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Shear deformation by the stress-induced martensitic transformation in shape memory alloys under the polycrystalline constraint
Scripta M E T A L L U R G I C A et M A T E R I A L I A
Vol.
24, pp. 2269-2272, 1990 Printed in the U.S.A.
Pergamon Press plc All rights reserved
SHEAR D E F O R M A T I O N BY THE S T R E S S - I N D U C E D 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 IN SHAPE MEMORY ALLOYS UNDER THE P O L Y C R Y S T A L L I N E C O N S T R A I N T Noboru Ono and Hiroshi Shimanuki* F a c u l t y of Engineering, Tohoku U n i v e r s i t y Aramaki-Aoba, Aoba, Sendai, 980, Japan. *Now at R & D L a b o r a t o r i e s - II, Nippon Steel Corporation, Fuchinobe, Sagamihara, 229, Japan. (Received August 6, 1990) (Revised September 18, 1990) Introduction Since 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 has a c r y s t a l l o g r a p h i c aspect, the performance of shape m e m o r y alloys(SMA) is influenced by texture. S i m i l a r l y to the case of slip deformation, the a n a l y s i s based on a p p r o p r i a t e p o l y c r y s t a l models serves for p r e d i c t i n g the effect of texture. Recently, the Taylor polycrystal model has been adopted to analyze the d e f o r m a t i o n of SMA by the formation of s t r e s s - i n d u c e d m a r t e n s i t e s ( S I M ) with the m o d i f i c a t i o n to include the volume change(I). The c a l c u l a t i o n based on this scheme was c o n d u c t e d for the case of uniaxial d e f o r m a t i o n in some r e p r e s e n t a t i v e SMA's(2). The c o m p a r i s o n of the calc u l a t i o n to e x p e r i m e n t s in Cu-base SMA's d e m o n s t r a t e d the q u a n t i t a t i v e reliability of the scheme(3,4). In many p r a c t i c a l applications, SMA is used in the form of a coil-spring, in which the principal d e f o r m a t i o n mode is shear. In this work, thus, the similar c a l c u l a t i o n of the m o d i f i e d Taylor factor M' and its o r i e n t a t i o n d e p e n d e n c e was c o n d u c t e d for the case of shear d e f o r m a t i o n of SMA's in two major categories; i.e., a TiNi alloy d e f o r m e d with 81'-SIM and Cu-base SMA's with either 81' or yI'-SIM. The results will be p r e s e n t e d and d i s c u s s e d in relation to the effect of texture on the p e r f o r m a n c e of SMA wire d e f o r m e d in twist. C a l c u l a t i o n Method The method for c a l c u l a t i n g M' is e s s e n t i a l l y the same as that d e s c r i b e d in (2) except that the p r e s c r i b e d d e v i a t o r i c strain is pure shear in the present work. That is, by using the vector notation, the strain is w r i t t e n as
{~'i}
= {0,0,0,0,~'5,0}-
[1 ]
A d e v i a t o r i c corner stress {0'i} is o b t a i n e d for a grain o r i e n t a t i o n as the solution of the l i n e a r - p r o g r a m m i n g p r o b l e m which calculates the stress that satisfies the yield c o n d i t i o n and, at the same time, can yield the p r e s c r i b e d strain. With this {o'i}, the m o d i f i e d Taylor factor M' for shear is defined as M' = nO'5/W' ,
[2]
where ~ and W' are the m a g n i t u d e of the shape strain of SIM(5) and the work done by the d e v i a t o r i c stress upon the formation of SIM of the unit volume, respectively. M' also relates the strain E' 5 with the volume fraction of SIM v in a grain as M' = nv/e' 5.
[3]
In the case of shear, the yield stress of a polycrystal ~5 is o b t a i n e d by simply a v e r a g i n g 0' 5 . This is related to the total work n e c e s s a r y to form the unit volume of SIM, W*, as follows. First, note that W' is given as W' = W* - Om~ ,
[4]
where the second term is the product of the h y d r o s t a t i c stress o m and the volu-
2269 0 0 3 6 - 9 7 4 8 / 9 0 $3.00 + .00 C o p y r i g h t (c) 1990 Pergamon P r e s s p l c
2270
SHEAR D E F O R M A T I O N AND 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
Vol.
24, No.
12
metric strain of SIM ~, representing the work done by °m- Substitute this to eqn.[2] and average it for 0' 5 to obtain ~'5 = M'W*/n - S'Om~/q.
[5]
We are considering the case that the deviatoric strain of pure shear as given in eqn.[1] is prescribed and there is no constraint upon the resulting m a c r o s c o p i c zolume change. That is, the e x t e r n a l l y applied stress {oi} is pure shear so that Gm=0. With this, the average in the second term of eqn.[5] is reduced to ~M'~Om, where AM' and Ao m denote the deviations from their averages. Ao m is caused by the d i f f e r e n c e among the volumetric strains of individual grains, which results from the n o n - u n i f o r m i t y in v as implied by eqn.[3]. Adopting the strategy given in (I) to estimate this, eqn.[5] is reduced to ~5 = M'W*/n
+ G'(~/n)2~S'2e' 5.
[6]
where G''is a factor depending on elastic coefficients. Since 3'5=0 upon the zielding, the shear yield stress for SIM is given with the average Taylor factor MEM' as 55 : MW~/n.
[7]
M' was calculated for 351 d i f f e r e n t l y oriented grains, the prescribed shear plane normals of which are evenly d i s t r i b u t e d in a unit stereographic triangle. In every grain, 23 shear directions were assigned on its shear plane. A more accurate result would be obtained when the number of the shear directions is increased. The alloys and martensites considered are the same as those in (2), i.e., 8 1 ' - m a r t e n s i t e in Cu-Zn-AI, 81'- and y 1 ' - m a r t e n s i t e in Cu-AI-Ni, and 81' m a r t e n s i t e in Ti-Ni. Refer to (2) for the c r y s t a l l o g r a p h i c data of these martensitic transformations. The small d i s a g r e e m e n t between the experimental data and the predictions from the c r y s t a l l o g r a p h i c p h e n o m e n o l o g i c a l theory found in the reports for YI' in CuAiNi(5) and for 81' in TiNi(6) influences the calculated value of M' only slightly as shown in (2). Thus, only the parameters based on the p h e n o m e n o l o g i c a l theory are used here. Results and Discussion The relations between M' and the mechanical properties of SMA's uniaxially d e f o r m e d by the SIM f o r m a t i o ~ have been d i s c u s s e d in (2) and they are valid also in the case of the shear deformation. An additional comment, however, may be w o r t h w h i l e on the effect of texture in the twisting of p o l y c r y s t a l l i n e wire. In the twisting of wire, shear occurs in the c i r c u m f e r e n t i a l direction. The shear in a set of grains with a p a r t i c u l a r shear plane normal n, therefore, occurs evenly in every d i r e c t i o n in the shear plane, provided that there is no preferred o r i e n t a t i o n in the c i r c u m f e r e n t i a l d i r e c t i o n among these grains. This c o n d i t i o n is fulfilled at least approximately, since the o r i e n t a t i o n distribution of grains in textured wire is g e n e r a l l y considered to be random with respect to the rotation around the wire axis(7). In other words, the relative population of the grains sheared on a p a r t i c u l a r plane is influenced by texture, but the d i s t r i b u t i o n of shear d i r e c t i o n in these grains is always uniform. In FIG.I, presented are M'maX(n), i.e., the maximum M' for a given shear plane normal n. The results for 81'-SIM's in CuZnAI and CuAINi are q u a l i t a t i v e l y identical so that only the result for the former is given. In the Cu-base alloys, the m i n i m u m of M'max(n) appears at n=<111> and the maximum of that, i.e., the overall maximum of M', M 'max, appears at n = < I 0 0 > In TiNi, the m i n i m u m of M'maX(n) appears between <100> and <111> and M 'max near <101> on the <101><111> line. By putting v=l in eqn.[3], the maximum possible strain produced by the SIM formation in the grain with M' is given as e5max = ~/M'.
