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Effect of stacking fault energy on the mutual cross slip of unlike dislocations—I. Coalescence mode
EFFECT
OF STACKING FAULT ENERGY UNLIKE DISLOCATIONS-I. M. J. MARCINKOWSKII,
ON THE MUTUAL CROSS COALESCENCE MODE*?
K. SADANANDAS
SLIP OF
and R. J. TAUNT:
A detailed numerical analysis has been made of the cross slip behavior associated with extended screw t,ype dislocations of opposite sign in face centered cubic metals and alloys which pass one another on parallel slip planes. It is shown that below some critical interplanar spacing, the mutual interaction between the extended dislocations is sufficient to bring about coalescence, followed by spontaneous cross slip and subsequent mutual annihilation. Just above this critical interplanar spacing, the estended dialocations may pass one another with a maximum interaction and thus a maximum stress. This maximum stress, i.e. strengthening, deoreases with increasing stacking fault energy and provides an alternate hypothesis to the pile-up model for Stage III work hardening behavior in face-centered cubic metals and alloys. EFFET
DE L’ENERGIE DE DEFAUT D’EMPILEMER’T MUTUEL DE DISLOCATIONS DIFFERENTES-I.
SUR LE GLISSEMEST PAR RECOMBIBSISOZi
DE1-IE
Une analyse numerique detaillee a et6 faite du glissement d&vie associe au passage dans des plans de glissement paralleles de dislocations vis dissociees de signe oppose, dens les metaux et alliages cubiques a faces centrees. On montre qu’en dessous d’un Qcartement,critique des plans de glissement, les interactions mutuelles entre les dislocations dissociees sont suffisantes pour produire leur recombinaison, suivie d’un glissement d&vie spontane et d’une annihilation mutuelle. Juste au dessus de cet Ccartement critique, les dislocations dissociees peuvent passer l’une p&s de l’autre avec un maximum d’interartion et done un maximum d’effort. Cet effort maximal, c’est-a-dire ce durcissement, decroit lorsque l’energie de defaut d’empilement croit; ceci donne une explication du durcissement des metaux et alliages rubiques a faces centrees dans le stade III, differente de celle que fournit le modirle de l’empilement. EINFLUO DER STAPELFEHLERENERGIE AUF DIE QUERGLEITUNG VON VERSETZUSGES ENTGEGENGESETZTEN VORZEICHENS I-KOALESZENZ-MODUS Das Quergleitverhalten aufgespaltener Schraubenversetzungen entgegengesetzten Vorzeichens in kubisch-fliichenzentrierten Metallen und Legierungen, die auf paralellen Gleitebenen aneinander vorbeigleiten, wurde ausftihrlich numerisch analysiert. Es wird gezeigt, da9 unterhalb eines kritischen Gleitebenenabstandes die gegenseitige Wechselwirkung zwischen den aufgespaltenen Schraubenversetzungen ausreicht, urn eine Koaleszen mit naohfolgender spontaner Quergleitung und gegenseitiger Annihilierung herbeizufiihren. Gerade oberhalb dieses kritischen Gleitebenenabstandes k&men die aufgespaltenen Versetzungen aneinander mit maximaler Wechselwirkung und somit maximaler Schubspannung aneinander vorbeigleiten. Diese maximale Schubspannung, d.h. Verfestigung, nimmt mit. zunehmender Stapelfehlerenergie ab und liefert eine Alternativhypothese zum Model1 der Versetzungsaufstauungen fur das Stufe-III-Verfestigungsverhalten von kubisch-flachenzentrierten Metallen und Legierungen. INTRODUCTION
One of the most important aspects of dislocation behavior concerns the process of cross slip from one plane to another. In the case of unextended screw type dislocations, the process is relatively simple.(l) However, in the case of extended dislocations separated by a stacking fault, the problem is much more complex. Seeger and associates(2-4) were the first to consider this problem in any detail. In particular, they postulated that under a sufficiently large stress concentration, due to say a pile-up of dislocations, the blocked lead dislocat,ion would have its extension reduced from d, its equilibrium value in the free state, to some smaller value d’. If d’ becomes sufficiently small, i.e. of the order of a Burgers vector, the extended dislocation may be considered as having coalesced into a single pure screw type dislocation * Received January 17, 1974; revised April 22. 1974. t The present research effort was supported by The United States At,omic Energy Commission under Contract Number AT-(40-1)3935. $ Engineering Materials Group, and Department of Mechanical Engineering. University of Maryland, College Park, Maryland 20742, U.S.A. ACTA METALLURGICA,
VOL. 22, NOVEMBER
1974
which can then easily cross slip under some relatively small externally applied stress. An alternative model to the coalescence mechanism was proposed by Fleisoherc5) in which he postulated the individual cross slip of each of the two partial dislocations comprising the extended screw type This particular process involved the dislocation. formation of stair-rod dislocations. The energetics of the stair-rod mechanism were analyzed in detail for single blocked extended screw type dislocations in face-centered cubic (f.c.c.) crystals and compared with the coalescence model.‘6) It was shown that the relative ease with which each of these processes occurred depended upon the orientation of the applied stress, but that the stair rod mechanism was generally more favorable. It is conceivable however that these results, although interesting from an academic point of view, may not necessarily represent the same processes occurring during Stage III of plastic deformation. It will be recalled that during t,his stage, the rate of work hardening is relatively low, indicating that very significant dynamic recovery is taking place.
1405
ACTA
1406
?IIETALLURGICA,
One of the most important of all dynamic recovery processes, from the dislocation point of view, is the mutual annihilation of screw type dislocations of opposite sign. The behavior of such dislocations, as they pass one another, has already been treated in detail for both ordered(r*s) as well as disordered@) alloys. Furthermore, the critical conditions for cross slip have been determined for passing screw type superlattice dislocations of opposite sign.@~lO) The much more complex case involving the conditions for cross slip of a pair of extended passing dislocations in a disordered close-packed lattice has yet to be analyzed. Such an analysis is important in understanding the Stage III work hardening behavior in f.c.c. metals and alloys and thus provides the motivation for the present study. Due to the complexity of this analysis however, it will be carried out in two separate parts. Part I will deal with the cross slip mode involving dislocation coalescence, while Part II will concentrate on the cross slip mode requiring stair-rod formation.
VOL.
OF
THE
1954
0)
L,/,,,~
C)
FORMULATION
22,
*/i/,//i/, b_
di
PROBLEM
Figure 1 shows the stacking sequence of the closepacked (111) planes in an f.c.c. st’ructure. In particular, the stacking periodicity normal to the drawing in ABCABC, etc. Six different types of slip vector are also shown in Fig. 1 and they are labelled as 0, 1,2,3, 4 and 5. It is apparent from inspection of t,his figure that Burgers vectors 3 and 4, as well as 0 and 1 correspond to the movement of at,oms in the C layer
Fxo. 1. Various slip modes via partial disiocetions on the close packed plane in a face-centered cubic structure.
