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A MODEL FOR ULTRASONIC EFFECTS DURING
AN FCC-HCP MARTENSITIC TRANSFORMATION
A. Granato, R. Schwarz, G. Kneezel
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque 6'5, suppZ6ment au nolO, Tome 4 2 , octobre 1981 page 65-1055
A MODEL FOR ULTRASONIC EFFECTS DURING AN FCC-HCP MARTENSITIC
TRANSFORMATION
*
=+
**
A.V. Granato, R.B. Schwarz and G.A. Kneezel
Physics Department and Materials Research Laboratory, University o f I l Z i n o i s a t Urbana-Champaign, Urbana, IL 61801, U.S.A.
Abstract.
-
It is proposed that the KC-HCP martensitic phase transformationmay be studied ultrasonically by measuring the effect of "optical" vibrations of extended dislocations on the [Ill] shear sound velocity and attenuation. A simple model for the behavior of such dislocations predicts a dip in velocity and a rise in attenuation as the transformation is approached.
ltt is generally believed that dislocations play a role in martensitic transformations, but the details of the role are obscure. A simple model for the FCC-HCP case is developed here which should be susceptible to experimental checks by ultrasonic measurements.
Dislocations in fcc metals are extended1 with a width that depends on the stack- ing fault energy. In a material which transforms from fcc to hcp, these extended
dislocations are likely nucleation centers as first proposed by ~hristian.' When the
free energy difference between the phases decreases, the extended dislocations (each boundary a two-layer hcp fault) widens. ~ r i c s o n ~ has observed such widening via electron microscopy in cobalt and cobalt-nbckel. His measurements were not made
close to the transformation temperature
s.
He suggested that the stacking faultenergy of an isolated extended dislocation is linear in temperature but does not go
to zero at Ms. Olson and cohen4 proposed subsequently that the nucleatiing defects
are overlapping extended dislocations spaced every other layer to produce a thicker hcp fault with lower total energy than an equal number of two layer faults. In both ~ c h e m e s ~ , ~ it is suggested that extended dislocations gradually widen as the trans- formation is approached and then propagate across the plane as the free energy dif- ference becomes negative.
Because the partial dislocations bounding the fault have anti-parallel compo- nents, it may be possible (using a suitably oriented stress wave) to excite an "optical mode" of vibration in which the partials move in opposite directions. As
the binding force between the partials decreases, the resonant frequency of the optical mode decreases. Ultrasonic measurements of the velocity and attenuation of
*Work supported by the National Science Foundation, under grant NSF DMR77-10556
*present address : Materials Science Dept., Argonne National Lab., Argonne, IL 60439,
U.S.A.
%present address : Xerox Research Laboratory, Webster, NY 14580, U.S.A.
C5-1056 JOURNAL DE PHYSIQUE
s t r e s s waves e x c i t i n g t h e o p t i c a l mode may t h e r e f o r e be a means of d e t e c t i n g t h e d i s l o c a t i o n widening as t h e t r a n s f o r m a t i a n temperature i s approached.
A r e c t a n g u l a r loop model can be used t o f i n d t h e behavior of t h e e q u i l i b r i u m
width and o p t i c a l mode resonant frequency ( N g u r e 1 ) . The extended d i s l o c a t i o n has a l e n g t h L and width w. The energy terms depending on w a r e Eg ( t h e r e p u l s i v e i n t e r - a c t i o n of t h e p a r t i a l s ) , ET (The l i n e energy of t h e v e r t i c a l segments which l e n g t h e n a s w i n c r e a s e s ) , E ( t h e s t r a i n energy i n t e r a c t i o n with e x t e r n a l s t r e s s ) , and Ey ( t h e s t a c k i n g f a u l t energy). Point d e f e c t pinning is n e g l e c t e d , but t h i s model does ' i n - c o r p o r a t e ET and E which simpler t r e a t m e n t s neglect=/.
a' The r e p u l s i v e energy
5
f o r an i s o t r o p i c s o l i d is o b t a i n e d by i n t e g r a t i n g t h e r a d i a l component of t h e i n t e r a c t i o n f o r c e given by G b 2~ s i n e s i n e 2] _ ,
whereel
= 6-
30° ----E [ C O S ~ ~ C O S ~ ~+
-
2 mF i g . 1
-
A. The r e c t a n g u l a r loop model f o r a n Ic---L-extended d i s l o c a t i o n of l e n g t h L and width w. The B2rgers v e c t o r of t h e complete d i s l o c a -
t i o n l5 i s assumed t o be a t an a n g l e 6 from W
t
t h e d i s l z c a t i o n d i r e c t i o n , w h i l e t h e a n g l e
between b and t h e s h e a r s t r e s s on t h e p l a n e A i s
a.
B. The f o u r energy terms v e r s u s width w. Eo and Ey may b e p o s i t i v e o r nega- t i v e and a r e grouped t o g e t h e r f o r t h e s e p o s s i b i l i t i e s .
C. The t o t a l energy v e r s u s width w. For n e g a t i v e E y + E g t h e e q u i l i b r i u m width i s l a r g e r and t h e c u r v a t u r e a t t h e minimum i s s m a l l e r .
+ +
and O2 = 6
+
30° a r e t h e a n g l e s between b l , b and t h e segments of t h e d i s l o c a t i o n 2l i n e h o r i z o n t a l i n Ng. 1. The l i n e energy ET i s obtained by summing over t h e f o u r segments v e r t i c a l i n Mg. 1 of
w i t h
el
= B-
1 2 0 ° ,e2
=+
60°, 9 = 8+
120°, and O4 = B-
60°. The s t r a i n energy 3ET a r i s i n g from t h e s e p a r a t i o n of t h e p a r t i a l s is found from t h e f o r c e Fi on t h e par- t i a l s h o r i z o n t a l i n Fig. 1 given by Fi = a b
L
cos$ where $i i s t h e angle betweenp
-,
i:t h e component of s t r e s s on t h e s l i p plane and bl o r b 2 , = a
+
30 andg2
= a-
30. This y i e l d s Ea = -obpL(w/2)[cos(a+
30)-
c o s ( a-
3011. The r e s u l t s a r eE = a Lb f (a)w
,
anda E p 3 ( 3 )
where b i s t h e partial, d i s l o c a t i o n Burgers v e c t o r , B i s t h e angle between t h e Burgers
$
v e c t o r b of t h e complete d i s l o c a t i o n and its d i r e c t i o n , aE is t h e component of e x t e r -
n a l s t r e s s on t h e (111) t r a n s f o r m a t i o n and s l i p p l a n e , a i s t h e a n g l e
between aE and
g,
and y is t h e s t a c k i n g f a u l t energy. I n t h e above f l , f 2 and f 3 a r e f a c t o r s depending upon t h e o r i e n t a t i o n of t h e Eurgers v e c t o r s , but a r e independent ofW, a , and y :
f ( a ) = ( s i n a l l 2 3
For v = 113, a s 6 goes from 0 t o 91°, fl v a r i e s between 3/8 and 7/8, while f 2 / ! i n ( ~ / b ) v a r i e s between 1114 and 914.
