A non-invariant plane model for the interface in CuAlNi single crystal shape memory alloys

Transcription

1 Journal of the Mechanics and Physics of Solids 48 () A non-invariant plane model for the interface in CuAlNi single crystal shape memory alloys Zhang Xiangyang a, Sun Qingping a,*, Yu Shouwen b a Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, SAR, People s Republic of China b Department of Engineering Mechanics, Tsinghua University, Beijing, 184, People s Republic of China Received 19 March 1999; received in revised form 1 November 1999 Abstract A non-invariant plane model for the austenite-martensitic (A-M) interface in CuAlNi single crystal shape memory alloys (SMAs) is proposed in this paper. The model is based on the crystallography of martensitic transformation and the recent uniaxial tensile Moiré tests on Cu-14%Al-4.1%Ni (wt%) single crystals. The two types of specimens used have the same tensile axis orientation but have different transition temperatures. One exhibits Shape Memory Effect (SME) and the other exhibits Superelasticity (SE). In the case of SME, the plane invariant nature of the A-M interface is well verified by the Moiré test. On the contrary, the A-M interface in the case of SE was identified to be a non-invariant plane. A crystallographybased model is proposed to explain the formation of this non-plane-invariant A-M interface in SE and is used to predict the resultant interface structure and the transformation strain of the stress-induced b 1 b 1 transformation in CuAlNi. The comparison between the theoretical calculations and the experimental results support the proposed model. Elsevier Science Ltd. All rights reserved. Keywords: Single crystal; Shape memory alloys; Austenite-martensite interface; Non-invariant plane; Martensitic phase transformation * Corresponding author. Tel.: ; fax: address: meqpsun@usthk.ust.hk (S. Qingping). -596//$ - see front matter Elsevier Science Ltd. All rights reserved. PII: S -596(99)1-7

2 164 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () Introduction The stress-induced martensitic transformation in CuAlNi single crystal shape memory alloys (SMAs) generally exhibit two kinds of behavior. When the single crystal Cu-Al-Ni SMAs are stressed at a temperature higher than A f (austenite finish temperature), they transform from b 1 parent phase to b 1 martensitic phase. The b 1 martensite variant is 18R type long period stacking order structure with stacking faults as the internal defects (Otsuka et al. 1974a, 1979; Saburi and Nenno, 1981). The martensite reverts to the parent phase (austenite) on the removal of stress because the stress free b 1 phase is unstable in this temperature range (Horikawa et al., 1988; Jiang and Xu, 199; Shield, 1995). This phenomenon is known as Superelasticity (SE) or Pseudoelasticity (PE). When the SMAs are stressed at the temperature near the M s (martensite start temperature) point, they transform from b 1 phase to g 1 martensitic phase. The g 1 martensite is H type structure with internal twins (Otsuka and Shimizu, 1974b; Saburi and Nenno, 1981; Okamoto et al., 1986). The martensite variant persists upon removal of the applied stress and the alloy must be heated to induce the reverse transformation (Oishi and Brown, 1971; Okamoto et al., 1986; Miyazaki and Otsuka, 1989). This phenomenon is known as Shape Memory Effect (SME). Significant progress in the research of SMAs has been made since the relationship between the properties of SMAs and the thermoelastic martensitic transformation was revealed (see Miyazaki and Otsuka, 1989). The deformations of stress-induced martensitic transformation in both SE and SME are strongly dependent on the crystal orientation as well as on the temperature. The investigation on the orientation dependence includes: (1) the orientation dependence of the Austenite-Martensite (A-M) interfaces, and () the orientation dependence of the transformation strain. Some crystallographic theories of the martensitic transformation were proposed to deal with these problems. They are mainly the Phenomenological Theory of Martensite Crystallography (PTMC, by Wechsler, Lieberman and Read, 1953, known as WLR Theory; and by Bowles and Mackenzie, 1954, known as BM Theory) and the Crystallographic Theory of Martensite (CTM, Ball and James, 1987). All these theories are based on such a hypothesis: there exists an undistorted invariant plane (called the habit plane) between the martensite and the parent phase. Based on this hypothesis, the interface normal, the substructure of martensite and the transformation strains are predicted. For the case of SME, much literature showed that the theoretical predictions of the above theories agreed well with the experimental results on habit plane normal, transformation strains and orientation relationship. For example, such agreement can be found in the b 1 g 1 stress-induced martensitic transformation in CuAlNi SMAs (Otsuka et al., 1974a; Okamoto et al., 1986; Shield, 1995; Zhang et al., 1999). However, for the b 1 b 1 stress-induced martensitic transformation in CuAlNi SMAs in the case of SE, the non-invariant plane of the A-M interface was recently identified by high sensitivity Moiré tests (Sun et al. 1997, 1999). This discovery raised doubt on the invariant plane hypothesis of Phenomenological Theory of Martensite Crystallography (PTMC) that was used to interpret the A-M interface in almost all theoretical models. So far, a satisfactory crystallography-based continuum mechanics theory

