498:
133:
286:
892:
3432:
36:
567:
875:. By convention, the strain is set to the horizontal axis and stress is set to vertical axis. Note that for engineering purposes we often assume the cross-section area of the material does not change during the whole deformation process. This is not true since the actual area will decrease while deforming due to elastic and plastic deformation. The curve based on the original cross-section and gauge length is called the
2155:
1564:
1907:
1725:
1383:, true stress is larger than engineering stress and true strain is less than engineering strain. Thus, a point defining true stress–strain curve is displaced upwards and to the left to define the equivalent engineering stress–strain curve. The difference between the true and engineering stresses and strains will increase with
563:. The necking phase is indicated by a reduction in cross-sectional area of the specimen. Necking begins after the ultimate strength is reached. During necking, the material can no longer withstand the maximum stress and the strain in the specimen rapidly increases. Plastic deformation ends with the fracture of the material.
3568:
A popular misconception is that all materials that bend are "weak" and those that do not are "strong". In reality, many materials that undergo large elastic and plastic deformations, such as steel, are able to absorb stresses that would cause brittle materials, such as glass, with minimal plastic
1391:
deformation), the differences between the two is negligible. As for the tensile strength point, it is the maximal point in engineering stress–strain curve but is not a special point in true stress–strain curve. Because engineering stress is proportional to the force applied along the sample, the
256:
so that the original shape (dashed lines) has changed (deformed) into one with bulging sides. The sides bulge because the material, although strong enough to not crack or otherwise fail, is not strong enough to support the load without change. As a result, the material is forced out laterally.
1225:
519:
This type of deformation is not undone simply by removing the applied force. An object in the plastic deformation range, however, will first have undergone elastic deformation, which is undone simply be removing the applied force, so the object will return part way to its original shape. Soft
265:
Depending on the type of material, size and geometry of the object, and the forces applied, various types of deformation may result. The image to the right shows the engineering stress vs. strain diagram for a typical ductile material such as steel. Different deformation modes may occur under
3194:
This figure also shows the dependency of the necking strain at different temperature. In case of FCC metals, both of the stress-strain curve at its derivative are highly dependent on temperature. Therefore, at higher temperature, necking starts to appear even under lower strain value.
1430:
2150:{\displaystyle {\begin{aligned}\sigma _{T}={\frac {F}{A_{f}}}&={\frac {F}{A_{i}}}\times {\frac {A_{i}}{A_{f}}}\\&=\sigma _{e}\times {\frac {l_{f}}{l_{i}}}\\&=\sigma _{E}\times {\frac {l_{i}+\delta l}{l_{i}}}\\&=\sigma _{E}(1+\varepsilon _{E})\end{aligned}}}
2311:
3439:
On the figure (a), there is only concave upward
Considere plot. It indicates that there is no yield drop so the material will be suffered from fracture before it yields. On the figure (b), there is specific point where the tangent matches with secant line at point where
233:. Elasticity in materials occurs when applied stress does not surpass the energy required to break molecular bonds, allowing the material to deform reversibly and return to its original shape once the stress is removed. The linear relationship for a material is known as
1586:
2457:
664:
A break occurs after the material has reached the end of the elastic, and then plastic, deformation ranges. At this point forces accumulate until they are sufficient to cause a fracture. All materials will eventually fracture, if sufficient forces are applied.
1828:
Since we disregard the change of area during deformation above, the true stress and strain curve should be re-derived. For deriving the stress strain curve, we can assume that the volume change is 0 even if we deformed the materials. We can assume that:
2519:
Additionally, based on the true stress-strain curve, we can estimate the region where necking starts to happen. Since necking starts to appear after ultimate tensile stress where the maximum force applied, we can express this situation as below:
3157:
1374:
273:
Permanent deformation is irreversible; the deformation stays even after removal of the applied forces, while the temporary deformation is recoverable as it disappears after the removal of applied forces. Temporary deformation is also called
3318:
2929:
3190:
3) Based on the true stress-strain curve and its derivative form, we can estimate the strain necessary to start necking. This can be calculated based on the intersection between true stress-strain curve as shown in right.
848:
680:
are approximations to the internal state that may be determined from the external forces and deformations of an object, provided that there is no significant change in size. When there is a significant change in size, the
1093:
343:
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
2689:
2923:
1896:
1795:
1281:
1024:
1912:
1559:{\displaystyle {\begin{aligned}&\delta F=\sigma _{\text{t}}\,\delta A+A\,\delta \sigma _{\text{t}}=0\\&-{\frac {\delta A}{A}}={\frac {\delta \sigma _{\mathrm {t} }}{\sigma _{\mathrm {t} }}}\end{aligned}}}
2694:
It indicates that the necking starts to appear where reduction of area becomes much significant compared to the stress change. Then the stress will be localized to specific area where the necking appears.
3435:
Considere Plot. (a) True stress-strain curve without tangents. There is neither necking nor drawing. (b) With one tangent. There is only necking. (c) With two tangents. There are both necking and drawing.
2761:
2602:
2173:
973:
713:
pulling at the ends so the bar is under tension. The material is experiencing a stress defined to be the ratio of the force to the cross sectional area of the bar, as well as an axial elongation:
1720:{\displaystyle {\begin{aligned}\sigma _{\mathrm {t} }&={\frac {F}{A_{\mathrm {neck} }}}\\\varepsilon _{\mathrm {t} }&=\ln \left({\frac {A_{0}}{A_{\mathrm {neck} }}}\right)\end{aligned}}}
1591:
1435:
2322:
482:
767:
2995:
3504:
3471:
419:
3426:
3240:
3201:
4) A graphical method, so-called "Considere construction", can help determine the behavior of stress-strain curve whether necking or drawing happens on the sample. By setting
2514:
3473:. After this value, the slope becomes smaller than the secant line where necking starts to appear. On the figure (c), there is point where yielding starts to appear but when
544:. Hard thermosetting plastics, rubber, crystals, and ceramics have minimal plastic deformation ranges. An example of a material with a large plastic deformation range is wet
3775:
3185:
1580:
After the formation of necking, the sample undergoes heterogeneous deformation, so equations above are not valid. The stress and strain at the necking can be expressed as:
1082:
3558:
3531:
2701:
1) True strain and stress curve can be expressed by the approximate linear relationship by taking a log on true stress and strain. The relation can be expressed as below:
2487:
1423:
3393:
3367:
1292:
3341:
2959:
3198:
All of these properties indicate the importance of calculating the true stress-strain curve for further analyzing the behavior of materials in sudden environment.
3248:
3066:
3055:
3035:
3015:
2844:
2824:
2804:
2784:
2849:
2) In reality, stress is also highly dependent on the rate of strain variation. Thus, we can induce the empirical equation based on the strain rate variation.
