2317:
25:
2773:– This theory assumes that the stored energy associated with elastic deformation at the point of yield is independent of the specific stress tensor. Thus yield occurs when the strain energy per unit volume is greater than the strain energy at the elastic limit in simple tension. For a 3-dimensional stress state this is given by:
2949:
2377:
A yield criterion often expressed as yield surface, or yield locus, is a hypothesis concerning the limit of elasticity under any combination of stresses. There are two interpretations of yield criterion: one is purely mathematical in taking a statistical approach while other models attempt to provide
3003:
When a metal is subjected to large plastic deformations the grain sizes and orientations change in the direction of deformation. As a result, the plastic yield behavior of the material shows directional dependency. Under such circumstances, the isotropic yield criteria such as the von Mises yield
972:
Microscopic material failure is defined in terms of crack initiation and propagation. Such methodologies are useful for gaining insight in the cracking of specimens and simple structures under well defined global load distributions. Microscopic failure considers the initiation and propagation of a
1915:. Linear elastic fracture mechanics predicts that a crack will extend when the stress intensity factor at the crack tip is greater than the fracture toughness of the material. Therefore, the critical applied stress can also be determined once the stress intensity factor at a crack tip is known.
1030:
Five general levels are considered, at which the meaning of deformation and failure is interpreted differently: the structural element scale, the macroscopic scale where macroscopic stress and strain are defined, the mesoscale which is represented by a typical void, the microscale and the atomic
927:
which separate "failed" states from "unfailed" states. A precise physical definition of a "failed" state is not easily quantified and several working definitions are in use in the engineering community. Quite often, phenomenological failure criteria of the same form are used to predict brittle
2488:(1850). Yield occurs when the largest principal stress exceeds the uniaxial tensile yield strength. Although this criterion allows for a quick and easy comparison with experimental data it is rarely suitable for design purposes. This theory gives good predictions for brittle materials.
3094:
of a ductile material. Several models for predicting the ultimate strength have been used by the engineering community with varying levels of success. For metals, such failure criteria are usually expressed in terms of a combination of porosity and strain to failure or in terms of a
952:. In structural problems, where the structural response may be beyond the initiation of nonlinear material behaviour, material failure is of profound importance for the determination of the integrity of the structure. On the other hand, due to the lack of globally accepted
2778:
1348:
Numerous other phenomenological failure criteria can be found in the engineering literature. The degree of success of these criteria in predicting failure has been limited. Some popular failure criteria for various type of materials are:
1340:
2625:
2766:
1260:
985:, microvoids nucleate and grow until a local plastic neck or fracture of the intervoid matrix occurs, which causes the coalescence of neighbouring voids. Such a model, proposed by Gurson and extended by Tvergaard and
3004:
criterion are unable to predict the yield behavior accurately. Several anisotropic yield criteria have been developed to deal with such situations. Some of the more popular anisotropic yield criteria are:
2472:
The following represent the most common yield criterion as applied to an isotropic material (uniform properties in all directions). Other equations have been proposed or are used in specialist situations.
2278:
1550:
2036:
1739:
973:
crack. Failure criteria in this case are related to microscopic fracture. Some of the most popular failure models in this area are the micromechanical failure models, which combine the advantages of
1009:
Macroscopic material failure is defined in terms of load carrying capacity or energy storage capacity, equivalently. Li presents a classification of macroscopic failure criteria in four categories:
2530:
182:
1031:
scale. The material behavior at one level is considered as a collective of its behavior at a sub-level. An efficient deformation and failure model should be consistent at every level.
1666:
3048:
3044:
920:, loading rate) most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile.
997:. Both models form a modification of the von Mises yield potential by introducing a scalar damage quantity, which represents the void volume fraction of cavities, the porosity
2944:{\displaystyle \sigma _{1}^{2}+\sigma _{2}^{2}+\sigma _{3}^{2}-2\nu \left(\sigma _{1}\sigma _{2}+\sigma _{2}\sigma _{3}+\sigma _{1}\sigma _{3}\right)\leq \sigma _{y}^{2}.\,\!}
2467:
2438:
2409:
1898:
2980:) strain energy. It is proposed that yield occurs when the distortion component exceeds that at the yield point for a simple tensile test. This theory is also known as the
1860:
2174:
1821:
2691:
1769:
1190:
1163:
1136:
1109:
2663:
1593:
1485:
923:
In mathematical terms, failure theory is expressed in the form of various failure criteria which are valid for specific materials. Failure criteria are functions in
1278:
1937:
approach has proved quite useful for such situations. The strain energy release rate for a mode I crack which runs through the thickness of a plate is defined as
3043:. Models for the evolution of the yield surface with increasing strain, temperature, and strain rate are used in conjunction with the above failure criteria for
3266:
2142:
1636:
944:
is the loss of load carrying capacity of a material unit. This definition introduces to the fact that material failure can be examined in different scales, from
2301:
2119:
2099:
2079:
2059:
1789:
1613:
1573:
2546:
3096:
2541:
reaches the strain corresponding to the yield point during a simple tensile test. In terms of the principal stresses this is determined by the equation:
3500:
860:
2303:
is the Young's modulus. If an initial crack size is known, then a critical stress can be determined using the strain energy release rate criterion.
1111:
in a material element exceeds the uniaxial tensile strength of the material. Alternatively, the material will fail if the minimum principal stress
2698:
1198:
731:
3078:
3259:
2144:
is the crack length for plane cracks. The crack is expected to propagate when the strain energy release rate exceeds a critical value
89:
1272:
has a similar form except that the principal strains are compared with experimentally determined uniaxial strains at failure, i.e.,
61:
2338:
2197:
42:
1493:
3252:
3084:
853:
68:
1943:
1686:
964:
Material failure can be distinguished in two broader categories depending on the scale in which the material is examined:
75:
3556:
3158:
3072:
2382:
qualities they can be described on the basis of three principal directions, in the case of stress these are denoted by
1423:
3066:
2493:
2364:
1138:
is less than the uniaxial compressive strength of the material. If the uniaxial tensile strength of the material is
846:
724:
108:
2346:
57:
956:
criteria, the determination of the structure's damage, due to material failure, is still under intensive research.
1791:
is a dimensionless factor that depends on the geometry, material properties, and loading condition. The quantity
3527:
138:
4009:
3940:
3628:
2342:
1388:
697:
46:
3989:
3373:
3060:
2991:
corresponding to these criteria have a range of forms. However, most isotropic yield criteria correspond to
1437:
1395:
1345:
The maximum principal stress and strain criteria continue to be widely used in spite of severe shortcomings.