3 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () to predict this non-invariant plane A-M interface in SE and the associated transformation strains has not been available in the literature. The idea of the non-invariant plane concept can be traced back to the earlier work of Bowles and Mackenzie (1954). Actually in B M theory an isotropic dilatation of the interface is admitted. The use of a dilatation parameter gives the B M theory an additional degree of flexibility (Wayman, 1964). By using such an idea, recently Kato (1998) analyzed the Cubic 18R(9R) martensitic transformation in copperbased shape memory alloys. The resulting transformation strain is therefore not a plane invariant strain. However simply introducing an isotropic dilatation into the habit plane could not provide more understanding of the microstructure of the interface. Buchheit and Wert (1996) proposed a detwinning model in which the predicted transformation strain is not a plane invariant strain. But only detwinning of the twinned g 1 martensite is not enough to form the stacking faulted b 1 martensite since the lattice structure and the lattice parameters of b 1 martensite are different from those of g 1 martensite (Okamoto et al., 1986; Horikawa et al., 1988; Miyazaki and Otsuka, 1989). The purpose of this article is to give a crystallographic prediction of b 1 b 1 martensitic transformation in CuAlNi single crystal shape memory alloys. In Section the experimental observations on the invariant-plane (IP) and non-invariant-plane (NIP) A-M interfaces in CuAlNi SMA are briefly reported. Based on the experimental observations a model is proposed to predict the non-plane-invariant interface in the case of SE. The model incorporates three basic operations of (1) detwinning of g 1, () the lattice distortion from g 1 to b 1 martensite, and (3) the stacking faults (lattice invariant slip shear) within the b 1 martensite. The A-M interface normal and the transformation strain of b 1 b 1 martensitic transformation are calculated. In Section 3 theoretical predictions are compared with the experimental results of both SME and SE. The conclusions are given in Section 4.. Experimental observations The shape and size of the uniaxial tensile single crystal specimen is shown in Fig. 1. The specimens were differently heat treated to get different transition temperatures. Fig. 1. The shape and size of the uniaxial tensile specimen.

4 166 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () Table 1 The orientation of the specimen (in degree) [1] [1] [1] X Y Z For the SME specimen the transition temperatures are M s =79 K, M f =76 K, A s =38 K, A f =318 K (by Differential Scanning Calorimeter DSC). For the SE specimen the characteristic temperatures are M s =36 K, M f =5 K, A s =36 K, A f =53 K. The orientation of the specimen was determined by the X-ray Back Laue Method. The relationships between the lattice axes and the space axes (with Y axis as the loading axis) are listed in Table 1. The results of the experiments are also summarized in Table 1 (see Zhang et al., 1997; Sun et al., 1999 for detailed information on the test)..1. Stress-induced b 1 g 1 transformation at T A s (shape memory effect) Fig. is the typical measured nominal tensile stress-strain curve for the SME specimen at T=93 K on MTS machine under displacement control. The loading speed is.1 mm/s and the gauge length is 5 mm. The serrations in the curve are due to the sudden formation of stress-induced martensite plate with sharp A-M interface. From the unloading curve, it is seen that the stress-induced martensite is stable at room temperature and remains on removal of the stress. The full-field deformation around the A-M interface is measured by the Moiré Fig.. The macroscopic nominal stress-strain curve under uniaxial tension for SME specimen.

