2266:
71:
1879:
2261:{\displaystyle {\begin{aligned}{\boldsymbol {l}}&={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}=\left({\dot {\boldsymbol {F}}}^{e}\cdot {\boldsymbol {F}}^{p}+{\boldsymbol {F}}^{e}\cdot {\dot {\boldsymbol {F}}}^{p}\right)\cdot \left\\&={\dot {\boldsymbol {F}}}^{e}\cdot ({\boldsymbol {F}}^{e})^{-1}+{\boldsymbol {F}}^{e}\cdot \cdot ({\boldsymbol {F}}^{e})^{-1}\,.\end{aligned}}}
1746:
1098:. Rock plasticity theories also use a similar concept except that the requirement of pressure-dependence of the yield surface requires a relaxation of the above assumption. Instead, it is typically assumed that the plastic strain increment and the normal to the pressure-dependent yield surface have the same direction, i.e.,
1439:
3085:
2955:
2374:
1632:
1015:
1163:
1345:
17:
752:
is indicated if the state of stress is on the yield surface and the stress increment is directed toward the outside of the yield surface; this occurs if the inner product of the stress increment and the outward normal of the yield surface is positive,
3192:
819:
931:
2462:
2681:
2779:
502:
1505:
2859:
1850:
569:
735:
3300:
1080:
1624:
2976:
2874:
2277:
418:
1741:{\displaystyle df={\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}:d{\boldsymbol {\sigma }}+{\frac {\partial f}{\partial {\boldsymbol {\varepsilon }}_{p}}}:d{\boldsymbol {\varepsilon }}_{p}=0\,.}
1451:
material, the yield surface can expand with increasing stress. We assume
Drucker's second stability postulate which states that for an infinitesimal stress cycle this plastic work is positive, i.e.,
1510:
The above quantity is equal to zero for purely elastic cycles. Examination of the work done over a cycle of plastic loading-unloading can be used to justify the validity of the associated flow rule.
1299:
1884:
1337:
3363:
660:
955:
1261:
1104:
186:
Typical flow plasticity theories for unidirectional loading (for small deformation perfect plasticity or hardening plasticity) are developed on the basis of the following requirements:
1434:{\displaystyle d{\boldsymbol {\sigma }}:{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}=0\quad {\text{and}}\quad d{\boldsymbol {\sigma }}:d{\boldsymbol {\varepsilon }}_{p}=0\,.}
328:
3126:
260:
224:
1094:
In metal plasticity, the assumption that the plastic strain increment and deviatoric stress tensor have the same principal directions is encapsulated in a relation called the
126:
99:
764:
1195:
876:
595:
360:
292:
2385:
180:
153:
1543:
2507:
2708:
863:
1572:
673:. For strain hardening materials, the yield surface evolves with increasing plastic strain and the elastic limit changes. The evolving yield surface has the form
426:
1223:
46:
in a body can be decomposed additively (or multiplicatively) into an elastic part and a plastic part. The elastic part of the strain can be computed from a
74:
Stress-strain curve showing typical plastic behavior of materials in uniaxial compression. The strain can be decomposed into a recoverable elastic strain (
1457:
2798:
1803:
534:
3462:
3406:
Bilby, B. A.; Bullough, R.; Smith, E. (1955), "Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry",
684:
3211:
3080:{\displaystyle {\boldsymbol {M}}={\frac {\partial W}{\partial {\boldsymbol {E}}^{e}}}=J\,{\frac {dU}{dJ}}+2\mu \,{\text{dev}}({\boldsymbol {E}}^{e})}
1039:
1577:
1444:
Hence, both the normal to the yield surface and the plastic strain tensor are perpendicular to the stress tensor and must have the same direction.
1301:, i.e., the yield surface remains constant under increasing plastic deformation. This implies that the increment of elastic strain is also zero,
2950:{\displaystyle {\boldsymbol {M}}:={\tfrac {1}{2}}({\boldsymbol {C}}^{e}\cdot {\boldsymbol {S}}+{\boldsymbol {S}}\cdot {\boldsymbol {C}}^{e})}
2369:{\displaystyle {\boldsymbol {l}}={\boldsymbol {l}}^{e}+{\boldsymbol {F}}^{e}\cdot {\boldsymbol {L}}^{p}\cdot ({\boldsymbol {F}}^{e})^{-1}\,.}
1783:
The concept of multiplicative decomposition of the deformation gradient into elastic and plastic parts was first proposed independently by
1775:
The first assumption was widely used for numerical simulations of metals but has gradually been superseded by the multiplicative theory.
155:. For strain hardening materials (as shown in the figure) the yield stress increases with increasing plastic deformation to a value of
368:
3408:
294:. If loading takes the stress state to the plastic domain then the increment of plastic strain is always greater than zero, i.e.,
2699:
1266:
3203:
1304:
3554:
3316:
512:
The above requirements can be expressed in three dimensional states of stress and multidirectional loading as follows.