FIG. 2. Various combinat’ions of ert.ended screw type dislocation pairs of opposite sign in a face-centered cubic crystal.
relative to those in the A layer, while Burgers vectors 1 and 2, along with 4 and 5 correspond to the motion of atoms in the B layer relative to those in A. From Fig. 1, and from what has been said above, it is possible to construct the four combinations of extended screw dislocations lying on parallel planes as shown in Fig. 2. Two views are shown in each case; the top view being along the dislocation line, i.e. parallel to the slip plane, while the bottom view is normal to the slip plane. The letters B and C shown between each extended dislocation pair simply indicates that the B and C layers in Fig. 1 have slipped relative to A as discussed earlier. Inspection of Fig. 2 shows that all four configurations are not unique and that those illustrated in Figs. 2(a and d) are equivalent. Figure 3 shows the details of the stacking disorder associated with the dislocation configuration of Fig. 2(a). The primed letters in t,his figure show the new stacking sequence brought about by the pair of extended dislocations. Note that extended pair l-2 has altered the stacking sequence of the planes above the dislocation with respect to the original stacking, It is next of interest to analyse the behavior of the pairs of extended dislocations shown in Fig. 2 as they pass one another. For simplicity the passing modes illustrated in Fig. 2(a and d) will be referred to as mode 1, while those shown in Fig. 2(b and c) will be termed
3IARCINKOWSKI
MUTUAL
et al.:
CROSS
-c-s
A-B-A
-B’-A
c-c-c
-tin’-----_
B-S-S
-C’-
B
8’ -
A
-ti-C
s
-C’a-------C’-A
~
i-B’-----c
-&-C
B’-
A
A-C’-A
El__, = E2_-4 = E In 2n
2
-c-c
B------n’-
b 72 --b1s2 + (1 ” r)
S-E------_
PbE2
A-C’-A-A-A
27r(l -
-
FIG. 3. More detailed view of Fig. 2(a) parallel to the slip planes showing stacking disorder resulting from the pair of extended dislocations.
mode 2. A more detailed schematic illustration of the mode 1 passing configuration is shown in Fig. 4(a). The total energy associated with this dislocation array may be writt.en as follows:
E I-4 = -&ln
V)
[@ -t (Q2 +
_ Pb8 Y)
co92 (n/2 -
cos (x/2 -
4- -L
+ E,-,
+ L-4 -4-G-a
-/- J-L, i-8,
-4-E,
(1)
where E,,, etc. are the interaction energies between dislocations 1 and 2, etc. while E, represents the total stacking fault energy, whereas E, is the potential energy due tSothe applied stress 7. Also, from symmetry considerations, El_, = E,, and E,_, = E,_,. The general expression for the interaction energy between a pair of infinite straight dislocations in t‘he isotropic approximation is given byfin
X
-
InR
E
2-2
W -
--&In -
X)2 + $1””
- rUb? 2741 -
Y)
b &s2
C
COS (x/2
+
f
1
2
-
(1 ZY)
as) cos (7r/2 -
aa)
GW where b,, and blE correspond t,o the screw and edge components respectively of the partial dislocations with Burgers vector b = &,(112), where a, is the edge length of the cubic unit cell. Inspection of Figs. 1 and 2 show that b,, = b cos 30” while blK = b sin 30”. In addition a1 =
tan-r
a2
tan-l
=I
a3 =
2 [3 ?/1 1 y
-
tan-’
Y
(d _ 5)
(44
W
.
[
a
%r(l ” Y)R‘J
1 (34
[ (d + 2) R
(3b)
(x2)cos (n/2 -i- a,)
R,
E, = E,-,
arr)
!J2Pe b 2 j b2 Is z [ fl - 4
4l
277(1 -
1 134 1
@’ + ‘!!2)“2 5 a _ bE2 R I.3 [ (1 - v) Q
A
A
i
1407
DISLOCATIOSS-I
2~
7
c-B’-C
UP;LIKE
E,_, = E,_, = k. In -!R,
c/,,,,/,,, c’ s
B------A’-8
OF
interaction energy associated with the screw component of each partial dislocation while the second two terms correspond to the in~raetioR energy contributions from the edge components. The symbols p and v denot,e the shear modulus and Poisson’s ratio respectively, while R, represents the distance of the dislocation from the surface of the cryst,al. The dislocation self energies have been omitt,ed from equation (I) since they are neither functions of x nor of d and thus do not enter into the present. calculations. Applying equation (2) to Fig. 4(a), it follows that
C-C-t-&-C
B-A-S
SLIP
The last two terms in equation (1) can be written as E, = 2dy
[(b, x 5) * RI
where bI and b, are t’he Burgers vectors associated with dislocations 1 and 2, respectively, E, is their tangent vector, while R is the distance between the two dislocations. The first term in equation (2) is the
(5)
where y is the stacking fault energy, and E 7 = -rb,,[r
-
d -t_ (x + d)].
(6)
In obtaining the above equation, it has been assumed, for simplicity, that r acts only on the screw components of the partial dislocations as well as acting only
ACTA
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METALLURGICA,
VOL.
22,
1974
in the primary slip plane of the extended dislocations. The condition of equilibrium for a fixed value of x is given by the condition aE,/&? = 0, which when applied to equation (1) gives
F
rUb2Y2@ + 4 41 - 4w + x)2 + y212
-
2d(d - N2 + Y21 4 b1s2 + Pu(d 4 - 4 7 lubm2y2(d 77(1 -
V)[(d - x)2 + y2]2
b t1
2
‘_” 1 v)
+ 2y = 0.
(7)
A similar analysis to that shown above can be carried out for mode 2 dislocation passing behavior. In fact, equation (7) applies equally well to both. In particular, in those cases where a term is preceded by a f or i, the top sign applies to mode 1 passing behavior, while the bottom sign corresponds to mode 2 passing. In order t’o obtain the stress 7 necessary to maintain the equilibrium configuration shown in Fig. 4(a),it is necessary to reconstruct this drawing as shown in Fig. P(b). In particular, dislocation 3 is perturbed from its equilibrium position by some small amount R. This immediately destroys the symmetry exhibited in Fig. 4(a), so that a new set of interaction energies
F9
FIG. 4(b). Schematic illustration used to determine the externally applied stress required to maintain a pair of extended passing dislocat,ions in statio equilibrium.
must be determined.
In particular
@aI
E 3-4 = &hi7 El_,
=
g
ln
(d2 +
a
b
RI
-b1s2
+
2
(1 ‘“VI
1 @b)
bl: - F. ‘(x- ‘i ’ y211’2 V
n
PblE2
_
277(1 -
Y)
cos2 (n/2 -
al)
(SC)
C(x - d, - 4) + Y2Y2
E 2-4
R,
x L2 -
b 3
”
(1
v)
1
PblE2 cos2 (77/2 $
27r(l -
P)
a4)
(3d)
[(x -+ d,)2 + y2p2
E 1-4
&
-
PblE2 2x(1 -
v)
1
cos (x/2 -
az) cos (7~/2 + az) (se)
E 2-3 -- $1,
x
FIG. 4(a). Schematic illustration used to determine the equilibrium configuration of extended passing dislocations.
[(d, -
z + R)2 -,- y2]l12 Rll
bls2 + -
w
(1 - 7)1
PblE2 2n(l -
Y)
cos (7r/2 + a3) cos (r/2 -
0c.J.
m
MARCINKOWSKI
et
al.:
MUTUAL
In addition
CROSS
(3) E,, = yR + yd, + $1
and E, = --k,[(&
+ 5) + (z -
R) -
41.
(10)
Expressions S-10 may then be substituted into equation (l), followed by the operation aE,/aR = 0. As R -+ 0, d, N d, and we obtain -FF,,
1 2 P” - 2x[x2 L2 =F & -+ $1 1 1 T ,4.2~2~ 2y) +%r[(d b1s2 It (1 '_ -x)2 +$1 - 1
aET P = aR =2nd
-b,s
6 2 + (1 ‘” y) b
?T(l -
Y) [x2 + $12
b
P(d - 2)
i
rUbm2Y2(d - 4 ?7(1 -
Y) [(d - x)2 + y2]2
+ y + T&s = 0.
