5 IntroductiontoPlasticity 5.1INTRODUCTION Afterahighenoughstressisreached,thestrainnolongerdisappearsonthe releaseofstress.Theremainingpermanentstrainiscalleda‘‘plastic’’strain (Fig.5.1).Additionalincrementalplasticstrainsmayalsobeaccumulated onsubsequentloadingandunloading,andthesecanleadultimatelyto failure.Insomecases,thedimensionalandshapechangesassociatedwith plasticitymayleadtolossoftolerance(s)andprematureretirementofa structureorcomponentfromservice.Anunderstanding ofplasticityis, therefore,importantinthedesignandanalysisof engineeringstructures andcomponents. Thischapter presentsabasicintroductiontothemechanismsand mechanicsofplasticityinmonolithicmaterials.Following a simple review of the physical basis for plasticity in different classes of monolithic materials (ceramics, metals, intermetallics, and polymers), empirical plastic flow rules are introduced along with multiaxial yield criteria. Constitutive equations of plasticity are then presented in the final section of the chapter. Copyright © 2003 Marcel Dekker, Inc. 5.2PHYSICALBASISFORPLASTICITY 5.2.1PlasticityinCeramics Mostceramicsonlyundergoonlyelasticdeformationpriortotheonsetof catastrophicfailureatroomtemperature.Hence,mostreportsonthe mechanicalpropertiesofceramicsareoftenlimitedtoelasticproperties. Furthermore,mostceramistsreportflexuralpropertiesobtainedunder three-orfour-pointbending.Typicalstrengthpropertiesofselectedceramic materialsarepresentedinTable5.1.Notethatceramicsarestronger(almost 15 times stronger) in compression than in tension. Also, the flexural strengths are intermediate between the compressive and tensile strength levels. Reasons for these load-dependent properties will be discussed in subsequent chapters. For now, it is simply sufficient to state that the trends are due largely to the effects of pre-existing defects such as cracks in the ceramic structures. The limited capacity of ceramic materials for plastic deformation is due largely to the limited mobility of dislocations in ceramic structures. The latter may be attributed to their large Burgers (slip) vectors and unfavorable (for plastic deformation) ionically/covalently bonded crystal structures. Plastic deformation in ceramics is, therefore, limited to very small strains (typically < 0.1–1%), except at elevated temperatures where thermally acti- vated dislocation motion and grain boundary sliding are possible. In fact, the extent of plasticity at elevated temperatures may be very significant in FIGURE 5.1 Schematic illustration of plastic strain after unloading. Copyright © 2003 Marcel Dekker, Inc. ceramicsdeformedatelevatedtemperature,andsuperplasticity(strainlevels upto1000%plasticstrain)hasbeenshowntooccurduetocreepphenom- enainsomefine-grainedceramicsproduced. However,inmostceramics,theplasticstrainstofailurearerelatively small(<1%),especiallyundertensileloadingwhichtendstoopenuppre- existingcracksthataregenerallypresentafterprocessing.Also,sinceinci- pientcracksinceramicstendtocloseupundercompressiveloading,the strengthlevelsandthetotalstraintofailureincompressionareoftengreater thanthoseintension.Furthermore,verylimitedplasticity(permanent strainsonremovalofappliedstresses)mayoccurinsomeceramicsorcera- micmatrixcompositesbymicrocrackingorstress-inducedphasetransfor- mations. MicrocrackinggenerallyresultsinareductioninYoung’smodulus,E, whichmaybeusedasaglobal/scalarmeasureofdamage(Fig.5.2).Ifwe