8 Dislocation Strengthening Mechanisms 8.1 INTRODUCTION The dislocation strengthening of metals and their alloys is perhaps one of the major technological accomplishments of the last 100 years. For example, the strength of pure metals such as aluminum and nickel have been improved by factors of 10–50 by the use of defects that restrict dislocation motion in a crystal subjected to stress. The defects may be point defects (solutes or interstitials), line defects (dislocations), surface defects (grain boundaries or twin boundaries), and volume defects (precipitates or disper- sions). The strain fields that surround such defects can impede the motion of dislocations, thus making it necessary to apply higher stresses to promote the movement of dislocations. Since yielding and plastic flow are associated primarily with the movement of dislocations, the restrictions give rise ulti- mately to intrinsic strengthening. The basic mechanisms of intrinsic strengthening are reviewed in this chapter, and examples of technologically significant materials that have been strengthened by the use of the strengthening concepts are presented. The strengthening mechanisms that will be considered include: 1. Solid solution strengthening (dislocation interactions with solutes or interstitials). Copyright © 2003 Marcel Dekker, Inc. 2. Dislocation strengthening which is also known as work/strain hardening (dislocation interactions with other dislocations). 3. Boundary strengthening (dislocation interactions with grain boundaries or stacking faults). 4. Precipitation strengthening (dislocation interactions with preci- pitates). 5. Dispersion strengthening (dislocation interactions with dispersed phases). Note the above sequence of dislocation interactions with: zero-dimensional point defects (solutes or interstitials); one-dimensional line defects (other dislocations; two-dimensional defects (grain boundaries or stacking faults), and three-dimensional defects (precipitates or dispersoids). 8.2 DISLOCATION INTERACTIONS WITH OBSTACLES Before presenting the specific details of individual dislocation strengthening mechanisms, it is important to examine the interactions of dislocations with arrays of obstacles such as solutes/interstitials and particles/precipitates (Fig. 8.1). When dislocations encounter such arrays as they glide through a lattice under an applied stress, they are bent through an angle, , before they can move on beyond the cluster of obstacles (note that 0 <<1808Þ. The angle, , is a measur e of the stre ngth of the obstacle, with weak obsta- cles having values of close to 1808, and strong obstacles having obstacles close to 0 8. It is also common to define the strength of a dislocation interaction by the angle, 0 ¼ 180 À through which the interaction turns the dislocation (Fig. 8.1). Furthermore, the number of obstacles per unit length (along the dislocation) depends strongly on . For weak obstacles with $ 1808, the FIGURE 8.1 Dislocation interactions with a random array of particles. Copyright © 2003 Marcel Dekker, Inc. numberofobstaclesperunitlengthmaybefoundbycalculatingthenumber ofparticleintersectionswitharandomstraightline.Also,asdecreases,the dislocationssweepoveralargerarea,andhenceinteractwithmoreparticles. Finally,inthelimit,thenumberofintersectionsisclosetothesquarerootof thenumberofparticlesthatintersectarandomplane. Thecriticalstress, c ,requiredforadislocationtobreakawayfroma clusterofobstaclesdependsontheparticlesize,thenumberofparticlesper unitvolume,andthenatureoftheinteraction.Ifthecriticalbreakaway angleis c ,thenthecriticalstressatwhichbreakawayoccursisgivenby c ¼ Gb L cos c 2 ð8:1Þ Equation(8.1)maybederivedbyapplyingforcebalancetothegeometryof Fig.8.1.However,forstrongobstacles,breakawaymaynotoccur,evenfor ¼0.Hence,insuchcases,thedislocationbowstothesemi-circular Frank–Readconfigurationanddislocationmultiplicationoccurs,leavinga smallloop(Orowanloop)aroundtheunbrokenobstacle.Thecriticalstress requiredforthistooccurisobtainedbysubstitutingr¼L=2and¼0into Eq.(8.1).Thisgives c ¼ Gb L ð8:2Þ Hence,themaximumstrengththatcanbeachievedbydislocation interactionsisindependentofobstaclestrength.Thiswasfirstshownby Orowan(1948).Theaboveexpressions[Eqs(8.1)and(8.2)]providesimple order-of-magnitudeestimatesofthestrengtheningthatcanbeachievedby dislocationinteractionswithstrongorweakobstacles.Theyalsoprovidea qualitativeunderstandingofthewaysinwhichobstaclesofdifferenttypes canaffectarangeofstrengtheninglevelsincrystallinematerials. 8.3SOLIDSOLUTIONSTRENGTHENING Whenforeignatomsaredissolvedinacrystallinelattice,theymayresidein eitherinterstitialorsubstitutionalsites( Fig.8.2).Dependingontheirsizes relativetothoseoftheparentatoms.Foreignatomswithradiiupto57%of the parent atoms may reside in interstitial sites, while those that are within Æ15% of the host atom radii substitute for solvent atoms, i.e. they form solid solutions. The rules governing the formation of solid solutions are called the Hume–Rothery rules. These state that solid solutions are most likely to form between atoms with similar radii, valence, electronegativity, and chemical bonding type. Copyright © 2003 Marcel Dekker, Inc. Since the foreign atoms have different shear moduli and sizes from the parent atoms, they impose additional strain fields on the lattice of the sur- rounding matrix. These strain fields have the overall effect of restricting dislocation motion through the parent lattice, Fig. 8.2(b). Additional applied stresses must, therefore, be applied to the dislocations to enable them to overcome the solute stress fields. These additional stresses represent what is commonly known as solid solution strengthening. The effectiveness of solid solution strengthening depends on the size and modulus mismatch between the foreign and parent atoms. The size mismatch gives rise to misfit (hydrostatic) strains that may be symmetric or asymmetric (Fleischer, 1961, 1962). The resulting misfit strains, are pro- portional to the change in the lattice parameter, a, per unit concentration, c. This gives " b ¼ 1 a da dc ð8:3Þ Similarly, because the solute/interstitial atoms have different moduli from the parent/host atoms, a modulus mismatch strain, " G , may be defined as " G ¼ 1 G dG dc ð8:4Þ In general, however, the overall strain, " s , due to the combined effect of the misfit and modulus mismatch, may be estimated from " s ¼j" 0 G À " b jð8:5Þ FIGURE 8.2 Interstitial and solute atoms in a crystalline lattice: (a) schematic of interstitial and solute atoms; (b) effects on dislocation motion. [(a) Adapted from Hull and Bacon (1984) and (b) adapted from Courtney (1990). Reprinted with permission from Pergamon Press.] Copyright © 2003 Marcel Dekker, Inc. where is a constant close to 3 " 0 G ¼ " G =ð1 þð1=2Þj" G jÞ, and " b is given by equation 8.3. The increase in the shear yield strength, Á s , due to the solid solution strengthening may now be estimated from Á s ¼ G" 3=2 s c 1=2 700 ð8:6Þ where G is the shear modulus, " s is given by Eq. (8.5), and c is the solute concentration specified in atomic fractions. Also, Á s may be converted into Á s by multiplying by the appropriate Schmid factor. Several models have been proposed for the estimation of solid solution strengthening. The most widely accepted models are those of Fleischer (1961 and 1962). They include the effects of Burgers vector mismatch and size mismatch. However, in many cases, it is useful to obtain simple order-of- magnitude estimates of solid solution strengthening, Á s , from expressions of the form: Á s ¼ k s c 1=2 ð8:7Þ where k s is a solid solution strengthening coefficient, and c is the concentra- tion of solute in atomic fractions. Equation (8.7) has been shown to provide reasonable fits to experimental data for numerous alloys. Examples of the c 1=2 dependence of yield strength are presented in Fig. 8.3. In summary, the extent of solid solution strengthening depends on the nature of the foreign atom (interstitial or solute) and the symmetry of the stress field that surrounds the foreign atoms. Since symmetrical stress fields FIGURE 8.3 Dependence of solid solution strengthening on c 1=2 . (Data taken from Fleischer (1963). Reprinted with permission from Acta Metall.) Copyright © 2003 Marcel Dekker, Inc. interact only with edge dislocations, the amount of strengthening that can be achieved with solutes with symmetrical stress fields is very limited (between G=100 and G=10). In contrast, asymmetric stress fields around solutes inter- act with both edge and screw dislocations, and their interactions give rise to very significant levels of strengthening ($ 2G À 9G), where G is the shear modulus. However, dislocation/solute interactions may also be associated with strain softening, especially at elevated temperature. 8.4 DISLOCATION STRENGTHENING Strengthening can also occur as a result of dislocation interactions with each other. These may be associated with the interactions of individual disloca- tions with each other, or dislocation tangles that impede subsequent dislo- cation motion (Fig. 8.4). The actual overall levels of strengthening will also depend on the spreading of the dislocation core, and possible dislocation reactions that can occur during plastic deformation. Nevertheless, simple estimates of the dislocation strengthening may be obtained by considering the effects of the overall dislocation density, , which is the line length, ‘,of dislocation per unit volume , ‘ 3 . The dislocation density, , therefore scales with ‘=‘ 3 . Conversely, the average separation, ‘, between dislocations may be estimated from ‘ ¼ 1 À1=2 ð8:8Þ FIGURE 8.4 Strain hardening due to interactions between multiple disloca- tions: (a) interactions between single dislocations; (b) interactions with forest dislocations. Copyright © 2003 Marcel Dekker, Inc. The shear strengthening associated with the pinned dislocation seg- ments is given by Á d ¼ Gb ‘ ð8:9Þ where is a prop ortionality constant, and all the other variables have their usual meaning. We may also substitute Eq. (8.8) into Eq. (8.9) to obtain the following expression for the shear strengthening due to dislocation interac- tions with each other: Á d ¼ Gb 1=2 ð8:10Þ Once again, we may convert from shear stress increments into axial stress increments by multiplying by the appropriate Schmid factor, m. This gives the strength increment, Á d , as (Taylor, 1934): Á d ¼ mGb 1=2 ¼ k d 1=2 ð8:11Þ where k d ¼ m Gb and the other variables have their usual meaning. Equations (8.10) and (8.11) have been shown to apply to a large number of metallic materials. Typical results are presented in Fig. 8.5. These show FIGURE 8.5 Dependence of shear yield strength on dislocation density. (From Jones and Conrad, 1969. Reprinted with permission from TMS-AIME.) Copyright © 2003 Marcel Dekker, Inc. thatthelineardependenceofstrengtheningonthesquarerootofdislocation densityprovidesareasonablefittotheexperimentaldata. ItisimportanttonoteherethatEq.(8.11)doesnotapplytodisloca- tionstrengtheningwhencellstructuresareformedduringthedeformation process(Fig.8.6).Insuchcases,theaveragecellsize,s,isthelengthscale thatcontrolstheoverallstrengtheninglevel.Thisgives Á 0 d ¼k 0 d ðsÞ À1=2 ð8:12Þ whereÁ 0 d isthestrengtheningduetodislocationcellwalls,k 0 d isthedis- locationstrengtheningcoefficientforthecellstructure,andsistheaverage sizeofthedislocationcells. 8.5GRAINBOUNDARYSTRENGTHENING Grainboundariesalsoimpededislocationmotion,andthuscontributeto thestrengtheningofpolycrystalline materials (Fig. 8.7). However, the strengthening provided by grain boundaries depends on grain boundary FIGURE8.6DislocationcellstructureinaNb–Al–Tibased alloy. (Courtesy of Dr. Seyed Allameh.) Copyright © 2003 Marcel Dekker, Inc. structureandthemisorientationbetweenindividualgrains.Thismaybe understoodbyconsideringthesequenceofeventsinvolvedintheinitiation ofplasticflowfromapointsource(withinagrain)inthepolycrystalline aggregateshownschematicallyinFig.8.8. Duetoanappliedshearstress, app dislocationsareemittedfroma pointsource(possiblyaFrank–Readsource)inoneofthegrainsinFig.8.8. Thesedislocationsencounteralatticefrictionstress, i ,astheyglideona slipplanetowardsthegrainboundaries.Theeffectiveshearstress, eff ,that contributestotheglideprocessis,therefore,givenby eff ¼ app À i ð8:13Þ However,sincethemotionofthedislocationsisimpededbygrainbound- aries,dislocationswillgenerallytendtopile-upatgrainboundaries.The stressconcentrationassociatedwiththispile-uphasbeenshownbyEshelby etal.(1951)tobe$ðd=4rÞ 1=2 ,wheredisthegrainsizeandristhedistance fromthesource.Theeffectiveshearstressis,therefore,scaledbythisstress concentrationfactor.Thisresultsinashearstress, 12 ,atthegrainbound- aries,thatisgivenby FIGURE8.7Dislocationinteractionswithgrainboundaries.(FromAshbyand Jones, 1996. Reprinted with permission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. 12 ¼ app À i ÀÁ d 4r 1=2 ð8:14Þ If we now consider bulk yielding to correspond to the condition slip for transmission to adjacent grains when a critical 12 is reached, then we may rearrange Eq. (8.14) to obtain the following expression for app at the onset of bulk yielding: app ¼ i þ 4r d 1=2 12 ð8:15Þ The magnitude of the critical shear stress, 12 , required for slip trans- mission to adjacent grains may be considered as a constant. Also, the aver- age distance, r, for the dislocations in the pile-up is approximately constant . Hence, (4r) 1=2 12 is a constant, k 0 y , and Eq. (8.15) reduces to y ¼ i þ k 0 y d À1=2 ð8:16Þ Once again, we may convert from shear stress into axial stress by multi- plying Eq. (8.16) by the appropriate Schmid factor, m. This gives the follow- ing relationship, which was first proposed by Hall (1951) and Petch (1953): y ¼ 0 þ k y d À1=2 ð8:17Þ where 0 is the yield strength of a single crystal, k y is a microstructure/grain boundary strengthening parameter, and d is the grain size. The reader should note that Eq. (8.17) shows that yield strength increases with decreas- ing grain size. Furthermore, ( o may be affected by solid solution alloying effects and dislocation substructures. FIGURE 8.8 Schematic illustration of dislocation emission from a source. (From Knott, 1973. Reprinted with permission from Butterworth.) Copyright © 2003 Marcel Dekker, Inc. [...]... and Jones, D.R.H ( 199 6) Engineering Materials: An Introduction to the Properties and Applications vol 1 Pergamon Press, New York Bacon, D.J ( 196 7) Phys Stat Sol vol 23, p 527 Brown, L.M and Ham, R.K ( 196 7) Dislocation–particle interactions In: A Kelly and R.B Nicholson, eds Strengthening Methods in Crystals Applied Science, London, pp 9 135 Courtney, T.H ( 199 0) Mechanical Behavior of Materials McGraw-Hill,... Materials McGraw-Hill, New York Dundurs, J and Mura, T ( 196 9) J Mech Phys Solids vol 17, p 4 59 Copyright © 2003 Marcel Dekker, Inc Eshelby, J.D., Frank, F.C., and Nabarro, F.R.N ( 195 1) The equilibrium of linear arrays of dislocations Phil Mag vol 42, p 351 Fleischer, R.L ( 196 1) Solution hardening Acta Metall vol 9, pp 99 6–1000 Fleischer, R.L ( 196 2) Solution hardening by tetragonal distortions: application... Fleischer, R.L ( 196 3) Acta Metall., Vol II, p 203 Foreman, A.J.E ( 196 7) Phil Mag vol 15, p 1011 Gleiter, H and Hornbogen, E ( 196 7) Precipitation hardening by coherent particles Mater Sci Eng vol 2, pp 285–302 Hall, E.O ( 195 1) Proc Phys Soc B vol 64, p 747 Hertzberg, R.W ( 199 6) Deformation and Fracture Mechanics of Engineering Materials 4th ed John Wiley, New York Hu, H and Cline, R.S ( 196 8) TMS-AIME vol... Bacon, D.J ( 198 4) Introduction to Dislocations 3rd ed Pergamon Press, New York Jones, R.L and Conrad, H ( 196 9) TMS-AIME vol 245, p 7 79 Knott, J.F ( 197 3) Fundamentals of Fracture Mechanics Butterworth, London Mitchell, T.E and Smialek, R.L ( 196 8) Proceedings of Chicago Work Hardening Symposium (AIME) Orowan, E ( 194 8) Discussion in the Symposium on Internal Stresses in Metals and Alloys Institute of Metals,... occupies the whole of the slip plane area of the precipitate and the energy increase is $ r 2 (APBE) The increase in energy is linear with the position of the dislocation in the particle Thus, Fmax ¼ r 2 (APBE)/2r ¼ r (APBE)/2 (From Courtney, 199 0 Reprinted with permission from McGraw-Hill.) isms, especially when the nature of the particle boundaries permit dislocation entry into the particles, as shown... discovered them The first set of FIGURE 8.15 (a) Left-hand section of the Al–Cu phase diagram; (b) aging heat treatment schedule (From Courtney, 199 0 Reprinted with permission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc FIGURE 8.16 Role of Guinier–Preston zones and precipitates in the strengthening of Al–Cu alloys aged at 1308 and 190 8C (From Courtney, 199 0 Reprinted with permission... FIGURE 8 .9 Hall–Petch dependence of yield strength (From Hu and Cline, 196 8 Original data presented by Armstrong and Jindal, 196 8 Reprinted with permission from TMS-AIME.) Evidence of Hall–Petch behavior has been reported in a large number of crystalline materials An example is presented in Fig 8 .9 Note that the microstructural strengthening term, ky , may vary significantly for different materials. .. increases (for the same volume fraction of particles), the strength dependence due to dislocation looping and dislocation shearing will be of forms shown schematically in Fig 8.14 It should be clear from Fig 8.14 that the stresses required for particle shearing are lower than those required for particle looping when the particle FIGURE 8.14 Schematic of the role of precipitation hardening mechanisms... shape A statistical treatment of the possible effects of these variables is, therefore, needed to develop a more complete understanding of precipitation strengthening Nevertheless, the idealized presentation (based on average particle sizes and distribution and simple particle geometries) is an essential first step in the development of a basic understanding of the physics of precipitation strengthening... interfaces This is because of the need to apply additional stresses to overcome the coherency strains/stresses associated with lattice mismatch As discussed earlier, particle shearing of disordered particles results in the formation of a slip step on entry, and another slip step on exit from the particle For a particle volume fraction, f , and similar crystal structures in the matrix and particle, it can be . Inc. isms,especiallywhenthenatureoftheparticleboundariespermitdisloca- tionentryintotheparticles,asshownschematicallyinFigs8.13(a)and 8.13(b)forcoherentandsemicoherentinterfaces(notethatcoherentinter- faceshavematchingprecipitateandmatrixatomsattheinterfaces,while semicoherentinterfaceshaveonlypartialmatchingofatoms).Incontrast, dislocationentry(intotheprecipitate)isdifficultwhentheinterfacesare incoherent,i.e.,thereislittleornomatchingbetweenthematrixandpre- cipitateatomsattheinterfaces,Fig.8.13(c).Dislocationentryinto,orexit from,particlesmayalsobedifficultwhenthemisfitstrain,",inducedasa resultoflatticemismatch(betweenthematrixandprecipitateatoms)is significantinsemicoherentorcoherentinterfaces.Thisisbecauseofthe needtoapplyadditionalstressestoovercomethecoherencystrains/stresses associatedwithlatticemismatch. Asdiscussedearlier,particleshearingofdisorderedparticlesresultsin theformationofaslipsteponentry,andanotherslipsteponexitfromthe particle.Foraparticlevolumefraction,f,andsimilarcrystalstructuresin thematrixandparticle,itcanbeshownthattheshearstrengtheningpro- videdbyparticleshearingofdisorderedprecipitatesisgivenby(Gleiterand Hornbogen, 196 7): Á ps ¼ 3Gðb p Àb m Þ b r d ð8:22Þ FIGURE8.12Viewofanedgedislocationpenetratinganorderedprecipitate (the. Inc. whereistheanglebetweenthedislocationlineandtheBurgersvector,AðÞ andBðÞarebothfunctionsof,L 0 istheeffectiveparticleseparation, L 0 ¼LÀ2r,bistheBurgersvector,andristheparticleradius.Thefunction AðÞhasbeendeterminedbyWeeksetal.( 196 9)forcriticalconditions correspondingtotheinstabilityconditionintheFrank–Readmechanism. Thespecialresultforthisconditionis c ¼AðÞ Gb 2L 0 ln L 0 r ð8:21Þ wherethefunctionAðÞ¼1forinitialedgedislocationsegmentsorAðÞ¼ 1=ð1ÀÞforinitialscrewdislocationsegments.Criticalstresseshavebeen calculatedfordifferenttypesofbowingdislocationconfigurations(Bacon, 196 7;Foreman, 196 7;MitchellandSmialek, 196 8). AverageeffectivevaluesofAandBhavebeencomputedforthedis- locationconfigurationssincethevaluesofvaryalongthedislocationlines. Forscrewdislocationswithhorizontalsidearms,Fig.8.11(a),B¼À1:38, whileA¼À0 :92 forcorrespondingedgedislocationconfigurations. Similarly,forbowingscrewandedgedislocationswithverticalsidearms,B ¼0:83and0.32,respectively.Asthereadercanimagine,differenteffective valuesofAandBhavebeenobtainedforawiderangeofdislocation configurations.ThesearediscussedindetailinpapersbyForeman( 196 7) andBrownandHam( 196 7). 8.6.2StrengtheningbyDislocationShearingor CuttingofPrecipitates Inadditiontobowingbetweenprecipitates,dislocationsmayshearorcut throughprecipitates.Thismayresultintheformationofledgesatthe interfacesbetweentheparticleandthematrix,intheregionswheredisloca- tionentryorexitoccurs(Fig.8.10).Alternatively,sincedislocationcutting oforderedprecipitatesbysingledislocationswillresultinthedisruptionof theorderedstructure,thepassageofaseconddislocationisoftenneededto restoretheorderedstructure(Fig.8.12).Suchpairsofdislocationsaregen- erally. Gistheaverageshearmodulus,b p istheparticleBurgersvector,b m is thematrixBurgersvector,bistheaverageBurgersvector,disthedistance traveledbythedislocationalongtheparticle,andristheparticleradius.In caseswherethemisfitstrain(duetolatticemismatchbetweenthematrixand particle)issignificant,theoverallstrengtheningisthestressrequiredto movethedislocationsthroughthestress/strainfieldsattheparticlebound- aries. Theincreaseinshearstrengthisthengivenby Á m ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 27Á4ÁbÁ" 3 E 3 Tð1þÞ 3 s f 4=6 r 1=2 ð8:23Þ whereEistheYoung’smodulus,bistheBurgersvector,"isthemisfit strain,Tisthelineenergyofthedislocation,isPoisson’sratio,fisthe precipitatevolumefraction,andristheprecipitateradius.Themismatch strain,",isnowgivenby "¼ 3KðÁa=aÞ 3Kþ2=1þÞ ð8:24Þ whereKisthebulkmodulus,isPoisson’sratio,andtheotherconstants havetheirusualmeanings. ItisimportanttonoteherethatEq.(8.23)maybeusedgenerallyin precipitationstrengtheningduetoanytypeofmisfitstrain.Hence,for example,"mayrepresentmisfitstrainsduetothermalexpansionmismatch. Incaseswheretheshearedparticlesareordered,e.g.,intermetalliccom- poundsbetweenmetalsandothermetals(Fig.8.12),theshearingofthe particlesoftenresultsinthecreationofSFsandAPBs.Theshearstrength- FIGURE8.13Schematicofthedifferenttypesofinterfaces:(a)coherentinter- face; (b)