5 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () interference technique. A high frequency crossed-line grating (1 lines/mm) was replicated on the surface of the specimen by a very thin layer of epoxy cement to ensure the grating deforms together with the specimen beneath. Uniaxial loading is realized by a specially designed loading frame. Figs. 3a and b respectively show the amplified u (along x-direction) and v (along y-direction) displacement fringe patterns Fig. 3. Fringe patterns of u (a) and v (b) displacement fields across the A-M interface for SME specimen.

6 168 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () across the A-M interface. These fringe patterns respectively represent contours of in-plane displacement u and v, from which the in-plane strain components can be calculated directly (Post et al., 1994; Zhang et al., 1997). It can be seen that both the strains in the martensitic region (with dense fringes) and in the parent phase (with sparse fringes) are uniform. It is also seen that the interface is straight and inclined at an angle of 55.5 with the x-axis in the x-y plane. There is a clear strain jump across the interface. After unloading, the fringe patterns in austenite totally disappear and the uniform transformation strain in martensite remains, which directly indicates that the A-M interface is indeed an undistorted invariant plane (IP). The martensite consists of twinned variants and the twinning morphology can be clearly observed under Atomic Force Microscope... The stress-induced b 1 b 1 transformation at T A f (superelasticity) Fig. 4 is the measured typical nominal stress strain curve of an SE specimen in an MTS test machine. It is seen that the macroscopic transformation strain of martensite ( 6%, where the whole gauge length is occupied by martensite) is much larger than that in SME. Fig. 5 shows the fringe patterns of the displacement fields at the initial stage of nucleation where many tiny narrow bands can be observed on the specimen surface. These bands appeared with an inclined angle of 5.3 with the x- axis in x-y plane. There are weak interactions between the austenite and martensite bands and among the narrow martensite bands at this stage. In addition, the jump of strain across the A-M interface is not as clear as in the case of SME (see Fig. 3). With further loading, these bands begin to merge into a single band and strains inside the bands increase with loading. The deformation afterward is mainly accomplished by the growth of a single martensite band via steady state propagation of the A-M interface (like Luder s band propagation in metals), which corresponds to the plateau in the curve of Fig. 4. Macroscopic direct observation on the specimen Fig. 4. The macroscopic nominal stress-strain curve under uniaxial tension for SE specimen.

7 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () Fig. 5. Fringe patterns of u (a) and v (b) displacement fields at the initial nucleation of b 1 martensite. surface without grating showed that the A-M interface is sharp and straight. The corresponding fringe patterns of the displacement fields across the A-M interface in this stage by Moiré are shown in Fig. 6. In sharp contrast with the case of SME, there is a high strain concentration in austenite near the A-M interface and no jump in the total strain can be observed across the physical A-M interface. The size of the region has the same dimension as the interface. This indicates that the A-M interface in superelasticity is strongly deformed and no longer an invariant plane. Far away from the interface the strains in both martensite and austenite regions are quite uniform and not disturbed. The uniform fringe patterns in the transformed zone are shown in Fig. 7. Experimental observation (Otsuka et al., 1974a; Saburi and Nenno, 1981; Horikawa et al., 1988) revealed that the A-M interface consists of stacking fault and no twinning can be observed in the b 1 martensite. 3. Non-invariant plane interface model It is seen that two kinds of interfaces invariant plane (IP) and non-invariantplane (NIP) are identified in single crystal CuAlNi and their different deformation features are revealed. The experimental results demonstrate that the invariant plane assumption on the A-M interface agrees very well with the Moiré data for the case of SME but does not hold in the case of SE at temperature T A f. For the noninvariant plane (in the case of SE) so far there is no satisfactory theoretical modelling to predict its formation and the corresponding transformation strain. In the modelling aspects, for the IP interface in the case of SME, there is a well known CTM theory (Ball and James, 1987) to model its structure and property (Ball and James, 1987; Bhattacharya 1991, 199). For the interface in the case of SE, it

