Mechanical Properties of Engineered Materials 2008 Part 8 docx

In contrast, a jog is produced by disloca-tion motion out of the glide plane as the rest of the dislocation line.. Kinksand jogs may exist in edge and screw dislocations, Figs 7.2a–7.2d.

Upon some reflection, you will probably come to the conclusion thatthe response of the copper must be associated with internal changes thatoccur in the metal during bending In fact, the strength of the copper, andthe progressive hardening of the copper, are associated with the movement

of dislocations, and their interactions with defects in the crystalline copperlattice This is hard to imagine However, it is the basis for crystallineplasticity in most metallic materials and their alloys

This chapter presents an overview of how dislocation motion anddislocation interactions contribute to plastic deformation in crystallinematerials We begin with a qualitative description of how individual dis-locations move, interact, and multiply The contributions of individualdislocations to bulk plastic strain are then considered within a simple con-

tinuum framework This is followed by an introduction to the phy of slip in hexagonal and cubic materials The role that dislocations play

crystallogra-in the deformation of scrystallogra-ingle crystals and polycrystals is then explacrystallogra-ined

7.2 DISLOCATION MOTION IN CRYSTALS

As discussed inChap 6,dislocations tend to glide on close-packed planesalong close-packed directions This is due to the relatively low lattice frictionstresses in these directions, Eq (6.8a) or (6.8b) Furthermore, the motion ofdislocations along a glide plane is commonly referred to as conservativemotion This is because the total number of atoms across the glide planeremains constant (conserved) in spite of the atomic interactions associatedwith dislocation glide (Fig 6.9) In contrast to conservative dislocationmotion by glide, nonconservative dislocation motion may also occur byclimb mechanisms (Fig 6.11) These often involve the exchange of atomswith vacancies Since the atom/vacancy exchanges may be assisted by bothstress and temperature, dislocation climb is more likely to occur duringloading at elevated temperature

So far, our discussion of dislocation motion has focused mostly onstraight dislocations Furthermore, it is presumed that the dislocations lie inthe positions of lowest energy within the lattice, i.e., energy valleys/troughs(Fig 6.10).However, in many cases, kinked dislocations are observed (Fig.7.1) These have inclined straight or curved line segments that all lie on thesame glide plane(Fig 7.1).The shape of the kinked dislocation segment isdependent on the magnitude of the energy difference between the energypeaks and energy valleys in the crystalline lattice In cases where the energydifference is large, dislocations can minimize their overall line energies byminimizing their line lengths in the higher energy peak regime This givesrise to sharp kinks (A in Fig 7.1) that enable dislocations to minimize theirline lengths in the high-energy regions It also maximizes the dislocation linelengths in the low-energy valley regions

In contrast, when the energy difference (between the peaks and thevalleys) is small, a diffuse kink is formed (C in Fig 7.1) The diffuse kink hassignificant fractions of its length in the low-energy valleys and high-energypeak regions In this way, a diffuse kink can also minimize the overall lineenergy 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 regionsalong the kink tend to move faster than those along the low-energy valleyswhich have to overcome a larger energy barrier Once the barriers are over-come, kink nucleation and propagation mechanisms may be likened to the

FIGURE 7.1 Schematic of kinked dislocation configurations between peaksand valleys in a crystalline lattice Note that sharp kink is formed when energydifference 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.)

snapping motion of a whip As in the case of a snapped whip, this may giverise to faster kink propagation than that of a straight dislocation The over-all mobility of a kink will also depend on the energy difference between thepeaks and the valleys, and the orientation of the dislocation with respect tothe lattice.

