Mechanical Properties of Engineered Materials 2008 Part 8 docx

7 Dislocations and Plastic Deformation 7.1 INTRODUCTION Let us begin this chapter by performing the following thought experiment. Imagine picking up a piece of copper tubing that can be bent easily, at least the first time you try to bend it. Now think about what really happens when you bend the piece of copper a few times. You will probably remember from past experience that it becomes progressively harder to bend the piece of copper tubing after each bend. However, you have probably never asked yourself why. Upon some reflection, you will probably come to the conclusion that the response of the copper must be associated with internal changes that occur in the metal during bending. In fact, the strength of the copper, and the progressive hardening of the copper, are associated with the movement of dislocations, and their interactions with defects in the crystalline copper lattice. This is hard to imagine. However, it is the basis for crystalline plasticity in most metallic materials and their alloys. This chapter presents an overview of how dislocation motion and dislocation interactions contribute to plastic deformation in crystalline materials. We begin with a qualitative description of how individual dis- locations move, interact, and multiply. The contributions of individual dislocations to bulk plastic strain are then considered within a simple con- Copyright © 2003 Marcel Dekker, Inc. tinuumframework.Thisisfollowedbyanintroductiontothecrystallogra- phyofslipinhexagonalandcubicmaterials.Therolethatdislocationsplay inthedeformationofsinglecrystalsandpolycrystalsisthenexplained. 7.2DISLOCATIONMOTIONINCRYSTALS AsdiscussedinChap.6,dislocationstendtoglideonclose-packedplanes alongclose-packeddirections.Thisisduetotherelativelylowlatticefriction stressesinthesedirections,Eq.(6.8a)or(6.8b).Furthermore,themotionof dislocationsalongaglideplaneiscommonlyreferredtoasconservative motion.Thisisbecausethetotalnumberofatomsacrosstheglideplane remainsconstant(conserved)inspiteoftheatomicinteractionsassociated withdislocationglide(Fig.6.9).Incontrasttoconservativedislocation motionbyglide,nonconservativedislocationmotionmayalsooccurby climbmechanisms(Fig.6.11).Theseofteninvolvetheexchangeofatoms withvacancies.Sincetheatom/vacancyexchangesmaybeassistedbyboth stressandtemperature,dislocationclimbismorelikelytooccurduring loadingatelevatedtemperature. Sofar,ourdiscussionofdislocationmotionhasfocusedmostlyon straightdislocations.Furthermore,itispresumedthatthedislocationsliein thepositionsoflowestenergywithinthelattice,i.e.,energyvalleys/troughs (Fig.6.10).However,inmanycases,kinkeddislocationsareobserved(Fig. 7.1).Thesehaveinclinedstraightorcurvedlinesegmentsthatalllieonthe sameglideplane(Fig.7.1).Theshapeofthekinkeddislocationsegmentis dependent on the magnitude of the energy difference between the energy peaks and energy valleys in the crystalline lattice. In cases where the energy difference is large, dislocations can minimize their overall line energies by minimizing their line lengths in the higher energy peak regime. This gives rise to sharp kinks (A in Fig. 7.1) that enable dislocations to minimize their line lengths in the high-energy regions. It also maximizes the dislocation line lengths in the low-energy valley regions. In contrast, when the energy difference (between the peaks and the valleys) is small, a diffuse kink is formed (C in Fig. 7.1). The diffuse kink has significant fractions of its length in the low-energy valleys and high-energy peak regions. In this way, a diffuse kink can also minimize the overall line energy of the dislocation. The motion of kinked dislocations is somewhat complex, and will only be discussed briefly in this section. In general, the higher energy regions along the kink tend to move faster than those along the low-energy valleys which have to overcome a larger energy barrier. Once the barriers are over- come, kink nucleation and propagation mechanisms may be likened to the Copyright © 2003 Marcel Dekker, Inc. FIGURE 7.1 Schematic of kinked dislocation configurations between peaks and valleys in a crystalline lattice. Note that sharp kink is formed when energy difference is large, diffuse kink is formed when energy difference is small (B), and most kinks are between the two extremes. (From Hull and Bacon, 1984. Reprinted with permission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. snappingmotionofawhip.Asinthecaseofasnappedwhip,thismaygive risetofasterkinkpropagationthanthatofastraightdislocation.Theover- allmobilityofakinkwillalsodependontheenergydifferencebetweenthe peaksandthevalleys,andtheorientationofthedislocationwithrespectto thelattice. Beforeconcludingthissection,itisimportanttonoteherethatthereis adifferencebetweenasharpkink[AinFig.7.1andFigs7.2(a)and7.2(b)] and a jog, Figs 7.2(a) and 7.2(b). A kink has all its segments on the same plane as the glide plane (Fig. 7.1). In contrast, a jog is produced by disloca- tion motion out of the glide plane as the rest of the dislocation line. Kinks and jogs may exist in edge and screw dislocations, Figs 7.2(a)–7.2(d). However, kinked dislocations tend to move in a direction that is perpendi- cular to the dislocation line, from one valley position to the other. Furthermore, kinks may also move faster than straight dislocation seg- ments, while jogged dislocation segments are generally not faster than the rest of the dislocation line. In fact, they may be less mobile than the rest of the dislocation line, depending on the directions of their Burgers vectors relative to those of the unjogged segments. FIGURE 7.2 (a), (b) Kinks in edge and screw dislocations; (c), (d) jogs in edge and screw dislocations. The slip planes are shaded. (From Hull and Bacon, 1984. Reprinted with permission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. 7.3 DISLOCATION VELOCITY When the shear stress that is applied to a crystal exceeds the lattice friction stress, dislocations move at a velocity that is dependent on the magnitude of the applied shear stress. This has been demonstrated for LiF crystals by Johnston and Gilman (1959). By measuring the displacement of etch pits in crystals with low dislocation densities, they were able to show that the dislocation velocity is proportional to the applied shear stress. Their results are presented in Fig. 7.3 for both screw and edge dislocations. Note that, at the same stress level, edge dislocations move at faster speeds (up to 50 times faster) than screw dislocations. Also, the velocities of dislocations extend over 12 orders of magnitude on the log–log plot shown FIGURE 7.3 Dependence of dislocation velocity on applied shear stress. (From Johnston and Gilman, 1959. Reprinted with permission from J. Appl. Phys.) Copyright © 2003 Marcel Dekker, Inc. inFig.7.3.However,foruniformdislocationmotion,thelimitingvelocity forbothscrewandedgedislocationscorrespondstothevelocityoftrans- verseshearwaves.Also,dampingforcesincreasinglyopposethemotionof dislocationsatvelocitiesabove10 3 cm/s. Dislocationvelocitiesinawiderangeofcrystalshavebeenshownto bestronglydependentonthemagnitudeoftheappliedshearstress(Fig. 7.4),althoughthedetailedshapesofthedislocationvelocityversusstress curvesmayvarysignificantly,asshowninFig.7.4.Forthestraightsections ofthedislocation–velocitycurves,itispossibletofitthemeasuredvelocity datatopower-lawequationsoftheform: v¼AðÞ m ð7:1Þ wherevisthedislocationvelocity,istheappliedshearstress,Aisa materialconstant,andmisaconstantthatincreaseswithdecreasingtem- perature.Anincreaseindislocationvelocitywithdecreasingtemperature hasalsobeendemonstratedbySteinandLow(1960)inexperimentsonFe– 3.25Sicrystals(Fig.7.5).Thisincreaseisassociatedwiththereduceddamp- ing forces due to the reduced scattering (phonons) of less frequent lattice vibrations at lower temperatures. FIGURE 7.4 Dependence of dislocation velocity on applied shear stress. The data are for 208C except for Ge (4508C) and Si (8508C). (From Haasen, 1988. Reprinted with permission from Cambridge University Press.) Copyright © 2003 Marcel Dekker, Inc. 7.4DISLOCATIONINTERACTIONS Thepossibleinteractionsbetweenscrewandedgedislocationswillbedis- cussedinthissection.Considertheedgedislocations(Burgersvectorsper- pendiculartothedislocationlines)ABandXYwithperpendicularBurgers vectors,b 1 andb 2 ,showninFig7.6.ThemovingdislocationXY[Fig.7.6(a)] glidesonaslipplanethatisastationarydislocationAB.Duringtheinter- section,ajogPP 0 correspondingtoonelatticespacingisproducedasdis- locationXYcutsdislocationAB,Fig.7.6(b).SincethejoghasaBurgers vectorthatisperpendiculartoPP 0 ,itisanedgejog.Also,sincetheBurgers vectorofPP 0 isthesameasthatoftheoriginaldislocation,AB,thejogwill continuetoglidealongwiththerestofthedislocation,ifthereisalarge enoughcomponentofstresstodriveitalongtheslipplane,whichisper- pendiculartothatoflinesegmentsAPorP 0 B,Fig.7.6(b). Letusnowconsidertheinteractionsbetweentwoedgedislocations (XYandAB)withparallelBurgersvectors,Fig.7.7(a).Inthiscase,disloca- FIGURE7.5Dependenceofdislocationvelocityontemperatureandapplied shear stress in Fe–3.25Si Crystals. (From Stein and Low, 1960. Reprinted with permission from J. Appl. Phys.) Copyright © 2003 Marcel Dekker, Inc. tionXYintersectsdislocationAB,andproducestwoscrewjogsPP 0 and QQ 0 .ThejogsPP 0 andQQ 0 arescrewinnaturebecausetheyareparallelto theBurgersvectorsb 1 andb 2 ,respectively,Figs7.7(a)and7.7(b).Sincethe joggedscrewdislocationsegmentshavegreatermobilitythantheedgedis- locationstowhichtheybelong,theywillnotimpedetheoveralldislocation motion.Hence,interactionsbetweenedgedislocationsdonotsignificantly affectdislocationmobility. Thisisnottrueforinteractionsinvolvingscrewdislocations.For example,inthecaseofaright-handedscrewdislocationthatintersectsa movingedgedislocation[Fig.7.8(a)],thedislocationsegmentPP 0 glides FIGURE7.7InteractionsbetweentwoedgedislocationswithparallelBurgers vectors:(a)beforeintersection;(b)afterintersection.(FromHullandBacon, 1984.ReprintedwithpermissionfromPergamonPress.) FIGURE7.6Interactionsbetweentwoedgedislocationswithperpendicular Burgers vectors: (a) before intersection; (b) after intersection. (From Hull and Bacon, 1984. Reprinted with permission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. downonelevel(fromoneatomicplanetotheother)followingaspiralpath (staircase)alongthedislocationlineXY,asitcutsthescrewdislocationXY, Fig.7.8(b).ThisproducesajogPP 0 inAB,andajogQQ 0 inXY.Hence,the segmentsAP 0 andPBlieondifferentplanes,Fig.7.8(b).Furthermore,since theBurgersvectorsofthedislocationlinesegmentsPP 0 andQQ 0 areper- pendiculartotheirlinesegments,thejogsareedgeincharacter.Therefore, theonlywaythejogcanmoveconservativelyisalongtheaxisofthescrew dislocation,asshowninFig.7.9.Thisdoesnotimpedethemotionofthe screwdislocation,providedthejogglidesontheplane(PP 0 RR 0 ). However,sinceedgedislocationcomponentscanonlymoveconserva- tivelybyglideonplanescontainingtheirBurgersvectorsandlinesegments, themovementoftheedgedislocationtoA 0 QQ 0 B(Fig.7.9)wouldrequire nonconservativeclimbmechanismsthatinvolvestress-andthermally assistedprocesses.Thiswillleavebehindatrailofvacanciesorinterstitials, dependingonthedirectionofmotion,andthesignofthedislocation.Thisis illustratedinFig.7.10forajoggedscrewdislocationthatproducesatrailof vacancies.Notethatthedislocationsegmentsbetweenthejogsarebowed duetotheeffectsoflinetension.Bowingofdislocationsduetolinetension effectswillbediscussedinthenextsection.Inclosing,however,itisimpor- tanttonoteherethattheinteractionsbetweentwoscrewdislocations(Fig. 7.11)cangiverisetosimilarphenomenatothosediscussedabove.Itisa useful exercise to try to work out the effects of such interactions. FIGURE 7.8 Interactions between right-handed screw dislocation and edge dislocations: (a) before intersection; (b) after intersection. (From Hull and Bacon, 1984. Reprinted with permission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. FIGURE 7.9 Movement of edge jog on a screw dislocation; conservative motion of jog only possible on plane PP 0 RR. Motion of screw dislocation to A 0 QQ 0 B would require climb of the jog along plane PQQ 0 P. (From Hull and Bacon, 1984. Reprinted with permission from Pergamon Press.) FIGURE 7.10 Schematic illustration of trail of vacancies produced by glide of screw dislocation. (From Hull and Bacon, 1984—reprinted with permission from Pergamon Press.) FIGURE 7.11 Interactions between two screw dislocations: (a) before intersec- tion; (b) after intersection. (From Hull and Bacon, 1984. Reprinted with per- mission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. [...]... in the field of crystal plasticity One possible mechanism by which the fifth strain component may be accommodated involves a mechanism of deformation-induced twinning This occurs by the co-ordinated movement of several dislocations (Fig 7. 18) However, further work is still needed to develop a fundamental understanding of the role of twinning in titanium and other h.c.p metals/alloys 7 .8. 5 Partial or Extended... explanation of the above hardening behavior has been presented by Kuhlmann-Wilsdorf (1962, 19 68) She attributes the low levels of Stage I hardening to heterogeneous slip of a low density of dislocations In this theory, Stage II slip corresponds to the onset of significant dislocation– dislocation interactions, but not necessarily the onset of multiple slip This results ultimately in the formation of a dislocation... plastic bending of a rod, as shown schematically in Fig 7.32 The initial configuration of the rod of length l and width t is shown in Fig 7.32(a) On the application of a bending moment, a curved profile with a radius of curvature, r, is produced The length of the outer surface is increased from l to l þ l, and the length of the inner surface is decreased from l to l À l Hence, the deformation of the outer... orientation factor must also account for the stronger effects of less favorably oriented grains This was first considered by Taylor (19 38) for " the deformation of f.c.c crystals, which were shown to have values of m of " $ 3:1 More recent simulations have also shown that the values of m are close to 3.0 for b.c.c crystals However, due to the large number of possible slip systems in b.c.c metals, the simulations... readily by f.c.c and b.c.c crystals In the case of f.c.c crystals, Taylor (19 38) has shown that only five of the 12 possible {111} h110i slip systems are independent, although there are 384 combinations of five slip systems that can result in any given strain Similar results have been reported by Groves and Kelly (1963) for b.c.c crystals in which 384 sets of five {110} h111i slip systems can be used to... compatibility in the presence of stress gradients, e.g., near grain boundaries The total number of GNDs is given simply by 2l=b The density of GNDs, G , may also be expressed as the ratio of the number of GNDs divided by the area ðltÞ This gives   d" 2Ál 1 dt ¼ ¼ ð7:20Þ G ¼ bðltÞ rb b The total density of dislocations, tot , in a polycrystal is, therefore, given by the sum of the statistically stored... schematic of a typical shear stress versus shear strain response of a single crystal is shown in Fig 7. 28 During the early stages of deformation, Stage I slip occurs by easy glide in a single slip direction along a single slip plane There is limited interaction between dislocations, and the extent of hardening is limited However, due to the constraints imposed by the specimen grips, the slipped segments of. .. can interact, rapid Stage II hardening may be observed in some crystals at the onset of plastic deformation FIGURE 7. 28 Three stages of plastic deformation in a single crystal Copyright © 2003 Marcel Dekker, Inc FIGURE 7.29 Schematic illustration of the effects of deformation constraint on the deformation behavior of a single crystal: (a) before deformation; (b) deformation without grip constraint;... occurs as a result of the combined effects of several dislocations that glide on multiple slip planes For simplicity, let us consider the glide of a single dislocation, as illustrated schematically in Fig 7.16 The crystal of height, h, is displaced by a horizontal distance, b, the Burgers vector, due to the glide of a single dislocation across distance, L, on the glide plane Hence, partial slip across... zig-zag motion of atoms required for slip in the h110i directions may not be energetically favorable since the movement of dislocations requires somewhat difficult motion of the ‘‘white’’ atoms over the ‘‘shaded’’ atoms in Fig 7.20 The ordinary h110i dislocations may, therefore, dissociate into partial dislocations with lower overall energies than those of the original h110i type dislocations The partial dislocations . (phonons) of less frequent lattice vibrations at lower temperatures. FIGURE 7.4 Dependence of dislocation velocity on applied shear stress. The data are for 208C except for Ge (4508C) and Si (85 08C) Bacon, 1 984 . Reprinted with permission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. snappingmotionofawhip.Asinthecaseofasnappedwhip,thismaygive risetofasterkinkpropagationthanthatofastraightdislocation.Theover- allmobilityofakinkwillalsodependontheenergydifferencebetweenthe peaksandthevalleys,andtheorientationofthedislocationwithrespectto thelattice. Beforeconcludingthissection,itisimportanttonoteherethatthereis adifferencebetweenasharpkink[AinFig.7.1andFigs7.2(a)and7.2(b)] and. and Bacon, 1 984 . Reprinted with permission from Pergamon Press.) FIGURE 7.10 Schematic illustration of trail of vacancies produced by glide of screw dislocation. (From Hull and Bacon, 1 984 —reprinted