8 17 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () Fig. 6. Fringe patterns of u (a) and v (b) displacement fields across the A-M interface during the steady state propagation of b 1 martensite. appears that the following two questions must be answered from the deformation mechanism and crystallographic points of view: (1) how to determine the A-M interface orientation in the b 1 b 1 martensitic transformation, and () what is the transformation strain of b 1. A quantitative, microstructure-based correlation between macroscopic observed quantities and the underlying structure parameters of the

9 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () Fig. 7. Fringe patterns of u (a) and v (b) displacement fields in the fully transformed zone. material is highly desired. Experimental observation on b 1 b 1 transition showed that the martensite consists of only single crystal b 1 (no twinning) with stacking fault (slip shear) at the A-M interface (Otsuka et al., 1974a; Saburi and Nenno, 1981; Horikawa et al., 1988) and that the A-M interface normal of the b 1 martensite is almost the same as the habit plane normal of the g 1 martensite (referring to Section ). Inspired by this, we propose that the process of b 1 b 1 is realized through the following steps. The b 1 parent phase is first transformed into g 1 twinned martensite; this transient or intermediate state of twinned martensite is further detwinned into a single correspondence variant by lattice invariant shear. Then g 1 single crystal is transformed into b 1 single crystal by a lattice distortion and finally slip shear (also lattice invariant shear) happens inside the b 1 lattice. Although the above steps might not exactly correspond to the real transformation processes, they do describe the final deformation state of the b 1 martensite. In the following the results for the SME habit plane case (i.e., b 1 g 1 twinned martensite) by using the well-known CTM

10 17 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () theory (Ball and James, 1987) will be given first. Based on this, a non-plane-invariant model is established From b 1 to the twinned g 1 martensite The six lattice correspondences of b 1 g 1 martensitic transformation for CuAlNi single crystal are listed in Table (Okamoto et al., 1986). The deformation gradient matrix F 1 of variant 1, for instance, is F 1 a a b a a a c c a a where a is the lattice parameter for the cubic b 1 parent phase and a, b, c are the parameters for the orthorhombic g 1 martensite. The gradient matrix F 1 can be polar decomposed as F 1 R 1 U 1 () where R 1 is a rigid rotation satisfying R T 1 R 1 and U 1 is the symmetric part of F 1 where U 1 b a+g a g a g a+g a a, b b, g c (4) a a a There are, in total, six independent symmetric matrices U i (i =1,,, 6) for b 1 g 1 martensitic transformation (see Bhattacharya, 1991; Shield, 1995; Zhang et al., (1) (3) Table The six lattice corresponding variants in g 1 Variant [1] m.[11] p.[1 1] p.[11] p.[11 ] p.[11] p.[1 1] p [1] m [1 ] p [1 ] p [1 ] p [1 ] p [1 ] p [1 ] p [1] m.[1 1] p.[1 1 ] p.[11 ] p.[1 1 ] p.[1 1] p.[1 1 ] p

11 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () and the Appendix) and each matrix represents one lattice correspondence variant. A sketch of the twinned martensite and the habit plane is shown in Fig. 8. The matrices F A and F B (A and B are integers in {1,, 6}) are respectively the deformation gradients of the two correspondence variants in the twin and the identity matrix I is the deformation gradient of the parent phase. The plane P-P represents the habit plane (A-M interface) and the vector n is the twinning plane normal and m is the habit plane normal. Assuming the volume fractions of variant A and B in the twinning system are respectively (1 f) and f, the average deformation for the martensite is F M R h (ff B (1 f)f A ) (5) where R h is the relative rotation between the twinned martensite and the parent phase. The kinematic compatibility between the two correspondence variants across the twinning plane requires (Ball and James, 1987; Bhattacharya 1991, 199) F B F A a n (6) and that between the martensite and the parent phase across the habit plane requires (Ball and James, 1987; Bhattacharya 1991, 199) F M I b m (7) where vector a is the twinning shear and b is the macroscopic shear of the martensite. Eqs. (5) and (6) can be rewritten in terms of the symmetric matrices U i as (Ball and James, 1987; Bhattacharya 1991, 199), F M R h (U A fa n) (8) R AB U B U A a n (9) where R AB is an orthogonal tensor and represents the relative rotation between the two twins. Eq. (8) can also be expressed as F M R h (I fa n U 1 A ) U A R h S g1 U A (8a) A sketch of the A-M interface (invariant plane) between the austenite and the twinned g 1 marten- Fig. 8. site.

