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ANALYSIS OF MULTIPLE SLIP IN COPPER TRICRYSTALS
T. Ohashi
To cite this version:
T. Ohashi. ANALYSIS OF MULTIPLE SLIP IN COPPER TRICRYSTALS. Journal de Physique
Colloques, 1990, 51 (C1), pp.C1-593-C1-598. �10.1051/jphyscol:1990193�. �jpa-00230362�
COLLOQUE DE PHYSIQUE
Colloque Cl, supplement au nol, Tome 51, janvier 1990
ANALYSIS OF MULTIPLE SLIP IN COPPER TRICRYSTALS
T. OHASHI
Hitachi Research Laboratory, Hitachi, Ltd. 4 0 2 6 , Kuji, Hitachi, 3 1 9 - 1 2 , Japan
A b s t r a c t - . . -..-- Non-uniform m u l t i p l e s l i p i n t r i c r y s t a l s i s analysed by a method o f continuum mechanics. A n a l y s i s r e s u l t s show t h a t compat i b i l i t y requirements on t h e g r a i n boundary planes and on the e n t i r e t r i c r y s t a l specimen lead t o d i f f e r e n t types o f m u l t i p l e s l i p . A mechanism f o r m u l t i p l e s l i p i s discussed from t h e view p o i n t o f excess s t r e s s which i s generated by t h e s l i p on t h e primary system.
1. I n t r o d u c t i o n
-.-p. - -. . - .
M u l t i p l e s l i p near g r a i n boundaries p l a y s an Important r o l e I n d e t e r n i n a t i c n o f the mechanical p r o p e r t i e s o f p o l y c r y s t a l s . So f a r , t h e m u l t i p l e s l i p phenomenon has been s t u d i e d mainly w i t h bicrystals/l/-/4/; but, s l i p near j u n c t ~ o n s o f g r a i n boundaries, such as g r a i n boundary t r i p l e l i n e s o r quadruple p o i n t s , i s more complicated and d i v e r s i f i e d than t h a t i n b i c r y s t a l s / 5 / , / 6 / . I n t h e present paper, s l i p deformation o f copper t r i c r y s t a l s i s n u m e r i c a l l y analysed t o examine shear s t r a i n d i s t r i b u t i o n on p r i m a r y and secondary s l i p systems. S t r e s s f i e l d caused by non-uniform s l i p i s discussed, too.
2. Method o f numerical analys_ls
- -- --
The Schmid's law i s used as t h e a c t ~ v a t i o n c o n d i t i o n f o r twelve s l i p systems Cl113 - i110). I f t h e s t r e s s components i n the g l o b a l c o o r d ~ n a t e system a r e denoted as o
,
a n d ' t h e c r i t i c a l resolved shear s t r e s s (abbreviated as CUSS) f o r the s l i p system n i s w r i t t e n as 8'"',
the Schmid c o n d i t i o n I S given by t h e f o l l o w i n g equations.where
P,':: ,,,
= 112 {67;
(kl + U':!b'rt
( h ,I
( 3 )S u b s c r i p t s and s u p e r s c r i p t s i n parentheses denote t h e g r a i n number and s l i p system, r e s p e c t ~ v e l y . I f r o t a t i o n o f t h e c r y s t a l o r i e n t a t i o n i s neglected, which i s acceptable w h i l e the deformation i s small, the c o n s t i t u t i v e equation f o r each c r y s t a l g r a i n i s w r i t t e n as f o l l o w s / W , / ? / .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990193
Cl-594 COLLOQUE DE PHYSIQUE
Here, S,
,'EJ,
denotes e l a s t i c compl iance.GPmJ
I S t h e s t r a i n hardenrns c o e f f i c i e n t which d e f i n e s t h e r e l a t i o n between increments o f t h e CRSS and p l a s t i c shear s t r a i n on s l i p systems.The CRSS i s assumed t o be a f u n c t i o n o f d i s l o c a t i o n d e n s i t r e s on t h e twelve s l i p systems p'"' ( m= I -12
1.
