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INTERNAL FRICTION MEASUREMENTS AND THERMODYNAMICAL ANALYSIS OF A
MARTENSITIC TRANSFORMATION
S. Koshimizu, W. Benoit
To cite this version:
S. Koshimizu, W. Benoit. INTERNAL FRICTION MEASUREMENTS AND THERMODYNAMI-
CAL ANALYSIS OF A MARTENSITIC TRANSFORMATION. Journal de Physique Colloques, 1982,
43 (C4), pp.C4-679-C4-684. �10.1051/jphyscol:19824110�. �jpa-00221966�
JOURNAL DE PHYSIQUE
CoZloque C4, suppZdment au n o 12, Tome 43, ddcembre 1982 page C4-679
I N T E R N A L F R I C T I O N MEASUREMENTS AND THERMODYNAMICAL A N A L Y S I S O F A M A R T E N S I T I C TRANSFORMATION
S. ~oshimizu* and W. Benoit
T n s t i t u t de Ce'nie Atomique, EcoZe PoZytechnique Fdde'raZe de Lausanne, 1015 Lausanne, SwitzerZand
(Accepted 9 August 1982)
Abstract.- Internal friction measurements as a function of temperature and strain amplitude have been performed on polycrystalline CuZnAl and T i Ni al- loys. These measurements show a typical "critical" behaviour round the tempe- rature of the martensitic transformation : discontinuities in the internal fric- tion curves, important decrease of some elastic constants and a thermal hystere- sis between heating and cooling.
In order to explain these results, theoretical calculations based on a Landau theory of first order phase transition have been performed. An important strain amplitude dependence is taken into account. It is assumed that the order para- meter has the same dynamical behaviour as the one of the dislocations when they break away from pinning points. The thermal hysteresis is then explained without references to the influence of internal stresses.
Introduction.- The experimental features of the internal friction spectrum observed -
during martensitic transformations have been widely studied in some Fe-based alloys (I), C o Ni (21, T i Ni ( 3 ) . Cu Z n Al(4)and so on. In these studies the internal fric- tion has been measured as a function of the temperature for different frequencies (j3, different amplitudes of vibration (E) and different cooling or heating rates ($). An internal friction maximum has systematically been observed at a temperature corres- ponding to the temperature of the martensitic transformation. The main features of this internal friction maximum can be listed as follows :
a) The peak height ),,:Q( varies linearly with the heating or cooling rate T.
b) Even for T = 0, the maximum is still observed.
c)
Q ~ A ~
varies linearly with the inverse of the frequency fd) The internal friction is strongly dependent on the strain amplitude E for E > lo-' The effects of T and f o n the internal friction spectrum were first described by Belko et a1 (2) and Delorme et a1 (1) while the influence of E was formulated by De Jonghe et a1 (5). The internal friction during a martensitic transformation may then be writ- ten as : (1-'
3;
= a
7
+ Q-' (T,E)i
(1)In this expression the first term (or ) is only significant in the low frequency ran- ge (Hz range). In this paper we will mainly be concerned with the second term and as a consequence, the results presented have been obtained in the kHz range where the first term is very small and then t-he internal friction depends neither on the fre- quency nor o n the temperature rate T.
First of all, let us consider the temperature dependence of the internal friction spectrum (6). Figure 1 shows discontinuities in the internal friction and frequency curves as a function of the temperature. These discontinuities lead us to the hypo-
'Present address : Centro de Metalurgia Nuclear, IPEN,
Caixo Postal 11049
-
Pinheiros-
CEP 01000-SP- BrazilArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19824110
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thesis that the internal friction due to nlartensitic transformation is a critical phenomenon.
- , --.__
--\. -
:i ---. -.._ '.
.'. .. . Figure 1 : Internal friction and fre-
-
.!-
? , . . ,-
." quency measured as a func-1 .I _
----.._
tion of temperature in a1.
:
' (:uZnAl alloy.:: 1:
I
In consequence a theoretical calculation of the internal friction, based on a Lan- dau approximation of a first order phase transition ( 7 ) , will be presented and com- pared with the experimental results.
Fig. 2 : Strain amplitude dependence of Fig. 3 : ~ r a n a t o - ~ u c k e plot : the internal friction measured during
log E x Q-I versus E-I heating in CuZnA1 alloy
Moreover, (Fig.2) shows that for very low value of the amplitude of vibration, the internal friction is strongly amplitude dependent and that this dependence fol- follows the Granato-Licke theory ( 8 ) o f d i s l o ~ a t i o n b r e a k a w a ~ (4)(Fig.3). In order to take account of these results, the relaxation hypothesis used in the first calcula- tion will be replaced by an hysteretical approximation similar to the one used in the case of dislocation damping.
Internal friction during first order phase transitl*.
The underlying idea behind such a thermodynamic approach is that a pseudo equili- brium can be established within the solid. Assuming that the temperature T is cons- tant, the Gibbs function g may be expanded as a Taylor series in the variables 0
(stress) and
5
(internal variable or order parameter). For a first order phase tran- sition Landau proposes ( 7 )whereJ is the unrelnxed compliance and with y < 0, K > 0 and 6 = a ('1'4' )
The equilibrium value of
5
is given by A = 0, witere A = - is the affinity.ag
a,TIt follows that three different temperatures should be considered (6, 10) : Tc, 7' = T
+ I 2
and I A = T+ k r f
< ? < Teq c 16 Ka
!*,
c I Ka Fig. 4 : c Gibbs funcr eq h i o n y for d; rfrrent--A
( a ) ID) IC) values of the temperature (a)
Figure 4 shows the shape of the g function for different values of the temperature.
The equilibrium temperature T corresponds LO the case where the g function (Fig.
4c) has two minima with the same depth. 'J'he temperature Tc corresponds to the con- q dition where the central minimum (fig. 4b) disappears during coo- ling and could be considered as the limit of n~etastability of the high temperature phase. O n the contrary Th corresponds to the condition where the lateral minimum
(Fig. 4d) disappears during heating and could be considered as the limit of metas- tability of the low temperature phase.
Fig. 5 : Schematic representation of
5
(o=O) as a function of the temperature. See also ref.(lO)Figure 5 shows the possible value of
5
in the stable or metastable state. According to this figure the phase transition occurs either at Teq witliout thermal hysteresis or at '!' > Yeq during heating and 5' < Teq during cooling and in this case a thermal hysteresis is observed. 'The internal friction will be calculated on the two bran- ches that are for ? ' < T I , in the low temperature phase and for T > i", in the high temperature phase.From the Eq. (2) the internal friction is calculated using additionally the relaxa- tion hypothesis
= MA ( 3 )
and it has been shown (6) that
Q-1 =
k
-- . d L for T < T ~ , ( 4 )J u u 2 + K
These results are ploted on fig. 6. The second derivative of 3 with regard to
5
maybe interpreted as an elastic constant, the frequency being proportional to thesqua- re root of this constant (Fig. 6b).
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C U Z N A L ( 2 0 . 6 ) O U E N C H E O C
i r 1 2 K I M I N
- 3 511-
I C O O L I N G 2 H E A T I N G - 3 . ~ 2 -
3 3 0
--
--,- 2 9 680 120 160 2 0 0 2 1 0 2 8 0
F i g . 7 : Q-' a n d f a s a f u n c t i o n o f P d u r i n g m a r t e n s i t i c t r a n s - f o r m a t i o n i n a CuZnAL a l l o y ( ~ = 4 , 4 . lo-')
C u r v e 1 : C o o l i n g C u r v e 2 : H e a t i n g
2%
F i g . 6 : I n t e r n a l f r i c t i o n ( a ) a n d -
a c
( h ) a s a f u n c t i o n o f Lhe tent- p e r a t u r e .
A c c o r d i n g t o F i g . 6 a t h e i n t e r n a l f r i c t i o n s h o u l d show a s n l a l l d i s ~ o n t i n u i t y a t T = 3 ' ~ ~ when t h e d i s c o n t i n ~ ~ i t i e s a r c much more i m p o r t a n t a t Th o r T,.. M o r e o v e r F i g . 6 shows t h a t t h e c r i t i c a l b e h a v i a u r
(9 "
c, g o e s t h r o u g h z e r o ) o c c u r s Ynly a t i = ltC o n c o o l i n g and 'i' = 'Ph o n h e a t i n g .So F i g . 1 a n d
rip,.
6 a r e v e r y s i m i l a r i f i t i s assumed t h a t t h e t r a n s i t i o n o c c u r s a t T = Th d u r i n g h e a t i n g . I n c o n s e q u e n c e t h e t h e r m a l h y s t e r e s i s o b s e r v e d b e t w e e n h e a t i n g and c o o l i n g c o u l d b e e x p l a i n e d ( F i g . 7 ) .I n s h o r t , t h e c a l c u l a t i o n p r e s e n L e ( 1 g i v e s a v c r y good d e s c r i p t i o n o f c h e t e m p e r a t u r e d e p e n d e n c e o f e i t h e r t h c d a m p i n g o r t h e f r e q u e n c y o b s e r v e d d u r i n g h e a t i n g . I t i s t h e n o b v i o u s t h a t t h e m a r t e n s i t i c t r a n s f o r m a t i o n c o u l d o c c u r a t T h d u r i n g h e a t i n g a n d yc d u r i n g c o o l i n g . A l t h o u g h a n e l a s ~ i c : s o f t e n i n g mode i s o b s e r v e d on c o o l i n g , t h e i n t e r - n a l F r i c t i o n s p c c l r u r ; ~ d o e s n o t v e r i f y e x a c t l y t h e theoretical p r e d i c t i o n s . T h i s f a c t c o u l d be c o r r e l a t e d w i t h t h e i n t r o d u c t i o n o f i n t e r n a l s t r e s s e s d u r i n g l l l a r t e n s i c i c t r a n s f o r m a t i o n c f f c c t , wliicll h a s L O b e t a k e n i n t o a c c o u n t i n t h e e x p r e s s i o n o f t h e g f u n c t i o n .
F r e q u e n c y a n d s t r a i n a m p l i t u d e d e p e n d e n c e
-
--A s y s t e m a t i c sLudy o f t h e s t r a i n amplitude d e p e n d e n c e o f t h e i n t e r n a l f r i . c t i o n s p e c - t r u m , c a r r i e d o u t i n a w i d e a m p l i t ~ l d e a n d f r e q u e n c y r a n g e , h a s p o i n t e d o u t t h e f o l l o - w i n g r e s u l t s :
a) the damping is frequency independent
b) Even in the low strain amplitude range, the damping depends stronglyonthe strain (Fig. 2) up to an amplitude where a plateau appears, a new increase of the damping being observed at higher amplitudes. The strain amplitude dependence is schematical- ly represented on Fig. 8 where 3 domains are considered.
Fig. 8 : Schematic representation of the strain amplitude depen- dence
The stage A corresponds to the case where martensite is induced by the stress and this effect has been described by De Jonghe (4).As a1,readydescribed in the introduction the stage C verifies quite well the so called Granato ~ u c k e plot given on Fig. 3. In order to take account of this fact, the calculation summa- rised previously has to be changed.
More precisely the relaxation hypothesis (Eq. 3) should be replaced by a dyna- mica1 behaviour of the order parameter similar to the behaviour of the disloca- t ions when breakaway occurs.
The calculation ( 6 ) yields to :
for T > Tc
This equation sfiows about the same temperature dependence as Eq. ( 4 ) , but no fre- quency dependence is considered and the amplitude dependence corresponds exactly to the Granato-Lucke plot of the fig. 3.
The purpose of this paper was to show how the internal friction spectrum measured during martensitic transformation can be explained using thermodynamical analysis.
It also shows that internal f r i c ~ i o n measurenents can be used for carrying out in- forniation about martensitic transformations. Two types of results have been con- sidered : 'The internal friction and frequency spectrum measured as a function of the temperature or of the strain amplitude.
Using a Landau phenomonolugical theory developed in the case of a first order pha- se transition, important features of the internal friction and frequency spectrum have been explained.
A fundamental point in the application of the Landau theory is the choice of the Gibbs function g andthis choice could be the cause of partial agreement between experiments and theory. In any case for the [irst order phase transition, the cri-
tical behaviour does not appear at the equilibrium temperature hut at t w o diffe- rent temperatures Tc and Th during cooling and heatin& respectively.
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This result is quite in agreement withot~rexperimental datas.
In the case of a phenomenological Landau theory, the problem of the physical meaning of the so called order parameter
5
still remains. The strain amplitude dependence of the internal friction gives an important information on the dynami- cal behaviour of this parameter.In the case of Martensitic transformation the motion equationof5 around an equilibrium position is not given by a classical relaxation equation ( F , = M A ) , but is the same as the one of the dislocations when they break away from pinning points. In consequence our interpretation shows that the martensitic transforma- tion is controlled by a mechanism similar or identical to the motion of disloca- tions (9).
References
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(1971) C2-101(21 BELKO V.N., DARINSKIY R.H., lJOSTNIKOV V.S. and SHARSHAKOV I.M., l'hysics of Metals and Metallurgy,
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(1979), 1467 (4) UE JONGHE W., UELAEY I.., DE BATIST R. and HUMBEECK J.V., Metal Science,11 (1977) 523 -
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(1976) 1125 (6) KOSHIMIZU S., Thesis, EPF-Lausanne (1981)(7) GAUTIER F., in "Solid State Phase Transformations in Metals and ~ l l o y s " , Ecole d'Et6 d'Aussois (1978), p. 223, Les Editions de Physique.
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LUCKE
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(1980) 1773 and contributions at this conferenceAcknowledgements
The partial support or the Swiss National Science Fund through subsidy No. 2.835-0.80 is gratefully acknowledged.
