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.
7.5 DISLOCATION BOWING DUE TO LINE TENSION
It should be clear from the above discussion that interactions between dis-
locations can give rise to pinned dislocation segments, e.g., dislocation line
segments that are pinned by jogs, solutes, interstitials, or precipitates. When
a crystal is subjected to a shear stress, the so-called line tension that develops
in a pinned dislocation segment can give rise to a form of dislocation bowing
that is somewhat analogous to the bowing of a string subjected to line
tension, T. In the case of a dislocation, the line tension has a magnitude
$ Gb
2
. The bowing of dislocation is illustrated schematically in Fig 7.12.
Let us now consider the free body diagram of the bowed dislocation
configuration in Fig. 7.12(b). For force equilibrium in the y direction, we
may write:
2T sin


2

¼ bL ð7:2Þ
where  is the applied shear stress, b is the Burgers vector, L is the disloca-
tion line length, and the other parameters are shown in Fig. 7.12. For small
curvatures, sin (=2Þ$=2, and so Eq. (7.2) reduces to
T  ffi bL ð7:3Þ
Recalling that T $ Gb
2
and that  ¼ L=R from Fig. 7.12, we may simplify
Eq. (7.3) to give
Gb
2
L
R
¼ bL ð7:4aÞ
or
 ¼
Gb
R
ð7:4bÞ
FIGURE 7.12 Schematics of (a) pinned dislocation segment and (b) bowed
dislocation configuration due to applied shear stress, . Note that T is the
line tension $ Gb
2
.
Copyright © 2003 Marcel Dekker, Inc.
ThecriticalshearstressdescribedbyEq.(7.4b)issufficienttocausethe
pinneddislocationtocontinuetobowinastablemanneruntilitreachesa

semicircularconfigurationwithr¼L=2.Thisbowingformsthebasisofone
ofthemostpotentmechanismsfordislocationmultiplication,whichisdis-
cussedinthenextsection.
7.6DISLOCATIONMULTIPLICATION
Thediscerningreaderisprobablywonderinghowplasticdeformationcan
actuallycontinueinspiteofthenumerousinteractionsbetweendislocations
thatarelikelytogiverisetoareductioninthedensityofmobiledisloca-
tions.Thisquestionwillbeaddressedinthissection.However,before
answeringthequestion,letusstartbyconsideringthesimplestcaseofa
well-annealedcrystal.Suchcrystalscanhavedislocationdensitiesaslowas
10
8
À10
12
m/m
3
.Whenannealedcrystalsaredeformed,theirdislocation
densitiesareknowntoincreaseto$10
16
À10
18
m/m
3
.Howcanthishappen
whentheinteractionsbetweendislocationsarereducingthedensityof
mobiledislocations?
ThisquestionwasfirstansweredbyFrankandReadinadiscussion
thatwasheldinapubinPittsburgh.Theirconversationledtothemechan-
ismofdislocationbreedingthatisillustratedschematicallyinFig.7.13.The
schematicsshowonepossiblemechanismbywhichdislocationscanmultiply

whenashearstressisappliedtoadislocationthatispinnedatbothends.
Underanappliedshearstress,thepinneddislocation[Fig.7.13(a)]bows
intoacirculararcwithradiusofcurvature,r¼L=2,showninFig.7.13(b).
Thebowingofthecurveddislocationiscausedbythelinetension,T,as
discussedinSect.7.5(Fig.7.12).Thiscausesthedislocationtobowina
stable manner until it reaches the circular configuration illustrated schema-
tically in Fig. 7.13(b). From Eq. (7.4b), the critical shear stress required for
this to occur is $ Gb=L.
Beyond the circular configuration of Fig. 7.13(b), the dislocation bows
around the pinned ends, as shown in Fig. 7.13(c). This continues until the
points labeled X and X
0
come into contact, Fig. 7.13(d). Since these disloca-
tion segments are opposite in sign, they annihilate each other. A new loop is,
therefore, produced as the cusped dislocation [Fig. 7.13(e)] snaps back to the
original straight configuration, Fig. 7.13(a).
Note that the shaded areas in Fig. 7.13 correspond to the regions of
the crystal that have been sheared by the above process. They have, there-
fore, been deformed plastically. Furthermore, subsequent bowing of the
pinned dislocation AB may continue, and the newly formed dislocation
loop will continue to sweep through the crystal, thereby causing further
Copyright © 2003 Marcel Dekker, Inc.
plasticdeformation.Newloopsarealsoformed,asthedislocationAB
repeatstheaboveprocessundertheapplicationofashearstress.This
leadsultimatelytoalargeincreaseindislocationdensity(Read,1953).
However,sincethedislocationloopsproducedbytheFrank–Readsources
interactwitheachother,orotherlatticedefects,backstressesaresoonset
up.ThesebackstresseseventuallyshutdowntheFrank–Readsources.
ExperimentalevidenceoftheoperationofFrank–Readsourceshasbeen
presentedbyDash(1957)forslipinsiliconcrystals(Fig.7.14).

Asecondmechanismthatcanbeusedtoaccountfortheincreasein
dislocationdensityisillustratedinFig.7.15.Thisinvolvestheinitialactiva-
tionofaFrank–Readsourceonagivenslipplane.Screwdislocationseg-
mentsthencross-slipontoadifferentslipplanewhereanewFrank–Read
FIGURE7.13BreedingofdislocationataFrank–Readsource:(a)initialpinned
dislocationsegment;(b)dislocationbowstocircularconfigurationdueto
appliedshearstress;(c)bowingaroundpinnedsegmentsbeyondloop
instabilitycondition;(d)annihilationofoppositedislocationsegmentsX
andX
0
,(e)loopexpandsoutandcuspeddislocationAXBreturnstoinitial
configurationtorepeatcycle.(AdaptedfromRead,1953.Reprintedwithper-
mission from McGraw-Hill.)
Copyright © 2003 Marcel Dekker, Inc.
FIGURE 7.14 Photograph of Frank–Read source in a silicon crystal. (From Dash
1957. Reprinted with permission from John Wiley.)
FIGURE 7.15 Dislocation multiplication by multiple cross-slip mechanism.
(From Low and Guard, 1959. Reprinted with permission from Acta Metall.)
Copyright © 2003 Marcel Dekker, Inc.
sourceisinitiated.Theaboveprocessmaycontinuebysubsequentcross-slip
andFrank–Readsourcecreation,givingrisetoalargeincreaseinthedis-
locationdensityondifferentslipplanes.Thehighdislocationdensity
(10
15
À10
18
m/m
3
)thatgenerallyresultsfromtheplasticdeformationof
annealedcrystals(withinitialdislocationdensitiesof$10

8
À10
10
m/m
3
)
may,therefore,beexplainedbythebreedingofdislocationsatsingle
Frank–Readsources(Fig.7.12),ormultipleFrank–Readsourcesproduced
bycross-slipprocesses(Fig.7.15).
7.7 CONTRIBUTIONS FROM DISLOCATION DENSITY
TO MACROSCOPIC STRAIN
The macroscopic strain that is developed due to dislocation motion 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 a distance, x, along the glide plane results in a displace-
ment that is a fraction, x=L, of the Burgers vector, b. Therefore, the overall
displacement due to N dislocations shearing different glide planes is given
FIGURE 7.16 Macroscopic strain from dislocation motion: (A) before slip; (B)
slip steps of one Burgers vector formed after slip; (C) displacement due to
glide through distance Áx . (From Read-Hill and Abbaschian, 1994. Reprinted
with permission from McGraw-Hill.)
Copyright © 2003 Marcel Dekker, Inc.
by
Á ¼
X
N
i¼1

x
i
h
i

b ð7:5Þ
For small displacements, we may assume that the shear strain, ,is
$ Á=h. Hence, from Eq. (7.4), we may write:
 ¼
Á
h
¼
X
N
i¼1
x
i
L

b
h
ð7:6Þ
If we also note that the dislocation density, , is given by Nz=ðhzL), we may
rewrite Eq. (7.6) as
 ¼
X
h
i¼1
bx
i

N
ð7:7Þ
Assuming that the average displacement of each dislocation is
x, Eq.
(7.7) may now be written as
 ¼ b
x ð7:8Þ
The shear strain rate,
_
, may also be obtained from the time derivative
of Eq. (7.8). This gives
_
 ¼
d
dt
¼ b
v ð7:9Þ
where
v is the average velocity of disl ocations, which is given by dx=dt.Itis
important to note here that  in the above equations may correspond either
to the overall dislocation density, 
tot
, or to the density of mobile disloca-
tions, 
m
, provided that x and v apply to the appropriate dislocation con-
figurations (mobile or total). Hence, 
m
x
m

¼ 
tot
x
tot
or 
m
v
m
¼ 
tot
v
tot
in
Eqs (7.8) and (7.9).
Finally in this section, it is important to note that Eqs (7.8) and (7.9)
have been obtained for straight dislocations that extend completely across
the crystal width, z. However, the same results may be derived for curved
dislocations with arbitrary configurations across multiple slip planes. This
may be easily realized by recognizing that the sheared area fraction of the
glide plane, dA=A, corresponds to the fraction of the Burgers vector, b,in
the expression for the displacement due to glide of curved dislocations.
Hence, for glide by curved dislocations, the overall displacement, Á,is
now given by
Copyright © 2003 Marcel Dekker, Inc.
Á¼
X
N
i¼1
ÁA
i

A
bð7:10Þ
Asbefore[Eq.(7.6)],theshearstrain,,duetotheglideofcurveddisloca-
tionsisalsogivenbyÁ=h.Hence,
¼
Á
h
X
N
i¼1
ÁA
i
A

b
h
ð7:11Þ
Similarly,theshearstrainratemaybeobtainedfromthetimederivativeof
Eq.(7.11).
7.8CRYSTALSTRUCTUREANDDISLOCATION
MOTION
InChap.6,welearnedthatthePeierl’s(latticefriction)stress[Eq.
(6.8)]isminimizedbysmallBurgersvectors,b,andlargelatticespacings,
a. Hence,dislocationmotionincubiccrystals tendstooccuronclosed
packed(orclosestpacked) planesinwhichthemagnitudesofthe Burgers
vectors,b,areminimized,and theverticallatticespacings,a,aremaximized.
Sincethelattice frictionstressesareminimizedonsuch planes,dislocation
motionismostlikely tooccuronclosedpackedplanesalongclosedpacked
directions(Table7.1).
7.8.1SlipinFace-CenteredCubicStructures

Close-packedplanesinface-centeredcubic(f.c.c.)materialsareofthe{111}
type.Anexampleof a(111)planeisshowninFig.7.17(a).All the atoms
touch within the closed packed (111) plane. Also, the possible {111} slip
planes form an octahedron if all the {111} planes in the eight possible
quadrants are considered. Furthermore, the closed packed directions corre-
spond to the h110i directions along the sides of a {111} triangle in the
octahedron. Hence, in the case of f.c.c. materials, slip is most likely to
occur on octahedral {111} planes along h110i directions. Since there are
four slip planes with three slip directions in the f.c.c. structure, this indicates
that there are 12 (four slip lanes  three slip directions) possible {111} h110i
slip systems (Table 7.1).
7.8.2 Slip in Body-Centered Cubic Structures
In the case of body-centered cubic (b.c.c.) structures, there are no close-
packed planes in which all the atoms touch, although the {101} planes are
the closest packed. The close-packed directions in b.c.c. structures are the
Copyright © 2003 Marcel Dekker, Inc.
h111idirections.Slipinb.c.c.structuresismostlikelytooccuron{101}
planesalongh111idirections,Fig.7.17(b).However,slipmayalsooccuron
{110},{112},and{123}planesalongh111idirections.Whenallthepossible
slipsystemsarecounted,thereare48suchsystemsinb.c.c.structures(Table
7.1).Thisratherlargenumbergivesrisetowavyslipinb.c.c.structures.
Nevertheless, the large number of possible slip systems in b.c.c. crystals
(four times more than those in f.c.c. materials) do not necessarily promote
improved ductility since the lattice friction (Peierls–Nabarro) stresses are
generally higher in b.c.c. structures.
FIGURE 7.17 Closed packed planes and directions in (a) face-centered cubic
structure, (b) body-centered cubic crystal, and (c) hexagonal closed packed
structure. (From Hertzberg, 1996. Reprinted with permission from John
Wiley.)
Copyright © 2003 Marcel Dekker, Inc.

7.8.3SlipinHexagonalClosedPackedStructures
Thebasal(0001)planeistheclosedpackedplaneinhexagonalclosed
packed(h.c.p.)structures.Withinthisplane,slipmayoccuralongclosed
packedh11
20idirections,Fig.7.17(c).Dependingonthec=aratios,slipmay
alsooccuronnonbasal(10
10)and(1011)planesalongh1120idirections
(Table7.1).ThisisalsoillustratedschematicallyinFig.7.18.Nonbasal
slipismorelikelytooccurinh.c.p.metalswithc=aratioscloseto1.63,
whichistheexpectedvalueforidealclosepacking.Also,pyramidal(10
11)
slipmayberepresentedbyequivalentcombinationsofbasal(0001)and
prismatic(1010)slip.
7.8.4ConditionforHomogeneousPlastic
Deformation
Theabilityofamaterialtoundergoplasticdeformation(permanentshape
change)dependsstronglyonthenumberofindependentslipsystemsthat
canoperateduringdeformation.Anecessary(butnotsufficient)condition
forhomogeneousplasticdeformationwasfirstproposedbyVonMises
(1928).Notingthatsixindependentcomponentsofstrainwouldrequire
sixindependentslipcomponentsforgrainboundarycompatibilitybetween
twoadjacentcrystals(Fig.7.19),hesuggestedthatsinceplasticdeformation
occurs at constant volume, then ÁV=V ¼ "
xx
þ "
yy
þ "
zz
¼ 0. Thi s reduces
the number of grain boundary compatibility equations by one. Hence, only

five independent slip systems are required for homogeneous plastic defor-
TABLE 7.1 Summary of Slip Systems in Cubic and Hexagonal Crystals
Crystal
Structure
Slip
Plane
Slip
direction
Number of
nonparallel
planes
Slip
directions
per plane
Number
of slip
systems
Face-centered
cubic {111} h1
10i 4312¼ð4  3Þ
Body-centered
cubic {110} h
111i 6212¼ð6 Â2Þ
{112} h11
1i 12 1 12 ¼ð12  1Þ
{123} h11
1i 24 1 24 ¼ð24  1Þ
Hexagonal
close-packed {0001} h11
20i 133¼ð1  3Þ

{10
10} h1120i 313¼ð3  1Þ
{10
11} h1120i 616¼ð6  1Þ
Copyright © 2003 Marcel Dekker, Inc.
FIGURE 7.18 Planes in a hexagonal closed packed structure, with a common
[11
20 direction. (From Hull and Bacon, 1984. Reprinted with permission from
Pergamon Press.)
FIGURE 7.19 Strain conditions for slip compatibility at adjacent crystals. (From
Courtney, 1990. Reprinted with permission from McGraw-Hill.)
Copyright © 2003 Marcel Dekker, Inc.
mation.Thisisanecessary(butnotsufficient)conditionforhomogeneous
plasticdeformationinpolycrystals.
Theso-calledVonMisesconditionforhomogeneousplasticdeforma-
tionissatisfiedreadilybyf.c.c.andb.c.c.crystals.Inthecaseoff.c.c.
crystals,Taylor(1938)hasshownthatonlyfiveofthe12possible{111}
h110islipsystemsareindependent,althoughthereare384combinationsof
fiveslipsystemsthatcanresultinanygivenstrain.Similarresultshavebeen
reportedbyGrovesandKelly(1963)forb.c.c.crystalsinwhich384setsof
five{110}h111islipsystemscanbeusedtoaccountforthesamestrain.A
muchlargernumberofindependentslipsystemsisobservedinb.c.c.struc-
tureswhenpossibleslipsinthe{112}h111iand{123}h111isystemsare
considered.Thelargenumberofpossibleslipsystemsinthiscasehavebeen
identifiedusingcomputersimulationsbyChinandcoworkers(1967,1969).
Incontrasttob.c.c.andf.c.c.crystals,itisdifficulttoshowtheexis-
tenceoffiveindependentslipsystemsinh.c.p.metals/alloysinwhichslip
mayoccuronbasal,prismatic,andpyramidalplanes,Fig.7.17(c).However,
onlytwoofthe{0001}h11
20islipsystemsinthebasalplaneareindepen-

dent.Similarly,onlytwooftheprismatic{10
20}h1120itypesystemsare
independent.Furthermore,allthepyramidalslipsystemscanbereproduced
bycombinationsofbasalandprismaticslip.Thereare,therefore,onlyfour
independentslipsystemsinh.c.p.metals.So,howthencanhomogeneous
plasticdeformationoccurinh.c.p.metalssuchastitanium?Well,theanswer
tothisquestionremainsanunsolvedpuzzleinthefieldofcrystalplasticity.
Onepossiblemechanismbywhichthefifthstraincomponentmaybe
accommodatedinvolvesamechanismofdeformation-inducedtwinning.
Thisoccursbytheco-ordinatedmovementofseveraldislocations(Fig.
7.18).However,furtherworkisstillneededtodevelopafundamentalunder-
standingoftheroleoftwinningintitaniumandotherh.c.p.metals/alloys.
7.8.5PartialorExtendedDislocations
Inf.c.c.crystals,thezig-zagmotionofatomsrequiredforslipintheh110i
directionsmaynotbeenergeticallyfavorablesincethemovementofdislo-
cationsrequiressomewhatdifficultmotionofthe‘‘white’’atomsoverthe
‘‘shaded’’atomsinFig.7.20.Theordinaryh110idislocationsmay,there-
fore,dissociateintopartialdislocationswithloweroverallenergiesthan
thoseoftheoriginalh110itypedislocations.
Thepartialdislocationsmaybedeterminedsimplybyvectoraddition,
asshownschematicallyinFig.7.21.NotethatB
1
C ¼ b
2
¼ 1=6½121.
Similarly, CB
2
can be shown to be given by CB
2
¼ b

3
¼ 1=6½211]. The ordin-
ary dislocation b
1
¼ 1=2½110] may, therefore, be shown by vector addition to
be given by
Copyright © 2003 Marcel Dekker, Inc.
b
1
¼ b
2
þ b
3
ð7:12aÞ
or
1
2
½
110¼
1
6
½
211þ
1
6
½
121ð7:12bÞ
The partial dislocations, b
2
and b

3
, are generally referred to as Shockley
partials. They are formed because the elastic energies of the ordinary dis-
locations of type b
1
are greater than the sum of the line energies of the
Shockley partials. Hence,
Gðb
1
Þ
2
> Gðb
2
Þ
2
þ Gðb
3
Þ
2
ð7:13aÞ
or
G
4
ðÀ1Þ
2
þð1Þ
2
þð0Þ
2
hi

>
G
36
ðÀ1Þ
2
þð2Þ
2
þðÀ1Þ
2
hi
þ
G
36
ðÀ2Þ
2
þð1Þ
2
þð1Þ
2
hi
ð7:13bÞ
or
G
2
>
G
3
ð7:13cÞ
The above dislocation reaction is, therefore, likely to proceed since it is
energetically favorable. This separation occurs because the net force on

FIGURE 7.20 Zig-zag motion of atoms required for slip in face-centered cubic
crystals. Note that ‘‘white’’ atoms are in a row above the ‘‘shaded’’ atoms.
(From Read-Hill and Abbaschian, 1991. Reprinted with permission from
McGraw-Hill.)
Copyright © 2003 Marcel Dekker, Inc.
thepartialsisrepulsive.Asthepartialsseparate,theregularABCstacking
ofthef.c.c.latticeisdisturbed.Theseparationcontinuesuntilanequili-
briumconditionisreachedwherethenetrepulsiveforceisbalancedbythe
stackingfaultenergy(Fig.7.22).Theequilibriumseparation,d,betweenthe
twopartialshasbeenshownbyCottrell(1953)tobe
d¼
Gb
2
b
3
2
ð7:14Þ
whereGistheshearmodulus,isthestackingfaultenergy,andb
2
andb
3
correspondtotheBurgersvectorsofthepartialdislocations.Stackingfaults
ribbonscorrespondingtobandsofpartialdislocationsarepresentedinFig.
7.22(b).Typicalvaluesofthestackingfaultenergiesforvariousmetalsare
alsosummarizedinTable7.2.Notethatthestackingfaultenergiesvary
widelyfordifferentelementsandtheiralloys.Theseparationsofthepartial
dislocationsmay,therefore,varysignificantly,dependingonalloycomposi-
tion,atomicstructure,andelectronicstructure.
Thevariationsinstackingfaultenergyhavebeenfoundtohavea
strongeffectonslipplanarity,orconversely,thewavinessofslipinmetals

andtheiralloysthatcontainpartial(extended)dislocations.Thisisbecause
themovementextendeddislocationsisgenerallyconfinedtotheplaneofthe
stackingfault.Thepartialdislocationsmust,therefore,recombinebefore
cross-slipcanoccur.Forthisreason,metals/alloyswithhigherstackingfault
energieswillhavenarrowstackingfaults[Eq.(7.14)],thusmakingrecombi-
FIGURE7.21Pathofwhole(ordinary)andpartial(Shockley)dislocations.
(From Courtney, 1990. Reprinted with permission from McGraw-Hill.)
Copyright © 2003 Marcel Dekker, Inc.
nation and cross-slip easier. This reduces the stress required for recombina-
tion.
Conversely, metals and alloys with low stacking fault energies have
wide separations (stacking faults) between the partial dislocations. It is,
therefore, difficult for cross-slip to occur, since the recombination of partial
dislocations is difficult. Furthermore, because the movement of uncombined
FIGURE 7.22 (a) Shockley b
2
and b
3
surrounding stacking fault region A; (b)
stacking fault ribbons in a stainless steel. (From Michelak, 1976. Reprinted
with permission from John Wiley.)
TABLE 7.2 Stacking Fault Energies for
Face Centered Cubic Metals and Alloys
Metal
Stacking faulty energy
(mJ/m
2
¼ ergs/cm
2
)

Brass < 10
Stainless steel < 10
Ag $ 25
Au $ 75
Cu $ 90
Ni $ 200
Al $ 250
Source
: Hertzberg (1996). Reprinted with
permission from John Wiley.
Copyright © 2003 Marcel Dekker, Inc.
partialdislocationsisconfinedtoplanescontainingthestackingfaults,
materialswithlowerstackingfaultenergies(wideseparationsofpartials)
tendtoexhibithigherlevelsofstrainhardening(Table7.3).
7.8.6Superdislocations
Sofar,ourdiscussionhasfocusedondislocationmotionindisordered
structuresinwhichthesoluteatomscanoccupyanypositionwithinthe
crystalstructure.However,insomeintermetallicsystems(intermetallics
arecompoundsbetweenmetalsandmetals),orderedcrystalstructuresare
formedinwhichtheatomsmustoccupyspecificsiteswithinthecrystal
structure.Oneexampleofanorderedf.c.c.structureistheNi
3
Alcrystal
showninFig.7.23.Thenickelandaluminumatomsoccupyspecificposi-
tionsinthestructure,whichmustberetainedafterdislocationglidethrough
{111}planes.However,themovementofasingledislocationontheglide
planedisturbstheorderedarrangementofatoms,givingrisetoanenerge-
ticallyunfavorablearrangementofatoms,Fig.7.24(a).
A favorable arrangement is restored by the passage of a second dis-
location, which restores the lower energy ordered crystal structure, Fig.

TABLE 7.3 Stacking Faults and Strain Hardening Exponents
Metal
Stacking faulty energy
(mJ/M
2
)
Strain-hardening
coefficient
Slip
character
Stainless steel <10 –0.45 Planar
Cu $ 90 $ 0:3 Planar/wavy
Al $ 250 $ 0:15 Wavy
Source: Hertzberg (1996). Reprinted with permission from John Wiley.
FIGURE 7.23 Ordered face-centered cubic structure of Ni
3
Al. (From Hertzberg,
1996. Reprinted with permission from John Wiley.)
Copyright © 2003 Marcel Dekker, Inc.