6 IntroductiontoDislocation Mechanics 6.1INTRODUCTION Earlyinthe20thcentury,anumberofscientiststriedtopredictthetheore- ticalstrengthofacrystallinesolidbyestimatingtheshearstressrequiredto moveoneplaneofatomsoveranother(Fig.6.1).Theyfoundthatthe predictedtheoreticalstrengthsweremuchgreaterthanthemeasured strengthsofcrystallinesolids.Thelargediscrepancy(anorderofmagnitude ortwo)betweenthetheoreticalandmeasuredshearstrengthspuzzledmany scientistsuntilOrowan,Polanyi,andTaylor(1934)independentlypublished theirseparateclassicalpapersondislocations(linedefects). Themeasuredstrengthswerefoundtobelowerthanthepredicted theoreticallevelsbecauseplasticityoccurredprimarilybythemovement oflinedefectscalleddislocations.Thestresslevelsrequiredtoinducedis- locationmotionwerelowerthanthoserequiredtoshearcompleteatomic planesovereachother(Fig.6.1).Hence,themovementofdislocations occurredpriortotheshearofatomicplanesthatwaspostulatedbyearlier workerssuchasFrenkel(1926). Since1934,numerouspapershavebeenpublishedontheroleofdis- locationsincrystallineplasticity.Anumberofbooks(HirthandLothe, 1982;HullandBacon,1984;WeertmanandWeertman,1992)havealso beenwrittenonthesubject.Thischapterwill,therefore,notattemptto Copyright © 2003 Marcel Dekker, Inc. presentacomprehensiveoverviewofdislocations.Instead,thefundamental ideasindislocationmechanicsrequiredforabasicunderstandingofcrystal- lineplasticitywillbepresentedatanintroductorylevel.Theinterested readerisreferredtopapersandmoreadvancedtextsthatarelistedinthe bibliographyattheendofthechapter. 6.2THEORETICALSHEARSTRENGTHOFA CRYSTALLINESOLID Frenkel(1926)obtainedausefulestimateofthetheoreticalshearstrengthof acrystallinesolid.Heconsideredtheshearstressrequiredtocauseshearof onerowofcrystalsovertheother(Fig.6.1).Theshearstrain,,associated withasmalldisplacement,x,isgivenby ¼ x a ð6:1Þ Hence,forsmallstrains,theshearstress,,maybeobtainedfrom ¼G¼G x a ð6:2Þ whereGistheshearmodulus.Similarly,wemayalsouseanapproximate sinusoidalpotentialfunctiontoobtainanexpressionforthevariationinthe appliedshearstress,,asafunctionofdisplacement,x(Fig.6.2).Thisgives ¼ max sin 2x b ð6:3Þ FIGURE6.1Shearofonerowofatomsoveranotherinaperfectcrystal. Copyright © 2003 Marcel Dekker, Inc. where max isthemaximumshearstressintheapproximatelysinusoidal versusxcurveshowninFig.6.2;xisthedisplacement,andbistheintera- tomicspacing(Fig.6.1).Forsmalldisplacements,sin(2x=bÞ$2x=b. Hence,isgivenby ¼ max 2x b ð6:4Þ WemaynowequateEqs(6.4)and(6.2)toobtainanexpressionfor max . Notingthatforcubiccrystalsbthisgives max ¼ G 2 ð6:5Þ Equation6.5providesanapproximatemeasureofthetheoreticalshear strengthofacrystallinesolid.Morerigorousanalysisusingmorerepresen- tativeinteratomicpotentials(Fig.6.2)givesestimatesofthetheoretical shearstrengthtobe$G=30.However,mostestimatesofthetheoretical shearstrengthareaboutoneortwoordersofmagnitudegreaterthanthe measuredvaluesobtainedfromactualcrystallinesolids. Thisdiscrepancybetweenthemeasuredandtheoreticalstrengthsled Orowan,Polanyi,andTaylor(1934)torecognizetheroleoflinedefects (dislocations)incrystalplasticity.However,theseauthorswerenotthefirst toproposetheideaofdislocations.Dislocationstructureswerefirstpro- posedbyVolterra(1907),whosepurelymathematicalworkwasunknownto Orowan,Polanyi,andTaylorin1934whentheypublishedtheiroriginal papersondislocations. FIGURE6.2Schematicillustrationofshearstressvariations.Thedashedcurve corresponds to more precise shear stress – displacement function. Copyright © 2003 Marcel Dekker, Inc. Sincetheearlyideasondislocations,considerableexperimentaland analyticalworkhasbeendonetoestablishtheroleofdislocationsincrystal plasticity.Materialswithlowdislocation/defectcontent(whiskersand fibers)havealsobeenproducedbyspecialprocessingtechniques.Such materialshavebeenshowntohavestrengthlevelsthatareclosertotheore- ticalstrengthlevelsdiscussedearlier(Kelly,1986).Theconceptofdisloca- tionshasalsobeenusedtoguidethedevelopmentofstrongeralloyssince muchofwhatweperceiveasstrengtheningisduelargelytotherestrictionof dislocationmotionbydefectsincrystallinesolids. 