1202:
1791:
and extended to continuum plasticity by
Erasmus Lee. The decomposition assumes that the total deformation gradient (
624:
1010:{\displaystyle d{\boldsymbol {\varepsilon }}=d{\boldsymbol {\varepsilon }}_{e}+d{\boldsymbol {\varepsilon }}_{p}\,.}
1158:{\displaystyle d{\boldsymbol {\varepsilon }}_{p}=d\lambda \,{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}}
1519:
59:
3117:
2472:
1235:
613:). The elastic limit is defined by a yield surface that does not depend on the plastic strain and has the form
3559:
193:
The material has an elastic limit defined as the stress at which plastic deformation first takes place, i.e.,
3187:{\displaystyle {\boldsymbol {D}}^{p}={\dot {\lambda }}\,{\frac {\partial f}{\partial {\boldsymbol {M}}}}\,.}
297:
55:
43:
36:
1198:
1761:
232:
196:
2695:
814:{\displaystyle d{\boldsymbol {\sigma }}:{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}\geq 0\,.}
51:
1226:
926:{\displaystyle d{\boldsymbol {\sigma }}:{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}<0\,.}
104:
77:
70:
3539:
3511:
3477:
3417:
2457:{\displaystyle {\boldsymbol {L}}^{p}:={\dot {\boldsymbol {F}}}^{p}\cdot ({\boldsymbol {F}}^{p})^{-1}}
1868:
1768:
1756:
Large deformation flow theories of plasticity typically start with one of the following assumptions:
32:
1171:
2865:
576:
336:
268:
265:
Loading is defined as the situation under which increments of stress are greater than zero, i.e.,
3502:
Anand, L. (1979), "On H. Hencky's approximate strain-energy function for moderate deformations",
2789:
2785:
2676:{\displaystyle {\boldsymbol {D}}^{p}={\tfrac {1}{2}}~,~~{\boldsymbol {W}}^{p}={\tfrac {1}{2}}\,.}
1788:
333:
Unloading is defined as the situation under which increments of stress are less than zero, i.e.,
158:
131:
2774:{\displaystyle {\boldsymbol {C}}^{e}:=({\boldsymbol {F}}^{e})^{T}\cdot {\boldsymbol {F}}^{e}\,.}
1525:
1095:
505:
47:
497:{\displaystyle d\sigma \,d\varepsilon =d\sigma \,(d\varepsilon _{e}+d\varepsilon _{p})\geq 0}
3519:
3485:
3425:
1522:
is needed to close the set of constitutive equations and to eliminate the unknown parameter
1448:
944:: The additive decomposition of the strain into elastic and plastic parts can be written as
842:
35:
behavior of materials. Flow plasticity theories are characterized by the assumption that a
362:. The material is elastic during unloading and no additional plastic strain is accumulated.
2967:
1548:
28:
54:
constitutive model. However, determination of the plastic part of the strain requires a
3515:
3481:
3421:
523:). In the linear elastic regime the stresses and strains in the material are related by
39:
exists that can be used to determine the amount of plastic deformation in the material.
1208:
520:
3548:
1232:
The above flow rule is easily justified for perfectly plastic deformations for which
610:
229:
Beyond the elastic limit the stress state always remains on the yield surface, i.e.,
420:. The plastic part cannot be recovered while the elastic part is fully recoverable.
1500:{\displaystyle d{\boldsymbol {\sigma }}:d{\boldsymbol {\varepsilon }}_{p}\geq 0\,.}
2854:{\displaystyle {\boldsymbol {E}}^{e}:={\tfrac {1}{2}}\ln {\boldsymbol {C}}^{e}\,.}
2686:
Typically, the plastic spin is ignored in most descriptions of finite plasticity.
1845:{\displaystyle {\boldsymbol {F}}={\boldsymbol {F}}^{e}\cdot {\boldsymbol {F}}^{p}}
564:{\displaystyle {\boldsymbol {\sigma }}={\mathsf {D}}:{\boldsymbol {\varepsilon }}}
365:
The total strain is a linear combination of the elastic and plastic parts, i.e.,
1784:
1771:
tensor can be multiplicatively decomposed in an elastic part and a plastic part.
730:{\displaystyle f({\boldsymbol {\sigma }},{\boldsymbol {\varepsilon }}_{p})=0\,.}
16:
3295:{\displaystyle {\dot {\lambda }}\geq 0~,~~f\leq 0~,~~{\dot {\lambda }}\,f=0\,.}
2271:
where a superposed dot indicates a time derivative. We can write the above as
1764:
tensor can be additively decomposed into an elastic part and a plastic part, or
1075:{\displaystyle d{\boldsymbol {\sigma }}:d{\boldsymbol {\varepsilon }}\geq 0\,.}
2868:
tensor is a convenient stress measure for finite plasticity and is defined as
3442:
Kröner, E. (1958), "Kontinuumstheorie der
Versetzungen und Eigenspannungen",
2694:
The elastic behavior in the finite strain regime is typically described by a
1619:{\displaystyle f({\boldsymbol {\sigma }},{\boldsymbol {\varepsilon }}_{p})=0}
3429:
3310:
The consistency condition is identical to that for the small strain case,
3120:
leads, in the absence of a plastic spin, to the finite strain flow rule
2970:. A possible hyperelastic model in terms of the logarithmic strain is
3523:
3489:
3202:
The loading-unloading conditions can be shown to be equivalent to the
508:
postulate and eliminates the possibility of strain softening behavior.
423:
The work done of a loading-unloading cycle is positive or zero, i.e.,
1545:
from the system of equations. The consistency condition states that
3108:
is a modulus, and "dev" indicates the deviatoric part of a tensor.
1197:
is a hardening parameter. This form of the flow rule is called an
823:
The above equation, when it is equal to zero, indicates a state of
2698:
model. The elastic strain can be measured using an elastic right
69:
413:{\displaystyle d\varepsilon =d\varepsilon _{e}+d\varepsilon _{p}}
839:: A similar argument is made for unloading for which situation
1867:
is the plastic (unrecoverable) part of the deformation. The
42:
In flow plasticity theories it is assumed that the total
2887:
2818:
2612:
2527:
1294:{\displaystyle d{\boldsymbol {\varepsilon }}_{p}>0}
1201:
and the assumption of co-directionality is called the
3319:
3214:
3129:
2979:
2877:
2801:
2711:
2510:
2388:
2280:
1882:
1806:
1635:
1580:
1551:
1528:
1460:
1348:
1307:
1269:
1238:
1211:
1174:
1107:
1042:
958:
879:
845:
827:
where the stress state moves along the yield surface.
