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On the achievement of strength at high temperature in binary alloys
Scripta
METALLURGICA
Vol. 23, pp. 721-726, 1989 Printed in the U.S.A.
Pergamon
Press
plc
ON THE A C H I E V E M E N T OF STRENGTH AT HIGH TEMPERATURE IN BINARY ALLOYS
N. P. Louat* and M. A. Imam Naval Research Laboratory Washington, DC 20375-5000 (Received D e c e m b e r 28, 1987) (Revised F e b r u a r y 21, 1989) INTRODUCTION It has recently been p o i n t e d out [i] that the strengthening encountered in polycrystals, the Hall-Petch effect, can also be expected in materials which are not n e c e s s a r i l y p o l y c r y s t a l l i n e but, insofar as the effect is concerned, are similar to them. Specifically considered were two-phase alloys in which the m a t r i x forms the minor constituent and in which the other, major phase, is particulate. Such alloys are to be considered as equivalent to p o l y c r y s t a l s if the matrix is so thin that, at the stress involved, it cannot, independent of the particles, deform plastically. This restriction is also available in circumstances where the particles are replaced by rods or plates, effectively infinite in length. The operation of the H a l l - P e t c h effect in such materials in which the particle size is now disposable in practice down to diameters conveniently m e a s u r e d in nanometers, allows the p o ss i b i l i t y of material strengths ap p r o a c h i n g the t h e o r e t i c a l limit. It has also been n o t e d [I] that the p h e n o m e n o n of dilatancy which is responsible for the firmness of wet beach sand [2] may have metallurgical applications; a direct analogy being a packed dispersion of particles (each contacts ~ six others) in a metal matrix. If this analogue is to have important c o n s e q u e n c e s at temperatures so high that the matrix approaches its melting point or is a c t u a l l y molten it is necessary that the particles be strong at these t e m p e r a t u r e s and be very much smaller than ordinary sand grains. The essential feature of dilatancy is that the total volume of the interstices b e t w e e n particles, which are essentially densely packed and individually rigid, expands upon deformation. When this volume is occupied by a continuous fluid, such expansion must result either: in the formation of voids; a reduction in (compressive) pressure or; in an inward flow from an external surface. W h e n the fluid is liquid, formation of voids is resisted by its cohesion; inward flow by the development at the outer surface of a pressure: p ~ 2y cose/r h
(I)
where ~ is the surface energy of contact between particle and liquid, e the contact angle and r h the radius of the hole through w h i c h the liquid must flow. Purely elastic b e h a v i o r of the whole follows from that of the *On-site contractor for Geo-Centers, Ft. Washington, MD 20744
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p a r t i c l e s at all a p p l i e d s t r e s s e s less t h a n p. To e x a m i n e t h e m a g n i t u d e of p, w e s u p p o s e y to b e of t h e o r d e r o f t h e s u r f a c e e n e r g y o f t h e m a t r i x a n d ~a -i0 w h e r e ~ is s h e a r m o d u l u s a n d 'a' t h e l a t t i c e p a r a m e t e r d i s t a n c e . Then, (I) w i t h 8 ~ 0: p ~ ~ / I 0 4 ,r h ~ 1 ~m; p ~ ~ / i 0 2 ,r h ~ .01 ~Lm. thus t o b e about:
from
The l a t t e r s t r e s s is t e c h n i c a l l y l a r g e in m o s t cases. T h e p o s s i b i l i t y that m a t e r i a l s c o u l d h a v e s u c h s t r e n g t h n e a r or a b o v e a r e l e v a n t m e l t i n g p o i n t is unique to this approach. In p r a c t i c e t h e a c h i e v e m e n t of p a c k i n g so c l o s e as to s a t i s f y t h e r e q u i r e m e n t s c i t e d a b o v e m a y b e d i f f i c u l t to a c h i e v e . T h i s is t h e first c o n c l u s i o n of a n a t t e m p t at e x p e r i m e n t a l c o n f i r m a t i o n of t h e s e ideas. The s y s t e m e m p l o y e d w a s Fe-Hg. T h e r e s u l t s o b t a i n e d a r e d e t a i l e d below. H e r e we n o t e t h a t t h e l a r g e s t p a c k i n g a c h i e v e d (~66% Fe) was i n s u f f i c i e n t to a c h i e v e a r i g i d p a r t i c l e s k e l e t o n so t h a t d e f o r m a t i o n of t h e m a t r i x a l o n e w a s possible. D e f o r m a t i o n u n d e r t h e s e c i r c u m s t a n c e s is a c c o r d i n g l y t h e c o n c e r n of t h e a c c o m p a n y i n g a n a l y s i s . ANALYSIS F o r s i m p l i c i t y w e s e p a r a t e the d i s c u s s i o n i n t o t w o realms, c h a r a c t e r i s a b l e b y t e m p e r a t u r e , a n d d e a l f i r s t w i t h t h e e f f e c t s of p a c k i n g d e f i c i e n c y w h e n t h e t e m p e r a t u r e is so l o w t h a t t h e e f f e c t s d u e to d i f f u s i o n can be n e g l e c t e d . In t h e s a i d c i r c u m s t a n c e s we can s u p p o s e t h a t i n d e p e n d e n t p l a s t i c d e f o r m a t i o n of t h e m a t r i x can o c c u r b y b e n d i n g d i s l o c a t i o n s i n t o s e m i c i r c l e s of r a d i u s L/2. T h i s w i l l occur, p r o v i d e d t h e r e e x i s t s an a p p l i e d s h e a r stress:
~b o =
--.
(2)
L
Intuitively,
it
is
clear
that
L will
decrease
with
decreasing
particle
r a d i u s a n d w i t h i n c r e a s i n g d e n s i t y of p a c k i n g . T h e m a n n e r of t h i s d e c r e a s e can b e e x p e c t e d to d e p e n d on t h e s e p a r a m e t e r s in a c o m p l e x way. Rather than u n d e r t a k e a d e t a i l e d a n a l y s i s o f t h i s q u e s t i o n , w e r e c o g n i z e t h a t for o u r present purposes, namely, to recognize trends, a simplified approach dealing o n l y w i t h a v e r a g e v a l u e s is s u f f i c i e n t . F o l l o w i n g O r o w a n w e i d e n t i f y L as the s p a c e b e t w e e n p a r t i c l e s a n d a s s u m e t h e p a r t i c u l a t e p h a s e to be s p h e r i c a l w i t h r a d i u s r a n d to r e p r e s e n t a v o l u m e f r a c t i o n f. We then find that the m e a n d e n s i t y of p l a n a r c i r c u l a r i n t e r s e c t i o n s h a v i n g a v e r a g e a r e a 2Kr2/3 is 3f/2Kr 2 with average
Kr r a d i u s ~-.
Thence,
we obtain the mean
interparticle
spacing:
L-r[
~2~ ~-~]~
N u m e r i c a l v a l u e s o b t a i n e d f r o m t h e u s e of t h i s f o r m u l a a r e to b e a c c e p t e d o n l y w i t h c a u t i o n , b u t it w o u l d s e e m to b e c o r r e c t in p r e d i c t i n g a l i n e a r r e l a t i o n b e t w e e n L a n d r a n d in i n d i c a t i n g t h a t L << r at v o l u m e f r a c t i o n s w h i c h a p p r o a c h t h a t of c l o s e p a c k e d s p h e r e s : f=K/3~2. Accepting its v a l i d i t y w e f i n d w h e n f = . 6 6 t h a t L / r = 0 . 2 a n d so w e h a v e f r o m (2) 0
~
~/1400,
However, same modulus.
r =
2 ~Lm
(3)
it is a s s u m e d in e q u a t i o n (2) t h a t m a t r i x a n d p a r t i c l e Here, w h e r e it is to b e e x p e c t e d t h a t t h e m o d u l i a r e
h a v e the
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s i g n i f i c a n t l y d i f f e r e n t , a c o r r e c t i o n is n e c e s s a r y . S e e m i n g l y , an a p p r e c i a t i o n of its m a g n i t u d e can be o b t a i n e d s i m p l y f r o m a c o n s i d e r a t i o n of the b e h a v i o r of four s t r a i g h t p a r a l l e l s c r e w d i s l o c a t i o n s , the o u t e r p a i r of s e p a r a t i o n , 2 L - L 1 r e p r e s e n t the n e a r i m a g e s of the i n n e r pair, s e p a r a t i o n LI, in the s u r f a c e s h a v i n g s e p a r a t i o n L at w h i c h the e l a s t i c m o d u l u s c h a n g e s from ~I to ~2 as shown in Fig. i. W e c o n s i d e r the e q u i l i b r i u m of the i n n e r p a i r u n d e r the a c t i o n ~ib 2~
[L~ +
of a s h e a r ~(2L-LI) ] L(L-LI)
stress ~ and write = ~; e = ~I-B____~2 i. ~i+~2
(4)
Then, p r o v i d i n g the s e p a r a t i o n of a d i s l o c a t i o n a n d its image is m u c h g r e a t e r t h a n its core radius, it is e a s i l y s e e n that the s t r e s s n e c e s s a r y for p a s s a g e is g i v e n by -- ~ b a =~whence
(i
+ ~(2L-LI) ) ~ ~ib
L1
L(L_LI )
for ~ = 1 we
find:
(5)
L1 L 1 =3L/4
so that
(3) b e c o m e s
c = ~/i000
Here, we h a v e n e g l e c t e d t h e e f f e c t s of all but first i m a g e s in the nearest interfaces. The e r r o r so i n v o l v e d w o u l d s e e m small, c e r t a i n l y a c c e p t a b l e in t h e c u r r e n t c o n s i d e r a t i o n of g r o s s m a g n i t u d e s , w h e n as here the r a t i o of the s p a c i n g b e t w e e n d i s l o c a t i o n s a n d i n t e r f a c e s to t h a t b e t w e e n i n t e r f a c e s is s m a l l (~1/8). The e f f e c t s due to m u l t i p l e l a y e r i n g (implicit in p r e s e n c e of p a r t i c l e s ) can a l s o be e x p e c t e d to be s m a l l [3]. A p a r t f r o m m i s f i t s h e a r s t r e s s e s w h i c h are p r o b a b l y of h y d r o s t a t i c o r i g i n ~ n e g l e c t e d s i n c e t h e y w o u l d fall off r a p i d l y (inverse cube of d i s t a n c e ) , t h e s e c o n s i d e r a t i o n s a p p e a r to e x h a u s t the p o s s i b i l i t i e s of direct i n h i b i t i o n of d i s l o c a t i o n m o t i o n . T h e r e r e m a i n s the p o s s i b i l i t y of a s p e c i a l type of w o r k h a r d e n i n g . The q u e s t i o n of the h a r d e n i n g i n d u c e d by the p r e s e n c e of p a r t i c l e s w h i c h are e s s e n t i a l l y u n p e n e t r a b l e to l a t t i c e d i s l o c a t i o n s has b e e n e x a m i n e d p r e v i o u s l y , e.g. B r o w n [4], but o n l y for the case of d i l u t e d i s p e r s i o n s . In t h e s e c i r c u m s t a n c e s c r o s s s l i p w a s f o u n d to be i m p o r t a n t in r e d u c i n g h a r d e n i n g b e l o w its p o t e n t e n t i a l . The e f f i c a c y of the m e c h a n i s m i n v o k e d may be t r a c e d to t h e fact that the n e c e s s a r y i n c r e a s e in d i s l o c a t i o n l e n g t h i n v o l v e d in b y - p a s s i n g a p a r t i c l e is r e l a t i v e l y small. S u c h s h o u l d not be so here, w h e r e t h e p a r t i c l e c o n c e n t r a t i o n is high. R a t h e r we w o u l d e x p e c t that p l a s t i c d e f o r m a t i o n of the m a t r i x w o u l d by a n d large, i n d u c e d by c i r c u m s c r i b i n g d i s l o c a t i o n , r e s u l t in an e l a s t i c d e f o r m a t i o n of the p a r t i c l e s of like amount. T h e a c c o m p a n y i n g r e d u c t i o n in s t r e s s in the m a t r i x m a y be r e g a r d e d as a w o r k h a r d e n i n g effect. For t h i s h y p o t h e s i s to be v a l i d it is at least n e c e s s a r y that the p a r t i c l e s b e h a v e in a q u a s i - i m p e n e t r a b l e fashion. To e x a m i n e this p o s s i b i l i t y we c o n s i d e r t h e c o n d i t i o n s n e c e s s a r y for t h e p r o p a g a t i o n of d e f o r m a t i o n f r o m t h e m a t r i x to t h e p a r t i c l e s . This we can e s t i m a t e u s i n g the Hall-Petch relation =
~o
+
k
d -I/2 .
(6)
To e s t i m a t e k w e r e f e r to the e x p r e s s i o n g i v e n b y L o u a t a c t i n g on t h e h e a d of a p i l e - u p l e n g t h w h i c h t e r m i n a t e s e l a s t i c m o d u l u s ~i t o ~216Em2~ 2 a Thus,
F =
~ B~2a, ~ (l+cos2~m)
1 where m = ~
cos-le
[I] for the force at a d i s c o n t i n u i t y
in
724
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Substitution the s t r e s s e s ~2) n a m e l y ~e =
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o f t h i s e x p r e s s i o n in t h e e q u a t i o n g i v e n in t h e s a m e p a p e r for n e c e s s a r y for a c t i v a t i o n of s l i p in t h e s e c o n d m a t e r i a l (modulus
~/PlP2 ~aa'
Oe = g2
when,
OF B I N A R Y
w h e r e A is a c o n s t a n t
results
" PFe
as here,
m - ~
in: (7)
E ~
~Fe"
is small. A s i g n i f i c a n t f e a t u r e of (7) is t h a t t h i s c r i t i c a l s t r e s s is i n d e p e n d e n t of t h e m o d u l u s of m e r c u r y . A c c o r d i n g l y , we m a y e m p l o y (6) w i t h a v a l u e for k a p p r o p r i a t e t o iron. D o i n g so a n d s e t t i n g a in (6) as 2 ~ m we find that
ac ~ 160
a value
very much larger than that
given
in
(3).
W e n o w t u r n to t h e c o n s e q u e n c e s o f i m p o r t a n t d i f f u s i o n a l e f f e c t s . First, d i s l o c a t i o n climb. T h e r a p i d w o r k h a r d e n i n g we h a v e i n v o k e d c a n n o t be e x p e c t e d w h e n d i s l o c a t i o n s c l i m b readily. We therefore anticipate s u b s t a n t i a l d e c r e a s e s in s t r e n g t h as t h e t e s t t e m p e r a t u r e p r o g r e s s i v e l y e x c e e d s a b o u t .50% h o m o l o g o u s . T u r n i n g t o d i r e c t e f f e c t s o f d i f f u s i o n we r e m e m b e r t h a t d i f f u s i o n a l c r e e p r e s t s on t h e t r a n s p o r t of m a t t e r b e t w e e n g r a i n b o u n d a r i e s a n d that this p r o c e s s c a n b e e x p e c t e d [1] to s e l f - t e r m i n a t e b y t h e a g g r e g a t i o n of p a r t i c l e s into r i g i d a s s e m b l i e s at b o u n d a r i e s w h i c h act as s i n k s for v a c a n c i e s . This has b e e n e s t i m a t e d [I] t o o c c u r at a strain! ~c = 2 A r / G w h e r e G r e p r e s e n t s g r a i n d i a m e t e r a n d A is t h e c h a n g e in m a t r i x c o n c e n t r a t i o n n e c e s s a r y to 74-66 achieve close Packing. Here, A ~ i0------~" M e t a l l o g r a p h i c d i f f i c u l t i e s h a v e p r e v e n t e d a m e a s u r e m e n t of G in t h e c u r r e n t experiment. G e n e r a l e x p e r i e n c e w o u l d s u g g e s t t h a t G is m u c h l a r g e r t h a n 2 ~ m so t h a t ec s h o u l d n o t b e large. For values which may be thought of a typical: G > 100 ~ m w e h a v e ec < - 0.3%. EXPERIMENTAL
The s y s t e m F e - H g w a s c h o s e n as t h e v e h i c l e for a f i r s t t e s t o f s u c h materials because: m e r c u r y is i m m i s c i b l e w i t h i r o n a n d m e l t s n e a r r o o m temperature. A f a c t o r of e q u a l i m p o r t a n c e is t h a t i r o n p a r t i c l e s c a n be i n t r o d u c e d i n t o m e r c u r y w i t h r e l a t i v e ease. T h u s i r o n p a r t i c l e s (4 ~m) of 99.5% p u r i t y w e r e i m m e r s e d in a w e a k l y a c i d u l a t e d (HCI) s o l u t i o n t o g e t h e r w i t h a q u a n t i t y of m e r c u r y . The subsequent addition of mercurous chloride allowed the wetting and absorption of the p a r t i c l e s i n t o t h e m e r c u r y . T h e r e s u l t a n t , e s s e n t i a l l y solid, m a t e r i a l w a s t h e n c o m p a c t e d b y h a n d a n d frozen. In t h i s w a y s m a l l s p e c i m e n s lcc c o n t a i n i n g r o u g h l y e q u a l v o l u m e f r a c t i o n s o f t h e t w o c o n s t i t u e n t s w e r e produced. T h e f l o w c h a r a c t e r i s t i c s of t h e s e H g - F e a l l o y s w e r e d e t e r m i n e d u s i n g the s o - c a l l e d i m p r e s s i o n t e s t m e t h o d [5] a n d its a s s o c i a t e d a n a l y s i s . Here, s t r e s s - s t r a i n r e l a t i o n s are o b t a i n e d t h r o u g h s i m u l t a n e o u s a n d p r e c i s e m e a s u r e m e n t s o f a p p l i e d l o a d a n d the r e s u l t a n t p e n e t r a t i o n of a f l a t - e n d e d circular indentor into a specimen. T h e (tungsten carbide) i n d e n t o r is c o n s t r a i n e d t o m o v e in a d i r e c t i o n n o r m a l to t h e (flat) s u r f a c e of t h e specimen. T e s t s w e r e c a r r i e d o u t o v e r a r a n g e of t e m p e r a t u r e s d e r i v e d f r o m t h e use of d r y ice a n d l i q u i d n i t r o g e n on s m a l l s p e c i m e n s in a h y d r a u l i c t e s t i n g
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machine. The d i s p l a c e m e n t of t h e p u n c h was f o u n d f r o m the o u t p u t o f a LVDT (linear-variable-differential-transformer) a t t a c h e d b e t w e e n t h e l o a d cell and specimen. This a n d t h e c o r r e s p o n d i n g l o a d s i g n a l w e r e d i s p l a y e d on an x - y recorder. The i n d e n t o r s p e e d w a s c h o s e n to b e 8.5 x 10 -7 m/s. This s p e e d is e q u i v a l e n t to a t r u e s t r a i n [5] rate o f 1 . 2 7 5 x 10-3/s. T h e o r y [6] a n d e x p e r i m e n t [5,7,8] are in a c c o r d in p r o v i d i n g a p p r o x i m a t e c o r r e l a t i o n s b e t w e e n t r u e t e n s i l e stress, z, a n d load, P, on t h e one h a n d and true strain, £, a n d p e n e t r a t i o n ~ on t h e other, t h e s e are, w i t h e r r o r less than 10%: ~ = P/3; £ = 6/d, w h e r e d is the d i a m e t e r of the i n d e n t o r . Fig. 2 s h o w s a p l o t of t r u e s t r e s s v e r s u s t r u e s t r a i n for r e p r e s e n t a t i v e data t a k e n at -II0°C a n d at -62°C u s i n g t h e s e c o r r e l a t i o n s . From these curves we f i n d a y i e l d s t r e s s of ~y = 43.5 M P a (-I10°C) a n d 2 4 . 1 3 M P a (-62°C). DISCUSSION
AND CONCLUSIONS
It is a p p a r e n t f r o m t h e w i d e v a r i a t i o n in s t r e n g t h b e t w e e n r o o m t e m p e r a t u r e a n d -II0°C s h o w n by t h i s m a t e r i a l that the p r i n c i p l e of d e t a i l e d c o n t i n u i t y of d e f o r m a t i o n p r e v i o u s l y e l a b o r a t e d [I] d o e s not a p p l y at all temperatures. Thus, w h e n the m e r c u r y is m o l t e n the r e s t r a i n t a f f o r d e d b y the iron p a r t i c l e s is s m a l l b u t l a r g e e n o u g h for the s p e c i m e n s to s t a n d alone. The p r o c e s s i n v o l v e d h e r e is not c l e a r but it w o u l d s e e m that d i f f u s i o n w o u l d be of d o m i n a t i n g i m p o r t a n c e at t e m p e r a t u r e s just b e l o w t h e m e l t i n g point. W i t h p r o g r e s s i v e l y l o w e r t e m p e r a t u r e s the s t r a i n rate a c h i e v a b l e b y d i f f u s i o n a l o n e w i l l e v e n t u a l l y fall b e l o w the rate i m p r e s s e d . In this r e a l m the n e e d for d e t a i l e d c o n t i n u i t y of d e f o r m a t i o n will be i n c r e a s i n g l y i m p o r t a n t . T o w a r d s a n a p p r e c i a t i o n of the s i g n i f i c a n c e of t h e r e s u l t s of t h e s e m e a s u r e m e n t s we r e l a t e t h e m to t h e e l a s t i c m o d u l i of t h e c o n s t i t u e n t s . E f f o r t s to f i n d d a t a r e l a t i n g to m e r c u r y h a v e b e e n u n a v a i l i n g . However, h a v i n g r e g a r d to t h e g e n e r a l l y s y m b a t i c c o r r e l a t i o n b e t w e e n m e l t i n g p o i n t and e l a s t i c m o d u l u s in m e t a l s we e x p e c t that, that of m e r c u r y w i l l b e less t h a n that of I n d i u m s i n c e t h e l a t t e r m e l t s at 156.4°C t h e f o r m e r at -389°C. The m o d u l u s o f i n d i u m is ii x 103 M P a t h a t of m e r c u r y m i g h t w e l l be ~ 6.89 x 103 M P a w h i c h w o u l d g i v e a g r e e m e n t w i t h i n a f a c t o r of two b e t w e e n the v a l u e of ~y at -ll0°C a n d t h e e s t i m a t e of ~c from ~g. S u p p o r t for t h e u s a g e of s u c h a v a l u e is a v a i l a b l e f r o m the m e a s u r e d v a l u e of t h e s l o p e of t h e l i n e a r p o r t i o n of s t r e s s - s t r a i n r e l a t i o n f o u n d at I10°C. thus, t h i s s l o p e is 10.3 x 103 MPa, in a c c o r d w i t h a c r u d e formulation using'a rule of mixtures. On this b a s i s t h e m o d u l i are d i s p a r a t e ; the i n d i c a t e d r a t i o is 30. No s u c h r a t i o a p p e a r s to e x i s t b e t w e e n a n y two m a t e r i a l s w h i c h m i g h t b e i n c l u d e d as p r a c t i c a l . H o w e v e r , for the case of Ni-W, w h i l e o n l y 6:1, this ratio is n e v e r t h e l e s s s i g n i f i c a n t l y l a r g e r t h a n unity. The i n a b i l i t y to f i n d an exact m a t c h in this r e g a r d is p r o b a b l y not i m p o r t a n t . The f e a t u r e of s i g n i f i c a n c e is that p l a s t i c d e f o r m a t i o n lies in t h e r e a l m of h i g h t e m p e r a t u r e for m e r c u r y a n d low for iron. T h i s d i s p a r i t y is s h a r e d b y the c o m b i n a t i o n N i - w t e s t e d at ~ 1300°C. Thus, w e s u p p o s e that the b e h a v i o r of a l l o y s c o n s t i t u t e d r e s p e c t i v e l y of n i c k e l a n d m e r c u r y is the same w h e n the h o m o l o g o u s t e m p e r a t u r e a n d i m p o s e d e l a s t i c s t r a i n s are the same. S u c h an i d e n t i t y of e l a s t i c s t r a i n s i n d i c a t e s the use of a s c a l i n g f a c t o r o f = 30 (~Ni/~Hg) in a s s e s s i n g y i e l d stress. Again, at the h i g h e r of the two, the h o m o l o g o u s t e m p e r a t u r e w a s 0.9 (Hg). A p p l y i n g t h e s e f a c t o r s we f i n d a t e n s i l e y i e l d s t r e s s o f (9.71 x 108 Pa) at 1280°C for a h y p o t h e t i c a l N i - W a l l o y in w h i c h the t u n g s t e n is p a r t i c u l a t e at 4 ~ m d i a m e t e r a n d c o n s t i t u t e s = 6 6 % b y volume. Such s t r e s s e s : ~ ~ / 2 0 0 at 90% of the m e l t i n g p o i n t of t h e m a t r i x are significantly higher than those previously achieved through more conventional approaches.
726
STRENGTH OF BINARY ALLOYS
Vol.
23,
No.
REFERENCES 1.
2. 3. 4. 5.
6. 7. 8.
N. P. Louat, Acta. Met., 33, 59 (1985). O. R e y n o l d s , N a t u r e , 33, 429 (1886). S. V. Kamat, J. P. H i r t h a n d B. C a r n a h a n , S c r i p t a Met. 21, 1587 (1987). L. M. B r o w n a n d w. M. Stobbs, Phil. Mag. (GB) Vol. 23, No. 185, 1201-33, M a y 1971. H. Y. Yu, M. A. Imam, a n d B. B. Rath, J. Mat. Sc., 20, 636 (1985). D. Tabor, Proc. R. Soc. A192, 247 (1948). H. O ' N e i l l a n d H. G r e e n w o o d , J. Inst. Met., 48, 47 (1932). E. C. Yu a n d J. C. M. Li, Phil. Mag. 32, 811 (1977).
4-
+
<
Lt -----~
L
Fig.
1
Four straight
"
parallel
screw dislocations.
-110°C
50
-I~.oc
i
'm w-
3o
W
20
10
o
~'6
STRAIN (10-3) Fig.
2
True
stress
versus
true
strain
curves obtained
by impression
test.
5
METALLURGICA
Vol. 23, pp. 721-726, 1989 Printed in the U.S.A.
Pergamon
Press
plc
ON THE A C H I E V E M E N T OF STRENGTH AT HIGH TEMPERATURE IN BINARY ALLOYS
N. P. Louat* and M. A. Imam Naval Research Laboratory Washington, DC 20375-5000 (Received D e c e m b e r 28, 1987) (Revised F e b r u a r y 21, 1989) INTRODUCTION It has recently been p o i n t e d out [i] that the strengthening encountered in polycrystals, the Hall-Petch effect, can also be expected in materials which are not n e c e s s a r i l y p o l y c r y s t a l l i n e but, insofar as the effect is concerned, are similar to them. Specifically considered were two-phase alloys in which the m a t r i x forms the minor constituent and in which the other, major phase, is particulate. Such alloys are to be considered as equivalent to p o l y c r y s t a l s if the matrix is so thin that, at the stress involved, it cannot, independent of the particles, deform plastically. This restriction is also available in circumstances where the particles are replaced by rods or plates, effectively infinite in length. The operation of the H a l l - P e t c h effect in such materials in which the particle size is now disposable in practice down to diameters conveniently m e a s u r e d in nanometers, allows the p o ss i b i l i t y of material strengths ap p r o a c h i n g the t h e o r e t i c a l limit. It has also been n o t e d [I] that the p h e n o m e n o n of dilatancy which is responsible for the firmness of wet beach sand [2] may have metallurgical applications; a direct analogy being a packed dispersion of particles (each contacts ~ six others) in a metal matrix. If this analogue is to have important c o n s e q u e n c e s at temperatures so high that the matrix approaches its melting point or is a c t u a l l y molten it is necessary that the particles be strong at these t e m p e r a t u r e s and be very much smaller than ordinary sand grains. The essential feature of dilatancy is that the total volume of the interstices b e t w e e n particles, which are essentially densely packed and individually rigid, expands upon deformation. When this volume is occupied by a continuous fluid, such expansion must result either: in the formation of voids; a reduction in (compressive) pressure or; in an inward flow from an external surface. W h e n the fluid is liquid, formation of voids is resisted by its cohesion; inward flow by the development at the outer surface of a pressure: p ~ 2y cose/r h
(I)
where ~ is the surface energy of contact between particle and liquid, e the contact angle and r h the radius of the hole through w h i c h the liquid must flow. Purely elastic b e h a v i o r of the whole follows from that of the *On-site contractor for Geo-Centers, Ft. Washington, MD 20744
721 0 0 3 6 - 9 7 4 8 / 8 9 $3.00
+ .00
Inc.
722
STRENGTH
OF B I N A R Y
ALLOYS
Vol.
23, No.
p a r t i c l e s at all a p p l i e d s t r e s s e s less t h a n p. To e x a m i n e t h e m a g n i t u d e of p, w e s u p p o s e y to b e of t h e o r d e r o f t h e s u r f a c e e n e r g y o f t h e m a t r i x a n d ~a -i0 w h e r e ~ is s h e a r m o d u l u s a n d 'a' t h e l a t t i c e p a r a m e t e r d i s t a n c e . Then, (I) w i t h 8 ~ 0: p ~ ~ / I 0 4 ,r h ~ 1 ~m; p ~ ~ / i 0 2 ,r h ~ .01 ~Lm. thus t o b e about:
from
The l a t t e r s t r e s s is t e c h n i c a l l y l a r g e in m o s t cases. T h e p o s s i b i l i t y that m a t e r i a l s c o u l d h a v e s u c h s t r e n g t h n e a r or a b o v e a r e l e v a n t m e l t i n g p o i n t is unique to this approach. In p r a c t i c e t h e a c h i e v e m e n t of p a c k i n g so c l o s e as to s a t i s f y t h e r e q u i r e m e n t s c i t e d a b o v e m a y b e d i f f i c u l t to a c h i e v e . T h i s is t h e first c o n c l u s i o n of a n a t t e m p t at e x p e r i m e n t a l c o n f i r m a t i o n of t h e s e ideas. The s y s t e m e m p l o y e d w a s Fe-Hg. T h e r e s u l t s o b t a i n e d a r e d e t a i l e d below. H e r e we n o t e t h a t t h e l a r g e s t p a c k i n g a c h i e v e d (~66% Fe) was i n s u f f i c i e n t to a c h i e v e a r i g i d p a r t i c l e s k e l e t o n so t h a t d e f o r m a t i o n of t h e m a t r i x a l o n e w a s possible. D e f o r m a t i o n u n d e r t h e s e c i r c u m s t a n c e s is a c c o r d i n g l y t h e c o n c e r n of t h e a c c o m p a n y i n g a n a l y s i s . ANALYSIS F o r s i m p l i c i t y w e s e p a r a t e the d i s c u s s i o n i n t o t w o realms, c h a r a c t e r i s a b l e b y t e m p e r a t u r e , a n d d e a l f i r s t w i t h t h e e f f e c t s of p a c k i n g d e f i c i e n c y w h e n t h e t e m p e r a t u r e is so l o w t h a t t h e e f f e c t s d u e to d i f f u s i o n can be n e g l e c t e d . In t h e s a i d c i r c u m s t a n c e s we can s u p p o s e t h a t i n d e p e n d e n t p l a s t i c d e f o r m a t i o n of t h e m a t r i x can o c c u r b y b e n d i n g d i s l o c a t i o n s i n t o s e m i c i r c l e s of r a d i u s L/2. T h i s w i l l occur, p r o v i d e d t h e r e e x i s t s an a p p l i e d s h e a r stress:
~b o =
--.
(2)
L
Intuitively,
it
is
clear
that
L will
decrease
with
decreasing
particle
r a d i u s a n d w i t h i n c r e a s i n g d e n s i t y of p a c k i n g . T h e m a n n e r of t h i s d e c r e a s e can b e e x p e c t e d to d e p e n d on t h e s e p a r a m e t e r s in a c o m p l e x way. Rather than u n d e r t a k e a d e t a i l e d a n a l y s i s o f t h i s q u e s t i o n , w e r e c o g n i z e t h a t for o u r present purposes, namely, to recognize trends, a simplified approach dealing o n l y w i t h a v e r a g e v a l u e s is s u f f i c i e n t . F o l l o w i n g O r o w a n w e i d e n t i f y L as the s p a c e b e t w e e n p a r t i c l e s a n d a s s u m e t h e p a r t i c u l a t e p h a s e to be s p h e r i c a l w i t h r a d i u s r a n d to r e p r e s e n t a v o l u m e f r a c t i o n f. We then find that the m e a n d e n s i t y of p l a n a r c i r c u l a r i n t e r s e c t i o n s h a v i n g a v e r a g e a r e a 2Kr2/3 is 3f/2Kr 2 with average
Kr r a d i u s ~-.
Thence,
we obtain the mean
interparticle
spacing:
L-r[
~2~ ~-~]~
N u m e r i c a l v a l u e s o b t a i n e d f r o m t h e u s e of t h i s f o r m u l a a r e to b e a c c e p t e d o n l y w i t h c a u t i o n , b u t it w o u l d s e e m to b e c o r r e c t in p r e d i c t i n g a l i n e a r r e l a t i o n b e t w e e n L a n d r a n d in i n d i c a t i n g t h a t L << r at v o l u m e f r a c t i o n s w h i c h a p p r o a c h t h a t of c l o s e p a c k e d s p h e r e s : f=K/3~2. Accepting its v a l i d i t y w e f i n d w h e n f = . 6 6 t h a t L / r = 0 . 2 a n d so w e h a v e f r o m (2) 0
~
~/1400,
However, same modulus.
r =
2 ~Lm
(3)
it is a s s u m e d in e q u a t i o n (2) t h a t m a t r i x a n d p a r t i c l e Here, w h e r e it is to b e e x p e c t e d t h a t t h e m o d u l i a r e
h a v e the
5
Vol.
23,
No.
5
STRENGTH
OF B I N A R Y
ALLOYS
723
s i g n i f i c a n t l y d i f f e r e n t , a c o r r e c t i o n is n e c e s s a r y . S e e m i n g l y , an a p p r e c i a t i o n of its m a g n i t u d e can be o b t a i n e d s i m p l y f r o m a c o n s i d e r a t i o n of the b e h a v i o r of four s t r a i g h t p a r a l l e l s c r e w d i s l o c a t i o n s , the o u t e r p a i r of s e p a r a t i o n , 2 L - L 1 r e p r e s e n t the n e a r i m a g e s of the i n n e r pair, s e p a r a t i o n LI, in the s u r f a c e s h a v i n g s e p a r a t i o n L at w h i c h the e l a s t i c m o d u l u s c h a n g e s from ~I to ~2 as shown in Fig. i. W e c o n s i d e r the e q u i l i b r i u m of the i n n e r p a i r u n d e r the a c t i o n ~ib 2~
[L~ +
of a s h e a r ~(2L-LI) ] L(L-LI)
stress ~ and write = ~; e = ~I-B____~2 i. ~i+~2
(4)
Then, p r o v i d i n g the s e p a r a t i o n of a d i s l o c a t i o n a n d its image is m u c h g r e a t e r t h a n its core radius, it is e a s i l y s e e n that the s t r e s s n e c e s s a r y for p a s s a g e is g i v e n by -- ~ b a =~whence
(i
+ ~(2L-LI) ) ~ ~ib
L1
L(L_LI )
for ~ = 1 we
find:
(5)
L1 L 1 =3L/4
so that
(3) b e c o m e s
c = ~/i000
Here, we h a v e n e g l e c t e d t h e e f f e c t s of all but first i m a g e s in the nearest interfaces. The e r r o r so i n v o l v e d w o u l d s e e m small, c e r t a i n l y a c c e p t a b l e in t h e c u r r e n t c o n s i d e r a t i o n of g r o s s m a g n i t u d e s , w h e n as here the r a t i o of the s p a c i n g b e t w e e n d i s l o c a t i o n s a n d i n t e r f a c e s to t h a t b e t w e e n i n t e r f a c e s is s m a l l (~1/8). The e f f e c t s due to m u l t i p l e l a y e r i n g (implicit in p r e s e n c e of p a r t i c l e s ) can a l s o be e x p e c t e d to be s m a l l [3]. A p a r t f r o m m i s f i t s h e a r s t r e s s e s w h i c h are p r o b a b l y of h y d r o s t a t i c o r i g i n ~ n e g l e c t e d s i n c e t h e y w o u l d fall off r a p i d l y (inverse cube of d i s t a n c e ) , t h e s e c o n s i d e r a t i o n s a p p e a r to e x h a u s t the p o s s i b i l i t i e s of direct i n h i b i t i o n of d i s l o c a t i o n m o t i o n . T h e r e r e m a i n s the p o s s i b i l i t y of a s p e c i a l type of w o r k h a r d e n i n g . The q u e s t i o n of the h a r d e n i n g i n d u c e d by the p r e s e n c e of p a r t i c l e s w h i c h are e s s e n t i a l l y u n p e n e t r a b l e to l a t t i c e d i s l o c a t i o n s has b e e n e x a m i n e d p r e v i o u s l y , e.g. B r o w n [4], but o n l y for the case of d i l u t e d i s p e r s i o n s . In t h e s e c i r c u m s t a n c e s c r o s s s l i p w a s f o u n d to be i m p o r t a n t in r e d u c i n g h a r d e n i n g b e l o w its p o t e n t e n t i a l . The e f f i c a c y of the m e c h a n i s m i n v o k e d may be t r a c e d to t h e fact that the n e c e s s a r y i n c r e a s e in d i s l o c a t i o n l e n g t h i n v o l v e d in b y - p a s s i n g a p a r t i c l e is r e l a t i v e l y small. S u c h s h o u l d not be so here, w h e r e t h e p a r t i c l e c o n c e n t r a t i o n is high. R a t h e r we w o u l d e x p e c t that p l a s t i c d e f o r m a t i o n of the m a t r i x w o u l d by a n d large, i n d u c e d by c i r c u m s c r i b i n g d i s l o c a t i o n , r e s u l t in an e l a s t i c d e f o r m a t i o n of the p a r t i c l e s of like amount. T h e a c c o m p a n y i n g r e d u c t i o n in s t r e s s in the m a t r i x m a y be r e g a r d e d as a w o r k h a r d e n i n g effect. For t h i s h y p o t h e s i s to be v a l i d it is at least n e c e s s a r y that the p a r t i c l e s b e h a v e in a q u a s i - i m p e n e t r a b l e fashion. To e x a m i n e this p o s s i b i l i t y we c o n s i d e r t h e c o n d i t i o n s n e c e s s a r y for t h e p r o p a g a t i o n of d e f o r m a t i o n f r o m t h e m a t r i x to t h e p a r t i c l e s . This we can e s t i m a t e u s i n g the Hall-Petch relation =
~o
+
k
d -I/2 .
(6)
To e s t i m a t e k w e r e f e r to the e x p r e s s i o n g i v e n b y L o u a t a c t i n g on t h e h e a d of a p i l e - u p l e n g t h w h i c h t e r m i n a t e s e l a s t i c m o d u l u s ~i t o ~216Em2~ 2 a Thus,
F =
~ B~2a, ~ (l+cos2~m)
1 where m = ~
cos-le
[I] for the force at a d i s c o n t i n u i t y
in
724
STRENGTH
Substitution the s t r e s s e s ~2) n a m e l y ~e =
ALLOYS
Vol.
