2507:
1716:
2527:
47:
1949:
1158:
6075:
5296:
4451:
4036:
4968:
5849:
2502:{\displaystyle {\begin{bmatrix}x_{1}(X_{1},X_{2},X_{3},t)\\x_{2}(X_{1},X_{2},X_{3},t)\\x_{3}(X_{1},X_{2},X_{3},t)\end{bmatrix}}={\begin{bmatrix}Q_{11}(t)&Q_{12}(t)&Q_{13}(t)\\Q_{21}(t)&Q_{22}(t)&Q_{23}(t)\\Q_{31}(t)&Q_{32}(t)&Q_{33}(t)\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}+{\begin{bmatrix}c_{1}(t)\\c_{2}(t)\\c_{3}(t)\end{bmatrix}}}
1711:{\displaystyle {\begin{bmatrix}x_{1}(X_{1},X_{2},X_{3},t)\\x_{2}(X_{1},X_{2},X_{3},t)\\x_{3}(X_{1},X_{2},X_{3},t)\end{bmatrix}}={\begin{bmatrix}F_{11}(t)&F_{12}(t)&F_{13}(t)\\F_{21}(t)&F_{22}(t)&F_{23}(t)\\F_{31}(t)&F_{32}(t)&F_{33}(t)\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}+{\begin{bmatrix}c_{1}(t)\\c_{2}(t)\\c_{3}(t)\end{bmatrix}}}
5093:
4268:
3863:
4761:
4650:
4235:
2827:
2538:. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration
2993:
6070:{\displaystyle {\boldsymbol {F}}\cdot \mathbf {e} _{2}=F_{12}\mathbf {e} _{1}+F_{22}\mathbf {e} _{2}=\gamma \mathbf {e} _{1}+\mathbf {e} _{2}\quad \implies \quad {\boldsymbol {F}}\cdot (\mathbf {e} _{2}\otimes \mathbf {e} _{2})=\gamma \mathbf {e} _{1}\otimes \mathbf {e} _{2}+\mathbf {e} _{2}\otimes \mathbf {e} _{2}}
51:
The deformation of a thin straight rod into a closed loop. The length of the rod remains almost unchanged during the deformation, which indicates that the strain is small. In this particular case of bending, displacements associated with rigid translations and rotations of material elements in the
2649:
is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field. In general, the displacement
3431:
2569:
If after a displacement of the continuum there is a relative displacement between particles, a deformation has occurred. On the other hand, if after displacement of the continuum the relative displacement between particles in the current configuration is zero, then there is no deformation and a
3830:
3676:
5291:{\displaystyle {\boldsymbol {F}}={\boldsymbol {R}}\cdot {\boldsymbol {U}}={\begin{bmatrix}\cos \theta &\sin \theta &0\\-\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}{\begin{bmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&1\end{bmatrix}}}
4446:{\displaystyle {\begin{aligned}\mathbf {U} (\mathbf {x} ,t)&=\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\\\nabla _{\mathbf {x} }\mathbf {U} &=\mathbf {I} -\nabla _{\mathbf {x} }\mathbf {X} \\\nabla _{\mathbf {x} }\mathbf {U} &=\mathbf {I} -\mathbf {F} ^{-1}\end{aligned}}}
4456:
4041:
4031:{\displaystyle {\begin{aligned}\mathbf {u} (\mathbf {X} ,t)&=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \\\nabla _{\mathbf {X} }\mathbf {u} &=\nabla _{\mathbf {X} }\mathbf {x} -\mathbf {I} \\\nabla _{\mathbf {X} }\mathbf {u} &=\mathbf {F} -\mathbf {I} \end{aligned}}}
1944:
2653:
5084:
910:
It is convenient to identify a reference configuration or initial geometric state of the continuum body which all subsequent configurations are referenced from. The reference configuration need not be one the body actually will ever occupy. Often, the configuration at
5634:
4963:{\displaystyle {\boldsymbol {F}}=F_{11}\mathbf {e} _{1}\otimes \mathbf {e} _{1}+F_{12}\mathbf {e} _{1}\otimes \mathbf {e} _{2}+F_{21}\mathbf {e} _{2}\otimes \mathbf {e} _{1}+F_{22}\mathbf {e} _{2}\otimes \mathbf {e} _{2}+\mathbf {e} _{3}\otimes \mathbf {e} _{3}}
2832:
5844:
902:
Deformation is the change in the metric properties of a continuous body, meaning that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement. If none of the curves changes length, it is said that a
6194:
1874:
1125:
3278:
3681:
3527:
3214:
860:. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations may remain, and these exist even after stresses have been removed. One type of irreversible deformation is
6129:
3520:
3100:
1879:
5723:
4973:
5509:
3273:
5761:
5462:
deformation is defined as an isochoric plane deformation in which there is a set of line elements with a given reference orientation that do not change length and orientation during the deformation.
6134:
4461:
4273:
4046:
3868:
4645:{\displaystyle {\begin{aligned}U_{J}&=\delta _{Ji}x_{i}-X_{J}=x_{J}-X_{J}\\{\frac {\partial U_{J}}{\partial x_{k}}}&=\delta _{Jk}-{\frac {\partial X_{J}}{\partial x_{k}}}\end{aligned}}}
4230:{\displaystyle {\begin{aligned}u_{i}&=x_{i}-\delta _{iJ}X_{J}=x_{i}-X_{i}\\{\frac {\partial u_{i}}{\partial X_{K}}}&={\frac {\partial x_{i}}{\partial X_{K}}}-\delta _{iK}\end{aligned}}}
2822:{\displaystyle \mathbf {u} (\mathbf {X} ,t)=\mathbf {b} (\mathbf {X} ,t)+\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=\alpha _{iJ}b_{J}+x_{i}-\alpha _{iJ}X_{J}}
5406:
165:
5449:
1798:
1049:
1016:
The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.
4736:, is one where the deformation is restricted to one of the planes in the reference configuration. If the deformation is restricted to the plane described by the basis vectors
2988:{\displaystyle \mathbf {U} (\mathbf {x} ,t)=\mathbf {b} (\mathbf {x} ,t)+\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=b_{J}+\alpha _{Ji}x_{i}-X_{J}}
5756:
951:. Additionally, time is not considered when analyzing deformation, thus the sequence of configurations between the undeformed and deformed configurations are of no interest.
3123:
97:
6080:
3449:
3029:
5090:, the deformation gradient, up to a change of coordinates, can be decomposed into a stretch and a rotation. Since all the deformation is in a plane, we can write
5639:
4673:
4663:
3426:{\displaystyle \mathbf {u} (\mathbf {X} ,t)=u_{i}\mathbf {e} _{i}=u_{i}(\alpha _{iJ}\mathbf {E} _{J})=U_{J}\mathbf {E} _{J}=\mathbf {U} (\mathbf {x} ,t)}
3825:{\displaystyle \mathbf {U} (\mathbf {x} ,t)=\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=\delta _{Ji}x_{i}-X_{J}=x_{J}-X_{J}}
3671:{\displaystyle \mathbf {u} (\mathbf {X} ,t)=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=x_{i}-\delta _{iJ}X_{J}=x_{i}-X_{i}}
998:
There are two methods for analysing the deformation of a continuum. One description is made in terms of the material or referential coordinates, called
3221:
4668:
714:
1778:
A rigid body motion is a special affine deformation that does not involve any shear, extension or compression. The transformation matrix
4659:
Homogeneous (or affine) deformations are useful in elucidating the behavior of materials. Some homogeneous deformations of interest are
1939:{\displaystyle {\boldsymbol {Q}}\cdot {\boldsymbol {Q}}^{T}={\boldsymbol {Q}}^{T}\cdot {\boldsymbol {Q}}={\boldsymbol {\mathit {1}}}}
5345:
5411:
5079:{\displaystyle {\boldsymbol {F}}={\begin{bmatrix}F_{11}&F_{12}&0\\F_{21}&F_{22}&0\\0&0&1\end{bmatrix}}}
17:
6362:
Three-Dimensional
Continuum Instabilities and Effects of Finite Strain Tensor, chapter 11 in "Stability of Structures", 3rd ed
992:
of a particle in the deformed configuration, taken with respect to the spatial coordinate system of reference, are called the
3847:
4252:
1039:(such as rotation, shear, extension and compression) and a rigid body translation. Affine deformations are also called
6370:
5629:{\displaystyle F_{11}\mathbf {e} _{1}+F_{21}\mathbf {e} _{2}=\mathbf {e} _{1}\quad \implies \quad F_{11}=1~;~~F_{21}=0}
890:
deformation. In the case of elastic deformations, the response function linking strain to the deforming stress is the
6536:
6515:
6494:
6473:
6454:
6412:
6391:
6319:
5728:
4250:
Similarly, the partial differentiation of the displacement vector with respect to the spatial coordinates yields the
707:
3435:
It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in
971:
of a particle in the reference configuration, taken with respect to the reference coordinate system, are called the
6596:
6225:
5839:{\displaystyle {\boldsymbol {F}}={\begin{bmatrix}1&\gamma &0\\0&1&0\\0&0&1\end{bmatrix}}}
1013:
The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
121:
6557:
6433:
2535:
2521:
680:
6586:
6189:{\displaystyle {\boldsymbol {F}}={\boldsymbol {\mathit {1}}}+\gamma \mathbf {e} _{1}\otimes \mathbf {e} _{2}}
381:
218:
6581:
3845:
The partial differentiation of the displacement vector with respect to the material coordinates yields the
3840:
700:
421:
307:
6240:
5087:
376:
285:
168:
71:
866:, which occurs in material bodies after stresses have attained a certain threshold value known as the
6230:
292:
31:
5474:
is the fixed reference orientation in which line elements do not deform during the deformation then
1155:
is the translation. In matrix form, where the components are with respect to an orthonormal basis,
6591:
6218:
1869:{\displaystyle \mathbf {x} (\mathbf {X} ,t)={\boldsymbol {Q}}(t)\cdot \mathbf {X} +\mathbf {c} (t)}
1789:
1120:{\displaystyle \mathbf {x} (\mathbf {X} ,t)={\boldsymbol {F}}(t)\cdot \mathbf {X} +\mathbf {c} (t)}
1002:. A second description of deformation is made in terms of the spatial coordinates it is called the
875:
587:
582:
371:
364:
197:
78:
891:
760:
650:
645:
314:
3004:
are the direction cosines between the material and spatial coordinate systems with unit vectors
2517:
842:
202:
4715:
2578:
1036:
1031:
815:
776:
625:
6235:
5321:
4755:
862:
784:
463:
280:
260:
248:
192:
4689:
Linear or longitudinal deformations of long objects, such as beams and fibers, are called
837:
or because of some changes in the conditions of the body. The relation between stress and
8:
4698:
1003:
999:
871:
792:
744:
732:
665:
513:
406:
112:
6270:
6265:
3209:{\displaystyle u_{i}=\alpha _{iJ}U_{J}\qquad {\text{or}}\qquad U_{J}=\alpha _{Ji}u_{i}}
838:
803:
685:
319:
275:
270:
6601:
6553:
6532:
6511:
6490:
6469:
6450:
6429:
6408:
6387:
6366:
6315:
4720:
1785:
850:
302:
253:
6124:{\displaystyle \mathbf {e} _{i}\otimes \mathbf {e} _{i}={\boldsymbol {\mathit {1}}}}
6576:
6205:
4708:
Plane deformations are also of interest, particularly in the experimental context.