Before concluding this section, it is important to note here that there is

a difference between a sharp kink [A inFig 7.1and Figs 7.2(a) and 7.2(b)]and a jog, Figs 7.2(a) and 7.2(b) A kink has all its segments on the sameplane 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 Kinksand 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 therest of the dislocation line In fact, they may be less mobile than the rest ofthe dislocation line, depending on the directions of their Burgers vectorsrelative to those of the unjogged segments

FIGURE7.2 (a), (b) Kinks in edge and screw dislocations; (c), (d) jogs in edgeand screw dislocations The slip planes are shaded (From Hull and Bacon,

1984 Reprinted with permission from Pergamon Press.)

7.3 DISLOCATION VELOCITY

When the shear stress that is applied to a crystal exceeds the lattice frictionstress, dislocations move at a velocity that is dependent on the magnitude ofthe applied shear stress This has been demonstrated for LiF crystals byJohnston and Gilman (1959) By measuring the displacement of etch pits

in crystals with low dislocation densities, they were able to show that thedislocation velocity is proportional to the applied shear stress Their resultsare presented in Fig 7.3 for both screw and edge dislocations

Note that, at the same stress level, edge dislocations move at fasterspeeds (up to 50 times faster) than screw dislocations Also, the velocities ofdislocations extend over 12 orders of magnitude on the log–log plot shown

FIGURE7.3 Dependence of dislocation velocity on applied shear stress (FromJohnston and Gilman, 1959 Reprinted with permission from J Appl Phys.)

inFig 7.3.However, for uniform dislocation motion, the limiting velocityfor both screw and edge dislocations corresponds to the velocity of trans-verse shear waves Also, damping forces increasingly oppose the motion ofdislocations at velocities above 103cm/s.

Dislocation velocities in a wide range of crystals have been shown to

be strongly dependent on the magnitude of the applied shear stress (Fig.7.4), although the detailed shapes of the dislocation velocity versus stresscurves may vary significantly, as shown in Fig 7.4 For the straight sections

of the dislocation–velocity curves, it is possible to fit the measured velocitydata to power-law equations of the form:

where v is the dislocation velocity,

material constant, and m is a constant that increases with decreasing perature An increase in dislocation velocity with decreasing temperaturehas also been demonstrated by Stein and Low (1960) in experiments on Fe–3.25Si crystals(Fig 7.5).This increase is associated with the reduced damp-ing forces due to the reduced scattering (phonons) of less frequent latticevibrations at lower temperatures

tem-FIGURE7.4 Dependence of dislocation velocity on applied shear stress Thedata are for 208C except for Ge (4508C) and Si (8508C) (From Haasen, 1988.Reprinted with permission from Cambridge University Press.)

7.4 DISLOCATION INTERACTIONS

The possible interactions between screw and edge dislocations will be cussed in this section Consider the edge dislocations (Burgers vectors per-pendicular to the dislocation lines) AB and XY with perpendicular Burgersvectors, b1and b2, shown inFig 7.6.The moving dislocation XY [Fig 7.6(a)]glides on a slip plane that is a stationary dislocation AB During the inter-section, a jog PP0 corresponding to one lattice spacing is produced as dis-location XY cuts dislocation AB, Fig 7.6(b) Since the jog has a Burgersvector that is perpendicular to PP0, it is an edge jog Also, since the Burgersvector of PP0is the same as that of the original dislocation, AB, the jog willcontinue to glide along with the rest of the dislocation, if there is a largeenough component of stress to drive it along the slip plane, which is per-pendicular to that of line segments AP or P0B, Fig 7.6(b)

dis-Let us now consider the interactions between two edge dislocations(XY and AB) with parallel Burgers vectors,Fig 7.7(a) In this case, disloca-

FIGURE 7.5 Dependence of dislocation velocity on temperature and appliedshear stress in Fe–3.25Si Crystals (From Stein and Low, 1960 Reprinted withpermission from J Appl Phys.)

tion XY intersects dislocation AB, and produces two screw jogs PP0 and

QQ0 The jogs PP0and QQ0 are screw in nature because they are parallel tothe Burgers vectors b1 and b2, respectively, Figs 7.7(a) and 7.7(b) Since thejogged screw dislocation segments have greater mobility than the edge dis-locations to which they belong, they will not impede the overall dislocationmotion Hence, interactions between edge dislocations do not significantlyaffect dislocation mobility

This is not true for interactions involving screw dislocations Forexample, in the case of a right-handed screw dislocation that intersects amoving edge dislocation [Fig 7.8(a)], the dislocation segment PP0 glides

FIGURE7.7 Interactions between two edge dislocations with parallel Burgersvectors: (a) before intersection; (b) after intersection (From Hull and Bacon,

1984 Reprinted with permission from Pergamon Press.)

FIGURE 7.6 Interactions between two edge dislocations with perpendicularBurgers vectors: (a) before intersection; (b) after intersection (From Hulland Bacon, 1984 Reprinted with permission from Pergamon Press.)

down one level (from one atomic plane to the other) following a spiral path(staircase) along the dislocation line XY, as it cuts the screw dislocation XY,Fig 7.8(b) This produces a jog PP0in AB, and a jog QQ0in XY Hence, thesegments AP0and PB lie on different planes, Fig 7.8(b) Furthermore, sincethe Burgers vectors of the dislocation line segments PP0 and QQ0 are per-pendicular to their line segments, the jogs are edge in character Therefore,the only way the jog can move conservatively is along the axis of the screwdislocation, as shown in Fig 7.9.This does not impede the motion of thescrew dislocation, provided the jog glides on the plane (PP0RR0).

However, since edge dislocation components can only move tively by glide on planes containing their Burgers vectors and line segments,the movement of the edge dislocation to A0QQ0B (Fig 7.9) would requirenonconservative climb mechanisms that involve stress- and thermallyassisted processes This will leave behind a trail of vacancies or interstitials,depending on the direction of motion, and the sign of the dislocation This isillustrated inFig 7.10for a jogged screw dislocation that produces a trail ofvacancies Note that the dislocation segments between the jogs are boweddue to the effects of line tension Bowing of dislocations due to line tensioneffects will be discussed in the next section In closing, however, it is impor-tant to note here that the interactions between two screw dislocations (Fig.7.11) can give rise to similar phenomena to those discussed above It is auseful exercise to try to work out the effects of such interactions

conserva-FIGURE 7.8 Interactions between right-handed screw dislocation and edgedislocations: (a) before intersection; (b) after intersection (From Hull andBacon, 1984 Reprinted with permission from Pergamon Press.)

FIGURE 7.9 Movement of edge jog on a screw dislocation; conservativemotion of jog only possible on plane PP0RR Motion of screw dislocation to

A0QQ0B would require climb of the jog along plane PQQ0P (From Hull andBacon, 1984 Reprinted with permission from Pergamon Press.)

FIGURE7.10 Schematic illustration of trail of vacancies produced by glide ofscrew dislocation (From Hull and Bacon, 1984—reprinted with permissionfrom Pergamon Press.)

FIGURE7.11 Interactions between two screw dislocations: (a) before tion; (b) after intersection (From Hull and Bacon, 1984 Reprinted with per-mission from Pergamon Press.)

intersec-7.5 DISLOCATION BOWING DUE TO LINE TENSION

It should be clear from the above discussion that interactions between locations can give rise to pinned dislocation segments, e.g., dislocation linesegments that are pinned by jogs, solutes, interstitials, or precipitates When

dis-a crystdis-al is subjected to dis-a shedis-ar stress, the so-cdis-alled line tension thdis-at develops

in a pinned dislocation segment can give rise to a form of dislocation bowingthat is somewhat analogous to the bowing of a string subjected to linetension, T In the case of a dislocation, the line tension has a magnitude

 Gb2 The bowing of dislocation is illustrated schematically in Fig 7.12.Let us now consider the free body diagram of the bowed dislocationconfiguration in Fig 7.12(b) For force equilibrium in the y direction, wemay write:

2T sin 

2

 

ð7:2Þwhere

tion line length, and the other parameters are shown in Fig 7.12 For smallcurvatures, sin (=2Þ  =2, and so Eq (7.2) reduces to

The critical shear stress described by Eq (7.4b) is sufficient to cause thepinned dislocation to continue to bow in a stable manner until it reaches asemicircular configuration with r ¼ L=2 This bowing forms the basis of one

of the most potent mechanisms for dislocation multiplication, which is cussed in the next section

dis-7.6 DISLOCATION MULTIPLICATION

The discerning reader is probably wondering how plastic deformation canactually continue in spite of the numerous interactions between dislocationsthat are likely to give rise to a reduction in the density of mobile disloca-tions This question will be addressed in this section However, beforeanswering the question, let us start by considering the simplest case of awell-annealed crystal Such crystals can have dislocation densities as low as

This question was first answered by Frank and Read in a discussionthat was held in a pub in Pittsburgh Their conversation led to the mechan-ism of dislocation breeding that is illustrated schematically inFig 7.13 Theschematics show one possible mechanism by which dislocations can multiplywhen a shear stress is applied to a dislocation that is pinned at both ends.Under an applied shear stress, the pinned dislocation [Fig 7.13(a)] bowsinto a circular arc with radius of curvature, r ¼ L=2, shown in Fig 7.13(b).The bowing of the curved dislocation is caused by the line tension, T, asdiscussed in Sect 7.5 (Fig 7.12) This causes the dislocation to bow in astable manner until it reaches the circular configuration illustrated schema-tically in Fig 7.13(b) From Eq (7.4b), the critical shear stress required forthis to occur is  Gb=L

Beyond the circular configuration of Fig 7.13(b), the dislocation bowsaround the pinned ends, as shown in Fig 7.13(c) This continues until thepoints labeled X and X0come 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 theoriginal straight configuration, Fig 7.13(a)

Note that the shaded areas in Fig 7.13 correspond to the regions ofthe crystal that have been sheared by the above process They have, there-fore, been deformed plastically Furthermore, subsequent bowing of thepinned dislocation AB may continue, and the newly formed dislocationloop will continue to sweep through the crystal, thereby causing further

plastic deformation New loops are also formed, as the dislocation ABrepeats the above process under the application of a shear stress Thisleads ultimately to a large increase in dislocation density (Read, 1953).However, since the dislocation loops produced by the Frank–Read sourcesinteract with each other, or other lattice defects, back stresses are soon set

up These back stresses eventually shut down the Frank–Read sources.Experimental evidence of the operation of Frank–Read sources has beenpresented by Dash (1957) for slip in silicon crystals (Fig 7.14)

A second mechanism that can be used to account for the increase indislocation density is illustrated inFig 7.15 This involves the initial activa-tion of a Frank–Read source on a given slip plane Screw dislocation seg-ments then cross-slip on to a different slip plane where a new Frank–Read

FIGURE7.13 Breeding of dislocation at a Frank–Read source: (a) initial pinneddislocation segment; (b) dislocation bows to circular configuration due toapplied shear stress; (c) bowing around pinned segments beyond loopinstability condition; (d) annihilation of opposite dislocation segments Xand X0, (e) loop expands out and cusped dislocation AXB returns to initialconfiguration to repeat cycle (Adapted from Read, 1953 Reprinted with per-mission from McGraw-Hill.)

FIGURE7.14 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.)

source is initiated The above process may continue by subsequent cross-slipand Frank–Read source creation, giving rise to a large increase in the dis-location density on different slip planes The high dislocation density(10151018

m/m3) that generally results from the plastic deformation ofannealed crystals (with initial dislocation densities of  1081010

m/m3)may, therefore, be explained by the breeding of dislocations at singleFrank–Read sources (Fig 7.12), or multiple Frank–Read sources produced

by cross-slip processes (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 onmultiple slip planes For simplicity, let us consider the glide of a singledislocation, as illustrated schematically in Fig 7.16 The crystal of height,

h, is displaced by a horizontal distance, b, the Burgers vector, due to theglide 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 overalldisplacement due to N dislocations shearing different glide planes is given

FIGURE7.16 Macroscopic strain from dislocation motion: (A) before slip; (B)slip steps of one Burgers vector formed after slip; (C) displacement due toglide through distancex (From Read-Hill and Abbaschian, 1994 Reprintedwith permission from McGraw-Hill.)

For small displacements, we may assume that the shear strain, , is

 =h Hence, from Eq (7.4), we may write:

h ¼XN

i¼1

xiL

The shear strain rate, _, may also be obtained from the time derivative

of Eq (7.8) This gives

disloca-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 acrossthe crystal width, z However, the same results may be derived for curveddislocations with arbitrary configurations across multiple slip planes Thismay be easily realized by recognizing that the sheared area fraction of theglide plane, dA=A, corresponds to the fraction of the Burgers vector, b, inthe expression for the displacement due to glide of curved dislocations.Hence, for glide by curved dislocations, the overall displacement, , isnow given by

7.8.1 Slip in Face-Centered Cubic Structures

Close-packed planes in face-centered cubic (f.c.c.) materials are of the {111}type An example of a (111) plane is shown inFig 7.17(a).All the atomstouch within the closed packed (111) plane Also, the possible {111} slipplanes form an octahedron if all the {111} planes in the eight possiblequadrants are considered Furthermore, the closed packed directions corre-spond to the h110i directions along the sides of a {111} triangle in theoctahedron Hence, in the case of f.c.c materials, slip is most likely tooccur on octahedral {111} planes along h110i directions Since there arefour slip planes with three slip directions in the f.c.c structure, this indicatesthat there are 12 (four slip lanes
three slip directions) possible {111} h110islip 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 packed planes in which all the atoms touch, although the {101} planes arethe closest packed The close-packed directions in b.c.c structures are the

close-h111i directions Slip in b.c.c structures is most likely to occur on {101}planes along h111i directions, Fig 7.17(b) However, slip may also occur on{110}, {112}, and {123} planes along h111i directions When all the possibleslip systems are counted, there are 48 such systems in b.c.c structures (Table7.1) This rather large number gives rise to wavy slip in b.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 promoteimproved ductility since the lattice friction (Peierls–Nabarro) stresses aregenerally higher in b.c.c structures.

FIGURE7.17 Closed packed planes and directions in (a) face-centered cubicstructure, (b) body-centered cubic crystal, and (c) hexagonal closed packedstructure (From Hertzberg, 1996 Reprinted with permission from JohnWiley.)

7.8.3 Slip in Hexagonal Closed Packed Structures

The basal (0001) plane is the closed packed plane in hexagonal closedpacked (h.c.p.) structures Within this plane, slip may occur along closedpacked h1120i directions, Fig 7.17(c) Depending on the c=a ratios, slip mayalso occur on nonbasal (1010) and (1011) planes along h1120i directions(Table 7.1) This is also illustrated schematically in Fig 7.18 Nonbasalslip is more likely to occur in h.c.p metals with c=a ratios close to 1.63,which is the expected value for ideal close packing Also, pyramidal (1011)slip may be represented by equivalent combinations of basal (0001) andprismatic (1010) slip

7.8.4 Condition for Homogeneous Plastic

Deformation

The ability of a material to undergo plastic deformation (permanent shapechange) depends strongly on the number of independent slip systems thatcan operate during deformation A necessary (but not sufficient) conditionfor homogeneous plastic deformation was first proposed by Von Mises(1928) Noting that six independent components of strain would requiresix independent slip components for grain boundary compatibility betweentwo adjacent crystals(Fig 7.19), he suggested that since plastic deformationoccurs at constant volume, thenV=V ¼ "xxþ "yyþ "zz¼ 0 This reducesthe number of grain boundary compatibility equations by one Hence, onlyfive independent slip systems are required for homogeneous plastic defor-

TABLE7.1 Summary of Slip Systems in Cubic and Hexagonal Crystals

Crystal

Structure

SlipPlane

Slipdirection

Number ofnonparallelplanes

Slipdirectionsper plane

Number

of slipsystemsFace-centered

FIGURE7.18 Planes in a hexagonal closed packed structure, with a common[1120 direction (From Hull and Bacon, 1984 Reprinted with permission fromPergamon Press.)

FIGURE7.19 Strain conditions for slip compatibility at adjacent crystals (FromCourtney, 1990 Reprinted with permission from McGraw-Hill.)

mation This is a necessary (but not sufficient) condition for homogeneousplastic deformation in polycrystals.

The so-called Von Mises condition for homogeneous plastic tion is satisfied readily by f.c.c and b.c.c crystals In the case of f.c.c.crystals, Taylor (1938) has shown that only five of the 12 possible {111}h110i slip systems are independent, although there are 384 combinations offive slip systems that can result in any given strain Similar results have beenreported by Groves and Kelly (1963) for b.c.c crystals in which 384 sets offive {110} h111i slip systems can be used to account for the same strain Amuch larger number of independent slip systems is observed in b.c.c struc-tures when possible slips in the {112} h111i and {123} h111i systems areconsidered The large number of possible slip systems in this case have beenidentified using computer simulations by Chin and coworkers (1967, 1969)

deforma-In contrast to b.c.c and f.c.c crystals, it is difficult to show the tence of five independent slip systems in h.c.p metals/alloys in which slipmay occur on basal, prismatic, and pyramidal planes,Fig 7.17(c) However,only two of the {0001} h1120i slip systems in the basal plane are indepen-dent Similarly, only two of the prismatic {1020} h1120i type systems areindependent Furthermore, all the pyramidal slip systems can be reproduced

exis-by combinations of basal and prismatic slip There are, therefore, only fourindependent slip systems in h.c.p metals So, how then can homogeneousplastic deformation occur in h.c.p metals such as titanium? Well, the answer

to this question remains an unsolved puzzle in the field of crystal plasticity.One possible mechanism by which the fifth strain component may beaccommodated 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 under-standing of the role of twinning in titanium and other h.c.p metals/alloys

7.8.5 Partial or Extended Dislocations

In f.c.c crystals, the zig-zag motion of atoms required for slip in the h110idirections may not be energetically favorable since the movement of dislo-cations requires somewhat difficult motion of the ‘‘white’’ atoms over the

‘‘shaded’’ atoms inFig 7.20 The ordinary h110i dislocations may, fore, dissociate into partial dislocations with lower overall energies thanthose of the original h110i type dislocations

there-The partial dislocations may be determined simply by vector addition,

as shown schematically in Fig 7.21 Note that B1C ¼ b2 ¼ 1=6½121.Similarly, CB2can be shown to be given by CB2¼ b3¼ 1=6½211] The ordin-ary dislocation b1¼ 1=2½110] may, therefore, be shown by vector addition to

be given by

b1¼ b2þ b3 ð7:12aÞor

the partials is repulsive As the partials separate, the regular ABC stacking

of the f.c.c lattice is disturbed The separation continues until an brium condition is reached where the net repulsive force is balanced by thestacking fault energy(Fig 7.22) The equilibrium separation, d, between thetwo partials has been shown by Cottrell (1953) to be

equili-d ¼Gb2b3

where G is the shear modulus,  is the stacking fault energy, and b2 and b3correspond to the Burgers vectors of the partial dislocations Stacking faultsribbons corresponding to bands of partial dislocations are presented in Fig.7.22(b) Typical values of the stacking fault energies for various metals arealso summarized in Table 7.2 Note that the stacking fault energies varywidely for different elements and their alloys The separations of the partialdislocations may, therefore, vary significantly, depending on alloy composi-tion, atomic structure, and electronic structure

The variations in stacking fault energy have been found to have astrong effect on slip planarity, or conversely, the waviness of slip in metalsand their alloys that contain partial (extended) dislocations This is becausethe movement extended dislocations is generally confined to the plane of thestacking fault The partial dislocations must, therefore, recombine beforecross-slip can occur For this reason, metals/alloys with higher stacking faultenergies will have narrow stacking faults [Eq (7.14)], thus making recombi-

FIGURE 7.21 Path of whole (ordinary) and partial (Shockley) dislocations.(From Courtney, 1990 Reprinted with permission from McGraw-Hill.)