23, No.
o f t h i s e x p r e s s i o n in t h e e q u a t i o n g i v e n in t h e s a m e p a p e r for n e c e s s a r y for a c t i v a t i o n of s l i p in t h e s e c o n d m a t e r i a l (modulus
~/PlP2 ~aa'
Oe = g2
when,
OF B I N A R Y
w h e r e A is a c o n s t a n t
results
" PFe
as here,
m - ~
in: (7)
E ~
~Fe"
is small. A s i g n i f i c a n t f e a t u r e of (7) is t h a t t h i s c r i t i c a l s t r e s s is i n d e p e n d e n t of t h e m o d u l u s of m e r c u r y . A c c o r d i n g l y , we m a y e m p l o y (6) w i t h a v a l u e for k a p p r o p r i a t e t o iron. D o i n g so a n d s e t t i n g a in (6) as 2 ~ m we find that
ac ~ 160
a value
very much larger than that
given
in
(3).
W e n o w t u r n to t h e c o n s e q u e n c e s o f i m p o r t a n t d i f f u s i o n a l e f f e c t s . First, d i s l o c a t i o n climb. T h e r a p i d w o r k h a r d e n i n g we h a v e i n v o k e d c a n n o t be e x p e c t e d w h e n d i s l o c a t i o n s c l i m b readily. We therefore anticipate s u b s t a n t i a l d e c r e a s e s in s t r e n g t h as t h e t e s t t e m p e r a t u r e p r o g r e s s i v e l y e x c e e d s a b o u t .50% h o m o l o g o u s . T u r n i n g t o d i r e c t e f f e c t s o f d i f f u s i o n we r e m e m b e r t h a t d i f f u s i o n a l c r e e p r e s t s on t h e t r a n s p o r t of m a t t e r b e t w e e n g r a i n b o u n d a r i e s a n d that this p r o c e s s c a n b e e x p e c t e d [1] to s e l f - t e r m i n a t e b y t h e a g g r e g a t i o n of p a r t i c l e s into r i g i d a s s e m b l i e s at b o u n d a r i e s w h i c h act as s i n k s for v a c a n c i e s . This has b e e n e s t i m a t e d [I] t o o c c u r at a strain! ~c = 2 A r / G w h e r e G r e p r e s e n t s g r a i n d i a m e t e r a n d A is t h e c h a n g e in m a t r i x c o n c e n t r a t i o n n e c e s s a r y to 74-66 achieve close Packing. Here, A ~ i0------~" M e t a l l o g r a p h i c d i f f i c u l t i e s h a v e p r e v e n t e d a m e a s u r e m e n t of G in t h e c u r r e n t experiment. G e n e r a l e x p e r i e n c e w o u l d s u g g e s t t h a t G is m u c h l a r g e r t h a n 2 ~ m so t h a t ec s h o u l d n o t b e large. For values which may be thought of a typical: G > 100 ~ m w e h a v e ec < - 0.3%. EXPERIMENTAL
The s y s t e m F e - H g w a s c h o s e n as t h e v e h i c l e for a f i r s t t e s t o f s u c h materials because: m e r c u r y is i m m i s c i b l e w i t h i r o n a n d m e l t s n e a r r o o m temperature. A f a c t o r of e q u a l i m p o r t a n c e is t h a t i r o n p a r t i c l e s c a n be i n t r o d u c e d i n t o m e r c u r y w i t h r e l a t i v e ease. T h u s i r o n p a r t i c l e s (4 ~m) of 99.5% p u r i t y w e r e i m m e r s e d in a w e a k l y a c i d u l a t e d (HCI) s o l u t i o n t o g e t h e r w i t h a q u a n t i t y of m e r c u r y . The subsequent addition of mercurous chloride allowed the wetting and absorption of the p a r t i c l e s i n t o t h e m e r c u r y . T h e r e s u l t a n t , e s s e n t i a l l y solid, m a t e r i a l w a s t h e n c o m p a c t e d b y h a n d a n d frozen. In t h i s w a y s m a l l s p e c i m e n s lcc c o n t a i n i n g r o u g h l y e q u a l v o l u m e f r a c t i o n s o f t h e t w o c o n s t i t u e n t s w e r e produced. T h e f l o w c h a r a c t e r i s t i c s of t h e s e H g - F e a l l o y s w e r e d e t e r m i n e d u s i n g the s o - c a l l e d i m p r e s s i o n t e s t m e t h o d [5] a n d its a s s o c i a t e d a n a l y s i s . Here, s t r e s s - s t r a i n r e l a t i o n s are o b t a i n e d t h r o u g h s i m u l t a n e o u s a n d p r e c i s e m e a s u r e m e n t s o f a p p l i e d l o a d a n d the r e s u l t a n t p e n e t r a t i o n of a f l a t - e n d e d circular indentor into a specimen. T h e (tungsten carbide) i n d e n t o r is c o n s t r a i n e d t o m o v e in a d i r e c t i o n n o r m a l to t h e (flat) s u r f a c e of t h e specimen. T e s t s w e r e c a r r i e d o u t o v e r a r a n g e of t e m p e r a t u r e s d e r i v e d f r o m t h e use of d r y ice a n d l i q u i d n i t r o g e n on s m a l l s p e c i m e n s in a h y d r a u l i c t e s t i n g
5
Vol.
23,
No.
5
STRENGTH
OF B I N A R Y
ALLOYS
725
machine. The d i s p l a c e m e n t of t h e p u n c h was f o u n d f r o m the o u t p u t o f a LVDT (linear-variable-differential-transformer) a t t a c h e d b e t w e e n t h e l o a d cell and specimen. This a n d t h e c o r r e s p o n d i n g l o a d s i g n a l w e r e d i s p l a y e d on an x - y recorder. The i n d e n t o r s p e e d w a s c h o s e n to b e 8.5 x 10 -7 m/s. This s p e e d is e q u i v a l e n t to a t r u e s t r a i n [5] rate o f 1 . 2 7 5 x 10-3/s. T h e o r y [6] a n d e x p e r i m e n t [5,7,8] are in a c c o r d in p r o v i d i n g a p p r o x i m a t e c o r r e l a t i o n s b e t w e e n t r u e t e n s i l e stress, z, a n d load, P, on t h e one h a n d and true strain, £, a n d p e n e t r a t i o n ~ on t h e other, t h e s e are, w i t h e r r o r less than 10%: ~ = P/3; £ = 6/d, w h e r e d is the d i a m e t e r of the i n d e n t o r . Fig. 2 s h o w s a p l o t of t r u e s t r e s s v e r s u s t r u e s t r a i n for r e p r e s e n t a t i v e data t a k e n at -II0°C a n d at -62°C u s i n g t h e s e c o r r e l a t i o n s . From these curves we f i n d a y i e l d s t r e s s of ~y = 43.5 M P a (-I10°C) a n d 2 4 . 1 3 M P a (-62°C). DISCUSSION
AND CONCLUSIONS
It is a p p a r e n t f r o m t h e w i d e v a r i a t i o n in s t r e n g t h b e t w e e n r o o m t e m p e r a t u r e a n d -II0°C s h o w n by t h i s m a t e r i a l that the p r i n c i p l e of d e t a i l e d c o n t i n u i t y of d e f o r m a t i o n p r e v i o u s l y e l a b o r a t e d [I] d o e s not a p p l y at all temperatures. Thus, w h e n the m e r c u r y is m o l t e n the r e s t r a i n t a f f o r d e d b y the iron p a r t i c l e s is s m a l l b u t l a r g e e n o u g h for the s p e c i m e n s to s t a n d alone. The p r o c e s s i n v o l v e d h e r e is not c l e a r but it w o u l d s e e m that d i f f u s i o n w o u l d be of d o m i n a t i n g i m p o r t a n c e at t e m p e r a t u r e s just b e l o w t h e m e l t i n g point. W i t h p r o g r e s s i v e l y l o w e r t e m p e r a t u r e s the s t r a i n rate a c h i e v a b l e b y d i f f u s i o n a l o n e w i l l e v e n t u a l l y fall b e l o w the rate i m p r e s s e d . In this r e a l m the n e e d for d e t a i l e d c o n t i n u i t y of d e f o r m a t i o n will be i n c r e a s i n g l y i m p o r t a n t . T o w a r d s a n a p p r e c i a t i o n of the s i g n i f i c a n c e of t h e r e s u l t s of t h e s e m e a s u r e m e n t s we r e l a t e t h e m to t h e e l a s t i c m o d u l i of t h e c o n s t i t u e n t s . E f f o r t s to f i n d d a t a r e l a t i n g to m e r c u r y h a v e b e e n u n a v a i l i n g . However, h a v i n g r e g a r d to t h e g e n e r a l l y s y m b a t i c c o r r e l a t i o n b e t w e e n m e l t i n g p o i n t and e l a s t i c m o d u l u s in m e t a l s we e x p e c t that, that of m e r c u r y w i l l b e less t h a n that of I n d i u m s i n c e t h e l a t t e r m e l t s at 156.4°C t h e f o r m e r at -389°C. The m o d u l u s o f i n d i u m is ii x 103 M P a t h a t of m e r c u r y m i g h t w e l l be ~ 6.89 x 103 M P a w h i c h w o u l d g i v e a g r e e m e n t w i t h i n a f a c t o r of two b e t w e e n the v a l u e of ~y at -ll0°C a n d t h e e s t i m a t e of ~c from ~g. S u p p o r t for t h e u s a g e of s u c h a v a l u e is a v a i l a b l e f r o m the m e a s u r e d v a l u e of t h e s l o p e of t h e l i n e a r p o r t i o n of s t r e s s - s t r a i n r e l a t i o n f o u n d at I10°C. thus, t h i s s l o p e is 10.3 x 103 MPa, in a c c o r d w i t h a c r u d e formulation using'a rule of mixtures. On this b a s i s t h e m o d u l i are d i s p a r a t e ; the i n d i c a t e d r a t i o is 30. No s u c h r a t i o a p p e a r s to e x i s t b e t w e e n a n y two m a t e r i a l s w h i c h m i g h t b e i n c l u d e d as p r a c t i c a l . H o w e v e r , for the case of Ni-W, w h i l e o n l y 6:1, this ratio is n e v e r t h e l e s s s i g n i f i c a n t l y l a r g e r t h a n unity. The i n a b i l i t y to f i n d an exact m a t c h in this r e g a r d is p r o b a b l y not i m p o r t a n t . The f e a t u r e of s i g n i f i c a n c e is that p l a s t i c d e f o r m a t i o n lies in t h e r e a l m of h i g h t e m p e r a t u r e for m e r c u r y a n d low for iron. T h i s d i s p a r i t y is s h a r e d b y the c o m b i n a t i o n N i - w t e s t e d at ~ 1300°C. Thus, w e s u p p o s e that the b e h a v i o r of a l l o y s c o n s t i t u t e d r e s p e c t i v e l y of n i c k e l a n d m e r c u r y is the same w h e n the h o m o l o g o u s t e m p e r a t u r e a n d i m p o s e d e l a s t i c s t r a i n s are the same. S u c h an i d e n t i t y of e l a s t i c s t r a i n s i n d i c a t e s the use of a s c a l i n g f a c t o r o f = 30 (~Ni/~Hg) in a s s e s s i n g y i e l d stress. Again, at the h i g h e r of the two, the h o m o l o g o u s t e m p e r a t u r e w a s 0.9 (Hg). A p p l y i n g t h e s e f a c t o r s we f i n d a t e n s i l e y i e l d s t r e s s o f (9.71 x 108 Pa) at 1280°C for a h y p o t h e t i c a l N i - W a l l o y in w h i c h the t u n g s t e n is p a r t i c u l a t e at 4 ~ m d i a m e t e r a n d c o n s t i t u t e s = 6 6 % b y volume. Such s t r e s s e s : ~ ~ / 2 0 0 at 90% of the m e l t i n g p o i n t of t h e m a t r i x are significantly higher than those previously achieved through more conventional approaches.
726
STRENGTH OF BINARY ALLOYS
Vol.
23,
No.
REFERENCES 1.
2. 3. 4. 5.
6. 7. 8.
N. P. Louat, Acta. Met., 33, 59 (1985). O. R e y n o l d s , N a t u r e , 33, 429 (1886). S. V. Kamat, J. P. H i r t h a n d B. C a r n a h a n , S c r i p t a Met. 21, 1587 (1987). L. M. B r o w n a n d w. M. Stobbs, Phil. Mag. (GB) Vol. 23, No. 185, 1201-33, M a y 1971. H. Y. Yu, M. A. Imam, a n d B. B. Rath, J. Mat. Sc., 20, 636 (1985). D. Tabor, Proc. R. Soc. A192, 247 (1948). H. O ' N e i l l a n d H. G r e e n w o o d , J. Inst. Met., 48, 47 (1932). E. C. Yu a n d J. C. M. Li, Phil. Mag. 32, 811 (1977).
4-
+
<
Lt -----~
L
Fig.
1
Four straight
"
parallel
screw dislocations.
-110°C
50
-I~.oc
i
'm w-
3o
W
20
10
o
~'6
STRAIN (10-3) Fig.
2
True
stress
versus
true
strain
curves obtained
by impression
test.
5