E and E may be p o s i t i v e o r n e g a t i v e , depending on t h e temperature and t h e di-
Y
r e c t i o n of t h e a p p l i e d s t r e s s . They a r e t h e r e f o r e p l o t t e d t o g e t h e r i n Figure 1. ET is always p o s i t i v e , a s i t always t a k e s energy t o c r e a t e t h e e x t r a l e n g t h of d i s l o c a - t i o n n e c e s s a r y t o widen t h e f a u l t . The c r e a t i o n of t h e e x t r a l e n g t h of d i s l o c a t i o n is one source of t h e h y s t e r e s i s observed i n t h e transformation. The f r e e energy d i f f e r - ence between t h e two phases mst not merely go t o zero but mst become s u f f i c i e n t l y n e g a t i v e t o minimize t h e t o t a l energy of t h e f a u l t as it grows. The term E s u g g e s t s t h a t w i t h p r o p e r l y a p p l i e d e x t e r n a l s t r e s s one may favor o r hinder t h e widening of t h e f a u l t , and i n t h i s way narrow o r widen t h e h y s t e r e s i s of t h e transformation. Weston
81
C5-1058 JOURNAL DE PHYSIQUE
The e q u i l i b r i u m width wo i s found by s e t t i n g (dE/dw) = 0:
For i s o l a t e d extended d i s l o c a t i o n s t h e second term i n t h e denominator ( c o r r e s - ponding t o t h e l i n e energy b a r r i e r ET) s u g g e s t s t h a t t h e t r a n s f o r m a t i o n begins a t t h e l o n g e s t d i s l o c a t i o n and is d r i v e n by a n e g a t i v e s t a c k i n g f a u l t o r s t r e s s energy term. Since not a l l d i s l o c a t i o n s a r e t h e same l e n g t h t h e t r a n s f o r m a t i o n occurs over a range of temperatures r a t h e r than a t a s i n g l e temperature.
Based on t h e r e c t a n g u l a r loop model we c a l c u l a t e t h e decrement and f r a c t i o n a l change i n v e l o c i t y f o r an applied s t r e s s of amplitude o and frequency w. With no s t a t i c a p p l i e d s t r e s s , t h e equation of motion f o r t h e o p t i c a l mode i s
Aj;
+ Bjr
+
Dy = bo cos o t ( 6 )where A is t h e e f f e c t i v e d i s l o c a t i o n mass, B i s t h e damping c o n s t a n t , and D is a s p r i n g c o n s t a n t given by
The r e s u l t i n g change i n shear modulus and t h e decrement a r e
0
.SX
I X10-4 I J X I O - ~
Fig. 2
- The f r a c t i o n a l change i n s h e a r
modulus -(AG/G) and t h e decrement Awhere uy( = (D/LI)'/~
-
.3v/wo, v2 = G/p, and A is the density of extended dislocations AGhaving an optical mode resonant frequency wR. In Figure 2,
-
and A are plottedG
versus wR assuming A = lo4cm2 and B = in cgs units. Because y is approximately
linear in temperature near the transformation, it is seen from Eqs. 5 and 7 that
the % axis may be viewed as a temperature axis. Fbr above the transformation
temperature MS, the equilibrium width of the fault is small and o is much larger than
R
the measurement frequency w. For Y( large compared to the measurement frequency, the
2
dislocation motion is in phase with the applied stress, but small, with AV/V
-
11%.
For w small compared to the measurement frequency the dislocation motion is out of R
2
phase with the applied stress and small again, with Av/v
-
%
.
In between, a maximumin the velocity change is obtained. The decrement meanwhile increases from its pre-
transformation value by an amount (4nAv )/v.
max
The model thus predict a dip in velocity and a corresponding rise in decrement as Ms is approached. These effects must be superposed on the background (a linear increase in velocity with decreasing temperature and a temperature independent decre-
ment). Mnally as the transformation occurs, the quasi-static picture of the model is
no longer applicable. Large numbers of dislocations are believed to move with veloci- ties approaching the velocity of sound>/ There will also be scattering due to the presence of two phases with different elastic constants leading to a dramatic increase in decrement and decrease in velocity.
In summary, the optical mode vibration model predicts a dip in velocity and a corresponding rise in decrement as the Ms temperature is approached. What is more, these precursor effects can in principle be observed reversibly without transforming the sample.
References
1) R. D. Heidenreich and W. Shockley, in Report of Bristol Conference on the Strength of Solids (The Physical Society, London, 1948).
2) J. W. Christian, Proc. Roy. Soc. Lond. A*, 51 (1951). 3) T. Ericsson, Acta. Met.
14,
853 (1966).4) G. B. Olson and M. Cohen, Met. Trans. A l , 1897 (1976).
5) J. W. Christian, The Theory of Transformations in Metals and Alloys (Pergamon, New York, 1965).
6) J. P. Hirth and J. Lothe, Theory of Dislocations, (McGraw-Hill, New York, 1968).
7) H. Kronmuller, Phys. Stat. Sol. B