Here, 9 o i s a constant and '""'is a constant m a t r i x which i s determined i n accordance w i t h t h e v a r i a t i o n o f r e a c t i o n between d i s l o c a t i o n s on s l i p systems n and m. The f u n c t i o n G i s g i v e n b y :
Here,
-
b i s t h e magnitude o f Burgers vector, i s e l a s t i c shear modulus and a i s a numerical f a c t o r o f o r d e r 0. l.I t i s assumed t h a t d i s l o c a t i o n sources on a c t i v e s l i p systems e m i t d i s l o c a t i o n loops, and they move f r e e l y i n t h e manner o f f r e e f l i g h t motion u n t i l they a r e trapped by o b s t a c l e s /8/. I f i t i s a l s o assumed t h a t the shape o f the trapped d i s l o c a t i o n loops IS r e c t a n g l a r w i t h aspect r a t i o a , then t h e increment o f shear s t r a i n and d i s l o c a t i o n d e n s i t y on t h e s l i p system a r e c o r r e l a t e d as f o l lows.
L denotes t h e mean f r e e f l i g h t d i s t a n c e o f t h e d i s l o c a t i o n segments. For t h e mean f r e e f l i g h t distance, Seeger's model i s used w i t h a smal l m o d i f i c a t i o n .
- . . - - - s i n g l e s l i p region
(10) A / { Z 7'"' - ( 7'-
A
/ L , )1 ...
m u l t i p l e s l i p regionHere, CO and A a r e constants. 7 D denotes t h e shear s t r a i n a t which m u l t i p l e s l i p occurs. From equat rons (7)-(9)
h"''
i s given as :The f i n i t e element method i s used f o r numerical analysis. Fundamental equations f o r the method are.
c
Kl {h
={h. C K 1
= Z Ckl,where.
;
and denote increments o f nodal displacement and nodal force. M a t r i x B i s a shape f u n c t i o n o f each element. D i s t h e matrrx expression o fD,:,;
which i s given by equation (5).Propagation o f s l i p i s analysed by an incremental procedure.
Fig. l Geometries o f t r i c r y s t a l s . (a) Symmetric type and (b) pseudo-compat i b l e type.
Fig. 2(a) Crystal o r i e n t a t ions for symmetric t r i c r y s t a l . Primary and c r i t i c a l s l i p
A systems are indicated by sets o f black and open symbols.
Fig. 2(b) Crystal o r i e n t a t ions f o r pseuao-compat i b l e t r i c r y s t a l S. Primary and conjugate s l i p systems are indicated by s e t s o f black and open symbols.
3. T r i c r y s t a l models
------p
Two special t r i c r y s t a l s are considered. The f i r s t one i s shown i n f i g u r e l(a). Crystal o r i e n t a t i o n s f o r i t s grains a r e given i n f i g u r e 2(a). The c r y s t a l o r i e n t a t i o n f o r g r a i n 2 has m i r r o r symmetry w i t h g r a i n I, w i t h respect t o the yz plane. Normal vector o f the s l i p plane and s l i p d i r e c t i o n f o r the primary s l i p syste-ms p of grains 1 and 2 l i e on the x y plane. Moreover,
Here, U , and b , denote u n i t vectors o f the s l i p plane normal and s l i p d i r e c t on, respect lvely. When o n l y the'primary s l i p system i s active. s t r a i n component is,
Cl-596 COLLOQUE DE PHYSIQUE
According t o equations (3) and (14) a l l s t r a i n components a r e i d e n t i c a l f o r b o t h g r a i n s when 7
:!:
=( 0 1
7 This means t h a t the compat i b i l i t y requl rement i s n a t u r a l l y s a t i s f i e d and no mutual c o n s t r a i n t o f deformation between g r a i n s 1 and 2 w i l l occur.
C r y s t a l o r i e n t a t i o n o f g r a i n 3 i s t h a t f o r a n / 2 r o t a t i o n o f g r a i n 1 about the y axis.
S l i p plane normal and s l ~ p d i r e c t i o n a r e p a r a l l e l t o t h e yz plane. Therefore, t h e primary system i n g r a i n 3 i s n e i t h e r continuous t o t h a t i n the g r a i n 1, nor symmetric t o i t w i t h respect t o the g r a i n boundary between g r a i n s 1 and 3 ( a b b r e v i a t e d as g r a i n boundary 1-3, h e r e a f t e r ). Therefore, c o m p a t i b i l i t y requirements w i l l cause m u l t i p l e s l i p i n t h e v i c i n i t i y o f t h i s g r a i n boundary plane.
The same w i l l occur between g r a i n s 3 and 2. On t h e o t h e r hand, as a whole, t h e geometrical
r e l a t i o n s h i p of t h e 3 p r i m a r y s l i p systems i n each g r a i n have a symmetric r e l a t i o n s h i p w i t h respect t o yz plane. Then, t h i s t r i c r y s t a l i s r e f e r r e d t o as a symmetric type.