767:
687:
627:
579:
537:
429:
371:
339:
300:
271:
235:
199:
161:
134:
107:
80:
1332:{\displaystyle d{\boldsymbol {\varepsilon }}_{e}=0}
3358:{\displaystyle {\dot {\lambda }}\,{\dot {f}}=0\,.}
3357:
3294:
3186:
3079:
2949:
2853:
2773:
2675:
2475:) stress-free configuration. The symmetric part (
2456:
2368:
2260:
1844:
1740:
1618:
1566:
1537:
1499:
1433:
1331:
1293:
1255:
1217:
1189:
1157:
1074:
1009:
925:
857:
813:
729:
654:
589:
563:
496:
412:
354:
322:
286:
254:
218:
174:
147:
120:
93:
3463:"Elastic-Plastic Deformation at Finite Strains"
655:{\displaystyle f({\boldsymbol {\sigma }})=0\,.}
8:
865:, the material is in the elastic domain, and
1256:{\displaystyle d{\boldsymbol {\sigma }}=0}
1028:: The stability postulate is expressed as
20:Plastic deformation of a thin metal sheet.
3351:
3334:
3333:
3332:
3321:
3320:
3318:
3288:
3278:
3267:
3266:
3216:
3215:
3213:
3180:
3172:
3158:
3157:
3146:
3145:
3136:
3131:
3128:
3068:
3063:
3054:
3053:
3024:
3023:
3008:
3003:
2988:
2980:
2978:
2938:
2933:
2924:
2916:
2907:
2902:
2886:
2878:
2876:
2847:
2841:
2836:
2817:
2808:
2803:
2800:
2767:
2761:
2756:
2746:
2736:
2731:
2718:
2713:
2710:
2669:
2660:
2650:
2645:
2632:
2627:
2611:
2602:
2597:
2575:
2565:
2560:
2547:
2542:
2526:
2517:
2512:
2509:
2445:
2435:
2430:
2417:
2406:
2405:
2395:
2390:
2387:
2362:
2353:
2343:
2338:
2325:
2320:
2310:
2305:
2295:
2290:
2281:
2279:
2250:
2241:
2231:
2226:
2207:
2197:
2192:
2179:
2168:
2167:
2154:
2149:
2136:
2126:
2121:
2108:
2097:
2096:
2071:
2061:
2056:
2040:
2030:
2025:
2002:
1991:
1990:
1980:
1975:
1965:
1960:
1950:
1939:
1938:
1920:
1915:
1900:
1899:
1887:
1883:
1881:
1836:
1831:
1821:
1816:
1807:
1805:
1734:
1722:
1717:
1701:
1696:
1681:
1673:
1659:
1645:
1634:
1601:
1596:
1587:
1579:
1550:
1527:
1493:
1481:
1476:
1464:
1459:
1427:
1415:
1410:
1398:
1389:
1374:
1360:
1352:
1347:
1317:
1312:
1306:
1279:
1274:
1268:
1242:
1237:
1210:
1173:
1147:
1133:
1132:
1117:
1112:
1106:
1068:
1057:
1046:
1041:
1003:
997:
992:
979:
974:
962:
957:
919:
905:
891:
883:
878:
844:
807:
793:
779:
771:
766:
748:. For general states of stress, plastic
723:
708:
703:
694:
686:
648:
634:
626:
581:
580:
578:
556:
547:
546:
538:
536:
479:
463:
452:
436:
428:
404:
388:
370:
338:
308:
299:
270:
246:
234:
210:
198:
166:
160:
139:
133:
112:
106:
85:
79:
190:The material has a linear elastic range.
15:
3374:
3173:
3132:
3064:
3004:
2981:
2934:
2925:
2917:
2903:
2879:
2837:
2804:
2757:
2732:
2714:
2646:
2628:
2598:
2561:
2543:
2513:
2431:
2408:
2391:
2339:
2321:
2306:
2291:
2282:
2227:
2193:
2170:
2150:
2122:
2099:
2057:
2026:
1993:
1976:
1961:
1941:
1916:
1902:
1888:
1832:
1817:
1808:
1779:Kinematics of multiplicative plasticity
1718:
1697:
1674:
1660:
1597:
1588:
1477:
1465:
1411:
1399:
1375:
1353:
1313:
1275:
1243:
1148:
1113:
1058:
1047:
993:
975:
963:
906:
884:
794:
772:
704:
695:
635:
557:
539:
1861:is the elastic (recoverable) part and
1339:, because of Hooke's law. Therefore,
582:
548:
323:{\displaystyle d\varepsilon _{p}>0}
3094:is a strain energy density function,
7:
31:theory that is used to describe the
2471:and is defined in an intermediate (
255:{\displaystyle \sigma =\sigma _{y}}
219:{\displaystyle \sigma =\sigma _{0}}
128:). The stress at initial yield is
3409:Proceedings of the Royal Society A
3169:
3161:
2999:
2991:
1692:
1684:
1656:
1648:
1371:
1363:
1144:
1136:
902:
894:
790:
782:
14:
121:{\displaystyle \varepsilon _{p}}
94:{\displaystyle \varepsilon _{e}}
2700:Cauchy-Green deformation tensor
2491:while the skew-symmetric part (
1787:, E. Kröner, in the context of
1394:
1388:
3074:
3059:
2944:
2898:
2792:tensor may then be defined as
2743:
2727:
2666:
2657:
2641:
2623:
2581:
2572:
2556:
2538:
2442:
2426:
2350:
2334:
2238:
2222:
2216:
2204:
2188:
2163:
2133:
2117:
2068:
2052:
2037:
2021:
1607:
1584:
1190:{\displaystyle d\lambda >0}
714:
691:
639:
631:
485:
453:
1:
3386:, Courier Dover Publications.