830:
740:
640:
615:
528:
503:
498:
453:
6245:
856:
Deformations which cease to exist after the stress field is removed are termed as
27:
Transformation of a body from a reference configuration to a current configuration
6547:
6528:
Plasticity: A Treatise on Finite
Deformation of Heterogeneous Inelastic Materials
6526:
6505:
6484:
6444:
6423:
6402:
6381:
6360:
6275:
3443:
887:
822:
799:
630:
554:
518:
468:
399:
388:
333:
235:
3515:{\displaystyle \mathbf {E} _{J}\cdot \mathbf {e} _{i}=\delta _{Ji}=\delta _{iJ}}
3095:{\displaystyle \mathbf {E} _{J}\cdot \mathbf {e} _{i}=\alpha _{Ji}=\alpha _{iJ}}
6260:
846:
635:
493:
458:
359:
265:
2526:
1009:
There is continuity during deformation of a continuum body in the sense that:
6570:
6250:
4702:
675:
508:
882:
mechanisms at the atomic level. Another type of irreversible deformation is
818:), or changes in temperature, moisture content, or chemical reactions, etc.
6255:
5459:
4678:
660:
655:
620:
352:
59:
2577:
in the undeformed configuration and deformed configuration is called the
879:
670:
573:
46:
5718:{\displaystyle F_{11}F_{22}-F_{12}F_{21}=1\quad \implies \quad F_{22}=1}
4683:
943:
For deformation analysis, the reference configuration is identified as
904:
807:
764:
592:
488:
6209:
564:
559:
393:
5758:
Then, the deformation gradient in simple shear can be expressed as
780:
543:
448:
428:
414:
52:
rod are much greater than displacements associated with straining.
6213:
811:
752:
728:
297:
2534:
A change in the configuration of a continuum body results in a
748:
438:
6365:. Singapore, New Jersey, London: World Scientific Publishing.
3268:{\displaystyle \mathbf {e} _{i}=\alpha _{iJ}\mathbf {E} _{J}}
834:
756:
342:
6383:
Continuum
Mechanics: Elasticity, Plasticity, Viscoelasticity
2650:
field is expressed in terms of the material coordinates as
1133:
is the position of a point in the deformed configuration,
5332:
If the deformation is isochoric (volume preserving) then
478:
1029:
is a deformation that can be completely described by an
5778:
5216:
5126:
4990:
2423:
2366:
2159:
1958:
1632:
1575:
1368:
1167:
6137:
6083:
5852:
5764:
5731:
5642:
5512:
5414:
5348:
5096:
4976:
4764:
4459:
4271:
4044:
3866:
3684:
3530:
3452:
3281:
3224:
3126:
3032:
2835:
2656:
1952:
1882:
1801:
1161:
1052:
124:
81:
6466:Rheology: principles, measurement and applications
6188:
6123:
6069:
5838:
5750:
5717:
5628:
5443:
5400:
5290:
5078:
4962:
4644:
4445:
4229:
4030:
3824:
3670:
3514:
3425:
3267:
3208:
3094:
2987:
2821:
2570:rigid-body displacement is said to have occurred.
2501:
1938:
1868:
1710:
1119:
159:
91:
6568:
3834:
2573:The vector joining the positions of a particle
6131:we can also write the deformation gradient as
5327:
1139:is the position in a reference configuration,
1046:Therefore, an affine deformation has the form
1000:material description or Lagrangian description
6421:
6358:
6294:
897:
708:
6524:
6428:. London and New York: J. Wiley & Sons.
4718:; the relative volume deformation is called
775:configuration, excluding the body's average
6400:
5401:{\displaystyle F_{11}F_{22}-F_{12}F_{21}=1}
2511:
1004:spatial description or Eulerian description
918:is considered the reference configuration,
160:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
6359:Bazant, Zdenek P.; Cedolin, Luigi (2010).
6297:The non-linear field theories of mechanics
5961:
5957:
5697:
5693:
5580:
5576:
5444:{\displaystyle \lambda _{1}\lambda _{2}=1}
3678:or in terms of the spatial coordinates as
2829:or in terms of the spatial coordinates as
715:
701:
45:
6531:. Cambridge: Cambridge University Press.
6504:Mase, G. Thomas; Mase, George E. (1999).
6204:The deformation of long elements such as
1035:. Such a transformation is composed of a
6503:
6288:
2525:
1795:A rigid body motion can be described by
932:. The configuration at the current time
6463:
6442:
6401:Hutter, Kolumban; Jöhnk, Klaus (2004).
6335:
6333:
6331:
6139:
5964:
5854:
5766:
5114:
5106:
5098:
4978:
4766:
2552:to a current or deformed configuration
1922:
1908:
1893:
1884:
1828:
1079:
841:(relative deformation) is expressed by
14:
6569:
6545:
6404:Continuum Methods of Physical Modeling
4714:is a uniform scaling due to isotropic
759:(m). It is quantified as the residual
84:
6549:Introduction to Mechanics of Continua
6339:
6299:(3rd ed.). Springer. p. 48.
3848:material displacement gradient tensor
2530:Figure 1. Motion of a continuum body.
1020:
6482:
6379:
6328:
5636:Since the deformation is isochoric,
4727:
4253:spatial displacement gradient tensor
1773:
975:. On the other hand, the components
886:, which is the irreversible part of
798:A deformation can occur because of
3442:, and the direction cosines become
1788:in order to allow rotations but no
947:, and the current configuration as
24:
6352:
6312:Continuum Mechanics and Plasticity
6309:
6148:
6116:
4622:
4607:
4565:
4550:
4393:
4372:
4340:
4191:
4176:
4150:
4135:
3988:
3959:
3935:
2611:in the Lagrangian description, or
1931:
25:
6613:
6507:Continuum Mechanics for Engineers
6422:Jirasek, M; Bazant, Z.P. (2002).
6303:
4732:A plane deformation, also called
973:material or reference coordinates
6425:Inelastic Analysis of Structures
6295:Truesdell, C.; Noll, W. (2004).
6176:
6161:
6101:
6086:
6057:
6042:
6027:
6012:
5991:
5976:
5946:
5931:
5913:
5888:
5863:
5565:
5550:
5525:
4950:
4935:
4920:
4905:
4880:
4865:
4840:
4825:
4800:
4785:
4426:
4417:
4405:
4398:
4384:
4377:
4364:
4352:
4345:
4322:
4314:
4306:
4285:
4277:
4020:
4012:
4000:
3993:
3979:
3971:
3964:
3947:
3940:
3926:
3909:
3901:
3880:
3872:
3727:
3719:
3711:
3694:
3686:
3582:
3565:
3557:
3540:
3532:
3470:
3455:
3410:
3402:
3388:
3360:
3319:
3291:
3283:
3255:
3227:
3050:
3035:
2903:
2895:
2887:
2870:
2862:
2845:
2837:
2733:
2716:
2708:
2691:
2683:
2666:
2658:
1853:
1845:
1811:
1803:
1104:
1096:
1062:
1054:
6342:Non-linear Elastic Deformations
5962:
5956:
5751:{\displaystyle \gamma :=F_{12}}
5698:
5692:
5581:
5575:
5453:
3746:
3740:
3592:
3586:
3169:
3163:
2922:
2916:
2743:
2737:
6001:
5971:
5958:
5694:
5577:
4332:
4318:
4295:
4281:
3919:
3905:
3890:
3876:
3737:
3723:
3704:
3690:
3575:
3561:
3550:
3536:
3420:
3406:
3370:
3342:
3301:
3287:
2913:
2899:
2880:
2866:
2855:
2841:
2726:
2712:
2701:
2687:
2676:
2662:
2522:Displacement field (mechanics)
2488:
2482:
2465:
2459:
2442:
2436:
2350:
2344:
2329:
2323:
2308:
2302:
2285:
2279:
2264:
2258:
2243:
2237:
2220:
2214:
2199:
2193:
2178:
2172:
2140:
2095:
2078:
2033:
2016:
1971:
1863:
1857:
1838:
1832:
1821:
1807:
1720:The above deformation becomes
1697:
1691:
1674:
1668:
1651:
1645:
1559:
1553:
1538:
1532:
1517:
1511:
1494:
1488:
1473:
1467:
1452:
1446:
1429:
1423:
1408:
1402:
1387:
1381:
1349:
1304:
1287:
1242:
1225:
1180:
1149:is the linear transformer and
1114:
1108:
1089:
1083:
1072:
1058:
795:of all particles of the body.
30:For usage in engineering, see
13:
1:
6281:
5302:is the angle of rotation and
4697:; derived quantities are the
3102:and the relationship between
2642:in the Eulerian description.
743:or size of an object. It has
92:{\displaystyle {\mathsf {L}}}
6489:. McGraw-Hill Professional.
3841:Displacement gradient tensor
3835:Displacement gradient tensor
7:
6510:(2nd ed.). CRC Press.
6380:Dill, Ellis Harold (2006).
6241:Infinitesimal strain theory
6226:EulerâBernoulli beam theory
6198:
5328:Isochoric plane deformation
5088:polar decomposition theorem
4654:
4245:deformation gradient tensor
802:, intrinsic activity (e.g.
