50:
2095:
472:
2317:
1655:
697:
1838:
1849:
1011:
309:
2106:
599:
298:
667:
1470:
1260:
1372:
226:
1150:
1672:
1644:
2090:{\displaystyle E_{1}(\omega )={\frac {E\eta ^{2}\omega ^{2}}{\eta ^{2}\omega ^{2}+E^{2}}}={\frac {(\eta /E)^{2}\omega ^{2}}{(\eta /E)^{2}\omega ^{2}+1}}E={\frac {\tau ^{2}\omega ^{2}}{\tau ^{2}\omega ^{2}+1}}E}
683:
The model is usually applied to the case of small deformations. For the large deformations we should include some geometrical non-linearity. For the simplest way of generalizing the
Maxwell model, refer to the
889:
467:{\displaystyle {\frac {d\varepsilon _{\mathrm {Total} }}{dt}}={\frac {d\varepsilon _{\rm {D}}}{dt}}+{\frac {d\varepsilon _{\rm {S}}}{dt}}={\frac {\sigma }{\eta }}+{\frac {1}{E}}{\frac {d\sigma }{dt}}}
821:
1557:
303:
where the subscript D indicates the stressāstrain in the damper and the subscript S indicates the stressāstrain in the spring. Taking the derivative of strain with respect to time, we obtain:
2312:{\displaystyle E_{2}(\omega )={\frac {\omega E^{2}\eta }{\eta ^{2}\omega ^{2}+E^{2}}}={\frac {(\eta /E)\omega }{(\eta /E)^{2}\omega ^{2}+1}}E={\frac {\tau \omega }{\tau ^{2}\omega ^{2}+1}}E}
152:
2484:
111:
2354:
851:
2438:
2397:
1517:
760:
733:
521:
2514:
1185:
1292:
we released the material, then the deformation of the elastic element would be the spring-back deformation and the deformation of the viscous element would not change:
29:
material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. It is named for
1290:
881:
232:
37:
equation lacks motivation from more microscopic models and does not comply with the concept of material objectivity. However, these criteria are fulfilled by the
610:
1380:
1193:
1016:
Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below:
65:
spring connected in series, as shown in the diagram. If, instead, we connect these two elements in parallel, we get the generalized model of a solid
1298:
160:
1482:
If a small stress is applied for a sufficiently long time, then the irreversible strains become large. Thus, Maxwell material is a type of liquid.
1022:
1833:{\displaystyle E^{*}(\omega )={\frac {1}{1/E-i/(\omega \eta )}}={\frac {E\eta ^{2}\omega ^{2}+i\omega E^{2}\eta }{\eta ^{2}\omega ^{2}+E^{2}}}}
1566:
1006:{\displaystyle \varepsilon _{\mathrm {back} }=-{\frac {\sigma (t_{1})}{E}}=\varepsilon _{0}\exp \left(-{\frac {E}{\eta }}t_{1}\right).}
776:
1524:
2722:
116:
2450:
33:
who proposed the model in 1867. It is also known as a
Maxwell fluid. A generalization of the scalar relation to a
2616:
75:
2758:
2556:
685:
38:
1187:, then the elastic element would suddenly deform and the viscous element would deform with a constant rate:
2768:
2738:
2601:
2551:
2325:
826:
594:{\displaystyle {\frac {1}{E}}{\frac {d\sigma }{dt}}+{\frac {\sigma }{\eta }}={\frac {d\varepsilon }{dt}}}
2409:
2368:
1493:
738:
2763:
705:
505:
711:
2541:
2536:
676:
or to the uniform tension in a material. In the former case, the viscosity corresponds to that for a
66:
2496:
1163:
2664:
2714:
2322:
The picture shows relaxational spectrum for
Maxwell material. The relaxation time constant is
680:. In the latter case, it has a slightly different meaning relating stress and rate of strain.