6.3TYPESOFDISLOCATIONS Therearebasicallytwotypesofdislocations.Thefirsttypeofdislocation thatwasproposedin1934istheedgedislocation.Theothertypeofdis- locationisthescrewdislocation,whichwasproposedlaterbyBurgers (1939).Bothtypesofdislocationswillbeintroducedinthissectionbefore discussingtheideaofmixeddislocations,i.e.,dislocationswithbothedge andscrewcomponents. 6.3.1EdgeDislocations ThestructureofanedgedislocationisillustratedschematicallyinFig.6.3. Thisshowscolumnsofatomsinacrystallinesolid.Notethelineofatomsat whichthehalf-filledcolumnterminates.Thislinerepresentsadiscontinuity intheotherwiseperfectstackingofatoms.Itisalinedefectthatisgenerally referredtoasanedgedislocation.Thecharacterofanedgedislocationmay alsobedescribedbydrawingaso-calledright-handedBurgerscircuit aroundthedislocation,asshowninFig.6.4(a).NotethatSinFig.6.4 corresponds to the start of the Burgers circuit, while F corresponds to the finish. The direction of the circuit in this case is also chosen to be right- handed, although there is no general agreement on the sign convention in the open literature. In any case, we may now proceed to draw the same Burgers circuit in a perfect reference crystal, Fig. 6.4(b). Note that the finish position, F, is different from the start position, S, due to the absence of the edge dislocation in the perfect reference crystal. We may, therefore, define a vector to connect the finish position, F,to the start position, S, in Fig. 6.4(b). This vector is called the Burgers vector.It is often denoted by the letter, b, and it corresponds to one atomic spacing for a single edge dislocation. It is important to remember that we have used a right-handed finish-to-start definition in the above discussion. However, this is not always used in the open literature. For consistency, however, we Copyright © 2003 Marcel Dekker, Inc. will retain the current sign convention, i.e., the finish-to-start (F=S) right- handed rule. Finally in this section, it is important to note that we may define the sense vector, s, of an edge dislocation in a direction along the dislocation line (into the page). The sense of an edge dislocation, s, is therefore perpen- dicular to the Burgers vector, b. Hence, we may describe an edge dislocation FIGURE 6.3 Schematic of edge dislocation. (Taken from Hirthe and Lothe, 1982. Reprinted with permission from John Wiley.) FIGURE 6.4 Finish to start (F=S) right-handed Burgers circuits: (a) around edge dislocation; (b) in a perfect reference crystal. (Taken from Hirthe and Lothe, 1982. Reprinted with permission from John Wiley.) Copyright © 2003 Marcel Dekker, Inc. asalinedefectwithasensevector,s,thatisperpendiculartotheBurgers vector,b,i.e.b.s¼0. 6.3.2ScrewDislocations Thestructureofascrewdislocationmaybevisualizedbyconsideringthe sheardisplacementoftheupperhalfofacrystaloverthelowerhalf,as showninFig.6.5(a).Iftheatomsintheupperhalfofthecrystalaredenoted asopencircles,whilethoseinthelowerhalfaredenotedasfilledcircles[Fig. 6.5(a)],thentherelativedisplacementsbetweentheopenandfilledcircles maybeusedtodescribethestructureofascrewdislocation.Thearrange- mentoftheatomsaroundthedislocationlineABfollowsaspiralpaththat issomewhatsimilartothepaththatonemightfollowalongaspiralstair- case.ThisisillustratedclearlyinFig.6.5(b)foraright-handedscrewdis- location. Asbefore,wemayalsodefineaBurgersvectorforascrewdislocation usingafinish-to-startright-handedscrewrule.Thisisshownschematically