2435:
2024:
2964:
4024:
2430:{\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{bmatrix}}={\begin{bmatrix}\varepsilon _{xx}&{\tfrac {1}{2}}\gamma _{xy}&{\tfrac {1}{2}}\gamma _{xz}\\{\tfrac {1}{2}}\gamma _{yx}&\varepsilon _{yy}&{\tfrac {1}{2}}\gamma _{yz}\\{\tfrac {1}{2}}\gamma _{zx}&{\tfrac {1}{2}}\gamma _{zy}&\varepsilon _{zz}\\\end{bmatrix}}}
2536:
3652:
5105:
2959:{\displaystyle {\begin{aligned}\mathrm {length} (ab)&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&={\sqrt {dx^{2}\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+dx^{2}\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\&=dx~{\sqrt {\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\end{aligned}}}
5219:
4019:{\displaystyle {\begin{aligned}\tan \alpha &={\frac {{\tfrac {\partial u_{y}}{\partial x}}dx}{dx+{\tfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\tfrac {\partial u_{y}}{\partial x}}{1+{\tfrac {\partial u_{x}}{\partial x}}}}\\\tan \beta &={\frac {{\tfrac {\partial u_{x}}{\partial y}}dy}{dy+{\tfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\tfrac {\partial u_{x}}{\partial y}}{1+{\tfrac {\partial u_{y}}{\partial y}}}}\end{aligned}}}
2445:
4857:
1642:
4599:
3374:
1430:
5100:{\displaystyle {\frac {\Delta V}{V_{0}}}={\frac {\left(1+\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}+\varepsilon _{11}\cdot \varepsilon _{33}+\varepsilon _{22}\cdot \varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}\right)\cdot a^{3}-a^{3}}{a^{3}}}}
937:
4404:
972:
is the reference position of material points of the body; displacement has units of length and does not distinguish between rigid body motions (translations and rotations) and deformations (changes in shape and size) of the body. The spatial derivative of a uniform translation is zero, thus strains
5212:
1937:
3199:
3164:
3484:
4244:
1787:
5218:
4346:
4841:
837:
4730:
1637:{\displaystyle {\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta l}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln(\lambda )\\&=\ln(1+e)\\&=e-{\frac {e^{2}}{2}}+{\frac {e^{3}}{3}}-\cdots \end{aligned}}}
4119:
5110:
1080:
where strains and rotations are both small. In this case, the undeformed and deformed configurations of the body can be assumed identical. The infinitesimal strain theory is used in the analysis of deformations of materials exhibiting
1805:
4594:{\displaystyle \gamma _{yz}=\gamma _{zy}={\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\quad ,\qquad \gamma _{zx}=\gamma _{xz}={\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}}
3023:
980:
quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the
1986:(Mechanics), as a "tensor quantity representing the deformation of matter caused by stress. Strain tensor is symmetric and has three linear strain and three shear strain (Cartesian) components." ISO 80000-4 further defines
3387:
1112:
and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%; thus other more complex definitions of strain are required, such as
1260:
is defined as the tangent of that angle, and is equal to the length of deformation at its maximum divided by the perpendicular length in the plane of force application, which sometimes makes it easier to calculate.
768:
configuration. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the
1648:
is the engineering strain. The logarithmic strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path.
4154:
1666:
4249:
1425:
1231:
3369:{\displaystyle \varepsilon _{x}={\frac {\text{extension}}{\text{original length}}}={\frac {\mathrm {length} (ab)-\mathrm {length} (AB)}{\mathrm {length} (AB)}}={\frac {\partial u_{x}}{\partial x}}}
4638:
5395:
3657:
2541:
1435:
1144:, is expressed as the ratio of total deformation to the initial dimension of the material body on which forces are applied. In the case of a material line element or fiber axially loaded, its
1108:
is the most common definition applied to materials used in mechanical and structural engineering, which are subjected to very small deformations. On the other hand, for some materials, e.g.,
4029:
1976:
3645:
187:
4739:
2531:
1316:
1277:(symbol λ) is an alternative measure related to the extensional or normal strain of an axially loaded differential line element. It is defined as the ratio between the final length
1363:, which can sustain stretch ratios of 3 or 4 before they fail. On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios.
1256:
is defined as the change in the angle (in radians) between two material line elements initially perpendicular to each other in the undeformed or initial configuration. The
1354:
1174:
of the line element or fibers (in meters per meter). The normal strain is positive if the material fibers are stretched and negative if they are compressed. Thus, we have
932:{\displaystyle {\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}},}
3018:
2991:
736:
119:
1004:, radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions.
1000:, which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the
5207:{\displaystyle 1\gg \varepsilon _{ii}\gg \varepsilon _{ii}\cdot \varepsilon _{jj}\gg \varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}}
1038:, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of the
1932:{\displaystyle \varepsilon _{E}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{l^{2}}}\right)={\tfrac {1}{2}}\left(1-{\frac {1}{\lambda ^{2}}}\right)}
1389:
1177:
5338:
1994:
as the "quotient of parallel displacement of two surfaces of a layer and the thickness of the layer". Thus, strains are classified as either
3159:{\displaystyle \mathrm {length} (ab)\approx dx\left(1+{\frac {\partial u_{x}}{\partial x}}\right)=dx+{\frac {\partial u_{x}}{\partial x}}dx}
3479:{\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}}
1979:
5517:
5419:
5375:
1023:
Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories:
729:
5512:
3608:
1356:
This equation implies that when the normal strain is zero, so that there is no deformation, the stretch ratio is equal to unity.
5325:
2478:
2467:
1288:
702:
5425:
5381:
403:
240:
5502:
4239:{\displaystyle \alpha \approx {\frac {\partial u_{y}}{\partial x}}~;~~\beta \approx {\frac {\partial u_{x}}{\partial y}}}
1782:{\displaystyle \varepsilon _{G}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{L^{2}}}\right)={\tfrac {1}{2}}(\lambda ^{2}-1)}
5497:
722:
443:
329:
4736:, it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions
398:
4610:
1947:
1061:
989:, within a deforming body. This could be applied by elongation, shortening, or volume changes, or angular distortion.
307:
4341:{\displaystyle \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}}
190:
5243:
4836:{\displaystyle a\cdot (1+\varepsilon _{11})\times a\cdot (1+\varepsilon _{22})\times a\cdot (1+\varepsilon _{33})}
1959:
1327:
1082:
314:
143:
85:
1042:
are significantly different and a clear distinction has to be made between them. This is commonly the case with
5507:
609:
604:
219:
393:
386:
4725:{\displaystyle \delta ={\frac {\Delta V}{V_{0}}}=I_{1}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}}
996:
of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the
1085:
behavior, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel.
758:
672:
667:
336:
1145:
831:
224:
4617:
The volumetric strain, also called bulk strain, is the relative variation of the volume, as arising from
647:
265:
5238:
A strain field associated with a displacement is defined, at any point, by the change in length of the
4114:{\displaystyle {\frac {\partial u_{x}}{\partial x}}\ll 1~;~~{\frac {\partial u_{y}}{\partial y}}\ll 1}
5271:
5233:
1797:
1658:
1047:
1027:
485:
302:
282:
270:
214:
97:
1039:
1011:; otherwise, if there is reduction or compression in the length of the material line, it is called
993:
777:
765:
687:
535:
428:
134:
985:, and the amount of distortion associated with the sliding of plane layers over each other is the
5461:
5259:
4619:
827:
804:
707:
341:
292:
5247:
1359:
The stretch ratio is used in the analysis of materials that exhibit large deformations, such as
5415:
5371:
5334:
5275:
5263:
812:
808:
324:
275:
5411:
Basic
Engineering Plasticity: An Introduction with Engineering and Manufacturing Applications
5367:
Basic
Engineering Plasticity: An Introduction with Engineering and Manufacturing Applications
5492:
5251:
662:
637:
550:
525:
520:
475:
5439:
Hencky, H. (1928). "Über die Form des
Elastizitätsgesetzes bei ideal elastischen Stoffen".