293:{\displaystyle \varepsilon _{\mathrm {Total} }=\varepsilon _{\rm {D}}+\varepsilon _{\rm {S}}}
1268:
859:
62:
662:{\displaystyle {\frac {\dot {\sigma }}{E}}+{\frac {\sigma }{\eta }}={\dot {\varepsilon }}}
8:
2582:
1476:
1465:{\displaystyle \varepsilon _{\mathrm {irreversible} }=t_{1}{\frac {\sigma _{0}}{\eta }}.}
30:
2732:
2707:
2628:
2595:
1255:{\displaystyle \varepsilon (t)={\frac {\sigma _{0}}{E}}+t{\frac {\sigma _{0}}{\eta }}}
2718:
2684:
2646:
767:
2676:
2638:
2531:
498:
2546:
1663:
763:
677:
486:
700:
Dependence of dimensionless stress upon dimensionless time under constant strain
485:
is the material coefficient of viscosity. This model describes the damper as a
490:
1367:{\displaystyle \varepsilon _{\mathrm {reversible} }={\frac {\sigma _{0}}{E}},}
221:{\displaystyle \sigma _{\mathrm {Total} }=\sigma _{\rm {D}}=\sigma _{\rm {S}}}
2752:
2688:
2650:
1145:{\displaystyle \varepsilon _{\mathrm {irreversible} }=\varepsilon _{0}\left.}
2590:. Cambridge, MA 02139: Massachusetts Institute of Technology. pp. 8ā11.
2680:
2642:
673:
72:
In
Maxwell configuration, under an applied axial stress, the total stress,
26:
1654:
1639:{\displaystyle \sigma (t)=\eta {\dot {\varepsilon }}(1-e^{-Et/\eta })}
696:
58:
2633:
49:
34:
2617:"Viscoelastic flows of Maxwell fluids with conservation laws"
883:, then the elastic element will spring back by the value of
1490:
If a
Maxwell material is subject to a constant strain rate
1519:
then the stress increases, reaching a constant value of
735:, then the stress decays on a characteristic timescale of
2669:
704:
If a
Maxwell material is suddenly deformed and held to a
1160:
If a
Maxwell material is suddenly subjected to a stress
1843:
Thus, the components of the dynamic modulus are :
816:{\displaystyle {\frac {\sigma (t)}{E\varepsilon _{0}}}}
2499:
2453:
2412:
2371:
2328:
2109:
1852:
1675:
1569:
1527:
1496:
1383:
1301:
1271:
1196:
1166:
1025:
892:
862:
829:
779:
773:
The picture shows dependence of dimensionless stress
741:
714:
613:
524:
312:
235:
163:
119:
78:
2621:
ESAIM: Mathematical
Modelling and Numerical Analysis
1485:
1479:since it models strain as linear function of time.
2706:
2508:
2478:
2432:
2391:
2348:
2311:
2089:
1832:
1638:
1552:{\displaystyle \sigma =\eta {\dot {\varepsilon }}}
1551:
1511:
1464:
1366:
1284:
1254:
1179:
1144:
1005:
875:
845:
815:
754:
727:
661:
593:
466:
292:
220:
146:
105:
691:
2750:
511:and their rates of change with respect to time
147:{\displaystyle \varepsilon _{\mathrm {Total} }}
2700:
2698:
2576:
2574:
2572:
2479:{\displaystyle {\frac {E_{2}}{\omega \eta }}}
57:The Maxwell model is represented by a purely
1155:
2713:. London, W1X6BA: Academic Press. pp.
2704:
2358:
2695:
2569:
1658:Relaxational spectrum for Maxwell material
672:The equation can be applied either to the
106:{\displaystyle \sigma _{\mathrm {Total} }}
2632:
2580:
1653:
695:
48:
2614:
515:are governed by equations of the form:
2751:
2665:"IV. On the dynamical theory of gases"
2349:{\displaystyle \tau \equiv \eta /E}
1475:The Maxwell model does not exhibit
846:{\displaystyle {\frac {E}{\eta }}t}
13:
2433:{\displaystyle {\frac {E_{2}}{E}}}
2392:{\displaystyle {\frac {E_{1}}{E}}}
1649:
1512:{\displaystyle {\dot {\epsilon }}}
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
908:
905:
902:
899:
755:{\displaystyle {\frac {\eta }{E}}}
401:
369:
337:
334:
331:
328:
325:
284:
269:
254:
251:
248:
245:
242:
212:
197:
182:
179:
176:
173:
170:
138:
135:
132:
129:
126:
97:
94:
91:
88:
85:
14:
2780:
2615:Boyaval, SĆ©bastien (1 May 2021).