(11)
The above expression is simply the requirement that the force on dislocation 3 in the x direction be equal Identical relationships were obtained to zero. previously@) using the expressions for the elastic stress fields about a dislocation, rather than the energy expression given by equation (2). As was the case in equation (7)) equation ( 11) can also be used for passing mode 2 by employing the bottom of the & or F signs wherever they occur. The importance of equation (11) lies in the fact that, it can be employed to determine 7 for a fixed value of x, after first determining d for this same value of x from equation (7). It is also interest’ing to note that equation (7) could have been obtained from relationships of the type given by equation (11) by forming combinations of t.he type given by Fz3 + F,, = 0 or Fz2 + F,, = 0. It is obvious that such combinations eliminate the term containing r, since it is of opposite sign in the two different force expressions. These matters have been discussed in great,er detail in Ref. 10. NUMERICAL
SLIP
OF
USLIKE
1409
DISLOCATIOSS-I
Although equation (i) could be solved numerically for d as a function of x, it is much more desirable to minimize the energy given by equation (1) directly.02) This was done w&h the aid of a minimizat,ion subrout,ine entitled FUNFIT employing an 1108 digital computer.(la) Figure 5 shows the result’s obtained from such calculations for disordered il’i,RIn for two distinct values of y and for passing modes 1 and 2. The mode 1 results are indicated by solid lines, whereas the mode 2 curves are drawn dashed. These results show that the dislocation extension ri undergoes a marked oscillation as bhe dislocat.ions pass one another, and reaches a minimum at x = 0. This minimum is further reduced as y decreases, as can be seen more clearly in Fig. 6 for the case of disordered Ni,Mn in passing mode 1. From the st,andpoint of the present analysis, this observation is of greatest importance, since it shows-that for y sufficiently small, a pair of extended screw t,ype dislocations can be forced into coalescence. The dashed line extension of the curve in Fig. 6 below y = 5 A indicates t,he uncertainty of employing linear elasticity theory below these small dislocation spacings. As y approaches 00, d attains a value of 46.2 A. A second important feature of the results shoun in Fig. 5 is that at y = 50 A the dislocat,ion paths are completely reversible, irrespective of whether x is
IS0
160
I40:
:
ANALYSIS
Fixed stackin,gfault energy Because of the difficulties associated with carrying out the present calculations in reduced form, it was decided to perform them using the disordered N&n alloy as a specific reference material. The structure of this alloy is f.c.c. with values of ,u = 7.0 x lOi dyn/cm”, Y = 0.31, b = 1.46 x lo-* cm and y = 20 erg/cm z.(B) Also for convenience R, in equation (2) wa,s chosen as 1 cm.
01
0
Fro. 6.
I
50
I
100
I
I
I
I50 x dl
200
250
300
Varietion of dislocation extension with passing distancefor four distinct asses.
ACTA
1410
36:
I
r;-
I
/
1
L
METALLURGICA,
I
,
4
I/ :
i/ 2
1
/
IO
20
I
I
30 40 Y(iil
L
,
50
60
/
J
FIG. 6. Effect
extension
of interplanar spacing y on dislocation in the mode 1 passing configuration for x = 0.
increasing or decreasing. Such is not the case for y = 25 _% however. In particular, as z is increased from x = 0, d increases linearly with z, i.e. the dislocations become uncoupled, in turn generating two ribbons of stacking fault. On the other hand, as z is decreased from very large values, d increases gradually until at some critical value of x, d increases discontinuous@ to that value possessed by the curve corresponding to increasing x. The arrows affixed t,o various port,ions of the curves indicate the path along which x and cl are simultaneously changing. It is thus clear that for sufficiently high values of x, two solutions exist, for d, while for small x, only one solution is possible. The reason for this is shown more clearly by observing the variation of E,, given by equation (1 ), with d for various fixed values of 5. This is shown in Fig. i, where only those energy terms in equation (1) that depend on d are included, i.e. AET = El_, -j- E3_&+ E,, + E2_-3_t E,. All minima are designated by upward pointing arrows, while the maxima are indicated by downward pointing arrows. It is clear from reference to Fig. 5 t’hat the two minima in the energy curves of Fig. 7 for x = 200 and 300 A correspond to the two stable disloration eon~~rations at high x, while t,he single energ?- minimum for x = 100 A corresponds to the single stable dislocation configuration at lotv values of z. The maximum in energy in Fig. 7 separating the two stable dislocat)ion configurations is almost imperceptibly small, in turn making the rightmost
VQL.
22,
19i4
minimum very shallow. Because of the shallowness of the minimum corresponding t,o the uncoupled dislocat~ions, it is difficult to determine using numerical techniques, and in searching for it, the leftmost. minimum in Fig. 7 is frequently obtained. This behavior is now believed to be responsible for the serrations in some of the d vs x and T vs x curves obtained in some of the earlier studies of t.his type.(7*8) The shallow energy minimum associated with the pair of uncoupled passing dislocations is also significant from a physical point of view. In particular. it is conceivable t,hat since the energy basrier between the coupled and uncoupled states in Fig. i is so small, very little energy is required to overcome t,his barrier. Thus, thermal energy within the crystal may be sufficient, to permit the dipole between dislocations 2 and 3 to be broken, in turn allowing the pair of extended partial dislocations to snap back into their coupled configurat,ions. The final point of interest in Fig. 5 concerns t,he difference in dislocation morphology bet\Teen passing modes 1 and 2. It is apparent that the curves in both cases are qualitat,ively t’he same, and in fact are even not significantly different from a quantit.ative point of view. Because of this similarity, nearly all of t.he future analyses will be restricted to the behavior of mode 1 type passing dislocations. The values of d shown in Fig. 5 can in turn be used in conjunction with equation (11) t.o obtain Tforafixed
Fm. 7. Variation of energy associated with passing mode I with dislocation ext.ension for several fixed values of 2.
MARCISKOWSKI
et ~2.:
MUTUAL
CROSS
2. These results are shown in Fig. 8, where as expected, the maximum passing stress increases with decreasing y and occurs for x values of the order of the value of y. In those cases where uncoupling occurs, i.e. at y = 25 A, T becomes equal to that stress necessary to generate stacking faults, i.e. 7 = y/b cos 30” = 15.9 dynes/cm2. It is important to emphasize at this point that no matter whether or not uncoupling occurs, the maximum in the T vs z curves in Fig. 8 continuously increase with decreasing y. This observation is important with respect to the section that follows.
SLIP
OF
UXLIKE
1411
DISLOCITIOBS-I
i
240 t
-
Variable stacking fault energy It is obvious at this point that the pair of extended dislocations shown in Fig. 4(a) may either pass one another or else cross slip with resulting mutual annihilation. If d is reduced to some critical value d’ the extended dislocation may be visualized as coalesced so that cross slip will occur. If, on the other hand, d remains greater than d’ the pair of extended dislocations can pass one another under a sufficiently high external stress. The maximum in the passing stress determines the amount of strengthening. The maximum degree of strengthening in turn depends on the smallest value of y which does not cause d to become equal to or less than d’, since this would insure cross slip, so that no resulting strengthening could occur. There is of course the third possibility that
45-
40-
35-
"E - 30: 2 5 250 (D 2 - zob 15-
IO-
5-
O-
I
I
50
100
I
150 XC%)
I
I
200
250
300
FIG. 8. Variation of applied stress with passing distance necessary to maintain the equilibrium configurations described in Fig. 5.
0:. 0
40
80
120
160
2oc
---
Y (erg/cm’)
FIG. 9. Variation of maximum passing stresswith stacking fault energy corresponding to two critical conditions of coalescence.
the applied stress may be sufficient’ly small and y sufficiently large so that neither cross slip nor passing occurs; but instead, the dislocations become trapped. From the point of view of the present study however. this is an uninteresting case in that. the applied stress can always be made sufficiently large so that, t’he dislocations pass, and it is this stress which det’ermines the maximum strengthening. A computer program was formulated which allowed y to be decremented in units of 1 or 2 A. At each of these values, d was determined for z = 0. When y became sufficiently small so t,hat d = d’ bhe maximum in the 7 vs x curve was determined and labeled as T,,,. This particular value of T corresponds to t’he smallest value of y below which spontaneous cross slip will occur by means of the dislocat,ion coalescence mechanism. It is difficult t,o decide upon the correct values for d’ since non linear elasticity theory must. be used at these small separat#ions. However for convenience, it was decided to use t.he values of d’ = 5 and 10 8. Furthermore, in order to ascertain the effect of stacking fault energy on rlLlaX.it was also employed as a variable, with all other constants being kept the same as those for disordered Ni,Mn. The results are shown in Fig. 9, where it is seen that the maximum passing stress decreases continuously with increasing stacking fault energy. It is apparent that this same variation is observed for the &age III flow stress in f.c.c. metals and alloys.(14) In addition, a pile-up mechanism, as required by Seeger’s theory(24) is not required in the present formulation,
1412
ACTA
BIETALLURGICA,
since the coalescence of the extended dislocations is brought about by their mutual interaction with one another. A number of important features can be deduced from t’he curves in Fig. 9 other than the decrease in rmBXwith y as discussed above, In the first place, there exists a critical value of y above which 7max vanishes. Reference to Fig. 10 shows the variation of y with y at which a given cl’ is attained. It can thus be seen that the value of y at which T,,, becomes zero corresponds to y -+ 00. This means that y alone is sufficient to attain the critical value of c?’ required for cross slip by partial dislocation coalescence. The condition for this to obtain may be found from equation (7) which reduces to the following simple relation as y + co.