[8]
e5max for M 'max, i.e., the minimum among e5max of e v e r y grain orientation, is a measure of the m a x i m u m a l l o w a b l e strain of SMA. M 'max and the c o r r e s p o n d i n g £5 max are summarized in TABLE I. As seen there, £5ma x is the smallest for YI' in CuAINi and the largest for TiNi. This is the same as in the case of uniaxial deformation(2).
Vol.
24,
No.
12
SHEAR
DEFORMATION
/Z•
(a)
AND
MARTENSITIC
TRANSFORMATION
~
"31
2271
.27
(b)
5.0 6.0
?.0 7.35-
531
4.90
(c)
24
FIG.I. The maximum M' in a shear plane, M'max(n), as a function of the shear plane normal. The minimum and the maximum in them are marked with dots and squares, respectively. (a) ~I' in CuZnA1. (b) YI' in CuAINi. (c) 81' in TiNi.
2.43
2.41
,~11
/•.60
r 6.6'~.
3.98
(c)
5.33
3.25
18/~ FIG.2. The average M' in a shear plane, M'(n), as a function of the shear plane normal. Dots and squares are the same as in FIG.I. (a) B I ' in CuZnAI. (b) YI' in CuAINi. (c) 81' in TiNi.
2272
SHEAR DEFORMATION AND MARTENSITIC TRANSFORMATION
Vol.
24,
No.
12
TABLE I. Some characteristic values of the modified Taylor factor M' and its averages. Internal stress and work-hardening caused by inhomogeneous volumetric strain are also included. Alloy
Marten- M,max site
e5max
~,(n)min ~,(n)max
~
AommaX
AM,2
(MPa) CuZnAl CuAINi TiNi
81' YI' 81' 81'
7.35 5.80 9.26 2.42
0.025 0.016 0.018 0.054
3.60 3.06 4.41 1.61
6.67 5.33 8.39 1.98
4.73 3.86 5.89 1.81
20.4 25.1 23.7 5.4
WorkHardening
(MPa) 1.57 1.02 2.59 0.08
0.23 0.42 0.31 0.02
Another imRortant parameter is the average M' over the shear directions on a shear plane, M'(n). From those d i s c u s s e d above, M'(n) represents the contribution from the grains with a given n to the overall response of p o l y c r y s t a l l i n e wire aginst twisting. The average of M'(n) for all n a s s u m i n g the uniform orientation d i s t r i b u t i o n gives the average Taylor factor M for texture-free polyc r y s t a l s for shear. This value, together with the m i n i m u m and the m a x i m u m of M'(n), M'(n) min and M'(n) max, is shown also in TABLE I. It is generally recommendable for SMA devices to be operated at stresses as low as possible to avoid the harmful processes such as plastic d e f o r m a t i o n by slip, fatigue or fracture. In wires used as coil-springs, therefore, the g[ain o r i e n t a t i o n with smaller M'(n) is more favorable than those with larger M'(n). As shown in FIG.2, the most favorable o r i e n t a t i o n in the C u - a l l o y s is <111>, and <110> may be equally good, <100> being the worst. The ratio of M'(n) max to M'(n) min is almost 2, so that significant improvement in the p e r f o r m a n c e of a SMA c o i l - s p [ i n g is e x p e c t e d if wire with the <111>- or <110>-texture is fabricated. In TiNi M'(n) mln is located near <113> and M'(n) tends to increase toward <100>, <111> and particularly <110>. The ratio of M'(n) max to M'(n) min is 1.23 so that the effect of texture may not be as marked as in the case of the Cu-base alloys. Finally, two other points of interest will be noted. TABLE I also includes the m a x i m u m of the h y d r o s t a t i c i n t e r n a l stress Ao m. This was c a l c u l a t e d from the d i f f e r e n c e between M 'max and M using the same elastic c o e f f i c i e n t s as those used in (2). At e5=0.01 , they are some 20MPa for the Cu-base alloys and 5MPa for TiNi. This stress is bound to assist the onset of fracture. The above values, however, are r e l a t i v e l y small in c o m p a r i s o n with the flow stresses of these alloys so that its c o n t r i b u t i o n may not be significant. The amount of the linear w o r k - h a r d e n i n g given by the second term of eqn.