2447:dimensionless apparent viscosity
2406:dimensionless modulus of losses
1666:of a Maxwell material would be:
1486:Effect of a constant strain rate
856:If we free the material at time
728:{\displaystyle \varepsilon _{0}}
2657:
2608:
2365:dimensionless elastic modulus
2232:
2217:
2209:
2195:
2126:
2120:
1996:
1981:
1960:
1945:
1869:
1863:
1735:
1726:
1692:
1686:
1633:
1600:
1579:
1573:
1206:
1200:
939:
926:
792:
786:
692:Effect of a sudden deformation
16:Model of viscoelastic material
1:
2562:
2557:Upper-convected Maxwell model
766:. The phenomenon is known as
686:upper-convected Maxwell model
53:Diagram of a Maxwell material
44:
39:Upper-convected Maxwell model
2509:{\displaystyle \omega \tau }
7:
2675:: 49ā88. 31 December 1867.
2584:Engineering Viscoelasticity
2552:Standard linear solid model
2525:
1180:{\displaystyle \sigma _{0}}
489:and models the spring with
481:is the elastic modulus and
154:can be defined as follows:
10:
2785:
2705:Christensen, R. M (1971).
2709:Theory of Viscoelasticity
2537:Generalized Maxwell model
1156:Effect of a sudden stress
25:is the most simple model
2737:: CS1 maint: location (
2600:: CS1 maint: location (
2581:Roylance, David (2001).
2493:dimensionless frequency
823:upon dimensionless time
497:In a Maxwell material,
2681:10.1098/rstl.1867.0004
2510:
2480:
2434:
2393:
2350:
2313:
2091:
1834:
1659:
1640:
1553:
1513:
1466:
1368:
1286:
1256:
1181:
1146:
1007:
877:
847:
817:
756:
729:
701:
663:
595:
468:
294:
222:
148:
113:and the total strain,
107:
54:
2542:KelvināVoigt material
2511:
2481:
2435:
2394:
2351:
2314:
2092:
1835:
1657:
1641:
1554:
1514:
1467:
1369:
1287:
1285:{\displaystyle t_{1}}
1257:
1182:
1147:
1008:
878:
876:{\displaystyle t_{1}}
848:
818:
757:
730:
699:
664:
604:or, in dot notation:
596:
469:
295:
223:
149:
108:
67:KelvināVoigt material
52:
2759:Non-Newtonian fluids
2643:10.1051/m2an/2020076
2497:
2451:
2410:
2369:
2326:
2107:
1850:
1673:
1567:
1525:
1494:
1381:
1299:
1269:
1194:
1164:
1023:
890:
860:
827:
777:
739:
712:
611:
522:
310:
233:
161:
117:
76:
61:damper and a purely
2769:James Clerk Maxwell
31:James Clerk Maxwell
2506:
2476:
2430:
2389:
2346:
2309:
2087:
1830:
1660:
1636:
1549:
1509:
1462:
1364:
1282:
1252:
1177:
1142:
1003:
873:
843:
813:
752:
725:
702:
659:
591:
464:
290:
218:
144:
103:
55:
2764:Materials science
2520:
2519:
2474:
2428:
2387:
2304:
2258:
2187:
2082:
2022:
1937:
1828:
1739:
1597:
1546:
1506:
1457:
1359:
1250:
1227:
1117:
983:
946:
838:
811:
768:stress relaxation
750:
656:
642:
629:
624:
589:
566:
553:
533:
462:
442:
429:
416:
384:
352:
2776:
2743:
2742:
2736:
2728:
2712:
2702:
2693:
2692:
2661:
2655:
2654:
2636:
2612:
2606:
2605:
2599:
2591:
2589:
2578:
2532:Burgers material
2515:
2513:
2512:
2507:
2485:
2483:
2482:
2477:
2475:
2473:
2465:
2464:
2455:
2439:
2437:
2436:
2431:
2429:
2424:
2423:
2414:
2398:
2396:
2395:
2390:
2388:
2383:
2382:
2373:
2359:
2355:
2353:
2352:
2347:
2342:
2318:
2316:
2315:
2310:
2305:
2303:
2296:
2295:
2286:
2285:
2275:
2267:
2259:
2257:
2250:
2249:
2240:
2239:
2227:
2215:
2205:
2193:
2188:
2186:
2185:
2184:
2172:
2171:
2162:
2161:
2151:
2147:
2146:
2133:
2119:
2118:
2096:
2094:
2093:
2088:
2083:
2081:
2074:
2073:
2064:
2063:
2053:
2052:
2051:
2042:
2041:
2031:
2023:
2021:
2014:
2013:
2004:
2003:
1991:
1979:
1978:
1977:
1968:
1967:
1955:
1943:
1938:
1936:
1935:
1934:
1922:
1921:
1912:
1911:
1901:
1900:
1899:
1890:
1889:
1876:
1862:
1861:
1839:
1837:
1836:
1831:
1829:
1827:
1826:
1825:
1813:
1812:
1803:
1802:
1792:
1788:
1787:
1769:
1768:
1759:
1758:
1745:
1740:
1738:
1725:
1711:
1699:
1685:
1684:
1645:
1643:
1642:
1637:
1632:
1631:
1627:
1599:
1598:
1590:
1558:
1556:
1555:
1550:
1548:
1547:
1539:
1518:
1516:
1515:
1510:
1508:
1507:
1499:
1471:
1469:
1468:
1463:
1458:
1453:
1452:
1443:
1441:
1440:
1428:
1427:
1426:
1373:
1371:
1370:
1365:
1360:
1355:
1354:
1345:
1340:
1339:
1338:
1291:
1289:
1288:
1283:
1281:
1280:
1265:If at some time
1261:
1259:
1258:
1253:
1251:
1246:
1245:
1236:
1228:
1223:
1222:
1213:
1186:
1184:
1183:
1178:
1176:
1175:
1151:
1149:
1148:
1143:
1138:
1134:
1133:
1129:
1128:
1127:
1118:
1110:
1083:
1082:
1070:
1069:
1068:
1012:
1010:
1009:
1004:
999:
995:
994:
993:
984:
976:
960:
959:
947:
942:
938:
937:
921:
913:
912:
911:
882:
880:
879:
874:
872:
871:
852:
850:
849:
844:
839:
831:
822:
820:
819:
814:
812:
810:
809:
808:
795:
781:
761:
759:
758:
753:
751:
743:
734:
732:
731:
726:
724:
723:
668:
666:
665:
660:
658:
657:
649:
643:
635:
630:
625:
617:
615:
600:
598:
597:
592:
590:
588:
580:
572:
567:
559:
554:
552:
544:
536:
534:
526:
473:
471:
470:
465:
463:
461:
453:
445:
443:
435:
430:
422:
417:
415:
407:
406:
405:
404:
390:
385:
383:
375:
374:
373:
372:
358:
353:
351:
343:
342:
341:
340:
314:
299:
297:
296:
291:
289:
288:
287:
274:
273:
272:
259:
258:
257:
227:
225:
224:
219:
217:
216:
215:
202:
201:
200:
187:
186:
185:
153:
151:
150:
145:
143:
142:
141:
112:
110:
109:
104:
102:
101:
100:
23:Maxwell material
2784:
2783:
2779:
2778:
2777:
2775:
2774:
2773:
2749:
2748:
2747:
2746:
2730:
2729:
2725:
2703:
2696:
2663:
2662:
2658:
2613:
2609:
2593:
2592:
2587:
2579:
2570:
2565:
2547:Oldroyd-B model
2528:
2522:
2498:
2495:
2494:
2466:
2460:
2456:
2454:
2452:
2449:
2448:
2419:
2415:
2413:
2411:
2408:
2407:
2378:
2374:
2372:
2370:
2367:
2366:
2338:
2327:
2324:
2323:
2291:
2287:
2281:
2277:
2276:
2268:
2266:
2245:
2241:
2235:
2231:
2223:
2216:
2201:
2194:
2192:
2180:
2176:
2167:
2163:
2157:
2153:
2152:
2142:
2138:
2134:
2132:
2114:
2110:
2108:
2105:
2104:
2069:
2065:
2059:
2055:
2054:
2047:
2043:
2037:
2033:
2032:
2030:
2009:
2005:
1999:
1995:
1987:
1980:
1973:
1969:
1963:
1959:
1951:
1944:
1942:
1930:
1926:
1917:
1913:
1907:
1903:
1902:
1895:
1891:
1885:
1881:
1877:
1875:
1857:
1853:
1851:
1848:
1847:
1821:
1817:
1808:
1804:
1798:
1794:
1793:
1783:
1779:
1764:
1760:
1754:
1750:
1746:
1744:
1721:
1707:
1703:
1698:
1680:
1676:
1674:
1671:
1670:
1664:dynamic modulus
1652:
1650:Dynamic modulus
1646:
1623:
1613:
1609:
1589:
1588:
1568:
1565:
1564:
1538:
1537:
1526:
1523:
1522:
1498:
1497:
1495:
1492:
1491:
1488:
1448:
1444:
1442:
1436:
1432:
1389:
1388:
1384:
1382:
1379:
1378:
1350:
1346:
1344:
1307:
1306:
1302:
1300:
1297:
1296:
1276:
1272:
1270:
1267:
1266:
1241:
1237:
1235:
1218:
1214:
1212:
1195:
1192:
1191:
1171:
1167:
1165:
1162:
1161:
1158:
1123:
1119:
1109:
1105:
1101:
1088:
1084:
1078:
1074:
1031:
1030:
1026:
1024:
1021:
1020:
989:
985:
975:
971:
967:
955:
951:
933:
929:
922:
920:
898:
897:
893:
891:
888:
887:
867:
863:
861:
858:
857:
830:
828:
825:
824:
804:
800:
796:
782:
780:
778:
775:
774:
764:relaxation time
762:, known as the
742:
740:
737:
736:
719:
715:
713:
710:
709:
694:
678:Newtonian fluid
648:
647:
634:
616:
614:
612:
609:
608:
581:
573:
571:
558:
545:
537:
535:
525:
523:
520:
519:
496:
487:Newtonian fluid
454:
446:
444:
434:
421:
408:
400:
399:
395:
391:
389:
376:
368:
367:
363:
359:
357:
344:
324:
323:
319:
315:
313:
311:
308:
307:
283:
282:
278:
268:
267:
263:
241:
240:
236:
234:
231:
230:
211:
210:
206:
196:
195:
191:
169:
168:
164:
162:
159:
158:
125:
124:
120:
118:
115:
114:
84:
83:
79:
77:
74:
73:
47:
17:
12:
11:
5:
2782:
2772:
2771:
2766:
2761:
2745:
2744:
2723:
2694:
2656:
2627:(3): 807ā831.
