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torsion of CuAlNi single crystal tube
Scripta Materialia 48 (2003) 1153–1159 www.actamat-journals.com
Stress induced martensitic transformations in tension/torsion of CuAlNi single crystal tube ittner P. S a
a,*
, K. Hashimoto b, M. Kato b, M. Tokuda
b
Department of Physics of Metals, Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, Prague 18221, Czech Republic b Faculty of Engineering, Mie University, Tsu, Mie 514, Japan Received 19 November 2002; received in revised form 26 November 2002; accepted 27 November 2002
Abstract A CuAlNi single crystal tube was loaded in tension while keeping the torsional displacements free. The tube extended and simultaneously twisted during the stress induced martensitic transformation. A cooperative activity of four habit plane/shear direction transformation systems appearing at the same time in different quadrants of the tube wall is considered to rationalize the unusual mechanical behavior. Ó 2003 Published by Elsevier Science Ltd. on behalf of Acta Materialia Inc. Keywords: Shape memory alloys (SMAs); Martensitic phase transformation; Single crystal tube; Tension test; Torsion
1. Introduction The reversible inelastic deformation of shape memory alloys (SMAs)––pseudoelasticity––has been intensively studied in proportional uniaxial tests both on polycrystalline and single crystalline specimens in the last 40 years [1]. Recently, when mechanics modelling of the shape memory thermomechanical behaviors became urgently required for the design of increasingly complex engineering applications of SMAs, pseudoelastic deformation under multiaxial loads have attracted a special attention. The stress induced martensitic transformation (SIMT) was studied in tension/torsion
* Corresponding author. Tel.: +420-266052657; fax: +420286890527. ittner). E-mail address: [email protected] (P. S
experiments on Cu-based [2–4] and NiTi [5,6,8] polycrystalline tubes. New information was found concerning e.g. the distorted transformation surfaces [7], unique responses in nonproportional tests [3,6,7] or localized deformation mode of NiTi microtubes that deform via propagation of martensite bi-spirals when loaded in tension [8]. In this respect, we thought it would be interesting to learn how a SMA single crystal tube would deform pseudoelastically. SIMT in SMA single crystals proceeds by the nucleation and motion of crystallographically well defined phase interfaces intersecting the whole volume of the specimen [1,9,10]. The SMA single crystals were mainly investigated in uniaxial tests to find out the crystallographic, kinematic and thermodynamic characteristics of the SIMT [1]. The constraints imposed in common tension or compression tests by axially aligned specimen grips on the general
1359-6462/03/$ - see front matter Ó 2003 Published by Elsevier Science Ltd. on behalf of Acta Materialia Inc. doi:10.1016/S1359-6462(02)00583-3
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shape change of the transforming single crystal are well known by experimentalists to affect the observed macroscopic stress–strain response. The effect is by far not negligible since the macroscopic shape change of the stretched SMA single crystal may be very different from uniaxial extension [11]. Consequently, usage of the long tensile specimens and special hinge like grips in the tensile tests or laterally shearable grips in the compression tests is preferred to suppress this constraint effect as much as possible. In the present work, a SIMT conforming with another type of external constraint is studied in pseudoelastic tests on CuAlNi single crystal tube that is free to twist when loaded in tension or free to deform axially when loaded in torsion.
2. Experimental Single crystals of Cu–14.3Al–4.1Ni (wt.%) were grown by seeded Bridgman technique in the form of a solid bar 18 mm in diameter and 200 mm in length. The bar axis was exactly aligned with [0 0 1] cubic austenite crystallographic direction. A tubular specimen (Fig. 1, length of the gauge part between two collars, l0 ¼ 37 mm, diameters, dint ¼ 5 mm, dext ¼ 7 mm) was machined from the single crystal. The specimen was given a standard thermal treatment by annealing at 900 °C for 2 h in an argon atmosphere and quenched into the ice water. Following that, the transformation temperatures were evaluated by DSC measurement (Fig. 2) as Ms ¼ 70 °C, Mf ¼ 78 °C, As ¼ 49 °C, Af ¼ 40 °C. The austenite crystal lattice orientation was evaluated using Laue X-ray diffraction measurement. The specimen was deformed at room temperature T ¼ 27 °C using a tension/torsion combined
Fig. 1. CuAlNi single crystal tube specimen, long axis orientation [0 0 1], gauge length l0 ¼ 37 mm, dint ¼ 5 mm, dext ¼ 7 mm.