3204:Karush-Kuhn-Tucker conditions
2968:second Piola-Kirchhoff stress
590:{\displaystyle {\mathsf {D}}}
355:{\displaystyle d\sigma <0}
287:{\displaystyle d\sigma >0}
3504:Journal of Applied Mechanics
3470:Journal of Applied Mechanics
3198:Loading-unloading conditions
2489:plastic rate of deformation
1520:Prager consistency condition
573:where the stiffness matrix
504:. This is also called the
175:{\displaystyle \sigma _{y}}
148:{\displaystyle \sigma _{0}}
101:) and an inelastic strain (
3576:
3118:Clausius-Duhem inequality
2469:plastic velocity gradient
1869:spatial velocity gradient
1538:{\displaystyle d\lambda }
3382:Lubliner, Jacob (2008),
1797:) can be decomposed as:
1752:Large deformation theory
671:Beyond the elastic limit
66:Small deformation theory
3430:10.1098/rspa.1955.0171
3359:
3296:
3188:
3081:
2951:
2855:
2775:
2677:
2458:
2370:
2262:
1846:
1742:
1620:
1568:
1539:
1501:
1435:
1333:
1295:
1257:
1219:
1191:
1159:
1076:
1011:
927:
859:
858:{\displaystyle f<0}
815:
731:
656:
591:
565:
498:
414:
356:
324:
288:
256:
220:
183:
176:
149:
122:
95:
21:
3360:
3306:Consistency condition
3297:
3189:
3082:
2952:
2856:
2776:
2696:hyperelastic material
2678:
2459:
2371:
2263:
1847:
1743:
1621:
1569:
1540:
1514:Consistency condition
1502:
1436:
1334:
1296:
1258:
1220:
1192:
1160:
1077:
1012:
928:
860:
816:
732:
657:
592:
566:
499:
415:
357:
325:
289:
257:
221:
177:
150:
123:
96:
73:
19:
3540:Plasticity (physics)
3317:
3212:
3127:
2977:
2875:
2799:
2709:
2508:
2386:
2278:
1880:
1804:
1769:deformation gradient
1633:
1578:
1567:{\displaystyle df=0}
1549:
1526:
1458:
1346:
1305:
1267:
1236:
1209:
1199:associated flow rule
1172:
1105:
1040:
956:
942:Strain decomposition
877:
843:
765:
685:
625:
577:
535:
427:
369:
337:
298:
269:
233:
197:
159:
132:
105:
78:
3555:Continuum mechanics
3516:1979JAM....46...78A
3482:1969JAM....36....1L
3461:Lee, E. H. (1969),
3422:1955RSPSA.231..263B
3396:Anandarajah (2010).
3116:Application of the
1762:rate of deformation
1203:normality condition
1026:Stability postulate
3355:
3292:
3184:
3077:
2947:
2896:
2851:
2827:
2771:
2673:
2621:
2536:
2454:
2366:
2258:
2256:
1842:
1789:crystal plasticity
1738:
1616:
1564:
1535:
1497:
1431:
1329:
1291:
1253:
1215:
1187:
1155:
1072:
1007:
923:
855:
811:
727:
652:
587:
561:
494:
410:
352:
320:
284:
252:
216:
184:
172:
145:
118:
91:
22:
3524:10.1115/1.3424532
3490:10.1115/1.3564580
3444:Erg. Angew. Math.
3416:(1185): 263–273,
3384:Plasticity Theory
3342:
3329:
3275:
3265:
3262:
3256:
3244:
3241:
3235:
3224:
3178:
3154:
3057:
3042:
3015:
2895:
2826:
2620:
2595:
2592:
2586:
2535:
2414:
2176:
2105:
1999:
1947:
1908:
1708:
1665:
1574:at yield because
1392:
1380:
1227:plastic potential
1225:is also called a
1218:{\displaystyle f}
1153:
911:
799:
506:Drucker stability
3567:
3527:
3526:
3499:
3493:
3492:
3467:
3458:
3452:
3451:
3439:
3433:
3432:
3403:
3397:
3394:
3388:
3387:
3379:
3364:
3362:
3361:
3356:
3344:
3343:
3335:
3331:
3330:
3322:
3301:
3299:
3298:
3293:
3277:
3276:
3268:
3263:
3260:
3254:
3242:
3239:
3233:
3226:
3225:
3217:
3193:
3191:
3190:
3185:
3179:
3177:
3176:
3167:
3159:
3156:
3155:
3147:
3141:
3140:
3135:
3086:
3084:
3083:
3078:
3073:
3072:
3067:
3058:
3055:
3043:
3041:
3033:
3025:
3016:
3014:
3013:
3012:
3007:
2997:
2989:
2984:
2956:
2954:
2953:
2948:
2943:
2942:
2937:
2928:
2920:
2912:
2911:
2906:
2897:
2888:
2882:
2864:The symmetrized
2860:
2858:
2857:
2852:
2846:
2845:
2840:
2828:
2819:
2813:
2812:
2807:
2780:
2778:
2777:
2772:
2766:
2765:
2760:
2751:
2750:
2741:
2740:
2735:
2723:
2722:
2717:
2682:
2680:
2679:
2674:
2665:
2664:
2655:
2654:
2649:
2637:
2636:
2631:
2622:
2613:
2607:
2606:
2601:
2593:
2590:
2584:
2580:
2579:
2570:
2569:
2564:
2552:
2551:
2546:
2537:
2528:
2522:
2521:
2516:
2497:) is called the
2463:
2461:
2460:
2455:
2453:
2452:
2440:
2439:
2434:
2422:
2421:
2416:
2415:
2407:
2400:
2399:
2394:
2375:
2373:
2372:
2367:
2361:
2360:
2348:
2347:
2342:
2330:
2329:
2324:
2315:
2314:
2309:
2300:
2299:
2294:
2285:
2267:
2265:
2264:
2259:
2257:
2249:
2248:
2236:
2235:
2230:
2215:
2214:
2202:
2201:
2196:
2184:
2183:
2178:
2177:
2169:
2159:
2158:
2153:
2144:
2143:
2131:
2130:
2125:
2113:
2112:
2107:
2106:
2098:
2088:
2084:
2080:
2079:
2078:
2066:
2065:
2060:
2048:
2047:
2035:
2034:
2029:
2012:
2008:
2007:
2006:
2001:
2000:
1992:
1985:
1984:
1979:
1970:
1969:
1964:
1955:
1954:
1949:
1948:
1940:
1928:
1927:
1919:
1910:
1909:
1901:
1891:
1851:
1849:
1848:
1843:
1841:
1840:
1835:
1826:
1825:
1820:
1811:
1747:
1745:
1744:
1739:
1727:
1726:
1721:
1709:
1707:
1706:
1705:
1700:
1690:
1682:
1677:
1666:
1664:
1663:
1654:
1646:
1625:
1623:
1622:
1617:
1606:
1605:
1600:
1591:
1573:
1571:
1570:
1565:
1544:
1542:
1541:
1536:
1506:
1504:
1503:
1498:
1486:
1485:
1480:
1468:
1440:
1438:
1437:
1432:
1420:
1419:
1414:
1402:
1393:
1390:
1381:
1379:
1378:
1369:
1361:
1356:
1338:
1336:
1335:
1330:
1322:
1321:
1316:
1300:
1298:
1297:
1292:
1284:
1283:
1278:
1262:
1260:
1259:
1254:
1246:
1224:
1222:
1221:
1216:
1205:. The function
1196:
1194:
1193:
1188:
1164:
1162:
1161:
1156:
1154:
1152:
1151:
1142:
1134:
1122:
1121:
1116:
1081:
1079:
1078:
1073:
1061:
1050:
1016:
1014:
1013:
1008:
1002:
1001:
996:
984:
983:
978:
966:
932:
930:
929:
924:
912:
910:
909:
900:
892:
887:
864:
862:
861:
856:
820:
818:
817:
812:
800:
798:
797:
788:
780:
775:
736:
734:
733:
728:
713:
712:
707:
698:
661:
659:
658:
653:
638:
596:
594:
593:
588:
586:
585:
570:
568:
567:
562:
560:
552:
551:
542:
503:
501:
500:
495:
484:
483:
468:
467:
419:
417:
416:
411:
409:
408:
393:
392:
361:
359:
358:
353:
329:
327:
326:
321:
313:
312:
293:
291:
290:
285:
261:
259:
258:
253:
251:
250:
225:
223:
222:
217:
215:
214:
181:
179:
178:
173:
171:
170:
154:
152:
151:
146:
144:
143:
127:
125:
124:
119:
117:
116:
100:
98:
97:
92:
90:
89:
3575:
3574:
3570:
3569:
3568:
3566:
3565:
3564:
3560:Solid mechanics
3545:
3544:
3536:
3531:
3530:
3501:
3500:
3496:
3465:
3460:
3459:
3455:
3441:
3440:
3436:
3405:
3404:
3400:
3395:
3391:
3381:
3380:
3376:
3371:
3315:
3314:
3308:
3210:
3209:
3200:
3168:
3160:
3130:
3125:
3124:
3114:
3062:
3034:
3026:
3002:
2998:
2990:
2975:
2974:
2932:
2901:
2873:
2872:
2835:
2802:
2797:
2796:
2755:
2742:
2730:
2712:
2707:
2706:
2692:
2656:
2644:
2626:
2596:
2571:
2559:
2541:
2511:
2506:
2505:
2441:
2429:
2404:
2389:
2384:
2383:
2349:
2337:
2319:
2304:
2289:
2276:
2275:
2255:
2254:
2237:
2225:
2203:
2191:
2166:
2148:
2132:
2120:
2095:
2086:
2085:
2067:
2055:
2036:
2024:
2020:
2016:
1989:
1974:
1959:
1937:
1936:
1932:
1914:
1892:
1878:
1877:
1830:
1815:
1802:
1801:
1781:
1754:
1716:
1695:
1691:
1683:
1655:
1647:
1631:
1630:
1595:
1576:
1575:
1547:
1546:
1524:
1523:
1516:
1475:
1456:
1455:
1409:
1370:
1362:
1344:
1343:
1311:
1303:
1302:
1273:
1265:
1264:
1234:
1233:
1207:
1206:
1170:
1169:
1143:
1135:
1111:
1103:
1102:
1092:
1038:
1037:
991:
973:
954:
953:
901:
893:
875:
874:
841:
840:
825:neutral loading
789:
781:
763:
762:
702:
683:
682:
623:
622:
575:
574:
533:
532:
475:
459:
425:
424:
400:
384:
367:
366:
335:
334:
304:
296:
295:
267:
266:
242:
231:
230:
206:
195:
194:
162:
157:
156:
135:
130:
129:
108:
103:
102:
81:
76:
75:
68:
60:hardening model
29:solid mechanics
25:Flow plasticity
12:
11:
5:
3573:
3571:
3563:
3562:
3557:
3547:
3546:
3543:
3542:
3535:
3532:
3529:
3528:
3494:
3453:
3434:
3398:
3389:
3373:
3372:
3370:
3367:
3366:
3365:
3354:
3350:
3347:
3341:
3338:
3328:
3325:
3307:
3304:
3303:
3302:
3291:
3287:
3284:
3281:
3274:
3271:
3259:
3253:
3250:
3247:
3238:
3232:
3229:
3223:
3220:
3199:
3196:
3195:
3194:
3183:
3175:
3171:
3166:
3163:
3153:
3150:
3144:
3139:
3134:
3113:
3110:
3088:
3087:
3076:
3071:
3066:
3061:
3052:
3049:
3046:
3040:
3037:
3032:
3029:
3022:
3019:
3011:
3006:
3001:
2996:
2993:
2987:
2983:
2958:
2957:
2946:
2941:
2936:
2931:
2927:
2923:
2919:
2915:
2910:
2905:
2900:
2894:
2891:
2885:
2881:
2862:
2861:
2850:
2844:
2839:
2834:
2831:
2825:
2822:
2816:
2811:
2806:
2782:
2781:
2770:
2764:
2759:
2754:
2749:
2745:
2739:
2734:
2729:
2726:
2721:
2716:
2691:
2690:Elastic regime
2688:
2684:
2683:
2672:
2668:
2663:
2659:
2653:
2648:
2643:
2640:
2635:
2630:
2625:
2619:
2616:
2610:
2605:
2600:
2589:
2583:
2578:
2574:
2568:
2563:
2558:
2555:
2550:
2545:
2540:
2534:
2531:
2525:
2520:
2515:
2487:is called the
2465:
2464:
2451:
2448:
2444:
2438:
2433:
2428:
2425:
2420:
2413:
2410:
2403:
2398:
2393:
2377:
2376:
2365:
2359:
2356:
2352:
2346:
2341:
2336:
2333:
2328:
2323:
2318:
2313:
2308:
2303:
2298:
2293:
2288:
2284:
2269:
2268:
2253:
2247:
2244:
2240:
2234:
2229:
2224:
2221:
2218:
2213:
2210:
2206:
2200:
2195:
2190:
2187:
2182:
2175:
2172:
2165:
2162:
2157:
2152:
2147:
2142:
2139:
2135:
2129:
2124:
2119:
2116:
2111:
2104:
2101:
2094:
2091:
2089:
2087:
2083:
2077:
2074:
2070:
2064:
2059:
2054:
2051:
2046:
2043:
2039:
2033:
2028:
2023:
2019:
2015:
2011:
2005:
1998:
1995:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1946:
1943:
1935:
1931:
1926:
1923:
1918:
1913:
1907:
1904:
1898:
1895:
1893:
1890:
1886:
1885:
1875:
1853:
1852:
1839:
1834:
1829:
1824:
1819:
1814:
1810:
1780:
1777:
1773:
1772:
1765:
1753:
1750:
1749:
1748:
1737:
1733:
1730:
1725:
1720:
1715:
1712:
1704:
1699:
1694:
1689:
1686:
1680:
1676:
1672:
1669:
1662:
1658:
1653:
1650:
1644:
1641:
1638:
1615:
1612:
1609:
1604:
1599:
1594:
1590:
1586:
1583:
1563:
1560:
1557:
1554:
1534:
1531:
1515:
1512:
1508:
1507:
1496:
1492:
1489:
1484:
1479:
1474:
1471:
1467:
1463:
1449:work hardening
1442:
1441:
1430:
1426:
1423:
1418:
1413:
1408:
1405:
1401:
1397:
1387:
1384:
1377:
1373:
1368:
1365:
1359:
1355:
1351:
1328:
1325:
1320:
1315:
1310:
1290:
1287:
1282:
1277:
1272:
1252:
1249:
1245:
1241:
1214:
1186:
1183:
1180:
1177:
1166:
1165:
1150:
1146:
1141:
1138:
1131:
1128:
1125:
1120:
1115:
1110:
1091:
1088:
1087:
1086:
1085:
1084:
1083:
1082:
1071:
1067:
1064:
1060:
1056:
1053:
1049:
1045:
1030:
1029:
1022:
1021:
1020:
1019:
1018:
1017:
1006:
1000:
995:
990:
987:
982:
977:
972:
969:
965:
961:
946:
945:
938:
937:
936:
935:
934:
933:
922:
918:
915:
908:
904:
899:
896:
890:
886:
882:
867:
866:
854:
851:
848:
833:
832:
831:
830:
829:
828:
821:
810:
806:
803:
796:
792:
787:
784:
778:
774:
770:
755:
754:
742:
741:
740:
739:
738:
737:
726:
722:
719:
716:
711:
706:
701:
697:
693:
690:
675:
674:
667:
666:
665:
664:
663:
662:
651:
647:
644:
641:
637:
633:
630:
615:
614:
603:
602:
601:
600:
599:
598:
584:
571:
559:
555:
550:
545:
541:
525:
524:
510:
509:
493:
490:
487:
482:
478:
474:
471:
466:
462:
458:
455:
451:
448:
445:
442:
439:
435:
432:
421:
407:
403:
399:
396:
391:
387:
383:
380:
377:
374:
363:
351:
348:
345:
342:
331:
319:
316:
311:
307:
303:
283:
280:
277:
274:
263:
249:
245:
241:
238:
227:
213:
209:
205:
202:
191:
169:
165:
142:
138:
115:
111:
88:
84:
67:
64:
48:linear elastic
13:
10:
9:
6:
4:
3:
2:
3572:
3561:
3558:
3556:
3553:
3552:
3550:
3541:
3538:
3537:
3533:
3525:
3521:
3517:
3513:
3509:
3505:
3498:
3495:
3491:
3487:
3483:
3479:
3475:
3471:
3464:
3457:
3454:
3449:
3445:
3438:
3435:
3431:
3427:
3423:
3419:
3415:
3411:
3410:
3402:
3399:
3393:
3390:
3385:
3378:
3375:
3368:
3352:
3348:
3345:
3339:
3336:
3326:
3323:
3313:
3312:
3311:
3305:
3289:
3285:
3282:
3279:
3272:
3269:
3257:
3251:
3248:
3245:
3236:
3230:
3227:
3221:
3218:
3208:
3207:
3206:
3205:
3197:
3181:
3164:
3151:
3148:
3142:
3137:
3123:
3122:
3121:
3119:
3111:
3109:
3107:
3103:
3102:
3097:
3093:
3069:
3050:
3047:
3044:
3038:
3035:
3030:
3027:
3020:
3017:
3009:
2994:
2985:
2973:
2972:
2971:
2969:
2965:
2964:
2939:
2929:
2921:
2913:
2908:
2892:
2889:
2883:
2871:
2870:
2869:
2867:
2866:Mandel stress
2848:
2842:
2832:
2829:
2823:
2820:
2814:
2809:
2795:
2794:
2793:
2791:
2790:Hencky strain
2787:
2768:
2762:
2752:
2747:
2737:
2724:
2719:
2705:
2704:
2703:
2701:
2697:
2689:
2687:
2670:
2661:
2651:
2638:
2633:
2617:
2614:
2608:
2603:
2587:
2576:
2566:
2553:
2548:
2532:
2529:
2523:
2518:
2504:
2503:
2502:
2500:
2496:
2495:
2490:
2486:
2485:
2480:
2479:
2474:
2470:
2449:
2446:
2436:
2423:
2418:
2411:
2401:
2396:
2382:
2381:
2380:
2379:The quantity
2363:
2357:
2354:
2344:
2331:
2326:
2316:
2311:
2301:
2296:
2286:
2274:
2273:
2272:
2251:
2245:
2242:
2232:
2219:
2211:
2208:
2198:
2185:
2180:
2173:
2160:
2155:
2145:
2140:
2137:
2127:
2114:
2109:
2102:
2092:
2090:
2081:
2075:
2072:
2062:
2049:
2044:
2041:
2031:
2017:
2013:
2009:
2003:
1996:
1986:
1981:
1971:
1966:
1956:
1951:
1944:
1933:
1929:
1924:
1921:
1911:
1905:
1896:
1894:
1876:
1874:
1873:
1872:
1871:is given by
1870:
1866:
1865:
1860:
1859:
1837:
1827:
1822:
1812:
1800:
1799:
1798:
1796:
1795:
1790:
1786:
1778:
1776:
1770:
1766:
1763:
1759:
1758:
1757:
1751:
1735:
1731:
1728:
1723:
1713:
1710:
1702:
1687:
1678:
1670:
1667:
1651:
1642:
1639:
1636:
1629:
1628:
1627:
1613:
1610:
1602:
1592:
1581:
1561:
1558:
1555:
1552:
1532:
1529:
1521:
1513:
1511:
1494:
1490:
1487:
1482:
1472:
1469:
1461:
1454:
1453:
1452:
1450:
1445:
1428:
1424:
1421:
1416:
1406:
1403:
1395:
1385:
1382:
1366:
1357:
1349:
1342:
1341:
1340:
1326:
1323:
1318:
1308:
1288:
1285:
1280:
1270:
1250:
1247:
1239:
1230:
1228:
1212:
1204:
1200:
1184:
1181:
1178:
1175:
1139:
1129:
1126:
1123:
1118:
1108:
1101:
1100:
1099:
1097:
1089:
1069:
1065:
1062:
1054:
1051:
1043:
1036:
1035:
1034:
1033:
1032:
1031:
1027:
1024:
1023:
1004:
998:
988:
985:
980:
970:
967:
959:
952:
951:
950:
949:
948:
947:
943:
940:
939:
920:
916:
913:
897:
888:
880:
873:
872:
871:
870:
869:
868:
852:
849:
846:
838:
835:
834:
826:
822:
808:
804:
801:
785:
776:
768:
761:
760:
759:
758:
757:
756:
751:
747:
744:
743:
724:
720:
717:
709:
699:
688:
681:
680:
679:
678:
677:
676:
672:
669:
668:
649:
645:
642:
628:
621:
620:
619:
618:
617:
616:
612:
611:Yield surface
608:
607:Elastic limit
605:
604:
572:
553:
543:
531:
530:
529:
528:
527:
526:
522:
518:
515:
514:
513:
507:
491:
488:
480:
476:
472:
469:
464:
460:
456:
449:
446:
443:
440:
437:
433:
430:
422:
405:
401:
397:
394:
389:
385:
381:
378:
375:
372:
364:
349:
346:
343:
340:
332:
317:
314:
309:
305:
301:
281:
278:
275:
272:
264:
247:
243:
239:
236:
228:
211:
207:
203:
200:
192:
189:
188:
187:
167:
163:
140:
136:
113:
109:
86:
82:
72:
65:
63:
61:
57:
53:
49:
45:
40:
38:
34:
30:
26:
18:
3510:(1): 78–82,
3507:
3503:
3497:
3473:
3469:
3456:
3447:
3443:
3437:
3413:
3407:
3401:
3392:
3383:
3377:
3309:
3201:
3115:
3105:
3100:
3099:
3095:
3091:
3089:
2962:
2961:
2959:
2863:
2783:
2702:defined as:
2693:
2685:
2499:plastic spin
2498:
2493:
2492:
2488:
2483:
2482:
2477:
2476:
2473:incompatible
2468:
2467:is called a
2466:
2378:
2270:
1863:
1862:
1857:
1856:
1854:
1793:
1792:
1782:
1774:
1755:
1626:, and hence
1517:
1509:
1446:
1443:
1231:
1167:
1093:
1025:
941:
836:
824:
749:
745:
670:
606:
597:is constant.