2607:
2567:
2566:
2564:
2561:
2560:
2559:
2554:
2549:
2544:
2539:
2534:
2527:
2524:
2518:
2517:
2505:
2502:
2491:
2487:
2486:
2472:
2469:
2463:
2459:
2445:
2441:
2440:
2427:
2422:
2418:
2404:
2400:
2399:
2386:
2381:
2377:
2363:
2345:
2341:
2337:
2334:
2331:
2320:
2319:
2308:
2302:
2299:
2294:
2290:
2284:
2280:
2274:
2271:
2265:
2262:
2256:
2253:
2248:
2244:
2238:
2234:
2230:
2226:
2222:
2219:
2214:
2211:
2208:
2204:
2200:
2197:
2191:
2183:
2179:
2175:
2170:
2166:
2160:
2156:
2150:
2145:
2141:
2137:
2131:
2128:
2125:
2122:
2117:
2113:
2098:
2097:
2086:
2080:
2077:
2072:
2068:
2062:
2058:
2050:
2046:
2040:
2036:
2029:
2026:
2020:
2017:
2012:
2008:
2002:
1998:
1994:
1990:
1986:
1983:
1976:
1972:
1966:
1962:
1958:
1954:
1950:
1947:
1941:
1933:
1929:
1925:
1920:
1916:
1910:
1906:
1898:
1894:
1888:
1884:
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1871:
1868:
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1797:
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1749:
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1416:
1413:
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1401:
1398:
1395:
1392:
1387:
1375:
1374:
1363:
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1353:
1349:
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1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1305:
1279:
1275:
1263:
1262:
1249:
1244:
1240:
1234:
1231:
1226:
1221:
1217:
1211:
1208:
1205:
1202:
1199:
1174:
1170:
1157:
1154:
1153:
1152:
1141:
1137:
1132:
1126:
1122:
1116:
1113:
1108:
1104:
1100:
1097:
1094:
1091:
1087:
1081:
1077:
1073:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1029:
1014:
1013:
1002:
998:
992:
988:
982:
979:
974:
970:
966:
963:
958:
954:
950:
945:
941:
936:
932:
928:
925:
919:
916:
910:
907:
904:
901:
896:
870:
866:
842:
837:
834:
807:
803:
799:
794:
791:
788:
785:
749:
746:
722:
718:
693:
690:
670:
669:
655:
652:
646:
641:
638:
633:
628:
623:
620:
602:
601:
587:
584:
579:
576:
570:
565:
562:
557:
551:
548:
543:
540:
532:
529:
475:
474:
460:
457:
452:
449:
441:
438:
433:
428:
425:
420:
414:
411:
403:
398:
394:
388:
382:
379:
371:
366:
362:
356:
350:
347:
339:
336:
333:
330:
327:
322:
318:
301:
300:
286:
281:
277:
271:
266:
262:
256:
253:
250:
247:
244:
239:
228:
214:
209:
205:
199:
194:
190:
184:
181:
178:
175:
172:
167:
140:
137:
134:
131:
128:
123:
99:
96:
93:
90:
87:
82:
46:
43:
15:
9:
6:
4:
3:
2:
2781:
2770:
2767:
2765:
2762:
2760:
2757:
2756:
2754:
2740:
2734:
2726:
2724:9780121742508
2720:
2716:
2711:
2710:
2701:
2699:
2690:
2686:
2682:
2678:
2674:
2670:
2666:
2660:
2652:
2648:
2644:
2640:
2635:
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2622:
2618:
2611:
2603:
2597:
2586:
2585:
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2568:
2558:
2555:
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2548:
2545:
2543:
2540:
2538:
2535:
2533:
2530:
2529:
2523:
2503:
2500:
2492:
2489:
2488:
2470:
2467:
2461:
2457:
2446:
2443:
2442:
2425:
2420:
2416:
2405:
2402:
2401:
2384:
2379:
2375:
2364:
2361:
2360:
2357:
2343:
2339:
2335:
2332:
2329:
2306:
2300:
2297:
2292:
2288:
2282:
2278:
2272:
2269:
2263:
2260:
2254:
2251:
2246:
2242:
2236:
2228:
2224:
2220:
2212:
2206:
2202:
2198:
2189:
2181:
2177:
2173:
2168:
2164:
2158:
2154:
2148:
2143:
2139:
2135:
2129:
2123:
2115:
2111:
2103:
2102:
2101:
2084:
2078:
2075:
2070:
2066:
2060:
2056:
2048:
2044:
2038:
2034:
2027:
2024:
2018:
2015:
2010:
2006:
2000:
1992:
1988:
1984:
1974:
1970:
1964:
1956:
1952:
1948:
1939:
1931:
1927:
1923:
1918:
1914:
1908:
1904:
1896:
1892:
1886:
1882:
1878:
1872:
1866:
1858:
1854:
1846:
1845:
1844:
1822:
1818:
1814:
1809:
1805:
1799:
1795:
1789:
1784:
1780:
1776:
1773:
1770:
1765:
1761:
1755:
1751:
1747:
1741:
1732:
1729:
1722:
1718:
1715:
1712:
1708:
1704:
1700:
1695:
1689:
1681:
1677:
1669:
1668:
1667:
1665:
1656:
1647:
1628:
1624:
1620:
1617:
1614:
1610:
1606:
1603:
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1576:
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1562:
1559:
1543:
1540:
1534:
1531:
1528:
1520:
1503:
1500:
1483:
1480:
1478:
1459:
1454:
1449:
1445:
1437:
1433:
1429:
1385:
1377:
1376:
1361:
1356:
1351:
1347:
1341:
1303:
1295:
1294:
1293:
1277:
1273:
1247:
1242:
1238:
1232:
1229:
1224:
1219:
1215:
1209:
1203:
1197:
1190:
1189:
1188:
1172:
1168:
1139:
1135:
1130:
1124:
1120:
1114:
1111:
1106:
1102:
1098:
1095:
1092:
1089:
1085:
1079:
1075:
1071:
1027:
1019:
1018:
1017:
1000:
996:
990:
986:
980:
977:
972:
968:
964:
961:
956:
952:
948:
943:
934:
930:
923:
917:
914:
894:
886:
885:
884:
868:
864:
854:
840:
835:
832:
805:
801:
797:
789:
783:
771:
769:
765:
747:
744:
720:
716:
707:
698:
689:
687:
681:
679:
675:
653:
650:
644:
639:
636:
631:
626:
621:
618:
607:
606:
605:
585:
582:
577:
574:
568:
563:
560:
555:
549:
546:
541:
538:
530:
527:
518:
517:
516:
514:
510:
507:
503:
500:
494:
492:
488:
484:
480:
458:
455:
450:
447:
439:
436:
431:
426:
423:
418:
412:
409:
396:
392:
386:
380:
377:
364:
360:
354:
348:
345:
320:
316:
306:
305:
304:
279:
275:
264:
260:
237:
229:
207:
203:
192:
188:
165:
157:
156:
155:
121:
80:
70:
68:
64:
60:
51:
42:
40:
36:
32:
28:
24:
19:
2708:
2672:
2668:
2659:
2624:
2620:
2610:
2583:
2521:
2444:Yellow curve
2321:
2099:
1842:
1662:The complex
1661:
1563:
1561:In general
1560:
1521:
1489:
1481:
1474:
1264:
1159:
1015:
855:
772:
703:
682:
674:shear stress
671:
603:
512:
508:
501:
495:
482:
478:
476:
302:
71:
56:
27:viscoelastic
22:
20:
18:
491:Hooke's law
2753:Categories
2634:2007.16075
2563:References
2403:Pink curve
2362:Blue curve
45:Definition
2733:cite book
2689:0261-0523
2651:0764-583X
2596:cite book
2504:τ
2501:ω
2471:η
2468:ω
2336:η
2333:≡
2330:τ
2289:ω
2279:τ
2273:ω
2270:τ
2243:ω
2221:η
2213:ω
2199:η
2165:ω
2155:η
2149:η
2136:ω
2124:ω
2067:ω
2057:τ
2045:ω
2035:τ
2007:ω
1985:η
1971:ω
1949:η
1915:ω
1905:η
1893:ω
1883:η
1867:ω
1806:ω
1796:η
1790:η
1777:ω
1762:ω
1752:η
1733:η
1730:ω
1716:−
1690:ω
1682:∗
1629:η
1615:−
1607:−
1595:˙
1592:ε
1586:η
1571:σ
1544:˙
1541:ε
1535:η
1529:σ
1504:˙
1501:ϵ
1455:η
1446:σ
1386:ε
1348:σ
1304:ε
1248:η
1239:σ
1216:σ
1198:ε
1169:σ
1115:η
1107:−
1099:
1093:−
1076:ε
1028:ε
981:η
973:−
965:
953:ε
924:σ
918:−
895:ε
836:η
802:ε
784:σ
745:η
717:ε
654:˙
651:ε
640:η
637:σ
622:˙
619:σ
578:ε
564:η
561:σ
542:σ
451:σ
427:η
424:σ
397:ε
365:ε
321:ε
280:ε
265:ε
238:ε
208:σ
193:σ
166:σ
122:ε
81:σ
2526:See also
63:elastic
59:viscous
2721:
2687:
2649:
2490:X-axis
706:strain
506:strain
499:stress
477:where
35:tensor
2717:ā20.
