122:) and associated material degradation mechanisms become dominant modes of structural failure. While these deformation and damage mechanisms originate at the microscale where discrete processes dominate, practical application of failure theories to macroscale components is most readily achieved using the formalism of continuum mechanics. In this context, microscopic damage is idealized as a continuous state variable defined at all points within a structure. State equations are defined which govern the time evolution of damage. These equations may be readily integrated into finite element codes to analyze the damage evolution in complex 3D structures and calculate how long a component may safely be used before failure occurs.
1780:
thermodynamically stable and grow via diffusion when exposed to elevated temperatures. As the precipitates coarsen, their ability to restrict dislocation motion decreases as the average spacing between particles increases, thus decreasing the required Orowan stress for bowing. In the case of grain boundary precipitates, precipitate growth means that fewer grain boundaries are impeded from grain boundary sliding. When cast into the damage mechanics formalism, precipitation coarsening and its effect on strain rate may be represented by the following equations.
2512:
795:
not justified. This limitation was remedied by researchers such as A.C.F. Cocks, M.F. Ashby, and B.F. Dyson, who proposed mechanistically informed strain and damage evolution equations. Extrapolation using such equations is justified if the dominant damage mechanism remains the same at the conditions of interest.
2316:
2645:
The preceding equations are valid under uniaxial tension only. When a multiaxial state of stress is present in the system, each equation must be adapted so that the driving multiaxial stress is considered. For void-growth by power-law creep, the relevant stress is the von Mises stress as this drives
794:
While easy to apply, the lumped damage model proposed by
Kachanov and Robotnov is limited by the fact that the damage state variable cannot be directly tied to a specific mechanism of strain and damage evolution. Correspondingly, extrapolation of the model beyond the original dataset of test data is
803:
In the power-law creep regime, global deformation is controlled by glide and climb of dislocations. If internal voids are present within the microstructure, global structural continuity requires that the voids must both elongate and expand laterally, further reducing the local section. When cast in
2609:
Note that both damage mechanisms are included in the creep strain rate equation. The precipitate coarsening damage mechanisms influences the void-growth damage mechanism as the void-growth mechanism depends on the global strain rate. The precipitate growth mechanisms is only time and temperature
1779:
Many modern steels and alloys are designed such that precipitates will precipitate either within the matrix or along grain boundaries during casting. These precipitates restrict dislocation motion and, if present on grain boundaries, grain boundary sliding during creep. Many precipitates are not
1188:
At very high temperature and/or low stresses, void growth on grain boundaries is primarily controlled by the diffusive flux of vacancies along the grain boundary. As matter diffuses away from the void and plates onto the adjacent grain boundaries, a roughly spherical void is maintained by rapid
19:
is concerned with the representation, or modeling, of damage of materials that is suitable for making engineering predictions about the initiation, propagation, and fracture of materials without resorting to a microscopic description that would be too complex for practical engineering analysis.
498:
In this simple case, the strain rate is governed by power-law creep with the stress enhanced by the damage state variable as damage accumulates. The damage term ω is interpreted as a distributed loss of load bearing area which results in an increased local stress at the microscale. The time to
1087:
1413:
2324:
1295:
1881:
958:
1522:
215:
2108:
Multiple damage mechanism can be combined to represent a broader range of phenomena. For instance, if both void-growth by power-law creep and precipitate coarsening are relevant mechanisms, the following combined set of equations may be used:
300:
27:, "It is often argued that the ultimate task of engineering research is to provide not so much a better insight into the examined phenomenon but to supply a rational predictive tool applicable in design". Damage mechanics is a topic of
2604:
2115:
1959:
709:
966:
1303:
2719:
Barbero, E.J., Cortes, D.H., A mechanistic model for transverse damage initiation, evolution, and stiffness reduction in laminated composites (2010) Composites Part B: Engineering, 41 (2), pp. 124-132.
2507:{\displaystyle {\dot {\omega }}_{1}={\dot {\epsilon }}_{0}\sigma ^{n}\left({\frac {1}{\left(1-\omega _{1}\right)^{n}}}-\left(1-\omega _{1}\right)\right)\left(1+K^{\prime \prime }\omega _{2}\right)^{n}}
1189:
diffusion of vacancies along the surface of the void. When cast in the damage mechanics formalism, the growth of internal voids by boundary diffusion can be represented by the following equations.