Fig. 2. DSC record of the CuAlNi single crystal tube (small specimen cut from the gauge part after failure).
load testing machine Shimadzu Autograph AG10. A special combined load extensometer was attached to the collars (Fig. 1) bordering the gauge length of the specimen. The elongation l l0 (used to evaluate the axial strain e) and torsion angle a (used to evaluate the shear strain c), Eq. (1), were measured using this extensometer. The axial stress, r, and shear stress, s, were evaluated from axial force F and torque M, respectively, Eq. (2). e¼
l l0 ; l0
r¼
4F ð1 þ eÞ; 2 d2 Þ pðdext int
c¼
adext 2l0
ð1Þ
s¼
16Mdext 4 d4 Þ pðdext int
ð2Þ
When doing combined tension/torsion experiments, time evolution of one of the two variables describing axial (e; r) and torsional (c; s) loads, respectively, is always prescribed and the remaining two variables are measured as a specimen response. Depending on the choice of the two prescribed variables, different modes of the experiment control are possible. In the present work, at first, tensile tests were carried out as follows. Time evolution of the axial strain (de=dt ¼ 0:0001 s1 ) and rate of the shear stress (torque) (ds=dt ¼ 0 MPa/s) were prescribed (Fig. 3a) and axial stress and shear strain were measured. Next, torsion tests were performed with prescribed constant shear strain rate (dc=dt ¼ 0:0001 s1 ) and axial stress rate (dr=dt ¼ 0 MPa/s). The polished
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Fig. 3. Tensile test on CuAlNi single crystal tube at T ¼ 300 K. Tensile strain rate de=dt ¼ 0:0001 s1 while the tube was let free to twist (ds=dt ¼ 0 MPa/s). (a) Time dependence of variables e, c, r, s; (b) axial stress–strain, r–e, curve; (c) strain path, e–c, response in strain space; (d) strain path response in torsion angle–axial displacement coordinates.
surface of the tube was observed in situ during the tests by an attached optical microscope.
Table 1 Lattice parameters of the austenite b1 and martensite c01 , b01 phases transforming martensitically in CuAlNi [1] Phase
3. Results Fig. 3 shows a result of the tensile pseudoelastic test made up to 6% of axial strain. Looking at Fig. 3b, it is found that the tube deforms pseudoelastically with a narrow stress hysteresis. However, the martensitic transformation observed during stress free cooling of this crystal (DSC measurement, Fig. 2) exhibited a relatively wide temperature hysteresis of DH ¼ Af Ms ¼ 38 °C. While the former evidences the cubic to monoclinic b1 ! b01 martensitic transformation, the latter is characteristic for cubic to orthorhombic b1 ! c01 transformation in CuAlNi [1,9]. In fact, thermomechanically loaded CuAlNi single crystals [9] exhibit martensitic transformations mainly between three solid phases b1 , b01 and c01 depending on the composition [1], temperature [1], stress level [1], load axis orientation [9] and sense of loading [9,10]. Lattice parameters of these phases are given in Table 1. Which transition is going to take place in thermomechanical test under an imposed stress–temperature path can be understood with the help of a nonequilibrium r–T diagram [9,10] specific for each load axis orientation. Particularly for the [0 0 1] oriented crystals, the b1 ! b01 transformation is observed in tension at temperatures far above Ms , while the b1 ! c01 in tension close above Ms [1], in compression [10] and upon stress free cooling [1,9,10]. In the present case, the [0 0 1] oriented CuAlNi tube was deformed far above the Ms temperature in tension and the pseudoelastic response
b1
c01
b01
Structure
Cubic
Monoclinic
Lattice param. ) (A
a ¼ 5:83
Orthorhombic a ¼ 4:38, b ¼ 5:34, c ¼ 4:22
a ¼ 4:43, b ¼ 5:33, c ¼ 38:19, b ¼ 89
is hence safely due to the stress induced b1 ! b01 martensitic transformation. The experimental conditions as the test temperature, load axis orientation, stress level and sense of load are the same or similar as those in the earlier bar experiments [1,9], but the topology of the specimen (tube) and the type of the external constraint are different (the tube is free to twist). The strain path response recorded in the pseudoelastic test (Fig. 3c) is mainly of interest here. The stretched tube twists even if no torque is applied. Looking more closely, the strain response is well proportional in the elastic range, and only after the SIMT starts and proceeds, significant torsion strains appear and increase in the forward and decrease in the reverse loading branch. The strain path response is also shown in coordinates of torque angle–axial displacement (Fig. 3d) measured at the specimen collars. The tensile experiment was repeated several times starting with small maximum strains 2% and subsequently increasing them up to 10%. Surprisingly, during the SIMT (e ¼ 3%–6%), only faint traces inclined to the load axis were observed optically on the surface all over the tube circumference. Similar tensile tests