The above relation is simply the expression for the extension of a single isolated dislocation comprised of a pair of partials separated by & stacking fault. It is apparent that as d’ is reduced from 10 to 5 8, the critical value of y’ at which 7_= becomes zero is doubled from 93 erg/cm2 to 186 erg~cm2, as can be seen in Fig. 9. The second important feature of Fig. 9 is that the values of r,,, are higher for smaller values of d’. The reason for this, as Fig. 10 shows, is that y must be reduced accordingly as d’ decreases, for /
,
I
f
36-
Oi
C
I
40
1
80
120 160 Y (erg/cm2)
1
200
I
240
1
10. ITariation of interplanar spacing for dislocation coelescence with stacking fault energy corresponding to two critic& conditions of coalescence.
FIG.
VOL.
22,
1974
the mutual interaction between all four partial dislocations in Fig. 3 to reduce d to the smaller value for d’. A t,hird in~r~ting aspect of Fig. 9 is that a stable solution exists for the dislocation configuration corresponding to x = 0 and y = 0. Such solutions were found to be unstable, i.e. d + 00 in those cases where dislocations l-4 of Fig. 3 were all of pure screw tvpe.(lQ It is thus obvious that the present L cor&guration is stabilized by the edge components of t,he array; in particular, the attract.ive forces which exist between the edge components of dislocations 1 and 2 as well as 3 and 4. A fourth and final interest,ing feature of Fig. 9 is that as p is reduced from it,s ralup of that for disorderedXi,Mn to half t~hisvalue, i.e. 3.5 x loll dyn/ cm2, a value which closely corresponds t,o that of copper, and while d ’ is simultaneously chosen as5 A, the r,,, vs y curve becomes identical to that obtained for P= i x 10” dynjcm2 and d’ = 10 A. At t,he same time Fig. 10 shows that the corresponding y vs y curves for the two cases are displaced along the y axis by a factor of close to two, The reasoning behind this behavior is related to the fact that the maximum passing stress associated with a pair of dissociated screw type dislocat.ions is given by(‘) 7n1as= pb/hy.
(13)
Thus to maintain the same T,,,,~ for p/2 as obtained for p, y must be changed t,o y/2 which is essentially what occurs in Fig. 9 when d’ is changed from 10 to 5 8. By employing arguments of this type it is possible to use the relat,ionships in Fig. 9 for any combination of ,u, b, y and d’. The downward pointing arrows associated with each of t,he two curves shown in Fig. 9 indicate the initial y value below which the extended dislocations become uncoupled in the sense described with the 21= 25 b curves of Fig. 5. It is apparent that the uncoupling has no effect on the shape of t,he curves, and is related to the fact that 7mar in the 7 vs x curves, as illustrated in Fig. 8, occurs at sufficiently small values of 2, so that uncoupling can be considered as not yet having begun. In concluding this se&ion, it, is important to emphasize again that, isotropic elasticity has been employed throughout the present study. It is quite likely t.hat anisotropic elasticity would produce a significant pert,urbation on the present findings,(16*17) alt,hough the same general conclusions are expected to obtain. SUMMARY
AND
CONCLUSIONS
The present study has been concerned with the passing behavior of extended screw t.ype dislocations
MARCIXKOWSKI
et al.:
MUTUAL
CROSS
of opposite sign in f.c.c. metals and alloys. In particular, it has been shown that, below some critical interplanar spacing, the dislocations become sufficiently coalesced by their mutual interaction to undergo cross slip with mutual annihilation. Under these conditions no strengthening can occur. If, on the other hand, the extended dislocations pass one another at distances greater than this critical interplanar spacing, no cross slip occurs and a maximum in passing stress obtains. This maximum becomes greatest just above the interplanar spacing where spontaneous cross slip occurs, and thus determines the maximum strengthening that can be imparted to an alloy. Detailed numerical analyses have shown, among other things, that the maximum strengthening as discussed above, decreases with increased stacking fault energy, in agreement with observed experimental results. The present treatment thus provides an alternate hypothesis to the pile-up mechanism for dislocation coalescence used in the interpretation of Stage III work hardening behavior in disordered f.c.c. metals and alloys. It should be emphasized however that the present analysis is strictly valid only at 0 K. At higher temperatures, the cross slip process would need to be reconsidered in terms of a finite length of dislocation so as to include thermal activation mechanisms. These more sophisticated models will be presented in future publications. The second part of the present study which follows will examine an alternat,e mode of mutual cross slip of a pair of extended screw type dislocations of opposite sign which involves the formation of stair rod t,ype dislocat,ions.
SLIP
OF
UNLIKE
DISLOCATIOSS-I
1413
The present research effort was supported by The United States Atomic Energy Commission under Contract No. AT.(40-I).3935
REFERENCES 1. J. C. M. LI, J. appZ. Phys. 32, 593 (1961). 2. G. SCHOECKand A. SEEGER.Rep. of the C&j. ON Dejects in. CrystQZZirae Solids, p. 340, H. H. Wills Physical Lsboratory, University of Bristol (July 13-27, 1954). The Physical Society, London (1955). 3. A. SEEOER,I&. Conj. ~~Z~~~~o~s end ~iec~~I~~eaZProperties of Cry&aZs, Lake Placid (September 6-8, 1%56), edited by J. C. FISHER, W. G. JOHRSTOS,R. Tnoxsos and T. VREELAND, JR., p. 243. Wiley. and H. KRONM~~LLER. Electrolt 4. A. SEEOER, S. MADER Microscopy and Strength of Crystals, edited by G. THOXAS and J. WASHBURN, p. 665, Interscience. (1963). 5. R. L. FLEISCIIER,Acla Met. 7, 134 (1959). 6. M. J. MARCINKOWSKI,K. SADANAXD.~and K. J. OLSOS, Crystal Lattice Defects, submitted for public&ion. 7. M. J. MARCIXKO~SKI and G. Lasso, J. appl. Phys. 58, 2124 (1967). 8. M. J. MARCIXKOWSKIand h’. J. OLSOS, Phil. Nag. 19, 1111 (196%). and M. J. M~crxxowssr, J. appt. P&8. 9. K. SADAXANDA 44, 4445 (1973). and K. S.~~.+X;~SD.*,J. appf. Phys. 10. M. J. MARCINXOWSKI 45, 2441 (1974). 11. JOHN PRICE HIRTH and JEXS LOTHE. Them-u ” of” L)isZoeations. McGraw-Hill (1968). 12. K. SADANANDAand M. J. MARCINKO~SEI,J. Appt. Phys. 82,260% (1972). 13. M. J. MARCINKOWSKI,Advances ila 3fateriaZa Research, edited by H. HERMAN, Vol. 5, p. 443. Wiley-Interscience (1971). 14. C. K. L. DAVIES, V. SAGAE and R. K. STEVESS, dcta Met. 21, 1343 (1973). 15. K. SADANANDAand M. J. MARCINKO~~KI,J. appl. Phya. 48,293 (1972). 16. A. J. MORTONand C. T. FORWOOD,Crystal Lattice Defects 4,166 (1973). C. T. FOR~OOD and A. J. MORTON, 17. L. M. CLARXBROUUH, Crystd L&tics Deject8 4, 17% (1973).