1195:
102:
function but this requires additional phenomenological parameters that must be found through experimentation, which is expensive, time consuming, and virtually no one does. On the other hand,
572:
2001:
94:. A damage activation criterion is needed to predict damage initiation. Damage evolution does not progress spontaneously after initiation, thus requiring a damage evolution model. In
1786:
1561:
1126:
810:
748:
368:
784:
473:
2071:
1421:
332:
437:
136:
2635:
2098:
1678:
599:
529:
1765:
1178:
221:
2021:
1698:
1581:
1146:
619:
388:
1651:
118:
When mechanical structures are exposed to temperatures exceeding one-third of the melting temperature of the material of construction, time-dependent deformation (
2520:
1604:
2041:
1738:
1718:
1624:
493:
408:
2311:{\displaystyle {\dot {\epsilon }}={\dot {\epsilon }}_{0}\sigma ^{n}\left(1+{\frac {2r_{h}^{0}}{d}}\left\right)\left(1+K^{\prime \prime }\omega _{2}\right)^{n}}
1889:
627:
1082:{\displaystyle {\dot {\omega }}={\dot {\epsilon }}_{0}\sigma ^{n}\left({\frac {1}{\left(1-\omega \right)^{n}}}-\left(1-\omega \right)\right)}
130:
L. M. Kachanov and Y. N. Rabotnov suggested the following evolution equations for the creep strain ε and a lumped damage state variable ω:
1408:{\displaystyle {\dot {\omega }}={\dot {\epsilon }}_{0}\phi _{0}\sigma {\frac {1}{\omega ^{1/2}\ln \left({\frac {1}{\omega }}\right)}}}
2646:
the global creep deformation; however, for void-growth by boundary diffusion, the maximum principal stress drives the vacancy flux.
804:
the damage mechanics formalism, the growth of internal voids by power-law creep can be represented by the following equations.
750:
and n are found by fitting the creep strain rate equation at zero damage to minimum creep rate measurements. Model parameters
2849:
2781:
1290:{\displaystyle {\dot {\epsilon }}={\dot {\epsilon }}_{0}\phi _{0}\sigma {\frac {2l}{d\ln \left({\frac {1}{\omega }}\right)}}}
2702:
Struik, L C E, Physical aging in amorphous polymers and other materials, Elsevier
Scientific Pub. Co.; New York, 1978,
534:
3042:
2981:
2920:
2707:
601:
is taken to be 1, this results in the following prediction for a structure loaded under a constant uniaxial stress
76:
1876:{\displaystyle {\dot {\epsilon }}={\dot {\epsilon }}_{0}\sigma ^{n}\left(1+K^{\prime \prime }\omega \right)^{n}}
3101:
953:{\displaystyle {\dot {\epsilon }}={\dot {\epsilon }}_{0}\sigma ^{n}\left(1+{\frac {2r_{h}^{0}}{d}}\left\right)}
1967:
1530:
1095:
717:
337:
3096:
1517:{\displaystyle \phi _{0}={\frac {2D_{B}\delta _{B}\Omega }{kTl^{3}}}{\frac {1}{{\dot {\varepsilon }}_{0}}}}
753:
442:
3106:
2046:
308:
413:
210:{\displaystyle {\dot {\epsilon }}={\dot {\epsilon }}_{0}\left({\frac {\sigma }{1-\omega }}\right)^{n}}
3075:
Dyson, B. F. (1992). "Materials Data
Requirements, Creep Damage Mechanisms, and Predictive Models".
499:
failure is determined by integrating the damage evolution equation from an initial undamaged state
106:
of damage formulations are able to predict both damage initiation and evolution without additional
23:
Damage mechanics illustrates the typical engineering approach to model complex phenomena. To quote
295:{\displaystyle {\dot {\omega }}={\dot {\omega }}_{0}\left({\frac {\sigma }{1-\omega }}\right)^{m}}
2665:
2655:
2613:
2076:
1656:
577:
502:
99:
1743:
1151:
2006:
1683:
1566:
1131:
604:
373:
55:
and remaining life of the material that is damaging as a result of thermomechanical load and
2599:{\displaystyle {\dot {\omega }}_{2}={\frac {K^{\prime }}{3}}\left(1-\omega _{2}\right)^{4}}
1629:
95:
8:
119:
107:
32:
3061:
3000:
2939:
1586:
2814:
2746:
2026:
1723:
1703:
1609:
478:
393:
2684:
Krajcinovic, D., Damage mechanics (1989) Mechanics of
Materials, 8 (2-3), pp. 117-197.
3027:
2966:
2905:
2878:
2845:
2818:
2777:
2750:
2703:
28:
24:
3014:
Cocks, A. C. F.; Ashby, M. F. (1 January 1982). "On creep fracture by void growth".
2953:
Cocks, A. C. F.; Ashby, M. F. (1 January 1982). "On creep fracture by void growth".
2892:
Cocks, A. C. F.; Ashby, M. F. (1 January 1982). "On creep fracture by void growth".
2865:
Cocks, A. C. F.; Ashby, M. F. (1 January 1982). "On creep fracture by void growth".
3057:
3023:
2996:
2962:
2935:
2901:
2874:
2837:
2806:
2769:
2738:
2660:
1954:{\displaystyle {\dot {\omega }}={\frac {K^{\prime }}{3}}\left(1-\omega \right)^{4}}
704:{\displaystyle t_{f}={\frac {1}{\left(m+1\right){\dot {\omega }}_{0}\sigma ^{m}}}}
2841:
2773:
80:
60:
40:
103:
2810:
2742:
84:
3090:
87:
90:
forces that motivate further damage. Initially the material is pristine, or
2797:
Kachanov, Lazar M. (1 April 1999). "Rupture Time Under Creep
Conditions".