[6] is also e v a l u a t e d and i n c l u d e d in TABLE I. In all the alloys and SIM's, it is less than IMPa at e5=0.01 and can be safely ignored as in the case of the uniaxial deformation(2). In the case of the slip deformation, the Taylor model predicts the yield surface of isotropic p o l y c r y s t a l s to be close to that of the yon Mises theory (8): e.g., the ratio of the yield stress for uniaxial loading to the shear yield stress should be /3. This ratio is c a l c u l a t e d from M in TABLE I and the average of M's for tension and c o m p r e s s i o n given in (2). The obtained values are 1.733, 1.829, 1.751 and 1.753 for 81' in CuZnAI, YI' and 81' in CuAINi, and 81' in TiNi, respectively. All these results a p p r o x i m a t e l y conform to the von Mises yield criterion. References 1. 2. 3. 4. 5.
N. Ono and A. Sato, Trans. JIM, 2__99, 267 (1988). N. Ono, A. Sato and H. Ohta, Mater. Trans. JIM, 3__00,756 (1989). N. Ono, Mater. Trans., JIM, 31, 381 (1990). N. Ono, Mater. Trans., JIM, 31, (1990), in press. K. Okamoto, S. Ichinose, K. Morii, K. Otsuka and K.Shimizu, Acta Metall. 34, 2065 (1986). 6. O. Matsumoto, S. Miyazaki, K. Otsuka and H. Tamura, Acta Metall. 35, 2137 (1987). 7. B.D. Cullity, "Elements of X-Ray Diffraction", Addison-Wesley, (1956). 8. J.F.W. B i s h o p and R. Hill, Phil. Mag. 42, 414, 1298 (1951).
Vol.
24, pp. 2269-2272, 1990 Printed in the U.S.A.
Pergamon Press plc All rights reserved
SHEAR D E F O R M A T I O N BY THE S T R E S S - I N D U C E D 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 IN SHAPE MEMORY ALLOYS UNDER THE P O L Y C R Y S T A L L I N E C O N S T R A I N T Noboru Ono and Hiroshi Shimanuki* F a c u l t y of Engineering, Tohoku U n i v e r s i t y Aramaki-Aoba, Aoba, Sendai, 980, Japan. *Now at R & D L a b o r a t o r i e s - II, Nippon Steel Corporation, Fuchinobe, Sagamihara, 229, Japan. (Received August 6, 1990) (Revised September 18, 1990) Introduction Since 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 has a c r y s t a l l o g r a p h i c aspect, the performance of shape m e m o r y alloys(SMA) is influenced by texture. S i m i l a r l y to the case of slip deformation, the a n a l y s i s based on a p p r o p r i a t e p o l y c r y s t a l models serves for p r e d i c t i n g the effect of texture. Recently, the Taylor polycrystal model has been adopted to analyze the d e f o r m a t i o n of SMA by the formation of s t r e s s - i n d u c e d m a r t e n s i t e s ( S I M ) with the m o d i f i c a t i o n to include the volume change(I). The c a l c u l a t i o n based on this scheme was c o n d u c t e d for the case of uniaxial d e f o r m a t i o n in some r e p r e s e n t a t i v e SMA's(2). The c o m p a r i s o n of the calc u l a t i o n to e x p e r i m e n t s in Cu-base SMA's d e m o n s t r a t e d the q