2586:
22:
1602:
541:
2298:
1195:
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1384:
2593:
The standard linear solid model combines aspects of the
Maxwell and KelvināVoigt models to accurately describe the overall behavior of a system under a given set of loading conditions. The behavior of a material applied to an instantaneous stress is shown as having an instantaneous component of the
2594:
response. Instantaneous release of a stress also results in a discontinuous decrease in strain, as is expected. The shape of the time-dependent strain curve is true to the type of equation that characterizes the behavior of the model over time, depending upon how the model is loaded.
1521:
2068:
1007:
2309:
142:
are used. These models often prove insufficient, however; the
Maxwell model does not describe creep or recovery, and the KelvināVoigt model does not describe stress relaxation. SLS is the simplest model that predicts both phenomena.
1206:
2597:
Although this model can be used to accurately predict the general shape of the strain curve, as well as behavior for long time and instantaneous loads, the model lacks the ability to accurately model material systems numerically.
170:. In contrast to the Maxwell and KelvināVoigt models, the SLS is slightly more complex, involving elements both in series and in parallel. Springs, which represent the elastic component of a viscoelastic material, obey
1834:
773:
1960:
899:
530:
409:
1395:
273:
1765:
704:
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471:
350:
2293:{\displaystyle \sigma (t)+{\frac {\eta }{E_{1}+E_{2}}}{\frac {d\sigma (t)}{dt}}={\frac {E_{1}E_{2}}{E_{1}+E_{2}}}\varepsilon (t)+{\frac {E_{1}\eta }{E_{1}+E_{2}}}{\frac {d\varepsilon (t)}{dt}}}
1591:
1190:{\displaystyle {\frac {d\varepsilon (t)}{dt}}={\frac {{\frac {E_{2}}{\eta }}\left({\frac {\eta }{E_{2}}}{\frac {d\sigma (t)}{dt}}+\sigma (t)-E_{1}\varepsilon (t)\right)}{E_{1}+E_{2}}}}
211:
2523:
2482:{\displaystyle \sigma +{\frac {\eta }{E_{1}+E_{2}}}{\dot {\sigma }}={\frac {E_{1}E_{2}}{E_{1}+E_{2}}}\varepsilon +{\frac {E_{1}\eta }{E_{1}+E_{2}}}{\dot {\varepsilon }}}
2062:
Using these relationships, their time derivatives, and the above stress-strain relationships for the spring and dashpot elements, the system can be modeled as follows:
1001:
Using these relationships, their time derivatives, and the above stress-strain relationships for the spring and dashpot elements, the system can be modeled as follows:
1693:
1640:
632:
579:
2057:
2030:
1660:
1548:
996:
969:
599:
216:
where Ļ is the applied stress, E is the Young's modulus of the material, and Īµ is the strain. The spring represents the elastic component of the model's response.
2679:
1379:{\displaystyle \sigma (t)+{\frac {\eta }{E_{2}}}{\frac {d\sigma (t)}{dt}}=E_{1}\varepsilon (t)+{\frac {\eta (E_{1}+E_{2})}{E_{2}}}{\frac {d\varepsilon (t)}{dt}}}
2003:
1983:
942:
922:
219:
Dashpots represent the viscous component of a viscoelastic material. In these elements, the applied stress varies with the time rate of change of the strain:
2601:
The fluid model equivalent to the standard linear solid model includes a dashpot in series with the KelvināVoigt model and is called the
Jeffreys model.
1773:
712:
1905:
844:
134:
material using a linear combination of springs and dashpots to represent elastic and viscous components, respectively. Often, the simpler
39:
476:
355:
86:
1516:{\displaystyle \sigma +{\frac {\eta }{E_{2}}}{\dot {\sigma }}=E_{1}\varepsilon +{\frac {\eta (E_{1}+E_{2})}{E_{2}}}{\dot {\varepsilon }}}
58:
2651:
65:
225:
2704:
2680:
http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/MIT3_11F99_visco.pdf
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643:
1842:
781:
72:
2531:
417:
296:
54:
2714:
105:
1556:
180:
43:
548:
This model consists of two systems in parallel. The first, referred to as the
Maxwell arm, contains a spring (
2741:
637:
These relationships help relate the various stresses and strains in the overall system and the
Maxwell arm:
1698:
These relationships help relate the various stresses and strains in the overall system and the Kelvin arm:
1609:
This model consists of two systems in series. The first, referred to as the Kelvin arm, contains a spring (
79:
2736:
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167:
139:
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32:
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2008:
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8:
152:
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1988:
1968:
927:
907:
2710:
2625:
2610:
2493:
163:
135:
290:
In order to model this system, the following physical relations must be realized:
151:
Materials undergoing strain are often modeled with mechanical components, such as
2630:
1527:
131:
171:
127:
2730:
2589:
Comparison of creep and stress relaxation for three and four element models
2653:
Acoustic wave equations and four ways media may perturb the speed of sound
1829:{\displaystyle \varepsilon _{tot}=\varepsilon _{k}+\varepsilon _{S_{1}}}
768:{\displaystyle \varepsilon _{tot}=\varepsilon _{m}=\varepsilon _{S_{1}}}
1955:{\displaystyle \varepsilon _{k}=\varepsilon _{D}=\varepsilon _{S_{2}}}
894:{\displaystyle \varepsilon _{m}=\varepsilon _{D}+\varepsilon _{S_{2}}}
279:
166:
while connecting a spring and damper in parallel yields a model of a
2059:
refer to Kelvin, dashpot, spring one, and spring two, respectively.
998:
refer to
Maxwell, dashpot, spring one and spring two, respectively.
525:{\displaystyle \varepsilon _{tot}=\varepsilon _{1}+\varepsilon _{2}}
404:{\displaystyle \varepsilon _{tot}=\varepsilon _{1}=\varepsilon _{2}}
21:
156:
2690:
Krystyn J. Van Vliet, MIT course 3.032 Lecture, October 23, 2006
2678:
David
Roylance, "Engineering Viscoelasticity" (October 24, 2001)
162:
Connecting a spring and damper in series yields a model of a
268:{\displaystyle \sigma _{D}=\eta {\frac {d\varepsilon }{dt}}}
1760:{\displaystyle \sigma _{tot}=\sigma _{k}=\sigma _{S_{1}}}
699:{\displaystyle \sigma _{tot}=\sigma _{m}+\sigma _{S_{1}}}
1662:) in parallel. The other system contains only a spring (
2692:
http://stellar.mit.edu/S/course/3/fa06/3.032/index.html
1892:{\displaystyle \sigma _{k}=\sigma _{D}+\sigma _{S_{2}}}
1601:
831:{\displaystyle \sigma _{m}=\sigma _{D}=\sigma _{S_{2}}}
601:) in series. The other system contains only a spring (
540:
2570:{\displaystyle {\bar {\tau }}={\frac {\eta }{E_{2}}}}
2534:
2502:
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2011:
1991:
1971:
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847:
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715:
646:
607:
587:
554:
479:
466:{\displaystyle \sigma _{tot}=\sigma _{1}=\sigma _{2}}
420:
358:
345:{\displaystyle \sigma _{tot}=\sigma _{1}+\sigma _{2}}
299:
228:
183:
544:
Standard Linear Solid model, Maxwell representation
46:. Unsourced material may be challenged and removed.
2569:
2517:
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2024:
1997:
1977:
1954:
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1605:Standard Linear Solid model, Kelvin representation
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2525:, is different for each material and is equal to
1550:, is different for each material and is equal to
2728:
1596:
1586:{\displaystyle \tau ={\frac {\eta }{E_{2}}}}
130:, is a method of modeling the behavior of a
2674:
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2668:
2666:
2709:. Springer Science & Business Media.
535:
206:{\displaystyle \sigma _{s}=E\varepsilon }
106:Learn how and when to remove this message
2584:
2580:
1600:
539:
2663:
1200:The equation can also be expressed as:
2729:
2706:Fluid Dynamics of Viscoelastic Liquids
2702:
2659:. Vol. 1.3. University of Oslo.
2649:
285:
44:adding citations to reliable sources
15:
2684:
13:
155:(restorative force component) and
14:
2753:
2703:Joseph, Daniel D. (2013-11-27).
20:
31:needs additional citations for
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2643:
2541:
2518:{\displaystyle {\bar {\tau }}}
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1:
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55:"Standard linear solid model"
7:
2604:
1597:Kelvin-Voigt representation
120:standard linear solid (SLS)
10:
2758:
282:of the dashpot component.
2616:Generalized Maxwell model
1642:) and dashpot (viscosity
581:) and dashpot (viscosity
293:For parallel components:
1688:{\displaystyle E=E_{1}}
1635:{\displaystyle E=E_{2}}
627:{\displaystyle E=E_{1}}
574:{\displaystyle E=E_{2}}
414:For series components:
2590:
2571:
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2483:
2294:
2053:
2026:
1999:
1979:
1956:
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992:
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832:
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595:
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545:
536:Maxwell representation
526:
467:
405:
346:
269:
207:
2650:Holm, Sverre (2024).
2621:KelvināVoigt material
2588:
2581:Model characteristics
2572:
2520:
2484:
2303:or, in dot notation:
2295:
2054:
2052:{\displaystyle S_{2}}
2027:
2025:{\displaystyle S_{1}}
2000:
1980:
1965:where the subscripts
1957:
1894:
1831:
1762:
1690:
1657:
1655:{\displaystyle \eta }
1637:
1604:
1588:
1545:
1543:{\displaystyle \tau }
1518:
1389:or, in dot notation:
1381:
1192:
993:
991:{\displaystyle S_{2}}
966:
964:{\displaystyle S_{1}}
939:
919:
904:where the subscripts
896:
833:
770:
701:
629:
596:
594:{\displaystyle \eta }
576:
543:
527:
468:
406:
347:
270:
208:
168:KelvināVoigt material
159:(damping component).
2742:Non-Newtonian fluids
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2500:
2310:
2069:
2036:
2009:
1989:
1969:
1906:
1843:
1774:
1705:
1666:
1646:
1613:
1557:
1534:
1396:
1207:
1008:
975:
948:
928:
908:
845:
782:
713:
644:
605:
585:
552:
477:
418:
356:
297:
226:
181:
122:, also known as the
40:improve this article
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2567:
2515:
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2290:
2049:
2022:
1995:
1975:
1952:
1889:
1826:
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1607:
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988:
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828:
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140:KelvināVoigt model
2737:Materials science
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1998:{\displaystyle D}
1978:{\displaystyle k}
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1510:
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1038:
937:{\displaystyle D}
917:{\displaystyle m}
286:Solving the model
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108:
90:
2749:
2721:
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2700:
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2626:Maxwell material
2611:Burgers material
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2494:retardation time
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2002:
2001:
1996:
1984:
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1503:
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1498:
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1470:
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1432:
1424:
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1406:
1385:
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1373:
1365:
1348:
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1343:
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1300:
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1270:
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1226:
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1113:
1111:
1103:
1086:
1084:
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1081:
1069:
1062:
1057:
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1047:
1044:
1039:
1037:
1029:
1012:
997:
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994:
989:
987:
986:
970:
968:
967:
962:
960:
959:
943:
941:
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935:
923:
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675:
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580:
578:
577:
572:
570:
569:
531:
529:
528:
523:
521:
520:
508:
507:
495:
494:
472:
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469:
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461:
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410:
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387:
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351:
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328:
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315:
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212:
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164:Maxwell material
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89:
48:
24:
16:
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2717:
2701:
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2644:
2639:
2631:Viscoelasticity
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2503:
2501:
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2138:
2121:
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2110:
2106:
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2092:
2087:
2070:
2067:
2066:
2043:
2039:
2037:
2034:
2033:
2016:
2012:
2010:
2007:
2006:
1990:
1987:
1986:
1970:
1967:
1966:
1944:
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1939:
1935:
1926:
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1913:
1909:
1907:
1904:
1903:
1881:
1877:
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1872:
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1859:
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1706:
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1702:
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1647:
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1643:
1626:
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1614:
1611:
1610:
1599:
1575:
1571:
1566:
1558:
1555:
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1535:
1532:
1531:
1528:relaxation time
1502:
1501:
1493:
1489:
1479:
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1453:
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1423:
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1397:
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1301:
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1234:
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1178:
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1087:
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1005:
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978:
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951:
949:
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945:
929:
926:
925:
909:
906:
905:
883:
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865:
861:
852:
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820:
816:
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811:
802:
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688:
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586:
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149:
112:
101:
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92:
49:
47:
37:
25:
12:
11:
5:
2755:
2745:
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128:Clarence Zener
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136:Maxwell model
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74:
71:
67:
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60:
57: ā
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
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2698:
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132:viscoelastic
123:
119:
117:
102:
93:
83:
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69:
62:
50:
38:Please help
33:verification
30:
278:where Ī· is
172:Hooke's law
124:Zener model
2731:Categories
2637:References
147:Definition
96:April 2017
66:newspapers
2553:η
2542:¯
2539:τ
2510:¯
2507:τ
2474:˙
2471:ε
2437:η
2418:ε
2358:˙
2355:σ
2322:η
2314:σ
2268:ε
2231:η
2203:ε
2126:σ
2090:η
2073:σ
1937:ε
1924:ε
1911:ε
1874:σ
1861:σ
1848:σ
1811:ε
1798:ε
1779:ε
1742:σ
1729:σ
1710:σ
1650:η
1569:η
1561:τ
1538:τ
1508:˙
1505:ε
1457:η
1448:ε
1429:˙
1426:σ
1408:η
1400:σ
1354:ε
1303:η
1285:ε
1249:σ
1228:η
1211:σ
1143:ε
1130:−
1118:σ
1092:σ
1071:η
1059:η
1018:ε
876:ε
863:ε
850:ε
813:σ
800:σ
787:σ
750:ε
737:ε
718:ε
681:σ
668:σ
649:σ
589:η
514:ε
501:ε
482:ε
455:σ
442:σ
423:σ
393:ε
380:ε
361:ε
334:σ
321:σ
302:σ
280:viscosity
252:ε
243:η
231:σ
201:ε
186:σ
2605:See also
157:dashpots
138:and the
153:springs
80:scholar
2713:
473:, and
352:, and
126:after
82:
75:
68:
61:
53:
2657:(PDF)
2032:,and
87:JSTOR
73:books
2711:ISBN
2492:The
1526:The
971:and
118:The
59:news
1695:).
634:).
42:by
2733::
2665:^
2496:,
2005:,
1985:,
1530:,
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532:.
411:.
174::
2719:.
2561:2
2557:E
2548:=
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1973:k
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1933:=
1928:D
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1802:k
1794:=
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1435:=
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1371:t
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1360:t
1357:(
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1272:=
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1098:t
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1089:d
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1065:(
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1035:t
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1027:)
1024:t
1021:(
1015:d
984:2
980:S
957:1
953:S
932:D
912:m
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881:S
872:+
867:D
859:=
854:m
822:2
818:S
809:=
804:D
796:=
791:m
759:1
755:S
746:=
741:m
733:=
728:t
725:o
722:t
690:1
686:S
677:+
672:m
664:=
659:t
656:o
653:t
620:1
616:E
612:=
609:E
567:2
563:E
559:=
556:E
518:2
510:+
505:1
497:=
492:t
489:o
486:t
459:2
451:=
446:1
438:=
433:t
430:o
427:t
397:2
389:=
384:1
376:=
371:t
368:o
365:t
338:2
330:+
325:1
317:=
312:t
309:o
306:t
260:t
257:d
249:d
240:=
235:D
198:E
195:=
190:s
109:)
103:(
98:)
94:(
84:Ā·
77:Ā·
70:Ā·
63:Ā·
36:.
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