The influence of an anisotrophic elastic medium on the motion of dislocations: Application to the martensitic transformation

Scripta M E T A L L U R G I C A

Vol. I0, pp. 1075-1080, 1976 Printed in the U n i t e d States

Pergamon Press.

Inc.

THE INFLUENCE OF AN A N I S O T R O P I C ELASTIC M E D I U M ON THE M O T I O N OF DISLOCATIONS: A P P L I C A T I O N TO THE 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 O. M e r c i e r and K.N. Melton Brown Boveri R e s e a r c h Centre, CH-5401 Baden, (Received August 26, 1976) (Revised O c t o b e r 28. 1976)

Switzerland

(1-4)

It has been r e p o r t e d by several authors , that in certain alloy systems showing a m a r t e n s i t i c transformation, usually w i t h the formation of a t h e r m o e l a s t i c martensite, the elastic constants of the m a t e r i a l change significantly in the v i c i n i t y of the t r a n s f o r m a t i o n temperature. In p a r t i c u l a r the shear elastic c o n s t a n t (CI]-C~9)/2 decreases towards zero, as a c o n s e q u e n c e of w h i c h the elastic a n i s o £ r o ~ 9 factor A = 2 C a / ~ ] I - C ] ~ ) increases and can become as high as 20 or 30. The purpose of t h 4 " p r ~ e n £ - p a p e r is to examine the effects of this a n i s o t r o p y on the lattice dislocations and the m a t e r i a l properties w h i c h depend on them. In the first part, the elastic energy of these d i s l o c a t i o n s will be d i s c u s s e d as a function of (C -C )/2 and in the second 1 12 part the effect of the change of this p a r a m e t e r on ~he internal friction, the yield stress and the n u c l e a t i o n of m a r t e n s i t e will be described. i. The Elastic Energy of Dislocations Usually, to calculate the elastic energy of a dislocation, the a p p r o x i m a tion of an isotropic m e d i u m is used, the elastic energy W per unit length depending only on the average elastic constant U. In the case of an anisotropic material, the 9x9 m a t r i x Cijkl of the elastic constants m u s t be used, Ci~kl H being defined by ~ij = CijklSkl

(i)

where i,j,k,l take the values 1,2,3 and the unit vectors of the frame Ox. are parallel to the directions. To calculate the d l s p l a c e m e n t aroug~%a dlslocation lying along any d i r e c t i o n a new m a t r i x C' .... may be defined" " in a i K frame Ox! w i t h an axis Ox" parallel to the line d i r ~ c % i o n t of the d i s l o c a t i o n (Fig. l)isuch that J •

1

C' ijkl = Q i j g h C g h m n Q m n k l where,, Q,mnKz_~ and Qijgh are the m a t r i c e s of t r a n s f o r m a t i o n OXlX2X 3 •

.

(2) from OXlX2X 3 to

With the help of these new elastic constants C' .... , Foreman (18) was able Kl to derive a formula for the elastic energy per unit ~ n g t h W a of dislocations in an a n i s o t r o p i c m e d i u m WA

=

uAb 2 in R 47 r

(3) O

b,R,r having their usual d e f i n i t i o n and U A r e p l a c i n g the elastic c o n s t a n t o and being called the energy factor. Only in some limiting cases,

can ~

A

be e s t i m a t e d analytically.

1075

In par-

1076

ANISOTROPIC ELASTICITY IN MARTENSITIC TRANSFORMATION

Vol.10, No. 12

ticular, if the plane x.x_' ' is' a reflection plane of the considered structure I z the following e l a s t i c constants equal zero C'u873 = 0

e,8,7 = 1,2

C'u333 = 0

U

= 1,2

(4)

A and if C' .... is reduced to the 6x6 matrix C'.., ~ for a dislocation lying parallel %~ 6xq, is given for a screw dislocation with b l lOx{, by (for a detailed calcuIation, see (5)), A s = (C,44C,55 _ C,452)I/2

(5)

and for an edge dislocation with bxl I Ox{, .bx2 l lOx~ and t l lOx{, by

a

:

gexl

~I

71/2

, -, , C66(Cli C12)

I

(6)

(Cl1+C12)

--I

I

U

!

(C11+C12+2C66)C22

A ~ex 2

___

,!

s --I _ ! C66(Cii C12)

u

(~"II+C12)

--!

l

I

l

1/2

(7)

!

(Cli+C12+2C66)Cli Cll-' being defined by -' C11

=

' ' (CII . C22)1/2

(8)

To compute exactly now the elastic energy of a dislocation of a given structure, the constant C'.. must be evaluated with the help of the matrix of transformation from the fraZ~e Ox. to Ox!. 1 1 The results will be given only for two types of dislocations of the F.C.C. structure. For other structures (B.C.C. or hexagonal), the calculations are much more complicated, but the type of results will be similar and the same general conclusions can be drawn. For a screw dislocation, lying in a direction and gliding in a {iii} plane (F.C.C. structure) or in a {ii0} plane (NaCl-type structure)

el

1/2

S

(9)

For an edge dislocation lying in a direction and gliding in plane a Pax I

A

(Cii+C12) [C44 (Cll_Cl2)J

= ~ex 2

~ (C11+C12+2C44) Cll ]I/2

1/2

a

{llO}

(lO)

For the edge component of a 60 ° mixed dislocation lying in a direction and gliding in a {111} plane

Vol.

lo,

A

v ex

No.

= 1

ANISOTROPIC ELASTICITY IN MARTENSITIC TRANSFORMATION

12

(i

1077

cll(cll+c12+2c44)11/2 2

+ C12L

1

(11)

c~~~[c~~~c~~+c~2+2c~~~11'2-c~2~

:

2

1

l/2

I L${[C,, (cll+c12+2c44)Jl'2+c12+2c44~ (cll+c12+2c 1 ' 44 J

2

A U ex

2

=

Cl1 (cll+c12+2c44)

(

2

[

l/2

I

+ Cl21-

‘44{ [C11(C11’C12+2c44,] 1’2-c121

l/2

2

I

(12)

For our present purposes, the important point concerning these equations is that for eq. 9 and 10, the energy factor is proportional to JC -Cl2 whereas eq. 11 and 12 have no terms containing functions of (C -C ).'if we now ) 2 approaches zero during ;he phase %aftZformation suppose that (C the energy factbh-c12~~ defined in eq. (9) and exlr UA ex defined in eq.'(lO) # will also go to zero, whereas uA exl, eq. (11)i iJex2r eq. tl2) will not. As a consequence, in the vicinity of MS, the martensite transformation temperature, dislocations inF.C.C. structures such as Fe-Ni, some Cu-al10ys(~' should show very different behavior to these in NaCl-type structures such as In-Tl. For the B.C.C. structures (CsCl type), results similar to these for NaCl-type structures were obtained for the 35016' mixed dislocations lying in irection and gliding in a IllO] plane and similar behavior can be for di ocations lying in directions, from calculations carried on by Head ‘ii, . 2. Consequences of the Anisotropy on Different Properties of the Material In this part, only the structures containing dislocations having an elastic energy going to zero during the martensitic transformation will be discussed. For materials having dislocations gliding in a clll) plane, similar effects should not be observed. Internal friction -1 resulting from dislocation motion in a material The internal friction Q under an alternating stress o = co exp (iwt) can often be described by a Granato-Liicketype equation (see for example(g)) (13) where g 2 is a geometric fa tar, Jo the compliance, K the "spring constant"and fi -tthe relaxation time ( = K, B being the damping constant).

1078

ANISOTROPIC

ELASTICITY

In a quasi-static K ~ 12 ~b2

IN MARTENSITIC

approximation,

TRANSFORMATION

Vol.

K is equal to (14)

If we now replace U by ~A, and consider the case ~T << i, e.g. torsional pendulum, eq. (13) becomes Q-I

I0+ No.12

~2 IZ4B~ = 144b2Jo(~A) i

in a (13i)

ie Q-I ~ (,.~)2. Thus if ~A has a m i n i m u m and approaches zero at the transformation temperature, then eq. (13i) predicts a corresponding m a x i m u m in internal friction (Fig. 2). Such a m a x i m u m has been observed (I0-12) • If we now consider the effect of frequency on the internal friction and assu~e that a peak is observed at a frequency Up such that ~pT = i, then since T = ~ , from eq. (14) ~p may be w r i t t e n ~P= 12 ~A b 2 B£ 2 and

(15)

(13) now becomes

-I g21£2 Qmax = 24 Jo ~A (13ii) These equations predict that the frequency ~p at which the m a x i m u m value of internal friction Qm~x is observed is a functlon of ~A. C o n s e q u e n t l y as the temperature (and hence ~A) is varied, Up should change and show a m i n i m u m at M s and furthermore the value of Qm~x should be a m a x i m u m at Ms, since ~A is a m i n i m u m (Fig. 3). There are insufficient published data to fully verify the predictions of eq. (13ii) and (15), but the results of Pace and Saunders (ii) over a limited range of frequencies and temperatures can be fully explained i.e. ~p decreases and Qmax-1 increases as the temperature is decreased to the t r a n s f o r m a t i o n temperature. Critical

stress

for d i s l o c a t i o n m u l t i p l i c a t i o n

The critical stress ~c for d i s l o c a t i o n m u l t i p l i c a t i o n by the Frank-Read m e c h a n i s m is given for a d i s l o c a t i o n of length i, by, (u being a constant) b ~c = 2~ ~ ~

(16)

Now if we replace ~ by ~A, the stress Oc will decrease and goes through a m i n i m u m during the martensitic transformation. A consequence of this is that d i s l o c a t i o n m u l t i p l i c a t i o n processes at the t r a n s f o r m a t i o n temperature will be efficient, allowing the shear transformation to occur easily. Furthermore Oc will be low in the martensitic phase, so that twinning of the m a r t e n s i t e should occur easily, since twinning can occur by a d i s l o c a t i o n process. Yield stress For materials showing a thermoelastic m a r t e n s i t i c transformation, the yield stress is a function of temperature near M s , since a stress induced m a r t e n s i t e is formed. As the temperature is increased above M s , the yield stress increases rapidly until a temperature M d is reached at which no martensite can be stress induced, and the material then shows normal temperature dependence of a plastic yield point. At M s itself, a m i n i m u m in yield stress is observed (13,14,15) • This m i n i m u m may be correlated with the low value of Cc at M s. Nucleation

of the m a r t e n s i t e phase

A criterion defining the unstable dislocation

in an anisotropic

crystal

Vol.

i0, No.

12

ANISOTROPIC ELASTICITY

IN 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

1079

was given by de Wit and K o e h l e r (16) . If W (8) is the line energy of a straight d i s l o c a t i o n as a function of e, the d i r e c t i o n of the d i s l o c a t i o n in its slip plane, then the d i s l o c a t i o n is unstable if it lies in a d i r e c t i o n for which W

d2W (8) + d e ~ < 0

(17)

Head (16) found that if the anisotropic constant A becomes large (20-30), d i s l o c a t i o n s remain stable only around 1 or 2 directions which contains always the d i r e c t i o n of the screw dislocation.

the

W h e n the m a t e r i a l approaches the temperature of the phase t r a n s i t i o n and (CII-C12)/2 approaches zero, the a n i s o t r o p y will increase and therefore the d i s l o c a t i o n s become unstable in certain directions. As a result, they will r e a r r a n g e and line up in the stable orientations forming walls of dislocations. In a theory of m a r t e n s i t e nucleation, M e n d e l s o n (17) analysed a model based on zonal d i s l o c a t i o n s at the m a r t e n s i t e / p a r e n t interface, but did not e x p l a i n the m e c h a n i s m w h e r e b y the d i s l o c a t i o n s m i g r a t e d to the interface. U s i n g the present ideas, it can be seen that there is a strong d r i v i n g force for dislocations to rearrange and the r e s u l t i n g network may act as the site for nucleation of the ma~tensite. If the d i s l o c a t i o n walls accommodate the lattice m i s m a t c h b e t w e e n parent and m a r t e n s i t e phase, then the fact that their elastic energy goes to zero means that they can move under low stresses, i.e. the interphase b o u n d a r y can be mobile, possibly r e s u l t i n g in the t h e r m o e l a s t i c effects. 3. C o n c l u s i o n In this paper, we have shown in a q u a l i t a t i v e way that the e l a s t i c energy and the s t a b i l i t y of dislocations can vary d r a s t i c a l l y during a m a r t e n s i t i c phase transformation. This is due to the shear elastic constant (CII-C12)/2 going through a m i n i m u m at these temperatures. The consequences of this are that for some structures (B.C.C., NaCl-type) the internal f r i c t i o n goes through a m a x i m u m at low frequencies and the yield stress through a minimum. The change in s t a b i l i t y also helps e x p l a i n the M a n d e l s o n (17) theory of martensite nucleation. For other structures, like the F.C.C. structure, only some d i s l o c a t i o n s have this behavior; other d i s l o c a t i o n s keep their e l a s t i c e n e r g y almost unchanged. This could contribute to the very high n u c l e a t i o n energy for the 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 of these alloys. REFERENCES i. 2. 3. 4. 5. 6. 7. 8. 9. i0. ii. 12.

L.C. C h a n g and T.A. Read, Trans. AIME, 189, 47 (1951). Z.S. B a s i n s k i and J.W. Christian, A c t a Met., 2, 148 (1954). W.J. Buehler, J.V. Gilfrich and R.C. Wiley, J. Appl. Phys., 34, 1475(1963) N. Nakanishi, Y. M u r a k a m i and S. Kachi, Scripta Met., 5, 433 (1971). J.P. Hirth and J. Lothe, Thoery of dislocations, M c G r a w Hill, (1968) p. 398 et seq. G.B. Brook, R.F. Iles and P.L. Brooks, in Shape M e m o r y Effects in Alloys, e d i t e d by Jeff Perkins, P l e n u m Press, 477 (1975). O. Mercier, to be published. A.K. Head, Phys. Stat. Solidi, 5, 51 (1964) ~ 6, 461 (1964). O. M e r c i e r and W. Benoit, R a d i a t i o n effects, 27, 207 (1976). R.R. H a s i g u t i and K. Iwasaki, J. Appl. Phys., 39, 5 (1968). N.G. Pace and G.A. Saunders, Phil. Mag., 22, 73 (1970). W. De Jonghe, R. De Batist, L. Deleay and M. De Bonte, in Shape M e m o r y Effects in alloys, edited by Jeff Perkins, P l e n u m Press, 451 (1975).

1080

ANISOTROPIC ELASTICITY IN MARTENSITIC TRANSFORMATIONS

Vol. 10,No. 12

13. H. Warlimont and L. Delaey, Martensitic transformations in copper, silver and gold-based alloys, Prog. Mat. Sol. Vol. 18, (1975). 14. W.B. Cross, A.H. Kariotis and F.J. Stimler, Nitinol Characterisation Study NASA-CR-1433 (1969). 15. S. Miura, M. Ito and N. Nakanishi, Scripta Met., i0, 87 (1976). 16. A.K. Head, Phys. Stat. Solidi, 19, 185 (1967). 17. S. Mendelson, in Shape memory effects in alloys, edited by Jeff Perkins, Plenum Press, 487 (1975). 18. A.J.F. Foreman, Acta met., 3, 32 (1955).

FIG. i i

lllustration of the relationship between the crystal axes Oxi and a new set of axes Ox'i, such that Ox' 3 is parallel to the dislocation line direction t . The screw component b s lies along Ox'3, while the edge component b e is in the plane Ox'ix' 2 and is resolved into bx, I along Ox' I and bx, 2 along Ox' 2 .

x3 • bt, 3 / I

,

,//\

t , /line dir.ction ,/

~

o, o ° . o o o ,

"

oo

~2

1/~,,

"x I

I

I I T

"T/

I

1"o

arbitrclry scall

FIG. 2 Schematic representation of the shear elastic constant C' = (CII-C12)/2, of the energy_~actor ~A and of the internal friction 0 as a function of temperature. The subscript M designates the martensitic phase and the subscript A the austenitic phase.

10 ~

0

,o / t0- 5

10-2

t0-I

10 0

101

t0 2

FIG. 3 Schematic representation of the normalised internal friction O-I/Q$ I as a function of the normalised frequency ~To for different values o~ the n~rmalised energy factor ~= ~'~o • Q is equal to ~ ! - - - a n d 1.o =B~2 12 ~ob~