398:
235:
1641:
1459:
is to estimate the amount of energy needed to grow a preexisting crack in a brittle material. The earliest
3575:
3483:
1430:
717:
438:
324:
4014:
3984:
3696:
3407:
3341:
809:
393:
302:
185:
4004:
3163:
2538:
1934:
924:
799:
3212:
Griffiths, A.A. 1920. The phenomena of rupture and flow in solids. Phil.Trans.Roy.Soc.Lond. A221, 163.
1929:
The linear elastic fracture mechanics method is difficult to apply for anisotropic materials (such as
3424:
2981:
2957:
2443:
2414:
2385:
1416:
1367:
814:
309:
82:
1865:
3994:
3855:
3850:
3733:
3701:
3616:
3356:
3346:
2327:
1830:
604:
599:
388:
381:
214:
2147:
1794:
3915:
3512:
3402:
3118:
3040:
3018:
2668:
2331:
1912:
1824:
1747:
1402:
1168:
1141:
1114:
1087:
819:
667:
662:
35:
2964:. – This theory proposes that the total strain energy can be separated into two components: the
3495:
3368:
2637:
1360:
789:
219:
1335:{\displaystyle \varepsilon _{c}<\varepsilon _{3}<\varepsilon _{1}<\varepsilon _{t}\,}
3900:
3478:
3224:
3143:
2647:
1578:
1470:
642:
260:
3963:
3637:
3541:
3536:
3507:
3473:
3453:
3438:
3419:
3390:
3331:
3133:
1354:
1074:
480:
297:
277:
265:
209:
3211:
8:
3935:
3787:
3726:
3706:
3522:
3468:
3351:
3310:
3276:
3123:
3013:
3008:
2620:{\displaystyle \sigma _{1}-\nu \left(\sigma _{2}+\sigma _{3}\right)\leq \sigma _{y}.\,\!}
1924:
1409:
1374:
1265:
Note that the convention that tension is positive has been used in the above expression.
1058:
982:
974:
909:
834:
804:
769:
682:
530:
423:
129:
2124:
1618:
3999:
3606:
3601:
3385:
3138:
3108:
2977:
2286:
2184:
2104:
2084:
2064:
2044:
1930:
1911:
The state of stress around cracks of various shapes can be expressed in terms of their
1905:
1901:
1774:
1673:
1598:
1558:
1464:
1460:
1456:
1450:
1070:
1047:
978:
917:
794:
774:
702:
336:
292:
287:
2378:
a justification based on established physical principles. Since stress and strain are
3654:
3591:
3570:
3565:
3490:
3458:
3182:
Besson J., Steglich D., Brocks W. (2003), Modelling of plain strain ductile rupture,
2101:
is the displacement at the point of application of the load due to crack growth, and
937:
877:
764:
319:
270:
1405:, an empirical failure criterion that is used for orthotropic materials such as wood
3890:
3799:
3792:
3782:
3758:
3623:
3153:
3148:
3091:
1082:
990:
989:, is known as GTN. Another approach, proposed by Rousselier, is based on continuum
885:
657:
632:
545:
520:
515:
470:
3905:
3840:
3817:
3753:
3738:
3721:
3671:
3659:
3546:
3517:
3434:
3412:
3395:
3361:
3295:
3290:
3052:
2992:
2485:
893:
881:
647:
571:
535:
485:
416:
405:
350:
252:
3244:
16:
Science of predicting if, when, and how a given material will fail under loading
3910:
3777:
3691:
3649:
3463:
3090:
There is another important aspect to ductile materials - the prediction of the
1381:
994:
986:
652:
510:
475:
376:
282:
1463:
approach for unstable crack growth is
Griffiths' theory. When applied to the
3978:
3958:
3925:
3822:
3686:
3551:
3128:
3036:
3030:
2988:
2761:{\displaystyle \tau ={\frac {\sigma _{1}-\sigma _{3}}{2}}\leq \tau _{y}.\,\!}
1069:
The failure criteria that were developed for brittle solids were the maximum
824:
692:
525:
3039:
of a ductile material usually changes as the material experiences increased
3930:
3920:
3895:
2641:
2631:
2188:
1677:
677:
672:
637:
369:
1255:{\displaystyle \sigma _{c}<\sigma _{3}<\sigma _{1}<\sigma _{t}\,}
3845:
2969:
1467:
opening of a crack, Griffiths' theory predicts that the critical stress (
1039:
Failure of brittle materials can be determined using several approaches:
1017:
949:
945:
913:
687:
590:
1933:) or for situations where the loading or the geometry are complex. The
3860:
3772:
3611:
3305:
609:
505:
3882:
3872:
3867:
3809:
3716:
3300:
759:
581:
576:
410:
2316:
24:
3832:
3743:
3711:
3336:
3113:
953:
901:
889:
784:
754:
560:
465:
445:
431:
3644:
3584:
905:
897:
779:
314:
3748:
2379:
455:
3950:
3679:
3446:
2273:{\displaystyle G_{\rm {Ic}}={\cfrac {1}{E}}~K_{\rm {Ic}}^{2}}
359:
2306:
1545:{\displaystyle \sigma ={\sqrt {\cfrac {2E\gamma }{\pi a}}}}
829:
3195:
Li, Q.M. (2001), Strain energy density failure criterion,
3765:
1192:, then the safe region for the material is assumed to be
495:
2537:– by St.Venant. Yield occurs when the maximum principal
2031:{\displaystyle G_{I}={\cfrac {P}{2t}}~{\cfrac {du}{da}}}
1734:{\displaystyle K_{\rm {Ic}}=Y\sigma _{c}{\sqrt {\pi a}}}
2644:. This assumes that yield occurs when the shear stress
2234:
2222:
2012:
1997:
1975:
1963:
1525:
1507:
896:. The failure of a material is usually classified into
2237:
2225:
2015:
2000:
1978:
1966:
1827:
and is determined experimentally. Similar quantities
1595:
is the surface energy per unit area of the crack, and
1528:
1510:
2781:
2701:
2671:
2650:
2549:
2496:
2446:
2417:
2388:
2289:
2200:
2150:
2127:
2107:
2087:
2067:
2047:
1946:
1868:
1833:
1797:
1777:
1750:
1689:
1644:
1621:
1601:
1581:
1561:
1496:
1473:
1281:
1201:
1171:
1144:
1117:
1090:
141:
1638:
is the crack length for plane cracks. The quantity
1444:
1064:
1034:
981:. Such models are based on the concept that during
49:. Unsourced material may be challenged and removed.