T h e s e c o n d c r y s t a l o r i e n t a t i o n c o n s i d e r e d h e r e i s s h o w n i n f i g u r e 2(b). I n t h i s c a s e , t h e o r i e n t a t i o n r e l a t i o n s h i p s between g r a i n s 1 and 2 and between 2 and 3 a r e symmetric w i t h respect t o t h e yz and zx planes, r e s p e c t i v e l y . The primary system f o r g r a i n 3 c o i n c i d e s w i t h t h a t f o r g r a i n 1.
Therefore, i f t h e s t r u c t u r e o f t r i c r y s t a l model i s chosen as i n f i g u r e 1 (b), compatibi l i t y c o n d i t i o n i s s a t i s f i e d on a l l g r a i n boundary planes. This t r i c r y s t a l i s r e f e r r e d t o as a pseudo- compat i b l e type.
The f o l l o w i n g data a r e used f o r b o t h types o f t r i c r y s t a l s .
E l a s t ~ c constants A = 4 p m
S,, = 5, S,, = 1.5 p . = 1 x 1 0 9 / m 2
s4*
= 13 X l o - " m ' / Ne o
=o
b
= 2.556 x 1 0 - ' ~ mR'""'=
l ( n,m = l...
12 )a = l L, = 1000 p m
The t r i c r y s t a l s a r e d i v i d e d i n t o 784 composite elements w i t h e i g h t nodes. The g r a i n boundary planes a r e t r e a t e d as zero t h i c k n e s s ~ n t e r f a c e s on which c o n t i n u i t y of displacement and t r a c t i o n a r e maintained. External load i s given t o t h e specimens by a p p l i c a t i o n o f u n i f o r m displacement i n t h e y d i r e c t i o n on t h e i r upper and bottom surfaces.
4.1 Symmetric t r ~ c r y s t a l a) Ear l y deformat ion
D i s t r i b u t i o n o f shear s t r a i n on t h e primary s l i p system i n an e a r l y stage o f deformation i s shown i n f i g u r e 3(a). The shear s t r a i n I S small i n t h e v i c i n i t y o f t h e t r i p l e j u n c t i o n and increases a l o n g t h e g r a i n boundary 1-2. The minimum and maximum o f t h e shear s t r a i n a r e 0.33x10-\and
2 . 5 7 ~ 1 0 , r e s p e c t i v e l y . The r a t i o o f t h e maximum t o t h e minimum i s about 7.8.
The process f o r f o r m a t i o n o f t h i s d i s t r i b u t i o n rs thought t o be as f o l l o w s ; on t h e g r a i n
boundaries 2-3 and 1-3, lack o f c o m p a t i b i l i t y causes supression o f a c t i v a t i o n o f t h e primary s l i p systems near t h e g r a i n boundaries and r e s u l t s i n d i s t r i b , u t i o n o f s m a l l e r shear s t r a i n around t h e t r i p l e j u n c t i o n . As compensation f o r t h i s . a b i g g e r shear occurs along t h e g r a i n boundary 1-2.
F i g u r e 3(b) shows d i s t r i b u t i o n o f von Mises's e q u i v a l e n t s t r e s s . This s t r e s s i s a measure o f e l a s t i c s t r a i n energy. Figures 3 (a) and (b) i n d i c a t e t h a t t h e von Mises's s t r e s s r e f l e c t s t h e e x t e n t o f shear s t r a i n g r a d i e n t on t h e primary s y s t e m .
b) Subsequent deformatlon
D ~ s t r i b u t ~ o n s o f shear s t r a l n on t h e prlmary and c r i t ~ c a l system a t a l a t e r stage o f deformatlon a r e glven ~n f l g u r e s 4(a) and (b). The q ~ a l ~ t a t i v e aspect of t h e shear s t r a ~ n
d ~ s t r ~ b u t i o n on t h e prlmary system remains unchanged. The minlmum and maxlmum o f t h e st:a,n a r e l. 04x10-" and 6 . 8 6 x 1 0 - ~
.
The r a t 1 0 of t h e max lmum t o the mlnlmum I s a b o u t 6.6.Regard~ng th e c r ~ t r c a l system w h ~ c h I S activated along t h e g r a i n b o u n d a r ~ e s 1-3 and 2-3, the magn~tude o f the shear s t r a ~ n I S one o r d e r s m a l l e r than t h a t on t h e prlmary system. Compar~son o f f ~ g u r e s 3(b) and 4(b) suggests t h a t a c t ~ v a t r o n o f t h e c r ~ t i c a l system p r o v ~ d e s a release mechanism f o r s t r a ~ n energy t h a t I S accumulated through the n o n - u n ~ f o r m s l ~ p on t h e prlmary systems.
No. Y 1 0.33 X 1 0 - '
2 0.58
3 0 . 8 3
4 1.08
5 1 . 3 3
6 1.58
7 1.83
8 2 . 0 7
9 2.32
10 2.57
No. U
1 1 . 5 1 HPa 2 1 . 5 2 3 l. 54
4 1.56
5 1 . 5 7
6 1.59
Fig.3 (a) Shear s t r a i n on t h e primary s l i p system and (b) e q u i v a l e n t s t r e s s i n the
symmetric t r i c r y s t a l a t an e a r l y stage of deformation. The mean s t r e s s and t h e mean s t r a i n i n t h e Y d i r e c t ion a r e 1.52 MPa and 1.31 X 10-" r e s p e c t i v e l y .
No. 7
1 1.04 X 1 0 . ~ 2 1 . 6 9
3 2.33
4 2 . 9 8
5 3.63
6 4 . 2 7 7 4 . 9 2 8 5 . 5 7 9 6. 21
10 6.86
No.
1 0.23TX 1 0 - ~
2 0.46
3 0.68
4 0 . 9 1 5 1 . 1 4 6 1 . 3 7
7 1.59
8 1.82
9 2.06
10 2. 28
Fig.4 (a) Shear s t r a i n on t h e primary s l i p system and (b) t h e c r i t i c a l s l i p system i n t h e symmetric t r i c r y s t a l a t a l a t e r stage o f deformatron. The mean s t r e s s and t h e mean s t r a i n i n t h e Y d i r e c t ion a r e 1.55 MPa and 2.49 X 10-5, respect i v e l y .
4 . 1 Pseudo-compatible t r i c r y s t a l
Figures 5(a) and (b) show d i s t r i b u t i o n s o f shear s t r a i n on t h e primary and conjugate system i n t h e pseudo-compatible type t r i c r y s t a l . As a l r e a d y mentioned, deformation c o m p a t i b i l i t y on t h e g r a i n boundary planes i s assured through t h e symmetry o r c o n t i n u i t y o f t h e primary s l i p systems
i n t h r e e c r y s t a l grains. The reason f o r t h e non-uniform d i s t r i b u t i o n o f t h e shear s t r a i n on t h e primary system and a c t i v a t i o n o f t h e secondary one i s discussed below.
Because t h e s l i p d i r e c t i o n s f o r t h e primary systems i n t h e t h r e e g r a i n s a r e p a r a l l e l t o t h e xy plane, on l y t h r e e s t r a i n components E *,, E and E
..
should be considered. When the shear s t r a i n on t h e p r i m a r y s l i p systems i n t h r e e g r a i n s are i d e n t i c a l , magnitudes o f t h e s t r a i n componentsE x i and E
..
,in each g r a i n a r e i d e n t i c a l ( r e f e r t o equat ions (3) and (15) ). But, from equation (15)and f i g u r e 2(b) E
..
i s d e r i v e d as:Generation o f t h i s d l s c o n t l n u l t y and t h e c o n d ~ t l o n t h a t t h e upper and lower surfaces o f t h e specimen a r e kept f l a t b r i n g s about the non-uniform slip on the primary system. W i t h a symmetr~c b ~ c r y s t a l , Hook and H i r t h / 3 / have a l r e a d y r e p o r t e d t h e e f f e c t o f t h e ~ n - p l a n e shear s t r a l n component. The,;
experimental r e s u l t s f o r s l i p I lne d r s t r i b u t Ion and the present r e s u l t s a r e essent ~ a l l y t h e same.
The d l s c o n t l n u l t y o f the in-plane shear s t r a ~ n causes a mechanrcal t n t e r a c t ~ o n betueen the component gralns, too. For d ~ s c u s s l o n , assume t h a t g r a i n 2 i s c u t out o f the specimen and u n i f o r m shear on t h e prlmary s i i p systems occurs I n t h r e e grains. Because t h e i n - p l a n e shear s t r a l n i n g r a l n 2 occurs In t h e opposrte d i r e c t r o n t o t h a t i n t h e remaining p a r t , then t h e shape o f g r a i n 2 i s not t h e same as t h e shape o f t h e space i t once o c c u p ~ e d next t o g r a i n s 1 and 3. Thrs disagreement I n shear d e f o r m a t ~ o n causes g r a r n 2 be cramped by g r a i n 1 and 3 when t h e t r ~ c r y s t a l I S deformed as one c o n t ~ n u o u s body. Thrs cramping e f f e c t produces an i n t e r n a l s t r e s s f i e l d .
COLLOQUE DE PHYSIQUE
No. 7
1 3 . 0 4 X 1 0 ' ~
2 3.32
3 3 . 6 0
4 3.88
5 4.16
6 4.44
7 4.72
8 5 . 0 0
9 5.28
No. 7
1 0.10 10.'
2 0.21
3 0.31
4 .0.41
5 0.52
6 0.62
7 0.73
8 0.83
9 0.93
Fig.5 (a) Shear s t r a i n on t h e primary s l i p system and (b) the conjugate s l i p system i n the pseudo-compatible t r i c r y s t a l . The mean s t r e s s and the mean s t r a i n i n the y
d i r e c t ion a r e 1.74 MPa and 2.31 X 10-5, respectively.
Fig.6 Stress d i s t r i b u t i o n i n t h e unloaded pseudo-compatible t r i c r y s t a l . (a) Tensile s t r e s s and (b) compress i v e stress.
To examine t h e i n t e r n a l stress, the specimen i s unloaded u n t i l the t o t a l f o r c e on the loading surfaces decreases t o zero. D i s t r i b u t i o n o f the maximum and minimum p r i n c i p a l stresses i n the unloaded specimen i s given i n f i g u r e 6. Through t h e t r i p l e j u n c t i o n there a r e s l a n t ~ n g bands i n which t e n s i l e s t r e s s p a r a l l e l t o the bands occur. The d i r e c t i o n of the i n t e r n a l s t r e s s i n the t e n s i l e band r o t a t e s toward t h e X and - X axes. On t h e other hand. Figure 2(b) shows t h a t i n g r a i n s 1 and 3, r o t a t i o n o f t h e s t r e s s a x i s toward t h e X a x i s favors the a c t i v a t i o n o f the conjugate system. And a c t u a l l y . i n the v i c ~ n i t i e s o f t h e t r i p l e line, the d i r e c t i o n o f t h e region o f double s l i p coincides w i t h t h e d i r e c t ion o f the band. These f a c t s suggest t h a t the secondary s l i p system i n t h i s t r i c r y s t a l i s a c t i v a t e d on account o f t h i s i n t e r n a l stress.
Zaoui and CO-workers/5/./6/ have reported a j u n c t i o n type m u l t i p l e s l i p regton which i s not generated along the g r a i n boundary plane, but propagates i n t o t h e g r a i n i n t e r i o r from g r a i n boundary junctions. The Region o f secondary s l i p shown i n f i g u r e 5(b) i s s i m i l a r t o the j u n c t i o n t y p e m u l t i p l e s l i p region. Thus, n o t o n l y t h e c o m p a t i b i l i t y requirementon g r a i n boundaries, but a l s o the cramping e f f e c t which causes p l a s t i c m u l t i p l e s l i p i n p o l y c r y s t a l s must be taken note of.
5. Reference - -. .
/l/ Livingston, J.D. and Chalmers,
B.,
Acta Met., 5(1957), 322./2/ Hauser, J.J., and Chalmers, B., ibid., 9(1961), 802.
/3/ Hook,
R.E.,
and H i r t h , J.P., ibid., 15(1967), 535./4/ Ohashi. T.. Trans. Japan I n s t . Met., 28(1987), 906.
/5/ Zaoui, A., Model l in9 smal l deformat ions o f p o l y c r y s t a l s , ed. Gi t t u s , j. and Zarka. J., El sev i e r Appl Sci. Pub., London(1986), 187.
/6/ Rey, C., Mussot. P., and Zaoui, A., Grain boundary s t r u c t u r e and r e l a t e d phenomena. Suppl.
Trans. Japan Inst. Met.. 27(1986). 867.
/7/ H i l l , R.. J. Mech. Phys. Sol.. 14(1966). 95.
/8/ Kuhlmann-Wi lsdorf, D.. Met. Trans.. 16A(1985). 2091.