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with free torsional displacements were earlier made on CuAlZnMn polycrystalline tubes [2,3]. However, the curious nonproportional torsion strains never appeared. The Young modulus was evaluated from the elastic part of the stress–strain curve. It is surprisingly low (Eexp ¼ 20:6 GPa) compared to the elastic modulus measured earlier on polycrystalline CuAlZnMn tubes (E ffi 60 GPa) the elastic constants of which are comparable. Nevertheless, the experimental value is quite close to the value E100 ¼ 24:4 GPa calculated for uniaxial loading of the crystal along the [0 0 1] direction by Eq. (3) using experimentally determined elastic constants of the CuAlNi alloy (C11 ¼ 142:8 GPa, C12 ¼ 126:2 GPa, C44 ¼ 95:9 GPa) [12]. It is due to the strong elastic anisotropy of CuAlNi (2C44 =ðC11 C12 Þ ¼ 11:6) that the single crystal is softest in the [1 0 0] direction but very hard in the [1 1 1] direction (E111 ¼ 214 GPa). 1 C11 þ C12 ¼ Ehkl ðC11 C12 ÞðC11 þ 2C12 Þ 1 1 2 Ahkl ; C11 C12 2C44
Ahkl ¼
h 2 k 2 þ h 2 l2 þ k 2 l2 ðh2 þ k 2 þ l2 Þ2
ð3Þ
ð4Þ
Accordingly, we tried to study the response of the single crystal tube in the torsion tests with free axial displacements. However, the observed response was only elastic (G ¼ 83 GPa). The loading was reversed at the peak shear stress 400 MPa, since another specimen had broken in the previous torsion test at s ¼ 450 MPa before the SIMT could start. No traces of stress induced martensite plates were seen on the surface. This was surprising since the earlier studied CuAlZnMn polycrystalline tubes [2,3] transforming in tension at similar stresses (rtr ¼ 320 MPa) deformed elastically in torsion with low shear modulus (G ¼ 40 GPa) and started to transform at far lower shear stresses str ¼ 240 MPa. The tube finally fractured near one of the collars in a tensile test at stress r ¼ 360 MPa and maximum strain e ¼ 0:1 when we unsuccess-
fully tried to find the end of the transformation plateau.
4. Discussion There are two curious phenomena reported above to be discussed: (i) why the strain path response of the single crystal tube in the tension test is not proportional to the applied stress path i.e. why the torsion strains appear in the tensile test and (ii) why we failed to stress induce the martensitic transformation in the torsion test? In order to rationalize these phenomena, we shall consider the crystallography of the SIMT, elastic anisotropy and the effect of the constraints imposed by the specimen grips on the shape change of the single crystal. The crystallography of the stress induced b1 ! b01 transformation is well known [1]. There are 24 possible habit plane/shear direction systems with four habit planes clustered around the {0 1 1} poles leading to 24 martensite variant particles. The macroscopic stress interacts with the shape strain associated with these particles (the system with the largest Schmid factor for the shape strain is selected by the applied uniaxial stress [1]). Due to the symmetry arguments, eight transformation systems with habit planes denoted by filled and empty circles in Fig. 4 are equally preferred in the case of single crystal stressed in tension along the [0 0 1] direction. In the case of a bar specimen, one or two of these transformation systems become commonly active [1,11]. Corresponding b01 martensite thin plates (habit plane inclined 45° to the bar axis) are nucleated by the applied tensile stress, stretch to the specimen surface and thicken with increasing tensile stress till the whole central part of the specimen transforms into a b01 martensite single crystal. The transformation yields theoretically a tensile elongation of about 8.5% [1,9]. The shape strain, however, is not compatible with the constraint imposed by the axially aligned specimen grips and different martensite variants have to be activated near the specimen heads to accommodate this incompatibility. The single crystal tube can theoretically deform in the same manner as the bar. However, the
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Fig. 4. CuAlNi single crystal tube deformed in the tensile test with free torsional displacements: (a) crystal orientation in stereographic projection along the load axis [0 0 1] with denoted habit plane poles a–d (filled circles), habit plane traces (dashed great circles) and shear directions a1–d1 (crosses) of the four transformation systems involved in the proposed cooperative deformation mode; (b) sketch of the martensite plate traces observed in situ on the surface of the stretched tube.