OF STACKING FAULT ENERGY UNLIKE DISLOCATIONS-I. M. J. MARCINKOWSKII,
ON THE MUTUAL CROSS COALESCENCE MODE*?
K. SADANANDAS
SLIP OF
and R. J. TAUNT:
A detailed numerical analysis has been made of the cross slip behavior associated with extended screw t,ype dislocations of opposite sign in face centered cubic metals and alloys which pass one another on parallel slip planes. It is shown that below some critical interplanar spacing, the mutual interaction between the extended dislocations is sufficient to bring about coalescence, followed by spontaneous cross slip and subsequent mutual annihilation. Just above this critical interplanar spacing, the estended dialocations may pass one another with a maximum interaction and thus a maximum stress. This maximum stress, i.e. strengthening, deoreases with increasing stacking fault energy and provides an alternate hypothesis to the pile-up model for Stage III work hardening behavior in face-centered cubic metals and alloys. EFFET
DE L’ENERGIE DE DEFAUT D’EMPILEMER’T MUTUEL DE DISLOCATIONS DIFFERENTES-I.
SUR LE GLISSEMEST PAR RECOMBIBSISOZi
DE1-IE
Une analyse numerique detaillee a et6 faite du glissement d&vie associe au passage dans des plans de glissement paralleles de dislocations vis dissociees de signe oppose, dens les metaux et alliages cubiques a faces centrees. On montre qu’en dessous d’un Qcartement,critique des plans de glissement, les interactions mutuelles entre les dislocations dissociees sont suffisantes pour produire leur recombinaison, suivie d’un glissement d&vie spontane et d’une annihilation mutuelle. Juste au dessus de cet Ccartement critique, les dislocations dissociees peuvent passer l’une p&s de l’autre avec un maximum d’interartion et done un maximum d’effort. Cet effort maximal, c’est-a-dire ce durcissement, decroit lorsque l’energie de defaut d’empilement croit; ceci donne une explication du durcissement des metaux et alliages rubiques a faces centrees dans le stade III, differente de celle que fournit le modirle de l’empilement. EINFLUO DER STAPELFEHLERENERGIE AUF DIE QUERGLEITUNG VON VERSETZUSGES ENTGEGENGESETZTEN VORZEICHENS I-KOALESZENZ-MODUS Das Quergleitverhalten aufgespaltener Schraubenversetzungen entgegengesetzten Vorzeichens in kubisch-fliichenzentrierten Metallen und Legierungen, die auf paralellen Gleitebenen aneinander vorbeigleiten, wurde ausftihrlich numerisch analysiert. Es wird gezeigt, da9 unterhalb eines kritischen Gleitebenenabstandes die gegenseitige Wechselwirkung zwischen den aufgespaltenen Schraubenversetzungen ausreicht, urn eine Koaleszen mit naohfolgender spontaner Quergleitung und gegenseitiger Annihilierung herbeizufiihren. Gerade oberhalb dieses kritischen Gleitebenenabstandes k&men die aufgespaltenen Versetzungen aneinander mit maximaler Wechselwirkung und somit maximaler Schubspannung aneinander vorbeigleiten. Diese maximale Schubspannung, d.h. Verfestigung, nimmt mit. zunehmender Stapelfehlerenergie ab und liefert eine Alternativhypothese zum Model1 der Versetzungsaufstauungen fur das Stufe-III-Verfestigungsverhalten von kubisch-flachenzentrierten Metallen und Legierungen. INTRODUCTION
One of the most important aspects of dislocation behavior concerns the process of cross slip from one plane to another. In the case of unextended screw type dislocations, the process is relatively simple.(l) However, in the case of extended dislocations separated by a stacking fault, the problem is much more complex. Seeger and associates(2-4) were the first to consider this problem in any detail. In particular, they postulated that under a sufficiently large stress concentration, due to say a pile-up of dislocations, the blocked lead dislocat,ion would have its extension reduced from d, its equilibrium value in the free state, to some smaller value d’. If d’ becomes sufficiently small, i.e. of the order of a Burgers vector, the extended dislocation may be considered as having coalesced into a single pure screw type dislocation * Received January 17, 1974; revised April 22. 1974. t The present research effort was supported by The United States At,omic Energy Commission under Contract Number AT-(40-1)3935. $ Engineering Materials Group, and Department of Mechanical Engineering. University of Maryland, College Park, Maryland 20742, U.S.A. ACTA METALLURGICA,
VOL. 22, NOVEMBER
1974
which can then easily cross slip under some relatively small externally applied stress. An alternative model to the coalescence mechanism was proposed by Fleisoherc5) in which he postulated the individual cross slip of each of the two partial dislocations comprising the extended screw type This particular process involved the dislocation. formation of stair-rod dislocations. The energetics of the stair-rod mechanism were analyzed in detail for single blocked extended screw type dislocations in face-centered cubic (f.c.c.) crystals and compared with the coalescence model.‘6) It was shown that the relative ease with which each of these processes occurred depended upon the orientation of the applied stress, but that the stair rod mechanism was generally more favorable. It is conceivable however that these results, although interesting from an academic point of view, may not necessarily represent the same processes occurring during Stage III of plastic deformation. It will be recalled that during t,his stage, the rate of work hardening is relatively low, indicating that very significant dynamic recovery is taking place.
1405
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One of the most important of all dynamic recovery processes, from the dislocation point of view, is the mutual annihilation of screw type dislocations of opposite sign. The behavior of such dislocations, as they pass one another, has already been treated in detail for both ordered(r*s) as well as disordered@) alloys. Furthermore, the critical conditions for cross slip have been determined for passing screw type superlattice dislocations of opposite sign.@~lO) The much more complex case involving the conditions for cross slip of a pair of extended passing dislocations in a disordered close-packed lattice has yet to be analyzed. Such an analysis is important in understanding the Stage III work hardening behavior in f.c.c. metals and alloys and thus provides the motivation for the present study. Due to the complexity of this analysis however, it will be carried out in two separate parts. Part I will deal with the cross slip mode involving dislocation coalescence, while Part II will concentrate on the cross slip mode requiring stair-rod formation.
VOL.
OF
THE
1954
0)
L,/,,,~
C)
FORMULATION
22,
*/i/,//i/, b_
di
PROBLEM
Figure 1 shows the stacking sequence of the closepacked (111) planes in an f.c.c. st’ructure. In particular, the stacking periodicity normal to the drawing in ABCABC, etc. Six different types of slip vector are also shown in Fig. 1 and they are labelled as 0, 1,2,3, 4 and 5. It is apparent from inspection of t,his figure that Burgers vectors 3 and 4, as well as 0 and 1 correspond to the movement of at,oms in the C layer
Fxo. 1. Various slip modes via partial disiocetions on the close packed plane in a face-centered cubic structure.
FIG. 2. Various combinat’ions of ert.ended screw type dislocation pairs of opposite sign in a face-centered cubic crystal.
relative to those in the A layer, while Burgers vectors 1 and 2, along with 4 and 5 correspond to the motion of atoms in the B layer relative to those in A. From Fig. 1, and from what has been said above, it is possible to construct the four combinations of extended screw dislocations lying on parallel planes as shown in Fig. 2. Two views are shown in each case; the top view being along the dislocation line, i.e. parallel to the slip plane, while the bottom view is normal to the slip plane. The letters B and C shown between each extended dislocation pair simply indicates that the B and C layers in Fig. 1 have slipped relative to A as discussed earlier. Inspection of Fig. 2 shows that all four configurations are not unique and that those illustrated in Figs. 2(a and d) are equivalent. Figure 3 shows the details of the stacking disorder associated with the dislocation configuration of Fig. 2(a). The primed letters in t,his figure show the new stacking sequence brought about by the pair of extended dislocations. Note that extended pair l-2 has altered the stacking sequence of the planes above the dislocation with respect to the original stacking, It is next of interest to analyse the behavior of the pairs of extended dislocations shown in Fig. 2 as they pass one another. For simplicity the passing modes illustrated in Fig. 2(a and d) will be referred to as mode 1, while those shown in Fig. 2(b and c) will be termed
3IARCINKOWSKI
MUTUAL
et al.:
CROSS
-c-s
A-B-A
-B’-A
c-c-c
-tin’-----_
B-S-S
-C’-
B
8’ -
A
-ti-C
s
-C’a-------C’-A
~
i-B’-----c
-&-C
B’-
A
A-C’-A
El__, = E2_-4 = E In 2n
2
-c-c
B------n’-
b 72 --b1s2 + (1 ” r)
S-E------_
PbE2
A-C’-A-A-A
27r(l -
-
FIG. 3. More detailed view of Fig. 2(a) parallel to the slip planes showing stacking disorder resulting from the pair of extended dislocations.
mode 2. A more detailed schematic illustration of the mode 1 passing configuration is shown in Fig. 4(a). The total energy associated with this dislocation array may be writt.en as follows:
E I-4 = -&ln
V)
[@ -t (Q2 +
_ Pb8 Y)
co92 (n/2 -
cos (x/2 -
4- -L
+ E,-,
+ L-4 -4-G-a
-/- J-L, i-8,
-4-E,
(1)
where E,,, etc. are the interaction energies between dislocations 1 and 2, etc. while E, represents the total stacking fault energy, whereas E, is the potential energy due tSothe applied stress 7. Also, from symmetry considerations, El_, = E,, and E,_, = E,_,. The general expression for the interaction energy between a pair of infinite straight dislocations in t‘he isotropic approximation is given byfin
X
-
InR
E
2-2
W -
--&In -
X)2 + $1””
- rUb? 2741 -
Y)
b &s2
C
COS (x/2
+
f
1
2
-
(1 ZY)
as) cos (7r/2 -
aa)
GW where b,, and blE correspond t,o the screw and edge components respectively of the partial dislocations with Burgers vector b = &,(112), where a, is the edge length of the cubic unit cell. Inspection of Figs. 1 and 2 show that b,, = b cos 30” while blK = b sin 30”. In addition a1 =
tan-r
a2
tan-l
=I
a3 =
2 [3 ?/1 1 y
-
tan-’
Y
(d _ 5)
(44
W
.
[
a
%r(l ” Y)R‘J
1 (34
[ (d + 2) R
(3b)
(x2)cos (n/2 -i- a,)
R,
E, = E,-,
arr)
!J2Pe b 2 j b2 Is z [ fl - 4
4l
277(1 -
1 134 1
@’ + ‘!!2)“2 5 a _ bE2 R I.3 [ (1 - v) Q
A
A
i
1407
DISLOCATIOSS-I
2~
7
c-B’-C
UP;LIKE
E,_, = E,_, = k. In -!R,
c/,,,,/,,, c’ s
B------A’-8
OF
interaction energy associated with the screw component of each partial dislocation while the second two terms correspond to the in~raetioR energy contributions from the edge components. The symbols p and v denot,e the shear modulus and Poisson’s ratio respectively, while R, represents the distance of the dislocation from the surface of the cryst,al. The dislocation self energies have been omitt,ed from equation (I) since they are neither functions of x nor of d and thus do not enter into the present. calculations. Applying equation (2) to Fig. 4(a), it follows that
C-C-t-&-C
B-A-S
SLIP
The last two terms in equation (1) can be written as E, = 2dy
[(b, x 5) * RI
where bI and b, are t’he Burgers vectors associated with dislocations 1 and 2, respectively, E, is their tangent vector, while R is the distance between the two dislocations. The first term in equation (2) is the
(5)
where y is the stacking fault energy, and E 7 = -rb,,[r
-
d -t_ (x + d)].
(6)
In obtaining the above equation, it has been assumed, for simplicity, that r acts only on the screw components of the partial dislocations as well as acting only
ACTA
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METALLURGICA,
VOL.
22,
1974
in the primary slip plane of the extended dislocations. The condition of equilibrium for a fixed value of x is given by the condition aE,/&? = 0, which when applied to equation (1) gives
F
rUb2Y2@ + 4 41 - 4w + x)2 + y212
-
2d(d - N2 + Y21 4 b1s2 + Pu(d 4 - 4 7 lubm2y2(d 77(1 -
V)[(d - x)2 + y2]2
b t1
2
‘_” 1 v)
+ 2y = 0.
(7)
A similar analysis to that shown above can be carried out for mode 2 dislocation passing behavior. In fact, equation (7) applies equally well to both. In particular, in those cases where a term is preceded by a f or i, the top sign applies to mode 1 passing behavior, while the bottom sign corresponds to mode 2 passing. In order t’o obtain the stress 7 necessary to maintain the equilibrium configuration shown in Fig. 4(a),it is necessary to reconstruct this drawing as shown in Fig. P(b). In particular, dislocation 3 is perturbed from its equilibrium position by some small amount R. This immediately destroys the symmetry exhibited in Fig. 4(a), so that a new set of interaction energies
F9
FIG. 4(b). Schematic illustration used to determine the externally applied stress required to maintain a pair of extended passing dislocat,ions in statio equilibrium.
must be determined.
In particular
@aI
E 3-4 = &hi7 El_,
=
g
ln
(d2 +
a
b
RI
-b1s2
+
2
(1 ‘“VI
1 @b)
bl: - F. ‘(x- ‘i ’ y211’2 V
n
PblE2
_
277(1 -
Y)
cos2 (n/2 -
al)
(SC)
C(x - d, - 4) + Y2Y2
E 2-4
R,
x L2 -
b 3
”
(1
v)
1
PblE2 cos2 (77/2 $
27r(l -
P)
a4)
(3d)
[(x -+ d,)2 + y2p2
E 1-4
&
-
PblE2 2x(1 -
v)
1
cos (x/2 -
az) cos (7~/2 + az) (se)
E 2-3 -- $1,
x
FIG. 4(a). Schematic illustration used to determine the equilibrium configuration of extended passing dislocations.
[(d, -
z + R)2 -,- y2]l12 Rll
bls2 + -
w
(1 - 7)1
PblE2 2n(l -
Y)
cos (7r/2 + a3) cos (r/2 -
0c.J.
m
MARCINKOWSKI
et
al.:
MUTUAL
In addition
CROSS
(3) E,, = yR + yd, + $1
and E, = --k,[(&
+ 5) + (z -
R) -
41.
(10)
Expressions S-10 may then be substituted into equation (l), followed by the operation aE,/aR = 0. As R -+ 0, d, N d, and we obtain -FF,,
1 2 P” - 2x[x2 L2 =F & -+ $1 1 1 T ,4.2~2~ 2y) +%r[(d b1s2 It (1 '_ -x)2 +$1 - 1
aET P = aR =2nd
-b,s
6 2 + (1 ‘” y) b
?T(l -
Y) [x2 + $12
b
P(d - 2)
i
rUbm2Y2(d - 4 ?7(1 -
Y) [(d - x)2 + y2]2
+ y + T&s = 0.