2729:
Kachanov, Lazar M. (1 April 1999). "Rupture Time Under Creep
Conditions".
786:
and m are found by fitting the above equation to creep rupture life data.
1768:
68:
48:
72:
52:
2073:
is a parameter linking the precipitation damage to the strain rate,
1740:
is the absolute temperatures. It is noted that factors present in
789:
56:
3043:"Creep and fracture of metals : mechanisms and mechanics"
2982:"Creep and fracture of metals : mechanisms and mechanics"
2921:"Creep and fracture of metals : mechanisms and mechanics"
2610:
dependent and hence does not depend on the void-growth damage
1180:
is the average initial void radius, and d is the grain size.
63:
may be measurable, e.g., crack density, or inferred from the
98:
like formulations, the damage evolution is controlled by a
410:
is the creep stress exponent of the material of interest,
1771:
pre-factors due to the similarity of the two mechanisms.
2693:
Dusan
Krajcinovic, Mechanics of Materials 8 (1989) 169.
1148:
is the applied stress, n is the creep stress exponent,
2478:
2282:
2058:
1854:
2616:
2523:
2327:
2118:
2079:
2049:
2029:
2009:
1970:
1892:
1789:
1746:
1726:
1706:
1686:
1659:
1632:
1612:
1589:
1569:
1533:
1424:
1306:
1198:
1154:
1134:
1098:
969:
813:
756:
720:
630:
607:
580:
537:
505:
481:
445:
416:
396:
376:
340:
311:
224:
139:
1183:
113:
2629:
2598:
2506:
2310:
2092:
2065:
2035:
2015:
1995:
1953:
1875:
1759:
1732:
1712:
1692:
1672:
1645:
1618:
1598:
1575:
1555:
1516:
1407:
1289:
1172:
1140:
1120:
1081:
952:
778:
742:
703:
613:
593:
566:
523:
487:
467:
431:
402:
382:
362:
326:
294:
209:
798:
3088:
567:{\displaystyle \left(\omega =\omega _{f}\right)}
2100:determines the rate of precipitate coarsening.
790:Mechanistically informed damage state variables
125:
2103:
1653:is the grain-boundary diffusion coefficient,
3013:
2952:
2891:
2864:
2832:Rabotnov, Y. N. (1969). "Creep rupture".
2764:Rabotnov, Y. N. (1969). "Creep rupture".
1774:
2831:
2796:
2763:
2728:
1996:{\displaystyle \ {\dot {\epsilon }}_{0}}
3089:
1606:is the center-to-center void spacing,
1556:{\displaystyle {\dot {\epsilon }}_{0}}
1121:{\displaystyle {\dot {\epsilon }}_{0}}
743:{\displaystyle {\dot {\epsilon _{0}}}}
363:{\displaystyle {\dot {\epsilon }}_{0}}
3074:
3040:
2979:
2918:
2640:
779:{\displaystyle {\dot {\omega _{0}}}}
468:{\displaystyle {\dot {\omega }}_{0}}
439:is the rate of damage accumulation,
2043:is the creep-rate stress exponent,
475:is the damage-rate multiplier, and
13:
3077:High Temperature Structural Design
2553:
2475:
2279:
2085:
2066:{\displaystyle K^{\prime \prime }}
2055:
1915:
1851:
1687:
1464:
327:{\displaystyle {\dot {\epsilon }}}
14:
3118:
3062:10.1051/rphysap:01988002304060500
3001:10.1051/rphysap:01988002304060500
2940:10.1051/rphysap:01988002304060500
2799:International Journal of Fracture
2731:International Journal of Fracture
1680:is the grain boundary thickness,
1184:Void-growth by Boundary Diffusion
432:{\displaystyle {\dot {\omega }}}
114:Creep Continuum Damage Mechanics
77:coefficient of thermal expansion
3068:
3034:
3007:
2973:
2946:
2912:
2885:
531:to a specified critical damage
495:is the damage stress exponent.
