' . / � ! -'.- . (�) ) - • \...r , -'

Indian Journal of Pure & Applied Physics Vol. 40, November 2002, pp. 775-779

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Temperature dependence of strain associated with domain walls in martensitic phase t�ansitiolJ from fcc to fct

. LA\DogartA tArslanr & .y jHavvatoglu\ \ KSU Fen-Edebiyat Fak�I;�si, Fizik Bolumu, 461 00, K.Maras, Turk�y '- . Received 27 June 200 1 ; revised 22 January 2002; accepted 5 April 2002

[in�TI alloy undergoes a martensitic transition from fcc to fct and is treated by a phenomenological theory based on Landau' s theory of phase transitions. Analytic expressions concerning the temperature aejJendence of the strain associated with the ??main wafls observed in this type of alloys are giVen

(�n the present study. ( " .

1 Introduction

Non-l inear differential equation systems associated with non- l inear structures in metal physics have become increasingly important with the introduction of new ideas from non-linear science . There is as yet no complete theory explaining all the features .of the martensit ic transformation. With a view to understand the elastic structure formation i n martensltlc transformations, attention is focussed on a l attice model which enables one to describe a phase transformation from cubic to tetragonal . A one­dimensional Landau theory for the martensit ic phase transi ti ons has been studied by Falk l . H i s model i s able t o predict qual i tati vely, in spite of the inherent over s impl ification, the observed phenomena in shape memory al loys such as , pseudoelasticity , ferroelasticity, l attice softening etcY. The phenomena in question are due to a first-order martensi t ic phase transition. The martensit ic phase transition in some al loys such as, Cu-base al loys i s from an ordered bcc crystal l attice with CsCI or Fe:AI superstructure into the orthorhombic symmetry with 9R, 1 8R, 3R or 2H stacking sequences. The martensitic phase transformation in these al loys is concerned wtih a shear deformation paral lel to the habit plane which i s near ( 1 1 0) i n direction . The Landau free energy density function associated with th is type of shear system has :

. . . ( 1 )

where a, � and y are posit i ve constant depending on

* Author for correspondence

the specific al loy . This function i s main ly focused on determining the strain of the domain boundary in the martensitic transformation I . On the other hand, for cubic-tetragonal phase trans ition, as in Bain d istortion, the free energy density function i n terms of the strain may be written as fol lows (Fig. I ) :

. . . (2)

where a, b and c are phenomenol ogical posItIve coefficients depending on the specific a l loys4.5.

The aim of th i s paper is to find analytical expressions for the macroscopic strain associated wi th the domain wall s observed as fin ite and un­bounded crystal as a function of the temperature parameter and posi tion using Landau' s free energy density function in Eq. (2) instead of Eg. ( I ) .

2 A Model For Fcc to Fct Martensite Phase Transformation

Including a strain gradient term which corresponds to the lattice curvature containing the domain wal l s, the Ginzburg-Landau free-energy can be written as :

. . . (3)

where 0 i s a posItive constant because of the curvature stabi l i ty of the lattice and x represents the stacking direction perpendicul ar to the habit p lane between Austeni te and Martens i te, which is near the p lane ( 1 1 0) . Replacement of Eq. ( l ) into Eq. (2) yields :

776 INDIAN J PURE & APPL PHYS, VOL 40, NOVEMBER 2002

- ? - 1 -4 e ( d- )2 G = ae - -be ' + ce + 8 ill . . . (4)

Here, a = a'(f -� ), and � IS defined as

martensite starting temperature. Although a, b and c coefficients in th is equation are temperature

dependent parameters, G i s affected markedly only by the coefficient a which is a combination of the second-order elastic constant and temperature. Eq. (4) can be written as :

4 1 ( I } ? '2 G = e - e' + I T+4" '- + e

. . . (5)

where

4 1 ( 1 } ? g = e - e' + l T +4" ,-

Al l the coeffic ients i n Eq. (5) were expressed in re­scaled units as fol lows :

c _ e = -e b

a'c (- - ) 1 T = Il T -� -4" . . . (6)

The discussion of the re-scaled Landau free­energy with respect to the order parameter, i .e . shear strain, for different temperature ranges should be considered, such as, T > 1 132, T <= 1 /32 and T < 1 /32. On the other hand, the austenitic phase corresponding to e = ° is stable for a temperature T > 1 /32. There is a region where the Austenite and Martensi te phases co-exist near the temperature T <=

1 /32. At th is temperature, the depths of the minima of Landau ' s free-energy wel l s are the same. From Eq. ( I ) , the pos ition of the martensit ic minimum can be calcul ated as :

. . . (7)

where T < 1 /32.

In case there is no external force appl ied to the specimen, Euler-Lagrange equations of the system are:

CJ - p' = 0 . . . ( 8)

where p' i s a derivat ive of the p = fJG/fJe' = 2e' .

From e" = (de'/de)e: and Eq. (8), an express ion can

be written as :

e'(de'/de)= CJ/2 . . . (9)

Integration of th is expression leads to the differential equation :

e' = de/dx = �g - go . . . ( 1 0)

Here, the lattice curvature e' condit ion i s :

e'(xl ) = �g(e(xl ))- go = �g(e(x2 ))- go = O , g > gll

. . . ( I I )

This equation states that the strains at the two surface points of the sample are such that, Landau' s free energies take the same value g'h namely :

. . . ( 1 2)

In order to sol ve Eq. ( 1 0) , define a function Q(e) by:

Q(e) = g - gll . . . ( 1 3)

On the other hand, an expression for th is function can be rewritten as in the form:

. . . ( 1 4)

where the values e l > e2, e, and e4 represent points of the constant gil with the curves g. Moreover, by assumming e l + e2 - (e, + e4) :!; 0, consider a new variable z to give:

p + qz e =--I + z . . . ( 1 5)

then, substitute th is expression into Eq. ( 1 4) . Using less algebra and taking Eq. ( 1 0) into account, one can obtain an expression for the strain of the domain wal l s associated with the Martensite-Martensite as in the form:

e(z ) = (q - p)

�G - � XP - � XP - � Xp - � ) dz

f �(I _ g 2 Z 2 XI - h2z2)

. . . ( 1 6)

DOGAN et at. : MARTENSITIC PHASE TRANSITION 777

0,04 1

"

.;. ·0,2 !,II ·0,02 � = � � -0,04 � '"

·0,06

·0,08

.0, 1 J Strain, e

Fig. I - Variation of Landau ' s free energy versus shear strain at several different temperatures in the Martensitic transformation from fcc to fct (Here, the energy and strain have dimensionless units)

u .5' g [fJ

·50 -30

0,4

o 5

2

0,1

-10 10 30 50 Stacking direction, x

Fig. 2 - The structure of Martensite-Martensite domain walls in a finite crystal for the temperature T = 0 (Here, the strain and position have dimensionless form)

in which:

h2 = (e, - qXq - e2 ) (p - e, Xp - eJ . . . ( 1 7)

-50 -40 -30 -20 -10 0 10 20 30 40 50 Stacking direction, x

Fig. 3 - The structure of Martensite-Martensite domain wal ls in an un-bounded crystal for the temperature T= 0 (Here, the strain and position have dimensionless form)

0 5

0 2

-50 -30 -10 10 30 50 Stacking direction, x

Fig. 4 - The structure of Austenite-Martensite domain wal l s in a finite crystal for the temperature T = 0 (Here, the strain and position have dimensionless form)

and

g2 = (e, - qXq- e2 ) (p - e3 Xp - e4 ) . . . ( 1 8)

Integration of Eq. ( 1 5) yields:

778 INDIAN J PURE & APPL PHYS, VOL 40, NOVEMBER 2002

ph + q sn({3x,k ) e (x) = -'-------''------'.:--'-..:.-

h + sn({3x.k ) . . . ( 1 9)

with the Jacobian e l l iptic function sn [Refs 6, 7 ] . Here, p , q, f3 and k were defined by:

function tan h. Thus, Egs ( 1 9) and (24) obtained in the present study can be re-written as :

) (ph + q tan hfJfJ) e(x = ...:..:....,._""'-....-_:....;....:.. (h + tan h{3x) . . . (27)

and

. . . (28)

0,24

. . . (20) 0,18

. . . (2 1 )

. . . (22)

and

. . . (23)

where e3* equals to e4 which is root of Eq. ( 1 4) .

Moreover, both e3 and e4 are i maginary roots.

On the other hand, for Martens ite-Austenite domain wal l s there is an arrangement among the roots e l , e l , e2, eo and e4 as el < e2 < e, < e4 and el + e2 = eJ + e4. Assumming the change of variable e = Z + [(e l + e2)/2] and following the calculation in the previous paragraph , a strain expression for the domain wal l s in question can be obtained as:

. . . (24)

in which

. . . (25)

and

k = (el - e2 [ /(el + e2 - 2e3 Xe, + e2 - 2e4 ) . . . (26)

In un-bounded crystals, k = 1 and the Jacobian e l lipt ic function sn is reduced to the trigonometric

� 0,12

d' .; 0 06 .l:l ' r/)

o -0,06

-0,12 ./==�:::::::::�---.---,-----, -400 -200 o 200 400

Stacking direction, x

Fig. 5 - The structure of Austenite-Martensite domain walls in an un-bounded crystal for the temperature T= 0 ( H ere. the strain and position have dimensionless form)

3 Results and Discussion

Some coeffic ients of the Gibbs free-energy In the al loy Au-30%Cu-47%Zn were determined i n Ref. I . On the other hand, for the al loy In-TI d isplaying martens itic phase transformat ion from fcc to fct, the coeffic ients corresponding to those above in Eq. (4) were experimentall y determined as a I = 2.5 x 1O� N/m2 deg, b = 25.6 x 1 09 N/m2 and c = 3 .4 1 X 1 01 1 N/m2 [Ref. 2, 8] . The strain gradient coeffic ient 0, in Eq. (4) has an approximate value of 1 0. 1 2 N/m2 [Ref. 9 ] . However, there are very l imited experimental data for these coeffic ients. For convenience of calcul at ion, all the coefficients and temperatures were changed to the d imensionless form. So, i t is noteworthy that, the coeffic ients were got rid of in the calcul ations. Considering th i s case, the positions of the martensit ic mini ma were determined from Eg. (7) and d isp layed in Fig. I . For a fin i te crystal at a temperature T = 0, Eq. ( 1 9) was p lotted in Fig. 2 as strain versus staking d i rection x,

DOGAN et al. : MARTENSITIC PHASE TRANSITION 779

which g ives usefu l information on the behaviour of the strain, associated with the domain wall s of Martensite-Martensite. In Fig. 2, a periodic arrangement of the martensltlc variants were depicted. B y using Eg. (27) which d isp lays domain walls associated with an un-bounded crystal, another figure was p lotted, i .e. kink solution (Fig. 3). On the other hand, the structure of Austenite­Martensite wall s in a fin ite crystal can be obtained from Eg. (24) . The graphs representing Egs (24) and (28) can be v isuali zed by Figs 4 and 5, respectively . Fig. 5 represents the structure of Austenite­Martensite wall s in an un-bounded crystal . Good simi larities between the shapes of the curves i n the present study and those of Ref. 10 can be readi l y seen .

To summarize, there are many common theories in solid state physics, such as, the strain dependence in terms of e4 or e(" which represent symmetry properti es of the martensit ic phase transitions. As far as the authors know, lots of them reflect the theoretical studies concerning the theory e(" and a large number of analytic relat ions dealing with these studies were reported 1 1 - 1 3 . However, in the present study, some analytic expressions for the strain, associated with the domain wall s were obtained for

the fin ite and un-bounded al loy as in In-TI, which represents a phase transit ion from fcc (m3m) to fct (4/mmm). The strain dependence i n the present model is in a fair ly good agreement with the results of some numerical studies 10.

References

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5 Koyama Y & Nittono 0, J Phys Colloq, C4 ( 1 982) 43, 1 45 .

6 Abramowitz M & Stegun I A, Handbook ()f mathematical junctions, 9th edition, (Pover Publ ications, New York) , 1 972.

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