inFig.6.6.NotethattheBurgersvectorisnowparalleltothesensevector, s,alongthedislocationline.Thisisincontrastwiththeedgedislocationfor whichtheBurgersvectorisperpendiculartothesensevector.Inanycase, wemaynowformallydescribearight-handedscrewdislocationasonewith b:s¼b.Aleft-handedscrewdislocationisonewithb:s¼Àb. 6.3.3MixedDislocations Inreality,mostdislocationshavebothedgeandscrewcomponents.Itis, therefore,necessarytointroducetheideaofamixeddislocation(onewith bothedgeandscrewcomponents).Atypicalmixeddislocationstructureis showninFig.6.7(a).Notethatthisdislocationstructureiscompletelyscrew in character at A, and completely edge in character at B. The segments of the dislocation line between A and B have both edge and screw components. They are, therefore, mixed dislocation segments. Other examples of mixed dislocation structures are presented in Figs 6.7(b) and 6.7(c). The screw components of the mixed dislocation segme nts, b s , may be obtained from the following expression: b s ¼ðb:sÞs ð6:6Þ Similarly, the edge components, b e , of the mixed dislocation segments may be obtained from b e ¼ s Âðb  sÞð6:7Þ Copyright © 2003 Marcel Dekker, Inc. FIGURE 6.5 Structure of a screw dislocation: (a) displacement of upper half of crystal over lower half; (b) spiral path along the dislocation line. (From Read, 1953. Reprinted with permission from McGraw-Hill.) Copyright © 2003 Marcel Dekker, Inc. 6.4MOVEMENTOFDISLOCATIONS Asdiscussedearlier,crystalplasticityiscausedlargelybythemovementof dislocations.Itis,therefore,importanttodevelopaclearunderstandingof howdislocationsmovethroughacrystal.However,dislocationsalso encounterlatticefrictionastheymovethroughalattice.Estimatesofthe latticefrictionstresswerefirstobtainedbyPeierls(1940)andNabarro (1947).Consideringthemotionofadislocationinalatticewithlattice parametersaandb(Fig.6.1),theyobtainedasimpleexpressionforthe latticefrictionstress,.Theso-calledPeierls–Nabarrolatticefrictionstress isgivenby f ¼Gexp À2a bð1À ð6:8aÞ or f ¼Gexp À2w b ð6:8bÞ whereaistheverticalspacingbetweenslipplanes,bistheslipdistanceor Burgersvector,Gistheshearmodulus,wisthedislocationwidth(Fig6.8), FIGURE6.6Right-handedBurgerscircuits:(a)aroundscrewdislocation;(b)in perfect reference crystal. (From Hull and Bacon, 1984. Reprinted with permis- sion from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. andisPoisson’sratio.Thelatticefractionstressisassociatedwiththe energyorthestressthatisneededtomovetheedgedislocationfromposi- tionAtopositionD(Fig.6.9).Notethatthedislocationlineenergy[Fig. 6.10(a)]andtheappliedshearstress[Fig.6.10(b)]varyinasinusoidalman- ner.Also,theshearstressincreasestoapeakvaluecorrespondingto f [Fig. 6.10(b)],thefrictionstress.Thelattermay,therefore,beconsideredasthe latticeresistancethatmustbeovercometoenabledislocationmotionto occurbetweenAandD(Fig.6.9).Itisimportanttonotethat f isgenerally muchlessthanthetheoreticalshearstrengthofaperfectlattice,whichis FIGURE6.7Structureofamixeddislocation:(a)quarterloop;(b)halfloop;(c) full loop. Copyright © 2003 Marcel Dekker, Inc. givenbyEq.(6.5)foracubiclattice.Slipis,therefore,morelikelytooccur bytheexchangeofbonds,thanthecompleteshearofatomicplanesover eachother,assuggestedbyFig.6.1. ThereadershouldexamineEqs(6.8a)and(6.8b)carefullysincethe dependenceofthelatticefrictionstress, f ,onlatticeparametersaandbhas someimportantimplications.Itshouldbereadilyapparentthatthefriction stressisminimizedonplaneswithlargeverticalspacings,a,andsmall horizontalspacings,b.Dislocationmotionis,therefore,mostlikelyto occuronclose-packedplaneswhichgenerallyhavethelargestvaluesofa andthesmallestvaluesofb.Dislocationmotionisalsomostlikelytooccur alongclose-packeddirectionswithsmallvaluesofb.Hence,close-packed