516:
511:
185:
52:hyperelastic
41:
24:
23:
2786:logarithmic
1785:B. A. Bilby
521:Hooke's law
3549:Categories
3476:(1): 1–6,
3369:References
517:Elasticity
3340:˙
3327:˙
3324:λ
3273:˙
3270:λ
3249:≤
3228:≥
3222:˙
3219:λ
3170:∂
3162:∂
3152:˙
3149:λ
3112:Flow rule
3051:μ
3000:∂
2992:∂
2930:⋅
2914:⋅
2833:
2753:⋅
2639:−
2447:−
2424:⋅
2412:˙
2355:−
2332:⋅
2317:⋅
2243:−
2220:⋅
2209:−
2186:⋅
2174:˙
2161:⋅
2138:−
2115:⋅
2103:˙
2073:−
2050:⋅
2042:−
2014:⋅
1997:˙
1987:⋅
1957:⋅
1945:˙
1922:−
1912:⋅
1906:˙
1828:⋅
1719:ε
1698:ε
1693:∂
1685:∂
1675:σ
1661:σ
1657:∂
1649:∂
1598:ε
1589:σ
1533:λ
1488:≥
1478:ε
1466:σ
1412:ε
1400:σ
1376:σ
1372:∂
1364:∂
1354:σ
1314:ε
1276:ε
1244:σ
1179:λ
1149:σ
1145:∂
1137:∂
1130:λ
1114:ε
1096:flow rule
1090:Flow rule
1063:≥
1059:ε
1048:σ
994:ε
976:ε
964:ε
907:σ
903:∂
895:∂
885:σ
837:Unloading
802:≥
795:σ
791:∂
783:∂
773:σ
705:ε
696:σ
636:σ
558:ε
540:σ
489:≥
477:ε
461:ε
450:σ
441:ε
434:σ
402:ε
386:ε
376:ε
344:σ
306:ε
276:σ
244:σ
237:σ
208:σ
201:σ
164:σ
137:σ
110:ε
83:ε
56:flow rule
37:flow rule
3534:See also
3512:Bibcode
3478:Bibcode
3450:: 1–179
3418:Bibcode
2966:is the
750:loading
746:Loading
33:plastic
3264:
3261:
3255:
3243:
3240:
3234:
3098:= det(
3090:where
2960:where
2594:
2591:
2585:
1855:where
1447:For a
1168:where
58:and a
44:strain
3466:(PDF)
2481:) of
1263:when
753:i.e.,
27:is a
2784:The
1767:the
1760:the
1518:The
1286:>
1182:>
914:<
850:<
347:<
315:>
279:>
3520:doi
3486:doi
3426:doi
3414:231
3104:),
3056:dev
2788:or
1391:and
50:or
3551::
3518:,
3508:46
3506:,
3484:,
3474:36
3472:,
3468:,
3446:,
3424:,
3412:,
2884::=
2830:ln
2815::=
2725::=
2501::
2402::=
1229:.
62:.
3522::
3514::
3488::
3480::
3448:5
3428::
3420::
3353:.
3349:0
3346:=
3337:f
3290:.
3286:0
3283:=
3280:f
3258:,
3252:0
3246:f
3237:,
3231:0
3182:.
3174:M
3165:f
3143:=
3138:p
3133:D
3106:μ
3101:F
3096:J
3092:W
3075:)
3070:e
3065:E
3060:(
3048:2
3045:+
3039:J
3036:d
3031:U
3028:d
3021:J
3018:=
3010:e
3005:E
2995:W
2986:=
2982:M
2963:S
2945:)
2940:e
2935:C
2926:S
2922:+
2918:S
2909:e
2904:C
2899:(
2893:2
2890:1
2880:M
2849:.
2843:e
2838:C
2824:2
2821:1
2810:e
2805:E
2769:.
2763:e
2758:F
2748:T
2744:)
2738:e
2733:F
2728:(
2720:e
2715:C
2671:.
2667:]
2662:T
2658:)
2652:p
2647:L
2642:(
2634:p
2629:L
2624:[
2618:2
2615:1
2609:=
2604:p
2599:W
2588:,
2582:]
2577:T
2573:)
2567:p
2562:L
2557:(
2554:+
2549:p
2544:L
2539:[
2533:2
2530:1
2524:=
2519:p
2514:D
2494:W
2484:L
2478:D
2450:1
2443:)
2437:p
2432:F
2427:(
2419:p
2409:F
2397:p
2392:L
2364:.
2358:1
2351:)
2345:e
2340:F
2335:(
2327:p
2322:L
2312:e
2307:F
2302:+
2297:e
2292:l
2287:=
2283:l
2252:.
2246:1
2239:)
2233:e
2228:F
2223:(
2217:]
2212:1
2205:)
2199:p
2194:F
2189:(
2181:p
2171:F
2164:[
2156:e
2151:F
2146:+
2141:1
2134:)
2128:e
2123:F
2118:(
2110:e
2100:F
2093:=
2082:]
2076:1
2069:)
2063:e
2058:F
2053:(
2045:1
2038:)
2032:p
2027:F
2022:(
2018:[
2010:)
2004:p
1994:F
1982:e
1977:F
1972:+
1967:p
1962:F
1952:e
1942:F
1934:(
1930:=
1925:1
1917:F
1903:F
1897:=
1889:l
1864:F
1858:F
1838:p
1833:F
1823:e
1818:F
1813:=
1809:F
1794:F
1736:.
1732:0
1729:=
1724:p
1714:d
1711::
1703:p
1688:f
1679:+
1671:d
1668::
1652:f
1643:=
1640:f
1637:d
1614:0
1611:=
1608:)
1603:p
1593:,
1585:(
1582:f
1562:0
1559:=
1556:f
1553:d
1530:d
1495:.
1491:0
1483:p
1473:d
1470::
1462:d
1429:.
1425:0
1422:=
1417:p
1407:d
1404::
1396:d
1386:0
1383:=
1367:f
1358::
1350:d
1327:0
1324:=
1319:e
1309:d
1289:0
1281:p
1271:d
1251:0
1248:=
1240:d
1213:f
1185:0
1176:d
1140:f
1127:d
1124:=
1119:p
1109:d
1070:.
1066:0
1055:d
1052::
1044:d
1005:.
999:p
989:d
986:+
981:e
971:d
968:=
960:d
921:.
917:0
898:f
889::
881:d
853:0
847:f
809:.
805:0
786:f
777::
769:d
725:.
721:0
718:=
715:)
710:p
700:,
692:(
689:f
650:.
646:0
643:=
640:)
632:(
629:f
609:(
583:D
554::
549:D
544:=
519:(
492:0
486:)
481:p
473:d
470:+
465:e
457:d
454:(
447:d
444:=
438:d
431:d
406:p
398:d
395:+
390:e
382:d
379:=
373:d
350:0
341:d
330:.
318:0
310:p
302:d
282:0
273:d
262:.
248:y
240:=
226:.
212:0
204:=
182:.
168:y
141:0
114:p
87:e
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