2996:
2969:
5294:
946:
652:
576:
490:
421:
355:
257:
540:
410:
2452:
Consider a two-dimensional, infinitesimal, rectangular material element with dimensions
5267:
5255:
5239:
4632:
3178:
657:
515:
480:
381:
287:
104:
5486:
5344:
5304:
5282:
3569:
3532:
3182:
2011:
792:
788:
770:
697:
530:
2021:
The strain tensor can then be expressed in terms of normal and shear components as:
1007:
If there is an increase in length of the material line, the normal strain is called
815:(sometimes called "microstrains" and "nanostrains", respectively), corresponding to
5278:
3559:
3494:
2444:
2015:
682:
677:
642:
374:
69:
5409:
5365:
5299:
1983:
1055:
973:
measure how much a given displacement differs locally from a rigid-body motion.
692:
595:
17:
1109:
816:
800:
614:
510:
5258:, states that, if the lengths of the tangent vectors fulfil the axioms of a
3174:
1360:
1043:
820:
750:
586:
581:
415:
1427:
the logarithmic strain is obtained by integrating this incremental strain:
5234:
Finite strain theory § Deformation tensors in curvilinear coordinates
2448:
Two-dimensional geometric deformation of an infinitesimal material element
565:
470:
450:
436:
5223:
In case of pure shear, we can see that there is no change of the volume.
3527:
3519:
2966:
For very small displacement gradients the squares of the derivative of
2463:
796:
319:
48:
1990:
as the "quotient of change in length of an object and its length" and
1104:
In each of these theories the strain is then defined differently. The
5266:, then the length of a vector is the square root of the value of the
5246:
curves passing through that point. A basic geometric result, due to
3536:
977:
781:
460:
1095:, which assumes small strains but large rotations and displacements.
3513:
2443:
2010:
is parallel to it. These definitions are consistent with those of
1051:
784:
364:
27:
Symmetric tensor quantity of the strain caused by stress in matter
1948:
Infinitesimal strain theory § Infinitesimal strain tensor
500:
1320:
The extension ratio λ is related to the engineering strain
949:. The displacement of a body may be expressed in the form
5396:
Encyclopædia
Britannica 2006 Ultimate Reference Suite DVD
1420:{\displaystyle \delta \varepsilon ={\frac {\delta l}{l}}}
1226:{\displaystyle e={\frac {\Delta L}{L}}={\frac {l-L}{L}}}
864:
852:
3981:
3946:
3904:
3858:
3803:
3768:
3726:
3680:
2380:
2353:
2324:
2282:
2253:
2226:
2207:
2051:
1885:
1823:
1746:
1684:
867:
855:
5113:
4860:
4742:
4641:
4407:
4252:
4157:
4032:
3655:
3611:
3390:
3202:
3026:
2999:
2972:
2539:
2481:
2027:
1962:
1808:
1669:
1433:
1392:
1330:
1291:
1180:
840:
146:
107:
4732:
Actually, if we consider a cube with an edge length
3196:-direction of the rectangular element is defined by
3542:
3526:
3506:
3501:
3185:will cause a normal strain. Normal strains produce
2475:. From the geometry of the adjacent figure we have
96:
80:
67:
57:
47:
37:
32:
5206:
5099:
4835:
4724:
4593:
4340:
4238:
4113:
4018:
3639:
3593:) is defined as the change in angle between lines
3478:
3368:
3158:
3012:
2985:
2958:
2525:
2429:
2006:is perpendicular to the face of an element, and a
1970:
1931:
1781:
1636:
1419:
1348:
1310:
1225:
931:
181:
113:
2462:, which, after deformation, takes the form of a
4611:Infinitesimal strain theory § Volumetric strain
1386:. Considering an incremental strain (Ludwik)
730:
8:
1971:{\displaystyle {\boldsymbol {\varepsilon }}}
791:of meter per meter (m/m). Hence strains are
3640:{\displaystyle \gamma _{xy}=\alpha +\beta }
182:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
737:
723:
570:
360:
203:
125:
5198:
5185:
5172:
5156:
5140:
5124:
5112:
5089:
5078:
5065:
5047:
5034:
5021:
5008:
4995:
4982:
4969:
4956:
4943:
4930:
4917:
4904:
4886:
4875:
4861:
4859:
4824:
4793:
4762:
4741:
4716:
4703:
4690:
4677:
4662:
4648:
4640:
4574:
4564:
4544:
4534:
4522:
4506:
4480:
4470:
4450:
4440:
4428:
4412:
4406:
4321:
4311:
4291:
4281:
4257:
4251:
4219:
4209:
4174:
4164:
4156:
4088:
4078:
4043:
4033:
4031:
4026:For small displacement gradients we have
3991:
3980:
3956:
3944:
3914:
3903:
3868:
3857:
3854:
3813:
3802:
3778:
3766:
3736:
3725:
3690:
3679:
3676:
3656:
3654:
3649:From the geometry of the figure, we have
3616:
3610:
3459:
3449:
3440:
3414:
3404:
3395:
3389:
3349:
3339:
3301:
3267:
3232:
3229:
3216:
3207:
3201:
3133:
3123:
3089:
3079:
3027:
3025:
3004:
2998:
2977:
2971:
2944:
2923:
2913:
2899:
2877:
2867:
2853:
2826:
2805:
2795:
2784:
2768:
2746:
2736:
2718:
2709:
2691:
2663:
2653:
2638:
2610:
2600:
2583:
2544:
2540:
2538:
2482:
2480:
2410:
2395:
2379:
2368:
2352:
2339:
2323:
2312:
2297:
2281:
2268:
2252:
2241:
2225:
2214:
2202:
2182:
2167:
2152:
2135:
2120:
2105:
2088:
2073:
2058:
2046:
2028:
2026:
1963:
1961:
1916:
1907:
1884:
1869:
1858:
1845:
1838:
1822:
1813:
1807:
1764:
1745:
1730:
1719:
1706:
1699:
1683:
1674:
1668:
1613:
1607:
1593:
1587:
1508:
1469:
1463:
1458:
1434:
1432:
1402:
1391:
1329:
1298:
1290:
1205:
1187:
1179:
921:
909:
895:
887:
872:
868:
856:
849:
841:
839:
159:
145:
106:
5333:(Revised ed.). Dover Publications.
2526:{\displaystyle \mathrm {length} (AB)=dx}
1245:is the original length of the fiber and
5316:
5242:representing the speeds of arbitrarily
2031:
1964:
1802:The Euler-Almansi strain is defined as
1311:{\displaystyle \lambda ={\frac {l}{L}}}
922:
910:
842:
594:
549:
499:
459:
363:
232:
206:
133:
5359:
5357:
4628:
3498:
2466:. The deformation is described by the
29:
5456:
5454:
5414:. Butterworth-Heinemann. p. 41.
5394:"Earth."Encyclopædia Britannica from
7:
3376:Similarly, the normal strain in the
5107:as we consider small deformations,
4864:
4651:
4582:
4567:
4552:
4537:
4488:
4473:
4458:
4443:
4329:
4314:
4299:
4284:
4227:
4212:
4182:
4167:
4096:
4081:
4051:
4036:
3999:
3984:
3964:
3949:
3922:
3907:
3876:
3861:
3821:
3806:
3786:
3771:
3744:
3729:
3698:
3683:
3467:
3452:
3422:
3407:
3357:
3342:
3317:
3314:
3311:
3308:
3305:
3302:
3283:
3280:
3277:
3274:
3271:
3268:
3248:
3245:
3242:
3239:
3236:
3233:
3141:
3126:
3097:
3082:
3043:
3040:
3037:
3034:
3031:
3028:
2931:
2916:
2885:
2870:
2813:
2798:
2754:
2739:
2671:
2656:
2618:
2603:
2560:
2557:
2554:
2551:
2548:
2545:
2498:
2495:
2492:
2489:
2486:
2483:
1980:International System of Quantities
1249:is the final length of the fiber.
1190:
1078:small displacement-gradient theory
869:
857:
25:
5441:Zeitschrift für technische Physik
5428:from the original on 2017-12-22.
5384:from the original on 2017-12-22.
5217:
4609:This section is an excerpt from
1663:The Green strain is defined as:
1170:per unit of the original length
896:
888:
873:
826:Strain can be formulated as the
4501:
4497:
3435:
3431:
795:and are usually expressed as a
4830:
4811:
4799:
4780:
4768:
4749:
3584:The engineering shear strain (
3330:
3321:
3296:
3287:
3261:
3252:
3056:
3047:
2573:
2564:
2511:
2502:
1776:
1757:
1568:
1556:
1537:
1531:
1154:engineering extensional strain
1:
1982:(ISQ), more specifically in
1349:{\displaystyle e=\lambda -1}
3020:are negligible and we have
1062:Infinitesimal strain theory
773:or its dual is considered.