2858:
2825:
2790:
2757:
2722:
2713:
2696:
2687:
2678:
2003:is the creep-rate multiplier,
1563:is the creep-rate multiplier,
1128:is the creep-rate multiplier,
799:Void-growth by Power-Law Creep
518:
506:
370:is the creep-rate multiplier,
1:
3016:Progress in Materials Science
2955:Progress in Materials Science
2894:Progress in Materials Science
2867:Progress in Materials Science
2671:
1720:is Boltzmann’s constant, and
3028:10.1016/0079-6425(82)90001-9
2967:10.1016/0079-6425(82)90001-9
2906:10.1016/0079-6425(82)90001-9
2879:10.1016/0079-6425(82)90001-9
2842:10.1007/978-3-642-85640-2_26
2774:10.1007/978-3-642-85640-2_26
126:Lumped damage state variable
7:
3050:Revue de Physique Appliquée
2989:Revue de Physique Appliquée
2928:Revue de Physique Appliquée
2649:
2630:{\displaystyle \omega _{1}}
2104:Combining Damage Mechanisms
2093:{\displaystyle K^{\prime }}
1673:{\displaystyle \delta _{B}}
594:{\displaystyle \omega _{f}}
524:{\displaystyle (\omega =0)}
79:, remaining life, etc. The
10:
3123:
334:is the creep strain rate,
1760:{\displaystyle \phi _{0}}
1173:{\displaystyle r_{h}^{0}}
1767:are very similar to the
2811:10.1023/A:1018671022008
2743:10.1023/A:1018671022008
2666:Critical plane analysis
2656:Lumped damage mechanics
2023:is the applied stress,
2016:{\displaystyle \sigma }
1693:{\displaystyle \Omega }
1583:is the applied stress,
1576:{\displaystyle \sigma }
1141:{\displaystyle \sigma }
614:{\displaystyle \sigma }
390:is the applied stress,
383:{\displaystyle \sigma }
31:that relies heavily on
2631:
2600:
2508:
2312:
2094:
2067:
2037:
2017:
1997:
1955:
1877:
1775:Precipitate Coarsening
1761:
1734:
1714:
1700:is the atomic volume,
1694:
1674:
1647:
1620:
1600:
1577:
1557:
1518:
1409:
1291:
1174:
1142:
1122:
1083:
954:
780:
744:
705:
615:
595:
568:
525:
489:
469:
433:
404:
384:
364:
328:
296:
211:
35:. Most of the work on
3102:Materials degradation
2632:
2601:
2509:
2313:
2095:
2068:
2038:
2018:
1998:
1956:
1878:
1762:
1735:
1715:
1695:
1675:
1648:
1646:{\displaystyle D_{B}}
1621:
1601:
1578:
1558:
1519:
1410:
1292:
1175:
1143:
1123:
1084:
955:
781:
745:
706:
616:
596:
569:
526:
490:
470:
434:
405:
385:
365:
329:
297:
212:
3041:Dyson, B.F. (1988).
2980:Dyson, B.F. (1988).
2919:Dyson, B.F. (1988).
2614:
2521:
2325:
2116:
2077:
2047:
2027:
2007:
1968:
1890:
1787:
1744:
1724:
1704:
1684:
1657:
1630:
1610:
1587:
1567:
1531:
1422:
1304:
1196:
1152:
1132:
1096:
967:
811:
754:
718:
628:
605:
578:
535:
503:
479:
443:
414:
394:
374:
338:
309:
222:
137:
3097:Continuum mechanics
2194:
1626:is the grain size,
1169:
889:
108:material properties
33:continuum mechanics
3107:Mechanical failure
2641:Multiaxial Effects
2627:
2596:
2504:
2308:
2180:
2090:
2063:
2033:
2013:
1993:
1951:
1873:
1757:
1730:
1710:
1690:
1670:
1643:
1616:
1599:{\displaystyle 2l}
1596:
1573:
1553:
1514:
1405:
1287:
1170:
1155:
1138:
1118:
1079:
950:
875:
776:
740:
701:
611:
591:
564:
521:
485:
465:
429:
400:
380:
360:
324:
292:
207:
71:property, such as
67:they have on some
2851:978-3-642-85642-6
2834:Applied Mechanics
2783:978-3-642-85642-6
2766:Applied Mechanics
2561:
2534:
2422:
2360:
2338:
2244:
2199:
2144:
2128:
2036:{\displaystyle n}
1984:
1973:
1923:
1902:
1815:
1799:
1733:{\displaystyle T}
