4673:
4665:
4663:
4658:
4657:
4654:
4638:
4636:
4635:
4630:
4628:
4627:
4624:
4611:
4609:
4608:
4603:
4601:
4600:
4571:
4569:
4568:
4563:
4559:
4557:
4534:
4519:
4517:
4516:
4511:
4507:
4505:
4497:
4496:
4487:
4482:
4481:
4478:
4462:
4460:
4459:
4454:
4452:
4451:
4448:
4435:
4433:
4432:
4427:
4425:
4424:
4395:
4393:
4392:
4387:
4384:
4383:
4371:
4370:
4367:
4353:
4351:
4350:
4345:
4343:
4332:
4331:
4319:
4318:
4302:
4301:
4281:
4280:
4264:
4263:
4237:
4235:
4234:
4229:
4226:
4221:
4220:
4211:
4206:
4201:
4196:
4195:
4192:
4178:
4176:
4175:
4170:
4168:
4157:
4156:
4140:
4139:
4119:
4118:
4106:
4105:
4089:
4088:
4062:
4060:
4059:
4054:
4051:
4049:
4048:
4039:
4038:
4025:
4020:
4007:
4002:
3993:
3988:
3987:
3984:
3970:
3968:
3967:
3962:
3960:
3949:
3948:
3936:
3935:
3919:
3918:
3894:
3893:
3867:
3865:
3864:
3859:
3856:
3853:
3848:
3832:
3827:
3815:
3814:
3805:
3804:
3791:
3786:
3777:
3772:
3771:
3768:
3754:
3752:
3751:
3746:
3744:
3733:
3732:
3716:
3715:
3691:
3690:
3664:
3662:
3661:
3656:
3654:
3652:
3648:
3647:
3646:
3637:
3636:
3624:
3623:
3608:
3607:
3598:
3597:
3585:
3584:
3569:
3568:
3559:
3558:
3546:
3545:
3528:
3520:
3518:
3513:
3512:
3509:
3495:
3493:
3492:
3487:
3478:
3477:
3465:
3464:
3452:
3451:
3429:
3427:
3426:
3421:
3419:
3417:
3413:
3411:
3406:
3393:
3388:
3375:
3370:
3350:
3346:
3345:
3344:
3335:
3334:
3322:
3321:
3306:
3305:
3296:
3295:
3283:
3282:
3267:
3266:
3257:
3256:
3244:
3243:
3226:
3218:
3216:
3211:
3210:
3207:
3193:No restrictions
3179:State of stress
3176:
3167:
3165:
3164:
3159:
3157:
3156:
3144:
3143:
3124:
3122:
3121:
3116:
3111:
3106:
3094:
3093:
3077:
3072:
3060:
3059:
3050:
3049:
3036:
3031:
3012:
3010:
3009:
3004:
2996:
2995:
2983:
2982:
2970:
2969:
2953:
2951:
2950:
2945:
2937:
2936:
2917:
2915:
2914:
2909:
2904:
2899:
2884:
2883:
2874:
2873:
2861:
2860:
2845:
2844:
2835:
2834:
2822:
2821:
2806:
2805:
2796:
2795:
2783:
2782:
2760:
2758:
2757:
2752:
2750:
2745:
2732:
2730:
2729:
2724:
2720:
2714:
2713:
2712:
2703:
2692:
2691:
2672:
2670:
2669:
2664:
2656:
2655:
2636:
2634:
2633:
2628:
2620:
2619:
2607:
2606:
2579:
2577:
2576:
2571:
2569:
2568:
2565:
2552:
2550:
2549:
2544:
2542:
2541:
2538:
2521:
2519:
2518:
2513:
2509:
2504:
2499:
2498:
2494:
2478:
2473:
2465:
2464:
2461:
2459:
2443:
2441:
2440:
2435:
2433:
2432:
2429:
2427:
2411:
2409:
2408:
2403:
2401:
2400:
2378:
2376:
2375:
2370:
2366:
2362:
2360:
2359:
2347:
2346:
2334:
2326:
2324:
2316:
2312:
2306:
2305:
2304:
2295:
2294:
2282:
2281:
2266:
2265:
2256:
2255:
2243:
2242:
2227:
2226:
2217:
2216:
2204:
2203:
2190:
2189:
2181:
2177:
2171:
2170:
2166:
2164:
2159:
2146:
2141:
2128:
2123:
2105:
2104:
2095:
2094:
2082:
2081:
2064:
2063:
2054:
2053:
2041:
2040:
2025:
2024:
2015:
2014:
2002:
2001:
1988:
1987:
1979:
1975:
1973:
1972:
1960:
1951:
1950:
1947:
1927:
1925:
1924:
1919:
1917:
1916:
1913:
1892:
1890:
1889:
1884:
1882:
1881:
1878:
1866:of the material
1861:
1859:
1858:
1853:
1851:
1850:
1830:
1828:
1827:
1822:
1818:
1817:
1814:
1805:
1804:
1785:
1783:
1782:
1777:
1769:
1768:
1756:
1755:
1737:
1736:
1711:
1709:
1708:
1703:
1701:
1700:
1688:
1687:
1671:
1669:
1668:
1663:
1655:
1654:
1626:
1624:
1623:
1618:
1616:
1615:
1606:
1597:
1596:
1587:
1585:
1584:
1579:
1574:
1569:
1556:
1554:
1553:
1548:
1532:
1530:
1529:
1524:
1522:
1521:
1509:
1508:
1496:
1488:
1483:
1479:
1478:
1474:
1472:
1467:
1454:
1449:
1436:
1431:
1411:
1410:
1401:
1400:
1388:
1387:
1372:
1371:
1362:
1361:
1349:
1348:
1333:
1332:
1323:
1322:
1310:
1309:
1292:
1284:
1278:
1273:
1270:
1250:
1248:
1247:
1242:
1240:
1239:
1220:
1218:
1217:
1212:
1210:
1209:
1194:
1193:
1177:
1172:
1150:
1148:
1147:
1142:
1140:
1138:
1137:
1125:
1120:
1119:
1107:
1106:
1087:
1085:
1084:
1079:
1077:
1076:
1057:
1055:
1054:
1049:
1047:
1041:
1040:
1039:
1030:
1007:
1005:
1004:
999:
984:
982:
981:
976:
972:
971:
959:
958:
927:
925:
924:
919:
917:
916:
907:
898:
897:
845:
843:
842:
837:
835:
834:
803:
801:
800:
795:
793:
792:
789:
768:
766:
765:
760:
758:
757:
754:
702:
700:
699:
694:
692:
691:
643:
636:
629:
475:
440:Gay-Lussac's law
430:Combined gas law
380:Capillary action
265:
108:
92:
90:
89:
84:
82:
80:
72:
64:
30:
21:
18:Von Mises stress
5261:
5260:
5256:
5255:
5254:
5252:
5251:
5250:
5221:
5220:
5219:
5209:
5208:
5204:
5192:
5188:
5158:
5157:
5153:
5143:
5142:
5138:
5128:
5127:
5120:
5103:
5091:
5090:
5086:
5076:
5075:
5071:
5062:
5060:
5058:
5043:
5042:
5038:
5020:
5019:
5012:
5002:
5000:
4995:
4994:
4990:
4986:
4981:
4922:
4888:
4878:
4863:
4855:
4854:
4796:
4783:
4776:
4775:
4768:
4736:
4711:
4706:
4705:
4674:
4649:
4644:
4643:
4619:
4614:
4613:
4592:
4587:
4586:
4538:
4522:
4521:
4498:
4488:
4473:
4468:
4467:
4443:
4438:
4437:
4416:
4411:
4410:
4404:
4375:
4362:
4357:
4356:
4341:
4340:
4323:
4310:
4303:
4293:
4290:
4289:
4272:
4265:
4255:
4246:
4245:
4212:
4187:
4182:
4181:
4166:
4165:
4148:
4141:
4131:
4128:
4127:
4110:
4097:
4090:
4080:
4071:
4070:
4040:
4030:
3979:
3974:
3973:
3958:
3957:
3940:
3927:
3920:
3910:
3907:
3906:
3895:
3885:
3876:
3875:
3806:
3796:
3763:
3758:
3757:
3742:
3741:
3724:
3717:
3707:
3704:
3703:
3692:
3682:
3673:
3672:
3638:
3628:
3615:
3599:
3589:
3576:
3560:
3550:
3537:
3533:
3529:
3504:
3499:
3498:
3469:
3456:
3443:
3438:
3437:
3361:
3357:
3336:
3326:
3313:
3297:
3287:
3274:
3258:
3248:
3235:
3231:
3227:
3202:
3197:
3196:
3174:
3148:
3135:
3130:
3129:
3085:
3051:
3041:
3018:
3017:
2987:
2974:
2961:
2956:
2955:
2928:
2923:
2922:
2875:
2865:
2852:
2836:
2826:
2813:
2797:
2787:
2774:
2766:
2765:
2739:
2738:
2704:
2683:
2678:
2677:
2644:
2639:
2638:
2611:
2598:
2593:
2592:
2560:
2555:
2554:
2533:
2528:
2527:
2486:
2479:
2454:
2449:
2448:
2422:
2417:
2416:
2389:
2384:
2383:
2364:
2363:
2348:
2335:
2314:
2313:
2296:
2286:
2273:
2257:
2247:
2234:
2218:
2208:
2195:
2191:
2179:
2178:
2096:
2086:
2073:
2072:
2068:
2055:
2045:
2032:
2016:
2006:
1993:
1989:
1977:
1976:
1964:
1952:
1942:
1933:
1932:
1908:
1903:
1902:
1899:
1873:
1868:
1867:
1842:
1837:
1836:
1809:
1796:
1791:
1790:
1760:
1747:
1728:
1723:
1722:
1719:
1692:
1679:
1674:
1673:
1646:
1641:
1640:
1633:
1607:
1590:
1589:
1563:
1562:
1539:
1538:
1510:
1497:
1422:
1418:
1402:
1392:
1379:
1363:
1353:
1340:
1324:
1314:
1301:
1297:
1293:
1260:
1259:
1231:
1226:
1225:
1201:
1185:
1159:
1158:
1129:
1111:
1098:
1093:
1092:
1068:
1063:
1062:
1031:
1018:
1017:
990:
989:
963:
950:
945:
944:
908:
891:
890:
883:
875:Heinrich Hencky
861:(Lord Kelvin).
859:William Thomson
826:
821:
820:
784:
779:
778:
749:
744:
743:
683:
678:
677:
647:
618:
617:
616:
536:
528:
527:
481:Viscoelasticity
472:
462:
461:
449:
399:
395:Surface tension
359:
262:
260:Fluid mechanics
252:
251:
250:
164:
162:Solid mechanics
154:
153:
105:
97:
73:
65:
46:
45:
28:
23:
22:
15:
12:
11:
5:
5259:
5257:
5249:
5248:
5243:
5241:Yield criteria
5238:
5233:
5223:
5222:
5218:
5217:
5202:
5186:
5167:(4): 323â334.
5151:
5136:
5118:
5102:Translated as
5084:
5069:
5056:
5036:
5010:
4987:
4985:
4982:
4980:
4979:
4974:
4972:3-D elasticity
4969:
4964:
4959:
4954:
4949:
4944:
4939:
4934:
4929:
4923:
4921:
4918:
4917:
4916:
4905:
4902:
4891:
4885:
4881:
4875:
4866:
4862:
4844:
4843:
4831:
4828:
4822:
4818:
4813:
4806:
4803:
4799:
4793:
4790:
4786:
4767:
4764:
4763:
4762:
4758:
4739:
4733:
4729:
4723:
4714:
4699:
4698:
4681:
4677:
4671:
4668:
4661:
4652:
4622:
4599:
4595:
4578:Arpad L. Nadai
4574:
4573:
4556:
4553:
4550:
4547:
4544:
4541:
4537:
4532:
4529:
4504:
4501:
4495:
4491:
4485:
4476:
4446:
4423:
4419:
4403:
4400:
4397:
4396:
4382:
4378:
4374:
4365:
4354:
4338:
4335:
4330:
4326:
4322:
4317:
4313:
4309:
4306:
4304:
4300:
4296:
4292:
4291:
4287:
4284:
4279:
4275:
4271:
4268:
4266:
4262:
4258:
4254:
4253:
4243:
4239:
4238:
4225:
4219:
4215:
4210:
4204:
4199:
4190:
4179:
4163:
4160:
4155:
4151:
4147:
4144:
4142:
4138:
4134:
4130:
4129:
4125:
4122:
4117:
4113:
4109:
4104:
4100:
4096:
4093:
4091:
4087:
4083:
4079:
4078:
4068:
4064:
4063:
4047:
4043:
4037:
4033:
4029:
4024:
4019:
4015:
4011:
4006:
4001:
3997:
3991:
3982:
3971:
3955:
3952:
3947:
3943:
3939:
3934:
3930:
3926:
3923:
3921:
3917:
3913:
3909:
3908:
3904:
3901:
3898:
3896:
3892:
3888:
3884:
3883:
3873:
3869:
3868:
3852:
3847:
3843:
3839:
3836:
3831:
3826:
3822:
3818:
3813:
3809:
3803:
3799:
3795:
3790:
3785:
3781:
3775:
3766:
3755:
3739:
3736:
3731:
3727:
3723:
3720:
3718:
3714:
3710:
3706:
3705:
3701:
3698:
3695:
3693:
3689:
3685:
3681:
3680:
3670:
3666:
3665:
3651:
3645:
3641:
3635:
3631:
3627:
3622:
3618:
3614:
3611:
3606:
3602:
3596:
3592:
3588:
3583:
3579:
3575:
3572:
3567:
3563:
3557:
3553:
3549:
3544:
3540:
3536:
3532:
3526:
3523:
3516:
3507:
3496:
3484:
3481:
3476:
3472:
3468:
3463:
3459:
3455:
3450:
3446:
3435:
3431:
3430:
3416:
3410:
3405:
3401:
3397:
3392:
3387:
3383:
3379:
3374:
3369:
3365:
3360:
3356:
3353:
3349:
3343:
3339:
3333:
3329:
3325:
3320:
3316:
3312:
3309:
3304:
3300:
3294:
3290:
3286:
3281:
3277:
3273:
3270:
3265:
3261:
3255:
3251:
3247:
3242:
3238:
3234:
3230:
3224:
3221:
3214:
3205:
3194:
3191:
3187:
3186:
3183:
3180:
3173:
3170:
3155:
3151:
3147:
3142:
3138:
3126:
3125:
3110:
3105:
3101:
3097:
3092:
3088:
3084:
3081:
3076:
3071:
3067:
3063:
3058:
3054:
3048:
3044:
3040:
3035:
3030:
3026:
3002:
2999:
2994:
2990:
2986:
2981:
2977:
2973:
2968:
2964:
2943:
2940:
2935:
2931:
2919:
2918:
2903:
2898:
2894:
2890:
2887:
2882:
2878:
2872:
2868:
2864:
2859:
2855:
2851:
2848:
2843:
2839:
2833:
2829:
2825:
2820:
2816:
2812:
2809:
2804:
2800:
2794:
2790:
2786:
2781:
2777:
2773:
2748:
2735:
2734:
2717:
2711:
2707:
2701:
2698:
2695:
2690:
2686:
2662:
2659:
2654:
2651:
2647:
2626:
2623:
2618:
2614:
2610:
2605:
2601:
2563:
2536:
2524:
2523:
2508:
2502:
2497:
2493:
2489:
2485:
2482:
2476:
2472:
2468:
2458:
2426:
2399:
2396:
2392:
2380:
2379:
2358:
2355:
2351:
2345:
2342:
2338:
2332:
2329:
2322:
2319:
2317:
2315:
2309:
2303:
2299:
2293:
2289:
2285:
2280:
2276:
2272:
2269:
2264:
2260:
2254:
2250:
2246:
2241:
2237:
2233:
2230:
2225:
2221:
2215:
2211:
2207:
2202:
2198:
2194:
2187:
2184:
2182:
2180:
2174:
2169:
2163:
2158:
2154:
2150:
2145:
2140:
2136:
2132:
2127:
2122:
2118:
2114:
2111:
2108:
2103:
2099:
2093:
2089:
2085:
2080:
2076:
2071:
2067:
2062:
2058:
2052:
2048:
2044:
2039:
2035:
2031:
2028:
2023:
2019:
2013:
2009:
2005:
2000:
1996:
1992:
1985:
1982:
1980:
1978:
1971:
1967:
1963:
1958:
1955:
1953:
1945:
1941:
1940:
1911:
1898:
1895:
1876:
1864:yield strength
1849:
1845:
1833:
1832:
1812:
1808:
1803:
1799:
1775:
1772:
1767:
1763:
1759:
1754:
1750:
1746:
1743:
1740:
1735:
1731:
1718:
1715:
1699:
1695:
1691:
1686:
1682:
1661:
1658:
1653:
1649:
1632:
1629:
1614:
1610:
1603:
1600:
1577:
1572:
1546:
1535:
1534:
1520:
1517:
1513:
1507:
1504:
1500:
1494:
1491:
1486:
1482:
1477:
1471:
1466:
1462:
1458:
1453:
1448:
1444:
1440:
1435:
1430:
1426:
1421:
1417:
1414:
1409:
1405:
1399:
1395:
1391:
1386:
1382:
1378:
1375:
1370:
1366:
1360:
1356:
1352:
1347:
1343:
1339:
1336:
1331:
1327:
1321:
1317:
1313:
1308:
1304:
1300:
1296:
1290:
1287:
1282:
1277:
1268:
1238:
1234:
1222:
1221:
1208:
1204:
1200:
1197:
1192:
1188:
1184:
1181:
1176:
1171:
1167:
1152:
1151:
1136:
1132:
1128:
1123:
1118:
1114:
1110:
1105:
1101:
1075:
1071:
1059:
1058:
1044:
1038:
1034:
1028:
1025:
997:
986:
985:
970:
966:
962:
957:
953:
915:
911:
904:
901:
882:
879:
833:
829:
787:
775:yield strength
752:
740:tensile stress
717:linear elastic
690:
686:
665:) states that
649:
648:
646:
645:
638:
631:
623:
620:
619:
615:
614:
609:
604:
599:
594:
589:
584:
579:
574:
569:
564:
559:
554:
549:
544:
538:
537:
534:
533:
530:
529:
524:
523:
522:
521:
516:
511:
503:
502:
496:
495:
494:
493:
488:
483:
473:
468:
467:
464:
463:
458:
457:
451:
450:
448:
447:
442:
437:
432:
427:
422:
417:
411:
408:
407:
401:
400:
398:
397:
392:
387:
385:Chromatography
382:
377:
371:
368:
367:
361:
360:
358:
357:
338:
337:
336:
317:
305:
300:
288:
275:
272:
271:
263:
258:
257:
254:
253:
249:
248:
243:
238:
237:
236:
226:
221:
216:
215:
214:
209:
199:
194:
189:
184:
183:
182:
172:
166:
165:
160:
159:
156:
155:
150:
149:
148:
147:
139:
138:
134:
133:
132:
131:
126:
121:
113:
112:
106:
103:
102:
99:
98:
93:
79:
76:
71:
68:
62:
59:
56:
53:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5258:
5247:
5244:
5242:
5239:
5237:
5234:
5232:
5229:
5228:
5226:
5213:
5206:
5203:
5200:
5199:0-07-451715-5
5196:
5190:
5187:
5182:
5178:
5174:
5170:
5166:
5162:
5155:
5152:
5147:
5140:
5137:
5132:
5125:
5123:
5119:
5114:
5110:
5106:
5099:
5095:
5088:
5085:
5080:
5077:Ford (1963).
5073:
5070:
5059:
5057:9780978722319
5053:
5049:
5048:
5040:
5037:
5033:(1): 582â592.
5032:
5028:
5024:
5017:
5015:
5011:
4998:
4992:
4989:
4983:
4978:
4975:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4938:
4935:
4933:
4930:
4928:
4927:Yield surface
4925:
4924:
4919:
4903:
4900:
4889:
4883:
4879:
4873:
4864:
4860:
4853:
4852:
4851:
4848:
4829:
4826:
4820:
4816:
4811:
4804:
4801:
4797:
4791:
4788:
4784:
4774:
4773:
4772:
4765:
4759:
4737:
4731:
4727:
4721:
4712:
4704:
4703:
4702:
4701:thus we have
4679:
4675:
4669:
4666:
4659:
4650:
4642:
4641:
4640:
4620:
4597:
4593:
4583:
4579:
4551:
4548:
4545:
4539:
4535:
4530:
4527:
4502:
4499:
4493:
4489:
4483:
4474:
4466:
4465:
4464:
4444:
4421:
4417:
4408:
4401:
4380:
4376:
4372:
4363:
4355:
4336:
4333:
4328:
4324:
4320:
4315:
4311:
4307:
4305:
4298:
4294:
4285:
4282:
4277:
4273:
4269:
4267:
4260:
4256:
4244:
4241:
4240:
4217:
4213:
4202:
4197:
4188:
4180:
4161:
4158:
4153:
4149:
4145:
4143:
4136:
4132:
4123:
4120:
4115:
4111:
4107:
4102:
4098:
4094:
4092:
4085:
4081:
4069:
4066:
4065:
4045:
4041:
4035:
4031:
4027:
4022:
4017:
4013:
4009:
4004:
3999:
3995:
3989:
3980:
3972:
3953:
3950:
3945:
3941:
3937:
3932:
3928:
3924:
3922:
3915:
3911:
3902:
3899:
3897:
3890:
3886:
3874:
3871:
3870:
3850:
3845:
3841:
3837:
3834:
3829:
3824:
3820:
3816:
3811:
3807:
3801:
3797:
3793:
3788:
3783:
3779:
3773:
3764:
3756:
3737:
3734:
3729:
3725:
3721:
3719:
3712:
3708:
3699:
3696:
3694:
3687:
3683:
3671:
3668:
3667:
3649:
3643:
3633:
3629:
3625:
3620:
3616:
3609:
3604:
3594:
3590:
3586:
3581:
3577:
3570:
3565:
3555:
3551:
3547:
3542:
3538:
3530:
3524:
3521:
3514:
3505:
3497:
3482:
3479:
3474:
3470:
3466:
3461:
3457:
3453:
3448:
3444:
3436:
3433:
3432:
3414:
3408:
3403:
3399:
3395:
3390:
3385:
3381:
3377:
3372:
3367:
3363:
3358:
3354:
3351:
3347:
3341:
3331:
3327:
3323:
3318:
3314:
3307:
3302:
3292:
3288:
3284:
3279:
3275:
3268:
3263:
3253:
3249:
3245:
3240:
3236:
3228:
3222:
3219:
3212:
3203:
3195:
3192:
3189:
3188:
3184:
3181:
3178:
3177:
3171:
3169:
3153:
3149:
3145:
3140:
3136:
3108:
3103:
3099:
3095:
3090:
3086:
3082:
3079:
3074:
3069:
3065:
3061:
3056:
3052:
3046:
3042:
3038:
3033:
3028:
3024:
3016:
3015:
3014:
3000:
2997:
2992:
2988:
2984:
2979:
2975:
2971:
2966:
2962:
2941:
2938:
2933:
2929:
2901:
2896:
2892:
2888:
2885:
2880:
2870:
2866:
2862:
2857:
2853:
2846:
2841:
2831:
2827:
2823:
2818:
2814:
2807:
2802:
2792:
2788:
2784:
2779:
2775:
2764:
2763:
2762:
2746:
2715:
2709:
2705:
2699:
2696:
2693:
2688:
2684:
2676:
2675:
2674:
2660:
2657:
2652:
2649:
2645:
2624:
2621:
2616:
2612:
2608:
2603:
2599:
2590:
2585:
2583:
2561:
2534:
2500:
2495:
2487:
2483:
2480:
2474:
2466:
2447:
2446:
2445:
2415:
2397:
2394:
2390:
2356:
2353:
2349:
2343:
2340:
2336:
2330:
2327:
2320:
2318:
2307:
2301:
2291:
2287:
2283:
2278:
2274:
2267:
2262:
2252:
2248:
2244:
2239:
2235:
2228:
2223:
2213:
2209:
2205:
2200:
2196:
2185:
2183:
2172:
2167:
2161:
2156:
2152:
2148:
2143:
2138:
2134:
2130:
2125:
2120:
2116:
2109:
2106:
2101:
2091:
2087:
2083:
2078:
2074:
2069:
2065:
2060:
2050:
2046:
2042:
2037:
2033:
2026:
2021:
2011:
2007:
2003:
1998:
1994:
1983:
1981:
1969:
1965:
1961:
1956:
1954:
1943:
1931:
1930:
1929:
1909:
1896:
1894:
1874:
1865:
1847:
1843:
1810:
1806:
1801:
1797:
1789:
1788:
1787:
1773:
1770:
1765:
1761:
1757:
1752:
1748:
1744:
1741:
1738:
1733:
1729:
1716:
1697:
1693:
1689:
1684:
1680:
1659:
1656:
1651:
1647:
1637:
1630:
1628:
1612:
1608:
1601:
1598:
1575:
1570:
1560:
1559:yield surface
1544:
1518:
1515:
1511:
1505:
1502:
1498:
1492:
1489:
1484:
1480:
1475:
1469:
1464:
1460:
1456:
1451:
1446:
1442:
1438:
1433:
1428:
1424:
1419:
1415:
1412:
1407:
1397:
1393:
1389:
1384:
1380:
1373:
1368:
1358:
1354:
1350:
1345:
1341:
1334:
1329:
1319:
1315:
1311:
1306:
1302:
1294:
1288:
1285:
1280:
1275:
1266:
1258:
1257:
1256:
1254:
1236:
1232:
1224:Substituting
1206:
1202:
1198:
1195:
1190:
1186:
1182:
1179:
1174:
1169:
1165:
1157:
1156:
1155:
1134:
1130:
1126:
1121:
1116:
1112:
1108:
1103:
1099:
1091:
1090:
1089:
1073:
1069:
1042:
1036:
1032:
1026:
1023:
1016:
1015:
1014:
1011:
995:
968:
964:
960:
955:
951:
943:
942:
941:
939:
931:
913:
909:
902:
899:
887:
880:
878:
876:
872:
868:
864:
860:
856:
851:
849:
831:
827:
818:
813:
811:
807:
785:
776:
772:
750:
741:
737:
733:
728:
726:
722:
718:
714:
710:
706:
688:
684:
676:
672:
668:
664:
660:
656:
644:
639:
637:
632:
630:
625:
624:
622:
621:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
585:
583:
580:
578:
575:
573:
570:
568:
565:
563:
560:
558:
555:
553:
550:
548:
545:
543:
540:
539:
532:
531:
520:
517:
515:
512:
510:
507:
506:
505:
504:
501:
497:
492:
489:
487:
484:
482:
479:
478:
477:
476:
471:
466:
465:
456:
452:
446:
443:
441:
438:
436:
433:
431:
428:
426:
425:Charles's law
423:
421:
418:
416:
413:
412:
410:
409:
406:
402:
396:
393:
391:
388:
386:
383:
381:
378:
376:
373:
372:
370:
369:
366:
362:
356:
353:
349:
346:
342:
339:
334:
333:non-Newtonian
331:
327:
323:
322:
321:
318:
316:
313:
309:
306:
304:
301:
299:
296:
292:
289:
287:
284:
280:
277:
276:
274:
273:
270:
266:
261:
256:
255:
247:
244:
242:
239:
235:
232:
231:
230:
227:
225:
222:
220:
219:Compatibility
217:
213:
210:
208:
207:Finite strain
205:
204:
203:
200:
198:
195:
193:
190:
188:
185:
181:
178:
177:
176:
173:
171:
168:
167:
163:
158:
157:
146:
143:
142:
141:
140:
135:
130:
127:
125:
122:
120:
117:
116:
115:
114:
111:Conservations
109:
101:
100:
96:
77:
74:
69:
66:
60:
57:
54:
51:
44:
43:
40:
36:
32:
31:
19:
5211:
5205:
5189:
5164:
5160:
5154:
5145:
5139:
5130:
5112:
5108:
5097:
5093:
5087:
5078:
5072:
5061:. Retrieved
5046:
5039:
5030:
5026:
5001:. Retrieved
4991:
4937:Henri Tresca
4849:
4845:
4769:
4700:
4575:
4405:
3127:
2920:
2736:
2589:shear stress
2586:
2525:
2381:
1900:
1863:
1862:reaches the
1834:
1720:
1536:
1223:
1153:
1060:
987:
935:
852:
814:
729:
725:viscoelastic
662:
658:
652:
500:Smart fluids
445:Graham's law
351:
344:
329:
315:Pascal's law
311:
294:
282:
137:Inequalities
4067:Pure shear
871:anticipated
736:engineering
711:. Prior to
519:Ferrofluids
420:Boyle's law
192:Hooke's law
170:Deformation
5225:Categories
5063:2017-06-11
5003:8 February
4984:References
727:behavior.