5534:
5231:
4608:
4121:For small rotations, i.e.
3492:
1945:
1795:
1656:
5370:. Butterworth-Heinemann.
3192:The normal strain in the
1239:engineering normal strain
1150:engineering normal strain
1074:small displacement theory
992:The state of strain at a
976:A strain is in general a
5518:Dimensionless quantities
5324:Lubliner, Jacob (2008).
1258:engineering shear strain
1163:or the change in length
1070:small deformation theory
1036:large deformation theory
241:Clausius–Duhem (entropy)
191:Fick's laws of diffusion
5513:Deformation (mechanics)
5214:therefore the formula.
4374:, it can be shown that
1281:and the initial length
757:is defined as relative
399:Navier–Stokes equations
337:Material failure theory
5208:
5101:
4837:
4726:
4629:first strain invariant
4595:
4342:
4240:
4115:
4020:
3641:
3480:
3370:
3160:
3014:
2987:
2960:
2527:
2449:
2431:
1972:
1933:
1783:
1638:
1421:
1350:
1312:
1285:of the material line.
1227:
933:
183:
115:
5209:
5102:
4838:
4727:
4596:
4343:
4241:
4116:
4021:
3642:
3481:
3371:
3161:
3015:
3013:{\displaystyle u_{x}}
2988:
2986:{\displaystyle u_{y}}
2961:
2528:
2447:
2432:
1973:
1946:Further information:
1934:
1784:
1639:
1422:
1351:
1313:
1228:
1093:large-rotation theory
1048:plastically-deforming
934:
394:Bernoulli's principle
387:Archimedes' principle
184:
116:
5503:Non-Newtonian fluids
5408:Rees, David (2006).
5364:Rees, David (2006).
5272:polarization formula
5111:
4858:
4740:
4639:
4405:
4250:
4155:
4030:
3653:
3609:
3388:
3384:-directions becomes
3200:
3177:material that obeys
3024:
2997:
2970:
2537:
2479:
2025:
1978:) is defined in the
1960:
1952:The (infinitesimal)
1806:
1798:Finite strain theory
1667:
1659:Finite strain theory
1431:
1390:
1328:
1289:
1178:
1050:materials and other
1028:Finite strain theory
838:
807:is also used, e.g.,
486:Cohesion (chemistry)
308:Infinitesimal strain
144:
105:
86:coord transformation
5498:Continuum mechanics
5270:associated, by the
4393:Similarly, for the
1468:
1161:relative elongation
1159:, which equals the
1066:small strain theory
1032:large strain theory
866:
854:
404:Poiseuille equation
135:Continuum mechanics
129:Part of a series on
5462:"ISO 80000-4:2019"
5204:
5097:
4833:
4722:
4591:
4338:
4236:
4111:
4016:
4014:
4007:
3972:
3930:
3884:
3829:
3794:
3752:
3706:
3637:
3476:
3366:
3156:
3010:
2983:
2956:
2954:
2523:
2468:displacement field
2450:
2427:
2421:
2389:
2362:
2333:
2291:
2262:
2235:
2193:
2041:
2037:
1968:
1929:
1894:
1832:
1779:
1755:
1693:
1634:
1632:
1454:
1417:
1373:logarithmic strain
1367:Logarithmic strain
1346:
1308:
1223:
1138:Engineering strain
1133:Engineering strain
1119:logarithmic strain
1106:engineering strain
1089:Large-displacement
1013:compressive strain
929:
878:
861:
828:spatial derivative
805:Parts-per notation
610:Magnetorheological
605:Electrorheological
342:Fracture mechanics
179:
111:
71:SI base units
5340:978-0-486-46290-5
5327:Plasticity Theory
5276:positive definite
5264:parallelogram law
5095:
4881:
4668:
4589:
4559:
4495:
4465:
4401:-planes, we have
4348:By interchanging
4336:
4306:
4234:
4202:
4199:
4193:
4189:
4103:
4077:
4074:
4068:
4058:
4010:
4006:
3971:
3939:
3929:
3883:
3832:
3828:
3793:
3761:
3751:
3705:
3582:
3581:
3474:
3429:
3364:
3334:
3224:
3223:
3220:
3148:
3104:
2950:
2938:
2892:
2852:
2832:
2820:
2761:
2697:
2678:
2625:
2440:Geometric setting
2388:
2361:
2332:
2290:
2261:
2234:
2030:
2029:
1922:
1893:
1875:
1831:
1754:
1736:
1692:
1622:
1602:
1516:
1482:
1415:
1306:
1254:true shear strain
1221:
1200:
1148:gives rise to an
880:
865:
853:
813:parts per billion
809:parts per million
747:
746:
622:
621:
556:
555:
325:Contact mechanics
248:
247:
177:
124:
123:
114:{\displaystyle 1}
16:(Redirected from
5525:
5477:
5476:
5474:
5473:
5458:
5449:
5448:
5436:
5430:
5429:
5405:
5399:
5392:
5386:
5385:
5361:
5352:
5351:
5349:
5343:. Archived from
5332:
5321:
5221:
5213:
5211:
5210:
5205:
5203:
5202:
5190:
5189:
5177:
5176:
5164:
5163:
5148:
5147:
5132:
5131:
5106:
5104:
5103:
5098:
5096:
5094:
5093:
5084:
5083:
5082:
5070:
5069:
5057:
5053:
5052:
5051:
5039:
5038:
5026:
5025:
5013:
5012:
5000:
4999:
4987:
4986:
4974:
4973:
4961:
4960:
4948:
4947:
4935:
4934:
4922:
4921:
4909:
4908:
4887:
4882:
4880:
4879:
4870:
4862:
4842:
4840:
4839:
4834:
4829:
4828:
4798:
4797:
4767:
4766:
4731:
4729:
4728:
4723:
4721:
4720:
4708:
4707:
4695:
4694:
4682:
4681:
4669:
4667:
4666:
4657:
4649:
4600:
4598:
4597:
4592:
4590:
4588:
4580:
4579:
4578:
4565:
4560:
4558:
4550:
4549:
4548:
4535:
4530:
4529:
4514:
4513:
4496:
4494:
4486:
4485:
4484:
4471:
4466:
4464:
4456:
4455:
4454:
4441:
4436:
4435:
4420:
4419:
4400:
4396:
4389:
4373:
4364:
4355:
4351:
4347:
4345:
4344:
4339:
4337:
4335:
4327:
4326:
4325:
4312:
4307:
4305:
4297:
4296:
4295:
4282:
4265:
4264:
4245:
4243:
4242:
4237:
4235:
4233:
4225:
4224:
4223:
4210:
4200:
4197:
4191:
4190:
4188:
4180:
4179:
4178:
4165:
4150:
4139:
4129:are ≪ 1 we have
4128:
4124:
4120:
4118:
4117:
4112:
4104:
4102:
4094:
4093:
4092:
4079:
4075:
4072:
4066:
4059:
4057:
4049:
4048:
4047:
4034:
4025:
4023:
4022:
4017:
4015:
4011:
4009:
4008:
4005:
3997:
3996:
3995:
3982:
3970:
3962:
3961:
3960:
3947:
3945:
3940:
3938:
3931:
3928:
3920:
3919:
3918:
3905:
3892:
3885:
3882:
3874:
3873:
3872:
3859:
3855:
3833:
3831:
3830:
3827:
3819:
3818:
3817:
3804:
3792:
3784:
3783:
3782:
3769:
3767:
3762:
3760:
3753:
3750:
3742:
3741:
3740:
3727:
3714:
3707:
3704:
3696:
3695:
3694:
3681:
3677:
3646:
3644:
3643:
3638:
3624:
3623:
3604:
3598:
3592:
3578:
3577:
3575:
3574:
3567:
3564:
3546:other quantities