2943:
2760:
2685:
2657:
2619:
2524:
2461:
2432:
2403:
2295:
2272:
2168:
2136:
2113:
2093:
2073:
2053:
2030:
1892:
1854:
1815:
1783:
1763:
1733:
1660:
1630:
1607:
1587:
1567:
1544:
1479:
1334:
1254:
1184:
1157:
1130:
1103:
176:
3274:
2940:
2757:
2682:
2654:
2616:
2521:
2458:
2429:
2400:
1668:is postulated as a material parameter called the
3976:
2187:and the critical strain energy release rate for
1771:is a critical value of the far field stress and
1370:or maximum elastic distortional energy criterion
2525:{\displaystyle \sigma _{1}\leq \sigma _{y}\,\!}
1081:assumes that a material fails when the maximum
3197:International Journal of Solids and Structures
1426:for high-rate deformations of isotropic solids
3260:
2998:
854:
725:
1487:) needed to propagate the crack is given by
959:
2476:
2345:. Unsourced material may be challenged and
177:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
3267:
3253:
861:
847:
732:
718:
2939:
2756:
2615:
2520:
2457:
2428:
2399:
2365:Learn how and when to remove this message
2307:Ductile material failure (yield) criteria
1331:
1251:
1165:and the uniaxial compressive strength is
109:Learn how and when to remove this message
3222:
1575:is the Young's modulus of the material,
912:). Depending on the conditions (such as
2121:is the crack length for edge cracks or
1918:
1615:is the crack length for edge cracks or
3977:
1661:{\displaystyle \sigma {\sqrt {\pi a}}}
1004:
967:
3248:
2343:adding citations to reliable sources
2310:
1353:criteria based on invariants of the
47:adding citations to reliable sources
18:
3184:International Journal of Plasticity
2178:critical strain energy release rate
931:
13:
3159:Size effect on structural strength
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2256:
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2207:
2160:
2157:
1884:
1881:
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1875:
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1843:
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1807:
1804:
1699:
1696:
1052:Elastic-plastic fracture mechanics
1016:Energy type failure (S-criterion,
14:
4026:
3079:Mechanical threshold stress model
2665:exceeds the shear yield strength
1457:linear elastic fracture mechanics
1445:Linear elastic fracture mechanics
1065:Phenomenological failure criteria
1043:Phenomenological failure criteria
1035:Brittle material failure criteria
876:is an interdisciplinary field of
3024:
3014:Generalized Hill yield criterion
3009:Hill's quadratic yield criterion
2954:Maximum distortion energy theory
2315:
1389:Bresler-Pister failure criterion
1382:Drucker-Prager failure criterion
23:
2535:Maximum principal strain theory
2482:Maximum principal stress theory
2462:{\displaystyle \sigma _{3}\,\!}
2433:{\displaystyle \sigma _{2}\,\!}
2404:{\displaystyle \sigma _{1}\,\!}
2081:is the thickness of the plate,
1396:Willam-Warnke failure criterion
34:needs additional citations for
3216:
3205:
3189:
3176:
2962:octahedral shear stress theory
1893:{\displaystyle K_{\rm {IIIc}}}
1424:Johnson–Holmquist damage model
1377:for cohesive-frictional solids
1375:Mohr-Coulomb failure criterion
1361:Tresca or maximum shear stress
1:
3169:
2640:, after the French scientist
1855:{\displaystyle K_{\rm {IIc}}}
1384:for pressure-dependent solids
2169:{\displaystyle G_{\rm {Ic}}}
1816:{\displaystyle K_{\rm {Ic}}}
1431:Hoek-Brown failure criterion
928:failure and ductile yields.
7:
3225:"What is von Mises Stress?"
3102:
3085:Preston-Tonks-Wallace model
2686:{\displaystyle \tau _{y}\!}
1764:{\displaystyle \sigma _{c}}
1185:{\displaystyle \sigma _{c}}
1158:{\displaystyle \sigma _{t}}
1131:{\displaystyle \sigma _{3}}
1104:{\displaystyle \sigma _{1}}
810:Metal-induced embrittlement
10:
4031:
3164:Concrete fracture analysis
3028:
2999:Anisotropic yield criteria
2771:Total strain energy theory
1935:strain energy release rate
1922:
1448:
1419:for anisotropic composites
800:Liquid metal embrittlement
3949:
3881:
3831:
3808:
3670:
3433:
3319:
3283:
3092:ultimate failure strength
3055:. Some such models are:
2982:von Mises yield criterion
2958:von Mises yield criterion
1417:Tsai-Wu failure criterion
960:Types of material failure
892:fail under the action of
815:Stress corrosion cracking
58:"Material failure theory"
3617:Compact tension specimen
3337:Conservation of momentum
2972:) strain energy and the
2477:Isotropic yield criteria
1913:stress intensity factors
1270:maximum strain criterion
1079:maximum stress criterion
1013:Stress or strain failure
747:Mechanical failure modes
236:Clausius–Duhem (entropy)
186:Fick's laws of diffusion
3697:Navier–Stokes equations
3597:Material failure theory
3585:Material failure theory
3223:sdcadmin (2022-05-05).
3119:Stress intensity factor
3073:Zerilli-Armstrong model
3019:Hosford yield criterion
2658:{\displaystyle \tau \!}
1825:stress intensity factor
1588:{\displaystyle \gamma }
1480:{\displaystyle \sigma }
1438:Cam-Clay failure theory
874:Material failure theory
820:Sulfide stress cracking
394:Navier–Stokes equations
332:Material failure theory
3369:Conservation of energy
3067:Steinberg-Guinan model
2960:) also referred to as
2945:
2762:
2687:
2659:
2638:Tresca yield criterion
2621:
2526:
2463:
2434:
2405:
2297:
2274:
2170:
2138:
2115:
2095:
2075:
2055:
2032:
1900:can be determined for
1894:
1856:
1817:
1785:
1765:
1735:
1662:
1632:
1609:
1589:
1569:
1546:
1481:
1455:The approach taken in
1412:for anisotropic solids
1336:
1256:
1186:
1159:
1132:
1105:
925:stress or strain space
886:predict the conditions
790:Hydrogen embrittlement
178:
4010:Materials degradation
3734:Archimedes' principle
3702:Bernoulli's principle
3144:Strength of materials
2946:
2763:
2688:
2660:
2622:
2527:
2464:
2435:
2406:
2298:
2275:
2171:
2139:
2116:
2096:
2076:
2061:is the applied load,
2056:
2033:
1895:
1857:
1818:
1786:
1766:
1736:
1663:
1633:
1610:
1590:
1570:
1547:
1482:
1337:
1257:
1187:
1160:
1133:
1106:
1059:Cohesive zone methods
389:Bernoulli's principle
382:Archimedes' principle
179:
3990:Plasticity (physics)
3964:William Prager Medal
3542:Rock mass plasticity
3439:Structural mechanics
3332:Conservation of mass
3320:Laws and Definitions
3134:Plasticity (physics)
2779:
2699:
2669:
2648:
2636:– Also known as the
2547:
2494:
2444:
2415:
2386:
2339:improve this section
2287:
2198:
2148:
2125:
2105:
2085:
2065:
2045:
1944:
1919:Energy-based methods
1908:loading conditions.