u a n t i t a t i v e reliability of the scheme(3,4). In many p r a c t i c a l applications, SMA is used in the form of a coil-spring, in which the principal d e f o r m a t i o n mode is shear. In this work, thus, the similar c a l c u l a t i o n of the m o d i f i e d Taylor factor M' and its o r i e n t a t i o n d e p e n d e n c e was c o n d u c t e d for the case of shear d e f o r m a t i o n of SMA's in two major categories; i.e., a TiNi alloy d e f o r m e d with 81'-SIM and Cu-base SMA's with either 81' or yI'-SIM. The results will be p r e s e n t e d and d i s c u s s e d in relation to the effect of texture on the p e r f o r m a n c e of SMA wire d e f o r m e d in twist. C a l c u l a t i o n Method The method for c a l c u l a t i n g M' is e s s e n t i a l l y the same as that d e s c r i b e d in (2) except that the p r e s c r i b e d d e v i a t o r i c strain is pure shear in the present work. That is, by using the vector notation, the strain is w r i t t e n as
{~'i}
= {0,0,0,0,~'5,0}-
[1 ]
A d e v i a t o r i c corner stress {0'i} is o b t a i n e d for a grain o r i e n t a t i o n as the solution of the l i n e a r - p r o g r a m m i n g p r o b l e m which calculates the stress that satisfies the yield c o n d i t i o n and, at the same time, can yield the p r e s c r i b e d strain. With this {o'i}, the m o d i f i e d Taylor factor M' for shear is defined as M' = nO'5/W' ,
[2]
where ~ and W' are the m a g n i t u d e of the shape strain of SIM(5) and the work done by the d e v i a t o r i c stress upon the formation of SIM of the unit volume, respectively. M' also relates the strain E' 5 with the volume fraction of SIM v in a grain as M' = nv/e' 5.
[3]
In the case of shear, the yield stress of a polycrystal ~5 is o b t a i n e d by simply a v e r a g i n g 0' 5 . This is related to the total work n e c e s s a r y to form the unit volume of SIM, W*, as follows. First, note that W' is given as W' = W* - Om~ ,
[4]
where the second term is the product of the h y d r o s t a t i c stress o m and the volu-
2269 0 0 3 6 - 9 7 4 8 / 9 0 $3.00 + .00 C o p y r i g h t (c) 1990 Pergamon P r e s s p l c
2270
SHEAR D E F O R M A T I O N AND 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
Vol.
24, No.
12
metric strain of SIM ~, representing the work done by °m- Substitute this to eqn.[2] and average it for 0' 5 to obtain ~'5 = M'W*/n - S'Om~/q.
[5]
We are considering the case that the deviatoric strain of pure shear as given in eqn.[1] is prescribed and there is no constraint upon the resulting m a c r o s c o p i c zolume change. That is, the e x t e r n a l l y applied stress {oi} is pure shear so that Gm=0. With this, the average in the second term of eqn.[5] is reduced to ~M'~Om, where AM' and Ao m denote the deviations from their averages. Ao m is caused by the d i f f e r e n c e among the volumetric strains of individual grains, which results from the n o n - u n i f o r m i t y in v as implied by eqn.[3]. Adopting the strategy given in (I) to estimate this, eqn.[5] is reduced to ~5 = M'W*/n
+ G'(~/n)2~S'2e' 5.
[6]
where G''is a factor depending on elastic coefficients. Since 3'5=0 upon the zielding, the shear yield stress for SIM is given with the average Taylor factor MEM' as 55 : MW~/n.
[7]
M' was calculated for 351 d i f f e r e n t l y oriented grains, the prescribed shear plane normals of which are evenly d i s t r i b u t e d in a unit stereographic triangle. In every grain, 23 shear directions were assigned on its shear plane. A more accurate result would be obtained when the number of the shear directions is increased. The alloys and martensites considered are the same as those in (2), i.e., 8 1 ' - m a r t e n s i t e in Cu-Zn-AI, 81'- and y 1 ' - m a r t e n s i t e in Cu-AI-Ni, and 81' m a r t e n s i t e in Ti-Ni. Refer to (2) for the c r y s t a l l o g r a p h i c data of these martensitic transformations. The small d i s a g r e e m e n t between the experimental data and the predictions from the c r y s t a l l o g r a p h i c p h e n o m e n o l o g i c a l theory found in the reports for YI' in CuAiNi(5) and for 81' in TiNi(6) influences the calculated value of M' only slightly as shown in (2). Thus, only the parameters based on the p h e n o m e n o l o g i c a l theory are used here. Results and Discussion The relations between M' and the mechanical properties of SMA's uniaxially d e f o r m e d by the SIM f o r m a t i o ~ have been d i s c u s s e d in (2) and they are valid also in the case of the shear deformation. An additional comment, however, may be w o r t h w h i l e on the effect of texture in the twisting of p o l y c r y s t a l l i n e wire. In the twisting of wire, shear occurs in the c i r c u m f e r e n t i a l direction. The shear in a set of grains with a p a r t i c u l a r shear plane normal n, therefore, occurs evenly in every d i r e c t i o n in the shear plane, provided that there is no preferred o r i e n t a t i o n in the c i r c u m f e r e n t i a l d i r e c t i o n among these grains. This c o n d i t i o n is fulfilled at least approximately, since the o r i e n t a t i o n distribution of grains in textured wire is g e n e r a l l y considered to be random with respect to the rotation around the wire axis(7). In other words, the relative population of the grains sheared on a p a r t i c u l a r plane is influenced by texture, but the d i s t r i b u t i o n of shear d i r e c t i o n in these grains is always uniform. In FIG.I, presented are M'maX(n), i.e., the maximum M' for a given shear plane normal n. The results for 81'-SIM's in CuZnAI and CuAINi are q u a l i t a t i v e l y identical so that only the result for the former is given. In the Cu-base alloys, the m i n i m u m of M'max(n) appears at n=<111> and the maximum of that, i.e., the overall maximum of M', M 'max, appears at n = < I 0 0 > In TiNi, the m i n i m u m of M'maX(n) appears between <100> and <111> and M 'max near <101> on the <101><111> line. By putting v=l in eqn.[3], the maximum possible strain produced by the SIM formation in the grain with M' is given as e5max = ~/M'.
[8]
e5max for M 'max, i.e., the minimum among e5max of e v e r y grain orientation, is a measure of the m a x i m u m a l l o w a b l e strain of SMA. M 'max and the c o r r e s p o n d i n g £5 max are summarized in TABLE I. As seen there, £5ma x is the smallest for YI' in CuAINi and the largest for TiNi. This is the same as in the case of uniaxial deformation(2).
Vol.
24,
No.
12
SHEAR
DEFORMATION
/Z•
(a)
AND
MARTENSITIC
TRANSFORMATION
~
"31
2271
.27
(b)
5.0 6.0
?.0 7.35-
531
4.90
(c)
24
FIG.I. The maximum M' in a shear plane, M'max(n), as a function of the shear plane normal. The minimum and the maximum in them are marked with dots and squares, respectively. (a) ~I' in CuZnA1. (b) YI' in CuAINi. (c) 81' in TiNi.
2.43
2.41
,~11
/•.60
r 6.6'~.
3.98
(c)
5.33
3.25
18/~ FIG.2. The average M' in a shear plane, M'(n), as a function of the shear plane normal. Dots and squares are the same as in FIG.I. (a) B I ' in CuZnAI. (b) YI' in CuAINi. (c) 81' in TiNi.
2272
SHEAR DEFORMATION AND MARTENSITIC TRANSFORMATION
Vol.
24,
No.
12
TABLE I. Some characteristic values of the modified Taylor factor M' and its averages. Internal stress and work-hardening caused by inhomogeneous volumetric strain are also included. Alloy
Marten- M,max site
e5max
~,(n)min ~,(n)max
~
AommaX
AM,2
(MPa) CuZnAl CuAINi TiNi
81' YI' 81' 81'
7.35 5.80 9.26 2.42
0.025 0.016 0.018 0.054
3.60 3.06 4.41 1.61
6.67 5.33 8.39 1.98
4.73 3.86 5.89 1.81
20.4 25.1 23.7 5.4
WorkHardening
(MPa) 1.57 1.02 2.59 0.08
0.23 0.42 0.31 0.02
Another imRortant parameter is the average M' over the shear directions on a shear plane, M'(n). From those d i s c u s s e d above, M'(n) represents the contribution from the grains with a given n to the overall response of p o l y c r y s t a l l i n e wire aginst twisting. The average of M'(n) for all n a s s u m i n g the uniform orientation d i s t r i b u t i o n gives the average Taylor factor M for texture-free polyc r y s t a l s for shear. This value, together with the m i n i m u m and the m a x i m u m of M'(n), M'(n) min and M'(n) max, is shown also in TABLE I. It is generally recommendable for SMA devices to be operated at stresses as low as possible to avoid the harmful processes such as plastic d e f o r m a t i o n by slip, fatigue or fracture. In wires used as coil-springs, therefore, the g[ain o r i e n t a t i o n with smaller M'(n) is more favorable than those with larger M'(n). As shown in FIG.2, the most favorable o r i e n t a t i o n in the C u - a l l o y s is <111>, and <110> may be equally good, <100> being the worst. The ratio of M'(n) max to M'(n) min is almost 2, so that significant improvement in the p e r f o r m a n c e of a SMA c o i l - s p [ i n g is e x p e c t e d if wire with the <111>- or <110>-texture is fabricated. In TiNi M'(n) mln is located near <113> and M'(n) tends to increase toward <100>, <111> and particularly <110>. The ratio of M'(n) max to M'(n) min is 1.23 so that the effect of texture may not be as marked as in the case of the Cu-base alloys. Finally, two other points of interest will be noted. TABLE I also includes the m a x i m u m of the h y d r o s t a t i c i n t e r n a l stress Ao m. This was c a l c u l a t e d from the d i f f e r e n c e between M 'max and M using the same elastic c o e f f i c i e n t s as those used in (2). At e5=0.01 , they are some 20MPa for the Cu-base alloys and 5MPa for TiNi. This stress is bound to assist the onset of fracture. The above values, however, are r e l a t i v e l y small in c o m p a r i s o n with the flow stresses of these alloys so that its c o n t r i b u t i o n may not be significant. The amount of the linear w o r k - h a r d e n i n g given by the second term of eqn.[6] is also e v a l u a t e d and i n c l u d e d in TABLE I. In all the alloys and SIM's, it is less than IMPa at e5=0.01 and can be safely ignored as in the case of the uniaxial deformation(2). In the case of the slip deformation, the Taylor model predicts the yield surface of isotropic p o l y c r y s t a l s to be close to that of the yon Mises theory (8): e.g., the ratio of the yield stress for uniaxial loading to the shear yield stress should be /3. This ratio is c a l c u l a t e d from M in TABLE I and the average of M's for tension and c o m p r e s s i o n given in (2). The obtained values are 1.733, 1.829, 1.751 and 1.753 for 81' in CuZnAI, YI' and 81' in CuAINi, and 81' in TiNi, respectively. All these results a p p r o x i m a t e l y conform to the von Mises yield criterion. References 1. 2. 3. 4. 5.
N. Ono and A. Sato, Trans. JIM, 2__99, 267 (1988). N. Ono, A. Sato and H. Ohta, Mater. Trans. JIM, 3__00,756 (1989). N. Ono, Mater. Trans., JIM, 31, 381 (1990). N. Ono, Mater. Trans., JIM, 31, (1990), in press. K. Okamoto, S. Ichinose, K. Morii, K. Otsuka and K.Shimizu, Acta Metall. 34, 2065 (1986). 6. O. Matsumoto, S. Miyazaki, K. Otsuka and H. Tamura, Acta Metall. 35, 2137 (1987). 7. B.D. Cullity, "Elements of X-Ray Diffraction", Addison-Wesley, (1956). 8. J.F.W. B i s h o p and R. Hill, Phil. Mag. 42, 414, 1298 (1951).