3544:Derivations from
3522:
3516:
3499:
3485:
3483:
3482:
3477:
3475:
3473:
3465:
3464:
3463:
3450:
3445:
3444:
3430:
3428:
3420:
3419:
3418:
3405:
3400:
3399:
3383:
3379:
3375:
3373:
3372:
3367:
3365:
3363:
3355:
3354:
3353:
3340:
3335:
3333:
3320:
3299:
3286:
3251:
3230:
3225:
3221:
3218:
3217:
3212:
3211:
3195:
3165:
3163:
3162:
3157:
3149:
3147:
3139:
3138:
3137:
3124:
3110:
3106:
3105:
3103:
3095:
3094:
3093:
3080:
3046:
3019:
3017:
3016:
3011:
3009:
3008:
2992:
2990:
2989:
2984:
2982:
2981:
2965:
2963:
2962:
2957:
2955:
2951:
2949:
2948:
2943:
2939:
2937:
2929:
2928:
2927:
2914:
2904:
2903:
2898:
2894:
2893:
2891:
2883:
2882:
2881:
2868:
2854:
2850:
2837:
2833:
2831:
2830:
2825:
2821:
2819:
2811:
2810:
2809:
2796:
2789:
2788:
2773:
2772:
2767:
2763:
2762:
2760:
2752:
2751:
2750:
2737:
2723:
2722:
2710:
2702:
2698:
2696:
2695:
2690:
2686:
2679:
2677:
2669:
2668:
2667:
2654:
2643:
2642:
2637:
2633:
2626:
2624:
2616:
2615:
2614:
2601:
2584:
2563:
2532:
2530:
2529:
2524:
2501:
2474:
2461:
2436:
2434:
2433:
2428:
2426:
2425:
2418:
2417:
2403:
2402:
2390:
2381:
2376:
2375:
2363:
2354:
2347:
2346:
2334:
2325:
2320:
2319:
2305:
2304:
2292:
2283:
2276:
2275:
2263:
2254:
2249:
2248:
2236:
2227:
2222:
2221:
2198:
2197:
2190:
2189:
2175:
2174:
2160:
2159:
2143:
2142:
2128:
2127:
2113:
2112:
2096:
2095:
2081:
2080:
2066:
2065:
2042:
1977:
1975:
1974:
1969:
1967:
1938:
1936:
1935:
1930:
1928:
1924:
1923:
1921:
1920:
1908:
1895:
1886:
1880:
1876:
1874:
1873:
1864:
1863:
1862:
1850:
1849:
1839:
1833:
1824:
1818:
1817:
1788:
1786:
1785:
1780:
1769:
1768:
1756:
1747:
1741:
1737:
1735:
1734:
1725:
1724:
1723:
1711:
1710:
1700:
1694:
1685:
1679:
1678:
1647:
1643:
1641:
1640:
1635:
1633:
1623:
1618:
1617:
1608:
1603:
1598:
1597:
1588:
1574:
1543:
1521:
1517:
1509:
1483:
1478:
1470:
1467:
1462:
1426:
1424:
1423:
1418:
1416:
1411:
1403:
1377:
1355:
1353:
1352:
1347:
1317:
1315:
1314:
1309:
1307:
1299:
1284:
1280:
1248:
1244:
1236:
1232:
1230:
1229:
1224:
1222:
1217:
1206:
1201:
1196:
1188:
1173:
1169:
1158:
1140:, also known as
971:
965:
944:
938:
936:
935:
930:
925:
917:
913:
904:
900:
899:
891:
881:
879:
877:
876:
862:
860:
850:
845:
797:decimal fraction
761:, compared to a
739:
732:
725:
571:
536:Gay-Lussac's law
526:Combined gas law
476:Capillary action
361:
204:
188:
186:
185:
180:
178:
176:
168:
160:
126:
120:
118:
117:
112:
88:
72:
30:
21:
18:Strain (physics)
5533:
5532:
5528:
5527:
5526:
5524:
5523:
5522:
5508:Solid mechanics
5483:
5482:
5481:
5480:
5471:
5469:
5460:
5459:
5452:
5438:
5437:
5433:
5422:
5407:
5406:
5402:
5393:
5389:
5378:
5363:
5362:
5355:
5347:
5341:
5330:
5323:
5322:
5318:
5313:
5295:Stress measures
5291:
5240:tangent vectors
5236:
5230:
5225:
5224:
5194:
5181:
5168:
5152:
5136:
5120:
5109:
5108:
5085:
5074:
5061:
5043:
5030:
5017:
5004:
4991:
4978:
4965:
4952:
4939:
4926:
4913:
4900:
4893:
4889:
4888:
4871:
4863:
4856:
4855:
4849:
4820:
4789:
4758:
4738:
4737:
4712:
4699:
4686:
4673:
4658:
4650:
4637:
4636:
4635:of the tensor:
4614:
4606:
4581:
4570:
4566:
4551:
4540:
4536:
4518:
4502:
4487:
4476:
4472:
4457:
4446:
4442:
4424:
4408:
4403:
4402:
4398:
4394:
4387:
4380:
4375:
4371:
4366:
4362:
4357:
4353:
4349:
4328:
4317:
4313:
4298:
4287:
4283:
4253:
4248:
4247:
4226:
4215:
4211:
4181:
4170:
4166:
4153:
4152:
4141:
4130:
4126:
4122:
4095:
4084:
4080:
4050:
4039:
4035:
4028:
4027:
4013:
4012:
3998:
3987:
3983:
3973:
3963:
3952:
3948:
3921:
3910:
3906:
3893:
3875:
3864:
3860:
3856:
3847:
3835:
3834:
3820:
3809:
3805:
3795:
3785:
3774:
3770:
3743:
3732:
3728:
3715:
3697:
3686:
3682:
3678:
3669:
3651:
3650:
3612:
3607:
3606:
3600:
3594:
3590:
3585:
3568:
3565:
3558:
3557:
3555:
3550:
3547:
3545:
3518:
3512:
3509:
3497:
3491:
3466:
3455:
3451:
3436:
3421:
3410:
3406:
3391:
3386:
3385:
3381:
3377:
3356:
3345:
3341:
3300:
3231:
3222:original length
3203:
3198:
3197:
3193:
3171:
3140:
3129:
3125:
3096:
3085:
3081:
3072:
3068:
3022:
3021:
3000:
2995:
2994:
2973:
2968:
2967:
2953:
2952:
2930:
2919:
2915:
2909:
2908:
2884:
2873:
2869:
2860:
2856:
2855:
2835:
2834:
2812:
2801:
2797:
2791:
2790:
2780:
2753:
2742:
2738:
2729:
2725:
2724:
2714:
2700:
2699:
2670:
2659:
2655:
2652:
2648:
2647:
2617:
2606:
2602:
2590:
2586:
2585:
2576:
2535:
2534:
2477:
2476:
2470:
2453:
2442:
2420:
2419:
2406:
2404:
2391:
2377:
2364:
2349:
2348:
2335:
2321:
2308:
2306:
2293:
2278:
2277:
2264:
2250:
2237:
2223:
2210:
2203:
2192:
2191:
2178:
2176:
2163:
2161:
2148:
2145:
2144:
2131:
2129:
2116:
2114:
2101:
2098:
2097:
2084:
2082:
2069:
2067:
2054:
2047:
2023:
2022:
1958:
1957:
1950:
1944:
1912:
1900:
1896:
1865:
1854:
1841:
1840:
1834:
1809:
1804:
1803:
1800:
1794:
1760:
1726:
1715:
1702:
1701:
1695:
1670:
1665:
1664:
1661:
1655:
1645:
1631:
1630:
1609:
1589:
1572:
1571:
1541:
1540:
1504:
1491:
1485:
1484:
1471:
1447:
1429:
1428:
1404:
1388:
1387:
1378:, also called,
1375:
1369:
1326:
1325:
1287:
1286:
1282:
1278:
1275:extension ratio
1267:
1246:
1242:
1234:
1207:
1189:
1176:
1175:
1171:
1164:
1156:
1135:
1102:
1100:Strain measures
1054:and biological
1021:
967:
950:
947:identity tensor
940:
908:
886:
882:
863:
851:
836:
835:
743:
714:
713:
712:
632:
624:
623:
577:Viscoelasticity
568:
558:
557:
545:
495:
491:Surface tension
455:
358:
356:Fluid mechanics
348:
347:
346:
260:
258:Solid mechanics
250:
249:
201:
193:
169:
161:
142:
141:
103:
102:
89:
84:
83:
82:Behaviour under
70:
60:
40:
28:
23:
22:
15:
12:
11:
5:
5531:
5529:
5521:
5520:
5515:
5510:
5505:
5500:
5495:
5485:
5484:
5479:
5478:
5450:
5431:
5420:
5400:
5387:
5376:
5353:
5350:on 2010-03-31.