1866:
1831:
1795:
1775:
1748:
1687:
1642:
1619:
1599:
1579:
1559:
1494:
1471:
1355:Cauchy stress tensor
1279:
1199:
1169:
1142:
1115:
1088:
1055:Energy-based methods
481:Cohesion (chemistry)
303:Infinitesimal strain
139:
43:improve this article
3707:Poiseuille equation
3484:Membrane elasticity
3469:Transverse isotropy
3311:Rigid body dynamics
3277:continuum mechanics
3124:Yield (engineering)
3049:kinematic hardening
3045:isotropic hardening
2935:
2832:
2814:
2796:
2269:
2236:
2224:
2014:
1999:
1977:
1965:
1925:Energy release rate
1527:
1509:
1410:Hill yield criteria
1403:Hankinson criterion
1005:Macroscopic failure
983:plastic deformation
975:continuum mechanics
968:Microscopic failure
805:Mechanical overload
399:Poiseuille equation
130:Continuum mechanics
124:Part of a series on
4015:Fracture mechanics
3985:Mechanical failure
3856:Electrorheological
3851:Magnetorheological
3607:Fracture mechanics
3374:Entropy inequality
3139:Structural failure
3109:Fracture mechanics
3061:Johnson-Cook model
2941:
2921:
2818:
2800:
2782:
2758:
2683:
2655:
2617:
2522:
2459:
2430:
2401:
2293:
2270:
2250:
2243:
2231:
2185:fracture toughness
2166:
2137:{\displaystyle 2a}
2134:
2111:
2091:
2071:
2051:
2028:
2024:
2009:
1987:
1972:
1890:
1852:
1823:is related to the
1813:
1781:
1761:
1731:
1674:fracture toughness
1670:fracture toughness
1658:
1631:{\displaystyle 2a}
1628:
1605:
1585:
1565:
1542:
1537:
1522:
1477:
1461:fracture mechanics
1451:Fracture mechanics
1332:
1252:
1182:
1155:
1128:
1101:
1048:fracture mechanics
979:fracture mechanics
888:under which solid
884:which attempts to
605:Magnetorheological
600:Electrorheological
337:Fracture mechanics
174:
4005:Materials science
3972:
3971:
3655:Bending of plates
3629:Johnson-Holmquist
3592:Drucker stability
3566:Contact mechanics
3513:Cauchy elasticity
3491:Equation of state
2738:
2375:
2374:
2367:
2296:{\displaystyle E}
2249:
2245:
2235:
2223:
2114:{\displaystyle a}
2094:{\displaystyle u}
2074:{\displaystyle t}
2054:{\displaystyle P}
2026:
2013:
1998:
1993:
1989:
1976:
1964:
1784:{\displaystyle Y}
1729:
1656:
1608:{\displaystyle a}
1568:{\displaystyle E}
1540:
1539:
1526:
1508:
1363:failure criterion
1026:Empirical failure
938:materials science
878:materials science
871:
870:
765:Corrosion fatigue
742:
741:
617:
616:
551:
550:
320:Contact mechanics
243:
242:
172:
119:
118:
111:
93:
4022:
3793:Combined gas law
3788:Gay-Lussac's law
3759:Capillary action
3624:Damage mechanics
3269:
3262:
3255:
3246:
3245:
3239:
3238:
3236:
3235:
3220:
3214:
3209:
3203:
3202:, pp. 6997–7013.
3193:
3187:
3180:
3154:Damage mechanics
3149:Ultimate failure
2995:yield surfaces.
2950:
2948:
2947:
2942:
2934:
2929:
2917:
2913:
2912:
2911:
2902:
2901:
2889:
2888:
2879:
2878:
2866:
2865:
2856:
2855:
2831:
2826:
2813:
2808:
2795:
2790:
2767:
2765:
2764:
2759:
2752:
2751:
2739:
2734:
2733:
2732:
2720:
2719:
2709:
2692:
2690:
2689:
2684:
2681:
2680:
2664:
2662:
2661:
2656:
2626:
2624:
2623:
2618:
2611:
2610:
2598:
2594:
2593:
2592:
2580:
2579:
2559:
2558:
2531:
2529:
2528:
2523:
2519:
2518:
2506:
2505:
2468:
2466:
2465:
2460:
2456:
2455:
2439:
2437:
2436:
2431:
2427:
2426:
2410:
2408:
2407:
2402:
2398:
2397:
2370:
2363:
2359:
2356:
2350:
2319:
2311:
2302:
2300:
2299:
2294:
2279:
2277:
2276:
2271:
2268:
2263:
2262:
2247:
2246:
2244:
2242:
2232:
2230:
2220:
2215:
2214:
2213:
2175:
2173:
2172:
2167:
2165:
2164:
2163:
2143:
2141:
2140:
2135:
2120:
2118:
2117:
2112:
2100:
2098:
2097:
2092:
2080:
2078:
2077:
2072:
2060:
2058:
2057:
2052:
2037:
2035:
2034:
2029:
2027:
2025:
2023:
2010:
2008:
1995:
1991:
1990:
1988:
1986:
1973:
1971:
1961:
1956:
1955:
1899:
1897:
1896:
1891:
1889:
1888:
1887:
1861:
1859:
1858:
1853:
1851:
1850:
1849:
1822:
1820:
1819:
1814:
1812:
1811:
1810:
1790:
1788:
1787:
1782:
1770:
1768:
1767:
1762:
1760:
1759:
1740:
1738:
1737:
1732:
1730:
1722:
1720:
1719:
1704:
1703:
1702:
1667:
1665:
1664:
1659:
1657:
1649:
1637:
1635:
1634:
1629:
1614:
1612:
1611:
1606:
1594:
1592:
1591:
1586:
1574:
1572:
1571:
1566:
1551:
1549:
1548:
1543:
1541:
1538:
1536:
1523:
1521:
1505:
1504:
1486:
1484:
1483:
1478:
1341:
1339:
1338:
1333:
1330:
1329:
1317:
1316:
1304:
1303:
1291:
1290:
1261:
1259:
1258:
1253:
1250:
1249:
1237:
1236:
1224:
1223:
1211:
1210:
1191:
1189:
1188:
1183:
1181:
1180:
1164:
1162:
1161:
1156:
1154:
1153:
1137:
1135:
1134:
1129:
1127:
1126:
1110:
1108:
1107:
1102:
1100:
1099:
1083:principal stress