(11)
The above expression is simply the requirement that the force on dislocation 3 in the x direction be equal Identical relationships were obtained to zero. previously@) using the expressions for the elastic stress fields about a dislocation, rather than the energy expression given by equation (2). As was the case in equation (7)) equation ( 11) can also be used for passing mode 2 by employing the bottom of the & or F signs wherever they occur. The importance of equation (11) lies in the fact that, it can be employed to determine 7 for a fixed value of x, after first determining d for this same value of x from equation (7). It is also interest’ing to note that equation (7) could have been obtained from relationships of the type given by equation (11) by forming combinations of t.he type given by Fz3 + F,, = 0 or Fz2 + F,, = 0. It is obvious that such combinations eliminate the term containing r, since it is of opposite sign in the two different force expressions. These matters have been discussed in great,er detail in Ref. 10. NUMERICAL
SLIP
OF
USLIKE
1409
DISLOCATIOSS-I
Although equation (i) could be solved numerically for d as a function of x, it is much more desirable to minimize the energy given by equation (1) directly.02) This was done w&h the aid of a minimizat,ion subrout,ine entitled FUNFIT employing an 1108 digital computer.(la) Figure 5 shows the result’s obtained from such calculations for disordered il’i,RIn for two distinct values of y and for passing modes 1 and 2. The mode 1 results are indicated by solid lines, whereas the mode 2 curves are drawn dashed. These results show that the dislocation extension ri undergoes a marked oscillation as bhe dislocat.ions pass one another, and reaches a minimum at x = 0. This minimum is further reduced as y decreases, as can be seen more clearly in Fig. 6 for the case of disordered Ni,Mn in passing mode 1. From the st,andpoint of the present analysis, this observation is of greatest importance, since it shows-that for y sufficiently small, a pair of extended screw t,ype dislocations can be forced into coalescence. The dashed line extension of the curve in Fig. 6 below y = 5 A indicates t,he uncertainty of employing linear elasticity theory below these small dislocation spacings. As y approaches 00, d attains a value of 46.2 A. A second important feature of the results shoun in Fig. 5 is that at y = 50 A the dislocat,ion paths are completely reversible, irrespective of whether x is
IS0
160
I40:
:
ANALYSIS
Fixed stackin,gfault energy Because of the difficulties associated with carrying out the present calculations in reduced form, it was decided to perform them using the disordered N&n alloy as a specific reference material. The structure of this alloy is f.c.c. with values of ,u = 7.0 x lOi dyn/cm”, Y = 0.31, b = 1.46 x lo-* cm and y = 20 erg/cm z.(B) Also for convenience R, in equation (2) wa,s chosen as 1 cm.
01
0
Fro. 6.
I
50
I
100
I
I
I
I50 x dl
200
250
300
Varietion of dislocation extension with passing distancefor four distinct asses.
ACTA
1410
36:
I
r;-
I
/
1
L
METALLURGICA,
I
,
4
I/ :
i/ 2
1
/
IO
20
I
I
30 40 Y(iil
L
,
50
60
/
J
FIG. 6. Effect
extension
of interplanar spacing y on dislocation in the mode 1 passing configuration for x = 0.
increasing or decreasing. Such is not the case for y = 25 _% however. In particular, as z is increased from x = 0, d increases linearly with z, i.e. the dislocations become uncoupled, in turn generating two ribbons of stacking fault. On the other hand, as z is decreased from very large values, d increases gradually until at some critical value of x, d increases discontinuous@ to that value possessed by the curve corresponding to increasing x. The arrows affixed t,o various port,ions of the curves indicate the path along which x and cl are simultaneously changing. It is thus clear that for sufficiently high values of x, two solutions exist, for d, while for small x, only one solution is possible. The reason for this is shown more clearly by observing the variation of E,, given by equation (1 ), with d for various fixed values of 5. This is shown in Fig. i, where only those energy terms in equation (1) that depend on d are included, i.e. AET = El_, -j- E3_&+ E,, + E2_-3_t E,. All minima are designated by upward pointing arrows, while the maxima are indicated by downward pointing arrows. It is clear from reference to Fig. 5 t’hat the two minima in the energy curves of Fig. 7 for x = 200 and 300 A correspond to the two stable disloration eon~~rations at high x, while t,he single energ?- minimum for x = 100 A corresponds to the single stable dislocation configuration at lotv values of z. The maximum in energy in Fig. 7 separating the two stable dislocat)ion configurations is almost imperceptibly small, in turn making the rightmost
VQL.
22,
19i4
minimum very shallow. Because of the shallowness of the minimum corresponding t,o the uncoupled dislocat~ions, it is difficult to determine using numerical techniques, and in searching for it, the leftmost. minimum in Fig. 7 is frequently obtained. This behavior is now believed to be responsible for the serrations in some of the d vs x and T vs x curves obtained in some of the earlier studies of t.his type.(7*8) The shallow energy minimum associated with the pair of uncoupled passing dislocations is also significant from a physical point of view. In particular. it is conceivable t,hat since the energy basrier between the coupled and uncoupled states in Fig. i is so small, very little energy is required to overcome t,his barrier. Thus, thermal energy within the crystal may be sufficient, to permit the dipole between dislocations 2 and 3 to be broken, in turn allowing the pair of extended partial dislocations to snap back into their coupled configurat,ions. The final point of interest in Fig. 5 concerns t,he difference in dislocation morphology bet\Teen passing modes 1 and 2. It is apparent that the curves in both cases are qualitat,ively t’he same, and in fact are even not significantly different from a quantit.ative point of view. Because of this similarity, nearly all of t.he future analyses will be restricted to the behavior of mode 1 type passing dislocations. The values of d shown in Fig. 5 can in turn be used in conjunction with equation (11) t.o obtain Tforafixed
Fm. 7. Variation of energy associated with passing mode I with dislocation ext.ension for several fixed values of 2.
MARCISKOWSKI
et ~2.:
MUTUAL
CROSS
2. These results are shown in Fig. 8, where as expected, the maximum passing stress increases with decreasing y and occurs for x values of the order of the value of y. In those cases where uncoupling occurs, i.e. at y = 25 A, T becomes equal to that stress necessary to generate stacking faults, i.e. 7 = y/b cos 30” = 15.9 dynes/cm2. It is important to emphasize at this point that no matter whether or not uncoupling occurs, the maximum in the T vs z curves in Fig. 8 continuously increase with decreasing y. This observation is important with respect to the section that follows.
SLIP
OF
UXLIKE
1411
DISLOCITIOBS-I
i
240 t
-
Variable stacking fault energy It is obvious at this point that the pair of extended dislocations shown in Fig. 4(a) may either pass one another or else cross slip with resulting mutual annihilation. If d is reduced to some critical value d’ the extended dislocation may be visualized as coalesced so that cross slip will occur. If, on the other hand, d remains greater than d’ the pair of extended dislocations can pass one another under a sufficiently high external stress. The maximum in the passing stress determines the amount of strengthening. The maximum degree of strengthening in turn depends on the smallest value of y which does not cause d to become equal to or less than d’, since this would insure cross slip, so that no resulting strengthening could occur. There is of course the third possibility that
45-
40-
35-
"E - 30: 2 5 250 (D 2 - zob 15-
IO-
5-
O-
I
I
50
100
I
150 XC%)
I
I
200
250
300
FIG. 8. Variation of applied stress with passing distance necessary to maintain the equilibrium configurations described in Fig. 5.
0:. 0
40
80
120
160
2oc
---
Y (erg/cm’)
FIG. 9. Variation of maximum passing stresswith stacking fault energy corresponding to two critical conditions of coalescence.
the applied stress may be sufficient’ly small and y sufficiently large so that neither cross slip nor passing occurs; but instead, the dislocations become trapped. From the point of view of the present study however. this is an uninteresting case in that. the applied stress can always be made sufficiently large so that, t’he dislocations pass, and it is this stress which det’ermines the maximum strengthening. A computer program was formulated which allowed y to be decremented in units of 1 or 2 A. At each of these values, d was determined for z = 0. When y became sufficiently small so t,hat d = d’ bhe maximum in the 7 vs x curve was determined and labeled as T,,,. This particular value of T corresponds to t’he smallest value of y below which spontaneous cross slip will occur by means of the dislocat,ion coalescence mechanism. It is difficult t,o decide upon the correct values for d’ since non linear elasticity theory must. be used at these small separat#ions. However for convenience, it was decided to use t.he values of d’ = 5 and 10 8. Furthermore, in order to ascertain the effect of stacking fault energy on rlLlaX.it was also employed as a variable, with all other constants being kept the same as those for disordered Ni,Mn. The results are shown in Fig. 9, where it is seen that the maximum passing stress decreases continuously with increasing stacking fault energy. It is apparent that this same variation is observed for the &age III flow stress in f.c.c. metals and alloys.(14) In addition, a pile-up mechanism, as required by Seeger’s theory(24) is not required in the present formulation,
1412
ACTA
BIETALLURGICA,
since the coalescence of the extended dislocations is brought about by their mutual interaction with one another. A number of important features can be deduced from t’he curves in Fig. 9 other than the decrease in rmBXwith y as discussed above, In the first place, there exists a critical value of y above which 7max vanishes. Reference to Fig. 10 shows the variation of y with y at which a given cl’ is attained. It can thus be seen that the value of y at which T,,, becomes zero corresponds to y -+ 00. This means that y alone is sufficient to attain the critical value of c?’ required for cross slip by partial dislocation coalescence. The condition for this to obtain may be found from equation (7) which reduces to the following simple relation as y + co.