1713:{\displaystyle k}
1619:{\displaystyle d}
1544:
1512:
1503:
1486:
1403:
1396:
1332:
1316:
1285:
1278:
1224:
1208:
1109:
1050:
995:
979:
932:
894:
839:
823:
773:
737:
714:Model parameters
699:
679:
488:{\displaystyle m}
456:
426:
403:{\displaystyle n}
351:
321:
280:
250:
234:
195:
165:
149:
43:to represent the
29:applied mechanics
25:Dusan Krajcinovic
3114:
3081:
3080:
3072:
3066:
3065:
3047:
3038:
3032:
3031:
3011:
3005:
3004:
2986:
2977:
2971:
2970:
2950:
2944:
2943:
2925:
2916:
2910:
2909:
2889:
2883:
2882:
2862:
2856:
2855:
2829:
2823:
2822:
2794:
2788:
2787:
2761:
2755:
2754:
2726:
2720:
2717:
2711:
2700:
2694:
2691:
2685:
2682:
2661:Failure analysis
2636:
2634:
2633:
2628:
2626:
2625:
2605:
2603:
2602:
2597:
2595:
2594:
2589:
2585:
2584:
2583:
2562:
2557:
2556:
2547:
2542:
2541:
2536:
2535:
2527:
2513:
2511:
2510:
2505:
2503:
2502:
2497:
2493:
2492:
2491:
2482:
2481:
2457:
2453:
2452:
2448:
2447:
2446:
2423:
2421:
2420:
2415:
2411:
2410:
2409:
2385:
2378:
2377:
2368:
2367:
2362:
2361:
2353:
2346:
2345:
2340:
2339:
2331:
2317:
2315:
2314:
2309:
2307:
2306:
2301:
2297:
2296:
2295:
2286:
2285:
2261:
2257:
2256:
2252:
2245:
2243:
2242:
2237:
2233:
2232:
2231:
2207:
2200:
2195:
2193:
2188:
2175:
2162:
2161:
2152:
2151:
2146:
2145:
2137:
2130:
2129:
2121:
2099:
2097:
2096:
2091:
2089:
2088:
2072:
2070:
2069:
2064:
2062:
2061:
2042:
2040:
2039:
2034:
2022:
2020:
2019:
2014:
2002:
2000:
1999:
1994:
1992:
1991:
1986:
1985:
1977:
1971:
1960:
1958:
1957:
1952:
1950:
1949:
1944:
1940:
1924:
1919:
1918:
1909:
1904:
1903:
1895:
1882:
1880:
1879:
1874:
1872:
1871:
1866:
1862:
1858:
1857:
1833:
1832:
1823:
1822:
1817:
1816:
1808:
1801:
1800:
1792:
1766:
1764:
1763:
1758:
1756:
1755:
1739:
1737:
1736:
1731:
1719:
1717:
1716:
1711:
1699:
1697:
1696:
1691:
1679:
1677:
1676:
1671:
1669:
1668:
1652:
1650:
1649:
1644:
1642:
1641:
1625:
1623:
1622:
1617:
1605:
1603:
1602:
1597:
1582:
1580:
1579:
1574:
1562:
1560:
1559:
1554:
1552:
1551:
1546:
1545:
1537:
1523:
1521:
1520:
1515:
1513:
1511:
1510:
1505:
1504:
1496:
1489:
1487:
1485:
1484:
1483:
1467:
1463:
1462:
1453:
1452:
1439:
1434:
1433:
1414:
1412:
1411:
1406:
1404:
1402:
1401:
1397:
1389:
1377:
1376:
1372:
1355:
1350:
1349:
1340:
1339:
1334:
1333:
1325:
1318:
1317:
1309:
1296:
1294:
1293:
1288:
1286:
1284:
1283:
1279:
1271:
1255:
1247:
1242:
1241:
1232:
1231:
1226:
1225:
1217:
1210:
1209:
1201:
1179:
1177:
1176:
1171:
1168:
1163:
1147:
1145:
1144:
1139:
1127:
1125:
1124:
1119:
1117:
1116:
1111:
1110:
1102:
1088:
1086:
1085:
1080:
1078:
1074:
1073:
1069:
1051:
1049:
1048:
1043:
1039:
1020:
1013:
1012:
1003:
1002:
997:
996:
988:
981:
980:
972:
959:
957:
956:
951:
949:
945:
944:
940:
933:
931:
930:
925:
921:
902:
895:
890:
888:
883:
870:
857:
856:
847:
846:
841:
840:
832:
825:
824:
816:
785:
783:
782:
777:
775:
774:
769:
768:
759:
749:
747:
746:
741:
739:
738:
733:
732:
723:
710:
708:
707:
702:
700:
698:
697:
696:
687:
686:
681:
680:
672:
668:
664:
645:
640:
639:
620:
618:
617:
612:
600:
598:
597:
592:
590:
589:
573:
571:
570:
565:
563:
559:
558:
557:
530:
528:
527:
522:
494:
492:
491:
486:
474:
472:
471:
466:
464:
463:
458:
457:
449:
438:
436:
435:
430:
428:
427:
419:
409:
407:
406:
401:
389:
387:
386:
381:
369:
367:
366:
361:
359:
358:
353:
352:
344:
333:
331:
330:
325:
323:
322:
314:
301:
299:
298:
293:
291:
290:
285:
281:
279:
265:
258:
257:
252:
251:
243:
236:
235:
227:
216:
214:
213:
208:
206:
205:
200:
196:
194:
180:
173:
172:
167:
166:
158:
151:
150:
142:
37:damage mechanics
17:Damage mechanics
3122:
3121:
3117:
3116:
3115:
3113:
3112:
3111:
3087:
3086:
3085:
3084:
3073:
3069:
3045:
3039:
3035:
3012:
3008:
2984:
2978:
2974:
2951:
2947:
2923:
2917:
2913:
2890:
2886:
2863:
2859:
2852:
2830:
2826:
2795:
2791:
2784:
2762:
2758:
2727:
2723:
2718:
2714:
2701:
2697:
2692:
2688:
2683:
2679:
2674:
2652:
2643:
2621:
2617:
2615:
2612:
2611:
2590:
2579:
2575:
2568:
2564:
2563:
2552:
2548:
2546:
2537:
2526:
2525:
2524:
2522:
2519:
2518:
2498:
2487:
2483:
2474:
2470:
2463:
2459:
2458:
2442:
2438:
2431:
2427:
2416:
2405:
2401:
2394:
2390:
2389:
2384:
2383:
2379:
2373:
2369:
2363:
2352:
2351:
2350:
2341:
2330:
2329:
2328:
2326:
2323:
2322:
2302:
2291:
2287:
2278:
2274:
2267:
2263:
2262:
2238:
2227:
2223:
2216:
2212:
2211:
2206:
2205:
2201:
2189:
2184:
2176:
2174:
2167:
2163:
2157:
2153:
2147:
2136:
2135:
2134:
2120:
2119:
2117:
2114:
2113:
2106:
2084:
2080:
2078:
2075:
2074:
2054:
2050:
2048:
2045:
2044:
2028:
2025:
2024:
2008:
2005:
2004:
1987:
1976:
1975:
1974:
1969:
1966:
1965:
1945:
1930:
1926:
1925:
1914:
1910:
1908:
1894:
1893:
1891:
1888:
1887:
1867:
1850:
1846:
1839:
1835:
1834:
1828:
1824:
1818:
1807:
1806:
1805:
1791:
1790:
1788:
1785:
1784:
1777:
1751:
1747:
1745:
1742:
1741:
1725:
1722:
1721:
1705:
1702:
1701:
1685:
1682:
1681:
1664:
1660:
1658:
1655:
1654:
1637:
1633:
1631:
1628:
1627:
1611:
1608:
1607:
1588:
1585:
1584:
1568:
1565:
1564:
1547:
1536:
1535:
1534:
1532:
1529:
1528:
1506:
1495:
1494:
1493:
1488:
1479:
1475:
1468:
1458:
1454:
1448:
1444:
1440:
1438:
1429:
1425:
1423:
1420:
1419:
1388:
1384:
1368:
1364:
1360:
1359:
1354:
1345:
1341:
1335:
1324:
1323:
1322:
1308:
1307:
1305:
1302:
1301:
1270:
1266:
1256:
1248:
1246:
1237:
1233:
1227:
1216:
1215:
1214:
1200:
1199:
1197:
1194:
1193:
1186:
1164:
1159:
1153:
1150:
1149:
1133:
1130:
1129:
1112:
1101:
1100:
1099:
1097:
1094:
1093:
1059:
1055:
1044:
1029:
1025:
1024:
1019:
1018:
1014:
1008:
1004:
998:
987:
986:
985:
971:
970:
968:
965:
964:
926:
911:
907:
906:
901:
900:
896:
884:
879:
871:
869:
862:
858:
852:
848:
842:
831:
830:
829:
815:
814:
812:
809:
808:
801:
792:
764:
760:
758:
757:
755:
752:
751:
728:
724:
722:
721:
719:
716:
715:
692:
688:
682:
671:
670:
669:
654:
650:
649:
644:
635:
631:
629:
626:
625:
606:
603:
602:
585:
581:
579:
576:
575:
553:
549:
542:
538:
536:
533:
532:
504:
501:
500:
480:
477:
476:
459:
448:
447:
446:
444:
441:
440:
418:
417:
415:
412:
411:
395:
392:
391:
375:
372:
371:
354:
343:
342:
341:
339:
336:
335:
313:
312:
310:
307:
306:
286:
269:
264:
260:
259:
253:
242:
241:
240:
226:
225:
223:
220:
219:
201:
184:
179:
175:
174:
168:
157:
156:
155:
141:
140:
138:
135:
134:
128:
116:
81:state variables
61:state variables
41:state variables
12:
11:
5:
3120:
3110:
3109:
3104:
3099:
3083:
3082:
3067:
3056:(4): 605–613.
3033:
3022:(3): 189–244.
3006:
2995:(4): 605–613.
2972:
2961:(3): 189–244.
2945:
2934:(4): 605–613.
2911:
2900:(3): 189–244.
2884:
2873:(3): 189–244.
2857:
2850:
2824:
2789:
2782:
2756:
2721:
2712:
2695:
2686:
2676:
2675:
2673:
2670:
2669:
2668:
2663:
2658:
2651:
2648:
2642:
2639:
2624:
2620:
2607:
2606:
2593:
2588:
2582:
2578:
2574:
2571:
2567:
2560:
2555:
2551:
2545:
2540:
2533:
2530:
2515:
2514:
2501:
2496:
2490:
2486:
2480:
2477:
2473:
2469:
2466:
2462:
2456:
2451:
2445:
2441:
2437:
2434:
2430:
2426:
2419:
2414:
2408:
2404:
2400:
2397:
2393:
2388:
2382:
2376:
2372:
2366:
2359:
2356:
2349:
2344:
2337:
2334:
2319:
2318:
2305:
2300:
2294:
2290:
2284:
2281:
2277:
2273:
2270:
2266:
2260:
2255:
2251:
2248:
2241:
2236:
2230:
2226:
2222:
2219:
2215:
2210:
2204:
2198:
2192:
2187:
2183:
2179:
2173:
2170:
2166:
2160:
2156:
2150:
2143:
2140:
2133:
2127:
2124:
2105:
2102:
2087:
2083:
2060:
2057:
2053:
2032:
2012:
1990:
1983:
1980:
1962:
1961:
1948:
1943:
1939:
1936:
1933:
1929:
1922:
1917:
1913:
1907:
1901:
1898:
1884:
1883:
1870:
1865:
1861:
1856:
1853:
1849:
1845:
1842:
1838:
1831:
1827:
1821:
1814:
1811:
1804:
1798:
1795:
1776:
1773:
1754:
1750:
1729:
1709:
1689:
1667:
1663:
1640:
1636:
1615:
1595:
1592:
1572:
1550:
1543:
1540:
1525:
1524:
1509:
1502:
1499:
1492:
1482:
1478:
1474:
1471:
1466:
1461:
1457:
1451:
1447:
1443:
1437:
1432:
1428:
1416:
1415:
1400:
1395:
1392:
1387:
1383:
1380:
1375:
1371:
1367:
1363:
1358:
1353:
1348:
1344:
1338:
1331:
1328:
1321:
1315:
1312:
1298:
1297:
1282:
1277:
1274:
1269:
1265:
1262:
1259:
1254:
1251:
1245:
1240:
1236:
1230:
1223:
1220:
1213:
1207:
1204:
1185:
1182:
1167:
1162:
1158:
1137:
1115:
1108:
1105:
1090:
1089:
1077:
1072:
1068:
1065:
1062:
1058:
1054:
1047:
1042:
1038:
1035:
1032:
1028:
1023:
1017:
1011:
1007:
1001:
994:
991:
984:
978:
975:
961:
960:
948:
943:
939:
936:
929:
924:
920:
917:
914:
910:
905:
899:
893:
887:
882:
878:
874:
868:
865:
861:
855:
851:
845:
838:
835:
828:
822:
819:
800:
797:
791:
788:
772:
767:
763:
736:
731:
727:
712:
711:
695:
691:
685:
678:
675:
667:
663:
660:
657:
653:
648:
643:
638:
634:
610:
588:
584:
562:
556:
552:
548:
545:
541:
520:
517:
514:
511:
508:
484:
462:
455:
452:
425:
422:
399:
379:
357:
350:
347:
320:
317:
303:
302:
289:
284:
278:
275:
272:
268:
263:
256:
249:
246:
239:
233:
230:
217:
204:
199:
193:
190:
187:
183:
178:
171:
164:
161:
154:
148:
145:
127:
124:
115:
112:
104:micromechanics
9:
6:
4:
3:
2:
3119:
3108:
3105:
3103:
3100:
3098:
3095:
3094:
3092:
3078:
3071:
3063:
3059:
3055:
3051:
3044:
3037:
3029:
3025:
3021:
3017:
3010:
3002:
2998:
2994:
2990:
2983:
2976:
2968:
2964:
2960:
2956:
2949:
2941:
2937:
2933:
2929:
2922:
2915:
2907:
2903:
2899:
2895:
2888:
2880:
2876:
2872:
2868:
2861:
2853:
2847:
2843:
2839:
2835:
2828:
2820:
2816:
2812:
2808:
2804:
2800:
2793:
2785:
2779:
2775:
2771:
2767:
2760:
2752:
2748:
2744:
2740:
2736:
2732:
2725:
2716:
2709:
2708:9780444416551
2705:
2699:
2690:
2681:
2677:
2667:
2664:
2662:
2659:
2657:
2654:
2653:
2647:
2638:
2622:
2618:
2591:
2586:
2580:
2576:
2572:
2569:
2565:
2558:
2549:
2543:
2538:
2531:
2528:
2517:
2516:
2499:
2494:
2488:
2484:
2471:
2467:
2464:
2460:
2454:
2449:
2443:
2439:
2435:
2432:
2428:
2424:
2417:
2412:
2406:
2402:
2398:
2395:
2391:
2386:
2380:
2374:
2370:
2364:
2357:
2354:
2347:
2342:
2335:
2332:
2321:
2320:
2303:
2298:
2292:
2288:
2275:
2271:
2268:
2264:
2258:
2253:
2249:
2246:
2239:
2234:
2228:
2224:
2220:
2217:
2213:
2208:
2202:
2196:
2190:
2185:
2181:
2177:
2171:
2168:
2164:
2158:
2154:
2148:
2141:
2138:
2131:
2125:
2122:
2112:
2111:
2110:
2101:
2081:
2051:
2030:
2010:
1988:
1981:
1978:
1946:
1941:
1937:
1934:
1931:
1927:
1920:
1911:
1905:
1899:
1896:
1886:
1885:
1868:
1863:
1859:
1847:
1843:
1840:
1836:
1829:
1825:
1819:
1812:
1809:
1802:
1796:
1793:
1783:
1782:
1781:
1772:
1770:
1752:
1748:
1727:
1707:
1665:
1661:
1638:
1634:
1613:
1593:
1590:
1570:
1548:
1541:
1538:
1507:
1500:
1497:
1490:
1480:
1476:
1472:
1469:
1459:
1455:
1449:
1445:
1441:
1435:
1430:
1426:
1418:
1417:
1398:
1393:
1390:
1385:
1381:
1378:
1373:
1369:
1365:
1361:
1356:
1351:
1346:
1342:
1336:
1329:
1326:
1319:
1313:
1310:
1300:
1299:
1280:
1275:
1272:
1267:
1263:
1260:
1257:
1252:
1249:
1243:
1238:
1234:
1228:
1221:
1218:
1211:
1205:
1202:
1192:
1191:
1190:
1181:
1165:
1160:
1156:
1135:
1113:
1106:
1103:
1075:
1070:
1066:
1063:
1060:
1056:
1052:
1045:
1040:
1036:
1033:
1030:
1026:
1021:
1015:
1009:
1005:
999:
992:
989:
982:
976:
973:
963:
962:
946:
941:
937:
934:
927:
922:
918:
915:
912:
908:
903:
897:
891:
885:
880:
876:
872:
866:
863:
859:
853:
849:
843:
836:
833:
826:
820:
817:
807:
806:
805:
796:
787:
770:
765:
761:
734:
729:
725:
693:
689:
683:
676:
673:
665:
661:
658:
655:
651:
646:
641:
636:
632:
624:
623:
622:
608:
586:
582:
560:
554:
550:
546:
543:
539:
515:
512:
509:
496:
482:
460:
453:
450:
423:
420:
397:
377:
355:
348:
345:
318:
315:
287:
282:
276:
273:
270:
266:
261:
254:
247:
244:
237:
231:
228:
218:
202:
197:
191:
188:
185:
181:
176:
169:
162:
159:
152:
146:
143:
133:
132:
131:
123:
121:
111:
109:
105:
101:
97:
93:
89:
88:thermodynamic
86:
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
21:
18:
3076:
3070:
3053:
3049:
3036:
3019:
3015:
3009:
2992:
2988:
2975:
2958:
2954:
2948:
2931:
2927:
2914:
2897:
2893:
2887:
2870:
2866:
2860:
2833:
2827:
2805:(1): 11–18.
2802:
2798:
2792:
2765:
2759:
2737:(1): 11–18.
2734:
2730:
2724:
2715:
2698:
2689:
2680:
2644:
2608:
2107:
1963:
1778:
1526:
1187:
1091:
802:
793:
713:
497:
304:
129:
117:
91:
64:
44:
36:
22:
16:
15:
2836:: 342–349.
2768:: 342–349.
1769:Coble creep
69:macroscopic
3091:Categories
2672:References
96:plasticity
2819:116979654
2751:116979654
2619:ω
2577:ω
2573:−
2554:′
2532:˙
2529:ω
2485:ω
2479:′
2476:′
2440:ω
2436:−
2425:−
2403:ω
2399:−
2371:σ
2358:˙
2355:ϵ
2336:˙
2333:ω
2289:ω
2283:′
2280:′
2247:−
2225:ω
2221:−
2155:σ
2142:˙
2139:ϵ
2126:˙
2123:ϵ
2086:′
2059:′
2056:′
2011:σ
1982:˙
1979:ϵ
1938:ω
1935:−
1916:′
1900:˙
1897:ω
1860:ω
1855:′
1852:′
1826:σ
1813:˙
1810:ϵ
1797:˙
1794:ϵ
1749:ϕ
1688:Ω
1662:δ
1571:σ
1542:˙
1539:ϵ
1501:˙
1498:ε
1465:Ω
1456:δ
1427:ϕ
1394:ω
1382:
1362:ω
1352:σ
1343:ϕ
1330:˙
1327:ϵ
1314:˙
1311:ω
1276:ω
1264:
1244:σ
1235:ϕ
1222:˙
1219:ϵ
1206:˙
1203:ϵ
1136:σ
1107:˙
1104:ϵ
1067:ω
1064:−
1053:−
1037:ω
1034:−
1006:σ
993:˙
990:ϵ
977:˙
974:ω
935:−
919:ω
916:−
850:σ
837:˙
834:ϵ
821:˙
818:ϵ
771:˙
762:ω
735:˙
726:ϵ
690:σ
677:˙
674:ω
609:σ
583:ω
551:ω
544:ω
510:ω
454:˙
451:ω
424:˙
421:ω
378:σ
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