991:damage mechanics
942:material failure
932:Material failure
863:
856:
849:
744:
743:
734:
727:
720:
566:
565:
531:Gay-Lussac's law
521:Combined gas law
471:Capillary action
356:
355:
199:
198:
183:
181:
180:
175:
173:
171:
163:
155:
121:
120:
114:
107:
103:
100:
94:
92:
51:
27:
19:
4030:
4029:
4025:
4024:
4023:
4021:
4020:
4019:
3995:Solid mechanics
3975:
3974:
3973:
3968:
3945:
3877:
3841:Viscoelasticity
3827:
3818:Acoustic theory
3804:
3754:Surface tension
3672:Fluid mechanics
3666:
3660:Sandwich theory
3552:Yield criterion
3547:Viscoplasticity
3518:Viscoelasticity
3479:hyperelasticity
3429:
3413:Antiplane shear
3396:Stress measures
3315:
3296:Fluid mechanics
3291:Solid mechanics
3279:
3273:
3243:
3242:
3233:
3231:
3221:
3217:
3210:
3206:
3194:
3190:
3181:
3177:
3172:
3105:
3053:viscoplasticity
3033:
3027:
3001:
2976:(distortion or
2951:
2930:
2925:
2907:
2903:
2897:
2893:
2884:
2880:
2874:
2870:
2861:
2857:
2851:
2847:
2846:
2842:
2827:
2822:
2809:
2804:
2791:
2786:
2780:
2777:
2776:
2768:
2747:
2743:
2728:
2724:
2715:
2711:
2710:
2708:
2700:
2697:
2696:
2676:
2672:
2670:
2667:
2666:
2649:
2646:
2645:
2627:
2606:
2602:
2588:
2584:
2575:
2571:
2570:
2566:
2554:
2550:
2548:
2545:
2544:
2532:
2514:
2510:
2501:
2497:
2495:
2492:
2491:
2486:William Rankine
2479:
2451:
2447:
2445:
2442:
2441:
2422:
2418:
2416:
2413:
2412:
2393:
2389:
2387:
2384:
2383:
2371:
2360:
2354:
2351:
2336:
2320:
2309:
2288:
2285:
2284:
2264:
2255:
2254:
2238:
2233:
2226:
2221:
2219:
2206:
2205:
2201:
2199:
2196:
2195:
2191:are related by
2156:
2155:
2151:
2149:
2146:
2145:
2126:
2123:
2122:
2106:
2103:
2102:
2086:
2083:
2082:
2066:
2063:
2062:
2046:
2043:
2042:
2016:
2011:
2001:
1996:
1994:
1979:
1974:
1967:
1962:
1960:
1951:
1947:
1945:
1942:
1941:
1927:
1921:
1874:
1873:
1869:
1867:
1864:
1863:
1839:
1838:
1834:
1832:
1829:
1828:
1803:
1802:
1798:
1796:
1793:
1792:
1776:
1773:
1772:
1755:
1751:
1749:
1746:
1745:
1721:
1715:
1711:
1695:
1694:
1690:
1688:
1685:
1684:
1648:
1643:
1640:
1639:
1620:
1617:
1616:
1600:
1597:
1596:
1580:
1577:
1576:
1560:
1557:
1556:
1529:
1524:
1511:
1506:
1503:
1495:
1492:
1491:
1472:
1469:
1468:
1453:
1447:
1433:for rock masses
1325:
1321:
1312:
1308:
1299:
1295:
1286:
1282:
1280:
1277:
1276:
1245:
1241:
1232:
1228:
1219:
1215:
1206:
1202:
1200:
1197:
1196:
1176:
1172:
1170:
1167:
1166:
1149:
1145:
1143:
1140:
1139:
1122:
1118:
1116:
1113:
1112:
1095:
1091:
1089:
1086:
1085:
1077:criteria. The
1067:
1046:Linear elastic
1037:
1007:
970:
962:
934:
882:solid mechanics
867:
738:
709:
708:
707:
627:
619:
618:
572:Viscoelasticity
563:
553:
552:
540:
490:
486:Surface tension
450:
353:
351:Fluid mechanics
343:
342:
341:
255:
253:Solid mechanics
245:
244:
196:
188:
164:
156:
154:
140:
137:
136:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
4028:
4018:
4017:
4012:
4007:
4002:
3997:
3992:
3987:
3970:
3969:
3967:
3966:
3961:
3955:
3953:
3947:
3946:
3944:
3943:
3938:
3933:
3928:
3923:
3918:
3913:
3908:
3903:
3898:
3893:
3887:
3885:
3879:
3878:
3876:
3875:
3870:
3865:
3864:
3863:
3858:
3853:
3843:
3837:
3835:
3829:
3828:
3826:
3825:
3820:
3814:
3812:
3806:
3805:
3803:
3802:
3796:
3795:
3790:
3785:
3780:
3775:
3769:
3768:
3762:
3761:
3756:
3751:
3746:
3741:
3736:
3731:
3730:
3729:
3724:
3714:
3709:
3704:
3699:
3694:
3692:Fluid dynamics
3689:
3683:
3682:
3676:
3674:
3668:
3667:
3665:
3664:
3663:
3662:
3657:
3652:
3650:Bending moment
3641:
3640:
3634:
3633:
3632:
3631:
3621:
3620:
3619:
3614:
3604:
3599:
3594:
3588:
3587:
3581:
3580:
3579:
3578:
3573:
3563:
3562:
3561:
3560:
3559:
3557:Bresler-Pister
3549:
3544:
3534:
3533:
3532:
3531:
3530:
3528:Concrete creep
3525:
3515:
3510:
3508:hypoelasticity
3505:
3504:
3503:
3498:
3488:
3487:
3486:
3476:
3471:
3466:
3461:
3450:
3449:
3443:
3441:
3431:
3430:
3428:
3427:
3422:
3417:
3416:
3415:
3405:
3400:
3399:
3398:
3393:
3382:
3381:
3377:
3376:
3371:
3366:
3365:
3364:
3359:
3354:
3349:
3344:
3334:
3328:
3327:
3323:
3321:
3317:
3316:
3314:
3313:
3308:
3303:
3298:
3293:
3287:
3285:
3281:
3280:
3272:
3271:
3264:
3257:
3249:
3241:
3240:
3215:
3204:
3188:
3174:
3173:
3171:
3168:
3167:
3166:
3161:
3156:
3151:
3146:
3141:
3136:
3131:
3126:
3121:
3116:
3111:
3104:
3101:
3088:
3087:
3081:
3075:
3069:
3063:
3029:Main article:
3026:
3023:
3022:
3021:
3016:
3011:
3000:
2997:
2989:yield surfaces
2938:
2933:
2928:
2924:
2920:
2916:
2910:
2906:
2900:
2896:
2892:
2887:
2883:
2877:
2873:
2869:
2864:
2860:
2854:
2850:
2845:
2841:
2838:
2835:
2830:
2825:
2821:
2817:
2812:
2807:
2803:
2799:
2794:
2789:
2785:
2775:
2755:
2750:
2746:
2742:
2737:
2731:
2727:
2723:
2718:
2714:
2707:
2704:
2695:
2679:
2675:
2653:
2614:
2609:
2605:
2601:
2597:
2591:
2587:
2583:
2578:
2574:
2569:
2565:
2562:
2557:
2553:
2543:
2517:
2513:
2509:
2504:
2500:
2490:
2478:
2475:
2454:
2450:
2425:
2421:
2396:
2392:
2373:
2372:
2323:
2321:
2314:
2308:
2305:
2292:
2281:
2280:
2267:
2261:
2258:
2253:
2241:
2229:
2218:
2212:
2209:
2204:
2162:
2159:
2154:
2133:
2130:
2110:
2090:
2070:
2050:
2039:
2038:
2022:
2019:
2007:
2004:
1985:
1982:
1970:
1959:
1954:
1950:
1923:Main article:
1920:
1917:
1886:
1883:
1880:
1877:
1872:
1848:
1845:
1842:
1837:
1809:
1806:
1801:
1780:
1758:
1754:
1742:
1741:
1728:
1725:
1718:
1714:
1710:
1707:
1701:
1698:
1693:
1680:is defined as
1672:. The mode I
1655:
1652:
1647:
1627:
1624:
1604:
1584:
1564:
1553:
1552:
1535:
1532:
1520:
1517:
1514:
1502:
1499:
1476:
1449:Main article:
1446:
1443:
1442:
1441:
1434:
1427:
1420:
1413:
1406:
1399:
1392:
1385:
1378:
1371:
1364:
1357:
1343:
1342:
1328:
1324:
1320:
1315:
1311:
1307:
1302:
1298:
1294:
1289:
1285:
1263:
1262:
1248:
1244:
1240:
1235:
1231:
1227:
1222:
1218:
1214:
1209:
1205:
1179:
1175:
1152:
1148:
1125:
1121:
1098:
1094:
1066:
1063:
1062:
1061:
1056:
1053:
1050:
1044:
1036:
1033:
1028:
1027:
1024:
1023:Damage failure
1021:
1014:
1006:
1003:
995:thermodynamics
977:and classical
969:
966:
961:
958:
933:
930:
894:external loads
869:
868:
866:
865:
858:
851:
843:
840:
839:
838:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
749:
748:
740:
739:
737:
736:
729:
722:
714:
711:
710:
706:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
629:
628:
625:
624:
621:
620:
615:
614:
613:
612:
607:
602:
594:
593:
587:
586:
585:
584:
579:
574:
564:
559:
558:
555:
554:
549:
548:
542:
541:
539:
538:
533:
528:
523:
518:
513:
508:
502:
499:
498:
492:
491:
489:
488:
483:
478:
476:Chromatography
473:
468:
462:
459:
458:
452:
451:
449:
448:
429:
428:
427:
408:
396:
391:
379:
366:
363:
362:
354:
349:
348:
345:
344:
340:
339:
334:
329:
328:
327:
317:
312:
307:
306:
305:
300:
290:
285:
280:
275:
274:
273:
263:
257:
256:
251:
250:
247:
246:
241:
240:
239:
238:
230:
229:
225:
224:
223:
222:
217:
212:
204:
203:
197:
194:
193:
190:
189:
184:
170:
167:
162:
159:
153:
150:
147:
144:
133:
132:
126:
125:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
4027:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3986:
3983:
3982:
3980:
3965:
3962:
3960:
3959:Eringen Medal
3957:
3956:
3954:
3952:
3948:
3942:
3939:
3937:
3934:
3932:
3929:
3927:
3924:
3922:
3919:
3917:
3914:
3912:
3909:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3888:
3886:
3884:
3880:
3874:
3871:
3869:
3866:
3862:
3859:
3857:
3854:
3852:
3849:
3848:
3847:
3844:
3842:
3839:
3838:
3836:
3834:
3830:
3824:
3823:Aeroacoustics
3821:
3819:
3816:
3815:
3813:
3811:
3807:
3801:
3798:
3797:
3794:
3791:
3789:
3786:
3784:
3783:Charles's law
3781:
3779:
3776:
3774:
3771:
3770:
3767:
3764:
3763:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3728:
3727:Non-Newtonian
3725:
3723:
3720:
3719:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3687:Fluid statics
3685:
3684:
3681:
3678:
3677:
3675:
3673:
3669:
3661:
3658:
3656:
3653:
3651:
3648:
3647:
3646:
3643:
3642:
3639:
3636:
3635:
3630:
3627:
3626:
3625:
3622:
3618:
3615:
3613:
3610:
3609:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3589:
3586:
3583:
3582:
3577:
3574:
3572:
3569:
3568:
3567:
3564:
3558:
3555:
3554:
3553:
3550:
3548:
3545:
3543:
3540:
3539:
3538:
3535:
3529:
3526:
3524:
3521:
3520:
3519:
3516:
3514:
3511:
3509:
3506:
3502:
3499:
3497:
3494:
3493:
3492:
3489:
3485:
3482:
3481:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3456:
3455:
3452:
3451:
3448:
3445:
3444:
3442:
3440:
3436:
3432:
3426:
3425:Compatibility
3423:
3421:
3418:
3414:
3411:
3410:
3409:
3406:
3404:
3401:
3397:
3394:
3392:
3391:Cauchy stress
3389:
3388:
3387:
3384:
3383:
3379:
3378:
3375:
3372:
3370:
3367:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3342:Navier-Stokes
3340:
3339:
3338:
3335:
3333:
3330:
3329:
3325:
3324:
3322:
3318:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3288:
3286:
3282:
3278:
3270:
3265:
3263:
3258:
3256:
3251:
3250:
3247:
3230:
3226:
3219:
3213:
3208:
3201:
3198:
3192:
3185:
3179:
3175:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3129:Yield surface