12 174 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () where S g1 I+fa n U 1 A is the lattice invariant shear that is equivalent to the twinning shear. The shear plane normal in the deformed configuration is n U 1 A, and the shearing vector is fa. A strategy on how to solve the problem of Eqs. (7) (9) were introduced in detail by Ball and James (1987). The solution to this problem is the quantities of a, n, b, m, R AB, R h and f. There are 1 choices for A and B in Eq. (9) and each pair can form either Type I twin or Type II twin. So there are 4 twinning systems of g 1 martensite in CuAlNi single crystal SMAs. Furthermore, each twinning system contains 4 habit plane variants, i.e., there are 4 solutions of Eqs. (7) and (8) for each a and n. Therefore, in total, 96 possible habit plane variants can be obtained for the CTM problem. See the references for more detailed discussion (Shield, 1995; Zhang et al., 1999). 3.. From the twinned g 1 martensite to the stacking faulted b 1 martensite The assumed structure of stacking faulted b 1 martensite is illustrated in Fig. 9. Compared with the structure of twinned g 1 martensite in Fig. 8, it is the product of detwinning (the minor variant in the g 1 twin changes into the major variant), lattice distortion and slip shear (abbreviated as DDS). It is seen that in this DDS model we can still use the A-M interface determined by CTM theory in Section 3.1 as the interface between the stacking faulted b 1 martensite and the parent phase. Thus the first question on the interface orientation in SE has been solved. The twelve lattice correspondences of b 1 b 1 martensitic transformation are listed in Table 3 (Horikawa et al., 1988). By comparing with Table, we can see that the variant 1 of g 1 is very similar to variants 1 and 1 of b 1, and so are the other variants, 3, 4, 5 and 6. The only difference is in the axis [1] m. Hence we may let the variant 1 of g 1 be changed to the variants 1 or 1 of b 1, variant of g 1 be changed to the variants or of b 1, and so on. This is just what happened in the lattice distortion step. The deformation gradients from cubic to monoclinic (b 1 martensite correspon- Fig. 9. A sketch of the strongly deformed A-M interface (non-invariant plane) between the austenite and the stacking faulted b 1 martensite.

13 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () Table 3 The 1 lattice corresponding variants in b 1 Vt [1] m.[11] p.[1 1 ] p.[1 1] p.[1 1] p.[11] p.[1 1 ] p.[11 ] p.[11 ] p.[11] p.[1 1 ] p.[1 1] p.[1 1] p [1] m [1 ] p [1] p [1 ] p [1] p [1 ] p [1] p [1 ] p [1] p [1 ] p [1] p [1 ] p [1] p [1] m [4 5] p [5 4] p [5 4 ] p [45] p [54 ] p [45 ] p [4 5 ] p [54] p [4 5] p [5 4] p [5 4 ] p [45] p dence variant) are denoted by G i and G i (i =1,,, 6). The deformation matrix of variant 1 of b 1, for instance, is: G 1 1a 9a 8a 9a b (1) a c c 9a 9a where a is the lattice parameter of the cubic b 1 parent phase and a, b, c are the parameters of the monoclinic b 1 martensite. To decompose G 1 into the similar form of F 1 as in the case of SME (Eq. ()), we divide the G 1 into two parts: First, let the lattice line [ 4 5] p change to 1[ 9 9] p and keep the other two lattice lines ( 1[11] p and [1 ] p ) unchanged. The deformation gradient for this step (variant 1 for instance) is: D (11) The matrices for the other variants are listed in the Appendix. Second, let the lattice correspondences be 1 [ 1 1] p [1 ] m,[1 ] p [ 1 ] m, 1 [ 9 9] p [ 1] m. In this step, the deformation gradient (variant 1 for instance) is