5339:
5315:
5314:
5312:
5309:
5308:
5307:
5302:
5297:
5290:
5287:
5268:quadratic form
5232:Main article:
5229:
5226:
5201:
5197:
5193:
5188:
5184:
5180:
5175:
5171:
5167:
5162:
5159:
5155:
5151:
5146:
5143:
5139:
5135:
5130:
5127:
5123:
5119:
5116:
5092:
5088:
5081:
5077:
5073:
5068:
5064:
5060:
5056:
5050:
5046:
5042:
5037:
5033:
5029:
5024:
5020:
5016:
5011:
5007:
5003:
4998:
4994:
4990:
4985:
4981:
4977:
4972:
4968:
4964:
4959:
4955:
4951:
4946:
4942:
4938:
4933:
4929:
4925:
4920:
4916:
4912:
4907:
4903:
4899:
4896:
4892:
4885:
4878:
4874:
4869:
4866:
4847:
4832:
4827:
4823:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4796:
4792:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4765:
4761:
4757:
4754:
4751:
4748:
4745:
4719:
4715:
4711:
4706:
4702:
4698:
4693:
4689:
4685:
4680:
4676:
4672:
4665:
4661:
4656:
4653:
4647:
4644:
4615:
4607:
4605:
4602:
4587:
4584:
4577:
4573:
4569:
4563:
4557:
4554:
4547:
4543:
4539:
4533:
4528:
4525:
4521:
4517:
4512:
4509:
4505:
4500:
4493:
4490:
4483:
4479:
4475:
4469:
4463:
4460:
4453:
4449:
4445:
4439:
4434:
4431:
4427:
4423:
4418:
4415:
4411:
4385:
4378:
4369:
4360:
4334:
4331:
4324:
4320:
4316:
4310:
4304:
4301:
4294:
4290:
4286:
4280:
4277:
4274:
4271:
4268:
4263:
4260:
4256:
4232:
4229:
4222:
4218:
4214:
4208:
4205:
4196:
4187:
4184:
4177:
4173:
4169:
4163:
4160:
4110:
4107:
4101:
4098:
4091:
4087:
4083:
4071:
4065:
4062:
4056:
4053:
4046:
4042:
4038:
4004:
4001:
3994:
3990:
3986:
3979:
3976:
3969:
3966:
3959:
3955:
3951:
3943:
3937:
3934:
3927:
3924:
3917:
3913:
3909:
3902:
3899:
3896:
3891:
3888:
3881:
3878:
3871:
3867:
3863:
3853:
3850:
3848:
3846:
3843:
3840:
3837:
3836:
3826:
3823:
3816:
3812:
3808:
3801:
3798:
3791:
3788:
3781:
3777:
3773:
3765:
3759:
3756:
3749:
3746:
3739:
3735:
3731:
3724:
3721:
3718:
3713:
3710:
3703:
3700:
3693:
3689:
3685:
3675:
3672:
3670:
3668:
3665:
3662:
3659:
3658:
3636:
3633:
3630:
3627:
3622:
3619:
3615:
3605:. Therefore,
3588:
3580:
3579:
3548:
3543:
3540:
3539:
3530:
3524:
3523:
3510:
3508:Common symbols
3507:
3504:
3503:
3490:
3487:
3472:
3469:
3462:
3458:
3454:
3448:
3443:
3439:
3434:
3427:
3424:
3417:
3413:
3409:
3403:
3398:
3394:
3362:
3359:
3352:
3348:
3344:
3338:
3332:
3329:
3326:
3323:
3319:
3316:
3313:
3310:
3307:
3304:
3298:
3295:
3292:
3289:
3285:
3282:
3279:
3276:
3273:
3270:
3266:
3263:
3260:
3257:
3254:
3250:
3247:
3244:
3241:
3238:
3235:
3228:
3215:
3210:
3206:
3170:
3167:
3155:
3152:
3146:
3143:
3136:
3132:
3128:
3122:
3119:
3116:
3113:
3109:
3102:
3099:
3092:
3088:
3084:
3078:
3075:
3071:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3045:
3042:
3039:
3036:
3033:
3030:
3007:
3003:
2980:
2976:
2947:
2942:
2936:
2933:
2926:
2922:
2918:
2912:
2907:
2902:
2897:
2890:
2887:
2880:
2876:
2872:
2866:
2863:
2859:
2849:
2846:
2843:
2840:
2838:
2836:
2829:
2824:
2818:
2815:
2808:
2804:
2800:
2794:
2787:
2783:
2779:
2776:
2771:
2766:
2759:
2756:
2749:
2745:
2741:
2735:
2732:
2728:
2721:
2717:
2713:
2708:
2705:
2703:
2701:
2694:
2689:
2685:
2682:
2676:
2673:
2666:
2662:
2658:
2651:
2646:
2641:
2636:
2632:
2629:
2623:
2620:
2613:
2609:
2605:
2599:
2596:
2593:
2589:
2582:
2579:
2577:
2575:
2572:
2569:
2566:
2562:
2559:
2556:
2553:
2550:
2547:
2543:
2542:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2500:
2497:
2494:
2491:
2488:
2485:
2441:
2438:
2424:
2416:
2413:
2409:
2405:
2401:
2398:
2394:
2387:
2384:
2378:
2374:
2371:
2367:
2360:
2357:
2351:
2350:
2345:
2342:
2338:
2331:
2328:
2322:
2318:
2315:
2311:
2307:
2303:
2300:
2296:
2289:
2286:
2280:
2279:
2274:
2271:
2267:
2260:
2257:
2251:
2247:
2244:
2240:
2233:
2230:
2224:
2220:
2217:
2213:
2209:
2208:
2206:
2201:
2196:
2188:
2185:
2181:
2177:
2173:
2170:
2166:
2162:
2158:
2155:
2151:
2147:
2146:
2141:
2138:
2134:
2130:
2126:
2123:
2119:
2115:
2111:
2108:
2104:
2100:
2099:
2094:
2091:
2087:
2083:
2079:
2076:
2072:
2068:
2064:
2061:
2057:
2053:
2052:
2050:
2045:
2040:
2036:
2033:
1966:
1943:
1940:
1927:
1919:
1915:
1911:
1906:
1903:
1899:
1892:
1889:
1883:
1879:
1872:
1868:
1861:
1857:
1853:
1848:
1844:
1837:
1830:
1827:
1821:
1816:
1812:
1796:Main article:
1793:
1792:Almansi strain
1790:
1778:
1775:
1772:
1767:
1763:
1759:
1753:
1750:
1744:
1740:
1733:
1729:
1722:
1718:
1714:
1709:
1705:
1698:
1691:
1688:
1682:
1677:
1673:
1657:Main article:
1654:
1651:
1629:
1626:
1621:
1616:
1612:
1606:
1601:
1596:
1592:
1586:
1583:
1580:
1577:
1575:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1544:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1520:
1515:
1512:
1507:
1503:
1500:
1497:
1494:
1492:
1490:
1487:
1486:
1481:
1477:
1474:
1466:
1461:
1457:
1453:
1450:
1448:
1446:
1443:
1440:
1437:
1436:
1414:
1410:
1407:
1401:
1398:
1395:
1368:
1365:
1345:
1342:
1339:
1336:
1333:
1305:
1302:
1297:
1294:
1266:
1263:
1220:
1216:
1213:
1210:
1204:
1199:
1195:
1192:
1186:
1183:
1134:
1131:
1127:Almansi strain
1101:
1098:
1097:
1096:
1086:
1064:, also called
1059:
1030:, also called
1020:
1019:Strain regimes
1017:
1009:tensile strain
994:material point
928:
924:
920:
916:
912:
907:
903:
898:
894:
890:
885:
875:
871:
859:
848:
844:
764:
745:
744:
742:
741:
734:
727:
719:
716:
715:
711:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
634:
633:
630:
629:
626:
625:
620:
619:
618:
617:
612:
607:
599:
598:
592:
591:
590:
589:
584:
579:
569:
564:
563:
560:
559:
554:
553:
547:
546:
544:
543:
538:
533:
528:
523:
518:
513:
507:
504:
503:
497:
496:
494:
493:
488:
483:
481:Chromatography
478:
473:
467:
464:
463:
457:
456:
454:
453:
434:
433:
432:
413:
401:
396:
384:
371:
368:
367:
359:
354:
353:
350:
349:
345:
344:
339:
334:
333:
332:
322:
317:
312:
311:
310:
305:
295:
290:
285:
280:
279:
278:
268:
262:
261:
256:
255:
252:
251:
246:
245:
244:
243:
235:
234:
230:
229:
228:
227:
222:
217:
209:
208:
202:
199:
198:
195:
194:
189:
175:
172:
167:
164:
158:
155:
152:
149:
138:
137:
131:
130:
122:
121:
110:
100:
94:
93:
90:
81:
78:
77:
74:
65:
64:
61:
58:
55:
54:
51:
45:
44:
41:
38:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5530:
5519:
5516:
5514:
5511:
5509:
5506:
5504:
5501:
5499:
5496:
5494:
5491:
5490:
5488:
5467:
5463:
5457:
5455:
5451:
5446:
5442:
5435:
5432:
5427:
5423:
5421:0-7506-8025-3
5417:
5413:
5412:
5404:
5401:
5397:
5391:
5388:
5383:
5379:
5377:0-7506-8025-3
5373:
5369:
5368:
5360:
5358:
5354:
5346:
5342:
5336:
5329:
5328:
5320:
5317:
5310:
5306:
5305:Strain tensor
5303:
5301:
5298:
5296:
5293:
5292:
5288:
5286:
5284:
5283:metric tensor
5280:
5277:
5273:
5269:
5265:
5261:
5257:
5253:
5249:
5245:
5241:
5235:
5228:Metric tensor
5227:
5222:
5220:
5215:
5199:
5195:
5191:
5186:
5182:
5178:
5173:
5169:
5165:
5160:
5157:
5153:
5149:
5144:
5141:
5137:
5133:
5128:
5125:
5121:
5117:
5114:
5090:
5086:
5079:
5075:
5071:
5066:
5062:
5058:
5054:
5048:
5044:
5040:
5035:
5031:
5027:
5022:
5018:
5014:
5009:
5005:
5001:
4996:
4992:
4988:
4983:
4979:
4975:
4970:
4966:
4962:
4957:
4953:
4949:
4944:
4940:
4936:
4931:
4927:
4923:
4918:
4914:
4910:
4905:
4901:
4897:
4894:
4890:
4883:
4876:
4872:
4867:
4853:
4846:
4825:
4821:
4817:
4814:
4808:
4805:
4802:
4794:
4790:
4786:
4783:
4777:
4774:
4771:
4763:
4759:
4755:
4752:
4746:
4743:
4735:
4717:
4713:
4709:
4704:
4700:
4696:
4691:
4687:
4683:
4678:
4674:
4670:
4663:
4659:
4654:
4645:
4642:
4634:
4630:
4626:
4622:
4621:
4612:
4604:Volume strain
4603:
4601:
4585:
4575:
4571:
4561:
4555:
4545:
4541:
4531:
4526:
4523:
4519:
4515:
4510:
4507:
4503:
4498:
4491:
4481:
4477:
4467:
4461:
4451:
4447:
4437:
4432:
4429:
4425:
4421:
4416:
4413:
4409:
4391:
4388:
4381:
4372:
4363:
4332:
4322:
4318:
4308:
4302:
4292:
4288:
4278:
4275:
4272:
4269:
4266:
4261:
4258:
4254:
4230:
4220:
4216:
4206:
4203:
4194:
4185:
4175:
4171:
4161:
4158:
4151:. Therefore,
4149:
4145:
4138:
4134:
4108:
4105:
4099:
4089:
4085:
4069:
4063:
4060:
4054:
4044:
4040:
4002:
3992:
3988:
3977:
3974:
3967:
3957:
3953:
3941:
3935:
3932:
3925:
3915:
3911:
3900:
3897:
3894:
3889:
3886:
3879:
3869:
3865:
3851:
3849:
3844:
3841:
3838:
3824:
3814:
3810:
3799:
3796:
3789:
3779:
3775:
3763:
3757:
3754:
3747:
3737:
3733:
3722:
3719:
3716:
3711:
3708:
3701:
3691:
3687:
3673:
3671:
3666:
3663:
3660:
3647:
3634:
3631:
3628:
3625:
3620:
3617:
3613:
3603:
3597:
3591:
3573:
3572:
3563:
3562:
3553:
3549:
3541:
3538:
3534:
3531:
3529:
3525:
3521:
3515:
3511:
3505:
3500:
3496:
3488:
3486:
3470:
3460:
3456:
3446:
3441:
3437:
3432:
3425:
3415:
3411:
3401:
3396:
3392:
3360:
3350:
3346:
3336:
3327:
3324:
3293:
3290:
3264:
3258:
3255:
3226:
3213:
3208:
3204:
3190:
3188:
3184:
3183:normal stress
3180:
3176:
3169:Normal strain
3168:
3166:
3153:
3150:
3144:
3134:
3130:
3120:
3117:
3114:
3111:
3107:
3100:
3090:
3086:
3076:
3073:
3069:
3065:
3062:
3059:
3053:
3050:
3005:
3001:
2978:
2974:
2945:
2940:
2934:
2924:
2920:
2910:
2905:
2900:
2895:
2888:
2878:
2874:
2864:
2861:
2857:
2847:
2844:
2841:
2839:
2827:
2822:
2816:
2806:
2802:
2792:
2785:
2781:
2777:
2774:
2769:
2764:
2757:
2747:
2743:
2733:
2730:
2726:
2719:
2715:
2711:
2706:
2704:
2692:
2687:
2683:
2680:
2674:
2664:
2660:
2649:
2644:
2639:
2634:
2630:
2627:
2621:
2611:
2607:
2597:
2594:
2591:
2587:
2580:
2578:
2570:
2567:
2520:
2517:
2514:
2508:
2505:
2473:
2469:
2465:
2460:
2456:
2446:
2439:
2437:
2422:
2414:
2411:
2407:
2399:
2396:
2392:
2385:
2382:
2372:
2369:
2365:
2358:
2355:
2343:
2340:
2336:
2329:
2326:
2316:
2313:
2309:
2301:
2298:
2294:
2287:
2284:
2272:
2269:
2265:
2258:
2255:
2245:
2242:
2238:
2231:
2228:
2218:
2215:
2211:
2204:
2199:
2194:
2186:
2183:
2179:
2171:
2168:
2164:
2156:
2153:
2149:
2139:
2136:
2132:
2124:
2121:
2117:
2109:
2106:
2102:
2092:
2089:
2085:
2077:
2074:
2070:
2062:
2059:
2055:
2048:
2043:
2038:
2034:
2019:
2017:
2013:
2012:normal stress
2009:
2005:
2004:normal strain
2001:
1997:
1993:
1989:
1988:linear strain
1985:
1981:
1955:
1954:strain tensor
1949:
1942:Strain tensor
1941:
1939:
1925:
1917:
1913:
1909:
1904:
1901:
1897:
1890:
1887:
1881:
1877:
1870:
1866:
1859:
1855:
1851:
1846:
1842:
1835:
1828:
1825:
1819:
1814:
1810:
1799:
1791:
1789:
1773:
1770:
1765:
1761:
1751:
1748:
1742:
1738:
1731:
1727:
1720:
1716:
1712:
1707:
1703:
1696:
1689:
1686:
1680:
1675:
1671:
1660:
1652:
1650:
1627:
1624:
1619:
1614:
1610:
1604:
1599:
1594:
1590:
1584:
1581:
1578:
1576:
1565:
1562:
1559:
1553:
1550:
1547:
1545:
1534:
1528:
1525:
1522:
1518:
1513:
1510:
1505:
1501:
1498:
1495:
1493:
1488:
1479:
1475:
1472:
1464:
1459:
1455:
1451:
1449:
1444:
1441:
1438:
1412:
1408:
1405:
1399:
1396:
1393:
1385:
1384:Hencky strain
1381:
1374:
1366:
1364:
1362:
1357:
1343:
1340:
1337:
1334:
1331:
1323:
1318:
1303:
1300:
1295:
1292:
1276:
1272:
1271:stretch ratio
1265:Stretch ratio
1264:
1262:
1259:
1255:
1250:
1240:
1218:
1214:
1211:
1208:
1202:
1197:
1193:
1184:
1181:
1168:
1162:
1155:
1151:
1147:
1143:
1142:Cauchy strain
1139:
1132:
1130:
1128:
1124:
1120:
1116:
1111:
1107:
1099:
1094:
1090:
1087:
1084:
1079:
1075:
1071:
1067:
1063:
1060:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1026:
1025:
1024:
1018:
1016:
1014:
1010:
1005:
1003:
999:
998:normal strain
995:
990:
988:
984:
983:normal strain
979:
974:
970:
963:
959:
958:
953:
948:
943:
926:
918:
914:
905:
901:
892:
883:
846:
833:
829:
824:
822:
818:
814:
810:
806:
802:
798:
794:
793:dimensionless
790:
789:SI base units
786:
783:
779:
774:
772:
771:metric tensor
767:
762:
760:
756:
752:
740:
735:
733:
728:
726:
721:
720:
718:
717:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
635:
628:
627:
616:
613:
611:
608:
606:
603:
602:
601:
600:
597:
593:
588:
585:
583:
580:
578:
575:
574:
573:
572:
567:
562:
561:
552:
548:
542:
539:
537:
534:
532:
529:
527:
524:
522:
521:Charles's law
519:
517:
514:
512:
509:
508:
506:
505:
502:
498:
492:
489:
487:
484:
482:
479:
477:
474:
472:
469:
468:
466:
465:
462:
458:
452:
449:
445:
442:
438:
435:
430:
429:non-Newtonian
427:
423:
419:
418:
417:
414:
412:
409:
405:
402:
400:
397:
395:
392:
388:
385:
383:
380:
376:
373:
372:
370:
369:
366:
362:
357:
352:
351:
343:
340:
338:
335:
331:
328:
327:
326:
323:
321:
318:
316:
315:Compatibility
313:
309:
306:
304:
303:Finite strain
301:
300:
299:
296:
294:
291:
289:
286:
284:
281:
277:
274:
273:
272:
269:
267:
264:
263:
259:
254:
253:
242:
239:
238:
237:
236:
231:
226:
223:
221:
218:
216:
213:
212:
211:
210:
207:Conservations
205:
197:
196:
192:
173:
170:
165:
162:
156:
153:
150:
147:
140:
139:
136:
132:
128:
127:
108:
101:
99:
95:
91:
87:
79:
75:
73:
66:
62:
56:
52:
50:
46:
43:Strain tensor
42:
36:
31:
19:
5470:. Retrieved
5468:. 2013-08-20
5465:
5444:
5440:
5434:
5410:
5403:
5390:
5366:
5345:the original
5326:
5319:
5279:bilinear map
5244:parametrized
5237:
5216:
4851:
4844:
4733:
4627:; it is the
4624:
4618:
4616:
4392:
4383:
4376:
4367:
4358:
4147:
4143:
4136:
4132:
3648:
3601:
3595:
3586:
3583:
3570:
3560:
3551:
3528:SI unit
3502:Shear strain
3495:Shear stress
3489:Shear strain
3191:
3186:
3172:
2471:
2458:
2454:
2451:
2020:
2016:shear stress
2008:shear strain
2007:
2003:
1999:
1995:
1992:shear strain
1991:
1987:
1953:
1951:
1801:
1662:
1653:Green strain
1383:
1379:
1372:
1370:
1358:
1321:
1319:
1274:
1270:
1268:
1257:
1253:
1251:
1238:
1166:
1160:
1153:
1149:
1141:
1137:
1136:
1126:
1123:Green strain
1122:
1118:
1114:
1105:
1103:
1092:
1088:
1077:
1073:
1069:
1065:
1035:
1031:
1022:
1012:
1008:
1006:
1002:shear strain
1001:
997:
991:
987:shear strain
986:
982:
975:
968:
961:
956:
955:
951:
941:
832:displacement
825:
775:
754:
748:
596:Smart fluids
541:Graham's law
447:
440:
425:
411:Pascal's law
407:
390:
378:
297:
233:Inequalities
49:SI unit
5300:Strain rate
5281:called the
5252:von Neumann
4625:compression
3179:Hooke's law
1984:ISO 80000-4
1380:true strain
1056:soft tissue
776:Strain has
759:deformation
615:Ferrofluids
516:Boyle's law
288:Hooke's law
266:Deformation
59:Other units
39:Other names
5487:Categories
5472:2023-08-28
5447:: 215–220.
5311:References
3493:See also:
1361:elastomers
1146:elongation
1110:elastomers
1044:elastomers
801:percentage
668:Gay-Lussac
631:Scientists
531:Fick's law
511:Atmosphere
330:frictional
283:Plasticity
271:Elasticity
5274:, with a
5196:ε
5192:⋅
5183:ε
5179:⋅
5170:ε
5166:≫
5154:ε
5150:⋅
5138:ε
5134:≫
5122:ε
5118:≫
5072:−
5059:⋅
5045:ε
5041:⋅
5032:ε
5028:⋅
5019:ε
5006:ε
5002:⋅
4993:ε
4980:ε
4976:⋅
4967:ε
4954:ε
4950:⋅
4941:ε
4928:ε
4915:ε
4902:ε
4865:Δ
4822:ε
4809:⋅
4803:×
4791:ε
4778:⋅
4772:×
4760:ε
4747:⋅
4714:ε
4701:ε
4688:ε
4652:Δ
4643:δ
4583:∂
4568:∂
4553:∂
4538:∂
4520:γ
4504:γ
4489:∂
4474:∂
4459:∂
4444:∂
4426:γ
4410:γ
4330:∂
4315:∂
4300:∂
4285:∂
4276:β
4270:α
4255:γ
4228:∂
4213:∂
4207:≈
4204:β
4183:∂
4168:∂
4162:≈
4159:α
4106:≪
4097:∂
4082:∂
4061:≪
4052:∂
4037:∂
4000:∂
3985:∂
3965:∂
3950:∂
3923:∂
3908:∂
3877:∂
3862:∂
3845:β
3842:
3822:∂
3807:∂
3787:∂
3772:∂
3745:∂
3730:∂
3699:∂
3684:∂
3667:α
3664:
3635:β
3629:α
3614:γ
3468:∂
3453:∂
3438:ε
3423:∂
3408:∂
3393:ε
3358:∂
3343:∂
3265:−
3219:extension
3205:ε
3187:dilations
3175:isotropic
3142:∂
3127:∂
3098:∂
3083:∂
3060:≈
2932:∂
2917:∂
2886:∂
2871:∂
2814:∂
2799:∂
2755:∂
2740:∂
2672:∂
2657:∂
2619:∂
2604:∂
2408:ε
2393:γ
2366:γ
2337:γ
2310:ε
2295:γ
2266:γ
2239:γ
2212:ε
2180:ε
2165:ε
2150:ε
2133:ε
2118:ε
2103:ε
2086:ε
2071:ε
2056:ε
2039:_
2035:_
2032:ε
1965:ε
1914:λ
1905:−
1852:−
1811:ε
1771:−
1762:λ
1713:−
1672:ε
1628:⋯
1625:−
1585:−
1554:
1535:λ
1529:
1502:
1489:ε
1473:δ
1456:∫
1445:ε
1442:δ
1439:∫
1406:δ
1397:ε
1394:δ
1341:−
1338:λ
1293:λ
1212:−
1191:Δ
1040:continuum
919:−
893:−
870:∂
858:∂
847:≐
843:ε
778:dimension
763:reference
751:mechanics
708:Truesdell
638:Bernoulli
587:Rheometer
582:Rheometry
422:Newtonian
416:Viscosity
166:φ
154:−
98:Dimension
5426:Archived
5382:Archived
5289:See also
5262:and the
4620:dilation
1956:(symbol
1233:, where
966:, where
915:′
766:position
566:Rheology
471:Adhesion
451:Pressure
437:Buoyancy
382:Dynamics
220:Momentum
5493:Tensors
5248:Fréchet
4854:, thus
3576:
3556:
3173:For an
2464:rhombus
1237:is the
1115:stretch
1083:elastic
945:is the
819:/m and
787:, with
653:Charles
461:Liquids
375:Statics
320:Bending
5418:
5374:
5337:
5256:Jordan
4397:- and
4201:
4198:
4192:
4076:
4073:
4067:
3537:radian
3380:- and
2851:
1996:normal
1644:where
1125:, and
1052:fluids
978:tensor
939:where
782:length
755:strain
703:Stokes
698:Pascal
688:Navier
683:Newton
673:Graham
648:Cauchy
551:Plasma
446:
444:Mixing
439:
424:
406:
389:
377:
365:Fluids
298:Strain
293:Stress
276:linear
225:Energy
92:tensor
33:Strain
5348:(PDF)
5331:(PDF)
4633:trace
4246:thus
3535:, or
2000:shear
1076:, or
799:or a
785:ratio
780:of a
678:Hooke
658:Euler
643:Boyle
501:Gases
5416:ISBN
5372:ISBN
5335:ISBN
5260:norm
5254:and
4843:and
4365:and
4356:and
4352:and
4142:tan
4131:tan
4125:and
3599:and
3181:, a
2993:and
2533:and
2014:and
2002:. A
1371:The
1269:The
1252:The
823:/m.
