THE PHENOMENOLOGICAL THEORIES OF MARTENSITE CRYSTALLOGRAPHY

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THE PHENOMENOLOGICAL THEORIES OF MARTENSITE CRYSTALLOGRAPHY

A. Crocker

To cite this version:

A. Crocker. THE PHENOMENOLOGICAL THEORIES OF MARTENSITE CRYSTALLOGRA- PHY. Journal de Physique Colloques, 1982, 43 (C4), pp.C4-209-C4-214. �10.1051/jphyscol:1982426�.

�jpa-00222140�

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JOURNAL DE PHYSIQUE

CoZloque C4, suppllment au n o 12, Tome 43, dlcembre 1982

THE'PHENOMENOLOGICAL T H E O R I E S OF M A R T E N S I T E CRYSTALLOGRAPHY

A.G. C r o c k e r

Department of Physics, University o f Surrey, Guildford, Surrey GUZ SXH, Eng Land

( A c c e p t e d 9 A u g u s t 1982)

Abstract.- The development of the basic phenomenological theories of martensite m o g r a p h y i n t o the generalised mu1 t i p l e l a t t i c e invariant shear analysis

of Acton e t a l . , referred to a s the

CRAB

theory i s summarised. Unfortunately the generality of the

CRAB

theory, and perhaps t h e concise notation adopted

i n

i t s presentation, appears t o have obscured the f a c t t h a t i t incorporates, as a special case, a new and very simple, d i r e c t and powerful formulation of the basic single shear theory. In order t o encourage wider use of t h i s method of predicting the crystallographic features of martensitic transformations, the single shear

CRAB

theory i s presented here, using elementary matrix algebra, f o r the case when the parent s t r u c t u r e i s referred t o an orthonormal basis.

The corresponding equations which a r i s e when a general parent basis i s adopted are a l s o given using t h e notation of the tensor calculus, in an appendix.

Introduction.- The well established phenomenological theories of the crystallography of martensitic transformations due t o Wechsler, Lieberman and Read ( 1 ) and t o Bowles and Mackenzie ( 2 ) were developed about 30 years ago. These theories, and the

s l i g h t l y l a t e r analysis of Bullough and Bil by ( 3 ) based on surface dislocation theory, a r e a1 1 e s s e n t i a l l y equivalent ( 4 ) , a1 though t o some extent the original formulations were r e s t r i c t e d t o p a r t i c u l a r transformations. In each case the principal assumption i s t h a t the t o t a l shape deformation i s an invariant plane s t r a i n , which i s represented here by the matrix

F.

The s i n g l e plane which i s l e f t both undistorted and unrotated by t h i s deformation-is taken t o be the i n t e r f a c e between the parent and product s t r u c t u r e s , and any volume change associated with the transformation i s assumed t o be accommodated by a s t r a i n normal to t h i s plane. The theories formally resolve the deformation E i n t o a l a t t i c e deformation D, and a l a t t i c e invariant deformation 3

( 5 ) .

When combined with any necessary atomic shuffling, the deformation I! converts the parent s t r u c t u r e i n t o the product. I t i s completely defined by the crystal s t r u c t u r e s of the two phases and a correspondence matrix C which s p e c i f i e s which parent u n i t c e l l i s deformed into a p a r t i c u l a r product unit c e l l . In practice i t i s convenient t o resolve into a rotation 8 and a pure s t r a i n p, which includes the volume change of the transformation. The deformation

S

i s assumed t o be a simple shear of magnitude g in a direction

a.

on a plane with normal m. I t does not change t h e crystal s t r u c t u r e and may therefore-be considered t o be s l i p ; twinning o r f a u l t i n g . The basic theories use the two crystal s t r u c t u r e s ,

C ,

m and a as data and give solutions f o r g and F.

There are i n general two possible values of

g ,

which defines f o r example the fraction of the product s t r u c t u r e which i s twinned. For each of these two values there a r e two solutions f o r F giving four in a l l . Each of these defines a possible habit plane of normal h and associated direction u and magnitude f of the deformation ( 4 ) .

The original formulations of these theories (1-3) of martensite crystallography were a l l based on rather complex and lengthy matrix algebra and, apart from a few special cases, solutions could only be obtained following tedious numerical comput- a t i o n s . Alternative formulations based on geometrical methods

( 6 )

and manipulations using stereographic projections

( 7 )

were therefore developed. In addition f o r some transformations the nature of the l a t t i c e invariant shear o r the correspondence or even the product crystal s t r u c t u r e was not known, although experimental information was available on the t o t a l shape deformation. Versions of the theory which could be used i n reverse t o determine information about the unknown mechanisms were therefore

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982426

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C4-210 JOURNAL DE PHYSIQUE

formulated (8,9). However e a r l y applications of the basic theories revealed t h a t the crystallographic features of some transformations could not be predicted s a t i s f a c t - o r i l y (4) so e f f o r t s were made t o introduce some f l e x i b i l i t y into the assumptions.

For example a d i l a t a t i o n parameter was introduced to allow f o r possible uniform s t r a i n s of the i n t e r f a c e ( 2 ) and attempts were made t o assess the e f f e c t of non- uniform s t r a i n s (10). In addition the influence of more complex l a t t i c e invariant shears was investigated by resolving the matrix

S

i n t o two o r more shears (11). If the group of shears share the same plane

m

or the same direction

a

they a r e auto- matically equivalent t o a single shear and can therefore be immediately accommodated in the theory. This i s also t r u e i f a l l the shear plane normals

mi

and shear d i r e c t - ions

-

.ti l i e in a common plane S, except t h a t i n t h i s case an additional rotation

-

about the normal t o

2

i s required (12). However none of these developments constituted any e s s e n t i a l change t o the basic formulation of the theory.

In

1970 Acton, Bevis, Crocker and Ross (13) published a generalised but very concise version of the martensite crystallography theories which allowed f o r independent multiple l a t t i c e invariant shears. This had been developed from a new analysis of the theory of the crystallography of deformation twinning (14,15). The need f o r t h i s theory had arisen from the discovery t h a t the basic d e f i n i t i o n of a twinning shear, being a homogeneous shear which generates the same l a t t i c e in a new orientation, allowed twinning modes with d i f f e r e n t c h a r a c t e r i s t i c s from those accepted previously. In p a r t i c u l a r modes in which both the shear plane and shear direction have i r r a t i o n a l indices had been discovered whilst investigating the phenomenon of double twinning ( 1 6 ) . Conventional twinning modes have a t l e a s t one of these elements r a t i o n a l . The extension of t h i s work t o produce a martensite c r y s t a l - lography theory with a double l a t t i c e invariant shear was carried out independently by Acton and Bevis (17) and by Ross and Crocker (18). The two approaches were however complementary, the former relying e n t i r e l y on algebraic manipulations while the l a t t e r was based on geometrical concepts. The two groups then collaborated to produce a general analysis which could be used t o investigate s i x types of l a t t i c e transfornation (13). These were s i n g l e and double twinninq shears, i n which the parent and product structures a r e the same, s i n g l e and double invariant plane transformations, w i t h a change of s t r u c t u r e but no l a t t i c e invariant shears, and f i n a l l y single and double shear martensitic transformations.

In the double shear martensite theory (17,18) the t o t a l shape deformation

F

i s resolved i n t o a rotation

R

a pure s t r a i n

P

and two l a t t i c e invariant shears and

S 2 .

Given the parent and product s t r u c t u r e s , the correspondence

C

the shear planes In, and rn, and directions

a,

^and

a,

^of

S,

^and

S,

and the shear magnitude g, of

S ,

a r g, of

S,,

the theory provides two possible values of g, o r g, f o r each of which there a r e two solutions f o r

F.

The analysis therefore provides an additional parameter, e i t h e r gl and g2, which can be used in an attempt t o explain the c h a r a c t e r i s t i c features of transformations which do not s a t i s f y the basic theory.

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U n f o r t u n a t e l y the g e n e r a l i t y o f the a n a l y s i s due t o Acton e t a l . (13), which here w i l l be r e f e r r e d t o as t h e CRAB theory, obscures t h e f a c t t h a t the new f o r m u l a t - i o n o f t h e b a s i c s i n g l e shear theory o f m a r t e n s i t e c r y s t a l l o g r a p h y , which i t i n c o r - porates, provides a very simple, d i r e c t and powerful method f o r o b t a i n i n g numerical s o l u t i o n s f o r t h e c h a r a c t e r i s t i c f e a t u r e s o f transformations, and a l s o gives a c l e a r i n s i g h t i n t o t h e mechanisms i n v o l v e d (19,20). I n t h e present paper a s i m p l i f i e d v e r s i o n o f t h e CRAB theory i n v o l v i n g o n l y a s i n g l e l a t t i c e i n v a r i a n t shear i s presented. I n most a p p l i c a t i o n s o f t h e theory i t i s p o s s i b l e t o make use o f simple m a t r i x algebra n o t a t i o n and t h i s has been adopted i n t h e main t e x t . The correspond-

i n g equations f o r the general case i n which the parent s t r u c t u r e i s u n r e s t r i c t e d are presented i n t h e Appendix. I n t h i s case t h e more convenient b u t perhaps l e s s f a m i l i a r n o t a t i o n o f t h e tensor c a l c u l u s has been used.

The S i n g l e Shear CRAB Theory of M a r t e n s i t e Crystallography.- L e t t h e parent s t r u c t u r e be deformed by an i n v a r i a n t plane s t r a i n o f magnitude f on a plane o f normal

h

i n t h e d i r e c t i o n

u.

^Here and o t h e r deformations w i l l be represented by (3x3) m a t r i c e s and h and u and o t h e r u n i t vectors by (3x1) column matrices. Hence

where

I

i s the u n i t m a t r i x and the s u p e r s c r i p t T i n d i c a t e s t r a n s p o s i t i o n . Resolve

F

i n t o an orthogonal r o t a t i o n m a t r i x

R,

a symmetric pure s t r a i n

P

and a l a t t i c e i n v a r i - a n t shear

S ,

so t h a t

F = R F " . E l i m i n a t e

5

by using t h e i d e n t i t y R T Fj =

I

t o o b t a i n

E T ~

=

s ~ ~ ~ s

⁼

^Q

i.e.

~ = F ' F - Q = o .

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The m a t r i x

X

i s symmetric so t h a t s u b s t i t u t i n g f o r

E

from ( 1 ) i n t o ( 4 ) s i x independent equations a r e obtained. These a r e

X,, = f 2 h f

+

2 f u,h,

+

1

-

Qll = 0

,

_(5a

and s i m i l a r l y f o r X,, and X33, and

X,, = f 2 hlh2

+

f(ulh2

+

u2hl)

-

^Q12⁼0

,

(5b

1

and s i m i l a r l y f o r X,, and X,,.

Equations ( 5 ) may be solved i n t h e f o l l o w i n g way f o r t h e f i v e unknowns f, hl/h3, h2/h3, u,/u, and u2/u3 which d e f i n e

F,

and a s i n g l e parameter contained i n

g.

F i r s t t h e t r a c e Xll

+

^X22

+

X 3 3 o f

X

gives

f 2 = Qll + Q2, + Q3,

-

2 F

-

1

,

( 6 ) where F i s t h e determinant

I F (

⁼1

+

f ( h l u ) o f F and i s t h e r e f o r e equal t o t h e known

- -

r a t i o o f t h e volumes o f corresponding c e l l s o f t h e two phases. Equation ( 6 ) gives two equal and o p p o s i t e values o f t h e magnitude f o f t h e i n v a r i a n t plane s t r a i n . Also t h e expression 2 hlh2X,,

- hi^,, -

h:~,, gives t h e q u a d r a t i c equation

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c4-212 JOURNAL DE PHYSIQUE

f o r hl/h2 and s i m i l a r expressions g i v e corresponding compatible q u a d r a t i c s f o r h2/h3 and h3/hl. I n general these equations g i v e r i s e t o two s o l u t i o n s f o r t h e i n v a r i a n t plane

2.

The equation Xll = 0 may now be w r i t t e n

2 f h1ul = Q,,

-

f2h:

-

¹ (8)

g i v i n g ul i n terms o f f and hl and s i m i l a r l y f o r u2 and u 3 . Thus t h e r e i s one displacement d i r e c t i o n f o r each f and b u t changing the s i g n o f f simply changes t h e s i g n o f u and does n o t provide a d i s t i n c t s o l u t i o n .

-

Hence i f Q i s known equat- i o n s ( 6 ) , ( 7 ) and (8) g i v e two s o l u t i o n s f o r the i n v a r i a n t plane s t r a i n

E.

A convenient method f o r determining Q i s t o i n v e r t equation ( 3 ) t o o b t a i n

S u b s t i t u t i n g i n ( 9 ) from

then gives

R 1 l = F - ~ ~ ~ u :

-

2 F-lf hlul

+

1

- 4;:

= 0

,

( l l a ) and s i m i l a r l y f o r

zZ2

and

x337

^and

and s i m i l a r l y f o r

X Z 3

and

f,,.

The t r a c e of

R

now gives

and e l i m i n a t i n g f 2 from ( 6 ) and (12) r e s u l t s i n t h e f o l l o w i n g r e s t r i c t i o n on

Q.

Now from ( 3 ) , =

s ~ P ~ ?

^where

S

i s a l a t t i c e i n v a r i a n t shear o f magnitude g on a plane o f normal m i n t h e d i r e c t i o n

_a,

and may thus be represented by

S = ; + g $ ? T

.

⁽¹⁴⁾

S u b s t i t u t i n g from (14) i n t o (13) the f o l l o w i n g q u a d r a t i c equation i s o b t a i n e d f o r g g 2 { F - l ( a P'Q) T

-

F ( ? ~ P - ~ ? )

> +

2 g { ~ - l ( ? ~ ~ 2 ; )

+

F(aTp-2,)

- -

1

+

IF-'T(P2)

-

F T(P-2)

-

F - l

+

F } = 0 (15)

where T(P2) = P2 ₁₁+ P i 2 + P i 3 + 2 P:2

+

2 Pz3

+

2 P:l i s t h e t r a c e o f 'P and s i m i l a r l y f o r T(P-2). Equation (15) gives two values f o r g which i n t u r n using (14) d e f i n e two p o s s i b l e shears

S

and hence two p o s s i b l e matrices Q.

The procedure f o r c a l c u l a t i n g t h e p r e d i c t e d c r y s t a l l o g r a p h i c f e a t u r e s o f a 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 u s i n g t h e CRAB f o r m u l a t i o n i s t h e r e f o r e as f o l l o w s .

( i ) Choose a correspondence r e l a t i n g t h e s t r u c t u r e s and c a l c u l a t e t h e pure s t r a i n Pand volume r a t i o 7 from t h i s correspondence and the l a t t i c e parameters.

( i i ) Choose a shear plane 3 and d i r e c t i o n % f o r t h e l a t t i c e i n v a r i a n t shear

S.

( i i i ) S u b s t i t u t e F,

e,

^m_^and^gi n t o (15) and s o l v e f o r t h e magnitude g o f

S.

( i v ) S u b s t i t u t e g i n t o (14) t o o b t a i n

5

and

P

and

2

i n t o ( 3 ) t o o b t a i n

Q.

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( v ) S u b s t i t u t e F and

q

i n t o ( 6 ) t o o b t a i n f,

9 h

and 0, f and

b

i n ( 8 ) t o o b t a i n u.

( v i ) S u b s t i t u t e f,

h

and

u

F.

I n g e n e r a l f o r each F, P, m and .L t h e r e w i l l be two values f o r g and f o u r v a l u e s f o r

F

Conclusions.- The phenomenological t h e o r i e s o f t h e c r y s t a l l o g r a p h y o f 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 s which were f i r s t developed i n t h e 7950s were o r i g i n a l l y p r e s e n t e d as l e n g t h y f o r m u l a t i o n s and t o some e x t e n t t h e analyses were r e s t r i c t e d t o p a r t i c u l a r t r a n s f o r m a t i o n s . A l t h o u g h s u c c e s s f u l l y p r e d i c t i n g t h e c r y s t a l l o g r a p h i c f e a t u r e s o f many t r a n s f o r m a t i o n s t h e r e have been n o t a b l e e x c e p t i o n s , so t h a t a t t e m p t s have been made t o g e n e r a l i s e t h e t h e o r i e s . Most o f these g e n e r a l i s a t i o n s have been t r i v i a l b u t t h e m u l t i p l e l a t t i c e i n v a r i a n t shear a n a l y s i s o f Acton e t a l . ( 1 3 ) c o n s t i t u t e d a genuine advance. However t h e most s i g n i f i c a n t f e a t u r e o f t h i s t h e o r y was p r o b a b l y t h e f a c t t h a t i t i n c o r p o r a t e d as a s p e c i a l case a much more d i r e c t and e l e g a n t

a n a l y s i s o f t h e t r a d i t i o n a l s i n g l e shear m a r t e n s i t e c r y s t a l l o g r a p h y t h e o r y . U n f o r t u n - a t e l y because o f i t s g e n e r a l i t y , and p o s s i b l y t h e c o n c i s e n o t a t i o n used i n i t s p r e s e n t a t i o n , t h i s p o w e r f u l a n a l y s i s has been l i t t l e used. The aim o f t h e p r e s e n t paper has t h e r e f o r e been t o p r o v i d e a s i m p l e v e r s i o n o f t h i s a n a l y s i s , r e f e r r e d t o a s t h e C?AB t h e o r y , f o r s i n g l e shear m a r t e n s i t e . ,!llthough r e c e n t developments i n t h e a n a l y s i s o f 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 s have tended t o c o n c e n t r a t e on mechanisms a s s o c i a t e d d i r e c t l y w i t h i n t e r f a c e s t r u c t u r e and d e f e c t s , i t i s i m p o r t a n t t h a t t h e macroscopic r e s t r i c t i o n s which f o r m t h e b a s i s o f t h e phenomenological t h e o r i e s s h o u l d a l s o be s a t i s f i e d . I t i s t h o u g h t t h a t t h e f o r m u l a t i o n o f t h e t h e o r y g i v e n h e r e w i l l p r o v i d e a s i m p l e t o o l w i t h which t o perform t h e necessary computations.

Yeferences

1. WECHSLER M.S., LIEBEQMAPI D.S. and READ T.A., Trans. AI?T(FI)E

x,

1503(1953).

2. BOLILES J.S. and IfACKEPIZIE J.K., A c t a Met.

2,

129(1954).

3. BULLOUGH R. and BILBY B.A., Proc. Phys. Soc. (Lond.) B69, 1276(1956).

4. CHRISTIAN J.W., The Theory o f T r a n s f o r m a t i o n s i n

etar rand

A1 l o y s , Pergamon Press, O x f o r d 1965; second e d i t i o n , p a r t I, 1975.

5. CHRISTIAN J.I.1. and CROCKEQ A.G., D i s l o c a t i o n s and L a t t i c e T r a n s f o r m a t i o n s , i n NABARRO F.9.N. ed., D i s l o c a t i o n s i n S o l i d s

3,

Moving D i s l o c a t i o n s , N o r t h Holland, Amsterdam, 1980.

6. CHRISTIAN J.W., J. I n s t . I l e t . 84, 343(1955-6).

7. WECHSLER H.S. ,READ T.A. and L I ~ E R ~ I A N D.S., Trans. AIWE

-

218, 202(1960).

8. LIEBERFIAN D.S., A c t a Net. 6, 680(1958).

9. ROSS N.D.H. and ABELL J.S.; Phys. S t a t . Sol. ( a )

1,

K33(1970).

10. WAYMAN C.N., Acta Net. 9, 912(1961).

11. CROCKER A.G. and BILBY

B.A.,

A c t a Met. 9, 678(1961).

12. CROCKER A.G., A c t a Met. 13, 815(1965). -

13. ACTON A.F., BEVIS : I . , CRXKEX A.G. and ROSS N.Q.H., Proc. Koy. Soc. (Lond.)=, 101 (1970).

14. BEVIS 11. and CROCKER A.G., Proc. 90y. Soc. (Lond. )A304, 123(1968).

15. BEVIS

H.

and CROCKER A.G., Proc. Eoy. Soc. ( ~ o n d . ) E , 509(1969).

16. CROCKEQ A.G., P h i l . Mag. 7, 1901 (1962).

17. ACTON A.F. and BEVIS PI., Mster. S c i . Eng.

5,

^19(1969)

18. ROSS N.D.H., and CROCKER A.G., Acta Met.,

18,

405(1970).

19. FLEllITT P.E.J., ASH P.J. and CSOCKER A.G., Acta Met.

g,

669(1976).

20. CROCKER A. G. and FLElsIITT P.E.J., The M i g r a t i o n o f I n t e r p h a s e Boundaries by Shear Mechanisms, i n SrIITH D.A. and CHADllICK G.A., I n t e r p h a s e Boundaries i n S o l i d s , Academic Press, London, i n t h e p r e s s .

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C4-214 JOURNAL DE PHYSIQUE

Appendix: The S i n g l e Shear CRAB Theory Using a General Co-ordinate System.- Using t h e n o t a t i o n of t h e t e n s o r c a l c u l u s (13), equations ( 1 ) - ( 1 5 ) o f the main t e x t may be w r i t t e n as f o l l o w s

Note t h a t although t h e product l a t t i c e basis pi i s n o t contained e x p l i c i t l y i n t h i s a n a l y s i s i t i s introduced i n the d e f i n i t i o n o f P I . which i s given by

.

. J

Pi PJ =

pijcikcja ,

'ij k a.

where

ci

i s t h e correspondence.

j