assume that the initial ‘‘undeformed’’ material has a damage state of zero, while the final state of damage at the point of catastrophic failure corre- sponds to a damage state of 1, we may estimate the state of damage using some simple damage rules. For an initial Young’s modulus of E 0 and an intermediate damage state, the damage variable, D, is given simply by D ¼ 1 ÀE=E 0 . Damage tensors may also be used to obtain more rigorous descriptions of damage (Lemaitre, 1991). Plasticity in ceramics may also occur by stress-induced phase transfor- mations. This has been observed in partially stabilized zirconia (ZrO 2 alloyed with CaO, Y 2 O 3, or CeO to stabilize the high-temperature tetragonal TABLE 5.1 Strength Properties of Selected Ceramic Materials Material Compressive strength (MPa (ksi)] Tensile strength [MPa (ksi)] Flexural strength [MPa (ksi)] Modulus of elasticity [GPa (10 6 psi)] Alumina (85% dense) 1620 (235) 125 (18) 295 (42.5) 220 (32) Alumina (99.8% dense) 2760 (400) 205 (30) 345 (60) 385 (56) Alumina silicate 275 (40) 17 (2.5) 62 (9) 55 (8) Transformation toughened zirconia 1760 (255) 350 (51) 635 (92) 200 (29) Partially stabilized zirconia þ9% MgO 1860 (270) — 690 (100) 205 (30) Cast Si 3 N 4 138 (20) 24 (3.5) 69 (10) 115 (17) Hot-pressed Si 3 N 4 3450 (500) — 860 (125) — Sources: After Hertzberg, 1996. Reprinted with permission from John Wiley. a Guide to Engineering Materials. vol. 1(1). ASM, Metals Park, OH, 1986, pp 16, 64, 65. Copyright © 2003 Marcel Dekker, Inc. phasedowntoroomtemperature).Undermonotonicloading,themeta- stabletetragonalphasecanundergostress-inducedphasetransformations fromthetetragonaltothemonoclinicphase.Thisstress-inducedphase transformationisassociatedwithavolumeincreaseof$4%,andcan giverisetoaformoftougheningknownastransformationtoughening, whichwillbediscussedinCh.13. Stress-inducedphasetransformationsoccurgraduallyinpartiallysta- bilizedzirconia,andtheygiverisetoagradualtransitionfromlinearityin theelasticregime,tothenonlinearsecondstageofthestress–straincurve showninFig.5.3.Thesecondstageendswhenthestress-inducedtransfor- mationspreadscompletelyacrossthegaugesectionofthespecimen.Thisis followedbythefinalstageinwhichrapidhardeningoccursuntilfailure.Itis importanttonotethatthetotalstraintofailureislimited,eveninpartially stabilizedzirconiapolycrystals.Also,asinconventionalplasticity,stress- inducedtransformationmaybeassociatedwithincreasing,level,ordecreas- ingstress–strainbehavior(Fig5.4). 5.2.2PlasticityinMetals Incontrasttoceramics,plasticdeformation inmetalsistypicallyassociated withrelativelylargestrainsbeforefinalfailure.ThisisillustratedinFig.5.5 usingdataobtained foranaluminumalloy.Ingeneral,thetotalplastic strainscanvarybetween5and100%inductilemetalsdeformedtofailure at room temperature. However, the elastic portion of the stress–strain curve is generally limited to strains below $ 0:1 to 1%. Furthermore, metals and their alloys may exhibit stress–strain characteristics with rising, level, or FIGURE5.2Schematicshowingthechangeinmodulusdue to damage during loading and unloading sequences. Copyright © 2003 Marcel Dekker, Inc. decreasing stress, as shown in Fig. 5.4. Materials in which the stress level remains constant with increasing strain [Fig. 5.4(b)] are known as elastic– perfectly plastic. Materials in which the stress level decreases with increasing strain are said to undergo strain softening [Fig. 5.4(c)], while those in which the stress level increases with increasing strain are described as strain hard- ening materials, Fig. 5.4(a). FIGURE 5.3 Schematic of the three stages of deformation in material under- going stress-induced phase transformation. (After Evans et al., 1981.) FIGURE 5.4 Types of stress–strain response: (a) strain hardening; (b) elastic– perfectly plastic deformation; (c) strain softening. Copyright © 2003 Marcel Dekker, Inc. Strainhardeningoccursasaresultofdislocationinteractionsinthe fullyplasticregime.Thesemayinvolveinteractionswithpointdefects (vacancies,interstitials,orsolutes),linedefects(screw,edge,ormixeddis- locations),surfacedefects(grainboundaries,twinboundaries,orstacking faults),andvolumedefects(porosity,entrappedgases,andinclusions).The dislocationinteractionsmaygiverisetohardeningwhenadditionalstresses mustbeappliedtoovercometheinfluenceofdefectsthatrestrictdislocation motion.Thismayresultinrisingstress–straincurvesthatarecharacteristic ofstrainhardeningbehavior,Fig.5.4(a). Asdiscussedearlier,thestress–straincurvesmayalsoremainlevel [Fig.5.4(b)],ordecreaseorincreasecontinuouslywithincreasingstrain, Fig.5.4(c).Thereasonsforsuchbehavioraregenerallycomplex,andnot fullyunderstoodatpresent.However,thereissomelimitedevidencethat suggeststhatelastic–perfectlyplasticbehaviorisassociatedwithslipplanar- ity,i.e.,sliponaparticularcrystallographicplane,whilestrainsoftening tendstooccurincaseswheresliplocalizesonaparticularmicrostructural featuresuchasaprecipitate.Theonsetofmacroscopicyielding,therefore, correspondstothestressneededtoshearthemicrostructuralfeature.Once theinitialresistancetoshearisovercome,thematerialmayofferdecreasing resistancetoincreasingdisplacement,givingriseultimatelytostrainsoft- eningbehavior,Fig.5.4(c). Sincethemovingdislocationsinteractwithsoluteclouds,serrated yieldingphenomenamaybeobservedinthestress–strainbehavior[Fig 5.6).Differenttypesofserratedyieldingphenomenahavebeenreported due to the interactions of dislocations with internal defects such as solutes and interstitials. The phenomenon is generally referred to as the Portevin– FIGURE 5.5 Stress–strain behavior in an aluminum alloy. (After Courtney, 1990. Reprinted with permission from McGraw-Hill.) Copyright © 2003 Marcel Dekker, Inc. LeChateliereffect,inhonorofthetwoFrenchmenwhofirstreportedit (PortevinandLeChatelier,1923).Theserrationsarecausedbythepinning andunpinningofgroupsofdislocationsfromsolutesthatdiffusetowardsit asitmovesthroughalattice.Themechanismsisparticularlyeffectiveat particularparametricrangesofstrain-rateandtemperature(Cottrell,1958). Finallyinthissection,itisimportanttodiscusstheso-calledanom- alousyieldphenomenathathasbeenreportedinsomeplaincarbonsteels (Fig.5.7).Thestress–strainplotsforsuchmaterialshavebeenobservedto exhibit double yield points in some annealed conditions, as shown in Fig. 5.7. The upper yield point (UYP) corresponds to the unpinning of disloca- tions from interstitial carbon clouds. Upon unpinning, the load drops to a lower yield point (LYP). Lu ¨ der’s bands (shear bands inclined at $ 458 degrees to the loading axis) are then observed to propagate across the FIGURE 5.6 Types of serrated yielding phenomena: (a) Type A; (b) Type B; (c) Type C; (d) Type S. (Types A–C After Brindley and Worthington, 1970; Type S After Pink, 1994. Reprinted with permission from Scripta Met.) Copyright © 2003 Marcel Dekker, Inc. gaugesectionsofthetensilespecimens,asthestrainisincreasedfurther (Fig.5.7).Notethatthestressremainsrelativelyconstantintheso-called Lu ¨ der’sstrainregime,althoughserrationsmaybeobservedwithsufficiently sensitiveinstrumentation.Thestrainattheendofthisconstantstressregime isknownastheLu ¨ dersstrain.Thiscorrespondstothepointatwhichthe Lu ¨ der’sbandshavespreadcompletelyacrossthegaugesectionofthespeci- men.Beyondthispoint,thestressgenerallyincreaseswithincreasingdueto themultipleinteractionsbetweendislocations,asdiscussedearlierforcon- ventionalmetallicmaterials(Fig.5.5). 5.2.3PlasticityinIntermetallics AsdiscussedinChap.1,intermetallicsarecompoundsbetweenmetalsand other metals.Duetotheirgenerallyordered structures,andpartiallycova- lentlyor ionicallybondedstructures,intermetallics generally exhibit only limited plasticity at room-temperature. Nevertheless, some ductility has been reported for ordered gamma-based titanium aluminide intermetallics FIGURE5.7Anomalousyieldingin1018 plaincarbonsteel.(After Courtney, 1990. Reprinted with permission from McGraw-Hill.) Copyright © 2003 Marcel Dekker, Inc. withduplex 2 þmicrostructures.Thesetwophaseintermetallicshave roomtemperatureplasticelongationstofailureofabout1–2%duetodefor- mationbyslipandtwinning(KimandDimiduk,1991).Theirlimitedroom- temperatureductilityhasbeenattributedtothesoakingupofinterstitial oxygenbythe 2 phase.Thisresultsinareductionininterstitialoxygen contentinthegammaphase,andtheincreaseddislocationmobilityofdis- locationsinthelatterwhichgivesrisetotheimprovedductilityintwo-phase gammatitaniumaluminides(Vasudevanetal.,1989). Niobiumaluminideintermetallicswithplasticelongationsof10–30% havealsobeendevelopedinrecentyears(Houetal.,1994;Yeet.al.,1998). TheductilityintheseB2(orderedbody-centeredcubicstructures)interme- tallicshasbeenattributedtothepartialorderintheirstructures.Similar improvementsinroom-temperature(10–50%)ductilityhavebeenreported inNi 3 Alintermetallicsthatarealloyedwithboron(AokiandIzumi,1979; Liuetal.,1983),andFe 3 Alintermetallicsalloyedwithboron(Liuand Kumar,1993). Theimprovementsintheroom-temperatureductilitiesofthenickel andironaluminideintermetallicshavebeenattributedtothecleaningupof thegrainboundariesbytheboronadditions.However,thereasonsforthe improvedductilityinorderedorpartiallyorderedintermetallicsarestillnot fullyunderstood,andareunderinvestigation.Similarly,anomalousyield- pointphenomena(increasingyieldstresswithincreasingtemperature)and thetransitionfrombrittlebehavioratroomtemperaturetoductilebehavior atelevatedtemperaturearestillunderinvestigation. 5.2.4PlasticityinPolymers Plasticityinpolymersisnotcontrolledbydislocations,althoughdisloca- tionsmayalsoexistinpolymericstructures.Instead,plasticdeformationin polymersoccurslargelybychainsliding,rotation,andunkinking(Figs1.7 and1.8).Suchchainslidingmechanismsdonotoccursoreadilyinthree- dimensional(thermoset)polymers(Fig.1.8).However,chainslidingmay occurrelativelyeasilyinlinear(thermoplastic)polymerswhentheslidingof polymerchainsisnothinderedsignificantlybyradicalsidegroupsorother sterichindrances.Theplasticdeformationofpolymersisalsoassociated withsignificantchangesinentropy,whichcanalterthelocaldrivingforce fordeformation. Elasticityandplasticity[Fig.5.8(a)]inrubberypolymersmayresultin strainlevelsthatarebetween 100and1000%atfracture.Such largestrains areassociatedwithchainsliding, unkinking,anduncoilingmechanisms. Furthermore, unloading does not result in a sudden load drop. Instead, unloading follows a time-dependent path, as shown in Fig. 5.8(b). Copyright © 2003 Marcel Dekker, Inc. Elasticityandplasticityinrubberypolymersare,therefore,oftentime dependent,sincetimeisoftenrequiredforthepolymerchainstoflowto andfromthedeformedconfigurations.Cyclicdeformationmayresultin hysterisisloopssincethestraingenerallylagsthestress(Fig.5.9),and anomalousstress–strainbehaviormayalsobeassociatedwithchaininter- actionswithdistributedsidegroupswhichareoftenreferredtoassteric hindrances. Crystallinepolymers(Fig.1.9)mayalsoexhibitinterestingstress– strainbehavior.Theminimuminthestress–straincurveisduetocold drawingandthecompetitionbetweenthebreakdownoftheinitialcrys- tallinestructure,andthereorganizationintoahighlyorientedchain structure. 5.3ELASTIC–PLASTICBEHAVIOR AgenericplotofstressversusstrainispresentedinFig.5.10.Thisshowsa transitionfromalinear‘‘elastic’’regimetoanonlinear‘‘plasticregime.’’ Thelinearelasticregimepersistsuptotheproportionallimit,atwhichthe deviationfromlinearelasticbehavioroccurs.However,theonsetofnon- linearstress–strainbehaviorisgenerallydifficulttodetermineexperimen- tally.Anengineeringoffsetyieldstrengthis,therefore,definedbydrawinga lineparalleltotheoriginallinearelasticline,butoffsetbyagivenstrain (usuallyanengineeringstrainlevelof0.002or0.2%). FIGURE5.8Elastic–plasticdeformationinrubberypolymers.(a)Rubberrand deformed at room temperature. (After Argon and McClintock, 1990) (b) Viscoelasticity in a rubbery polymer. (After Hertzberg, 1996. Reprinted with permission from John Wiley.) Copyright © 2003 Marcel Dekker, Inc. [...]... 3 16 403 410 431 AFC-77 PH 15-7Mo As-rolled As-rolled As-rolled Q þ T (2058C) Q þ T (4258C) Q þ T (65 08C) Q þ T (2058C) Q þ T (4258C) Q þ T (65 08C) Annealed plate Annealed plate Annealed plate Annealed plate Annealed bar Annealed bar Annealed bar Variable Variable 315 420 415 725 585 965 1590 1810 1150 1 260 62 0 800 167 5 1875 1 365 1470 855 965 275 725 240 565 310 65 5 250 565 275 515 275 515 65 5 860 560 – 160 5... 35 35 20 10– 26 2–35 61 40 17 35 51 66 38 44 60 — — — — — — — 32–74 — 805 950 925 1205 860 1000 995 1275 16 15 14 8 40 28 30 — 103–125 185–195 215– 260 220 290–295 295–315 9–12 4–9 4 6 — — — 250 345 395 415 145 475 360 485 475 485 240 530 17 18 10 13 23 11 — — — — — — TABLE 5.3 Continued Material 7075 7075 7178 Plastics ACBS Acetal Poly(tetra fluorethylene) Poly(vinylidene fluoride) Nylon 66 Polycarbonate... 560 – 160 5 835–2140 380–1450 895–1515 Titanium alloys Ti-5Al-2.5Sn Ti-8Al-I Mo-1V Ti-6Al-4V Ti-13V-11Cr-3Al Annealed Duplex annealed Annealed Solution + age Magnesium alloys AZ31B Annealed AZ80A Extruded bar ZK60A Artificially aged Aluminum alloys 2219 2024 2024 2014 60 61 7049 -T31, -T351 -T3 -T6, -T651 -T6, -T651 -T4, -T451 -T73 Copyright © 2003 Marcel Dekker, Inc 39 20 12 11 14 22 10 10 19 55 60 50... Treatment -T6 -T73 -T6 Yield strength (MPa) Reduction in area Tensile Elongation (1.28 cm diameter) in 5-cm strength (%) gauge (%) (MPa) 505 415 540 570 505 60 5 Medium impact Homopolymer — — 46 69 — — — — — Low density — — — — — — — 69 11 11 11 — — — 6 14 25–75 — — 14–48 100–450 — 35–48 59–83 55 69 7–21 41–54 — 100–300 60 –300 130 50–800 1.5–2.4 50–1000 — — — — — — Sources: After Hertzberg, 19 96 Reprinted... Argon, A S., (eds) (1 963 ) Mechanical Behavior of Materials, Addison Wesley, MA Nadai, A (1950) Theory of Flow and Fracture of Solids 2nd ed Pink (1994) Scripta Met vol 30, p 767 Portevin, A and LeChatelier, F (1923) Sur un Phenomene Observe lors de Pessai de traction d’alliages en cours de Transformation, Acad Sci Compt Rend vol 1 76, pp 507–510 Prager, W and Hodge, P.G., Jr (1951) Theory of Perfectly Plastic... most materials, however, the portion of the stress strain curve between the onset of bulk yielding (bulk yield stress) and the onset of necking (the ultimate tensile stress) tends to exhibit the type of rising stress–strain behavior shown in Fig 5.4(a) 5.4 EMPIRICAL STRESS–STRAIN RELATIONSHIPS It is currently impossible to develop ab initio methods for the prediction of the stress–strain behavior of materials. .. polymer The arbitrary offset strain level of 0.002 is recommended by the ASTM E-8 code for tensile testing for the characterization of stresses required for bulk yielding However, it is important to remember that the offset strain level is simply an arbitrary number selected by a group of experts with a considerable amount of combined experience in the area of tensile testing Above the offset yield strength,... magnitudes and directions of the local axial and shear components of stress To avoid unnecessary dependence on the choice of coordinate systems, stress invariants of the stress tensor are often defined for the local stress states These stress invariants are independent of the choice of co-ordinate system, and they can be used to develop yielding criteria that are independent of co-ordinate system Multiaxial... Alloys Institute of Metals, London, p 1 Courtney, T (1990) Mechanical Behavior of Materials McGraw-Hill, New York Hertzberg, R W (19 96) Deformation and Fracture Mechanics of Engineering Materials 4th ed., John Wiley, New York Hill, R (1950) The Mathematical Theory of Plasticity, Oxford University Press, Oxford, UK Hollomon, J H (1945) Trans AIME vol 162 , p 248 Hou, D H., Shyue, J., Yang, S S and Fraser,... ultimate FIGURE 5.11 Schematic illustration of gauge deformation in the elastic and plastic regimes Copyright © 2003 Marcel Dekker, Inc FIGURE 5.12 Hardening versus geometrical instability: (a) rate of hardening > rate of geometrical instability formation; (b) rate of hardening ¼ rate of geometrical instability formation (onset of necking); (c) rate of hardening < rate of geometrical instability formation . Inc. 5.2PHYSICALBASISFORPLASTICITY 5.2.1PlasticityinCeramics Mostceramicsonlyundergoonlyelasticdeformationpriortotheonsetof catastrophicfailureatroomtemperature.Hence,mostreportsonthe mechanicalpropertiesofceramicsareoftenlimitedtoelasticproperties. Furthermore,mostceramistsreportflexuralpropertiesobtainedunder three-orfour-pointbending.Typicalstrengthpropertiesofselectedceramic materialsarepresentedinTable5.1.Notethatceramicsarestronger(almost 15. dense) 162 0 (235) 125 (18) 295 (42.5) 220 (32) Alumina (99.8% dense) 2 760 (400) 205 (30) 345 (60 ) 385 ( 56) Alumina silicate 275 (40) 17 (2.5) 62 (9) 55 (8) Transformation toughened zirconia 1 760 (255). 350 (51) 63 5 (92) 200 (29) Partially stabilized zirconia þ9% MgO 1 860 (270) — 69 0 (100) 205 (30) Cast Si 3 N 4 138 (20) 24 (3.5) 69 (10) 115 (17) Hot-pressed Si 3 N 4 3450 (500) — 860 (125) — Sources: