CONTINUOUS-DISTRIBUTION DISLOCATION
MODEL OF INTERNAL FRICTION ASSOCIATED
WITH THE INHOMOGENEOUS SLIDING ALONG
HIGH-ANGLE GRAIN BOUNDARIES
Z. Sun, T. Kê
To cite this version:
Z. Sun, T. Kê. CONTINUOUS-DISTRIBUTION DISLOCATION MODEL OF INTERNAL FRICTION ASSOCIATED WITH THE INHOMOGENEOUS SLIDING ALONG HIGHANGLE GRAIN BOUNDARIES. Journal de Physique Colloques, 1981, 42 (C5), pp.C5-451C5-456. <10.1051/jphyscol:1981567>. <jpa-00221110>
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JOURNAL DE PHYSIQUE
CoZloque C5, supplgment au nOIO, Tome 42, octobre 1981
page C5-451
CONTINUOUS-DISTRIBUTION DISLOCATION MODEL O F INTERNAL FRICTION
ASSOCIATED WITH THE INHOMOGENEOUS SLIDING ALONG HIGH-ANGLE GRAIN
BOUNDARIES
2.9.
Sun and T.S.
KC*
I n s t i t u t e of Solid State Physics (Hefeiland I n s t i t u t e o f Metal Research
(Shenyang), Academia Sinica, China
Abstract.- Basing on a c o n t i n u o u s - d i s t r i b u t i o n d i s l o c a t i o n model
o f high-angle g r a i n boundaries, a n i n t e g r a l - d i f f e r e n t i a 1 e q u a t i o n
governing t h e inhomogeneous s l i d i n g a l o n g t h e boundaries was s e t
up. T h i s e q u a t i o n c o n s i s t s o f t h r e e terms. The f i r s t term i s t h e
s h e a r s t r e s s produced by t h e d i s l o c a t i o n s w i t h a given continuous
d i s t r i b u t i o n l i n e a r d e n s i t y , t h e second term i s t h e a p p l i e d s h e a r
s t r e s s and t h e t h i r d term i s t h e Newtonian v i s c o u s r e s i s t a n t
s t r e s s . T h i s e q u a t i o n w a s solved approximately and t h e approximate
formulae f o r t h e i n t e r n a l f r i c t i o n and modulus d e f e c t were o b t a i n ed. For h i g h - p u r i t y i s o t r o p i c m e t a l s , t h e v i s c o s i t y f o r g r a i n boundary s l i d i n g i s c o r r e l a t e d w i t h t h e d i f f u s i o n c o e f f i c i e n t
a l o n g grain-boundary, D , by an e x p r e s s i o n similar t o E i n s t e i n Stokes formula. The opt?mum temperature of grain-boundary i n t e r n a l
f r i c t i o n peak f o r a number o f pure m e t a l s c a l c u l a t e d a c c o r d i n g t o
t h i s model and s l i d i n g mechanism a r e f a i r l y c l o s e t o t h e c o r r e s ponding e x p e r i m e n t a l l y observed values. I t i s shown t h a t f o r i m p u r e m e t a l s , t h e v i s c o s i t y o f some g r a i n boundaries i s considera b l y changed by t h e s e l e c t i v e s e g r e g a t i o n o f i m p u r i t i e s a l o n g
them, s o t h a t a n o t h e r i n t e r n a l - f r i c t i o n peak ( t h e s o l u t e peak)
a p p e a r s a t a d i f f e r e n t temperature. Also, i n a n i s o t r o p i c pure
m e t a l s and i n t h e presence o f i n t e r n a l s t r e s s , t h e m i g r a t i o n o f
g r a i n boundaries may cause a n o t h e r h i g h temperature i n t e r n a l - f r i c t i o n peak o r a h i g h e r i n t e r n a l - f r i c t i o n background i n a d d i t i o n t o
t h e grain-boundary peak a s s o c i a t e d w i t h grain-boundary s l i d i n g .
I. I n t r o d u c t i o n . - Using a t o r s i o n pendulum, ~k observed i n p o l y c r y s t a l l i n e 99.991 A 1 an i n t e r n a l - f r i c t i o n (IF) peak around 2 8 5 ' ~ ( f = 0.8 H Z )
which w a s a b s e n t i n f u l l y annealed bamboo s t r u c t u r e " s i n g l e c r y s t a l s "
/ l / . He t h e r e f o r e a t t r i b u t e d t h i s peak ("grain-boundary peakt1) t o a
r e l a x a t i o n p r o c e s s i n t h e g r a i n boundaries. Recently, Woirgard e t a 1
/2/ r e p o r t e d t h a t a n o t a b l e r e l a x a t i o n e f f e c t i s p r e s e n t i n s l i g h t l y
s t r a i n e d Cu s i n g l e c r y s t a l s i n t h e same temperature range a s t h e g r a i n boundary (GB) peak i n p o l y c r y s t a l s and a l s o a very weak r e l a x a t i o n
e f f e c t i n u n s t r a i n e d A 1 s i n g l e c r y s t a l s . They concluded t h a t t h e GB
peaks must be e x p l a i n e d by mechanisms which a r e n o t s p e c i f i c o f t h e GB
i t s e l f , b u t i n v o l v e more g e n e r a l l y t h e climb and g l i d e o f l a t t i c e d i s l o c a t i o n s . S i m i l a r p r o p o s i t i o n b a s been made by Gondi e t a1 / 3 / . They
r e p o r t e d t h a t s i n g l e c r y s t a l s o r m a c r o c r y s t a l l i n e s h e e t s o f 99.6 % A 1
p
d ~ p tr e s e n t , Guest P r o f e s s o r o f P h y s i c s , Groupe d l E t u d e s de M Q t a l l u r g i e
Physique e t Physique d e s Materiaux, INSA de Lyon, V i l l e u r b a n n e , France.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981567
C5-452
JOURNAL DE PHYSIQUE
show I F i n s t a b i l i t i e s i n t h e temperature range of t h e K& peak, and
s l i g h t deformations of t h e s e specimens cause t h e K& peak t o appear. The
p o s s i b i l i t y i s considered t h a t t h e K8 peak depends on f r e e o r polygonized d i s l o c a t i o n s i n s i d e t h e grains. The p r e s e n t r e p o r t w i l l suggest a
concrete d i s l o c a t i o n model i l l u s t r a t i n g t h a t t h e GB peak ( o r the socan be a s s o c i a t e d with a r e l a x a t i o n process i n t h e
c a l l e d "K&
GB i t s e l f .
It i s evident t h a t any r e a l i s t i c mechanism of t h e GB peak must conform t o t h e s t r u c t u r e of t h e high-angle g r a i n boundaries. The coincidence s u p p e r l a t t i c e model of high-angle GB r e c e n t l y proposed w a s very
s u c c e s s f u l i n e x p l a i n i n g many p r o p e r t i e s of high-angle g r a i n boundaries
/4/. A c l o s e approach t o t h e coherency condition can be achieved f o r
high-angle boundaries with a p e r i o d i c segmented s t r u c t u r e and t h e r e i s
a r e g i o n where t h e " f i t 1 ' i s r e l a t i v e l y good and one where it i s poor.
T h i s i s b a s i c a l l y s i m i l a r t o t h e e a r l y models put forward by K& /5/ and
by Mott /6/ f o r e x p l a i n i n g t h e assumed viscous behavior of high-angle
g r a i n bomdary i n a t o m i s t i c terms. It h a s been pointed out t h a t t h e coherency condition i s achieved by t h e c o n s t r a i n t applied t o each l a t t i c e
by t h e r e g i o n s of *good f i t n between t h e two l a t t i c e s . Disturbance of
t h e r e g u l a r i t y of t h i s h i g h l y constrained s t r u c t u r e , by t h e i n t r o d u c t i o n
of " m i s f i t n segments, can be e f f e c t i v e l y described i n terms of dislocat i o n s with Burgers vector n o t of a l a t t i c e t r a n s l a t i o n r e c t o r but of
t h e a p p r o p r i a t e high index i n t e r p l a n e r apacings /7/. This l e a d s natur a l l y t o a continuous-distribution d i s l o c a t i o n model of high-angle
g r a i n boundaries.
-
11. Continuous-Distribution D i s l o c a t i o n Model of High-Angle Grain
Boundaries and t h e Mechanism of Inhomogeneous S l i d i n g along t h e Grain
Boundaries.- Let u s consider an a r b i t r a r y high-angle GB as shown i n
Fig. 1 i n which AB i s t h e GB plane s e p a r a t i n g g r a i n 1 and g r a i n 2.
Assume t h e t h i c k n e s s of GB l a y e r be d and ro t h e atomic r a d i u s . We can
imagine t h a t when g r a i n 1 and g r a i n 2 approach each o t h e r , t h e f r o n t i e r
atoms of one g r a i n w i l l be acted by a m i s f i t f o r c e from t h e f r o n t i e r
atoms of t h e o t h e r g r a i n , so t h a t t h e s e atoms w i l l be displaced through
( i r e p r e s e n t s t h e i - t h atom) and give r i s e t o a c e r t a i n inXli and
t e r n a l streser. Eventually a s t a b l e high-angle GB i s formed a s an e q u i l i brium s t a t e i s reached. The GB l a y e r i s t h e r e f o r e equivalent t o a cont i n u o u s d i s t r i b u t i o u of a l t e r n a t i v e l y +ve and -ve ( i n t h e sense of
s t a t i s t i c a l average) edge-type and screw-type d i s l o c a t i o n s .
According t o Kr8ner /a/, t h e continuous d i s t r i b u t i o n d i s l o c a t i o n
d e n s i t y t e n s o r i n t h e GB l a y e r may be expressed as
d = n X Grad (3 ,U,),
(1)
-
--
-
where g i s t h e normal u n i t vector. Although t h e s p e c i f i c form o f o(.i s
unknown, b u t t h e mean v a l u e o f 6 a lo n g GB should be z e r o a c c o r d i n g t o
simple p h y s i c a l arguments, s o t h a t & = 0. Moreover,o(
-depends on t h e
d i r e c t i o n and magnitude o f t h e B u r g e r s v e c t o r and t h e d i r e c t i o n o f d i s l o c a t i o n l i n e which a l l have a random d i s t r i b u t i o n . We may c l a s s i f y t h e
GB d i s l o c a t i o n s i n t o f o u r t y p i c a l t y p e s a s shown i n Fig. 2. ( a ) +ve and
-ve edge d i s l o c a t i o n s w i t h Burgers v e c t o r (B) a l o n g i and d i s l o c a t i o n
l i n e (D) a l o n g j ; ( b ) +ve and -ve screw d i s l o c a t i o n s with B and D both
a l o n g j; ( c ) +ve and -ve edge d i s l o c a t i o n s w i t h B and D a l o n g k a n d j;
( d ) GB v a c a n c i e s which can be considered as d i p o l e s composed o f a p a i r
o f d i s l o c a t i o n s o f +ve and -ve c-type.
It can be seen from Fig. 2 t h a t t h e elementary p r o c e s s f o r GB s l i d i n g i s t h e s l i p o f t h e d i s l o c a t i o n s o f a- o r b-type and t h a t f o r GB m i g r a t i o n i s t h e climb o f t h e d i s l o c a t i o n s o f a-type o r t h e s l i p o f ct y p e . The elementary p r o c e s s o f GB d i f f u s i o n i s t h e climb o f d i s l o c a t i o n
d i p o l e s o f d-type.
111. I n t e r n a l F r i c t i o n g i v e n r i s e by t h e Inhomo~eneousS l i d i n g a l o n g
Grain Boundaries.- When a s h e a r s t r e s s 7 i s a p p l i e d a l o n g t h e GB i n t h e
d i r e c t i o n X, some o f t h e elementary d i s l o c a t i o n s o f a-type o r b-type can
overcome t h e p o t e n t i a l b a r r i e r t o move i n t o a neighbouring v a l l e y by
t h e r m a l a c t i v a t i o n . T h i s w i l l f a c i l i t a t e t h e neighbouring elementary
d i s l o c a t i o n s t o overcome t h e p o t e n t i a l b a r r i e r , s o t h a t t h e s l i p t a k e s
p l a c e i n t h e form of an "avalancheH i n v o l v i n g t h e group motion o f a
c h a i n o f elernetary d i s l o c a t i o n s . Such a c o o p e r a t i v e p r o c e s s w a s r e c e n t l y proposed by I s h i t a e t a1 / 9 / b a s i n g on t h e i r s t u d i e s o f GB d i f f u s i o n
i n c o l l o i d p o l y s t y r e n e l a t e x and gold s o l c r y s t a l s .
Consider t h e inhomogeneous s l i d i n g o f a flat and smooth GB l a y e r
e x t e n d i n g i n d e f i n i t e l y a l o n g t h e y - d i r e c t i o n and having a width
( ~ i g .2). Consider o n l y t h e case o f i s o t r o p i c m e t a l s s o t h a t t h e s l i d i n g ~on b o t h s i d e s o f t h e boundary a r e e q u a l and opposite. The inhomogeneous s l i d e u ( x , t ) a t t h e p o i n t X a l o n g GB s a t i s f i e s t h e equation:
w h e r e h i s t h e s h e a r modulus,D i s t h e P o i s s o n r a t i o , COi s t h e a p p l i e d
i s t h e a n g u l a r frequency and
i s t h e assumed v i s s t r e s s amplitude,
c o s i t y c o e f f i c i e n t a s s o c i a t e d w i t h t h e GB s l i d i n g . The f i r s t term on t h e
left-hand s i d e of Eqn. ( 2 ) i s t h e s h e a r s t r e s s produced a t X by t h e d i s l o c a t i o n s w i t h a continuous d i s t r i b u t i o n l i n e a r d e n s i t y o f 3 ( 2 u ) b x , t h e
second term i a t h e a p p l i e d s t r e s s , and t h e - t h i r d term i s t h e Newtonian
viscotas r e s i s t a n t s t r e s s . The boundary c o n d i t i o n f o r u(x) i s u(*R/2)=0.
et
= 2x/R,
(3)
3
JOURNAL DE PHYSIQUE
CS-454
V = [~NT
l-~)J?%]ue
(
-imt ;
c = -n(l-')l)~~r//~d,
and
Eqn.
(
2
)
can
be
w
r
i
t
t
e
n
a
s
then
I d v dg'
l + C l i = 0 and V (21) = 0.
d.g' 3 5'
Now l e t u s t r y t o f i n d an approximate s o l u t i o n of t h e form
(4,5)
- -
V = q2/=/(1+~,,/
l - c i ).
(8)
Obviously t h i s s o l u t i o n s a t i s f i e s t h e boundary conditions given by Eqn.
(7). S u b s t i t u t i n g Eqn. (8) i n t o Eqn. ( 6 ) and l e t
= 0, we g e t on putt i n g t h e r e a l p a r t and t h e imaginary p a r t equal t o zero r e s p e c t i v e l y :
5
2~
+
43)) c,
( l / , , / 7 ) l o g c ( - t + p 2 + pJ, T ) / ( l + p 2 - p
( ~ + p ~ ) ' //( ~
i ~ + p c , , / T ) = q.
The a n e l a s t i c s t r a i n &a a s s o c i a t e d with t h e GB s l i d i n g i s
2udx = TT (1-2')q
&a
The t o t a l s t r a i n i s E
/v =
E; + E ; , where E, =
(9)
=
4
iio)
1
Vd?jeiWt.
z0/',( t h e
( 11)
elastic strain).
The i n t e r n a l f r i c t i o n and modulus d e f e c t can be obtained from t h e ima(I
l
g i n a r y and t h e r e a l p a r t r e s p e c t i v e l y from (f2/E1) = fa/(%/&) and We
have
Q-'
=
-1m(EaM/C,)
= [IT (I-y)/2
1
[
(4~-c)4/P2]
,
(12)
(
IS)
I n Fig. 3, Q-' and AM/M ( o r ~p/ht)axe p l o t t e d a s f u n c t i o n of C,
t h e u n i t of t h e o r d i n a t e has been chosen a s y( 1-21) /2. The d o t t e d curves
r e p r e s e n t t h e corresponding curves f o r s i n g l e r e l a x a t i o n time. It can
be seen t h a t a peak appeaxs a t
C =~(l-~),fu)~
= / 4.269,
~d
(34)
and t h e h e i g h t of t h e peak i s
Q
&
c 0.2971(1-V) /2.
(15)
The r e l a x a t i o n s t r e n g t h A l y can be determined from t h e extreme values
taken from t h e AM/M-C curve i n Fig. 3 a s
=AM/M( f o r GO) -AM/M ( f o r C=oo) = 0.5(1-Y) /2.
(16)
AM
Q,L
R o r t h i s r e get
= $ h M= 0.25 (I-Y) 12, which is close t o t h e
value determined d i r e c t l y from t h e Q-'-c curve.
1V.- Comparison with Experiments. Assuming t h a t only t h e GB d i s l o c a t i o n s
n e a r t h e GB vacancies a r e f e a s i b l e t o s l i d i n g , then t h e f o r c e a c t i n g on
a d i s l o c a t i o n segment of average l e n g t h a i s Z a b where b i s t h e t o t a l
Burgers vector of t h e s e mobile d i s l o c a t i o n s . I f we apply E i n s t e i n ' s
r e l a t i o n s h i p approxilnately t o t h e p r e s e n t case, t h e n we have t h e average
s l i d i n g r a t e v,, = ~ ~ Z a b / k Twhere
,
Db i s t h e c o e f f i c i e n t of GB d i f f u s i o n .
Since by d e f i n i t i o n , 7 = Z / ( v i l / d ) , we g e t f i n a l l y
= 'iTd/v,, = k T d / a b ~=~ kT/2roDb,
(17)
on assuming t h a t a = b = d = 2r0. Thus t h e v i s c o s i t y
f o r GB s l i d i n g i s
c o r r e l a t e d w i t h t h e d i f f u s i o n c o e f f i c i e n t along g r a i n boundaries. I f Dbo
i s t h e d i f f u s i o n c o n s t a n t and H t h e a c t i v a t i o n e n t h a l p y a s s o c i a t e d with
GB d i f f u s i o n , t h e n we have
Tpexp(H/Tp) = 4 . ~ 6 9 ~ ( 2 r o ) 2 ~ b o / ~k ( 2~n-f )~. )
(18)
where T i s t h e peak temperature o f t h e GB I F peak and f i s frequency o f
P
v i b r a t i o n . The T v a l u e s c a l c u l a t e d a c c o r d i n g t o Eqn. (18) f o r a number
D
o f pure m e t a l s &e summerieed i n Table 1 w i t h e x p e r i m e n t a l values. The
v a l u e s o f Dbo and H were i n g e n e r a l t a k e n from Ref./lO/. Data f o r Al,Cu,
Au were e s t i m a t e d a c c o r d i n g t o t h e e m p i r i c a l formula given i n ~ e. / lf l / .
I n c a s e s when 1 and f a r e n o t known, we assumed t h a t f = 1Hz and 1 =
0.05 mm t a k i n g account o f t h e f a c t t h a t Q may be s m a l l e r t h a n t h e a c t u a l
g r a i n s i z e i f t h e boundary i s n o t f l a t and smooth enough so t h a t l e d g e s
and p r o t r u s i o n s may a c t as o b s t r u c t i n g s i t e s f o r GB s l i d i n g . A s t h e o r i g i n s of t h e e x p e r i m e n t a l d a t a a r e q u i t e d i v e r s i f i e d and t h e e x p e r i m e n t a l
v a l u e s a r e v a r i e d a c c o r d i n g t o e x p e r i m e n t a l c o n d i t i o n s and specimen puri t y , t h e agreement between e s t i m a t e d v a l u e s and e x p e r i m e n t a l v a l u e s
shown i n Tab. 1 seems t o be q u i t e s a t i s f a c t o r y . For A l , Sn and some
o t h e r m e t a l s , t h e e s t i m a t e d v a l u e s a r e much lower t h a n t h e experimental
v a l u e s and t h e s e may be connected with an i m p u r i t y e f f e c t .
Tab. l . Comparison between e s t i m a t e d and e x p e r i m e n t a l v a l u e s o f T
P
Metals
Ag
estimated
426
T,
(K)experimental 440
Fe
638
704
Sn
139
740
W
Zn
MO
1587
1500
2'36
1365
1325
323
Ta
1590
.l400
A1
385
560
Cu
528
500
Au
478
455
V.- Discussion. I n m e t a l s c o n t a i n i n g s o l u b l e i m p u r i t i e s , it i s w e l l
known t h a t t h e r e i s a s e l e c t i v e s e g r e g a t i o n a l o n g c e r t a i n boundasies
/12/. If t h e v i s c o s i t y a s s o c i a t e d with t h e inhomogeneous s l i d i n g i n t h e
pure s o l v e n t r e g i o n of t h e boundary i s q l , and t h a t i n t h e s o l u t e r e g i o n
i s T 2 , t h e n Eqn. ( 6 ) s t i l l h o l d s but C w i l l depend upon 4 . T h i s w i l l
g i v e r i s e t o a s o l u t e I F peak a t a temperature above t h e o r i g i n a l s o l i s l a r g e r t h a n ql. The s o l u t e peak w i l l appear at a temv e n t peak i f
p e r a t u r e below t h a t of t h e s o l v e n t peak o n l y when p r e c i p i t a t i o n h a s
o c c u r r e d a t t h e g r a i n boundaries i n which c a s e q2 may be s m a l l e r t h a n
7,. T h i s may be t h e case observed i n Cu c o n t a i n i n g Bi /13/.
Although d i s l o c a t i o n s o f a-type and c-type shown i n Fig. 2 can
climb o r s l i p t o g i v e r i s e t o GB m i g r a t i o n , b u t a c t u a l l y t h e n e t migrat i o n i s z e r o s i n c e t h e r e - a r e e q u a l numbers o f p o s i t i v e and n e g a t i v e d i s l o c a t i o n s i n t h e boundaries. However, i n h i g h l y a n i s o t r o p i c pure m e t a l s
as Mg,Zn,etc.,or i n t h e presence o f i n t e r n a l s t r e s s (e.g.the occurrence
o f e x c e s s i v e d i s l o c a t i o n s n e a r GB l a y e r ) , t h e e l a s t i c modulus a d t h e
C5-456
JOURNAL DE PHYSIQUE
s t r e s s a r e d i f f e r e n t a t t h e two s i d e s o f t h e boundaries and it can be
shown t h a t t h e m i g r a t i o n r a t e o f t h e boundary toward t h e s i d e having a
s m a l l e r s h e a s modulus (e.g./$)
under t h e a c t i o n o f a s h e a r s t r e s s i s
'C2ab
v, = ( D b h T ) (1/M1-
* v/,
(l/,U1-
i/&)7<<v
.
(19)
T h i s s t r e s s - a s s i s t e d GB m i g r a t i o n r a t e v, may cause a h i g h temperature
I F background o r a n o t h e r GB peak which i s amplitude dependent. But i t
can be seen from Eqn.(lg) t h a t t h e c o n t r i b u t i o n o f GB m i g r a t i o n t o GB
i n t e r n a l f r i c t i o n should be much s m a l l e r t h a n t h a t o f GB s l i d i n g .
References
/ l / T.S. K k , Phys. Rev., 71,533(1947).
/2/ J. Woirgard, J. ~ m i r a r i tand J. de Pouquet, Proc. 5-ICIFUACS
(Aachen, 19751, Vol.1, p.392.
/3/ E. B o n e t t i , E. Evangelists, P. Gondi and R. T o m a t o , I1 Nuovo
Cimento, 3JB,408(1976).
/4/ H. G l e i t e r and B. Chalmers, High-Angle Grain Boundaries, i n "Prog r e s s i n M a t e r i a l s Science (Pergamon P r e s s , 1972), Vol. 16.
/5/ T.S. xQ, J. appl. Phys., 20, 274(19 9).
/6/ R.F. Mott, Proc. Phys. SO^, 60,991q1948).
/7/ ~ e f . / 4 / , p.12.
/8/ E. KriSner, Kontinumstheorie d e r Verzetzung und I n n e r Spannung
( s p r i n g e r Verlag, 1955).
/g/ P. I s h i d a , S. Okamoto and S. Hachiza, Acta m e t a l l . , 2,651 (1978).
/10/ ~ e f . / 4 / , p.93, p:222.
/ I l / N.A.
G j o s t e i n , Diffusion (ASM, Cleveland, Ohio,, 1973), p.234.
/12/ H. S a n t t e r , H. G l e i t e r and G. Baro, Acta m e t a l l . , 3 , 4 6 7 ( 1 9 7 7 ) .
/13/ T.S. K k ,
appl. Phys., 20,1227(1949).
J.
Fig.
An a r b i t r a r y high-angle
g r a i n boundary.
Fig. 2. Grain-boundary s l i d i n g
(upper) and f o u r t y p e s o f g r a i n
boundary d i s l o c a t i o n s (lower).
Fig. 3. Q-' v s C a n d a ;!/M
curve S.
vs C