14 176 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () a a a a T 1 b (1) a c c 9a 9a It is easy to verify that G 1 =T 1 D 1. If we define three constants a, b and g as a a, b b, g c, (13) a a 9a then the polar decomposition of T 1 is T 1 R 1 U 1 (14) where R 1 is the same as that in Eq. () and U 1 is expressed in terms of a, b and g as U1 b a +g a g. a g a +g Hence the deformation gradient G 1 can be written as G 1 R 1 U 1 D 1 R 1 V 1 (15) where V 1 U 1 D 1 b 1a +9g 8a 9g 18 (16) 1a 9g 8a +9g 18 V 1 is the deformation gradient of correspondence variant in SE (i.e., from b 1 b 1). Comparing Eq. (15) with Eq. (), it is seen that V 1 corresponds to U 1 in SME. In this paper, the slip shear in b 1 is treated as being equivalent to the twinning shear in the sense of average deformation. Therefore, we will use V 1 instead of U 1 in Eq. (8) to calculate the average transformation strain of b 1 martensite (without losing generality, let A=1 in this equation). Thus we have the average deformation of b 1 G M R h (V 1 f ã ñ) R h S b 1 V 1 (17) Similar to the case of twinning in SME, S b 1 =I+f ã ñ V 1 1 is the lattice invariant shear

15 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () due to slip (slip shear), f, ã and ñ are the counterparts to f, a and n in the case of b 1 b 1. There is no plane-invariant constraint on S b1. The observed lattice invariant shear in b 1 martensite in CuAlNi single crystal SMAs is {1 1} 11 (Otsuka et al. 1974a, 1979; Saburi and Nenno, 1981; Horikawa et al., 1988). These slip planes {11} are consistent with the twinning planes of the g 1 martensite variants with Type I Twin in CuAlNi single crystal SMAs (Shield, 1995; Zhang et al., 1999). Here we assume that the lattice invariant shear in the stacking faulted b 1 martensite has the same value as that of the twinned g 1 martensite. That is, the shearing vector of the simple shear in b 1 martensite is fa, and the shear plane normal is n (referring to Figs. 8 and 9). Theoretically S b1 should take such a value as to minimize the free energy of the system under applied stress. Note that n is defined in the undeformed reference and it is changed to n V 1 1 in the b 1 martensite after the phase transformation. Thus the lattice invariant shear in the b 1 martensitic transformation can be written as S b1 I fa n V 1 1 (18) and the average deformation of b 1 martensite is G M R h S b1 V 1 R h (V 1 fa n) (19) This equation can be obtained by just replacing U 1 in Eq. (8) by V 1. It is easy to verify that G M is not a plane invariant strain. This means that not only the individual b 1 correspondence variant is not coherent with b 1, but also the laminate of b 1 is still not coherent with b 1, leading to a strongly distorted macroscopic A-M interface as shown by the shaded area in Fig. 9. In the case of SME (Eqs. (7) (9)) there are 96 possible solutions. Compared with SE, the symmetric deformation U i in Eq. (3) can be replaced by either V i or V i, so there are in total 96 =19 possible solutions in this case. The present model extends the CTM theory from the plane invariant martensitic transformation to the non-plane-invariant martensitic transformation and thus we can call it a CTM+DDS model Comparison between theory and experiments The lattice parameters used in the calculation are from the literature (Otsuka et al., 1974a; Okamoto et al., 1986; Horikawa et al., 1988): Cubic b 1 phase: a =5.836Å Orthorhombic g 1 phase: a=4.38å, b=5.356å, c=4.å Monoclinic b 1 phase: a =4.38Å, b =5.356Å, c =38.Å The material studied here is single crystal CuAlNi shape memory alloys. The composition of the specimens (Cu-14wt%Al-4.1wt%Ni) is slightly different from those in the literature. Since the lattice parameters are not very sensitive to the composition, the above lattice parameters can still be used in the calculation. The con-

16 178 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () stants of a, b, g and a, b, g for b 1 g 1 and b 1 b 1 transformation are (see Eqs. (4) and (13)) a a 1.619, b b.9178, g g 1.31 () Thus U 1 =U 1. The deformation strains of b 1 b 1 martensitic transformation can be calculated with the interface orientation given by the calculation of the b 1 g 1 martensitic transformation. As described by Shield (1995), the work W i done by the applied stress s on the transformation strain e i can be used to select the variant. The work W i is given by W i s:e i (1) where i is an integer in {1,, 96} for the SME case and in {1,, 19} for SE case. The uniaxial tensile stress can be written as s te e () where t is the magnitude of the stress and the vector e is the uniaxial loading direction. The transformation strain is e i 1 (BT i B i I) (no summation) (3) where B i is the gradient matrix for the martensitic transformation. In the case of SME, B i equals F M in Eq. (7); in the case of SE, B i equals G M in Eq. (19). The variant with the largest value of W i will appear first under the applied stress. For the case of SME, the correspondence variant pair C.5-4 with Type I twin corresponds to the maximum work among the 96 g 1 variants. Here C.5-4 means that the twinning system consists of the lattice correspondence variants 5 and 4 and that variant 5 is the major part in the twin (see Table for reference). So, in Table 4 the habit plane normal and the transformation strain components of variant pair C.5-4 are used to compare with the experimental measurements of b 1 g 1 martensitic transformation. The data of the inclined angle and the transformation strain components e x, e y and g xy are obtained from the Moiré fringe patterns in Fig. 3. It is seen that Table 4 The comparison between theory and experiment for the case of SME Transformation strain The inclined angle of A-M interface in x-y plane e x e y g xy Experiment 4.7% 3.7%.75% 55.5 CTM Theory 4.47% 4.1%.66% 54.

17 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () the theoretical predictions by CTM theory agree quite well with experimental results on both habit plane normal and transformation strain. For the case of SE, the correspondence variant in the stacking faulted b 1 martensite is identified to be variant 5 (see Table 3 for reference). The comparison between experimental measurement and theoretical prediction is listed in Table 5. It is seen that the results of the present model agree well with the experimental measurement except the value of e x. 4. Conclusions Based on the recent experimental investigation on the deformation of A-M interface of single crystal Cu-Ni-Al shape memory alloys, modelling analysis is performed for two kinds of interfaces invariant plane and non-invariant plane. Microstructures are identified and their deformation is predicted. For the b 1 g 1 transformation in the case of SME, the A-M interface is indeed an undistorted plane, which is the essential concept in martensitic transformation theory. Based on the invariant plane (habit plane) assumption, the CTM theory (Ball and James, 1987) is used to predict the interface structure, habit plane normal, the transformation strain. The internal twin plays an important role in the theories. Many experimental results, including our present tests, agree well with the theories. For the b 1 b 1 transformation in the case of SE, The A-M interface was found to be a non-invariant plane by the high sensitivity Moiré tests. The interaction between the martensite and the parent phase can be seen clearly from the Moiré fringe patterns. This phenomenon could not be explained by the invariant-plane theory. Based on the detailed analyses of the twinned g 1 martensite and the stacking faulted b 1 martensite, a non-invariant plane model is proposed in this paper. In this model the b 1 is formed through the detwinning, distortion and slip shear (DDS) of the internal twinned g 1 martensite. The comparison between the model prediction and experimental results on both A-M interface normal and the transformation strain components supports the established model. Table 5 The comparison between theory and experiment for the case of SE Transformation strain The inclined angle of A-M interface in x-y plane e x e y g xy Experiment 6.37% 5.9% 3.34% 5.3 CTM+DDS Theory 4.31% 5.76% 3.9% 54.

18 18 Z. Xiangyang et al. / Journal of the Mechanics and Physics of Solids 48 () Acknowledgements The work reported in this paper is supported by the Hong Kong Research Grant Council (RGC) and the Natural Science Foundation of China (NSFC). The authors wish to acknowledge the helpful discussion with R. James and K. Bhattacharya on the interface deformation. Appendix A. The symmetric matrices and distortion matrices of the deformation gradients The six independent symmetric deformation matrices in b 1 g 1 martensitic transformation are U 1 b a+g a g U b a+g, a g a+g g a a+g g a b g a a+g g a a+g U3 a+g a g, b a g a+g, U4 a g U5 a+g g a, a g a+g U6 a+g b g a a+g b, The twelve independent distorting matrices in b 1 b 1 martensitic transformation are D D , D , 1 18 D 1 19, D D ,, 1 D , D ,, D5

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