1449:(the xz shear strain along the xz-GB plane must be equivalent for A and B). In addition, this GB constraint requires that five independent slip systems be activated per crystallite constituting the GB. Notably, because independent slip systems are defined as slip planes on which dislocation migrations cannot be reproduced by any combination of dislocation migrations along other slip systemâs planes, the number of geometrical slip systems for a given crystal system - which by definition can be constructed by slip system combinations - is typically greater than that of independent slip systems. Significantly, there is a maximum of five independent slip systems for each of the seven crystal systems, however, not all seven crystal systems acquire this upper limit. In fact, even within a given crystal system, the composition and Bravais lattice diversifies the number of independent slip systems (see the table below). In cases for which crystallites of a polycrystal do not obtain five independent slip systems, the GB condition cannot be met, and thus the time-independent deformation of individual crystallites results in cracks and voids at the GBs of the polycrystal, and soon fracture is realized. Hence, for a given composition and structure, a single crystal with less than five independent slip systems is stronger (exhibiting a greater extent of plasticity) than its polycrystalline form.
1338:) is low, representative of a small amount of applied shear stress necessary to induce a large amount of shear strain. Facile dislocation glide and corresponding flow is attributed to dislocation migration along parallel slip planes only (i.e. one slip system). Moderate impedance to dislocation migration along parallel slip planes is exhibited according to the weak stress field interactions between these dislocations, which heightens with smaller interplanar spacing. Overall, these migrating dislocations within a single slip system act as weak obstacles to flow, and a modest rise in stress is observed in comparison to the yield stress. During the linear hardening stage 2 of flow, the work hardening rate becomes high as considerable stress is required to overcome the stress field interactions of dislocations migrating on non-parallel slip planes (i.e. multiple slip systems), acting as strong obstacles to flow. Much stress is required to drive continual dislocation migration for small strains. The shear flow stress is directly proportional to the square root of the dislocation density (Ï
1346:), irrespective of the evolution of dislocation configurations, displaying the reliance of hardening on the number of dislocations present. Regarding this evolution of dislocation configurations, at small strains the dislocation arrangement is a random 3D array of intersecting lines. Moderate strains correspond to cellular dislocation structures of heterogeneous dislocation distribution with large dislocation density at the cell boundaries, and small dislocation density within the cell interior. At even larger strains the cellular dislocation structure reduces in size until a minimum size is achieved. Finally, the work hardening rate becomes low again in the exhaustion/saturation of hardening stage 3 of plastic flow, as small shear stresses produce large shear strains. Notably, instances when multiple slip systems are oriented favorably with respect to the applied stress, the Ï
1526:~d ). Therefore, the flow stress of a polycrystal, and hence the polycrystalâs strength, increases with small grain size. The reason for this is that smaller grains have a relatively smaller number of slip planes to be activated, corresponding to a fewer number of dislocations migrating to the GBs, and therefore less stress induced on adjacent grains due to dislocation pile up. In addition, for a given volume of polycrystal, smaller grains present more strong obstacle grain boundaries. These two factors provide an understanding as to why the onset of macroscopic flow in fine-grained polycrystals occurs at larger applied stresses than in coarse-grained polycrystals.
981:
1218:
1505:
governing single crystal time-independent yielding. Eventually, the activated slip planes within the grain interiors will permit dislocation migration to the GB where many dislocations then pile up as geometrically necessary dislocations. This pile up corresponds to strain gradients across individual grains as the dislocation density near the GB is greater than that in the grain interior, imposing a stress on the adjacent grain in contact. When considering the AB bicrystal as a whole, the most favorably oriented slip system in A will not be the that in B, and hence Ï
1323:
1633:
132:
1400:
cannot pass from one grain to another across the grain boundary. The following sections explore specific GB requirements for extensive plastic deformation of polycrystals prior to fracture, as well as the influence of microscopic yielding within individual crystallites on macroscopic yielding of the polycrystal. The critical resolved shear stress for polycrystals is defined by Schmidâs law as well (Ï
1757:
1540:
38:
1107:
modeling the foam as beams is only valid if the ratio of the density of the foam to the density of the matter is less than 0.3. This is because beams yield axially instead of bending. In closed cell foams, the yield strength is increased if the material is under tension because of the membrane that spans the face of the cells.
1106:
These materials plastically deform when the bending moment exceeds the fully plastic moment. This applies to open cell foams where the bending moment is exerted on the cell walls. The foams can be made of any material with a plastic yield point which includes rigid polymers and metals. This method of
1314:, however plastic flow will still occur due to thermally activated high temperature time-dependent plastic deformation mechanisms such as NabarroâHerring (NH) and Coble diffusional flow through the lattice and along the single crystal surfaces, respectively, as well as dislocation climb-glide creep.
997:
Most metals show more plasticity when hot than when cold. Lead shows sufficient plasticity at room temperature, while cast iron does not possess sufficient plasticity for any forging operation even when hot. This property is of importance in forming, shaping and extruding operations on metals. Most
993:
Plasticity in a crystal of pure metal is primarily caused by two modes of deformation in the crystal lattice: slip and twinning. Slip is a shear deformation which moves the atoms through many interatomic distances relative to their initial positions. Twinning is the plastic deformation which takes
1517:
between grains A and B is achieved, according to the GB constraint. Thus, for a given composition and structure, a polycrystal with five independent slip systems is stronger (greater extent of plasticity) than its single crystalline form. Correspondingly, the work hardening rate will be higher for
1424:
The GB constraint for polycrystals can be explained by considering a grain boundary in the xz plane between two single crystals A and B of identical composition, structure, and slip systems, but misoriented with respect to each other. To ensure that voids do not form between individually deforming
1083:
materials, the discussion of "dislocations" is inapplicable, since the entire material lacks long range order. These materials can still undergo plastic deformation. Since amorphous materials, like polymers, are not well-ordered, they contain a large amount of free volume, or wasted space. Pulling
1504:
Although the two crystallites A and B discussed in the above section have identical slip systems, they are misoriented with respect to each other, and therefore misoriented with respect to the applied force. Thus, microscopic yielding within a crystallite interior may occur according to the rules
1399:
Plasticity in polycrystals differs substantially from that in single crystals due to the presence of grain boundary (GB) planar defects, which act as very strong obstacles to plastic flow by impeding dislocation migration along the entire length of the activated slip plane(s). Hence, dislocations
1350:
for these systems may be similar and yielding may occur according to dislocation migration along multiple slip systems with non-parallel slip planes, displaying a stage 1 work-hardening rate typically characteristic of stage 2. Lastly, distinction between time-independent plastic deformation in
1012:
Crystalline materials contain uniform planes of atoms organized with long-range order. Planes may slip past each other along their close-packed directions, as is shown on the slip systems page. The result is a permanent change of shape within the crystal and plastic deformation. The presence of
934:
applied to a sample will cause it to behave in an elastic manner. Each increment of load is accompanied by a proportional increment in extension. When the load is removed, the piece returns to its original size. However, once the load exceeds a threshold â the yield strength â the
1139:
Inelastic deformations of rocks and concrete are primarily caused by the formation of microcracks and sliding motions relative to these cracks. At high temperatures and pressures, plastic behavior can also be affected by the motion of dislocations in individual grains in the microstructure.
941:, however, is an approximation and its quality depends on the time frame considered and loading speed. If, as indicated in the graph opposite, the deformation includes elastic deformation, it is also often referred to as "elasto-plastic deformation" or "elastic-plastic deformation".
2019:
Again, a visual representation of the yield surface may be constructed using the above equation, which takes the shape of an ellipse. Inside the surface, materials undergo elastic deformation. Reaching the surface means the material undergoes plastic deformations.
1188:
is the Schmid factor. The Schmid factor comprises two variables λ and Ï, defining the angle between the slip plane direction and the tensile force applied, and the angle between the slip plane normal and the tensile force applied, respectively. Notably, because
1045:
The presence of other defects within a crystal may entangle dislocations or otherwise prevent them from gliding. When this happens, plasticity is localized to particular regions in the material. For crystals, these regions of localized plasticity are called
952:, may need increasingly higher stresses to deform further. Generally, plastic deformation is also dependent on the deformation speed, i.e. higher stresses usually have to be applied to increase the rate of deformation. Such materials are said to deform
856:, a non-reversible change of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. In engineering, the transition from
1747:
may be constructed, which provides a visual representation of this concept. Inside of the yield surface, deformation is elastic. On the surface, deformation is plastic. It is impossible for a material to have stress states outside its yield surface.
1355:
Comparison between the time-independent plastic deformation of body centered cubic transition metals and face centered cubic metals, highlighting the critical resolved shear stress, work hardening rate, and necking strain during tensile testing.
1559:(of order d-1 in d dimensions) is a function of the strain tensor. Although this description is accurate when a small part of matter is subjected to increasing loading (such as strain loading), this theory cannot account for irreversibility.
1298:
remains so until region 3 is defined. Notably, in region 2 moderate temperature time-dependent plastic deformation (creep) mechanisms such as solute-drag should be considered. Furthermore, in the high temperature region 3
1650:
criteria are commonly used to determine whether a material has yielded. However, these criteria have proved inadequate for a large range of materials and several other yield criteria are also in widespread use.
1782:
under uniaxial loading, subtracting out hydrostatic stresses, and states that all effective stresses greater than that which causes material failure in uniaxial loading will result in plastic deformation.
1659:
The Tresca criterion is based on the notion that when a material fails, it does so in shear, which is a relatively good assumption when considering metals. Given the principal stress state, we can use
1646:
If the stress exceeds a critical value, as was mentioned above, the material will undergo plastic, or irreversible, deformation. This critical stress can be tensile or compressive. The Tresca and the
1260:*) shear stresses, arising from the stress required to move dislocations in the presence of other dislocations, and the resistance of point defect obstacles to dislocation migration, respectively. At
1717:
1416:
is the weighted Schmid factor. The weighted Schmid factor reflects the least favorably oriented slip system among the most favorably oriented slip systems of the grains constituting the GB.
1161:), initiating dislocation migration along parallel slip planes of a single slip system, thereby defining the transition from elastic to plastic deformation behavior in crystalline materials.
2014:
267:
1624:, uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase of deformation.
1121:
Soils, particularly clays, display a significant amount of inelasticity under load. The causes of plasticity in soils can be quite complex and are strongly dependent on the
984:
Plasticity under a spherical nanoindenter in (111) copper. All particles in ideal lattice positions are omitted and the color code refers to the von Mises stress field.
98:
1774:
The Huberâvon Mises criterion is based on the Tresca criterion but takes into account the assumption that hydrostatic stresses do not contribute to material failure.
74:
1290:* â 0, representing the elimination of point defect impedance to dislocation migration. Thus the temperature-independent critical resolved shear stress Ï
2339:
1518:
the polycrystal than the single crystal, as more stress is required in the polycrystal to produce strains. Importantly, just as with single crystal flow stress, Ï
816:
944:
Perfect plasticity is a property of materials to undergo irreversible deformation without any increase in stresses or loads. Plastic materials that have been
1247:
which is required to initiate dislocation glide and equivalently plastic flow. In region 1, the critical resolved shear stress has two components: athermal (
2619:
895:. Such defects are relatively rare in most crystalline materials, but are numerous in some and part of their crystal structure; in such cases,
1125:, chemical composition, and water content. Plastic behavior in soils is caused primarily by the rearrangement of clusters of adjacent grains.
2544:
2308:
2214:
2161:
2107:
2082:
1225:
There are three characteristic regions of the critical resolved shear stress as a function of temperature. In the low temperature region 1 (
2605:
2582:
2563:
2525:
2494:
2400:
2246:
MaaĂ, Robert; Derlet, Peter M. (January 2018). "Micro-plasticity and recent insights from intermittent and small-scale plasticity".
2189:
2136:
809:
1616:, roughly simultaneously, realized that the plastic deformation of ductile materials could be explained in terms of the theory of
2650:
1760:
The von Mises yield surfaces in principal stress coordinates circumscribes a cylinder around the hydrostatic axis. Also shown is
935:
extension increases more rapidly than in the elastic region; now when the load is removed, some degree of extension will remain.
1669:
782:
1330:
During the easy glide stage 1, the work hardening rate, defined by the change in shear stress with respect to shear strain (
483:
320:
1217:
1084:
these materials in tension opens up these regions and can give materials a hazy appearance. This haziness is the result of
2640:
1116:
1453:
The number of independent slip systems for a given composition (primary material class) and structure (Bravais lattice).
802:
523:
409:
478:
387:
1789:
270:
1567:
2044:
1769:
1663:
to solve for the maximum shear stresses our material will experience and conclude that the material will fail if
1647:
1150:
Time-independent plastic flow in both single crystals and polycrystals is defined by a critical/maximum resolved
853:
394:
223:
148:
2645:
1007:
904:
689:
684:
299:
473:
466:
136:
41:
2442:
2039:
1775:
752:
747:
416:
1513:. Paramount is the fact that macroscopic yielding of the bicrystal is prolonged until the higher value of Ï
1621:
1613:
1599:
1583:
304:
2124:
896:
727:
345:
167:
110:
2420:
2362:
2265:
1556:
1134:
1063:
857:
565:
382:
350:
294:
2593:
2029:
1641:
1544:
1221:
The three characteristic regions of the critical resolved shear stress as a function of temperature
1098:. The material may go from an ordered appearance to a "crazy" pattern of strain and stretch marks.
1022:
938:
861:
767:
615:
508:
214:
153:
45:
2480:
2333:
2281:
2255:
2049:
1095:
1030:
787:
421:
377:
372:
105:
77:
53:
2054:
1033:
such as
Nitinol wire also exhibit a reversible form of plasticity which is more properly called
1551:
There are several mathematical descriptions of plasticity. One is deformation theory (see e.g.
1322:
83:
2615:
2601:
2578:
2559:
2540:
2521:
2490:
2484:
2416:
2396:
2304:
2210:
2185:
2157:
2132:
2103:
2078:
834:
404:
355:
49:
59:
2513:
2370:
2273:
2034:
1779:
1660:
1059:
1034:
916:
742:
717:
630:
605:
600:
555:
161:
140:
1632:
1609:
1351:
body-centered cubic transition metals and face centered cubic metals is summarized below.
953:
931:
732:
656:
570:
501:
435:
337:
887:. However, the physical mechanisms that cause plastic deformation can vary widely. At a
620:
490:
2457:
2366:
2269:
1547:
showing elastic and plastic deformation regimes for the deformation theory of plasticity
1213:
Critical resolved shear stress dependence on temperature, strain rate, and point defects
2229:
Ziegenhain, Gerolf; and
Urbassek, Herbert M.; "Reversible Plasticity in fcc metals" in
1579:
1571:
1552:
1369:
Critical resolved shear stress = high (relatively) & strongly temperature-dependent
1122:
945:
876:
737:
595:
560:
461:
367:
121:
1522:~Ï, but is also inversely proportional to the square root of average grain diameter (Ï
980:
131:
2634:
2285:
1744:
1372:
Critical resolved shear stress = low (relatively) & weakly temperature-dependent
777:
610:
115:
2353:
Groves, Geoffrey W.; Kelly, Anthony (1963). "Independent Slip
Systems in Crystals".
1761:
1617:
1151:
969:
949:
762:
757:
722:
454:
2277:
1778:
was the first who proposed the criterion of shear energy. Von Mises solves for an
1169:
The critical resolved shear stress for single crystals is defined by Schmidâs law
903:
materials such as rock, concrete and bone, plasticity is caused predominantly by
2388:
2177:
1605:
1237:
900:
892:
888:
772:
675:
2234:
919:, plasticity is mainly a consequence of bubble or cell rearrangements, notably
30:"Plastic material" redirects here. For the material used in manufacturing, see
2374:
1047:
1026:
994:
place along two planes due to a set of forces applied to a given metal piece.
920:
908:
694:
590:
1756:
1587:
1539:
1080:
965:
666:
661:
495:
27:
Non-reversible deformation of a solid material in response to applied forces
17:
1326:
The three stages of time-independent plastic deformation of single crystals
1025:
metals is reversible, as long as there is no material transport in form of
1563:
880:
849:
645:
550:
530:
516:
37:
2445:(1904). "WĆaĆciwa praca odksztaĆcenia jako miara wytezenia materiaĆu".
1575:
1091:
1086:
927:
830:
399:
31:
2100:
Nonlinear Solid
Mechanics: Bifurcation Theory and Material Instability
540:
1743:
is the stress under which the material fails in uniaxial loading. A
1441:(the z-axial strain at the GB must be equivalent for A and B), and Δ
2260:
1144:
Time-independent yielding and plastic flow in crystalline materials
1755:
1631:
1538:
1321:
1216:
979:
912:
868:
846:
444:
1562:
Ductile materials can sustain large plastic deformations without
2425:
Nachrichten von der
Gesellschaft der Wissenschaften zu Göttingen
1433:(the x-axial strain at the GB must be equivalent for A and B), Δ
884:
872:
867:
Plastic deformation is observed in most materials, particularly
1058:
Microplasticity is a local phenomenon in metals. It occurs for
2303:(Second ed.). Long Grove, Illinois: Waveland Press, Inc.
580:
2421:"Mechanik der festen Körper im plastisch-deformablen Zustand"
1500:
Implications of the grain boundary constraint in polycrystals
1165:
Time-independent yielding and plastic flow in single crystals
998:
metals are rendered plastic by heating and hence shaped hot.
964:
The plasticity of a material is directly proportional to the
1425:
grains, the GB constraint for the bicrystal is as follows: Δ
1021:
On the nanoscale the primary plastic deformation in simple
1395:
Time-independent yielding and plastic flow in polycrystals
2458:"Specific Work of Strain as a Measure of Material Effort"
2180:; Ma, Guo-Wei; Qiang, Hong-Fu; Zhang, Yong-Qiang (2006).
1066:
domain while some local areas are in the plastic domain.
2326:
Deformation and
Fatigue of Hexagonal Close Packed Metals
891:
scale, plasticity in metals is usually a consequence of
1712:{\displaystyle \sigma _{1}-\sigma _{3}\geq \sigma _{0}}
1566:. However, even ductile metals will fracture when the
1286:) is defined, where the thermal shear stress component
2537:
Plasticity: Mathematical Theory and
Numerical Analysis
1812:
1318:
Stages of time-independent plastic flow, post yielding
2520:. Vol. 7. Oxford: Elsevier. pp. 7068â7071.
2516:(2001). "Plastic Deformation of Cellular Materials".
1792:
1672:
1636:
Comparison of Tresca criterion to Von Mises criterion
226:
86:
62:
1469:
Metal: 5, ceramic (covalent): 5, ceramic (ionic): 2
1461:Primary material class: # Independent slip systems
2008:
1711:
1094:are formed within the material in regions of high
261:
92:
68:
2518:Encyclopedia of Materials: Science and Technology
1590:of a worked piece, so that shaping can continue.
1013:dislocations increases the likelihood of planes.
1184:is the yield strength of the single crystal and
2009:{\displaystyle \sigma _{v}^{2}={\tfrac {1}{2}}}
1412:is the yield strength of the polycrystal and
1377:Work hardening rate = temperature-independent
810:
8:
1570:becomes large enoughâthis is as a result of
1380:Work hardening rate = temperature-dependent
1008:Slip (materials science) § Slip systems
1574:of the material, which causes it to become
262:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
2338:: CS1 maint: location missing publisher (
1620:. The mathematical theory of plasticity,
1388:Necking strain decreases with temperature
1268:*, the moderate temperature region 2 (0.25
1062:values where the metal is globally in the
817:
803:
650:
440:
283:
205:
2259:
1994:
1989:
1976:
1971:
1958:
1953:
1934:
1924:
1911:
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1872:
1856:
1846:
1833:
1811:
1802:
1797:
1791:
1703:
1690:
1677:
1671:
1420:Grain boundary constraint in polycrystals
1385:Necking strain increases with temperature
860:behavior to plastic behavior is known as
239:
225:
85:
61:
2626:. Massachusetts Institute of Technology.
2556:Fundamentals of the Theory of Plasticity
1451:
1353:
130:
36:
2573:Khan, Akhtar S.; Huang, Sujian (1995).
2065:
911:. In cellular materials such as liquid
674:
629:
579:
539:
443:
312:
286:
213:
2489:. New York: McGraw-Hill. p. 369.
2331:
2427:. Mathematisch-Physikalische Klasse.
2393:The Mathematical Theory of Plasticity
1361:Body-centered cubic transition metals
1310:) Î can be low, contributing to low Ï
7:
2539:(2nd ed.). New York: Springer.
2535:Han, Weimin; Reddy, B. Daya (2013).
2554:Kachanov, Lazar' Markovich (2004).
2620:"Mechanical Behavior of Materials"
2154:Limit Analysis and Soil Plasticity
1736:is the minimum normal stress, and
25:
2207:Plasticity in Reinforced Concrete
2486:History of Strength of Materials
2301:Mechanical Behavior of Materials
2129:Inelastic analysis of structures
2575:Continuum Theory of Plasticity
2231:Philosophical Magazine Letters
2102:. Cambridge University Press.
2003:
2000:
1946:
1931:
1904:
1892:
1865:
1853:
1826:
1823:
1729:is the maximum normal stress,
948:by prior deformation, such as
1:
2278:10.1016/j.actamat.2017.06.023
2235:DOI 10.1080/09500830903272900
1493:Metal: 2, ceramic (mixed): 2
1240:must be high to achieve high
1117:critical state soil mechanics
154:Yield strength (yield point)
2395:. Oxford University Press.
1764:'s hexagonal yield surface.
1364:Face-centered cubic metals
2667:
2598:Computational Inelasticity
1767:
1639:
1597:
1132:
1114:
1005:
29:
2375:10.1080/14786436308213843
2324:Partridge, Peter (1969).
2299:Courtney, Thomas (2005).
2045:Deformation (engineering)
1770:Von Mises yield criterion
1752:Huberâvon Mises criterion
1530:Mathematical descriptions
93:{\displaystyle \epsilon }
76:, shown as a function of
2443:Huber, Maksymilian Tytus
2073:Lubliner, Jacob (2008).
321:ClausiusâDuhem (entropy)
271:Fick's laws of diffusion
2651:Deformation (mechanics)
2624:MIT Course Number 3.032
2328:. University of Surrey.
2233:, 89(11):717-723, 2009
2131:. John Wiley and Sons.
2098:Bigoni, Davide (2012).
2040:Deformation (mechanics)
960:Contributing properties
479:NavierâStokes equations
417:Material failure theory
69:{\displaystyle \sigma }
2481:Timoshenko, Stephen P.
2355:Philosophical Magazine
2209:. J. Ross Publishing.
2205:Chen, Wai-Fah (2007).
2182:Generalized Plasticity
2156:. J. Ross Publishing.
2152:Chen, Wai-Fah (2008).
2010:
1765:
1713:
1637:
1622:flow plasticity theory
1614:Geoffrey Ingram Taylor
1600:Flow plasticity theory
1594:Flow plasticity theory
1548:
1543:An idealized uniaxial
1327:
1222:
985:
845:) is the ability of a
263:
203:
128:
94:
70:
2616:Van Vliet, Krystyn J.
2462:Archives of Mechanics
2447:Czasopismo Techniczne
2011:
1759:
1714:
1635:
1542:
1325:
1220:
1017:Reversible plasticity
983:
897:plastic crystallinity
852:to undergo permanent
474:Bernoulli's principle
467:Archimedes' principle
264:
134:
111:Proportionality limit
100:):
95:
71:
40:
2641:Plasticity (physics)
2594:Hughes, Thomas J. R.
1790:
1670:
1557:Cauchy stress tensor
1135:rock mass plasticity
566:Cohesion (chemistry)
388:Infinitesimal strain
224:
84:
60:
2367:1963PMag....8..877G
2270:2018AcMat.143..338M
2030:Yield (engineering)
1999:
1981:
1963:
1807:
1642:Yield (engineering)
1545:stress-strain curve
1485:Ceramic (ionic): 3
1474:Body centered cubic
1466:Face centered cubic
1454:
1357:
1279: < 0.7
1070:Amorphous materials
1031:Shape-memory alloys
1023:face-centered cubic
976:Physical mechanisms
939:Elastic deformation
843:plastic deformation
484:Poiseuille equation
215:Continuum mechanics
209:Part of a series on
137:stressâstrain curve
42:Stressâstrain curve
2417:von Mises, Richard
2006:
1985:
1967:
1949:
1821:
1793:
1766:
1709:
1638:
1549:
1535:Deformation theory
1452:
1354:
1328:
1223:
1129:Rocks and concrete
1102:Cellular materials
1096:hydrostatic stress
986:
917:biological tissues
690:Magnetorheological
685:Electrorheological
422:Fracture mechanics
259:
204:
129:
106:True elastic limit
90:
66:
2546:978-1-4614-5939-2
2514:Ashby, Michael F.
2310:978-1-57766-425-3
2216:978-1-932159-74-5
2163:978-1-932159-73-8
2125:BaĆŸant, ZdenÄk P.
2109:978-1-107-02541-7
2084:978-0-486-46290-5
2075:Plasticity theory
1820:
1497:
1496:
1392:
1391:
972:of the material.
954:visco-plastically
835:materials science
827:
826:
702:
701:
636:
635:
405:Contact mechanics
328:
327:
257:
177:Apparent stress (
149:Ultimate strength
50:nonferrous alloys
16:(Redirected from
2658:
2627:
2611:
2588:
2569:
2550:
2531:
2501:
2500:
2476:
2470:
2469:
2468:: 173â190. 2004.
2454:
2439:
2433:
2432:
2413:
2407:
2406:
2385:
2379:
2378:
2350:
2344:
2343:
2337:
2329:
2321:
2315:
2314:
2296:
2290:
2289:
2263:
2243:
2237:
2227:
2221:
2220:
2202:
2196:
2195:
2174:
2168:
2167:
2149:
2143:
2142:
2123:JirĂĄsek, Milan;
2120:
2114:
2113:
2095:
2089:
2088:
2070:
2035:Atterberg limits
2015:
2013:
2012:
2007:
1998:
1993:
1980:
1975:
1962:
1957:
1939:
1938:
1929:
1928:
1916:
1915:
1900:
1899:
1890:
1889:
1877:
1876:
1861:
1860:
1851:
1850:
1838:
1837:
1822:
1813:
1806:
1801:
1780:effective stress
1718:
1716:
1715:
1710:
1708:
1707:
1695:
1694:
1682:
1681:
1655:Tresca criterion
1586:can restore the
1455:
1358:
1303: â„ 0.7
1275: <
1035:pseudoelasticity
819:
812:
805:
651:
616:Gay-Lussac's law
606:Combined gas law
556:Capillary action
441:
284:
268:
266:
265:
260:
258:
256:
248:
240:
206:
162:Strain hardening
141:structural steel
99:
97:
96:
91:
75:
73:
72:
67:
44:showing typical
21:
2666:
2665:
2661:
2660:
2659:
2657:
2656:
2655:
2646:Solid mechanics
2631:
2630:
2614:
2608:
2592:Simo, Juan C.;
2591:
2585:
2572:
2566:
2558:. Dover Books.
2553:
2547:
2534:
2528:
2512:
2509:
2507:Further reading
2504:
2497:
2479:
2477:
2473:
2456:
2441:
2440:
2436:
2415:
2414:
2410:
2403:
2387:
2386:
2382:
2361:(89): 877â887.
2352:
2351:
2347:
2330:
2323:
2322:
2318:
2311:
2298:
2297:
2293:
2248:Acta Materialia
2245:
2244:
2240:
2228:
2224:
2217:
2204:
2203:
2199:
2192:
2176:
2175:
2171:
2164:
2151:
2150:
2146:
2139:
2122:
2121:
2117:
2110:
2097:
2096:
2092:
2085:
2072:
2071:
2067:
2063:
2055:Poisson's ratio
2026:
1930:
1920:
1907:
1891:
1881:
1868:
1852:
1842:
1829:
1788:
1787:
1772:
1754:
1742:
1735:
1728:
1699:
1686:
1673:
1668:
1667:
1657:
1644:
1630:
1610:Michael Polanyi
1602:
1596:
1537:
1532:
1525:
1521:
1516:
1512:
1508:
1502:
1458:Bravais lattice
1448:
1444:
1440:
1436:
1432:
1428:
1422:
1411:
1407:
1403:
1397:
1349:
1341:
1320:
1313:
1309:
1297:
1293:
1285:
1274:
1256:) and thermal (
1255:
1246:
1235:
1215:
1208:
1201:
1183:
1179:
1175:
1167:
1160:
1146:
1137:
1131:
1119:
1113:
1104:
1077:
1072:
1056:
1054:Microplasticity
1043:
1019:
1010:
1004:
991:
978:
962:
932:tensile loading
899:can result. In
841:(also known as
823:
794:
793:
792:
712:
704:
703:
657:Viscoelasticity
648:
638:
637:
625:
575:
571:Surface tension
535:
438:
436:Fluid mechanics
428:
427:
426:
340:
338:Solid mechanics
330:
329:
281:
273:
249:
241:
222:
221:
202:
191:Actual stress (
187:
173:
144:
127:
126:
82:
81:
58:
57:
35:
28:
23:
22:
15:
12:
11:
5:
2664:
2662:
2654:
2653:
2648:
2643:
2633:
2632:
2629:
2628:
2612:
2606:
2589:
2583:
2570:
2564:
2551:
2545:
2532:
2526:
2508:
2505:
2503:
2502:
2495:
2471:
2455:Translated as
2434:
2408:
2401:
2380:
2345:
2316:
2309:
2291:
2238:
2222:
2215:
2197:
2190:
2169:
2162:
2144:
2137:
2115:
2108:
2090:
2083:
2064:
2062:
2059:
2058:
2057:
2052:
2047:
2042:
2037:
2032:
2025:
2022:
2017:
2016:
2005:
2002:
1997:
1992:
1988:
1984:
1979:
1974:
1970:
1966:
1961:
1956:
1952:
1948:
1945:
1942:
1937:
1933:
1927:
1923:
1919:
1914:
1910:
1906:
1903:
1898:
1894:
1888:
1884:
1880:
1875:
1871:
1867:
1864:
1859:
1855:
1849:
1845:
1841:
1836:
1832:
1828:
1825:
1819:
1816:
1810:
1805:
1800:
1796:
1768:Main article:
1753:
1750:
1740:
1733:
1726:
1720:
1719:
1706:
1702:
1698:
1693:
1689:
1685:
1680:
1676:
1656:
1653:
1640:Main article:
1629:
1628:Yield criteria
1626:
1598:Main article:
1595:
1592:
1580:Heat treatment
1572:work hardening
1536:
1533:
1531:
1528:
1523:
1519:
1514:
1510:
1506:
1501:
1498:
1495:
1494:
1491:
1487:
1486:
1483:
1479:
1478:
1475:
1471:
1470:
1467:
1463:
1462:
1459:
1446:
1442:
1438:
1434:
1430:
1426:
1421:
1418:
1409:
1405:
1401:
1396:
1393:
1390:
1389:
1386:
1382:
1381:
1378:
1374:
1373:
1370:
1366:
1365:
1362:
1347:
1339:
1319:
1316:
1311:
1307:
1295:
1291:
1283:
1272:
1251:
1244:
1233:
1214:
1211:
1206:
1197:
1181:
1177:
1173:
1166:
1163:
1158:
1145:
1142:
1133:Main article:
1130:
1127:
1123:microstructure
1115:Main article:
1112:
1111:Soils and sand
1109:
1103:
1100:
1076:
1073:
1071:
1068:
1055:
1052:
1042:
1039:
1018:
1015:
1006:Main article:
1003:
1000:
990:
987:
977:
974:
961:
958:
825:
824:
822:
821:
814:
807:
799:
796:
795:
791:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
714:
713:
710:
709:
706:
705:
700:
699:
698:
697:
692:
687:
679:
678:
672:
671:
670:
669:
664:
659:
649:
644:
643:
640:
639:
634:
633:
627:
626:
624:
623:
618:
613:
608:
603:
598:
593:
587:
584:
583:
577:
576:
574:
573:
568:
563:
561:Chromatography
558:
553:
547:
544:
543:
537:
536:
534:
533:
514:
513:
512:
493:
481:
476:
464:
451:
448:
447:
439:
434:
433:
430:
429:
425:
424:
419:
414:
413:
412:
402:
397:
392:
391:
390:
385:
375:
370:
365:
360:
359:
358:
348:
342:
341:
336:
335:
332:
331:
326:
325:
324:
323:
315:
314:
310:
309:
308:
307:
302:
297:
289:
288:
282:
279:
278:
275:
274:
269:
255:
252:
247:
244:
238:
235:
232:
229:
218:
217:
211:
210:
201:
200:
189:
185:
174:
172:
171:
165:
159:
156:
151:
145:
125:
124:
122:yield strength
118:
113:
108:
102:
101:
89:
65:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2663:
2652:
2649:
2647:
2644:
2642:
2639:
2638:
2636:
2625:
2621:
2617:
2613:
2609:
2607:0-387-97520-9
2603:
2599:
2595:
2590:
2586:
2584:0-471-31043-3
2580:
2576:
2571:
2567:
2565:0-486-43583-0
2561:
2557:
2552:
2548:
2542:
2538:
2533:
2529:
2527:0-08-043152-6
2523:
2519:
2515:
2511:
2510:
2506:
2498:
2496:9780486611877
2492:
2488:
2487:
2482:
2475:
2472:
2467:
2463:
2459:
2452:
2448:
2444:
2438:
2435:
2431:(1): 582â592.
2430:
2426:
2422:
2418:
2412:
2409:
2404:
2402:0-19-850367-9
2398:
2394:
2390:
2384:
2381:
2376:
2372:
2368:
2364:
2360:
2356:
2349:
2346:
2341:
2335:
2327:
2320:
2317:
2312:
2306:
2302:
2295:
2292:
2287:
2283:
2279:
2275:
2271:
2267:
2262:
2257:
2253:
2249:
2242:
2239:
2236:
2232:
2226:
2223:
2218:
2212:
2208:
2201:
2198:
2193:
2191:3-540-25127-8
2187:
2183:
2179:
2173:
2170:
2165:
2159:
2155:
2148:
2145:
2140:
2138:0-471-98716-6
2134:
2130:
2126:
2119:
2116:
2111:
2105:
2101:
2094:
2091:
2086:
2080:
2076:
2069:
2066:
2060:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2027:
2023:
2021:
1995:
1990:
1986:
1982:
1977:
1972:
1968:
1964:
1959:
1954:
1950:
1943:
1940:
1935:
1925:
1921:
1917:
1912:
1908:
1901:
1896:
1886:
1882:
1878:
1873:
1869:
1862:
1857:
1847:
1843:
1839:
1834:
1830:
1817:
1814:
1808:
1803:
1798:
1794:
1786:
1785:
1784:
1781:
1777:
1771:
1763:
1758:
1751:
1749:
1746:
1745:yield surface
1739:
1732:
1725:
1704:
1700:
1696:
1691:
1687:
1683:
1678:
1674:
1666:
1665:
1664:
1662:
1661:Mohr's circle
1654:
1652:
1649:
1643:
1634:
1627:
1625:
1623:
1619:
1615:
1611:
1607:
1601:
1593:
1591:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1560:
1558:
1554:
1546:
1541:
1534:
1529:
1527:
1499:
1492:
1489:
1488:
1484:
1481:
1480:
1476:
1473:
1472:
1468:
1465:
1464:
1460:
1457:
1456:
1450:
1419:
1417:
1415:
1394:
1387:
1384:
1383:
1379:
1376:
1375:
1371:
1368:
1367:
1363:
1360:
1359:
1352:
1345:
1337:
1333:
1324:
1317:
1315:
1306:
1302:
1289:
1282:
1278:
1271:
1267:
1263:
1259:
1254:
1250:
1243:
1239:
1232:
1228:
1219:
1212:
1210:
1205:
1200:
1196:
1192:
1187:
1172:
1164:
1162:
1157:
1153:
1148:
1143:
1141:
1136:
1128:
1126:
1124:
1118:
1110:
1108:
1101:
1099:
1097:
1093:
1089:
1088:
1082:
1074:
1069:
1067:
1065:
1061:
1053:
1051:
1049:
1041:Shear banding
1040:
1038:
1036:
1032:
1028:
1024:
1016:
1014:
1009:
1001:
999:
995:
988:
982:
975:
973:
971:
967:
959:
957:
955:
951:
947:
942:
940:
936:
933:
929:
924:
922:
918:
914:
910:
906:
902:
898:
894:
890:
886:
882:
878:
874:
870:
865:
863:
859:
855:
851:
848:
844:
840:
836:
832:
820:
815:
813:
808:
806:
801:
800:
798:
797:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
715:
708:
707:
696:
693:
691:
688:
686:
683:
682:
681:
680:
677:
673:
668:
665:
663:
660:
658:
655:
654:
653:
652:
647:
642:
641:
632:
628:
622:
619:
617:
614:
612:
609:
607:
604:
602:
601:Charles's law
599:
597:
594:
592:
589:
588:
586:
585:
582:
578:
572:
569:
567:
564:
562:
559:
557:
554:
552:
549:
548:
546:
545:
542:
538:
532:
529:
525:
522:
518:
515:
510:
509:non-Newtonian
507:
503:
499:
498:
497:
494:
492:
489:
485:
482:
480:
477:
475:
472:
468:
465:
463:
460:
456:
453:
452:
450:
449:
446:
442:
437:
432:
431:
423:
420:
418:
415:
411:
408:
407:
406:
403:
401:
398:
396:
395:Compatibility
393:
389:
386:
384:
383:Finite strain
381:
380:
379:
376:
374:
371:
369:
366:
364:
361:
357:
354:
353:
352:
349:
347:
344:
343:
339:
334:
333:
322:
319:
318:
317:
316:
311:
306:
303:
301:
298:
296:
293:
292:
291:
290:
287:Conservations
285:
277:
276:
272:
253:
250:
245:
242:
236:
233:
230:
227:
220:
219:
216:
212:
208:
207:
198:
194:
190:
184:
180:
176:
175:
169:
166:
163:
160:
157:
155:
152:
150:
147:
146:
142:
138:
133:
123:
119:
117:
116:Elastic limit
114:
112:
109:
107:
104:
103:
87:
79:
63:
55:
51:
48:behavior for
47:
43:
39:
33:
19:
2623:
2600:. Springer.
2597:
2574:
2555:
2536:
2517:
2485:
2474:
2465:
2461:
2450:
2446:
2437:
2428:
2424:
2411:
2392:
2389:Hill, Rodney
2383:
2358:
2354:
2348:
2325:
2319:
2300:
2294:
2251:
2247:
2241:
2230:
2225:
2206:
2200:
2184:. Springer.
2181:
2178:Yu, Mao-Hong
2172:
2153:
2147:
2128:
2118:
2099:
2093:
2074:
2068:
2018:
1773:
1737:
1730:
1723:
1721:
1658:
1645:
1618:dislocations
1603:
1561:
1555:) where the
1550:
1503:
1482:Simple cubic
1423:
1413:
1408:/áč), where Ï
1398:
1343:
1335:
1331:
1329:
1304:
1300:
1287:
1280:
1276:
1269:
1265:
1261:
1257:
1252:
1248:
1241:
1230:
1226:
1224:
1203:
1198:
1194:
1190:
1185:
1170:
1168:
1155:
1152:shear stress
1149:
1147:
1138:
1120:
1105:
1085:
1078:
1057:
1044:
1020:
1011:
1002:Slip systems
996:
992:
970:malleability
963:
950:cold forming
943:
937:
925:
921:T1 processes
893:dislocations
866:
842:
838:
828:
676:Smart fluids
621:Graham's law
527:
520:
505:
491:Pascal's law
487:
470:
458:
362:
313:Inequalities
196:
192:
182:
178:
18:Plastic flow
2254:: 338â363.
2050:Plastometer
1776:M. T. Huber
1606:Egon Orowan
1553:Hooke's law
1238:strain rate
1180:/m, where Ï
1048:shear bands
909:microcracks
889:crystalline
854:deformation
695:Ferrofluids
596:Boyle's law
368:Hooke's law
346:Deformation
139:typical of
2635:Categories
2261:1704.07297
2061:References
1027:cross-slip
839:plasticity
748:Gay-Lussac
711:Scientists
611:Fick's law
591:Atmosphere
410:frictional
363:Plasticity
351:Elasticity
2577:. Wiley.
2334:cite book
2286:119387816
2077:. Dover.
1987:σ
1969:σ
1951:σ
1922:σ
1918:−
1909:σ
1883:σ
1879:−
1870:σ
1844:σ
1840:−
1831:σ
1795:σ
1701:σ
1697:≥
1688:σ
1684:−
1675:σ
1648:von Mises
1604:In 1934,
1588:ductility
1584:annealing
1490:Hexagonal
1477:Metal: 5
1081:amorphous
989:In metals
966:ductility
926:For many
788:Truesdell
718:Bernoulli
667:Rheometer
662:Rheometry
502:Newtonian
496:Viscosity
246:φ
234:−
88:ϵ
64:σ
2618:(2006).
2596:(1998).
2483:(1953).
2419:(1913).
2391:(1998).
2127:(2002).
2024:See also
1582:such as
1564:fracture
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