propagation of the stress induced martensite bands through the tube has to follow the more complex tube topology and the tube, moreover, possessed an additional degree of freedom compared to the bar––it could twist. Besides the deformation mode characterized by one active set of martensite variants discussed above, there are other options for the [0 0 1] oriented CuAlNi tube involving possibly activity of more (equally stressed) transformation systems. For example, let us consider simultaneous activation of four habit plane/transformation shear systems a–d in figure 4a activated selectively in the four quadrants of the tube wall. Martensite particles with habit plane a will be induced near the place where the [0 1 0] crystal direction points out of the tube wall, b where the [ 1 0 0] direction points out, etc. The key point is that the shape strains corresponding to each activated martensite variant particle yield the same axial elongation but differently oriented shears in the (0 0 1) plane. These shears, when distributed in the tube as suggested in Fig. 4, may result in the apparently curious torsional displacements (Fig. 3c,d). The tube loaded by the applied stress in tension hence starts to twist with the onset of the SIMT. The sense of the twist depends on how the four activated set of martensite variants happen to be distributed in the four tube quadrants. Clearly, the high symmetry of the deformation geometry in the case of the [0 0 1] oriented load axis of the tube is essential for the activation of the proposed cooperative deformation mode. The experimental ob-
servation of the only 45° inclined fine traces all over the surface of the stretched tube (but no 90° lines) is thought to be a supporting evidence for it. In contrast to the tension, the deformation of the single crystal tube in torsion cannot be even theoretically achieved through the activation of martensite plates corresponding to the single martensite variant due to the topological reasons. A single crystallographic shear direction cannot support the imposed shear deformation homogeneously everywhere along the tube circumference. A cooperative deformation mode, however, can be possibly activated in the torsion test equally well as in tension, based upon the analogous considerations involving possibly any of the eight available transformation systems. The twisted tube shall hence theoretically extend or contract when the SIMT starts. In reality, however, this was not observed. The deformation geometry of the [0 0 1] oriented tube in torsion is, meanwhile, unsuitable for both elastic and SIMT deformation mechanisms. The single crystal is elastically stiff against the shears applied in torsion due to the large C44 elastic constant of CuAlNi (C44 is defined as the resistance to f0 0 1gh1 0 0i shears but for cubic crystals it is equal to the resistance against any ð0 0 1Þhh k 0i shears). The experimentally measured shear modulus of the tube in the torsion test (83 GPa) is lower than the theoretical C44 value (95.9 GPa) but still rather high. Since the SIMT in Cu-base alloys couples with the f0 1 1gh0 1 1i shears (elastic constant
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C 0 ¼ ðC11 C12 Þ=2) and not with the shears in the (0 0 1) plane imposed in the torsion test (Fig. 4a), it will not be also easy to stress induce the b1 ! b01 transformation by applying torsion loads to the tube. The habit planes and the shear directions of the possibly activated transformation systems [1] are inclined either 45° or 90° (Fig. 4a) to the (0 0 1) plane. The crystallographic indexes of the shear direction imposed in torsion vary in the (0 0 1) zone along the tube circumference. Accordingly, the resolved shear stresses in the activated transformation systems vary too. In order to estimate the transformation start stress in the torsion test, we have calculated the critical shear stresses str needed to trigger the SIMT based on the value of the critical resolved shear stress srsts in the transformation system taken from the tensile test (srsts ¼ 0:48rtr ¼ 144 MPa). We may limit ourselves only to the transformation systems with habit planes inclined 45° marked in figure 4a. Considering that the shear direction in the (0 0 1) zone makes an angle u with the habit plane pole and k with the shear direction of the shape strain, the str can be calculated as str ¼ srsts =ðsin u cos kÞ. str changes periodically but stays in the range 300–473 MPa. Since the SIMT must proceed everywhere in the twisted tube in order to support the stable cooperative deformation mode in torsion, we obtain an approximate threshold for the transformation start shear stress in the torsion test str ¼ 473 MPa. Since this value was not reached in the torsion test, the cooperative deformation mode, even if geometrically possible, was not activated. This is, nevertheless, only a semiqualitative explanation. FEM modelling of the deformation of the CuAlNi single crystal tube via SIMT is in progress [13] to verify the idea of the cooperative deformation modes both in tension and torsion tests, as well as to calculate the stress–strain responses. It is interesting to see how the elastic and transformation anisotropies, affecting favorably the recoverable tensile deformation of the [0 0 1] oriented tube, apparently posed significant problems for its torsional deformation, even if the torsion mode is generally more effective to induce the shear stresses and hence the SIMT in torsion tests. Importance of this fact for torsion deformation of textured SMA polycrystal bars is evident.