materialsaremorelikelytobeductile,whilelessclose-packedmaterialssuch asceramicsaremorelikelytobebrittle. Wearenowpreparedtotackletheproblemofdislocationmotionin crystallinematerials.First,wewillconsiderthemovementofedgedisloca- tionsonclose-packedplanesinclose-packeddirections.Suchmovementis generallydescribedasconservativemotionsincethetotalnumberofatoms ontheslipplaneisconserved,i.e.,constant.However,wewillalsoconsider thenonconservativemotionofedgedislocationswhichisoftendescribedas FIGURE6.8Schematicof(a)wideand(b)narrowdislocations.(FromCottrell, 1957. Reprinted with permission from Institute of Mechanical Engineering.) Copyright © 2003 Marcel Dekker, Inc. [...]... Copyright © 2003 Marcel Dekker, Inc Cottrell, A.H (19 57) The Properties of Materials at High Rates of Strain Institute of Mechanical Engineering, London, UK Courtney, T.H (1990) Mechanical Behavior of Materials, John Wiley, New York Frenkel, J (1926) Z Phys vol 37, p 572 Gilman, J.J and Johnston, W.G (19 57) Dislocations and Mechanical Properties of Crystals In: J.C Fisher, W.G Johnston, R Thomson and... (1992), and Nabarro (19 67) The books by Nabarro (19 47) and Hirthe and Lothe (1982) provide comprehensive descriptions of almost all aspects of advanced dislocation theory The books by Hull and Bacon (1984) and Weertman and Weertman provide more of an introduction to basic concepts They are also lucid, and relatively easy to read Argon, A and McClintock, F.A (1966) Mechanical Behavior of Materials Addison... FIGURE 6.14 The Frank net (Taken from Cottrell, 19 57 Reprinted with permission from Institute of Mechanical Engineering.) The above expressions are, therefore, analogous to Kirchoff’s equations for current flow in electrical circuits 6.5 EXPERIMENTAL OBSERVATIONS OF DISLOCATIONS A large number of experimental techniques have been used to confirm the existence of dislocations They include: 1 2 3 4 5 Etch-pit... description of image forces between dislocations and free surfaces The dislocation mechanics topics covered in this chapter should provide the foundation for the development of a basic understanding of the plastic deformation of metals in Chap 7 BIBLIOGRAPHY The treatment of dislocation mechanics in this chapter is brief It is intended to serve as an introduction to the subject More detailed treatment of dislocation...FIGURE 6.9 Schematic of atomic rearrangements associated with edge dislocation motion: (a) atoms B and C equidistant from atom A along edge dislocation line at start of deformation; (b) greater attraction of C towards A as crystal is sheared; (c) subsequent motion of edge dislocation to the right; (d) formation of step of Burgers vector when dislocation reaches the edge of the crystal Copyright... the conventional movement of screw dislocations, and the cross-slip of screw dislocations 6.4.1 Movement of Edge Dislocations The movement of edge dislocations is relatively easy to visualize Let us start by considering the movement of the positive edge dislocation shown schematically in Fig 6.9 Prior to the application of shear stress to the crystal, the atom A at the center of the edge dislocation... compliant free surface offers no stress in opposition to the displacements of an approaching dislocation, the strain energy of a crystal will decrease as a dislocation approaches a free surface This tends to pull the dislocation towards the free surface to form a step of one interatomic distance The reduction in the strain energy of the crystal may also be expressed in terms of a ‘‘force’’ that pulls... observed to occur at elevated temperature 6.4.2 Movement of Screw Dislocations The movement of screw dislocations is a little more difficult to visualize Let us start by considering the effects of an applied shear stress on the screw dislocation shown in Fig 6.12 The shear stress on the upper part of the crystal displaces the atoms on one half of the crystal over the other, as shown in Fig 6.12 However,... R for polycrystals that contain several dislocations Nevertheless, the coefficient of Gb2 in Eq (6.31) may be approximated by unity for order -of- magnitude comparisons This gives the strain energy per unit length of screw dislocations as Us $ Gb 2 ð6:32Þ Similar calculations of the elastic strain energy per unit length of an edge dislocation may also be carried out However, the calculations are more... the resulting motion of dislocations may be considered to arise from the effects of ‘‘virtual’’ internal forces that act in directions that are perpendicular to segments on the dislocation line Let us start by considering the motion of the right-handed screw dislocation in the crystal shown in Fig 6.21 The external force applied to the surface of the crystal is given by the product of stress, xz , multiplied . Inc. givenbyEq.(6.5)foracubiclattice.Slipis,therefore,morelikelytooccur bytheexchangeofbonds,thanthecompleteshearofatomicplanesover eachother,assuggestedbyFig.6.1. ThereadershouldexamineEqs(6.8a)and(6.8b)carefullysincethe dependenceofthelatticefrictionstress, f ,onlatticeparametersaandbhas someimportantimplications.Itshouldbereadilyapparentthatthefriction stressisminimizedonplaneswithlargeverticalspacings,a,andsmall horizontalspacings,b.Dislocationmotionis,therefore,mostlikelyto occuronclose-packedplaneswhichgenerallyhavethelargestvaluesofa andthesmallestvaluesofb.Dislocationmotionisalsomostlikelytooccur alongclose-packeddirectionswithsmallvaluesofb.Hence,close-packed materialsaremorelikelytobeductile,whilelessclose-packedmaterialssuch asceramicsaremorelikelytobebrittle. Wearenowpreparedtotackletheproblemofdislocationmotionin crystallinematerials.First,wewillconsiderthemovementofedgedisloca- tionsonclose-packedplanesinclose-packeddirections.Suchmovementis generallydescribedasconservativemotionsincethetotalnumberofatoms ontheslipplaneisconserved,i.e.,constant.However,wewillalsoconsider thenonconservativemotionofedgedislocationswhichisoftendescribedas FIGURE6.8Schematicof(a)wideand(b)narrowdislocations.(FromCottrell, 19 57. . dis- location line at start of deformation; (b) greater attraction of C towards A as crystal is sheared; (c) subsequent motion of edge dislocation to the right; (d) formation of step of Burgers vector. Inc. dislocationclimb. * Sincedislocationclimbinvolvestheexchangeofatoms andvacanciesoutsidetheslipplane,thetotalnumberofatomsintheslip planeisgenerallynotconservedbydislocationclimbmechanisms. Followingthediscussionofedgedislocationmotionbyslipandclimb,we willthendiscusstheconventionalmovementofscrewdislocations,andthe cross-slipofscrewdislocations. 6.4.1MovementofEdgeDislocations Themovementofedgedislocationsisrelativelyeasytovisualize.Letusstart byconsideringthemovementofthepositiveedgedislocationshownsche- maticallyinFig.6.9.Priortotheapplicationofshearstresstothecrystal, theatomAatthecenteroftheedgedislocationisequidistantfromatomsB andC,Fig.6.9(a).Itis,therefore,equallyattractedtoatomsBandC. However,ontheapplicationofasmallshearstress,,tothetopandbottom facesofthecrystal,atomAisdisplacedslightlytotheright.Theslight asymmetrydevelopsinagreaterattractionbetweenAandC,compared tothatbetweenAandB.Iftheappliedshearstressisincreased,the increasedattractionbetweenatomsAandCmaybesufficienttocause thedisplacementofatomCandsurroundingatomstotheleftbyoneatomic spacing,b,Fig.6.9(b).Thehalfcolumnofatoms(positiveedgedislocation), FIGURE6.10Variationof(a)dislocationlineenergyand(b)stresswiththe position