1220:{\displaystyle \sigma _{\mathrm {t} }={\frac {F}{A}}={\frac {F}{A_{0}}}{\frac {A_{0}}{A}}={\frac {F}{A_{0}}}{\frac {L}{L_{0}}}=\sigma (1+\varepsilon )}
191:
are changes in the relative position between internals points on the object, excluding rigid transformations, causing the body to change shape or size.
773:
463:
This relationship only applies in the elastic range and indicates that the slope of the stress vs. strain curve can be used to find Young's modulus (
3855:
2613:
871:. The stress–strain curve for this material is plotted by elongating the sample and recording the stress variation with strain until the sample
1835:
1740:
1236:
979:
2855:
53:
100:
72:
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559:
region and finally, fracture (also called rupture). During strain hardening the material becomes stronger through the movement of
2306:{\displaystyle \varepsilon _{T}={\frac {dl}{l_{0}}}+{\frac {dl}{l_{1}}}+{\frac {dl}{l_{2}}}+\cdots =\sum _{i}{\frac {dl}{l_{i}}}}
2707:
79:
2526:
210:
change in shape of an infinitesimal cube of material relative to a reference configuration. Mechanical strains are caused by
3697:
936:
1035:
Here the dimensions are instantaneous values. Assuming volume of the sample conserves and deformation happens uniformly,
86:
17:
2452:{\displaystyle \int _{l_{0}}^{l_{i}}{\frac {dl}{l}}\,dx=\ln \left({\frac {l_{i}}{l_{0}}}\right)=\ln(1+\varepsilon _{E})}
252:
In the above figure, it can be seen that the compressive loading (indicated by the arrow) has caused deformation in the
501:
Swebor-brand high-strength low alloy steel plate, showing both sides, after plastic deformation from bringing to rest
3839:
3759:
3719:
3691:
467:). Engineers often use this calculation in tensile tests. The area under this elastic region is known as resilience.
119:
899:
In the above definitions of engineering stress and strain, two behaviors of materials in tensile tests are ignored:
305:
68:
3880:
732:
3579:
2964:
57:
917:
are defined differently than engineering stress and strain to account for these behaviors. They are given as
3323:
Therefore, the value of engineering stress can be expressed by the secant line from made by true stress and
2928:
3476:
3443:
395:
3506:, the drawing happens. After drawing, all the material will stretch and eventually show fracture. Between
3398:
3634:
3242:
as determinant, the true stress and strain can be expressed with engineering stress and strain as below:
317:
3604:
655:
456:
3204:
2492:
574:, showing the relationship between stress (force applied) and strain (deformation) of a ductile metal.
478:, and many polymers, respond in a nonlinear fashion. For these materials Hooke's law is inapplicable.
179:
are any change in position of a point on the object, including whole-body translations and rotations (
3624:
1570:
267:
170:
Occurrence of deformation in engineering applications is based on the following background concepts:
93:
3875:
3594:
3057:
is at the range of 0-0.1 at room temperature and as high as 0.8 when the temperature is increased.
1369:{\displaystyle \varepsilon _{\mathrm {t} }=\ln \left({\frac {L}{L_{0}}}\right)=\ln(1+\varepsilon )}
1041:
698:
595:
Loading a structural element or specimen will increase the compressive stress until it reaches its
237:. Above the yield point, some degree of permanent distortion remains after unloading and is termed
174:
3536:
3509:
3599:
2465:
1399:
571:
186:
136:
Compressive stress results in deformation which shortens the object but also expands it outwards.
46:
3372:
3346:
241:. The determination of the stress and strain throughout a solid object is given by the field of
3609:
1087:
The true stress and strain can be expressed by engineering stress and strain. For true stress,
868:
3326:
379:
Normal metals, ceramics and most crystals show linear elasticity and a smaller elastic range.
3829:
3683:
Basic
Engineering Plasticity: An Introduction with Engineering and Manufacturing Applications
3654:
1393:
556:
242:
3313:{\displaystyle \sigma _{T}=\sigma _{e}\times \lambda ,\qquad \varepsilon _{T}=\ln \lambda .}
3639:
3152:{\displaystyle \sigma _{T}=K''\times (\varepsilon _{T})^{n}({\dot {\varepsilon _{T}}})^{m}}
1388:
1384:
596:
514:
299:
257:
Internal forces (in this case at right angles to the deformation) resist the applied load.
238:
230:
225:
The relationship between stress and strain is generally linear and reversible up until the
180:
8:
3614:
3589:
3428:
diagram and secant line, we can determine whether the materials show drawing or necking.
3165:
879:, while the curve based on the instantaneous cross-section area and length is called the
600:
486:
253:
246:
217:
3560:, the material itself does not stretch but rather, only the neck starts to stretch out.
2939:
3849:
3644:
3619:
3040:
3020:
3000:
2829:
2809:
2789:
2769:
1731:
659:
639:
442:
234:
211:
194:
3807:
470:
Note that not all elastic materials undergo linear elastic deformation; some, such as
3835:
3755:
3715:
3687:
2997:
indicates the derivative of strain by the time, which is also known as strain rate.
497:
1734:
is commonly used to describe the relationship between true stress and true strain.
895:
The difference between true stress–strain curve and engineering stress–strain curve
552:
432:
316:, which are subjected to very small deformations. Engineering strain is modeled by
3681:
1816:
have a greater resistance to necking. Typically, metals at room temperature have
1380:
856:
446:
285:
132:
2698:
Additionally, we can induce various relation based on true stress-strain curve.
1574:
843:{\displaystyle \varepsilon ={\frac {L-L_{0}}{L_{0}}}={\frac {\Delta L}{L_{0}}}}
521:
475:
386:
308:
is applied to materials used in mechanical and structural engineering, such as
289:
Typical stress vs. strain diagram indicating the various stages of deformation.
1812:
is a measure of a material's work hardening behavior. Materials with a higher
3869:
3649:
3584:
864:
860:
207:
2684:{\displaystyle {\frac {d\sigma _{T}}{\sigma _{T}}}=-{\frac {dA_{i}}{A_{i}}}}
3659:
3629:
891:
524:
have a rather large plastic deformation range as do ductile metals such as
481:
365:
3060:
By combining the 1) and 2), we can create the ultimate relation as below:
1577:
effect is exactly balanced by the shrinking of section area at UTS point.
1891:{\displaystyle A_{i}\times \varepsilon _{i}=A_{f}\times \varepsilon _{f}}
1790:{\displaystyle \sigma _{\mathrm {t} }=K(\varepsilon _{\mathrm {t} })^{n}}
560:
545:
226:
141:
1276:{\displaystyle \delta \varepsilon _{\mathrm {t} }={\frac {\delta L}{L}}}
883:. Unless stated otherwise, engineering stress–strain is generally used.
3037:
is related to the resistance toward the necking. Usually, the value of
506:
502:
340:
157:
1019:{\displaystyle \varepsilon _{\mathrm {t} }=\int {\frac {\delta L}{L}}}
620:
541:
361:
3431:
3187:
is the global constant for relating strain, strain rate and stress.
2918:{\displaystyle \sigma _{T}=K'\times ({\dot {\varepsilon _{T}}})^{m}}
35:
872:
643:
471:
309:
616:
604:
599:. According to the properties of the material, failure modes are
566:
548:, which can be stretched to dozens of times its original length.
369:
2846:
is 1, we can express this material as perfect elastic material.
551:
Under tensile stress, plastic deformation is characterized by a
855:
Subscript 0 denotes the original dimensions of the sample. The
589:
529:
525:
373:
2932:
True stress-strain curve of FCC metal and its derivative form
631:
624:
608:
537:
313:
304:
The study of temporary or elastic deformation in the case of
156:. If the deformation is negligible, the object is said to be
3827:
612:
533:
2756:{\displaystyle \sigma _{T}=K\times (\varepsilon _{T})^{n}}
689:
can be derived from the instantaneous size of the object.
630:
In long, slender structural elements — such as columns or
2597:{\displaystyle dF=0=\sigma _{T}dA_{i}+A_{i}d\sigma _{T}}
2806:
is strain-hardening coefficient. Usually, the value of
1286:
Integrate both sides and apply the boundary condition,
376:. However, elasticity is nonlinear in these materials.
278:
deformation, while the permanent deformation is called
3539:
3512:
3479:
3446:
3401:
3375:
3349:
3329:
3251:
3207:
3168:
3069:
3043:
3023:
3003:
2967:
2942:
2858:
2832:
2826:
has range around 0.02 to 0.5 at room temperature. If
2812:
2792:
2772:
2710:
2616:
2529:
2495:
2468:
2325:
2176:
1910:
1838:
1743:
1589:
1433:
1402:
1295:
1239:
1096:
1044:
982:
968:{\displaystyle \sigma _{\mathrm {t} }={\frac {F}{A}}}
939:
776:
735:
398:
3017:is the strain-rate sensitivity. Moreover, value of
867:(1 pascal = 1 Pa = 1 N/m), and strain is
619:) or rupturing for brittle behavior (geomaterials,
60:. Unsourced material may be challenged and removed.
3552:
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1719:
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1275:
1219:
1076:
1018:
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842:
761:
413:
372:exhibit large elastic deformation ranges, as does
148:(the change in size or shape of an object) may be
3712:Fundamentals of Materials Science and Engineering
2961:is constant related to the material flow stress.
1901:Then, the true stress can be expressed as below:
266:different conditions, as can be depicted using a
3867:
692:
646:at lower stress than the compressive strength.
709:being subjected to equal and opposite forces
668:
588:Usually, compressive stress applied to bars,
3854:: CS1 maint: multiple names: authors list (
336:where strains and rotations are both small.
762:{\displaystyle \sigma ={\frac {F}{A_{0}}}}
3754:. Illinois: Waveland Press. p. 165.
2990:{\displaystyle {\dot {\varepsilon _{T}}}}
2462:Thus, we can induce the plot in terms of
2370:
1471:
1458:
886:
120:Learn how and when to remove this message
3749:
3731:
3430:
2927:
2607:so this form can be expressed as below:
890:
634:bars — an increase of compressive force
565:
496:
480:
284:
131:
27:Change in the shape or size of an object
3714:, John Wiley and Sons, 2nd ed. p. 184.
260:
14:
3868:
583:
492:
293:
3828:Rice, Peter and Dutton, Hugh (1995).
3736:. Waveland Press, Inc. pp. 6–13.
3686:. Butterworth-Heinemann. p. 41.
3499:{\displaystyle \lambda =\lambda _{d}}
3466:{\displaystyle \lambda =\lambda _{Y}}
1804:is the strain-hardening exponent and
1569:This analysis suggests nature of the
1387:deformation. At low strains (such as
906:compounding development of elongation
414:{\displaystyle \sigma =E\varepsilon }
3834:. Taylor & Francis. p. 33.
3745:
3743:
3679:
3421:{\displaystyle \sigma _{T}-\lambda }
58:adding citations to reliable sources
29:
921:True stress & strain equations
717:Eng. stress & strain equations
485:Difference in true and engineering
24:
3805:
2316:Then, we can express the value as
1771:
1750:
1701:
1698:
1695:
1692:
1651:
1633:
1630:
1627:
1624:
1600:
1544:
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1302:
1249:
1103:
989:
946:
821:
334:small displacement-gradient theory
25:
3892:
3740:
3563:
3752:Mechanical Behavior of Materials
3734:Mechanical behavior of materials
3700:from the original on 2017-12-22.
3235:{\displaystyle \lambda =L/L_{0}}
2509:{\displaystyle \varepsilon _{E}}
165:
34:
3284:
877:engineering stress–strain curve
45:needs additional citations for
3821:
3799:
3768:
3725:
3704:
3673:
3580:Artificial cranial deformation
3569:deformation ranges, to break.
3140:
3117:
3108:
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2906:
2883:
2744:
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2446:
2427:
2160:Additionally, the true strain
2140:
2121:
1778:
1762:
1363:
1351:
1214:
1202:
441:is a material constant called
69:"Deformation" engineering
13:
1:
3710:Callister, William D. (2004)
3666:
1823:
1808:is the strength coefficient.
1077:{\displaystyle A_{0}L_{0}=AL}
903:the shrinking of section area
693:Engineering stress and strain
3553:{\displaystyle \lambda _{d}}
3526:{\displaystyle \lambda _{Y}}
3395:. By analyzing the shape of
592:, etc. leads to shortening.
7:
3635:Planar deformation features
3572:
2482:{\displaystyle \sigma _{T}}
2167:can be expressed as below:
1418:{\displaystyle \delta F=0.}
697:Consider a bar of original
649:
318:infinitesimal strain theory
10:
3897:
3605:Deformation mechanism maps
3388:{\displaystyle \lambda =1}
3362:{\displaystyle \lambda =0}
2786:is stress coefficient and
1820:ranging from 0.02 to 0.5.
669:Types of stress and strain
656:Concrete fracture analysis
653:
578:
512:
383:Linear elastic deformation
297:
3750:Courtney, Thomas (2000).
3732:Courtney, Thomas (2005).
1571:ultimate tensile strength
339:For some materials, e.g.
330:small displacement theory
268:deformation mechanism map
3776:"True Stress and Strain"
3595:Deflection (engineering)
3336:{\displaystyle \lambda }
1396:formation can be set as
881:true stress–strain curve
326:small deformation theory
3881:Deformation (mechanics)
3600:Deformation (mechanics)
245:and for a structure by
229:and the deformation is
3808:"STRESS-STRAIN CURVES"
3610:Deformation monitoring
3554:
3527:
3500:
3467:
3436:
3422:
3389:
3363:
3337:
3314:
3236:
3181:
3153:
3051:
3031:
3011:
2991:
2955:
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2800:
2780:
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2598:
2510:
2483:
2453:
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1892:
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1721:
1560:
1419:
1370:
1277:
1221:
1078:
1020:
969:
896:
887:True stress and strain
844:
763:
575:
510:
489:
415:
290:
137:
3655:Strength of materials
3555:
3528:
3501:
3468:
3434:
3423:
3390:
3364:
3338:
3315:
3237:
3182:
3154:
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3012:
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2956:
2931:
2920:
2841:
2821:
2801:
2781:
2758:
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2599:
2511:
2484:
2454:
2308:
2152:
1893:
1792:
1722:
1561:
1420:
1371:
1278:
1222:
1079:
1021:
970:
894:
863:per square metre, or
845:
764:
569:
500:
484:
416:
298:Further information:
288:
243:strength of materials
181:rigid transformations
135:
3680:Rees, David (2006).
3640:Plasticity (physics)
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1042:
980:
937:
774:
733:
699:cross sectional area
597:compressive strength
515:Plasticity (physics)
487:stress-strain curves
396:
300:Elasticity (physics)
261:Types of deformation
54:improve this article
3615:Deformation retract
3590:Creep (deformation)
3180:{\displaystyle K''}
2354:
922:
718:
603:for materials with
584:Compressive failure
572:stress–strain curve
561:atomic dislocations
540:does, too, but not
493:Plastic deformation
322:small strain theory
294:Elastic deformation
247:structural analysis
239:plastic deformation
218:stress-strain curve
18:Elastic deformation
3620:Deformation theory
3550:
3523:
3496:
3463:
3437:
3418:
3385:
3359:
3333:
3310:
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3177:
3149:
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3027:
3007:
2987:
2954:{\displaystyle K'}
2951:
2934:
2915:
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2816:
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2776:
2753:
2681:
2594:
2506:
2479:
2449:
2326:
2303:
2280:
2147:
2145:
1888:
1787:
1732:empirical equation
1717:
1715:
1575:work strengthening
1556:
1554:
1415:
1366:
1273:
1217:
1074:
1016:
965:
920:
897:
840:
759:
716:
678:engineering strain
674:Engineering stress
660:Fracture mechanics
640:structural failure
576:
511:
490:
411:
350:logarithmic strain
306:engineering strain
291:
138:
3136:
3050:{\displaystyle m}
3030:{\displaystyle m}
3010:{\displaystyle m}
2984:
2902:
2839:{\displaystyle n}
2819:{\displaystyle n}
2799:{\displaystyle n}
2779:{\displaystyle K}
2679:
2644:
2516:as right figure.
2412:
2368:
2301:
2271:
2260:
2235:
2210:
2099:
2041:
1994:
1967:
1943:
1707:
1639:
1573:(UTS) point. The
1550:
1514:
1482:
1455:
1336:
1271:
1194:
1177:
1157:
1140:
1120:
1029:
1028:
1014:
963:
853:
852:
838:
813:
757:
455:is the resulting
212:mechanical stress
206:deformation, the
130:
129:
122:
104:
16:(Redirected from
3888:
3860:
3859:
3853:
3845:
3831:Structural glass
3825:
3819:
3818:
3812:
3803:
3797:
3796:
3794:
3793:
3787:
3781:. Archived from
3780:
3772:
3766:
3765:
3747:
3738:
3737:
3729:
3723:
3708:
3702:
3701:
3677:
3559:
3557:
3556:
3551:
3549:
3548:
3532:
3530:
3529:
3524:
3522:
3521:
3505:
3503:
3502:
3497:
3495:
3494:
3472:
3470:
3469:
3464:
3462:
3461:
3427:
3425:
3424:
3419:
3411:
3410:
3394:
3392:
3391:
3386:
3368:
3366:
3365:
3360:
3342:
3340:
3339:
3334:
3319:
3317:
3316:
3311:
3294:
3293:
3274:
3273:
3261:
3260:
3241:
3239:
3238:
3233:
3231:
3230:
3221:
3186:
3184:
3183:
3178:
3176:
3158:
3156:
3155:
3150:
3148:
3147:
3138:
3137:
3132:
3131:
3122:
3116:
3115:
3106:
3105:
3090:
3079:
3078:
3056:
3054:
3053:
3048:
3036:
3034:
3033:
3028:
3016:
3014:
3013:
3008:
2996:
2994:
2993:
2988:
2986:
2985:
2980:
2979:
2970:
2960:
2958:
2957:
2952:
2950:
2924:
2922:
2921:
2916:
2914:
2913:
2904:
2903:
2898:
2897:
2888:
2879:
2868:
2867:
2845:
2843:
2842:
2837:
2825:
2823:
2822:
2817:
2805:
2803:
2802:
2797:
2785:
2783:
2782:
2777:
2762:
2760:
2759:
2754:
2752:
2751:
2742:
2741:
2720:
2719:
2690:
2688:
2687:
2682:
2680:
2678:
2677:
2668:
2667:
2666:
2653:
2645:
2643:
2642:
2633:
2632:
2631:
2618:
2603:
2601:
2600:
2595:
2593:
2592:
2580:
2579:
2567:
2566:
2554:
2553:
2515:
2513:
2512:
2507:
2505:
2504:
2488:
2486:
2485:
2480:
2478:
2477:
2458:
2456:
2455:
2450:
2445:
2444:
2417:
2413:
2411:
2410:
2401:
2400:
2391:
2369:
2364:
2356:
2353:
2352:
2351:
2341:
2340:
2339:
2312:
2310:
2309:
2304:
2302:
2300:
2299:
2290:
2282:
2279:
2261:
2259:
2258:
2249:
2241:
2236:
2234:
2233:
2224:
2216:
2211:
2209:
2208:
2199:
2191:
2186:
2185:
2166:
2156:
2154:
2153:
2148:
2146:
2139:
2138:
2120:
2119:
2104:
2100:
2098:
2097:
2088:
2078:
2077:
2067:
2062:
2061:
2046:
2042:
2040:
2039:
2030:
2029:
2020:
2015:
2014:
1999:
1995:
1993:
1992:
1983:
1982:
1973:
1968:
1966:
1965:
1953:
1944:
1942:
1941:
1929:
1924:
1923:
1897:
1895:
1894:
1889:
1887:
1886:
1874:
1873:
1861:
1860:
1848:
1847:
1819:
1815:
1811:
1807:
1803:
1796:
1794:
1793:
1788:
1786:
1785:
1776:
1775:
1774:
1755:
1754:
1753:
1726:
1724:
1723:
1718:
1716:
1712:
1708:
1706:
1705:
1704:
1685:
1684:
1675:
1656:
1655:
1654:
1640:
1638:
1637:
1636:
1614:
1605:
1604:
1603:
1565:
1563:
1562:
1557:
1555:
1551:
1549:
1548:
1547:
1537:
1536:
1535:
1534:
1520:
1515:
1510:
1502:
1494:
1484:
1483:
1480:
1457:
1456:
1453:
1437:
1424:
1422:
1421:
1416:
1375:
1373:
1372:
1367:
1341:
1337:
1335:
1334:
1322:
1307:
1306:
1305:
1282:
1280:
1279:
1274:
1272:
1267:
1259:
1254:
1253:
1252:
1230:For the strain,
1226:
1224:
1223:
1218:
1195:
1193:
1192:
1180:
1178:
1176:
1175:
1163:
1158:
1153:
1152:
1143:
1141:
1139:
1138:
1126:
1121:
1113:
1108:
1107:
1106:
1083:
1081:
1080:
1075:
1064:
1063:
1054:
1053:
1025:
1023:
1022:
1017:
1015:
1010:
1002:
994:
993:
992:
974:
972:
971:
966:
964:
956:
951:
950:
949:
923:
919:
849:
847:
846:
841:
839:
837:
836:
827:
819:
814:
812:
811:
802:
801:
800:
784:
768:
766:
765:
760:
758:
756:
755:
743:
719:
715:
712:
708:
553:strain hardening
466:
454:
440:
430:
420:
418:
417:
412:
389:, which states:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
3896:
3895:
3891:
3890:
3889:
3887:
3886:
3885:
3876:Solid mechanics
3866:
3865:
3864:
3863:
3847:
3846:
3842:
3826:
3822:
3810:
3806:Roland, David.
3804:
3800:
3791:
3789:
3785:
3778:
3774:
3773:
3769:
3762:
3748:
3741:
3730:
3726:
3709:
3705:
3694:
3678:
3674:
3669:
3664:
3645:Poisson's ratio
3575:
3566:
3544:
3540:
3538:
3535:
3534:
3517:
3513:
3511:
3508:
3507:
3490:
3486:
3478:
3475:
3474:
3457:
3453:
3445:
3442:
3441:
3406:
3402:
3400:
3397:
3396:
3374:
3371:
3370:
3348:
3345:
3344:
3328:
3325:
3324:
3289:
3285:
3269:
3265:
3256:
3252:
3250:
3247:
3246:
3226:
3222:
3217:
3206:
3203:
3202:
3169:
3167:
3164:
3163:
3143:
3139:
3127:
3123:
3121:
3120:
3111:
3107:
3101:
3097:
3083:
3074:
3070:
3068:
3065:
3064:
3042:
3039:
3038:
3022:
3019:
3018:
3002:
2999:
2998:
2975:
2971:
2969:
2968:
2966:
2963:
2962:
2943:
2941:
2938:
2937:
2909:
2905:
2893:
2889:
2887:
2886:
2872:
2863:
2859:
2857:
2854:
2853:
2831:
2828:
2827:
2811:
2808:
2807:
2791:
2788:
2787:
2771:
2768:
2767:
2747:
2743:
2737:
2733:
2715:
2711:
2709:
2706:
2705:
2673:
2669:
2662:
2658:
2654:
2652:
2638:
2634:
2627:
2623:
2619:
2617:
2615:
2612:
2611:
2588:
2584:
2575:
2571:
2562:
2558:
2549:
2545:
2528:
2525:
2524:
2500:
2496:
2494:
2491:
2490:
2473:
2469:
2467:
2464:
2463:
2440:
2436:
2406:
2402:
2396:
2392:
2390:
2386:
2357:
2355:
2347:
2343:
2342:
2335:
2331:
2330:
2324:
2321:
2320:
2295:
2291:
2283:
2281:
2275:
2254:
2250:
2242:
2240:
2229:
2225:
2217:
2215:
2204:
2200:
2192:
2190:
2181:
2177:
2175:
2172:
2171:
2165:
2161:
2144:
2143:
2134:
2130:
2115:
2111:
2102:
2101:
2093:
2089:
2073:
2069:
2068:
2066:
2057:
2053:
2044:
2043:
2035:
2031:
2025:
2021:
2019:
2010:
2006:
1997:
1996:
1988:
1984:
1978:
1974:
1972:
1961:
1957:
1952:
1945:
1937:
1933:
1928:
1919:
1915:
1911:
1909:
1906:
1905:
1882:
1878:
1869:
1865:
1856:
1852:
1843:
1839:
1837:
1834:
1833:
1826:
1817:
1813:
1809:
1805:
1801:
1781:
1777:
1770:
1769:
1765:
1749:
1748:
1744:
1742:
1739:
1738:
1714:
1713:
1691:
1690:
1686:
1680:
1676:
1674:
1670:
1657:
1650:
1649:
1645:
1642:
1641:
1623:
1622:
1618:
1613:
1606:
1599:
1598:
1594:
1590:
1588:
1585:
1584:
1553:
1552:
1543:
1542:
1538:
1530:
1529:
1525:
1521:
1519:
1503:
1501:
1492:
1491:
1479:
1475:
1452:
1448:
1434:
1432:
1429:
1428:
1401:
1398:
1397:
1330:
1326:
1321:
1317:
1301:
1300:
1296:
1294:
1291:
1290:
1260:
1258:
1248:
1247:
1243:
1238:
1235:
1234:
1188:
1184:
1179:
1171:
1167:
1162:
1148:
1144:
1142:
1134:
1130:
1125:
1112:
1102:
1101:
1097:
1095:
1092:
1091:
1059:
1055:
1049:
1045:
1043:
1040:
1039:
1003:
1001:
988:
987:
983:
981:
978:
977:
955:
945:
944:
940:
938:
935:
934:
889:
857:SI derived unit
832:
828:
820:
818:
807:
803:
796:
792:
785:
783:
775:
772:
771:
751:
747:
742:
734:
731:
730:
710:
707:
701:
695:
671:
662:
652:
607:behavior (most
586:
581:
517:
495:
464:
452:
447:elastic modulus
443:Young's modulus
438:
431:is the applied
428:
397:
394:
393:
385:is governed by
368:metals such as
302:
296:
263:
235:Young's modulus
168:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3894:
3884:
3883:
3878:
3862:
3861:
3840:
3820:
3798:
3767:
3760:
3739:
3724:
3703:
3692:
3671:
3670:
3668:
3665:
3663:
3662:
3657:
3652:
3647:
3642:
3637:
3632:
3627:
3622:
3617:
3612:
3607:
3602:
3597:
3592:
3587:
3582:
3576:
3574:
3571:
3565:
3564:Misconceptions
3562:
3547:
3543:
3520:
3516:
3493:
3489:
3485:
3482:
3460:
3456:
3452:
3449:
3417:
3414:
3409:
3405:
3384:
3381:
3378:
3358:
3355:
3352:
3332:
3321:
3320:
3309:
3306:
3303:
3300:
3297:
3292:
3288:
3283:
3280:
3277:
3272:
3268:
3264:
3259:
3255:
3229:
3225:
3220:
3216:
3213:
3210:
3175:
3172:
3160:
3159:
3146:
3142:
3135:
3130:
3126:
3119:
3114:
3110:
3104:
3100:
3096:
3093:
3089:
3086:
3082:
3077:
3073:
3046:
3026:
3006:
2983:
2978:
2974:
2949:
2946:
2926:
2925:
2912:
2908:
2901:
2896:
2892:
2885:
2882:
2878:
2875:
2871:
2866:
2862:
2835:
2815:
2795:
2775:
2764:
2763:
2750:
2746:
2740:
2736:
2732:
2729:
2726:
2723:
2718:
2714:
2692:
2691:
2676:
2672:
2665:
2661:
2657:
2651:
2648:
2641:
2637:
2630:
2626:
2622:
2605:
2604:
2591:
2587:
2583:
2578:
2574:
2570:
2565:
2561:
2557:
2552:
2548:
2544:
2541:
2538:
2535:
2532:
2503:
2499:
2476:
2472:
2460:
2459:
2448:
2443:
2439:
2435:
2432:
2429:
2426:
2423:
2420:
2416:
2409:
2405:
2399:
2395:
2389:
2385:
2382:
2379:
2376:
2373:
2367:
2363:
2360:
2350:
2346:
2338:
2334:
2329:
2314:
2313:
2298:
2294:
2289:
2286:
2278:
2274:
2270:
2267:
2264:
2257:
2253:
2248:
2245:
2239:
2232:
2228:
2223:
2220:
2214:
2207:
2203:
2198:
2195:
2189:
2184:
2180:
2163:
2158:
2157:
2142:
2137:
2133:
2129:
2126:
2123:
2118:
2114:
2110:
2107:
2105:
2103:
2096:
2092:
2087:
2084:
2081:
2076:
2072:
2065:
2060:
2056:
2052:
2049:
2047:
2045:
2038:
2034:
2028:
2024:
2018:
2013:
2009:
2005:
2002:
2000:
1998:
1991:
1987:
1981:
1977:
1971:
1964:
1960:
1956:
1951:
1948:
1946:
1940:
1936:
1932:
1927:
1922:
1918:
1914:
1913:
1899:
1898:
1885:
1881:
1877:
1872:
1868:
1864:
1859:
1855:
1851:
1846:
1842:
1825:
1822:
1798:
1797:
1784:
1780:
1773:
1768:
1764:
1761:
1758:
1752:
1747:
1728:
1727:
1711:
1703:
1700:
1697:
1694:
1689:
1683:
1679:
1673:
1669:
1666:
1663:
1660:
1658:
1653:
1648:
1644:
1643:
1635:
1632:
1629:
1626:
1621:
1617:
1612:
1609:
1607:
1602:
1597:
1593:
1592:
1567:
1566:
1546:
1541:
1533:
1528:
1524:
1518:
1513:
1509:
1506:
1500:
1497:
1495:
1493:
1490:
1487:
1478:
1474:
1470:
1467:
1464:
1461:
1451:
1447:
1444:
1441:
1438:
1436:
1414:
1411:
1408:
1405:
1392:criterion for
1377:
1376:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1340:
1333:
1329:
1325:
1320:
1316:
1313:
1310:
1304:
1299:
1284:
1283:
1270:
1266:
1263:
1257:
1251:
1246:
1242:
1228:
1227:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1191:
1187:
1183:
1174:
1170:
1166:
1161:
1156:
1151:
1147:
1137:
1133:
1129:
1124:
1119:
1116:
1111:
1105:
1100:
1085:
1084:
1073:
1070:
1067:
1062:
1058:
1052:
1048:
1033:
1032:
1027:
1026:
1013:
1009:
1006:
1000:
997:
991:
986:
975:
962:
959:
954:
948:
943:
931:
930:
927:
908:
907:
904:
888:
885:
859:for stress is
851:
850:
835:
831:
826:
823:
817:
810:
806:
799:
795:
791:
788:
782:
779:
769:
754:
750:
746:
741:
738:
727:
726:
723:
705:
694:
691:
670:
667:
651:
648:
585:
582:
580:
577:
522:thermoplastics
494:
491:
476:gray cast iron
461:
460:
450:
436:
422:
421:
410:
407:
404:
401:
358:Almansi strain
320:, also called
295:
292:
262:
259:
223:
222:
192:
184:
167:
164:
128:
127:
110:September 2008
42:
40:
33:
26:
9:
6:
4:
3:
2:
3893:
3882:
3879:
3877:
3874:
3873:
3871:
3857:
3851:
3843:
3841:0-419-19940-3
3837:
3833:
3832:
3824:
3816:
3809:
3802:
3788:on 2018-01-27
3784:
3777:
3771:
3763:
3761:9780073228242
3757:
3753:
3746:
3744:
3735:
3728:
3721:
3720:0-471-66081-7
3717:
3713:
3707:
3699:
3695:
3693:0-7506-8025-3
3689:
3685:
3684:
3676:
3672:
3661:
3658:
3656:
3653:
3651:
3650:Strain tensor
3648:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3585:Buff strength
3583:
3581:
3578:
3577:
3570:
3561:
3545:
3541:
3518:
3514:
3491:
3487:
3483:
3480:
3458:
3454:
3450:
3447:
3433:
3429:
3415:
3412:
3407:
3403:
3382:
3379:
3376:
3356:
3353:
3350:
3330:
3307:
3304:
3301:
3298:
3295:
3290:
3286:
3281:
3278:
3275:
3270:
3266:
3262:
3257:
3253:
3245:
3244:
3243:
3227:
3223:
3218:
3214:
3211:
3208:
3199:
3196:
3192:
3188:
3173:
3170:
3144:
3133:
3128:
3124:
3112:
3102:
3098:
3091:
3087:
3084:
3080:
3075:
3071:
3063:
3062:
3061:
3058:
3044:
3024:
3004:
2981:
2976:
2972:
2947:
2944:
2930:
2910:
2899:
2894:
2890:
2880:
2876:
2873:
2869:
2864:
2860:
2852:
2851:
2850:
2847:
2833:
2813:
2793:
2773:
2748:
2738:
2734:
2727:
2724:
2721:
2716:
2712:
2704:
2703:
2702:
2699:
2696:
2674:
2670:
2663:
2659:
2655:
2649:
2646:
2639:
2635:
2628:
2624:
2620:
2610:
2609:
2608:
2589:
2585:
2581:
2576:
2572:
2568:
2563:
2559:
2555:
2550:
2546:
2542:
2539:
2536:
2533:
2530:
2523:
2522:
2521:
2517:
2501:
2497:
2474:
2470:
2441:
2437:
2433:
2430:
2424:
2421:
2418:
2414:
2407:
2403:
2397:
2393:
2387:
2383:
2380:
2377:
2374:
2371:
2365:
2361:
2358:
2348:
2344:
2336:
2332:
2327:
2319:
2318:
2317:
2296:
2292:
2287:
2284:
2276:
2272:
2268:
2265:
2262:
2255:
2251:
2246:
2243:
2237:
2230:
2226:
2221:
2218:
2212:
2205:
2201:
2196:
2193:
2187:
2182:
2178:
2170:
2169:
2168:
2135:
2131:
2127:
2124:
2116:
2112:
2108:
2106:
2094:
2090:
2085:
2082:
2079:
2074:
2070:
2063:
2058:
2054:
2050:
2048:
2036:
2032:
2026:
2022:
2016:
2011:
2007:
2003:
2001:
1989:
1985:
1979:
1975:
1969:
1962:
1958:
1954:
1949:
1947:
1938:
1934:
1930:
1925:
1920:
1916:
1904:
1903:
1902:
1883:
1879:
1875:
1870:
1866:
1862:
1857:
1853:
1849:
1844:
1840:
1832:
1831:
1830:
1821:
1782:
1766:
1759:
1756:
1745:
1737:
1736:
1735:
1733:
1709:
1687:
1681:
1677:
1671:
1667:
1664:
1661:
1659:
1646:
1619:
1615:
1610:
1608:
1595:
1583:
1582:
1581:
1578:
1576:
1572:
1539:
1526:
1522:
1516:
1511:
1507:
1504:
1498:
1496:
1488:
1485:
1476:
1472:
1468:
1465:
1462:
1459:
1449:
1445:
1442:
1439:
1427:
1426:
1425:
1412:
1409:
1406:
1403:
1395:
1390:
1386:
1382:
1360:
1357:
1354:
1348:
1345:
1342:
1338:
1331:
1327:
1323:
1318:
1314:
1311:
1308:
1297:
1289:
1288:
1287:
1268:
1264:
1261:
1255:
1244:
1240:
1233:
1232:
1231:
1211:
1208:
1205:
1199:
1196:
1189:
1185:
1181:
1172:
1168:
1164:
1159:
1154:
1149:
1145:
1135:
1131:
1127:
1122:
1117:
1114:
1109:
1098:
1090:
1089:
1088:
1071:
1068:
1065:
1060:
1056:
1050:
1046:
1038:
1037:
1036:
1031:
1030:
1011:
1007:
1004:
998:
995:
984:
976:
960:
957:
952:
941:
933:
932:
928:
925:
924:
918:
916:
912:
905:
902:
901:
900:
893:
884:
882:
878:
874:
870:
866:
862:
858:
833:
829:
824:
815:
808:
804:
797:
793:
789:
786:
780:
777:
770:
752:
748:
744:
739:
736:
729:
728:
724:
721:
720:
714:
704:
700:
690:
688:
684:
679:
675:
666:
661:
657:
647:
645:
641:
637:
633:
628:
626:
622:
618:
614:
610:
606:
602:
598:
593:
591:
573:
570:Diagram of a
568:
564:
562:
558:
555:region and a
554:
549:
547:
543:
539:
535:
531:
527:
523:
516:
508:
504:
499:
488:
483:
479:
477:
473:
468:
458:
451:
448:
444:
437:
434:
427:
426:
425:
408:
405:
402:
399:
392:
391:
390:
388:
384:
380:
377:
375:
371:
367:
363:
359:
355:
351:
347:
342:
337:
335:
331:
327:
323:
319:
315:
311:
307:
301:
287:
283:
282:deformation.
281:
277:
271:
269:
258:
255:
250:
248:
244:
240:
236:
232:
228:
220:
219:
213:
209:
208:dimensionless
205:
202:
198:
197:
193:
190:
189:
185:
182:
178:
177:
176:Displacements
173:
172:
171:
166:Main concepts
163:
161:
160:
155:
151:
147:
143:
134:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
3830:
3823:
3814:
3801:
3790:. Retrieved
3783:the original
3770:
3751:
3733:
3727:
3711:
3706:
3682:
3675:
3660:Wood warping
3630:Malleability
3567:
3438:
3343:value where
3322:
3200:
3197:
3193:
3189:
3161:
3059:
2935:
2848:
2765:
2700:
2697:
2693:
2606:
2518:
2461:
2315:
2159:
1900:
1827:
1799:
1729:
1579:
1568:
1381:tension test
1378:
1285:
1229:
1086:
1034:
914:
910:
909:
898:
880:
876:
854:
702:
696:
686:
682:
677:
673:
672:
663:
635:
629:
594:
587:
550:
518:
469:
462:
423:
382:
381:
378:
366:shape memory
357:
354:Green strain
353:
349:
345:
338:
333:
329:
325:
321:
303:
279:
275:
272:
264:
251:
224:
215:
203:
200:
195:
187:
175:
169:
158:
153:
149:
145:
139:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
915:true strain
911:True stress
687:true strain
683:true stress
546:chewing gum
503:projectiles
387:Hooke's law
227:yield point
188:Deformation
146:deformation
142:engineering
3870:Categories
3792:2018-05-15
3667:References
3625:Elasticity
1824:Discussion
654:See also:
513:See also:
507:ballistics
362:Elastomers
341:elastomers
80:newspapers
3850:cite book
3542:λ
3515:λ
3488:λ
3481:λ
3455:λ
3448:λ
3416:λ
3413:−
3404:σ
3377:λ
3351:λ
3331:λ
3305:λ
3302:
3287:ε
3279:λ
3276:×
3267:σ
3254:σ
3209:λ
3134:˙
3125:ε
3099:ε
3092:×
3072:σ
2982:˙
2973:ε
2900:˙
2891:ε
2881:×
2861:σ
2735:ε
2728:×
2713:σ
2650:−
2636:σ
2625:σ
2586:σ
2547:σ
2498:ε
2471:σ
2438:ε
2425:
2384:
2328:∫
2273:∑
2266:⋯
2179:ε
2132:ε
2113:σ
2083:δ
2064:×
2055:σ
2017:×
2008:σ
1970:×
1917:σ
1880:ε
1876:×
1854:ε
1850:×
1767:ε
1746:σ
1668:
1647:ε
1596:σ
1540:σ
1527:σ
1523:δ
1505:δ
1499:−
1477:σ
1473:δ
1460:δ
1450:σ
1440:δ
1404:δ
1361:ε
1349:
1315:
1298:ε
1262:δ
1245:ε
1241:δ
1212:ε
1200:σ
1099:σ
1005:δ
999:∫
985:ε
942:σ
873:fractures
822:Δ
790:−
778:ε
737:σ
638:leads to
627:, etc.).
621:cast iron
542:cast iron
409:ε
400:σ
3698:Archived
3573:See also
3174:″
3088:″
2948:′
2877:′
1379:So in a
869:unitless
650:Fracture
644:buckling
617:plastics
601:yielding
509:testing.
472:concrete
310:concrete
254:cylinder
204:internal
201:relative
1394:necking
1389:elastic
1385:plastic
929:Strain
926:Stress
865:pascals
861:newtons
725:Strain
722:Stress
642:due to
611:, some
605:ductile
590:columns
579:Failure
557:necking
370:Nitinol
346:stretch
280:plastic
276:elastic
231:elastic
199:is the
154:plastic
150:elastic
94:scholar
3838:
3758:
3718:
3690:
3162:Where
2936:Where
2766:Where
1800:Here,
609:metals
532:, and
530:silver
526:copper
457:strain
433:stress
429:σ
424:where
374:rubber
356:, and
196:Strain
96:
89:
82:
75:
67:
3811:(PDF)
3786:(PDF)
3779:(PDF)
632:truss
625:glass
613:soils
538:Steel
332:, or
314:steel
159:rigid
101:JSTOR
87:books
3856:link
3836:ISBN
3756:ISBN
3716:ISBN
3688:ISBN
3533:and
2489:and
913:and
685:and
676:and
658:and
615:and
534:gold
364:and
312:and
216:see
73:news
3815:MIT
3369:to
1730:An
505:in
445:or
152:or
140:In
56:by
3872::
3852:}}
3848:{{
3813:.
3742:^
3696:.
3299:ln
2422:ln
2381:ln
1665:ln
1413:0.
1346:ln
1312:ln
623:,
536:.
528:,
474:,
360:.
352:,
348:,
328:,
324:,
270:.
249:.
214:,
183:).
162:.
144:,
3858:)
3844:.
3817:.
3795:.
3764:.
3722:.
3546:d
3519:Y
3492:d
3484:=
3459:Y
3451:=
3408:T
3383:1
3380:=
3357:0
3354:=
3308:.
3296:=
3291:T
3282:,
3271:e
3263:=
3258:T
3228:0
3224:L
3219:/
3215:L
3212:=
3171:K
3145:m
3141:)
3129:T
3118:(
3113:n
3109:)
3103:T
3095:(
3085:K
3081:=
3076:T
3045:m
3025:m
3005:m
2977:T
2945:K
2911:m
2907:)
2895:T
2884:(
2874:K
2870:=
2865:T
2834:n
2814:n
2794:n
2774:K
2749:n
2745:)
2739:T
2731:(
2725:K
2722:=
2717:T
2675:i
2671:A
2664:i
2660:A
2656:d
2647:=
2640:T
2629:T
2621:d
2590:T
2582:d
2577:i
2573:A
2569:+
2564:i
2560:A
2556:d
2551:T
2543:=
2540:0
2537:=
2534:F
2531:d
2502:E
2475:T
2447:)
2442:E
2434:+
2431:1
2428:(
2419:=
2415:)
2408:0
2404:l
2398:i
2394:l
2388:(
2378:=
2375:x
2372:d
2366:l
2362:l
2359:d
2349:i
2345:l
2337:0
2333:l
2297:i
2293:l
2288:l
2285:d
2277:i
2269:=
2263:+
2256:2
2252:l
2247:l
2244:d
2238:+
2231:1
2227:l
2222:l
2219:d
2213:+
2206:0
2202:l
2197:l
2194:d
2188:=
2183:T
2164:T
2162:ε
2141:)
2136:E
2128:+
2125:1
2122:(
2117:E
2109:=
2095:i
2091:l
2086:l
2080:+
2075:i
2071:l
2059:E
2051:=
2037:i
2033:l
2027:f
2023:l
2012:e
2004:=
1990:f
1986:A
1980:i
1976:A
1963:i
1959:A
1955:F
1950:=
1939:f
1935:A
1931:F
1926:=
1921:T
1884:f
1871:f
1867:A
1863:=
1858:i
1845:i
1841:A
1818:n
1814:n
1810:n
1806:K
1802:n
1783:n
1779:)
1772:t
1763:(
1760:K
1757:=
1751:t
1710:)
1702:k
1699:c
1696:e
1693:n
1688:A
1682:0
1678:A
1672:(
1662:=
1652:t
1634:k
1631:c
1628:e
1625:n
1620:A
1616:F
1611:=
1601:t
1545:t
1532:t
1517:=
1512:A
1508:A
1489:0
1486:=
1481:t
1469:A
1466:+
1463:A
1454:t
1446:=
1443:F
1410:=
1407:F
1364:)
1358:+
1355:1
1352:(
1343:=
1339:)
1332:0
1328:L
1324:L
1319:(
1309:=
1303:t
1269:L
1265:L
1256:=
1250:t
1215:)
1209:+
1206:1
1203:(
1197:=
1190:0
1186:L
1182:L
1173:0
1169:A
1165:F
1160:=
1155:A
1150:0
1146:A
1136:0
1132:A
1128:F
1123:=
1118:A
1115:F
1110:=
1104:t
1072:L
1069:A
1066:=
1061:0
1057:L
1051:0
1047:A
1012:L
1008:L
996:=
990:t
961:A
958:F
953:=
947:t
834:0
830:L
825:L
816:=
809:0
805:L
798:0
794:L
787:L
781:=
753:0
749:A
745:F
740:=
711:F
706:0
703:A
636:F
465:E
459:.
453:ε
449:;
439:E
435:;
406:E
403:=
221:.
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