The above relation is simply the expression for the extension of a single isolated dislocation comprised of a pair of partials separated by & stacking fault. It is apparent that as d’ is reduced from 10 to 5 8, the critical value of y’ at which 7_= becomes zero is doubled from 93 erg/cm2 to 186 erg~cm2, as can be seen in Fig. 9. The second important feature of Fig. 9 is that the values of r,,, are higher for smaller values of d’. The reason for this, as Fig. 10 shows, is that y must be reduced accordingly as d’ decreases, for /
,
I
f
36-
Oi
C
I
40
1
80
120 160 Y (erg/cm2)
1
200
I
240
1
10. ITariation of interplanar spacing for dislocation coelescence with stacking fault energy corresponding to two critic& conditions of coalescence.
FIG.
VOL.
22,
1974
the mutual interaction between all four partial dislocations in Fig. 3 to reduce d to the smaller value for d’. A t,hird in~r~ting aspect of Fig. 9 is that a stable solution exists for the dislocation configuration corresponding to x = 0 and y = 0. Such solutions were found to be unstable, i.e. d + 00 in those cases where dislocations l-4 of Fig. 3 were all of pure screw tvpe.(lQ It is thus obvious that the present L cor&guration is stabilized by the edge components of t,he array; in particular, the attract.ive forces which exist between the edge components of dislocations 1 and 2 as well as 3 and 4. A fourth and final interest,ing feature of Fig. 9 is that as p is reduced from it,s ralup of that for disorderedXi,Mn to half t~hisvalue, i.e. 3.5 x loll dyn/ cm2, a value which closely corresponds t,o that of copper, and while d ’ is simultaneously chosen as5 A, the r,,, vs y curve becomes identical to that obtained for P= i x 10” dynjcm2 and d’ = 10 A. At t,he same time Fig. 10 shows that the corresponding y vs y curves for the two cases are displaced along the y axis by a factor of close to two, The reasoning behind this behavior is related to the fact that the maximum passing stress associated with a pair of dissociated screw type dislocat.ions is given by(‘) 7n1as= pb/hy.
(13)
Thus to maintain the same T,,,,~ for p/2 as obtained for p, y must be changed t,o y/2 which is essentially what occurs in Fig. 9 when d’ is changed from 10 to 5 8. By employing arguments of this type it is possible to use the relat,ionships in Fig. 9 for any combination of ,u, b, y and d’. The downward pointing arrows associated with each of t,he two curves shown in Fig. 9 indicate the initial y value below which the extended dislocations become uncoupled in the sense described with the 21= 25 b curves of Fig. 5. It is apparent that the uncoupling has no effect on the shape of t,he curves, and is related to the fact that 7mar in the 7 vs x curves, as illustrated in Fig. 8, occurs at sufficiently small values of 2, so that uncoupling can be considered as not yet having begun. In concluding this se&ion, it, is important to emphasize again that, isotropic elasticity has been employed throughout the present study. It is quite likely t.hat anisotropic elasticity would produce a significant pert,urbation on the present findings,(16*17) alt,hough the same general conclusions are expected to obtain. SUMMARY
AND
CONCLUSIONS
The present study has been concerned with the passing behavior of extended screw t.ype dislocations
MARCIXKOWSKI
et al.:
MUTUAL
CROSS
of opposite sign in f.c.c. metals and alloys. In particular, it has been shown that, below some critical interplanar spacing, the dislocations become sufficiently coalesced by their mutual interaction to undergo cross slip with mutual annihilation. Under these conditions no strengthening can occur. If, on the other hand, the extended dislocations pass one another at distances greater than this critical interplanar spacing, no cross slip occurs and a maximum in passing stress obtains. This maximum becomes greatest just above the interplanar spacing where spontaneous cross slip occurs, and thus determines the maximum strengthening that can be imparted to an alloy. Detailed numerical analyses have shown, among other things, that the maximum strengthening as discussed above, decreases with increased stacking fault energy, in agreement with observed experimental results. The present treatment thus provides an alternate hypothesis to the pile-up mechanism for dislocation coalescence used in the interpretation of Stage III work hardening behavior in disordered f.c.c. metals and alloys. It should be emphasized however that the present analysis is strictly valid only at 0 K. At higher temperatures, the cross slip process would need to be reconsidered in terms of a finite length of dislocation so as to include thermal activation mechanisms. These more sophisticated models will be presented in future publications. The second part of the present study which follows will examine an alternat,e mode of mutual cross slip of a pair of extended screw type dislocations of opposite sign which involves the formation of stair rod t,ype dislocat,ions.
SLIP
OF
UNLIKE
DISLOCATIOSS-I
1413
The present research effort was supported by The United States Atomic Energy Commission under Contract No. AT.(40-I).3935
REFERENCES 1. J. C. M. LI, J. appZ. Phys. 32, 593 (1961). 2. G. SCHOECKand A. SEEGER.Rep. of the C&j. ON Dejects in. CrystQZZirae Solids, p. 340, H. H. Wills Physical Lsboratory, University of Bristol (July 13-27, 1954). The Physical Society, London (1955). 3. A. SEEOER,I&. Conj. ~~Z~~~~o~s end ~iec~~I~~eaZProperties of Cry&aZs, Lake Placid (September 6-8, 1%56), edited by J. C. FISHER, W. G. JOHRSTOS,R. Tnoxsos and T. VREELAND, JR., p. 243. Wiley. and H. KRONM~~LLER. Electrolt 4. A. SEEOER, S. MADER Microscopy and Strength of Crystals, edited by G. THOXAS and J. WASHBURN, p. 665, Interscience. (1963). 5. R. L. FLEISCIIER,Acla Met. 7, 134 (1959). 6. M. J. MARCINKOWSKI,K. SADANAXD.~and K. J. OLSOS, Crystal Lattice Defects, submitted for public&ion. 7. M. J. MARCIXKO~SKI and G. Lasso, J. appl. Phys. 58, 2124 (1967). 8. M. J. MARCIXKOWSKIand h’. J. OLSOS, Phil. Nag. 19, 1111 (196%). and M. J. M~crxxowssr, J. appt. P&8. 9. K. SADAXANDA 44, 4445 (1973). and K. S.~~.+X;~SD.*,J. appf. Phys. 10. M. J. MARCINXOWSKI 45, 2441 (1974). 11. JOHN PRICE HIRTH and JEXS LOTHE. Them-u ” of” L)isZoeations. McGraw-Hill (1968). 12. K. SADANANDAand M. J. MARCINKO~SEI,J. Appt. Phys. 82,260% (1972). 13. M. J. MARCINKOWSKI,Advances ila 3fateriaZa Research, edited by H. HERMAN, Vol. 5, p. 443. Wiley-Interscience (1971). 14. C. K. L. DAVIES, V. SAGAE and R. K. STEVESS, dcta Met. 21, 1343 (1973). 15. K. SADANANDAand M. J. MARCINKO~~KI,J. appl. Phya. 48,293 (1972). 16. A. J. MORTONand C. T. FORWOOD,Crystal Lattice Defects 4,166 (1973). C. T. FOR~OOD and A. J. MORTON, 17. L. M. CLARXBROUUH, Crystd L&tics Deject8 4, 17% (1973).