2629:arXiv
2588:(PDF)
1477:creep
2739:link
2719:ISBN
2685:ISSN
2647:ISSN
2602:link
2100:and
2677:doi
2673:157
2639:doi
1096:exp
962:exp
708:of
2755::
2735:}}
2731:{{
2715:16
2697:^
2683:.
2671:.
2667:.
2645:.
2637:.
2625:55
2623:.
2619:.
2598:}}
2594:{{
2571:^
2516:.
2356:.
853::
770:.
688:.
504:,
493:.
69:.
41:.
21:A
2741:)
2727:.
2691:.
2679::
2653:.
2641::
2631::
2604:)
2462:2
2458:E
2426:E
2421:2
2417:E
2385:E
2380:1
2376:E
2344:E
2340:/
2307:E
2301:1
2298:+
2293:2
2283:2
2264:=
2261:E
2255:1
2252:+
2247:2
2237:2
2233:)
2229:E
2225:/
2218:(
2210:)
2207:E
2203:/
2196:(
2190:=
2182:2
2178:E
2174:+
2169:2
2159:2
2144:2
2140:E
2130:=
2127:)
2121:(
2116:2
2112:E
2085:E
2079:1
2076:+
2071:2
2061:2
2049:2
2039:2
2028:=
2025:E
2019:1
2016:+
2011:2
2001:2
1997:)
1993:E
1989:/
1982:(
1975:2
1965:2
1961:)
1957:E
1953:/
1946:(
1940:=
1932:2
1928:E
1924:+
1919:2
1909:2
1897:2
1887:2
1879:E
1873:=
1870:)
1864:(
1859:1
1855:E
1823:2
1819:E
1815:+
1810:2
1800:2
1785:2
1781:E
1774:i
1771:+
1766:2
1756:2
1748:E
1742:=
1736:)
1727:(
1723:/
1719:i
1713:E
1709:/
1705:1
1701:1
1696:=
1693:)
1687:(
1678:E
1634:)
1625:/
1621:t
1618:E
1611:e
1604:1
1601:(
1583:=
1580:)
1577:t
1574:(
1532:=
1460:.
1450:0
1438:1
1434:t
1430:=
1424:e
1421:l
1418:b
1415:i
1412:s
1409:r
1406:e
1403:v
1400:e
1397:r
1394:r
1391:i
1362:,
1357:E
1352:0
1342:=
1336:e
1333:l
1330:b
1327:i
1324:s
1321:r
1318:e
1315:v
1312:e
1309:r
1278:1
1274:t
1243:0
1233:t
1230:+
1225:E
1220:0
1210:=
1207:)
1204:t
1201:(
1173:0
1140:.
1136:]
1131:)
1125:1
1121:t
1112:E
1103:(
1090:1
1086:[
1080:0
1072:=
1066:e
1063:l
1060:b
1057:i
1054:s
1051:r
1048:e
1045:v
1042:e
1039:r
1036:r
1033:i
1001:.
997:)
991:1
987:t
978:E
969:(
957:0
949:=
944:E
940:)
935:1
931:t
927:(
915:=
909:k
906:c
903:a
900:b
869:1
865:t
841:t
833:E
806:0
798:E
793:)
790:t
787:(
748:E
721:0
645:=
632:+
627:E
586:t
583:d
575:d
569:=
556:+
550:t
547:d
539:d
531:E
528:1
513:t
509:Īµ
502:Ļ
483:Ī·
479:E
459:t
456:d
448:d
440:E
437:1
432:+
419:=
413:t
410:d
402:S
393:d
387:+
381:t
378:d
370:D
361:d
355:=
349:t
346:d
338:l
335:a
332:t
329:o
326:T
317:d
285:S
276:+
270:D
261:=
255:l
252:a
249:t
246:o
243:T
213:S
204:=
198:D
189:=
183:l
180:a
177:t
174:o
171:T
139:l
136:a
133:t
130:o
127:T
98:l
95:a
92:t
89:o
86:T
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