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3106:
3100:
3098:
3093:
3086:
3082:
3080:
3076:
3074:
3070:
3068:
3064:
3062:
3058:
3057:
3056:
3054:
3050:
3046:
3042:
3038:
3037:yield surface
3032:
3031:Yield surface
3025:Yield surface
3020:
3017:
3015:
3012:
3010:
3007:
3006:
3005:
2996:
2994:
2990:
2985:
2983:
2979:
2975:
2971:
2967:
2963:
2959:
2955:
2936:
2931:
2926:
2922:
2918:
2914:
2908:
2904:
2898:
2894:
2890:
2885:
2881:
2875:
2871:
2867:
2862:
2858:
2852:
2848:
2843:
2839:
2836:
2833:
2828:
2823:
2819:
2815:
2810:
2805:
2801:
2797:
2792:
2787:
2783:
2774:
2772:
2753:
2748:
2744:
2740:
2735:
2729:
2725:
2721:
2716:
2712:
2705:
2702:
2694:
2677:
2673:
2651:
2643:
2639:
2635:
2633:
2612:
2607:
2603:
2599:
2595:
2589:
2585:
2581:
2576:
2572:
2567:
2563:
2560:
2555:
2551:
2542:
2540:
2536:
2515:
2511:
2507:
2502:
2498:
2489:
2487:
2483:
2474:
2470:
2452:
2448:
2423:
2419:
2394:
2390:
2381:
2369:
2366:
2358:
2348:
2344:
2340:
2334:
2333:
2329:
2324:This section
2322:
2318:
2313:
2312:
2304:
2290:
2265:
2251:
2239:
2227:
2216:
2202:
2194:
2193:
2192:
2190:
2186:
2181:
2179:
2176:- called the
2152:
2131:
2128:
2108:
2088:
2068:
2048:
2020:
2017:
2005:
2002:
1983:
1980:
1968:
1957:
1952:
1948:
1940:
1939:
1938:
1936:
1932:
1926:
1916:
1914:
1909:
1907:
1903:
1870:
1835:
1826:
1799:
1778:
1756:
1752:
1726:
1723:
1716:
1712:
1708:
1705:
1691:
1683:
1682:
1681:
1679:
1675:
1671:
1653:
1650:
1645:
1625:
1622:
1602:
1582:
1562:
1533:
1530:
1518:
1515:
1512:
1500:
1497:
1490:
1489:
1488:
1474:
1466:
1462:
1458:
1452:
1439:
1435:
1432:
1428:
1425:
1421:
1418:
1414:
1411:
1407:
1404:
1400:
1397:
1393:
1390:
1386:
1383:
1379:
1376:
1372:
1369:
1365:
1362:
1358:
1356:
1352:
1351:
1350:
1346:
1326:
1322:
1318:
1313:
1309:
1305:
1300:
1296:
1292:
1287:
1283:
1275:
1274:
1273:
1271:
1266:
1246:
1242:
1238:
1233:
1229:
1225:
1220:
1216:
1212:
1207:
1203:
1195:
1194:
1193:
1177:
1173:
1150:
1146:
1123:
1119:
1096:
1092:
1084:
1080:
1076:
1072:
1060:
1057:
1054:
1051:
1049:
1045:
1042:
1041:
1040:
1032:
1025:
1022:
1019:
1015:
1012:
1011:
1010:
1002:
1000:
996:
992:
988:
984:
980:
976:
965:
957:
955:
951:
947:
943:
939:
929:
926:
921:
919:
915:
911:
907:
903:
899:
895:
891:
887:
883:
879:
875:
864:
859:
857:
852:
850:
845:
844:
842:
841:
836:
833:
831:
828:
826:
825:Thermal shock
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
752:
751:
750:
746:
745:
735:
730:
728:
723:
721:
716:
715:
713:
712:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
630:
623:
622:
611:
608:
606:
603:
601:
598:
597:
596:
595:
592:
589:
588:
583:
580:
578:
575:
573:
570:
569:
568:
567:
562:
557:
556:
547:
544:
543:
537:
534:
532:
529:
527:
524:
522:
519:
517:
516:Charles's law
514:
512:
509:
507:
504:
503:
501:
500:
497:
494:
493:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
463:
461:
460:
457:
454:
453:
447:
444:
440:
437:
433:
430:
425:
424:non-Newtonian
422:
418:
414:
413:
412:
409:
407:
404:
400:
397:
395:
392:
390:
387:
383:
380:
378:
375:
371:
368:
367:
365:
364:
361:
358:
357:
352:
347:
346:
338:
335:
333:
330:
326:
323:
322:
321:
318:
316:
313:
311:
310:Compatibility
308:
304:
301:
299:
298:Finite strain
296:
295:
294:
291:
289:
286:
284:
281:
279:
276:
272:
269:
268:
267:
264:
262:
259:
258:
254:
249:
248:
237:
234:
233:
232:
231:
227:
226:
221:
218:
216:
213:
211:
208:
207:
206:
205:
202:Conservations
201:
200:
192:
191:
187:
168:
165:
160:
157:
151:
148:
145:
142:
135:
134:
131:
128:
127:
123:
122:
113:
110:
102:
99:November 2014
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3846:Smart fluids
3739:Pascal's law
3596:
3571:Frictionless
3420:Large strain
3408:Small strain
3232:. Retrieved
3229:SDC Verifier
3228:
3218:
3207:
3199:
3196:
3191:
3183:
3178:
3089:
3034:
3002:
2986:
2973:
2965:
2961:
2953:
2952:
2770:
2769:
2642:Henri Tresca
2632:shear stress
2629:
2628:
2534:
2533:
2481:
2480:
2471:
2376:
2361:
2352:
2337:Please help
2325:
2282:
2189:plane stress
2182:
2177:
2040:
1928:
1910:
1743:
1678:plane strain
1669:
1554:
1454:
1398:for concrete
1391:for concrete
1347:
1344:
1269:
1267:
1264:
1078:
1068:
1038:
1029:
1008:
998:
971:
963:
941:
935:
922:
873:
872:
591:Smart fluids
536:Graham's law
442:
435:
420:
406:Pascal's law
402:
385:
373:
331:
228:Inequalities
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
3861:Ferrofluids
3778:Boyle's law
3464:Hooke's law
3403:Deformation
3380:Definitions
3099:parameter.
3041:deformation
2970:hydrostatic
1018:T-criterion
950:macroscopic
946:microscopic
916:, state of
914:temperature
610:Ferrofluids
511:Boyle's law
283:Hooke's law
261:Deformation
3979:Categories
3916:Gay-Lussac
3883:Scientists
3773:Atmosphere
3638:Structures
3612:J-integral
3576:Frictional
3537:Plasticity
3474:Orthotropy
3454:Elasticity
3357:Archimedes
3352:Poiseuille
3306:Vibrations
3275:Topics in
3234:2022-11-03
3170:References
2966:volumetric
1931:composites
993:(CDM) and
663:Gay-Lussac
626:Scientists
526:Fick's law
506:Atmosphere
325:frictional
278:Plasticity
266:Elasticity
69:newspapers
4000:Mechanics
3891:Bernoulli
3873:Rheometer
3868:Rheometry
3810:Acoustics
3722:Newtonian
3717:Viscosity
3347:Bernoulli
3301:Acoustics
3284:Divisions
2923:σ
2919:≤
2905:σ
2895:σ
2882:σ
2872:σ
2859:σ
2849:σ
2840:ν
2834:−
2820:σ
2802:σ
2784:σ
2745:τ
2741:≤
2726:σ
2722:−
2713:σ
2703:τ
2674:τ
2652:τ
2604:σ
2600:≤
2586:σ
2573:σ
2564:ν
2561:−
2552:σ
2512:σ
2508:≤
2499:σ
2449:σ
2420:σ
2391:σ
2355:June 2013
2326:does not
1906:model III
1753:σ
1724:π
1713:σ
1651:π
1646:σ
1583:γ
1531:π
1519:γ
1498:σ
1475:σ
1368:von Mises
1323:ε
1310:ε
1297:ε
1284:ε
1243:σ
1230:σ
1217:σ
1204:σ
1174:σ
1147:σ
1120:σ
1093:σ
987:Needleman
908:failure (
900:failure (
890:materials
760:Corrosion
703:Truesdell
633:Bernoulli
582:Rheometer
577:Rheometry
417:Newtonian
411:Viscosity
161:φ
149:−
3833:Rheology
3744:Pressure
3712:Buoyancy
3496:Hugoniot
3114:Fracture
3103:See also
2630:Maximum
1440:for soil
954:fracture
902:fracture
835:Yielding
785:Fracture
755:Buckling
561:Rheology
466:Adhesion
446:Pressure
432:Buoyancy
377:Dynamics
215:Momentum
3906:Charles
3749:Liquids
3645:Bending
3602:Fatigue
2347:removed
2332:sources
1902:mode II
906:ductile
898:brittle
780:Fouling
775:Fatigue
648:Charles
456:Liquids
370:Statics
315:Bending
83:scholar
3951:Awards
3941:Stokes
3936:Navier
3931:Newton
3926:Pascal
3901:Cauchy
3800:Plasma
3680:Fluids
3459:linear
3447:Solids
3386:Stress
3362:Pascal
3097:damage
3051:, and
2993:convex
2634:theory
2539:strain
2440:, and
2380:tensor
2283:where
2248:
2041:where
1992:
1744:where
1555:where
1465:mode I
1075:strain
1071:stress
918:stress
795:Impact
698:Stokes
693:Pascal
683:Navier
678:Newton
668:Graham
643:Cauchy
546:Plasma
441:
439:Mixing
434:
419:
401:
384:
372:
360:Fluids
293:Strain
288:Stress
271:linear
220:Energy
85:
78:
71:
64:
56:
3921:Hooke
3911:Euler
3896:Boyle
3766:Gases
3523:Creep
3435:Solid
3186:, 19.
2978:shear
2974:shape
2484:– by
948:, to
910:yield
904:) or
770:Creep
673:Hooke
653:Euler
638:Boyle
496:Gases
90:JSTOR
76:books
3437:and
3326:Laws
3083:the
3077:the
3071:the
3065:the
3059:the
3035:The
2987:The
2330:any
2328:cite
2183:The
1904:and
1862:and
1676:for
1436:the
1429:the
1422:the
1415:the
1408:the
1401:the
1394:the
1387:the
1380:the
1373:the
1366:the
1359:the
1319:<
1306:<
1293:<
1268:The
1239:<
1226:<
1213:<
880:and
830:Wear
688:Noll
658:Fick
210:Mass
195:Laws
62:news
3501:JWL
2341:by
936:In
45:by
3981::
3227:.
3200:38
3047:,
2984:.
2693::
2469:.
2411:,
2180:.
1001:.
940:,
3268:e
3261:t
3254:v
3237:.
2968:(
2956:(
2937:.
2932:2
2927:y
2915:)
2909:3
2899:1
2891:+
2886:3
2876:2
2868:+
2863:2
2853:1
2844:(
2837:2
2829:2
2824:3
2816:+
2811:2
2806:2
2798:+
2793:2
2788:1
2754:.
2749:y
2736:2
2730:3
2717:1
2706:=
2678:y
2613:.
2608:y
2596:)
2590:3
2582:+
2577:2
2568:(
2556:1
2516:y
2503:1
2453:3
2424:2
2395:1
2368:)
2362:(
2357:)
2353:(
2349:.
2335:.
2291:E
2266:2
2260:c
2257:I
2252:K
2240:E
2228:1
2217:=
2211:c
2208:I
2203:G
2161:c
2158:I
2153:G
2132:a
2129:2
2109:a
2089:u
2069:t
2049:P
2021:a
2018:d
2006:u
2003:d
1984:t
1981:2
1969:P
1958:=
1953:I
1949:G
1885:c
1882:I
1879:I
1876:I
1871:K
1847:c
1844:I
1841:I
1836:K
1808:c
1805:I
1800:K
1779:Y
1757:c
1727:a
1717:c
1709:Y
1706:=
1700:c
1697:I
1692:K
1654:a
1626:a
1623:2
1603:a
1563:E
1534:a
1516:E
1513:2
1501:=
1327:t
1314:1
1301:3
1288:c
1247:t
1234:1
1221:3
1208:c
1178:c
1151:t
1124:3
1097:1
1073:/
1020:)
999:f
862:e
855:t
848:v
733:e
726:t
719:v
443:·
436:·
426:)
421:·
415:(
403:·
386:·
374:·
169:x
166:d
158:d
152:D
146:=
143:J
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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