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Submitted on 1 Jan 1982
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LATTICE CORRESPONDANCE AND
CRYSTALLOGRAPHY OF MARTENSITES IN TITANIUM ALLOYS
K. Mukherjee, M. Kato
To cite this version:
K. Mukherjee, M. Kato. LATTICE CORRESPONDANCE AND CRYSTALLOGRAPHY OF
MARTENSITES IN TITANIUM ALLOYS. Journal de Physique Colloques, 1982, 43 (C4), pp.C4-
297-C4-302. �10.1051/jphyscol:1982441�. �jpa-00222155�
L A T T I C E CORRESPONDENCE AND CRYSTALLOGRAPHY O F M A R T E N S I T E S I N T I T A N I U M A L L O Y S
K. Mukherjee and M. Kato
Department of Metallurgy, Mechanics and Materials Science, Michigan State University, East Lansing, MI 48824, U.S.A.
(Accepted 9 August 1982)
A b s t r a c t . - L a t t i c e correspondence, c r y s t a l l o g r a p h y and i n t e r n a l s t r u c t u r e of v a r i o u s m a r t e n s i t e s i n t i t a n i u m a l l o y s a r e d i s c u s s e d i n a u n i f i e d manner.
S t r a i n e n e r g y m i n i m i z a t i o n c r i t e r i o n i s a p p l i e d t o d i s c u s s t h e o r i e n t a t i o n of h a b i t p l a n e s and l a t t i c e i n v a r i a n t s t r a i n s . A comparison of t h e p r e s e n t a n a l - y s i s w i t h e x p e r i m e n t a l o b s e r v a t i o n s h a s been c a r r i e d o u t .
I n t r o d u c t i o n . - Quenching from t h e h i g h - t e m p e r a t u r e 6 (b.c.c.) p h a s e of t i t a n i u m a l l o y s r e s u l t s i n t h e f o r m a t i o n of v a r i o u s k i n d s of m a r t e n s i t e s s u c h a s a ' ( h . c . p . ) ,
a" ( o r t h o r h o m b i c ) , f.c.0. ( f a c e - c e n t e r e d orthorhombic) and f . c . c . ( f a c e - c e n t e r e d
c u b i c ) m a r t e n s i t e s . Except f o r t h e f.c.c. m a r t e n s i t e , which i s c o n s i d e r e d t o occur o n l y i n t h i n f i l m s [ I ] , t h e o t h e r t h r e e m a r t e n s i t e s have been found i n b u l k t i t a n i u m a l l o y s . Although t h e r e a r e many s t u d i e s a v a i l a b l e f o r t h e c r y s t a l l o g r a p h y of t h e s e m a r t e n s i t e s , i t seems t h a t no u n i f i e d u n d e r s t a n d i n g o f 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 s i n t i t a n i u m a l l o y s h a s been developed.
I n t h i s s t u d y , t h e l a t t i c e c o r r e s p o n d e n c e and t h e c r y s t a l l o g r a p h y o f t h e a ' , a"
and f . c . 0 . m a r t e n s i t e s w i l l be i n v e s t i g a t e d s y s t e m a t i c a l l y by u s i n g e x p e r i m e n t a l o b s e r v a t i o n s a v a i l a b l e i n t h e l i t e r a t u r e . It w i l l a l s o be shown t h a t t h e s t r a i n e n e r g y m i n i m i z a t i o n c r i t e r i o n c a n b e a p p l i e d t o d i s c u s s t h e c r y s t a l l o g r a p h y and i n t e r n a l s t r u c t u r e ( l a t t i c e i n v a r i a n t s h e a r , h a b i t p l a n e , e t c . ) of t h e s e m a r t e n s i t e s . L a t t i c e Correspondence and C r y s t a l S t r u c -
t u r e Change.- F i g u r e 1 shows t h e f o u r
u n i t c e l l s o f t h e b . c . c . 6 - p h a s e a n d i t s X [ I f a c e - c e n t e r e d t e t r a g o n a l e q u i v a l e n t . It h a s b e e n demonstrated [2
-
41 t h a t a l l of t h e m a r t e n s i t e s ( a ' , a" and f.c.0.) c a n b e g e n e r a t e d from t h i s t e t r a g o n a l l a t t i c e . By u s i n g l a t t i c e parameter v a l u e s o f t h e p a r e n t 0-phase and t h e m a r t e n s i t e s , t h e l a t t i c e v a r i a n t s t r a i n (Bain s t r a i n ) c a n b e c a l c u l a t e d . T a b l e 1 shows t h e l a t t i c e p a r a m e t e r s o f t h e p a r e n t and m a r t e n s i t ep h a s e s [ I , 3 , 41 used i n t h e p r e s e n t
z
COO1J8s t u d y . From t h e s e v a l u e s , t h e Bain -..---
s t r a i n s based o n t h e x-y'-z' c o o r d i n a t e s y s t e m ( t - s y s t e m ) i n t h e t e t r a g o n a l c e l l i n F i g . 1 c a n b e c a l c u l a t e d and t h e s e a r e a l s o shown i n T a b l e 1. For t h e c a s e of t h e a' m a r t e n s i t e , t h e Bain s t r a i n i s b r o u g h t a b o u t by t h e well-known Burgers
mechanism where t h e o p e r a t i o n of t h e F i g . 1.- L a t t i c e c o r r e s p o n d e n c e of mar- {1121<111>0 s h e a r s y s t e m s followed by a t e n s i t e s showing t h e f o u r u n i t c e l l s of s m a l l volume i n c r e a s e r e s u l t s i n t h e t h e b.c.c. c r y s t a l and i t s f a c e - (OOO1)a' p l a n e from t h e (011)0 p l a n e . I n c e n t e r e d t e t r a g o n a l e q u i v a l e n t . a d d i t i o n t o t h e above Bain s t r a i n , a
a 6 [ 0 1 1 ] 0 / 6 s h u f f l i n g on e v e r y o t h e r ( 0 1 1 ) ~
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982441
JOURNAL
DE
PHYSIQUETable 1.- L a t t i c e param- e t e r s and t h e c a l c u l a t e d Bain s t r a i n s i n t i t a n i u m a l l o y s analyzed i n t h e p r e s e n t study.
l a t t i c e parame e r
/
of b p h a s e(1) I
3.2591 1
3 2 6 3I
m a x t e n s i t e a l l o y
a' Ti-5Mn [I]
l a t t i c e parameter
p l a n e i s n e c e s s a r y to c r e a t e t h e h.c.p. s t r u c t u r e a s shown i n Fig. 2 ( a ) . Bain s t r a i n
The formation of t h e a" m a r t e n s i t e can be explained i n e s s e n t i a l l y t h e same manner. The d i f f e r e n c e l i e s i n t h e f a c t t h a t a f t e r t h e o p e r a t i o n of t h e {1121<111>6 s h e a r s , t h e ( 0 1 1 ) ~ plane does not completely change i n t o t h e hexagonal close-packed p l a n e b u t s t o p s midway between t h e ( 0 1 1 ) ~ and t h e (0001),1 l e a v i n g t h e a n g l e s 81 and 8 2 i n Fig. 2(b) s l i g h t l y d i f f e r e n t from 120 degrees. Thus, t h e a" m a r t e n s i t e can be c o n s i d e r e d a s a " d i s t o r t e d " a ' m a r t e n s i t e . It has been r e p o r t e d [2, 3 , 5 1 t h a t a t r a n s i t i o n from t h e a' t o a" m a r t e n s i t e occurs with an i n c r e a s i n g 8 - s t a b i l i z i n g s o l u t e c o n t e n t . This i s s u e w i l l b e f u r t h e r d i s c u s s e d .
a"
TI-1OV-2Ee-3AI (31
a
-
2.950c
-
4.685The f.c.0. m a r t e n s i t e , on t h e o t h e r hand, i s a r a t h e r unique one found only i n a b u l k Ti-12.6% V a l l o y [ 4 ] . I n t h i s c a s e , no s h u f f l i n g i s n e c e s s a r y and t h e l a t t i c e correspondence i s very s i m i l a r t o t h a t i n t h e Au-Cd m a r t e n s i t e [6]. It should be noted, from Table 1, t h a t t h e components of t h e Bain s t r a i n f o r t h e f.c.0.
m a r t e n s i t e have t h e o p p o s i t e s i g n s t o t h o s e f o r t h e a ' and a" m a r t e n s i t e s . Thus, t h e f.c.0. m a r t e n s i t e cannot b e considered t o be a " d i s t o r t e d " a ' m a r t e n s i t e .
f . c . 0 . Ti-12.6V (41
c1
-
-0.0948c 2
-
0.1086c 3
-
0.0165Phenomenological Analysis Based on t h e S t r a i n Energy Minimization C r i t e r i o n . - Mura e t a l . 171 and Kato e t a l . [81 have shown t h a t t h e r e i s a one-to-one correspondence
a
-
3.01b
-
4.83c
-
4 . 6 2- -
- -between t h e phenomenological c r y s t a l l o g r a p h i c theory of 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 [9, 101 and t h e s t r a i n energy minimization approach based on t h e s m a l l deformation t h e o r y [ 7 , 81. According t o t h e i r a n a l y s i s , i f t h e t r a n s f o r m a t i o n s t r a i n
ET
(Bains t r a i n p l u s l a t t i c e i n v a r i a n t s t r a i n ) i n a n xl-xp-x3 c a r t e s i a n c o o r d i n a t e syAtem b
-
4.386c
-
4.467rl • -0.0652
C 2
-
0.0607C 3
-
0.0145shufflina
c1
-
0.0913C 2
-
-0.0495c 3
-
-0.0320Fig. 2.- C r y s t a l s t r u c t u r e change from b.c.c. t o
( a ) h.c.p. ( a ' ) and (b) o r thorhombic (a1').
( 0 1 planes
\
o
1st and 3 r d layer atoms 2nd layer atoms(a) (OOOI)a, planes
t h e s t r a i n energy becomes minimum ( z e r o ) and t h e c o n d i t i o n i n e q u a t i o n ( I ) , with t h e x3-axis a s an i n v a r i a n t p l a n e ( h a b i t plane) normal, i s e q u i v a l e n t t o t h e i n v a r i a n t p l a n e c o n d i t i o n i n t h e phenomenological theory. The advantage of u s i n g t h e small- deformation theory i s t h a t t h e c a l c u l a t i o n becomes much simpler and even i f t h e t h r e e p r i n c i p a l components of t h e Bain s t r a i n a r e d i f f e r e n t , such a s t h e c a s e i n t h e p r e s e n t m a r t e n s i t e s , t h e f i n a l s o l u t i o n s can be o b t a i n e d i n simple a n a l y t i c a l forms.
Let us now apply t h i s s t r a i n energy minimization c r i t e r i o n t o simultaneously a n a l y z e t h e c r y s t a l l o g r a p h y of t h e a ' , a" and t h e f.c.0. m a r t e n s i t e s . A s shown i n Table 1, t h e Bain s t r a i n i n t h e t-system f o r t h e a l l t h r e e m a r t e n s i t e s can b e w r i t t e n i n a form of
E O O
1
. l = C i j :2
,
tAS can b e seen i n Table 1, c 1 < 0 ( c o n t r a c t i o n ) and E 2 , E3 > 0 (expansion) f o r t h e a ' and a" m a r t e n s i t e s whereas EI > 0 and E2, € 3 < 0 f o r t h e f.c.0. m a r t e n s i t e . I f e q u a t i o n (2) i s viewed based on t h e x-y-z system i n t h e p a r e n t 8-phase (8-system), we have
It h a s been r e p o r t e d t h a t t h e l a t t i c e i n v a r i a n t s h e a r :or each m a r t e n s i t e occurs dominantly by twinning and t h e twinning p l a n e s a r e { l O 1 l } a ~ [1],{111)(111 [5] and
{ l l f ~ f . c
.,
[4]. I n t e r e s t i n g l y , a l l of t h e s e twinning p l a n e s correspond t o t h e ( l l O ) B of t h e p a r e n t 6-phase i n Fig. 1. Thus, by u s i n g t h e 6-system, t h e twinned v a r i a n t should have t h e Bain s t r a i n of+
E3)/2 0 (c2-
C3)/2o
1 O] ,
( v a r i a n t 2).-
s 3 ) / 2 0 (E2 f c 3 ) / 2 '3From e q u a t i o n s (3) and ( 4 ) , t h e t o t a l t r a n s f o r m a t i o n s t r a i n ,
rij,
can be w r i t t e n a swhere f i s t h e volume f r a c t i o n of t h e v a r i a n t 1 which is t o be determined. The h a b i t p l a n e and f can be o b t a i n e d i n t h e following manner.
L e t t h e u n i t v e c t o r
x,
normal t o t h e h a b i t p l a n e , b e w r i t t e n i n t h e $-system a sBy r o t a t i n g t h e c o o r d i n a t e system, we can f i n d a new XI-x2-x3 system i n which t h e t o t a l t r a n s f o r m a t i o n s t r a i n (5) s a t i s f i e s t h e i n v a r i a n t p l a n e c o n d i t i o n of e q u a t i o n (1). I n t h i s new system, $ becomes p a r a l l e l t o t h e x3-axis. It should be noted t h a t e q u a t i o n (1) a c t u a l l y g i v e s t h r e e simultaneous e q u a t i o n s w i t h f , 6 and 4 a s t h e
JOURNAL DE PHYSIQUE
t h r e e unknown p a r a m e t e r s . These can b e s o l v e d e a s i l y and t h e r e s u l t s become;
where
2 2 ( 1
-
£)El+
f ( E 2+
E3)t a n
6
=-
92fe1
+
( 1-
f ) ( E 2+
E3)It c a n e a s i l y b e shown t h a t s o l u t i o n s c o r r e s p o n d i n g t o t h e p o s i t i v e and nega- t i v e s i g n s i n e q u a t i o n (9) a r e c r y s t a l l o g r a p h i c a l l y e q u i v a l e n t . Moreover, i t s h o u l d b e n o t e d t h a t t h e r e a r e two i n d e p e n d e n t ( c r y s t a l l o g r a p h i c a l l y d i f f e r e n t ) s o l u t i o n s f o r t h e o r i e n t a t i o n of t h e h a b i t p l a n e , depending on t h e p o s i t i v e and n e g a t i v e v a l u e s of t a n $ i n e q u a t i o n (7).
By u s i n g t h e d a t a f o r t h e Bain s t r a i n i n T a b l e 1, t h e h a b i t p l a n e normal and t h e volume f r a c t i o n of t h e v a r i a n t 1 were c a l c u l a t e d from e q u a t i o n s (7) t h r o u g h ( 9 ) f o r e a c h o f t h e t h r e e m a r t e n s i t e s and t h e r e s u l t s a r e shown i n T a b l e 2. The h a b i t p l a n e n o r m a l s a r e a l s o p l o t t e d i n F i g . 3 where t h e s t e r e o g r a p h i c p r o j e c t i o n i s b a s e d on t h e i n d i c e s of t h e p a r e n t B-phase.
D i s c u s s i o n . - To i n v e s t i g a t e t h e a c c u r a c y of t h e p r e s e n t s m a l l d e f o r m a t i o n a n a l y s i s , t h e r e s u l t s of t h e n u m e r i c a l c a l c u l a t i o n of t h e h a b i t p l a n e normal f o r t h e a ' mar- t e n s i t e i n a Ti-Mn a l l o y by Knowles and Smith I l l ] , based on t h e phenomenological t h e o r y , a r e a l s o shown i n Fig. 3. S i n c e t h e same imput d a t a a r e a d o p t e d i n t h e p r e s e n t a n a l y s i s , i t i s n o t s u r p r i z i n g t o s e e from F i g . 3 t h a t b o t h t h e p r e s e n t a n a l y s i s and t h e phenomenological t h e o r y p r e d i c t e s s e n t i a l l y t h e same o r i e n t a t i o n of t h e h a b i t p l a n e nozmals; one of t h e s o l u t i o n s b e l o n g i n g t o t h e Class A (a- u+) and t h e o t h e r t o t h e C l a s s A (a+ wt.) i n t h e Mackenzie-Bowles n o t a t i o n [ I , 11, 121. From t h i s , we c a n s a y t h a t t h e s m a l l - d e f o r m a t i o n s t r a i n e n e r g y m i n i m i z a t i o n c r i t e r i o n c a n be u s e d i n l i e u o f t h e phenomenological t h e o r y t o d i s c u s s t h e c r y s t a l l o g r a p h y o f t h e t i t a n i u m a l l o y m a r t e n s i t e s . One main a d v a n t a g e i n t h e s m a l l d e f o r m a t i o n t h e o r y i s t h a t no n u m e r i c a l c a l c u l a t i o n i s i n v o l v e d and a s y s t e m a t i c i n v e s t i g a t i o n i s p o s s i b l e t o f i n d t h e change i n s o l u t i o n s due t o t h e change i n imput p a r a m e t e r s .
It h a s been found i n Ti-Mn a l l o y s
11,
111 t h a t t h e r e a r e two d i f f e r e n t t y p e s of a ' m a r t e n s i t e s , o n e w i t h t h e {3341B and t h e o t h e r w i t h t h e {3441B h a b i t p l a n e s . As c a n b e s e e n from F i g . 3 , none of t h e two s o l u t i o n s f o r t h e a' m a r t e n s i t e a c c u r a t e l y e x p l a i n s t h e above h a b i t p l a n e i n d i c e s . Hammond and K e l l y [ I ] were t h u s f o r c e d t o u s e a n a d j u s t a b l e " d i l a t a t i o n " p a r a m e t e r , 6, i n t h e phenomenological t h e o r y t o b r i n g t h e i r r a t i o n a l h a b i t p l a n e normals t o t h e o b s e r v e d i n d i c e s .Knowles and Smith [ l l ] t o o k a d i f f e r e n t approach. They a n a l y z e d t h e zig-zag T a b l e 2.- C a l c u l a t e d s o l u t i o n s f o r t i t a n i u m a l l o y m a r t e n s i t e s .
martensite volume fraction of variant 1, f
i:
a'' 0.699 a
'
0.703
f . c . 0 . 0.698
Fig. 3.- Calculated habit plane normals for titanium alloy martensites.
parent-martensite interface in the martensite on the basis of the Burgers vector minimization criterion of the interface dislocations. According to their calculation, the total magnitude of the Burgers vector averaged over the entire interface becomes zero. This physically means that no long-range strain field is created due to the transformation and the invariant plane condition in the macro- scopic habit plane is still satisfied in their model. Thus, the orientation of the calculated habit plane on the average remains the same as that predicted from the phenomenological analysis shown in Fig. 3.
The reason why the calculated orientations of the habit plane in Fig. 3 do not coincide with the observed (334l6 or (344) plane might simply be understood if we make the following analysis. Hammond and telly
[I]
as well as Knowles and Smith[ll] used the lattice parameter of the 6-phase of the Ti-5% Mn alloy (ag = 3.259
2
as shown in Table 1) measured at room temperature. Since the 6 + U' martensitic transformation takes place at a higher temperature, the lattice parameter should be compensated by taking into account the thermal dilatation. Davis et al. [5], on the other hand, have systematically obtained the lattice parameters of the 6-phase and the a' and a" martensites in Ti-Mo alloys taking into account the thermal dilatation as well as the change in the lattice parameter of the 6-phase due to the change in the molybdenum content. According to their observation, the a' martensite is formed in the alloy with as much as 4% Mo and the a" martensite is observed in Ti-6% Mo and Ti-8% Mo alloys. Similar tendency from h.c.p. to orthorhombic structure change as- sociated with increasing B-stabilizing
elements is also observed in other titanium
alloys
[2,
41. Table 3.- Bain strainsfor Ti-Mo martensites 151.
Table 3 shows the systematic change in the Bain strain due to the change in the molybdenum content obtained by Davis et al.
151. By using these data, trial calculation based on the strain energy minimization criterion was made to see the systematic change in the orientation of the habit plane with increasing molybdenum content.