693:Noll
663:Fick
215:Mass
200:Laws
5466:ISO
4631:or
4623:or
3839:tan
3661:tan
3517:or
1998:or
1382:or
1324:by
1273:or
1152:or
1091:or
830:of
811:or
749:In
76:m/m
68:In
5489::
5464:.
5453:^
5443:.
5424:.
5398:..
5380:.
5356:^
5285:.
5250:,
5200:33
5187:22
5174:11
5049:33
5036:22
5023:11
5010:33
4997:22
4984:33
4971:11
4958:22
4945:11
4932:33
4919:22
4906:11
4850:=
4826:33
4795:22
4764:11
4718:33
4705:22
4692:11
4399:xz
4395:yz
4390:.
4386:yx
4382:=
4379:xy
4146:≈
4140:,
4135:≈
3602:AB
3596:AC
3589:xy
3554:=
3189:.
2459:dy
2457:×
2455:dx
2018:.
1551:ln
1526:ln
1499:ln
1241:,
1129:.
1121:,
1117:,
1072:,
1068:,
1046:,
1034:,
1015:.
954:=
834::
821:nm
817:μm
803:.
753:,
5475:.
5445:9
5161:j
5158:j
5145:i
5142:i
5129:i
5126:i
5115:1
5091:3
5087:a
5080:3
5076:a
5067:3
5063:a
5055:)
5015:+
4989:+
4963:+
4937:+
4924:+
4911:+
4898:+
4895:1
4891:(
4884:=
4877:0
4873:V
4868:V
4852:a
4848:0
4845:V
4831:)
4818:+
4815:1
4812:(
4806:a
4800:)
4787:+
4784:1
4781:(
4775:a
4769:)
4756:+
4753:1
4750:(
4744:a
4734:a
4710:+
4697:+
4684:=
4679:1
4675:I
4671:=
4664:0
4660:V
4655:V
4646:=
4613:.
4586:z
4576:x
4572:u
4562:+
4556:x
4546:z
4542:u
4532:=
4527:z
4524:x
4516:=
4511:x
4508:z
4499:,
4492:y
4482:z
4478:u
4468:+
4462:z
4452:y
4448:u
4438:=
4433:y
4430:z
4422:=
4417:z
4414:y
4384:γ
4377:γ
4370:y
4368:u
4361:x
4359:u
4354:y
4350:x
4333:y
4323:x
4319:u
4309:+
4303:x
4293:y
4289:u
4279:=
4273:+
4267:=
4262:y
4259:x
4231:y
4221:x
4217:u
4195:;
4186:x
4176:y
4172:u
4148:β
4144:β
4137:α
4133:α
4127:β
4123:α
4109:1
4100:y
4090:y
4086:u
4070:;
4064:1
4055:x
4045:x
4041:u
4003:y
3993:y
3989:u
3978:+
3975:1
3968:y
3958:x
3954:u
3942:=
3936:y
3933:d
3926:y
3916:y
3912:u
3901:+
3898:y
3895:d
3890:y
3887:d
3880:y
3870:x
3866:u
3852:=
3825:x
3815:x
3811:u
3800:+
3797:1
3790:x
3780:y
3776:u
3764:=
3758:x
3755:d
3748:x
3738:x
3734:u
3723:+
3720:x
3717:d
3712:x
3709:d
3702:x
3692:y
3688:u
3674:=
3632:+
3626:=
3621:y
3618:x
3587:γ
3571:G
3566:/
3561:τ
3552:γ
3533:1
3520:ε
3514:γ
3471:z
3461:z
3457:u
3447:=
3442:z
3433:,
3426:y
3416:y
3412:u
3402:=
3397:y
3382:z
3378:y
3361:x
3351:x
3347:u
3337:=
3331:)
3328:B
3325:A
3322:(
3318:h
3315:t
3312:g
3309:n
3306:e
3303:l
3297:)
3294:B
3291:A
3288:(
3284:h
3281:t
3278:g
3275:n
3272:e
3269:l
3262:)
3259:b
3256:a
3253:(
3249:h
3246:t
3243:g
3240:n
3237:e
3234:l
3227:=
3214:=
3209:x
3194:x
3154:x
3151:d
3145:x
3135:x
3131:u
3121:+
3118:x
3115:d
3112:=
3108:)
3101:x
3091:x
3087:u
3077:+
3074:1
3070:(
3066:x
3063:d
3057:)
3054:b
3051:a
3048:(
3044:h
3041:t
3038:g
3035:n
3032:e
3029:l
3006:x
3002:u
2979:y
2975:u
2946:2
2941:)
2935:x
2925:y
2921:u
2911:(
2906:+
2901:2
2896:)
2889:x
2879:x
2875:u
2865:+
2862:1
2858:(
2848:x
2845:d
2842:=
2828:2
2823:)
2817:x
2807:y
2803:u
2793:(
2786:2
2782:x
2778:d
2775:+
2770:2
2765:)
2758:x
2748:x
2744:u
2734:+
2731:1
2727:(
2720:2
2716:x
2712:d
2707:=
2693:2
2688:)
2684:x
2681:d
2675:x
2665:y
2661:u
2650:(
2645:+
2640:2
2635:)
2631:x
2628:d
2622:x
2612:x
2608:u
2598:+
2595:x
2592:d
2588:(
2581:=
2574:)
2571:b
2568:a
2565:(
2561:h
2558:t
2555:g
2552:n
2549:e
2546:l
2521:x
2518:d
2515:=
2512:)
2509:B
2506:A
2503:(
2499:h
2496:t
2493:g
2490:n
2487:e
2484:l
2472:u
2423:]
2415:z
2412:z
2400:y
2397:z
2386:2
2383:1
2373:x
2370:z
2359:2
2356:1
2344:z
2341:y
2330:2
2327:1
2317:y
2314:y
2302:x
2299:y
2288:2
2285:1
2273:z
2270:x
2259:2
2256:1
2246:y
2243:x
2232:2
2229:1
2219:x
2216:x
2205:[
2200:=
2195:]
2187:z
2184:z
2172:y
2169:z
2157:x
2154:z
2140:z
2137:y
2125:y
2122:y
2110:x
2107:y
2093:z
2090:x
2078:y
2075:x
2063:x
2060:x
2049:[
2044:=
1926:)
1918:2
1910:1
1902:1
1898:(
1891:2
1888:1
1882:=
1878:)
1871:2
1867:l
1860:2
1856:L
1847:2
1843:l
1836:(
1829:2
1826:1
1820:=
1815:E
1777:)
1774:1
1766:2
1758:(
1752:2
1749:1
1743:=
1739:)
1732:2
1728:L
1721:2
1717:L
1708:2
1704:l
1697:(
1690:2
1687:1
1681:=
1676:G
1646:e
1620:3
1615:3
1611:e
1605:+
1600:2
1595:2
1591:e
1582:e
1579:=
1569:)
1566:e
1563:+
1560:1
1557:(
1548:=
1538:)
1532:(
1523:=
1519:)
1514:L
1511:l
1506:(
1496:=
1480:l
1476:l
1465:l
1460:L
1452:=
1413:l
1409:l
1400:=
1376:ε
1344:1
1335:=
1332:e
1322:e
1304:L
1301:l
1296:=
1283:L
1279:l
1247:l
1243:L
1235:e
1219:L
1215:L
1209:l
1203:=
1198:L
1194:L
1185:=
1182:e
1172:L
1167:L
1165:Δ
1157:e
1058:.
969:X
964:)
962:X
960:(
957:F
952:x
942:I
927:,
923:I
911:F
906:=
902:)
897:X
889:x
884:(
874:X
738:e
731:t
724:v
448:·
441:·
431:)
426:·
420:(
408:·
391:·
379:·
174:x
171:d
163:d
157:D
151:=
148:J
109:1
63:%
53:1
20:)
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