10:
6618:
6525:Nemat-Nasser, Sia (2006).
6443:Lubarda, Vlado A. (2001).
3838:
2515:
1143:is a time-like parameter,
898:Definition and formulation
29:
6231:Deformation (engineering)
70:
57:
44:
39:
32:Deformation (engineering)
6546:Prager, William (1961).
6483:Mase, George E. (1970).
2512:Background: displacement
1041:homogeneous deformations
945:undeformed configuration
874:, and are the result of
791:is a set containing the
219:ClausiusâDuhem (entropy)
169:Fick's laws of diffusion
6597:Deformation (mechanics)
6552:. Boston: Ginn and Co.
6464:Macosko, C. W. (1994).
6446:Elastoplasticity Theory
986:of the position vector
965:of the position vector
907:displacement occurred.
377:NavierâStokes equations
315:Material failure theory
6386:. Germany: CRC Press.
6190:
6125:
6071:
5840:
5752:
5719:
5630:
5445:
5402:
5292:
5080:
4964:
4646:
4447:
4231:
4032:
3826:
3672:
3516:
3427:
3269:
3210:
3096:
2989:
2823:
2531:
2518:Displacement (physics)
2503:
1940:
1870:
1712:
1121:
949:deformed configuration
843:constitutive equations
816:electromagnetic forces
763:of particles in a non-
161:
93:
18:Elongation (mechanics)
6407:. Germany: Springer.
6340:Ogden, R. W. (1984).
6191:
6126:
6072:
5841:
5753:
5720:
5631:
5446:
5403:
5293:
5081:
4965:
4647:
4448:
4232:
4033:
3827:
3673:
3517:
3428:
3270:
3211:
3097:
3026:, respectively. Thus
2990:
2824:
2529:
2504:
1941:
1871:
1713:
1122:
1037:linear transformation
1032:affine transformation
938:current configuration
833:field due to applied
739:is the change in the
372:Bernoulli's principle
365:Archimedes' principle
162:
94:
6587:Non-Newtonian fluids
6236:Finite strain theory
6135:
6081:
5850:
5762:
5729:
5640:
5510:
5412:
5346:
5094:
4974:
4762:
4756:deformation gradient
4457:
4269:
4042:
3864:
3682:
3528:
3450:
3279:
3222:
3124:
3030:
2833:
2654:
1950:
1880:
1799:
1159:
1050:
785:rigid transformation
464:Cohesion (chemistry)
286:Infinitesimal strain
122:
79:
6582:Continuum mechanics
6486:Continuum Mechanics
6216:forces is known as
5322:principal stretches
4699:relative elongation
4674:equibiaxial tension
2579:displacement vector
994:spatial coordinates
884:viscous deformation
863:plastic deformation
858:elastic deformation
771:configuration to a
733:continuum mechanics
382:Poiseuille equation
113:Continuum mechanics
107:Part of a series on
6468:. VCH Publishers.
6310:Wu, H.-C. (2005).
6271:Stress (mechanics)
6266:Strain (mechanics)
6186:
6121:
6067:
5836:
5830:
5748:
5715:
5626:
5441:
5398:
5288:
5282:
5205:
5076:
5070:
4960:
4712:Volume deformation
4642:
4640:
4443:
4441:
4227:
4225:
4028:
4026:
3822:
3668:
3512:
3423:
3265:
3206:
3092:
2985:
2819:
2647:displacement field
2532:
2499:
2493:
2409:
2355:
2145:
1936:
1866:
1708:
1702:
1618:
1564:
1354:
1117:
1027:affine deformation
1021:Affine deformation
804:muscle contraction
588:Magnetorheological
583:Electrorheological
320:Fracture mechanics
157:
89:
61:SI base units
5609:
5606:
5600:
4728:Plane deformation
4721:volumetric strain
4664:uniform extension
4636:
4579:
4205:
4164:
3744:
3590:
3167:
3120:is then given by
2920:
2741:
1786:proper orthogonal
1774:Rigid body motion
894:of the material.
892:compliance tensor
827:deformation field
725:
724:
600:
599:
534:
533:
303:Contact mechanics
226:
225:
155:
102:
101:
16:(Redirected from
6609:
6563:
6542:
6521:
6500:
6479:
6460:
6439:
6418:
6397:
6376:
6346:
6345:
6337:
6326:
6325:
6307:
6301:
6300:
6292:
6195:
6193:
6192:
6187:
6185:
6184:
6179:
6170:
6169:
6164:
6152:
6151:
6142:
6130:
6128:
6127:
6122:
6120:
6119:
6110:
6109:
6104:
6095:
6094:
6089:
6076:
6074:
6073:
6068:
6066:
6065:
6060:
6051:
6050:
6045:
6036:
6035:
6030:
6021:
6020:
6015:
6000:
5999:
5994:
5985:
5984:
5979:
5967:
5955:
5954:
5949:
5940:
5939:
5934:
5922:
5921:
5916:
5910:
5909:
5897:
5896:
5891:
5885:
5884:
5872:
5871:
5866:
5857:
5845:
5843:
5842:
5837:
5835:
5834:
5769:
5757:
5755:
5754:
5749:
5747:
5746:
5724:
5722:
5721:
5716:
5708:
5707:
5685:
5684:
5675:
5674:
5662:
5661:
5652:
5651:
5635:
5633:
5632:
5627:
5619:
5618:
5607:
5604:
5598:
5591:
5590:
5574:
5573:
5568:
5559:
5558:
5553:
5547:
5546:
5534:
5533:
5528:
5522:
5521:
5505:
5483:
5473:
5450:
5448:
5447:
5442:
5434:
5433:
5424:
5423:
5407:
5405:
5404:
5399:
5391:
5390:
5381:
5380:
5368:
5367:
5358:
5357:
5341:
5319:
5310:
5301:
5297:
5295:
5294:
5289:
5287:
5286:
5257:
5256:
5228:
5227:
5210:
5209:
5117:
5109:
5101:
5085:
5083:
5082:
5077:
5075:
5074:
5045:
5044:
5033:
5032:
5014:
5013:
5002:
5001:
4981:
4970:In matrix form,
4969:
4967:
4966:
4961:
4959:
4958:
4953:
4944:
4943:
4938:
4929:
4928:
4923:
4914:
4913:
4908:
4902:
4901:
4889:
4888:
4883:
4874:
4873:
4868:
4862:
4861:
4849:
4848:
4843:
4834:
4833:
4828:
4822:
4821:
4809:
4808:
4803:
4794:
4793:
4788:
4782:
4781:
4769:
4753:
4744:
4651:
4649:
4648:
4643:
4641:
4637:
4635:
4634:
4633:
4620:
4619:
4618:
4605:
4600:
4599:
4580:
4578:
4577:
4576:
4563:
4562:
4561:
4548:
4542:
4541:
4529:
4528:
4516:
4515:
4503:
4502:
4493:
4492:
4473:
4472:
4452:
4450:
4449:
4444:
4442:
4438:
4437:
4429:
4420:
4408:
4403:
4402:
4401:
4387:
4382:
4381:
4380:
4367:
4355:
4350:
4349:
4348:
4325:
4317:
4309:
4288:
4280:
4265:. Thus we have,
4264:
4242:
4236:
4234:
4233:
4228:
4226:
4222:
4221:
4206:
4204:
4203:
4202:
4189:
4188:
4187:
4174:
4165:
4163:
4162:
4161:
4148:
4147:
4146:
4133:
4127:
4126:
4114:
4113:
4101:
4100:
4091:
4090:
4075:
4074:
4058:
4057:
4037:
4035:
4034:
4029:
4027:
4023:
4015:
4003:
3998:
3997:
3996:
3982:
3974:
3969:
3968:
3967:
3950:
3945:
3944:
3943:
3929:
3912:
3904:
3883:
3875:
3860:. Thus we have:
3859:
3831:
3829:
3828:
3823:
3821:
3820:
3808:
3807:
3795:
3794:
3782:
3781:
3772:
3771:
3756:
3755:
3745:
3742:
3730:
3722:
3714:
3697:
3689:
3677:
3675:
3674:
3669:
3667:
3666:
3654:
3653:
3641:
3640:
3631:
3630:
3615:
3614:
3602:
3601:
3591:
3588:
3585:
3568:
3560:
3543:
3535:
3521:
3519:
3518:
3513:
3511:
3510:
3495:
3494:
3479:
3478:
3473:
3464:
3463:
3458:
3444:Kronecker deltas
3441:
3432:
3430:
3429:
3424:
3413:
3405:
3397:
3396:
3391:
3385:
3384:
3369:
3368:
3363:
3357:
3356:
3341:
3340:
3328:
3327:
3322:
3316:
3315:
3294:
3286:
3274:
3272:
3271:
3266:
3264:
3263:
3258:
3252:
3251:
3236:
3235:
3230:
3215:
3213:
3212:
3207:
3205:
3204:
3195:
3194:
3179:
3178:
3168:
3165:
3162:
3161:
3152:
3151:
3136:
3135:
3119:
3110:
3101:
3099:
3098:
3093:
3091:
3090:
3075:
3074:
3059:
3058:
3053:
3044:
3043:
3038:
3025:
3014:
3003:
2994:
2992:
2991:
2986:
2984:
2983:
2971:
2970:
2961:
2960:
2945:
2944:
2932:
2931:
2921:
2918:
2906:
2898:
2890:
2873:
2865:
2848:
2840:
2828:
2826:
2825:
2820:
2818:
2817:
2808:
2807:
2792:
2791:
2779:
2778:
2769:
2768:
2753:
2752:
2742:
2739:
2736:
2719:
2711:
2694:
2686:
2669:
2661:
2641:
2610:
2565:
2551:
2508:
2506:
2505:
2500:
2498:
2497:
2481:
2480:
2458:
2457:
2435:
2434:
2414:
2413:
2406:
2405:
2392:
2391:
2378:
2377:
2360:
2359:
2343:
2342:
2322:
2321:
2301:
2300:
2278:
2277:
2257:
2256:
2236:
2235:
2213:
2212:
2192:
2191:
2171:
2170:
2150:
2149:
2133:
2132:
2120:
2119:
2107:
2106:
2094:
2093:
2071:
2070:
2058:
2057:
2045:
2044:
2032:
2031:
2009:
2008:
1996:
1995:
1983:
1982:
1970:
1969:
1946:In matrix form,
1945:
1943:
1942:
1937:
1935:
1934:
1925:
1917:
1916:
1911:
1902:
1901:
1896:
1887:
1875:
1873:
1872:
1867:
1856:
1848:
1831:
1814:
1806:
1783:
1769:
1750:
1717:
1715:
1714:
1709:
1707:
1706:
1690:
1689:
1667:
1666:
1644:
1643:
1623:
1622:
1615:
1614:
1601:
1600:
1587:
1586:
1569:
1568:
1552:
1551:
1531:
1530:
1510:
1509:
1487:
1486:
1466:
1465:
1445:
1444:
1422:
1421:
1401:
1400:
1380:
1379:
1359:
1358:
1342:
1341:
1329:
1328:
1316:
1315:
1303:
1302:
1280:
1279:
1267:
1266:
1254:
1253:
1241:
1240:
1218:
1217:
1205:
1204:
1192:
1191:
1179:
1178:
1154:
1148:
1142:
1138:
1132:
1126:
1124:
1123:
1118:
1107:
1099:
1082:
1065:
1057:
991:
985:
970:
964:
935:
931:
917:
717:
710:
703:
549:
548:
514:Gay-Lussac's law
504:Combined gas law
454:Capillary action
339:
338:
182:
181:
166:
164:
163:
158:
156:
154:
146:
138:
104:
103:
98:
96:
95:
90:
88:
87:
62:
49:
37:
36:
21:
6617:
6616:
6612:
6611:
6610:
6608:
6607:
6606:
6592:Solid mechanics
6567:
6566:
6560:
6539:
6518:
6497:
6476:
6457:
6436:
6415:
6394:
6373:
6355:
6353:Further reading
6350:
6349:
6338:
6329:
6322:
6308:
6304:
6293:
6289:
6284:
6276:Stress measures
6201:
6180:
6175:
6174:
6165:
6160:
6159:
6147:
6146:
6138:
6136:
6133:
6132:
6115:
6114:
6105:
6100:
6099:
6090:
6085:
6084:
6082:
6079:
6078:
6061:
6056:
6055:
6046:
6041:
6040:
6031:
6026:
6025:
6016:
6011:
6010:
5995:
5990:
5989:
5980:
5975:
5974:
5963:
5950:
5945:
5944:
5935:
5930:
5929:
5917:
5912:
5911:
5905:
5901:
5892:
5887:
5886:
5880:
5876:
5867:
5862:
5861:
5853:
5851:
5848:
5847:
5829:
5828:
5823:
5818:
5812:
5811:
5806:
5801:
5795:
5794:
5789:
5784:
5774:
5773:
5765:
5763:
5760:
5759:
5742:
5738:
5730:
5727:
5726:
5703:
5699:
5680:
5676:
5670:
5666:
5657:
5653:
5647:
5643:
5641:
5638:
5637:
5614:
5610:
5586:
5582:
5569:
5564:
5563:
5554:
5549:
5548:
5542:
5538:
5529:
5524:
5523:
5517:
5513:
5511:
5508:
5507:
5504:
5497:
5485:
5481:
5475:
5472:
5466:
5456:
5429:
5425:
5419:
5415:
5413:
5410:
5409:
5408:Alternatively,
5386:
5382:
5376:
5372:
5363:
5359:
5353:
5349:
5347:
5344:
5343:
5333:
5330:
5318:
5312:
5309:
5303:
5299:
5281:
5280:
5275:
5270:
5264:
5263:
5258:
5252:
5248:
5246:
5240:
5239:
5234:
5229:
5223:
5219:
5212:
5211:
5204:
5203:
5198:
5193:
5187:
5186:
5181:
5170:
5155:
5154:
5149:
5138:
5122:
5121:
5113:
5105:
5097:
5095:
5092:
5091:
5069:
5068:
5063:
5058:
5052:
5051:
5046:
5040:
5036:
5034:
5028:
5024:
5021:
5020:
5015:
5009:
5005:
5003:
4997:
4993:
4986:
4985:
4977:
4975:
4972:
4971:
4954:
4949:
4948:
4939:
4934:
4933:
4924:
4919:
4918:
4909:
4904:
4903:
4897:
4893:
4884:
4879:
4878:
4869:
4864:
4863:
4857:
4853:
4844:
4839:
4838:
4829:
4824:
4823:
4817:
4813:
4804:
4799:
4798:
4789:
4784:
4783:
4777:
4773:
4765:
4763:
4760:
4759:
4752:
4746:
4743:
4737:
4730:
4657:
4639:
4638:
4629:
4625:
4621:
4614:
4610:
4606:
4604:
4592:
4588:
4581:
4572:
4568:
4564:
4557:
4553:
4549:
4547:
4544:
4543:
4537:
4533:
4524:
4520:
4511:
4507:
4498:
4494:
4485:
4481:
4474:
4468:
4464:
4460:
4458:
4455:
4454:
4440:
4439:
4430:
4425:
4424:
4416:
4409:
4404:
4397:
4396:
4392:
4389:
4388:
4383:
4376:
4375:
4371:
4363:
4356:
4351:
4344:
4343:
4339:
4336:
4335:
4321:
4313:
4305:
4298:
4284:
4276:
4272:
4270:
4267:
4266:
4261:
4256:
4238:
4224:
4223:
4214:
4210:
4198:
4194:
4190:
4183:
4179:
4175:
4173:
4166:
4157:
4153:
4149:
4142:
4138:
4134:
4132:
4129:
4128:
4122:
4118:
4109:
4105:
4096:
4092:
4083:
4079:
4070:
4066:
4059:
4053:
4049:
4045:
4043:
4040:
4039:
4025:
4024:
4019:
4011:
4004:
3999:
3992:
3991:
3987:
3984:
3983:
3978:
3970:
3963:
3962:
3958:
3951:
3946:
3939:
3938:
3934:
3931:
3930:
3925:
3908:
3900:
3893:
3879:
3871:
3867:
3865:
3862:
3861:
3856:
3851:
3843:
3837:
3816:
3812:
3803:
3799:
3790:
3786:
3777:
3773:
3764:
3760:
3751:
3747:
3741:
3726:
3718:
3710:
3693:
3685:
3683:
3680:
3679:
3662:
3658:
3649:
3645:
3636:
3632:
3623:
3619:
3610:
3606:
3597:
3593:
3587:
3581:
3564:
3556:
3539:
3531:
3529:
3526:
3525:
3503:
3499:
3487:
3483:
3474:
3469:
3468:
3459:
3454:
3453:
3451:
3448:
3447:
3436:
3409:
3401:
3392:
3387:
3386:
3380:
3376:
3364:
3359:
3358:
3349:
3345:
3336:
3332:
3323:
3318:
3317:
3311:
3307:
3290:
3282:
3280:
3277:
3276:
3259:
3254:
3253:
3244:
3240:
3231:
3226:
3225:
3223:
3220:
3219:
3200:
3196:
3187:
3183:
3174:
3170:
3164:
3157:
3153:
3144:
3140:
3131:
3127:
3125:
3122:
3121:
3117:
3112:
3108:
3103:
3083:
3079:
3067:
3063:
3054:
3049:
3048:
3039:
3034:
3033:
3031:
3028:
3027:
3024:
3016:
3013:
3005:
3001:
2996:
2979:
2975:
2966:
2962:
2953:
2949:
2940:
2936:
2927:
2923:
2917:
2902:
2894:
2886:
2869:
2861:
2844:
2836:
2834:
2831:
2830:
2813:
2809:
2800:
2796:
2787:
2783:
2774:
2770:
2761:
2757:
2748:
2744:
2738:
2732:
2715:
2707:
2690:
2682:
2665:
2657:
2655:
2652:
2651:
2640:
2632:
2612:
2609:
2601:
2581:
2558:
2553:
2545:
2539:
2524:
2516:Main articles:
2514:
2492:
2491:
2476:
2472:
2469:
2468:
2453:
2449:
2446:
2445:
2430:
2426:
2419:
2418:
2408:
2407:
2401:
2397:
2394:
2393:
2387:
2383:
2380:
2379:
2373:
2369:
2362:
2361:
2354:
2353:
2338:
2334:
2332:
2317:
2313:
2311:
2296:
2292:
2289:
2288:
2273:
2269:
2267:
2252:
2248:
2246:
2231:
2227:
2224:
2223:
2208:
2204:
2202:
2187:
2183:
2181:
2166:
2162:
2155:
2154:
2144:
2143:
2128:
2124:
2115:
2111:
2102:
2098:
2089:
2085:
2082:
2081:
2066:
2062:
2053:
2049:
2040:
2036:
2027:
2023:
2020:
2019:
2004:
2000:
1991:
1987:
1978:
1974:
1965:
1961:
1954:
1953:
1951:
1948:
1947:
1930:
1929:
1921:
1912:
1907:
1906:
1897:
1892:
1891:
1883:
1881:
1878:
1877:
1852:
1844:
1827:
1810:
1802:
1800:
1797:
1796:
1779:
1776:
1752:
1729:
1701:
1700:
1685:
1681:
1678:
1677:
1662:
1658:
1655:
1654:
1639:
1635:
1628:
1627:
1617:
1616:
1610:
1606:
1603:
1602:
1596:
1592:
1589:
1588:
1582:
1578:
1571:
1570:
1563:
1562:
1547:
1543:
1541:
1526:
1522:
1520:
1505:
1501:
1498:
1497:
1482:
1478:
1476:
1461:
1457:
1455:
1440:
1436:
1433:
1432:
1417:
1413:
1411:
1396:
1392:
1390:
1375:
1371:
1364:
1363:
1353:
1352:
1337:
1333:
1324:
1320:
1311:
1307:
1298:
1294:
1291:
1290:
1275:
1271:
1262:
1258:
1249:
1245:
1236:
1232:
1229:
1228:
1213:
1209:
1200:
1196:
1187:
1183:
1174:
1170:
1163:
1162:
1160:
1157:
1156:
1150:
1144:
1140:
1134:
1128:
1103:
1095:
1078:
1061:
1053:
1051:
1048:
1047:
1023:
987:
984:
976:
966:
963:
955:
954:The components
933:
925:
919:
912:
900:
829:results from a
823:continuous body
721:
692:
691:
690:
610:
602:
601:
555:Viscoelasticity
546:
536:
535:
523:
473:
469:Surface tension
433:
336:
334:Fluid mechanics
326:
325:
324:
238:
236:Solid mechanics
228:
227:
179:
171:
147:
139:
137:
123:
120:
119:
83:
82:
80:
77:
76:
60:
53:
35:
28:
23:
22:
15:
12:
11:
5:
6615:
6605:
6604:
6599:
6594:
6589:
6584:
6579:
6565:
6564:
6558:
6543:
6537:
6522:
6516:
6501:
6495:
6480:
6474:
6461:
6455:
6440:
6434:
6419:
6413:
6398:
6392:
6377:
6372:978-9814317030
6371:
6354:
6351:
6348:
6347:
6327:
6320:
6302:
6286:
6285:
6283:
6280:
6279:
6278:
6273:
6268:
6263:
6261:Shear strength
6258:
6253:
6248:
6243:
6238:
6233:
6228:
6223:
6200:
6197:
6183:
6178:
6173:
6168:
6163:
6158:
6155:
6150:
6145:
6141:
6118:
6113:
6108:
6103:
6098:
6093:
6088:
6064:
6059:
6054:
6049:
6044:
6039:
6034:
6029:
6024:
6019:
6014:
6009:
6006:
6003:
5998:
5993:
5988:
5983:
5978:
5973:
5970:
5966:
5960:
5953:
5948:
5943:
5938:
5933:
5928:
5925:
5920:
5915:
5908:
5904:
5900:
5895:
5890:
5883:
5879:
5875:
5870:
5865:
5860:
5856:
5833:
5827:
5824:
5822:
5819:
5817:
5814:
5813:
5810:
5807:
5805:
5802:
5800:
5797:
5796:
5793:
5790:
5788:
5785:
5783:
5780:
5779:
5777:
5772:
5768:
5745:
5741:
5737:
5734:
5714:
5711:
5706:
5702:
5696:
5691:
5688:
5683:
5679:
5673:
5669:
5665:
5660:
5656:
5650:
5646:
5625:
5622:
5617:
5613:
5603:
5597:
5594:
5589:
5585:
5579:
5572:
5567:
5562:
5557:
5552:
5545:
5541:
5537:
5532:
5527:
5520:
5516:
5502:
5495:
5479:
5470:
5455:
5452:
5440:
5437:
5432:
5428:
5422:
5418:
5397:
5394:
5389:
5385:
5379:
5375:
5371:
5366:
5362:
5356:
5352:
5329:
5326:
5316:
5307:
5285:
5279:
5276:
5274:
5271:
5269:
5266:
5265:
5262:
5259:
5255:
5251:
5247:
5245:
5242:
5241:
5238:
5235:
5233:
5230:
5226:
5222:
5218:
5217:
5215:
5208:
5202:
5199:
5197:
5194:
5192:
5189:
5188:
5185:
5182:
5180:
5177:
5174:
5171:
5169:
5166:
5163:
5160:
5157:
5156:
5153:
5150:
5148:
5145:
5142:
5139:
5137:
5134:
5131:
5128:
5127:
5125:
5120:
5116:
5112:
5108:
5104:
5100:
5073:
5067:
5064:
5062:
5059:
5057:
5054:
5053:
5050:
5047:
5043:
5039:
5035:
5031:
5027:
5023:
5022:
5019:
5016:
5012:
5008:
5004:
5000:
4996:
4992:
4991:
4989:
4984:
4980:
4957:
4952:
4947:
4942:
4937:
4932:
4927:
4922:
4917:
4912:
4907:
4900:
4896:
4892:
4887:
4882:
4877:
4872:
4867:
4860:
4856:
4852:
4847:
4842:
4837:
4832:
4827:
4820:
4816:
4812:
4807:
4802:
4797:
4792:
4787:
4780:
4776:
4772:
4768:
4750:
4741:
4729:
4726:
4687:
4686:
4681:
4676:
4671:
4666:
4656:
4653:
4632:
4628:
4624:
4617:
4613:
4609:
4603:
4598:
4595:
4591:
4587:
4584:
4582:
4575:
4571:
4567:
4560:
4556:
4552:
4546:
4545:
4540:
4536:
4532:
4527:
4523:
4519:
4514:
4510:
4506:
4501:
4497:
4491:
4488:
4484:
4480:
4477:
4475:
4471:
4467:
4463:
4462:
4436:
4433:
4428:
4423:
4419:
4415:
4412:
4410:
4407:
4400:
4395:
4391:
4390:
4386:
4379:
4374:
4370:
4366:
4362:
4359:
4357:
4354:
4347:
4342:
4338:
4337:
4334:
4331:
4328:
4324:
4320:
4316:
4312:
4308:
4304:
4301:
4299:
4297:
4294:
4291:
4287:
4283:
4279:
4275:
4274:
4259:
4220:
4217:
4213:
4209:
4201:
4197:
4193:
4186:
4182:
4178:
4172:
4169:
4167:
4160:
4156:
4152:
4145:
4141:
4137:
4131:
4130:
4125:
4121:
4117:
4112:
4108:
4104:
4099:
4095:
4089:
4086:
4082:
4078:
4073:
4069:
4065:
4062:
4060:
4056:
4052:
4048:
4047:
4022:
4018:
4014:
4010:
4007:
4005:
4002:
3995:
3990:
3986:
3985:
3981:
3977:
3973:
3966:
3961:
3957:
3954:
3952:
3949:
3942:
3937:
3933:
3932:
3928:
3924:
3921:
3918:
3915:
3911:
3907:
3903:
3899:
3896:
3894:
3892:
3889:
3886:
3882:
3878:
3874:
3870:
3869:
3854:
3839:Main article:
3836:
3833:
3819:
3815:
3811:
3806:
3802:
3798:
3793:
3789:
3785:
3780:
3776:
3770:
3767:
3763:
3759:
3754:
3750:
3739:
3736:
3733:
3729:
3725:
3721:
3717:
3713:
3709:
3706:
3703:
3700:
3696:
3692:
3688:
3665:
3661:
3657:
3652:
3648:
3644:
3639:
3635:
3629:
3626:
3622:
3618:
3613:
3609:
3605:
3600:
3596:
3584:
3580:
3577:
3574:
3571:
3567:
3563:
3559:
3555:
3552:
3549:
3546:
3542:
3538:
3534:
3524:Thus, we have
3509:
3506:
3502:
3498:
3493:
3490:
3486:
3482:
3477:
3472:
3467:
3462:
3457:
3422:
3419:
3416:
3412:
3408:
3404:
3400:
3395:
3390:
3383:
3379:
3375:
3372:
3367:
3362:
3355:
3352:
3348:
3344:
3339:
3335:
3331:
3326:
3321:
3314:
3310:
3306:
3303:
3300:
3297:
3293:
3289:
3285:
3262:
3257:
3250:
3247:
3243:
3239:
3234:
3229:
3203:
3199:
3193:
3190:
3186:
3182:
3177:
3173:
3160:
3156:
3150:
3147:
3143:
3139:
3134:
3130:
3115:
3106:
3089:
3086:
3082:
3078:
3073:
3070:
3066:
3062:
3057:
3052:
3047:
3042:
3037:
3020:
3009:
2999:
2982:
2978:
2974:
2969:
2965:
2959:
2956:
2952:
2948:
2943:
2939:
2935:
2930:
2926:
2915:
2912:
2909:
2905:
2901:
2897:
2893:
2889:
2885:
2882:
2879:
2876:
2872:
2868:
2864:
2860:
2857:
2854:
2851:
2847:
2843:
2839:
2816:
2812:
2806:
2803:
2799:
2795:
2790:
2786:
2782:
2777:
2773:
2767:
2764:
2760:
2756:
2751:
2747:
2735:
2731:
2728:
2725:
2722:
2718:
2714:
2710:
2706:
2703:
2700:
2697:
2693:
2689:
2685:
2681:
2678:
2675:
2672:
2668:
2664:
2660:
2636:
2628:
2605:
2597:
2556:
2543:
2513:
2510:
2496:
2490:
2487:
2484:
2479:
2475:
2471:
2470:
2467:
2464:
2461:
2456:
2452:
2448:
2447:
2444:
2441:
2438:
2433:
2429:
2425:
2424:
2422:
2417:
2412:
2404:
2400:
2396:
2395:
2390:
2386:
2382:
2381:
2376:
2372:
2368:
2367:
2365:
2358:
2352:
2349:
2346:
2341:
2337:
2333:
2331:
2328:
2325:
2320:
2316:
2312:
2310:
2307:
2304:
2299:
2295:
2291:
2290:
2287:
2284:
2281:
2276:
2272:
2268:
2266:
2263:
2260:
2255:
2251:
2247:
2245:
2242:
2239:
2234:
2230:
2226:
2225:
2222:
2219:
2216:
2211:
2207:
2203:
2201:
2198:
2195:
2190:
2186:
2182:
2180:
2177:
2174:
2169:
2165:
2161:
2160:
2158:
2153:
2148:
2142:
2139:
2136:
2131:
2127:
2123:
2118:
2114:
2110:
2105:
2101:
2097:
2092:
2088:
2084:
2083:
2080:
2077:
2074:
2069:
2065:
2061:
2056:
2052:
2048:
2043:
2039:
2035:
2030:
2026:
2022:
2021:
2018:
2015:
2012:
2007:
2003:
1999:
1994:
1990:
1986:
1981:
1977:
1973:
1968:
1964:
1960:
1959:
1957:
1933:
1928:
1924:
1920:
1915:
1910:
1905:
1900:
1895:
1890:
1886:
1865:
1862:
1859:
1855:
1851:
1847:
1843:
1840:
1837:
1834:
1830:
1826:
1823:
1820:
1817:
1813:
1809:
1805:
1775:
1772:
1705:
1699:
1696:
1693:
1688:
1684:
1680:
1679:
1676:
1673:
1670:
1665:
1661:
1657:
1656:
1653:
1650:
1647:
1642:
1638:
1634:
1633:
1631:
1626:
1621:
1613:
1609:
1605:
1604:
1599:
1595:
1591:
1590:
1585:
1581:
1577:
1576:
1574:
1567:
1561:
1558:
1555:
1550:
1546:
1542:
1540:
1537:
1534:
1529:
1525:
1521:
1519:
1516:
1513:
1508:
1504:
1500:
1499:
1496:
1493:
1490:
1485:
1481:
1477:
1475:
1472:
1469:
1464:
1460:
1456:
1454:
1451:
1448:
1443:
1439:
1435:
1434:
1431:
1428:
1425:
1420:
1416:
1412:
1410:
1407:
1404:
1399:
1395:
1391:
1389:
1386:
1383:
1378:
1374:
1370:
1369:
1367:
1362:
1357:
1351:
1348:
1345:
1340:
1336:
1332:
1327:
1323:
1319:
1314:
1310:
1306:
1301:
1297:
1293:
1292:
1289:
1286:
1283:
1278:
1274:
1270:
1265:
1261:
1257:
1252:
1248:
1244:
1239:
1235:
1231:
1230:
1227:
1224:
1221:
1216:
1212:
1208:
1203:
1199:
1195:
1190:
1186:
1182:
1177:
1173:
1169:
1168:
1166:
1116:
1113:
1110:
1106:
1102:
1098:
1094:
1091:
1088:
1085:
1081:
1077:
1074:
1071:
1068:
1064:
1060:
1056:
1022:
1019:
1018:
1017:
1014:
980:
959:
923:
899:
896:
851:linear elastic
800:external loads
774:
770:
723:
722:
720:
719:
712:
705:
697:
694:
693:
689:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
628:
623:
618:
612:
611:
608:
607:
604:
603:
598:
597:
596:
595:
590:
585:
577:
576:
570:
569:
568:
567:
562:
557:
547:
542:
541:
538:
537:
532:
531:
525:
524:
522:
521:
516:
511:
506:
501:
496:
491:
485:
482:
481:
475:
474:
472:
471:
466:
461:
459:Chromatography
456:
451:
445:
442:
441:
435:
434:
432:
431:
412:
411:
410:
391:
379:
374:
362:
349:
346:
345:
337:
332:
331:
328:
327:
323:
322:
317:
312:
311:
310:
300:
295:
290:
289:
288:
283:
273:
268:
263:
258:
257:
256:
246:
240:
239:
234:
233:
230:
229:
224:
223:
222:
221:
213:
212:
208:
207:
206:
205:
200:
195:
187:
186:
180:
177:
176:
173:
172:
167:
153:
150:
145:
142:
136:
133:
130:
127:
116:
115:
109:
108:
100:
99:
86:
74:
68:
67:
64:
55:
54:
50:
42:
41:
26:
9:
6:
4:
3:
2:
6614:
6603:
6600:
6598:
6595:
6593:
6590:
6588:
6585:
6583:
6580:
6578:
6575:
6574:
6572:
6561:
6555:
6551:
6550:
6544:
6540:
6538:0-521-83979-3
6534:
6530:
6529:
6523:
6519:
6517:0-8493-1855-6
6513:
6509:
6508:
6502:
6498:
6496:0-07-040663-4
6492:
6488:
6487:
6481:
6477:
6475:1-56081-579-5
6471:
6467:
6462:
6458:
6456:0-8493-1138-1
6452:
6449:. CRC Press.
6448:
6447:
6441:
6437:
6431:
6427:
6426:
6420:
6416:
6414:3-540-20619-1
6410:
6406:
6405:
6399:
6395:
6393:0-8493-9779-0
6389:
6385:
6384:
6378:
6374:
6368:
6364:
6363:
6357:
6356:
6343:
6336:
6334:
6332:
6323:
6321:1-58488-363-4
6317:
6314:. CRC Press.
6313:
6306:
6298:
6291:
6287:
6277:
6274:
6272:
6269:
6267:
6264:
6262:
6259:
6257:
6254:
6252:
6251:Shear modulus
6249:
6247:
6246:Moiré pattern
6244:
6242:
6239:
6237:
6234:
6232:
6229:
6227:
6224:
6221:
6220:
6215:
6211:
6207:
6203:
6202:
6196:
6181:
6171:
6166:
6156:
6153:
6143:
6111:
6106:
6096:
6091:
6062:
6052:
6047:
6037:
6032:
6022:
6017:
6007:
6004:
5996:
5986:
5981:
5968:
5951:
5941:
5936:
5926:
5923:
5918:
5906:
5902:
5898:
5893:
5881:
5877:
5873:
5868:
5858:
5831:
5825:
5820:
5815:
5808:
5803:
5798:
5791:
5786:
5781:
5775:
5770:
5743:
5739:
5735:
5732:
5712:
5709:
5704:
5700:
5689:
5686:
5681:
5677:
5671:
5667:
5663:
5658:
5654:
5648:
5644:
5623:
5620:
5615:
5611:
5601:
5595:
5592:
5587:
5583:
5570:
5560:
5555:
5543:
5539:
5535:
5530:
5518:
5514:
5506:. Therefore,
5501:
5494:
5490:
5489:
5478:
5469:
5463:
5461:
5451:
5438:
5435:
5430:
5426:
5420:
5416:
5395:
5392:
5387:
5383:
5377:
5373:
5369:
5364:
5360:
5354:
5350:
5339:
5338:
5325:
5323:
5315:
5306:
5283:
5277:
5272:
5267:
5260:
5253:
5249:
5243:
5236:
5231:
5224:
5220:
5213:
5206:
5200:
5195:
5190:
5183:
5178:
5175:
5172:
5167:
5164:
5161:
5158:
5151:
5146:
5143:
5140:
5135:
5132:
5129:
5123:
5118:
5110:
5102:
5089:
5071:
5065:
5060:
5055:
5048:
5041:
5037:
5029:
5025:
5017:
5010:
5006:
4998:
4994:
4987:
4982:
4955:
4945:
4940:
4930:
4925:
4915:
4910:
4898:
4894:
4890:
4885:
4875:
4870:
4858:
4854:
4850:
4845:
4835:
4830:
4818:
4814:
4810:
4805:
4795:
4790:
4778:
4774:
4770:
4758:has the form
4757:
4749:
4740:
4735:
4725:
4723:
4722:
4717:
4713:
4709:
4706:
4704:
4703:stretch ratio
4700:
4696:
4692:
4685:
4682:
4680:
4677:
4675:
4672:
4670:
4669:pure dilation
4667:
4665:
4662:
4661:
4660:
4652:
4630:
4626:
4615:
4611:
4601:
4596:
4593:
4589:
4585:
4583:
4573:
4569:
4558:
4554:
4538:
4534:
4530:
4525:
4521:
4517:
4512:
4508:
4504:
4499:
4495:
4489:
4486:
4482:
4478:
4476:
4469:
4465:
4434:
4431:
4421:
4413:
4411:
4368:
4360:
4358:
4329:
4326:
4310:
4302:
4300:
4292:
4289:
4263:
4255:
4254:
4248:
4246:
4241:
4218:
4215:
4211:
4207:
4199:
4195:
4184:
4180:
4170:
4168:
4158:
4154:
4143:
4139:
4123:
4119:
4115:
4110:
4106:
4102:
4097:
4093:
4087:
4084:
4080:
4076:
4071:
4067:
4063:
4061:
4054:
4050:
4016:
4008:
4006:
3975:
3955:
3953:
3922:
3916:
3913:
3897:
3895:
3887:
3884:
3858:
3850:
3849:
3842:
3832:
3817:
3813:
3809:
3804:
3800:
3796:
3791:
3787:
3783:
3778:
3774:
3768:
3765:
3761:
3757:
3752:
3748:
3734:
3731:
3715:
3707:
3701:
3698:
3663:
3659:
3655:
3650:
3646:
3642:
3637:
3633:
3627:
3624:
3620:
3616:
3611:
3607:
3603:
3598:
3594:
3578:
3572:
3569:
3553:
3547:
3544:
3522:
3507:
3504:
3500:
3496:
3491:
3488:
3484:
3480:
3475:
3465:
3460:
3445:
3439:
3433:
3417:
3414:
3398:
3393:
3381:
3377:
3373:
3365:
3353:
3350:
3346:
3337:
3333:
3329:
3324:
3312:
3308:
3304:
3298:
3295:
3260:
3248:
3245:
3241:
3237:
3232:
3218:Knowing that
3216:
3201:
3197:
3191:
3188:
3184:
3180:
3175:
3171:
3158:
3154:
3148:
3145:
3141:
3137:
3132:
3128:
3118:
3109:
3087:
3084:
3080:
3076:
3071:
3068:
3064:
3060:
3055:
3045:
3040:
3023:
3019:
3012:
3008:
3002:
2980:
2976:
2972:
2967:
2963:
2957:
2954:
2950:
2946:
2941:
2937:
2933:
2928:
2924:
2910:
2907:
2891:
2883:
2877:
2874:
2858:
2852:
2849:
2814:
2810:
2804:
2801:
2797:
2793:
2788:
2784:
2780:
2775:
2771:
2765:
2762:
2758:
2754:
2749:
2745:
2729:
2723:
2720:
2704:
2698:
2695:
2679:
2673:
2670:
2648:
2643:
2639:
2635:
2631:
2627:
2623:
2619:
2615:
2608:
2604:
2600:
2596:
2592:
2588:
2584:
2580:
2576:
2571:
2567:
2563:
2559:
2549:
2542:
2537:
2528:
2523:
2519:
2509:
2494:
2485:
2477:
2473:
2462:
2454:
2450:
2439:
2431:
2427:
2420:
2415:
2410:
2402:
2398:
2388:
2384:
2374:
2370:
2363:
2356:
2347:
2339:
2335:
2326:
2318:
2314:
2305:
2297:
2293:
2282:
2274:
2270:
2261:
2253:
2249:
2240:
2232:
2228:
2217:
2209:
2205:
2196:
2188:
2184:
2175:
2167:
2163:
2156:
2151:
2146:
2137:
2134:
2129:
2125:
2121:
2116:
2112:
2108:
2103:
2099:
2090:
2086:
2075:
2072:
2067:
2063:
2059:
2054:
2050:
2046:
2041:
2037:
2028:
2024:
2013:
2010:
2005:
2001:
1997:
1992:
1988:
1984:
1979:
1975:
1966:
1962:
1955:
1926:
1918:
1913:
1903:
1898:
1888:
1860:
1849:
1841:
1835:
1824:
1818:
1815:
1793:
1791:
1787:
1782:
1771:
1767:
1763:
1759:
1755:
1748:
1744:
1740:
1739:
1734:
1733:
1727:
1726:inhomogeneous
1723:
1718:
1703:
1694:
1686:
1682:
1671:
1663:
1659:
1648:
1640:
1636:
1629:
1624:
1619:
1611:
1607:
1597:
1593:
1583:
1579:
1572:
1565:
1556:
1548:
1544:
1535:
1527:
1523:
1514:
1506:
1502:
1491:
1483:
1479:
1470:
1462:
1458:
1449:
1441:
1437:
1426:
1418:
1414:
1405:
1397:
1393:
1384:
1376:
1372:
1365:
1360:
1355:
1346:
1343:
1338:
1334:
1330:
1325:
1321:
1317:
1312:
1308:
1299:
1295:
1284:
1281:
1276:
1272:
1268:
1263:
1259:
1255:
1250:
1246:
1237:
1233:
1222:
1219:
1214:
1210:
1206:
1201:
1197:
1193:
1188:
1184:
1175:
1171:
1164:
1153:
1147:
1137:
1131:
1111:
1100:
1092:
1086:
1075:
1069:
1066:
1044:
1042:
1038:
1034:
1033:
1028:
1015:
1012:
1011:
1010:
1007:
1005:
1001:
996:
995:
990:
983:
979:
974:
969:
962:
958:
952:
950:
946:
941:
939:
929:
922:
915:
908:
906:
895:
893:
889:
885:
881:
877:
873:
869:
868:elastic limit
865:
864:
859:
854:
852:
848:
844:
840:
836:
832:
828:
824:
819:
817:
813:
809:
805:
801:
796:
794:
790:
789:configuration
786:
782:
778:
772:
768:
766:
762:
758:
754:
750:
746:
742:
738:
734:
730:
718:
713:
711:
706:
704:
699:
698:
696:
695:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
613:
606:
605:
594:
591:
589:
586:
584:
581:
580:
579:
578:
575:
572:
571:
566:
563:
561:
558:
556:
553:
552:
551:
550:
545:
540:
539:
530:
527:
526:
520:
517:
515:
512:
510:
507:
505:
502:
500:
499:Charles's law
497:
495:
492:
490:
487:
486:
484:
483:
480:
477:
476:
470:
467:
465:
462:
460:
457:
455:
452:
450:
447:
446:
444:
443:
440:
437:
436:
430:
427:
423:
420:
416:
413:
408:
407:non-Newtonian
405:
401:
397:
396:
395:
392:
390:
387:
383:
380:
378:
375:
373:
370:
366:
363:
361:
358:
354:
351:
350:
348:
347:
344:
341:
340:
335:
330:
329:
321:
318:
316:
313:
309:
306:
305:
304:
301:
299:
296:
294:
293:Compatibility
291:
287:
284:
282:
281:Finite strain
279:
278:
277:
274:
272:
269:
267:
264:
262:
259:
255:
252:
251:
250:
247:
245:
242:
241:
237:
232:
231:
220:
217:
216:
215:
214:
210:
209:
204:
201:
199:
196:
194:
191:
190:
189:
188:
185:Conservations
184:
183:
175:
174:
170:
151:
148:
143:
140:
134:
131:
128:
125:
118:
117:
114:
111:
110:
106:
105:
75:
73:
69:
65:
63:
56:
48:
43:
38:
33:
19:
6548:
6527:
6506:
6485:
6465:
6445:
6424:
6403:
6382:
6361:
6341:
6311:
6305:
6296:
6290:
6256:Shear stress
6217:
5499:
5492:
5487:
5486:
5476:
5467:
5464:
5460:simple shear
5457:
5454:Simple shear
5342:and we have
5336:
5335:
5331:
5313:
5304:
4747:
4738:
4734:plane strain
4733:
4731:
4719:
4711:
4710:
4707:
4694:
4690:
4688:
4679:simple shear
4658:
4257:
4251:
4249:
4244:
4239:
3852:
3846:
3844:
3523:
3437:
3434:
3217:
3113:
3104:
3021:
3017:
3010:
3006:
2997:
2646:
2644:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2606:
2602:
2598:
2594:
2590:
2586:
2582:
2574:
2572:
2568:
2566:(Figure 1).
2561:
2554:
2547:
2540:
2536:displacement
2533:
1794:
1780:
1777:
1765:
1761:
1757:
1753:
1746:
1742:
1737:
1736:
1731:
1730:
1725:
1721:
1719:
1151:
1145:
1135:
1129:
1045:
1040:
1030:
1026:
1024:
1008:
997:
993:
988:
981:
977:
972:
967:
960:
956:
953:
948:
944:
942:
937:
927:
920:
913:
909:
901:
888:viscoelastic
883:
872:yield stress
867:
861:
857:
855:
853:materials.
826:
820:
797:
788:
761:displacement
736:
726:
574:Smart fluids
519:Graham's law
425:
418:
403:
389:Pascal's law
385:
368:
356:
243:
211:Inequalities
4716:compression
1790:reflections
880:dislocation
847:Hooke's law
808:body forces
777:translation
737:deformation
593:Ferrofluids
494:Boyle's law
266:Hooke's law
244:Deformation
40:Deformation
6571:Categories
6559:0486438090
6435:0471987166
6282:References
6219:deflection
4695:shortening
4691:elongation
4684:pure shear
1722:non-affine
905:rigid body
767:, from an
765:rigid body
646:Gay-Lussac
609:Scientists
509:Fick's law
489:Atmosphere
308:frictional
261:Plasticity
249:Elasticity
6172:⊗
6157:γ
6097:⊗
6053:⊗
6023:⊗
6008:γ
5987:⊗
5969:⋅
5959:⟹
5927:γ
5859:⋅
5787:γ
5733:γ
5695:⟹
5664:−
5578:⟹
5427:λ
5417:λ
5370:−
5250:λ
5221:λ
5179:θ
5176:
5168:θ
5165:
5159:−
5147:θ
5144:
5136:θ
5133:
5111:⋅
5086:From the
4946:⊗
4916:⊗
4876:⊗
4836:⊗
4796:⊗
4623:∂
4608:∂
4602:−
4590:δ
4566:∂
4551:∂
4531:−
4505:−
4483:δ
4432:−
4422:−
4394:∇
4373:∇
4369:−
4341:∇
4311:−
4212:δ
4208:−
4192:∂
4177:∂
4151:∂
4136:∂
4116:−
4081:δ
4077:−
4017:−
3989:∇
3976:−
3960:∇
3936:∇
3923:−
3810:−
3784:−
3762:δ
3716:−
3656:−
3621:δ
3617:−
3579:−
3501:δ
3485:δ
3466:⋅
3347:α
3242:α
3185:α
3142:α
3081:α
3065:α
3046:⋅
2973:−
2951:α
2892:−
2798:α
2794:−
2759:α
2730:−
1919:⋅
1889:⋅
1842:⋅
1093:⋅
810:(such as
793:positions
745:dimension
686:Truesdell
616:Bernoulli
565:Rheometer
560:Rheometry
400:Newtonian
394:Viscosity
144:φ
132:−
72:Dimension
6602:Geometry
6344:. Dover.
6199:See also
5320:are the
4701:and the
4655:Examples
845:, e.g.,
781:rotation
544:Rheology
449:Adhesion
429:Pressure
415:Buoyancy
360:Dynamics
198:Momentum
6577:Tensors
6214:bending
6212:due to
5725:Define
4243:is the
936:is the
812:gravity
769:initial
753:SI unit
729:physics
631:Charles
439:Liquids
353:Statics
298:Bending
6556:
6535:
6514:
6493:
6472:
6453:
6432:
6411:
6390:
6369:
6318:
6077:Since
5608:
5605:
5599:
5298:where
4754:, the
4237:where
2995:where
1876:where
1127:where
839:strain
835:forces
831:stress
787:). A
749:length
681:Stokes
676:Pascal
666:Navier
661:Newton
651:Graham
626:Cauchy
529:Plasma
424:
422:Mixing
417:
402:
384:
367:
355:
343:Fluids
276:Strain
271:Stress
254:linear
203:Energy
6210:studs
6206:beams
5846:Now,
5340:) = 1
3275:then
878:, or
821:In a
783:(its
773:final
757:metre
751:with
741:shape
656:Hooke
636:Euler
621:Boyle
479:Gases
6554:ISBN
6533:ISBN
6512:ISBN
6491:ISBN
6470:ISBN
6451:ISBN
6430:ISBN
6409:ISBN
6388:ISBN
6367:ISBN
6316:ISBN
5484:and
5334:det(
3111:and
3015:and
2624:) =
2593:) =
2520:and
876:slip
849:for
825:, a
779:and
731:and
671:Noll
641:Fick
193:Mass
178:Laws
6208:or
5482:= 1
5465:If
5173:cos
5162:sin
5141:sin
5130:cos
4693:or
4453:or
4038:or
3440:= 0
1784:is
1751:or
1728:if
1724:or
1025:An
916:= 0
870:or
814:or
806:),
755:of
747:of
727:In
58:In
6573::
6330:^
5907:22
5882:12
5744:12
5736::=
5705:22
5682:21
5672:12
5659:22
5649:11
5616:21
5588:11
5544:21
5519:11
5498:=
5458:A
5388:21
5378:12
5365:22
5355:11
5324:.
5311:,
5042:22
5030:21
5011:12
4999:11
4899:22
4859:21
4819:12
4779:11
4745:,
4724:.
4705:.
4247:.
3743:or
3589:or
3446::
3166:or
3000:Ji
2919:or
2740:or
2645:A
2340:33
2319:32
2298:31
2275:23
2254:22
2233:21
2210:13
2189:12
2168:11
1792:.
1770:.
1756:=
1735:=
1549:33
1528:32
1507:31
1484:23
1463:22
1442:21
1419:13
1398:12
1377:11
1043:.
1006:.
940:.
735:,
6562:.
6541:.
6520:.
6499:.
6478:.
6459:.
6438:.
6417:.
6396:.
6375:.
6324:.
6222:.
6182:2
6177:e
6167:1
6162:e
6154:+
6149:1
6144:=
6140:F
6117:1
6112:=
6107:i
6102:e
6092:i
6087:e
6063:2
6058:e
6048:2
6043:e
6038:+
6033:2
6028:e
6018:1
6013:e
6005:=
6002:)
5997:2
5992:e
5982:2
5977:e
5972:(
5965:F
5952:2
5947:e
5942:+
5937:1
5932:e
5924:=
5919:2
5914:e
5903:F
5899:+
5894:1
5889:e
5878:F
5874:=
5869:2
5864:e
5855:F
5832:]
5826:1
5821:0
5816:0
5809:0
5804:1
5799:0
5792:0
5782:1
5776:[
5771:=
5767:F
5740:F
5713:1
5710:=
5701:F
5690:1
5687:=
5678:F
5668:F
5655:F
5645:F
5624:0
5621:=
5612:F
5602:;
5596:1
5593:=
5584:F
5571:1
5566:e
5561:=
5556:2
5551:e
5540:F
5536:+
5531:1
5526:e
5515:F
5503:1
5500:e
5496:1
5493:e
5491:·
5488:F
5480:1
5477:λ
5471:1
5468:e
5439:1
5436:=
5431:2
5421:1
5396:1
5393:=
5384:F
5374:F
5361:F
5351:F
5337:F
5317:2
5314:λ
5308:1
5305:λ
5300:Ξ
5284:]
5278:1
5273:0
5268:0
5261:0
5254:2
5244:0
5237:0
5232:0
5225:1
5214:[
5207:]
5201:1
5196:0
5191:0
5184:0
5152:0
5124:[
5119:=
5115:U
5107:R
5103:=
5099:F
5072:]
5066:1
5061:0
5056:0
5049:0
5038:F
5026:F
5018:0
5007:F
4995:F
4988:[
4983:=
4979:F
4956:3
4951:e
4941:3
4936:e
4931:+
4926:2
4921:e
4911:2
4906:e
4895:F
4891:+
4886:1
4881:e
4871:2
4866:e
4855:F
4851:+
4846:2
4841:e
4831:1
4826:e
4815:F
4811:+
4806:1
4801:e
4791:1
4786:e
4775:F
4771:=
4767:F
4751:2
4748:e
4742:1
4739:e
4631:k
4627:x
4616:J
4612:X
4597:k
4594:J
4586:=
4574:k
4570:x
4559:J
4555:U
4539:J
4535:X
4526:J
4522:x
4518:=
4513:J
4509:X
4500:i
4496:x
4490:i
4487:J
4479:=
4470:J
4466:U
4435:1
4427:F
4418:I
4414:=
4406:U
4399:x
4385:X
4378:x
4365:I
4361:=
4353:U
4346:x
4333:)
4330:t
4327:,
4323:x
4319:(
4315:X
4307:x
4303:=
4296:)
4293:t
4290:,
4286:x
4282:(
4278:U
4262:U
4260:x
4258:â
4240:F
4219:K
4216:i
4200:K
4196:X
4185:i
4181:x
4171:=
4159:K
4155:X
4144:i
4140:u
4124:i
4120:X
4111:i
4107:x
4103:=
4098:J
4094:X
4088:J
4085:i
4072:i
4068:x
4064:=
4055:i
4051:u
4021:I
4013:F
4009:=
4001:u
3994:X
3980:I
3972:x
3965:X
3956:=
3948:u
3941:X
3927:X
3920:)
3917:t
3914:,
3910:X
3906:(
3902:x
3898:=
3891:)
3888:t
3885:,
3881:X
3877:(
3873:u
3857:u
3855:X
3853:â
3818:J
3814:X
3805:J
3801:x
3797:=
3792:J
3788:X
3779:i
3775:x
3769:i
3766:J
3758:=
3753:J
3749:U
3738:)
3735:t
3732:,
3728:x
3724:(
3720:X
3712:x
3708:=
3705:)
3702:t
3699:,
3695:x
3691:(
3687:U
3664:i
3660:X
3651:i
3647:x
3643:=
3638:J
3634:X
3628:J
3625:i
3612:i
3608:x
3604:=
3599:i
3595:u
3583:X
3576:)
3573:t
3570:,
3566:X
3562:(
3558:x
3554:=
3551:)
3548:t
3545:,
3541:X
3537:(
3533:u
3508:J
3505:i
3497:=
3492:i
3489:J
3481:=
3476:i
3471:e
3461:J
3456:E
3438:b
3421:)
3418:t
3415:,
3411:x
3407:(
3403:U
3399:=
3394:J
3389:E
3382:J
3378:U
3374:=
3371:)
3366:J
3361:E
3354:J
3351:i
3343:(
3338:i
3334:u
3330:=
3325:i
3320:e
3313:i
3309:u
3305:=
3302:)
3299:t
3296:,
3292:X
3288:(
3284:u
3261:J
3256:E
3249:J
3246:i
3238:=
3233:i
3228:e
3202:i
3198:u
3192:i
3189:J
3181:=
3176:J
3172:U
3159:J
3155:U
3149:J
3146:i
3138:=
3133:i
3129:u
3116:J
3114:U
3107:i
3105:u
3088:J
3085:i
3077:=
3072:i
3069:J
3061:=
3056:i
3051:e
3041:J
3036:E
3022:i
3018:e
3011:J
3007:E
2998:α
2981:J
2977:X
2968:i
2964:x
2958:i
2955:J
2947:+
2942:J
2938:b
2934:=
2929:J
2925:U
2914:)
2911:t
2908:,
2904:x
2900:(
2896:X
2888:x
2884:+
2881:)
2878:t
2875:,
2871:x
2867:(
2863:b
2859:=
2856:)
2853:t
2850:,
2846:x
2842:(
2838:U
2815:J
2811:X
2805:J
2802:i
2789:i
2785:x
2781:+
2776:J
2772:b
2766:J
2763:i
2755:=
2750:i
2746:u
2734:X
2727:)
2724:t
2721:,
2717:X
2713:(
2709:x
2705:+
2702:)
2699:t
2696:,
2692:X
2688:(
2684:b
2680:=
2677:)
2674:t
2671:,
2667:X
2663:(
2659:u
2638:J
2634:E
2630:J
2626:U
2622:t
2620:,
2618:x
2616:(
2614:U
2607:i
2603:e
2599:i
2595:u
2591:t
2589:,
2587:X
2585:(
2583:u
2575:P
2564:)
2562:B
2560:(
2557:t
2555:Îș
2550:)
2548:B
2546:(
2544:0
2541:Îș
2495:]
2489:)
2486:t
2483:(
2478:3
2474:c
2466:)
2463:t
2460:(
2455:2
2451:c
2443:)
2440:t
2437:(
2432:1
2428:c
2421:[
2416:+
2411:]
2403:3
2399:X
2389:2
2385:X
2375:1
2371:X
2364:[
2357:]
2351:)
2348:t
2345:(
2336:Q
2330:)
2327:t
2324:(
2315:Q
2309:)
2306:t
2303:(
2294:Q
2286:)
2283:t
2280:(
2271:Q
2265:)
2262:t
2259:(
2250:Q
2244:)
2241:t
2238:(
2229:Q
2221:)
2218:t
2215:(
2206:Q
2200:)
2197:t
2194:(
2185:Q
2179:)
2176:t
2173:(
2164:Q
2157:[
2152:=
2147:]
2141:)
2138:t
2135:,
2130:3
2126:X
2122:,
2117:2
2113:X
2109:,
2104:1
2100:X
2096:(
2091:3
2087:x
2079:)
2076:t
2073:,
2068:3
2064:X
2060:,
2055:2
2051:X
2047:,
2042:1
2038:X
2034:(
2029:2
2025:x
2017:)
2014:t
2011:,
2006:3
2002:X
1998:,
1993:2
1989:X
1985:,
1980:1
1976:X
1972:(
1967:1
1963:x
1956:[
1932:1
1927:=
1923:Q
1914:T
1909:Q
1904:=
1899:T
1894:Q
1885:Q
1864:)
1861:t
1858:(
1854:c
1850:+
1846:X
1839:)
1836:t
1833:(
1829:Q
1825:=
1822:)
1819:t
1816:,
1812:X
1808:(
1804:x
1781:F
1768:)
1766:t
1764:,
1762:X
1760:(
1758:c
1754:c
1749:)
1747:t
1745:,
1743:X
1741:(
1738:F
1732:F
1704:]
1698:)
1695:t
1692:(
1687:3
1683:c
1675:)
1672:t
1669:(
1664:2
1660:c
1652:)
1649:t
1646:(
1641:1
1637:c
1630:[
1625:+
1620:]
1612:3
1608:X
1598:2
1594:X
1584:1
1580:X
1573:[
1566:]
1560:)
1557:t
1554:(
1545:F
1539:)
1536:t
1533:(
1524:F
1518:)
1515:t
1512:(
1503:F
1495:)
1492:t
1489:(
1480:F
1474:)
1471:t
1468:(
1459:F
1453:)
1450:t
1447:(
1438:F
1430:)
1427:t
1424:(
1415:F
1409:)
1406:t
1403:(
1394:F
1388:)
1385:t
1382:(
1373:F
1366:[
1361:=
1356:]
1350:)
1347:t
1344:,
1339:3
1335:X
1331:,
1326:2
1322:X
1318:,
1313:1
1309:X
1305:(
1300:3
1296:x
1288:)
1285:t
1282:,
1277:3
1273:X
1269:,
1264:2
1260:X
1256:,
1251:1
1247:X
1243:(
1238:2
1234:x
1226:)
1223:t
1220:,
1215:3
1211:X
1207:,
1202:2
1198:X
1194:,
1189:1
1185:X
1181:(
1176:1
1172:x
1165:[
1152:c
1146:F
1141:t
1136:X
1130:x
1115:)
1112:t
1109:(
1105:c
1101:+
1097:X
1090:)
1087:t
1084:(
1080:F
1076:=
1073:)
1070:t
1067:,
1063:X
1059:(
1055:x
989:x
982:i
978:x
968:X
961:i
957:X
934:t
930:)
928:B
926:(
924:0
921:Îș
914:t
716:e
709:t
702:v
426:·
419:·
409:)
404:·
398:(
386:·
369:·
357:·
152:x
149:d
141:d
135:D
129:=
126:J
85:L
66:m
34:.
20:)
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