572:Gay-Lussac
535:Scientists
435:Fick's law
415:Atmosphere
234:frictional
187:Plasticity
175:Elasticity
4901:−
4890:σ
4827:≈
4738:σ
4713:τ
4651:τ
4621:τ
4576:In 1937
4552:ν
4377:σ
4364:σ
4325:σ
4312:σ
4295:σ
4274:σ
4257:σ
4242:Uniaxial
4214:σ
4189:σ
4150:σ
4133:σ
4112:σ
4099:σ
4082:σ
4042:σ
4032:σ
4028:−
4014:σ
3996:σ
3981:σ
3942:σ
3929:σ
3912:σ
3887:σ
3842:σ
3821:σ
3808:σ
3798:σ
3794:−
3780:σ
3765:σ
3726:σ
3709:σ
3684:σ
3630:σ
3626:−
3617:σ
3591:σ
3587:−
3578:σ
3552:σ
3548:−
3539:σ
3506:σ
3471:σ
3458:σ
3445:σ
3400:σ
3382:σ
3364:σ
3328:σ
3324:−
3315:σ
3289:σ
3285:−
3276:σ
3250:σ
3246:−
3237:σ
3204:σ
3150:σ
3146:−
3137:σ
3100:σ
3066:σ
3053:σ
3043:σ
3039:−
3025:σ
2989:σ
2976:σ
2963:σ
2930:σ
2893:σ
2867:σ
2863:−
2854:σ
2828:σ
2824:−
2815:σ
2789:σ
2785:−
2776:σ
2706:σ
2685:σ
2646:σ
2622:≠
2613:σ
2600:σ
2562:σ
2535:σ
2492:σ
2484:
2475:−
2471:σ
2457:σ
2425:σ
2288:σ
2284:−
2275:σ
2249:σ
2245:−
2236:σ
2210:σ
2206:−
2197:σ
2153:σ
2135:σ
2117:σ
2088:σ
2084:−
2075:σ
2047:σ
2043:−
2034:σ
2008:σ
2004:−
1995:σ
1944:σ
1910:σ
1875:σ
1844:σ
1811:σ
1798:σ
1762:σ
1749:σ
1739:≠
1730:σ
1694:σ
1681:σ
1648:σ
1609:σ
1461:σ
1443:σ
1425:σ
1394:σ
1390:−
1381:σ
1355:σ
1351:−
1342:σ
1316:σ
1312:−
1303:σ
1267:σ
1251:with the
1166:σ
1113:σ
1100:σ
1070:σ
1033:σ
910:σ
786:σ
751:σ
612:Truesdell
542:Bernoulli
491:Rheometer
486:Rheometry
326:Newtonian
320:Viscosity
70:φ
58:−
4920:See also
3190:General
667:yielding
470:Rheology
375:Adhesion
355:Pressure
341:Buoyancy
286:Dynamics
124:Momentum
5169:Bibcode
5100:. LwĂłw.
3172:Summary
671:ductile
557:Charles
365:Liquids
279:Statics
224:Bending
5197:
5054:
4967:Strain
4962:Stress
4761:shape.
4407:Hencky
2382:where
1537:where
1061:where
930:Tresca
709:metals
661:(also
657:, the
607:Stokes
602:Pascal
592:Navier
587:Newton
577:Graham
552:Cauchy
455:Plasma
350:
348:Mixing
343:
328:
310:
293:
281:
269:Fluids
202:Strain
197:Stress
180:linear
129:Energy
4830:0.577
1588:, or
1010:yield
988:Here
938:yield
723:, or
713:yield
669:of a
582:Hooke
562:Euler
547:Boyle
405:Gases
5195:ISBN
5052:ISBN
5031:1913
5005:2018
2954:and
734:and
597:Noll
567:Fick
119:Mass
104:Laws
5177:doi
4869:yld
4717:oct
4655:oct
4625:oct
2462:dev
2430:dev
1154:or
1008:is
730:In
653:In
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5175:.
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850:.
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5183:.
5179::
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5066:.
5007:.
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