In summary, the presented experimental result suggests that the SMA single crystal elements of more complex geometry may show very unusual nonproportional responses in mechanical or thermomechanical loads. The minimization of the deformation energy through the activation of specific cooperative deformation modes involving multiple martensitic variants yielding net macroscopic strains compatible with the external constraints is proposed to be the origin of the phenomena. Bhattacharya and James [14] studied theoretically constrained transformation of SMA single crystal films and predicted deformation modes for martensites that can be called cooperative in the sense introduced here. There might be many other cooperative deformation modes of SMA single crystals loaded both in austenite and martensite state, yet they are waiting to be explored. These deformation modes are not intrinsic properties of SMA single crystals (i.e. any CuAlNi single crystal tube will not generally show the nonproportional responses reported here) but originate from geometrical restrictions on the interfaces and shape changes involved in the SIMT or martensite twinning processes. The activity of a cooperative deformation mode of transforming SMA single crystal is a result of complex interplay between many distinct material and test parameters as elastic constants, crystallographic type of induced martensitic transformations, lattice parameters of austenite and martensite phases, load axis orientation and sense of loading, external constraint, etc. Since the modes will be activated only for particular sets of parameters describing the above conditions, modelling is clearly the most efficient way of looking for them. It may be reasonably expected that the SMA single crystals will deform via these cooperative deformation modes in a rather stable manner––i.e. for a large number of cycles without significant damage as usual for pseudoelasticity. Since it has recently become much easier to grow large single crystals of very good quality from various SMAs [15], there is even a reasonable prospect for development of engineering applications based on the peculiar deformation modes of SMA single crystals.
P. Sittner et al. / Scripta Materialia 48 (2003) 1153–1159
5. Conclusions Pseudoelastic tests were carried out on the [0 0 1] oriented CuAlNi single crystal tube that was free to twist when loaded in tension and free to deform axially when loaded in torsion. In the tension test, the tube simultaneously extended and twisted when SIMT took place. In order to rationalize this curious nonproportional strain path response, a transformation mode involving cooperative activity of four habit plane/shear direction systems is proposed based upon the symmetry arguments and results of the in situ observations of the tube surface. No pseudoelasticity was observed in the torsion test, likely due to the very large transformation stress required for it.
Acknowledgements P. Sittner would like to express sincere thanks to the Japanese Education Ministry and Mie University for supporting his research stay in Japan in 2002 as well as he thanks to the members of the technology group at IP ASCR (head P. Lejcek) who had kindly grown and machined the single crystal tube specimens. The support of the Grant
1159
Agency of the ASCR (contract no. A1048107) is acknowledged.
References [1] Otsuka K, Wayman CM. In: Otsuka K, Wayman CM, editors. Shape memory materials. Cambridge University Press; 1998. p. 1. ittner P, Hara Y, Tokuda M. Metall Mater Trans [2] S 1995;26A:2923. ittner P, Hara Y, Takakura M, Tokuda M. J Physique IV [3] S Iss C1 1996;6:357. [4] Rogueda C, Lexcellent C, Bocher L. Arch Mech 1996;48:1025. [5] Raniecki B, Tanaka K, Ziolkovski K. Mater Sci Res Int Special Tech Publ 2001;2:327. [6] Lim TJ, McDowell DL. J Eng Mater Technol 1999;121:9. [7] Lexcellent C, Vivet A, Bouvet C, Calloch S, Blanc P. J Mech Phys Sol 2002;50:2717. [8] Li ZQ, Sun QP. Int J Plast 2002;18:1481. ittner P, Novak V. Int J Plast 2000;16:1243. [9] S ittner P, Novak V, Zarubova N. Acta Mater [10] S 1998;46:1265. [11] Ichinose S, Funatsu K, Otsuka K. Acta Mater 1985;33: 1613. [12] Landa M, Plesek J, Urbanek P, Novak V. In: Proceedings of the 40th International Conference Experimental Stress Analysis, Prague, Czech Republic, June 3–6, 2002. p. 141. [13] Hall GJ, Govindjee S, Sittner P, Novak V. In preparation. [14] Bhattacharya K, James RD. J Mech Phys Sol 1999;47:531. [15] http://www.A2im.com.
Stress induced martensitic transformations in tension/torsion of CuAlNi single crystal tube ittner P. S a
a,*
, K. Hashimoto b, M. Kato b, M. Tokuda
b
Department of Physics of Metals, Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, Prague 18221, Czech Republic b Faculty of Engineering, Mie University, Tsu, Mie 514, Japan Received 19 November 2002; received in revised form 26 November 2002; accepted 27 November 2002
Abstract A CuAlNi single crystal tube was loaded in tension while keeping the torsional displacements free. The tube extended and simultaneously twisted during the stress induced martensitic transformation. A cooperative activity of four habit plane/shear direction transformation systems appearing at the same time in different quadrants of the tube wall is considered to rationalize the unusual mechanical behavior. Ó 2003 Published by Elsevier Science Ltd. on behalf of Acta Materialia Inc. Keywords: Shape memory alloys (SMAs); Martensitic phase transformation; Single crystal tube; Tension test; Torsion
1. Introduction The reversible inelastic deformation of shape memory alloys (SMAs)––pseudoelasticity––has been intensively studied in proportional uniaxial tests both on polycrystalline and single crystalline specimens in the last 40 years [1]. Recently, when mechanics modelling of the shape memory thermomechanical behaviors became urgently required for the design of increasingly complex engineering applications of SMAs, pseudoelastic deformation under multiaxial loads have attracted a special attention. The stress induced martensitic transformation (SIMT) was studied in tension/torsion
* Corresponding author. Tel.: +420-266052657; fax: +420286890527. ittner). E-mail address: [email protected] (P. S
experiments on Cu-based [2–4] and NiTi [5,6,8] polycrystalline tubes. New information was found concerning e.g. the distorted transformation surfaces [7], unique responses in nonproportional tests [3,6,7] or localized deformation mode of NiTi microtubes that deform via propagation of martensite bi-spirals when loaded in tension [8]. In this respect, we thought it would be interesting to learn how a SMA single crystal tube would deform pseudoelastically. SIMT in SMA single crystals proceeds by the nucleation and motion of crystallographically well defined phase interfaces intersecting the whole volume of the specimen [1,9,10]. The SMA single crystals were mainly investigated in uniaxial tests to find out the crystallographic, kinematic and thermodynamic characteristics of the SIMT [1]. The constraints imposed in common tension or compression tests by axially aligned specimen grips on the general
1359-6462/03/$ - see front matter Ó 2003 Published by Elsevier Science Ltd. on behalf of Acta Materialia Inc. doi:10.1016/S1359-6462(02)00583-3
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shape change of the transforming single crystal are well known by experimentalists to affect the observed macroscopic stress–strain response. The effect is by far not negligible since the macroscopic shape change of the stretched SMA single crystal may be very different from uniaxial extension [11]. Consequently, usage of the long tensile specimens and special hinge like grips in the tensile tests or laterally shearable grips in the compression tests is preferred to suppress this constraint effect as much as possible. In the present work, a SIMT conforming with another type of external constraint is studied in pseudoelastic tests on CuAlNi single crystal tube that is free to twist when loaded in tension or free to deform axially when loaded in torsion.
2. