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Analysis of pairwise dislocation interaction and its contribution to flow stress during magnesium crystal basal slip
Materials Science and Engineering, 32 ( 1 9 7 8 ) 113 - 119
113
© Elsevier S e q u o i a S.A., L a u s a n n e - P r i n t e d in t h e N e t h e r l a n d s
Analysis o f Pairwise Dislocation Interaction and its Contribution to Flow Stress During Magnesium Crystal Basal Slip
F. F. L A V R E N T E V , YU. A. P O K H I L a n d I. N. Z O L O T U K H I N A
Physico-Technical Institute o f Low Temperatures, Ukr. S. S.R. Academy o f Sciences, Kharkov ( U. S. S. R.) (Received in revised form J u l y 6, 1977 )
SUMMARY
The paper reports an analysis of conditions favouring reactions during attractive interaction between basal dislocations and basal with prismatic dislocations, based on the energy gain approximation, a coefficients are calculated which relate the flow stress to the secondary basal and prismatic dislocation density. The contribution to flow stress due to these types of dislocation interactions is determined by the energetic condition of the particular reactions. Calculated values of a are compared to experimental values.
INTRODUCTION
Experimental investigations of work hardening parameters in Mg single crystals strained in system (0001)(1120) (by basal slip) [1 - 6] revealed two stages in stress-strain curves, r - e , designated as A and B, and characterized by different work hardening coefficients dr/de. Based on structural studies, the authors [1 - 6] conclude that at stage A, work hardening is due to an interaction between basal dislocations, producing dipoles, multipoles, and junctions. In refs. 3 - 6 it was found that the A - B transition is caused by cross slip of screw basal dislocations and resulting prismatic " f o r e s t " dislocations in the (1010}(1120} slip system, due to relaxation of internal stresses in the primary slip system. The work hardening coefficient increase at the B stage, as against stage A, was shown to be related to basal-prismatic interaction, which entails recombination of basal dislocations with prismatic " f o r e s t " dislocations and thereby strongly hinders the former. A quantitative investigation of the dependence of the flow stress, r, on a particular dis-
location type density, Pi [6, 7], leads to the following relationship: r = t o + ~, a i G i b i x / p i ,
(1)
where the value of ai represents the contribution to the flow stress due to the i-th t y p e dislocation interaction. The authors of ref. 8 suggested a hypothesis on a direct relation of the value of ai to the energy gain during the attractive interaction of junction-constituting dislocations, which was supported by the experimental results [6, 7]. The hypothesis was then quantitatively supported and further developed in work [9] containing a theoretical analysis of the conditions favouring reactions attended by junction formation owing to basal-pyramidal dislocation interaction, based on an estimation of energy balance, and also on a calculation of the appropriate ai coefficients. The present work is a theoretical investigation of conditions conducive to dislocation reactions, and involves estimation of their contributions to flow stress for the kinds of interaction observable in Mg crystals during basal slip (basal-basal at stage A and basalprismatic at stage B).
RESULTS AND DISCUSSION
The junction size calculation method is described in detail in ref. 9, where the authors used the approach developed in refs. 10 - 12. The procedure employed the condition of maximum energy gain in the reaction due to decreasing total length of dislocation lines making up the recombination configuration. The energy gain, AE, should be calculated
114
with due account made for changes in the linear energy of the dislocations resulting from their reorientation: (2)
A E = Eo - - E ,
where Eo = Y~
4~(1 -- v) In
(1
- - V COS 2 7 i )
li
i=1,2
is the configuration energy before the reaction, and E = Z
4 . ( 1 - - v) In
(1 - - v cos 2 7'i) l~ +
i = 1,2
4 . ( 1 -- v) ,n
(1
--
P COS2 "[3) d
is the dislocation system energy following the reaction. In expressions for dislocation line energy, Gi and bi are the shear modulus and the Burgers vector for the particular dislocation; v, the Poisson coefficient; R, ro, the cut-off radii of the integration region of the dislocation and dislocation core stress fields; 7i and 7~ are the angles between the dislocation line and its Burgers vector before and after the reaction, respectively; li, Ii, are the lengths o f dislocation lines before and after the interaction; d = PQ is the junction length. The value of d at which function AE goes through the maximum determines the equilibrium junction length. The variation in this magnitude with the mutual dislocation orientation allows the analysis o f the dislocation reaction realization conditions. ( i) B a s a l - b a s a l i n t e r a c t i o n When basal dislocation Burgers vectors satisfy the Frank criterion (b I • b2 < 0), then the interaction is attractive and can cause junctions to develop. The reaction between basal dislocations can be written as:
b11"91 _ _
~- [1210] (0001)
+b21.91
b 1"4 3 1 ~- [11201 (0001)
Fig. 1. S c h e m e o f j u n c t i o n f o r m a t i o n during interact i o n o f t w o basal dislocations.
on the condition of reaction completeness, i.e., in the case of parallel dislocations involved. Figure 1 shows, schematically, the reaction according to eqn. (3). The mutual orientation of dislocations is given by the angles ¢ and ~, where ¢ is the angle included by dislocations 1 and 2 at the initial position and ~ is the angle of rotation of the junction about the X axis. As was shown in ref. 10, if reaction (3) takes place, the junction is always symmetrical with respect to the reacting dislocations. The X axis is oriented along the close packing direction, angles being/31 = 60 °, /32 = --60 °, t3a = 0 ° (see Fig. 1). In this case, the dislocation line length after the reaction is: Ii = x/l 2 + d 2 _ 21i d cos Oi * Ig x/1 + x 2 -- 2xi cos 0 i,
where xi=d/li;
0"=01--¢/;
*
02 = ~ - - ( 0 2 - ~ b ) ;
1 -
cos~i (3)
Here the subscripts show the magnitude and the direction o f the Burgers vectors of the dislocations and the planes they are in; the superscripts show the dislocation linear energy in units of E / l n (R/ro) × 10 5 erg/cm,
01=90°--(,p/2--tk);
02 = 9 0
0
+(¢/2+~).
(4b)
The dislocation now has turned by the angle ~:
._>
~- [21i-0] (0001)
(4a)
=
x i c o s O~
~/rl+x 2-2xicos0i
,.
(4c)
Angles 7 and 7' entering into eqn. (2) can be found from the scheme as follows: ~/1 = 90 ° -- ~/2 + ~ -- 60 ° = 30 ° -- ~/2 + ¢ ; 7~ = 30 °--~v/2 + ~ --~1; 3'2 = 90 ° + ~/2 + ~ -- 120 ° = - - 3 0 o + ¢/2 + ~;
115 t
72
=
- - 3 0 ° + ¢/2 + ¢ --~2.
Angle ¢ is always to be c o u n t e d counterclockwise from the first to the second dislocation• T h e formulae hold only at ¢ ~<180 °. If ¢ > 180 °, the situation is equivalent to alteration o f the dislocation numbering• In this case: ~1 = 0 2 - - 6 0 71 = 3 0
.I,G .,,it U ,t.,~
,,f,r
~4
+¢/2;
°=30°+¢
, t 0 ~ (~f"~
(5)
i ,90p
+~ +¢/2--~,
~t,SW, #0¶fOG•
p, 8Ot, lOOa
t
~2 = 01 + 60 o = 150 ° + ~ - - ¢ / 2 ;
4,R
~ = 150 o + ~ - - ¢ / 2 + }2;
or
(6)
~~ 70 #, it# a
1,~70o, t/to ~ . 6 0 e, ltO"
,,~o i|0 o ~,St~, taO"
" ~ , :*0 2 a*O" !~=30 ,/$0 ~-p:gO¶ I~"
V,JW,/,SO" k,H¶ a~ ° P~tO¶ / t O "
~ f ~ ~O "
t , O ' , /aO,
~2 = n --- (150 o + $ _ ¢/2); ~t
o
3'2=30
--q~ + ¢ / 2 - - ~ 2 .
T he friction stress for basal dislocation m o t i o n Tf~ = 5 p / m m 2 [ 9] . .,,,E
=o • ,,,# =¢00 °
0,/ QM a~
0,~
(a)
AE
,/0~8C'~ y~=20~ .,~=180~ /
/
~t 2,
65 I
~5 O
2
,,r
ae-
~to"
16o"
too-
r*o-
,so
s..o-
3. Dependence of the function, AF,, on the mutual basal dislocation orientation.
. IO'e,"oa
i
5
~, Fig•
#
6
8
I0
1,2 /4
/6
,'8
20
a'.~f
(b)
shows that, though the necessary condition bl • b 2 < 0 is met, the reaction appears realisable but not t h r o u g h o u t the range o f angles. F o r example, at ~ = 0 o the reaction can only occur when 80 ° < ¢ < 340 o. T he plot is asymmetrical with respect to the axis, ¢ = 180 °. The reason is that the cases of ¢ < 180 o and ¢ > 180 °, as was m e n t i o n e d above, differ in dislocation numbering, the three Burgers vectors remaining the same. The configuration formed at ~ < 90 ° and ¢ > 270 ° is c o m m o n l y called a horizontal j unct i on [11]. T he j unct i on line in this case is the bisectrix o f the blunt angle between the dislocations (Fig. 4(a)). If 90 ° < ¢ < 270 °, a vertical j unct i on is observed which halves the acute angle between the dislocations (Fig. 4(b)). T he difference in the reaction conditions for ~ -~ 0 °, 360 °, and 180 ° (see Fig. 3) is due to inequivalence o f the
Fig. 2• Energy gain, AE, as a function of the junction size, d, for two cases of mutual orientation of reacting dislocations.
Figure 2 presents function AE against the j u n ctio n size, d, for certain selected ¢'s and 's. Th e equilibrium value of d corresponds to the m a x i m u m o f this function. Figure 3 illustrates the energy gain as a function o f the angle between the dislocations, ¢, for different angles o f j unct i on rotation, $ , around the X axis. T he Figure
~az..gOO
~a~.270 o
X a
90<~
°
/80%9r<270 o
X #
Fig. 4. Difference between the "horizontal" (a) and the "vertical" (b) junctions.
116
fO.
.t ~: 50 ° v ¢20 e t ,t, , O O o O~ : 90 B
0,0. y, , ~O "
////// ........
Fig. 5. Difference in j u n c t i o n f o r m a t i o n conditions for angles (a) ~ -~ 0 o, (b) ~ -~ 360 °, and (c) ~ = 180 o
a~
iI i i
i
/ i
/
•
'
.d.., ~,TQ?t~o
t\\~.,t i, ilj
~,~ i
#'~1$$
.1 Fig. 7. Relative j u n c t i o n size, x, as a f u n c t i o n of angle ~0 b e t w e e n basal dislocations for different values o f ~. /2
ooo
I/Ittt\t III/I I/I Ill/V-ill/
~ , =so; ,,o. o ~V"
~o
: O?
/
/
/-
o
/
~Y/*,,o,"
,so" oo
#~0 #
$o
Fig. 6. J u n c t i o n size, d, as a f u n c t i o n o f the angle ~p b e t w e e n dislocations for different values o f param e t e r ~. The calculation did not go above d = 20/am.
vertical and the horizontal junctions in the case of the small angle included by the dislocations (Fig. 5). Figure 6 shows ~- and C-dependences o f the junction absolute size, assuming the maximum value, d = 20/am. As ~0 goes to 180 o the d value grows infinitely, which is typical of the complete pooling of two reacting dislocations into a resulting one. It will be noted that such large junctions have never been seen in the experiment, because it is difficult to find a quite separate pair o f interacting dislocations in a crystal; they are affected by stress fields of other dislocations and can be pinned at impurities or at dislocations of intersecting slip systems. Therefore, the dislocation b o w e d part appears essentially smaller than the calculated one determined by the lattice friction. As a result, calculated junctions also are overrated against experiment. Because of this, for comparison with the experiment, the equilibrium values of the relative junction extent, x = d/l, is to be determined, as was done in refs. 10 - 12, i.e., find the extremum of function A E / l = f ( x ). Figure 7 Shows the run of x (¢) for different angles, ~. The plot is asymmetrical with regard to the axis ~ = 180 ° and unlike the results for Zn in refs. 10 - 12, the curve dissociates into
separate branches for particular ~ angles over the entire range of ~ angles. (It is worth mentioning that in refs. 10 - 12 curves for different parameters of ~ merged into one at ~o < 180 °.) There are several experimental points on Fig. 7 obtained by virtue of measuring junction configurations in deformed Mg single crystals. Good agreement with the calculation is seen. The above analysis suggests that a dislocation reaction in the Mg crystal basal plane is realisable in a fairly wide range o f dislocation orientation angles. So, it is reasonable to expect a significant contribution to flow stress due to this dislocation reaction at stage A o f the Mg hardening curve if there are no intersecting slip planes. We shall try to estimate the contribution quantitatively. As was shown in refs. 3, 5 and 6, the flow stress at stage A of the hardening curve is determined by pairwise interaction of basal dislocations. The procedure described can help to find the energy gain resulting from reaction (3) for different cases of mutual dislocation orientation. However, the method of junction rupture stress calculation proposed in ref. 9 is inapplicable here, for dislocations of secondary basal systems cannot be thought of as rigid, neither is the resulting dislocation "sessile". Therefore, the rupture stress can only be estimated approximately, proceeding from the necessity to make up for the energy gain by applied external stress (r) work: T.b-S=AE,
(7)
where S is the average area per secondary dislocation, equal t o ~ l 2 , where l p f l / 2 , pf =
117
I ~00
/ ,oo C i
/
o ~=gO °
~00
Fig. 8. J u n c t i o n d i s r u p t i n g stress as a f u n c t i o n o f ~p f o r ~ = 0 ° a n d 180 °. l : p f = 106; 2: 0f = 107; 3: pf = 5 x 1 0 7 ; 4: pf = 1 0 g e m - 2 .
• ~,=90 o
~=60 o
100
being the average density of secondary basal dislocations. The mutual dislocation orientation in this case depends on the angles and ~. The stress values c o m p u t e d by eqn. (7) are shown in Fig. 8 as a function of the angles for ~ = 0 ° and 180 °. The plot has four curves for different secondary basal dislocation densities, pf: 106, 107, 5 × 107, and 10 s cm-2. As would have been expected, the shapes of the curves agree with the character of energy gain, AE, variation (see Fig. 3). The stress appropriate to disruption of junctions, brought about by pooled near-parallel dislocations (~ = 180 o), is high, and much higher than the flow stress values observed at stage A [6]. At small ~ angles (~ = 40 ° - 80 °, depending on @), the junctions are " w e a k " and are destroyed by stresses lower than experimental. The average weighted value of a (in r ~ a G b x / p f ) was obtained in this case by way of averaging within the ¢ angle range, where the dislocation reaction can proceed using the Gaussian normal distribution: ~/2
~2
1
f
---
e
= ~13 0
~= n/2
ff
(8)
1
ox/r~ e
20= d~ d@
~13 0
In view of the narrow range averaging (60 180 °), the dispersion value was selected at o = 0.8. Figure 9 presents curves for r(pf) for three particular cases of mutual dislocation orientation: near-parallel (¢ = 90 o, ~ = 170 °), two edge (~ = 0 o, ~ = 240 °), and two screw (¢ = 90 °, ~ = 60 o) dislocations. Clearly, the stress, r(x/p~), is well enough describable by
Fig. 9. T h e o r e t i c a l r e l a t i o n r ( p f ) f o r basal d i s l o c a t i o n interaction.
a linear law with the proportionality factor, a, ranging, for different angular configurations, from 0.1 to 1. Figure 9 also shows the dependence r (x/P f) with the averaged coefficient &, which was found to be 0.32. The experimental value of ~b at stage A, obtained from the dependence of r on the total basal dislocation density, is 0.2 [6]. However, for comparison with calculated &, it must be taken into consideration that secondary basal dislocations constitute a b o u t 2/3 of the total number, and only half of them can participate in attraction with primary system dislocations (because of two possible directions of the Burgers vectors). Therefore, to obtain correct experiment-calculation comparison, 1/3 of the total dislocation density must be plotted on the experimental graph for r(x/p), so that coefficient aexp is equal to 0.2 × x/3 = 0.35 in this case. This value is in fair agreement with the calculation of &. To evaluate the effect of parameter o selection on &, note that while o changed from 0.5 to 1, the value of ranged from 0.29 to 0.33. A comparison of the theoretical curve r(x/p~) for ~ = 0.32 with experiment [6], taking into account the correction for secondary basal dislocation density, shows that the plots have similar slopes and are displaced along the r axis with respect to each other by Ar -~ 50 p/mm 2. This value is adequate for the yield stress for real Mg crystal in the absence of secondary basal dislocations [5, 6]. Meanwhile, according to the calculated model, this line should pass through the
118
origin. Nevertheless, the coincidence of the coefficients shows that the model of crystal work hardening due to basal dislocation attraction with junction formation is well supported by the experimental r(p), that is to say, basal dislocation interaction may be referred to as dominant at stage A of Mg single-crystal work hardening during basal slip.
o
o
o
o
,,
,,
o
,,
o
o
o
o
o
o
~
~
o
~
o
o
o
180°
,,
~,
o
o
o
o
,,
~,
~. . . . . o
o
~
,~
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
~'
o
o
o
o
o
o
~ 5
I
f2
o. . . . . .
:i':, k' ""
""::
o
5
......
,:'"'
(ii) B a s a l - p r i s m a tic in terac tion
Development o f slip in prism I planes typical of stage B o f Mg crystal hardening for basal slip [3 - 6] can lead to the following reaction between basal and prismatic dislocations:
9
+
$ [12i-0](1010)
(9)
b~'9~ m~0j(000. " Analyze conditions favouring it. Figure 10 shows the scheme o f this interaction. The
\ \ \
!
Fig. 10. Junction formation during interaction of a basal (1) and a prismatic (2) dislocation (schematic).
mutual orientation of dislocations is given by the angles 01 and 02. Relations to calculate the parameters 7i, ~/i, Ii, l~, and Gi entering into eqn. (2) for the equilibrium value of d determination are supplied in ref. 9 (7i = /31 - Oi; 71 =/3i - - (Oi + ~i), where/3i is the angle between the dislocation Burgers vector and the junction). We list only a few of them: /31 = 1 2 0 ° ,
/32=0°,
138=60°;
bl=b 2=b~ =3.21A; G1 = G2 = G3 = 1.7 X 106 p/mm2; rfr~ = 5 p / m m 2,
o
o
~0
9
a4
o
o
o
gO,,~
Fig. 11. Junction size as a function of mutual orientation during basal-prismatic dislocation interaction.
-,
b~'_ [2ifo] (0001)
0 ~0 o
r ~ 2 = 9 0 p / m m 2.
The calculation results are shown in Fig. 11. The figures on the plot are the equilibrium
sizes of junctions at each point (01, 02). The lines connect points referring to the totality of angles with the same d values. The outer contour bounds the limiting angular orientations allowing this reaction. The open circles mark the absence of the junction (AE ~<0 for any d). The reaction realization range was rather wide, despite the high friction stress in prism plane r~2 (85 o to --85 ° for prismatic dislocation and 70 ° to --70 ° for basal dislocation). Such a wide range is due to a considerable energy gain during the reaction caused by the low linear energy of the resulting dislocation, as against, for example, the basal-pyramidal interaction [9]. The orientation range where the junction is large enough ( > 7 pm) is more contracted along axis 02, than along axis 01 (see Fig. 11, showing the contour for d = 20/am elongated along axis 01). This stems from the high mobility of the basal dislocations on account of the low friction in the (0001) planes. The plot of Fig. 11 is symmetrical with regard to axis 01, for the Burgers vector o f the prismatic dislocation, b2, is directed along the X axis, which is the origin of 0 angle counting. Therefore, the energy of this dislocation does not change with changing angle 02 for (--02) (because E ~ 1 -- v c o s 2 '~';
=/3-o). As the angles of slope of dislocation to the slip plane intersection line increase, the forming junctions become smaller. At angles 01 = 70 o and 02 -~ 80 o, junctions become so small that their contribution to long-range stress fields ceases to be significant and the contact
119
short-range action associated with intersection jogs formation becomes important. Consider the behaviour of configurations produced by reaction (9) under applied stress. As is seen from Fig. 10, the resulting dislocation can slip in the basal plane. However, bowing segments of the more heavily loaded primary dislocation 1 retard junction slipping by virtue of repulsive interaction of the dislocation segments. Moreover, the junction length is short as compared with the pinned dislocation segment, and this also impedes bowing of the junction. Therefore the model described in ref. 9 is suitable for finding the junction disruption stress.
8POOl
o 8, :30 ° 0~:0 ° 16OO
8, : 4 0 °
•
82: 0 " 1200;
C] ~ : 5 0 v -¢~: 0 ~
matic dislocations interact attractively with the given basal one, then the experimental value of ap becomes 1 × x/2 = 1.41. In this case we see satisfactory agreement between the experimental and the calculated dependences 7(pf). The experimental curves are, however, somewhat displaced in ref. 6 along the abscissa axis. The data disagree probably because a real crystal begins to "sense" the presence of "forest" prismatic dislocations no earlier than at a certain sufficient density, the flow stress at lower densities being due to a dominant coplanar system interaction. Though weaker, this interaction works at stage B t o o [6], being thus added to the intersecting system interaction. As a result, it is reasonable to conclude that the intersection mechanism of attracting prismatic " f o r e s t " against the background of coplanar basal-basal interaction is responsible for the flow stress of Mg single crystals at stage B of the stress-strain curve during basal slip.
0,: l,O °
a
REFERENCES
800
~:
.
Fig. 12. Theoretical relation T(pf) during basal-prismatic dislocation interaction.
Computation results are given in Fig. 12, where there is a number of curves for 7(x/p ~) a t p ~ = 1 0 6 , 1 0 7 , 5 × 1 0 7 , a n d 1 0 s c m -2 for various orientation configurations of reacting dislocations. For 01 > 30 o, dependence r(x/pf) can be quite adequately represented as a straight line. For 01 tending to zero, the disruption stress grows drastically, destroying the linear shape. This is consistent with experimental data for NaC1 [13]. However, the procedure developed in this paper for rupture stress averaging implies neglecting such cases (see the reasoning in ref. 9). The averaged value of coefficient ~ for this dependence is 1.34. The experimental dependence of 7 on the total prismatic dislocation density gives ap = 1 [6]. If one takes into consideration that only half of all the pris-
1 P. B. Hirsch and J. S. Lally, Philos. Mag., 12 (1965) 595. 2 J. V. Sharp, M. J. Makin and S. W. Christian, Phys. Status Solidi, 11 (1965) 845. 3 F. F. Lavrentev and Yu. A. Pokhil, Fiz. Metall. Metalloved., 34 (1972) 1270. 4 F. F. Lavrentev and Yu. A. Pokhil, Fiz. Tverd. Tela, 16 (1974) 2096. 5 Yu. A. Pokhil, Krist. Techn., 10 (1974) 1179. 6 F. F. Lavrentev and Yu. A. Pokhil, Mater. Sci. Eng., 18 (1975) 261. 7 F. F. Lavrentev and Yu. A. Pokhil, Phys. Status Solidi A, 32 (1975) 227. 8 F. F. Lavrentev, Collection, Fizika deformatsionnogo uprochneniya monokristallov (Single Crystal Work Hardening Physics), Naukova Dumka, Kiev, 1972, p. 107. 9 F. F. Lavrentev, Yu. A. Pokhil and I. N. Zolotukhina, Mater. Sci. Eng., 23 (1976) 69; Mater. Sci. Eng., to be published. 10 A. A. Predvoditelev, Collection, Dinamika dislokatsiy (Dislocation Dynamics), Phys.-Techn. Inst. Low Temp., Acad. Sci. Ukr.SSR, Kharkov, 1968, p. 311. 11 G. V. Bushueva, Synopsis of Thesis, Moscow State University, 1969. 12 A. A. Predvoditelev and O. A. Troitski, Dislokatsii i tochechnyye defekty v geksagonalnykh metallakh, Atomizdat, Moscow, 1973. 13 A. A. Predvoditelev, V. M. Stepanova and N. A. Nosov, Kristallografia, 11 (1966)632.
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© Elsevier S e q u o i a S.A., L a u s a n n e - P r i n t e d in t h e N e t h e r l a n d s
Analysis o f Pairwise Dislocation Interaction and its Contribution to Flow Stress During Magnesium Crystal Basal Slip
F. F. L A V R E N T E V , YU. A. P O K H I L a n d I. N. Z O L O T U K H I N A
Physico-Technical Institute o f Low Temperatures, Ukr. S. S.R. Academy o f Sciences, Kharkov ( U. S. S. R.) (Received in revised form J u l y 6, 1977 )
SUMMARY
The paper reports an analysis of conditions favouring reactions during attractive interaction between basal dislocations and basal with prismatic dislocations, based on the energy gain approximation, a coefficients are calculated which relate the flow stress to the secondary basal and prismatic dislocation density. The contribution to flow stress due to these types of dislocation interactions is determined by the energetic condition of the particular reactions. Calculated values of a are compared to experimental values.
INTRODUCTION
Experimental investigations of work hardening parameters in Mg single crystals strained in system (0001)(1120) (by basal slip) [1 - 6] revealed two stages in stress-strain curves, r - e , designated as A and B, and characterized by different work hardening coefficients dr/de. Based on structural studies, the authors [1 - 6] conclude that at stage A, work hardening is due to an interaction between basal dislocations, producing dipoles, multipoles, and junctions. In refs. 3 - 6 it was found that the A - B transition is caused by cross slip of screw basal dislocations and resulting prismatic " f o r e s t " dislocations in the (1010}(1120} slip system, due to relaxation of internal stresses in the primary slip system. The work hardening coefficient increase at the B stage, as against stage A, was shown to be related to basal-prismatic interaction, which entails recombination of basal dislocations with prismatic " f o r e s t " dislocations and thereby strongly hinders the former. A quantitative investigation of the dependence of the flow stress, r, on a particular dis-
location type density, Pi [6, 7], leads to the following relationship: r = t o + ~, a i G i b i x / p i ,
(1)
where the value of ai represents the contribution to the flow stress due to the i-th t y p e dislocation interaction. The authors of ref. 8 suggested a hypothesis on a direct relation of the value of ai to the energy gain during the attractive interaction of junction-constituting dislocations, which was supported by the experimental results [6, 7]. The hypothesis was then quantitatively supported and further developed in work [9] containing a theoretical analysis of the conditions favouring reactions attended by junction formation owing to basal-pyramidal dislocation interaction, based on an estimation of energy balance, and also on a calculation of the appropriate ai coefficients. The present work is a theoretical investigation of conditions conducive to dislocation reactions, and involves estimation of their contributions to flow stress for the kinds of interaction observable in Mg crystals during basal slip (basal-basal at stage A and basalprismatic at stage B).
RESULTS AND DISCUSSION
The junction size calculation method is described in detail in ref. 9, where the authors used the approach developed in refs. 10 - 12. The procedure employed the condition of maximum energy gain in the reaction due to decreasing total length of dislocation lines making up the recombination configuration. The energy gain, AE, should be calculated
114
with due account made for changes in the linear energy of the dislocations resulting from their reorientation: (2)
A E = Eo - - E ,
where Eo = Y~
4~(1 -- v) In
(1
- - V COS 2 7 i )
li
i=1,2
is the configuration energy before the reaction, and E = Z
4 . ( 1 - - v) In
(1 - - v cos 2 7'i) l~ +
i = 1,2
4 . ( 1 -- v) ,n
(1
--
P COS2 "[3) d
is the dislocation system energy following the reaction. In expressions for dislocation line energy, Gi and bi are the shear modulus and the Burgers vector for the particular dislocation; v, the Poisson coefficient; R, ro, the cut-off radii of the integration region of the dislocation and dislocation core stress fields; 7i and 7~ are the angles between the dislocation line and its Burgers vector before and after the reaction, respectively; li, Ii, are the lengths o f dislocation lines before and after the interaction; d = PQ is the junction length. The value of d at which function AE goes through the maximum determines the equilibrium junction length. The variation in this magnitude with the mutual dislocation orientation allows the analysis o f the dislocation reaction realization conditions. ( i) B a s a l - b a s a l i n t e r a c t i o n When basal dislocation Burgers vectors satisfy the Frank criterion (b I • b2 < 0), then the interaction is attractive and can cause junctions to develop. The reaction between basal dislocations can be written as:
b11"91 _ _
~- [1210] (0001)
+b21.91
b 1"4 3 1 ~- [11201 (0001)
Fig. 1. S c h e m e o f j u n c t i o n f o r m a t i o n during interact i o n o f t w o basal dislocations.
on the condition of reaction completeness, i.e., in the case of parallel dislocations involved. Figure 1 shows, schematically, the reaction according to eqn. (3). The mutual orientation of dislocations is given by the angles ¢ and ~, where ¢ is the angle included by dislocations 1 and 2 at the initial position and ~ is the angle of rotation of the junction about the X axis. As was shown in ref. 10, if reaction (3) takes place, the junction is always symmetrical with respect to the reacting dislocations. The X axis is oriented along the close packing direction, angles being/31 = 60 °, /32 = --60 °, t3a = 0 ° (see Fig. 1). In this case, the dislocation line length after the reaction is: Ii = x/l 2 + d 2 _ 21i d cos Oi * Ig x/1 + x 2 -- 2xi cos 0 i,
where xi=d/li;
0"=01--¢/;
*
02 = ~ - - ( 0 2 - ~ b ) ;
1 -
cos~i (3)
Here the subscripts show the magnitude and the direction o f the Burgers vectors of the dislocations and the planes they are in; the superscripts show the dislocation linear energy in units of E / l n (R/ro) × 10 5 erg/cm,
01=90°--(,p/2--tk);
02 = 9 0
0
+(¢/2+~).
(4b)
The dislocation now has turned by the angle ~:
._>
~- [21i-0] (0001)
(4a)
=
x i c o s O~
~/rl+x 2-2xicos0i
,.
(4c)
Angles 7 and 7' entering into eqn. (2) can be found from the scheme as follows: ~/1 = 90 ° -- ~/2 + ~ -- 60 ° = 30 ° -- ~/2 + ¢ ; 7~ = 30 °--~v/2 + ~ --~1; 3'2 = 90 ° + ~/2 + ~ -- 120 ° = - - 3 0 o + ¢/2 + ~;
115 t
72
=
- - 3 0 ° + ¢/2 + ¢ --~2.
Angle ¢ is always to be c o u n t e d counterclockwise from the first to the second dislocation• T h e formulae hold only at ¢ ~<180 °. If ¢ > 180 °, the situation is equivalent to alteration o f the dislocation numbering• In this case: ~1 = 0 2 - - 6 0 71 = 3 0
.I,G .,,it U ,t.,~
,,f,r
~4
+¢/2;
°=30°+¢
, t 0 ~ (~f"~
(5)
i ,90p
+~ +¢/2--~,
~t,SW, #0¶fOG•
p, 8Ot, lOOa
t
~2 = 01 + 60 o = 150 ° + ~ - - ¢ / 2 ;
4,R
~ = 150 o + ~ - - ¢ / 2 + }2;
or
(6)
~~ 70 #, it# a
1,~70o, t/to ~ . 6 0 e, ltO"
,,~o i|0 o ~,St~, taO"
" ~ , :*0 2 a*O" !~=30 ,/$0 ~-p:gO¶ I~"
V,JW,/,SO" k,H¶ a~ ° P~tO¶ / t O "
~ f ~ ~O "
t , O ' , /aO,
~2 = n --- (150 o + $ _ ¢/2); ~t
o
3'2=30
--q~ + ¢ / 2 - - ~ 2 .
T he friction stress for basal dislocation m o t i o n Tf~ = 5 p / m m 2 [ 9] . .,,,E
=o • ,,,# =¢00 °
0,/ QM a~
0,~
(a)
AE
,/0~8C'~ y~=20~ .,~=180~ /
/
~t 2,
65 I
~5 O
2
,,r
ae-
~to"
16o"
too-
r*o-
,so
s..o-
3. Dependence of the function, AF,, on the mutual basal dislocation orientation.
. IO'e,"oa
i
5
~, Fig•
#
6
8
I0
1,2 /4
/6
,'8
20
a'.~f
(b)
shows that, though the necessary condition bl • b 2 < 0 is met, the reaction appears realisable but not t h r o u g h o u t the range o f angles. F o r example, at ~ = 0 o the reaction can only occur when 80 ° < ¢ < 340 o. T he plot is asymmetrical with respect to the axis, ¢ = 180 °. The reason is that the cases of ¢ < 180 o and ¢ > 180 °, as was m e n t i o n e d above, differ in dislocation numbering, the three Burgers vectors remaining the same. The configuration formed at ~ < 90 ° and ¢ > 270 ° is c o m m o n l y called a horizontal j unct i on [11]. T he j unct i on line in this case is the bisectrix o f the blunt angle between the dislocations (Fig. 4(a)). If 90 ° < ¢ < 270 °, a vertical j unct i on is observed which halves the acute angle between the dislocations (Fig. 4(b)). T he difference in the reaction conditions for ~ -~ 0 °, 360 °, and 180 ° (see Fig. 3) is due to inequivalence o f the
Fig. 2• Energy gain, AE, as a function of the junction size, d, for two cases of mutual orientation of reacting dislocations.
Figure 2 presents function AE against the j u n ctio n size, d, for certain selected ¢'s and 's. Th e equilibrium value of d corresponds to the m a x i m u m o f this function. Figure 3 illustrates the energy gain as a function o f the angle between the dislocations, ¢, for different angles o f j unct i on rotation, $ , around the X axis. T he Figure
~az..gOO
~a~.270 o
X a
90<~
°
/80%9r<270 o
X #
Fig. 4. Difference between the "horizontal" (a) and the "vertical" (b) junctions.
116
fO.
.t ~: 50 ° v ¢20 e t ,t, , O O o O~ : 90 B
0,0. y, , ~O "
////// ........
Fig. 5. Difference in j u n c t i o n f o r m a t i o n conditions for angles (a) ~ -~ 0 o, (b) ~ -~ 360 °, and (c) ~ = 180 o
a~
iI i i
i
/ i
/
•
'
.d.., ~,TQ?t~o
t\\~.,t i, ilj
~,~ i
#'~1$$
.1 Fig. 7. Relative j u n c t i o n size, x, as a f u n c t i o n of angle ~0 b e t w e e n basal dislocations for different values o f ~. /2
ooo
I/Ittt\t III/I I/I Ill/V-ill/
~ , =so; ,,o. o ~V"
~o
: O?
/
/
/-
o
/
~Y/*,,o,"
,so" oo
#~0 #
$o
Fig. 6. J u n c t i o n size, d, as a f u n c t i o n o f the angle ~p b e t w e e n dislocations for different values o f param e t e r ~. The calculation did not go above d = 20/am.
vertical and the horizontal junctions in the case of the small angle included by the dislocations (Fig. 5). Figure 6 shows ~- and C-dependences o f the junction absolute size, assuming the maximum value, d = 20/am. As ~0 goes to 180 o the d value grows infinitely, which is typical of the complete pooling of two reacting dislocations into a resulting one. It will be noted that such large junctions have never been seen in the experiment, because it is difficult to find a quite separate pair o f interacting dislocations in a crystal; they are affected by stress fields of other dislocations and can be pinned at impurities or at dislocations of intersecting slip systems. Therefore, the dislocation b o w e d part appears essentially smaller than the calculated one determined by the lattice friction. As a result, calculated junctions also are overrated against experiment. Because of this, for comparison with the experiment, the equilibrium values of the relative junction extent, x = d/l, is to be determined, as was done in refs. 10 - 12, i.e., find the extremum of function A E / l = f ( x ). Figure 7 Shows the run of x (¢) for different angles, ~. The plot is asymmetrical with regard to the axis ~ = 180 ° and unlike the results for Zn in refs. 10 - 12, the curve dissociates into
separate branches for particular ~ angles over the entire range of ~ angles. (It is worth mentioning that in refs. 10 - 12 curves for different parameters of ~ merged into one at ~o < 180 °.) There are several experimental points on Fig. 7 obtained by virtue of measuring junction configurations in deformed Mg single crystals. Good agreement with the calculation is seen. The above analysis suggests that a dislocation reaction in the Mg crystal basal plane is realisable in a fairly wide range o f dislocation orientation angles. So, it is reasonable to expect a significant contribution to flow stress due to this dislocation reaction at stage A o f the Mg hardening curve if there are no intersecting slip planes. We shall try to estimate the contribution quantitatively. As was shown in refs. 3, 5 and 6, the flow stress at stage A of the hardening curve is determined by pairwise interaction of basal dislocations. The procedure described can help to find the energy gain resulting from reaction (3) for different cases of mutual dislocation orientation. However, the method of junction rupture stress calculation proposed in ref. 9 is inapplicable here, for dislocations of secondary basal systems cannot be thought of as rigid, neither is the resulting dislocation "sessile". Therefore, the rupture stress can only be estimated approximately, proceeding from the necessity to make up for the energy gain by applied external stress (r) work: T.b-S=AE,
(7)
where S is the average area per secondary dislocation, equal t o ~ l 2 , where l p f l / 2 , pf =
117
I ~00
/ ,oo C i
/
o ~=gO °
~00
Fig. 8. J u n c t i o n d i s r u p t i n g stress as a f u n c t i o n o f ~p f o r ~ = 0 ° a n d 180 °. l : p f = 106; 2: 0f = 107; 3: pf = 5 x 1 0 7 ; 4: pf = 1 0 g e m - 2 .
• ~,=90 o
~=60 o
100
being the average density of secondary basal dislocations. The mutual dislocation orientation in this case depends on the angles and ~. The stress values c o m p u t e d by eqn. (7) are shown in Fig. 8 as a function of the angles for ~ = 0 ° and 180 °. The plot has four curves for different secondary basal dislocation densities, pf: 106, 107, 5 × 107, and 10 s cm-2. As would have been expected, the shapes of the curves agree with the character of energy gain, AE, variation (see Fig. 3). The stress appropriate to disruption of junctions, brought about by pooled near-parallel dislocations (~ = 180 o), is high, and much higher than the flow stress values observed at stage A [6]. At small ~ angles (~ = 40 ° - 80 °, depending on @), the junctions are " w e a k " and are destroyed by stresses lower than experimental. The average weighted value of a (in r ~ a G b x / p f ) was obtained in this case by way of averaging within the ¢ angle range, where the dislocation reaction can proceed using the Gaussian normal distribution: ~/2
~2
1
f
---
e
= ~13 0
~= n/2
ff
(8)
1
ox/r~ e
20= d~ d@
~13 0
In view of the narrow range averaging (60 180 °), the dispersion value was selected at o = 0.8. Figure 9 presents curves for r(pf) for three particular cases of mutual dislocation orientation: near-parallel (¢ = 90 o, ~ = 170 °), two edge (~ = 0 o, ~ = 240 °), and two screw (¢ = 90 °, ~ = 60 o) dislocations. Clearly, the stress, r(x/p~), is well enough describable by
Fig. 9. T h e o r e t i c a l r e l a t i o n r ( p f ) f o r basal d i s l o c a t i o n interaction.
a linear law with the proportionality factor, a, ranging, for different angular configurations, from 0.1 to 1. Figure 9 also shows the dependence r (x/P f) with the averaged coefficient &, which was found to be 0.32. The experimental value of ~b at stage A, obtained from the dependence of r on the total basal dislocation density, is 0.2 [6]. However, for comparison with calculated &, it must be taken into consideration that secondary basal dislocations constitute a b o u t 2/3 of the total number, and only half of them can participate in attraction with primary system dislocations (because of two possible directions of the Burgers vectors). Therefore, to obtain correct experiment-calculation comparison, 1/3 of the total dislocation density must be plotted on the experimental graph for r(x/p), so that coefficient aexp is equal to 0.2 × x/3 = 0.35 in this case. This value is in fair agreement with the calculation of &. To evaluate the effect of parameter o selection on &, note that while o changed from 0.5 to 1, the value of ranged from 0.29 to 0.33. A comparison of the theoretical curve r(x/p~) for ~ = 0.32 with experiment [6], taking into account the correction for secondary basal dislocation density, shows that the plots have similar slopes and are displaced along the r axis with respect to each other by Ar -~ 50 p/mm 2. This value is adequate for the yield stress for real Mg crystal in the absence of secondary basal dislocations [5, 6]. Meanwhile, according to the calculated model, this line should pass through the
118
origin. Nevertheless, the coincidence of the coefficients shows that the model of crystal work hardening due to basal dislocation attraction with junction formation is well supported by the experimental r(p), that is to say, basal dislocation interaction may be referred to as dominant at stage A of Mg single-crystal work hardening during basal slip.
o
o
o
o
,,
,,
o
,,
o
o
o
o
o
o
~
~
o
~
o
o
o
180°
,,
~,
o
o
o
o
,,
~,
~. . . . . o
o
~
,~
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
~'
o
o
o
o
o
o
~ 5
I
f2
o. . . . . .
:i':, k' ""
""::
o
5
......
,:'"'
(ii) B a s a l - p r i s m a tic in terac tion
Development o f slip in prism I planes typical of stage B o f Mg crystal hardening for basal slip [3 - 6] can lead to the following reaction between basal and prismatic dislocations:
9
+
$ [12i-0](1010)
(9)
b~'9~ m~0j(000. " Analyze conditions favouring it. Figure 10 shows the scheme o f this interaction. The
\ \ \
!
Fig. 10. Junction formation during interaction of a basal (1) and a prismatic (2) dislocation (schematic).
mutual orientation of dislocations is given by the angles 01 and 02. Relations to calculate the parameters 7i, ~/i, Ii, l~, and Gi entering into eqn. (2) for the equilibrium value of d determination are supplied in ref. 9 (7i = /31 - Oi; 71 =/3i - - (Oi + ~i), where/3i is the angle between the dislocation Burgers vector and the junction). We list only a few of them: /31 = 1 2 0 ° ,
/32=0°,
138=60°;
bl=b 2=b~ =3.21A; G1 = G2 = G3 = 1.7 X 106 p/mm2; rfr~ = 5 p / m m 2,
o
o
~0
9
a4
o
o
o
gO,,~
Fig. 11. Junction size as a function of mutual orientation during basal-prismatic dislocation interaction.
-,
b~'_ [2ifo] (0001)
0 ~0 o
r ~ 2 = 9 0 p / m m 2.
The calculation results are shown in Fig. 11. The figures on the plot are the equilibrium
sizes of junctions at each point (01, 02). The lines connect points referring to the totality of angles with the same d values. The outer contour bounds the limiting angular orientations allowing this reaction. The open circles mark the absence of the junction (AE ~<0 for any d). The reaction realization range was rather wide, despite the high friction stress in prism plane r~2 (85 o to --85 ° for prismatic dislocation and 70 ° to --70 ° for basal dislocation). Such a wide range is due to a considerable energy gain during the reaction caused by the low linear energy of the resulting dislocation, as against, for example, the basal-pyramidal interaction [9]. The orientation range where the junction is large enough ( > 7 pm) is more contracted along axis 02, than along axis 01 (see Fig. 11, showing the contour for d = 20/am elongated along axis 01). This stems from the high mobility of the basal dislocations on account of the low friction in the (0001) planes. The plot of Fig. 11 is symmetrical with regard to axis 01, for the Burgers vector o f the prismatic dislocation, b2, is directed along the X axis, which is the origin of 0 angle counting. Therefore, the energy of this dislocation does not change with changing angle 02 for (--02) (because E ~ 1 -- v c o s 2 '~';
=/3-o). As the angles of slope of dislocation to the slip plane intersection line increase, the forming junctions become smaller. At angles 01 = 70 o and 02 -~ 80 o, junctions become so small that their contribution to long-range stress fields ceases to be significant and the contact
119
short-range action associated with intersection jogs formation becomes important. Consider the behaviour of configurations produced by reaction (9) under applied stress. As is seen from Fig. 10, the resulting dislocation can slip in the basal plane. However, bowing segments of the more heavily loaded primary dislocation 1 retard junction slipping by virtue of repulsive interaction of the dislocation segments. Moreover, the junction length is short as compared with the pinned dislocation segment, and this also impedes bowing of the junction. Therefore the model described in ref. 9 is suitable for finding the junction disruption stress.
8POOl
o 8, :30 ° 0~:0 ° 16OO
8, : 4 0 °
•
82: 0 " 1200;
C] ~ : 5 0 v -¢~: 0 ~
matic dislocations interact attractively with the given basal one, then the experimental value of ap becomes 1 × x/2 = 1.41. In this case we see satisfactory agreement between the experimental and the calculated dependences 7(pf). The experimental curves are, however, somewhat displaced in ref. 6 along the abscissa axis. The data disagree probably because a real crystal begins to "sense" the presence of "forest" prismatic dislocations no earlier than at a certain sufficient density, the flow stress at lower densities being due to a dominant coplanar system interaction. Though weaker, this interaction works at stage B t o o [6], being thus added to the intersecting system interaction. As a result, it is reasonable to conclude that the intersection mechanism of attracting prismatic " f o r e s t " against the background of coplanar basal-basal interaction is responsible for the flow stress of Mg single crystals at stage B of the stress-strain curve during basal slip.
0,: l,O °
a
REFERENCES
800
~:
.
Fig. 12. Theoretical relation T(pf) during basal-prismatic dislocation interaction.
Computation results are given in Fig. 12, where there is a number of curves for 7(x/p ~) a t p ~ = 1 0 6 , 1 0 7 , 5 × 1 0 7 , a n d 1 0 s c m -2 for various orientation configurations of reacting dislocations. For 01 > 30 o, dependence r(x/pf) can be quite adequately represented as a straight line. For 01 tending to zero, the disruption stress grows drastically, destroying the linear shape. This is consistent with experimental data for NaC1 [13]. However, the procedure developed in this paper for rupture stress averaging implies neglecting such cases (see the reasoning in ref. 9). The averaged value of coefficient ~ for this dependence is 1.34. The experimental dependence of 7 on the total prismatic dislocation density gives ap = 1 [6]. If one takes into consideration that only half of all the pris-
1 P. B. Hirsch and J. S. Lally, Philos. Mag., 12 (1965) 595. 2 J. V. Sharp, M. J. Makin and S. W. Christian, Phys. Status Solidi, 11 (1965) 845. 3 F. F. Lavrentev and Yu. A. Pokhil, Fiz. Metall. Metalloved., 34 (1972) 1270. 4 F. F. Lavrentev and Yu. A. Pokhil, Fiz. Tverd. Tela, 16 (1974) 2096. 5 Yu. A. Pokhil, Krist. Techn., 10 (1974) 1179. 6 F. F. Lavrentev and Yu. A. Pokhil, Mater. Sci. Eng., 18 (1975) 261. 7 F. F. Lavrentev and Yu. A. Pokhil, Phys. Status Solidi A, 32 (1975) 227. 8 F. F. Lavrentev, Collection, Fizika deformatsionnogo uprochneniya monokristallov (Single Crystal Work Hardening Physics), Naukova Dumka, Kiev, 1972, p. 107. 9 F. F. Lavrentev, Yu. A. Pokhil and I. N. Zolotukhina, Mater. Sci. Eng., 23 (1976) 69; Mater. Sci. Eng., to be published. 10 A. A. Predvoditelev, Collection, Dinamika dislokatsiy (Dislocation Dynamics), Phys.-Techn. Inst. Low Temp., Acad. Sci. Ukr.SSR, Kharkov, 1968, p. 311. 11 G. V. Bushueva, Synopsis of Thesis, Moscow State University, 1969. 12 A. A. Predvoditelev and O. A. Troitski, Dislokatsii i tochechnyye defekty v geksagonalnykh metallakh, Atomizdat, Moscow, 1973. 13 A. A. Predvoditelev, V. M. Stepanova and N. A. Nosov, Kristallografia, 11 (1966)632.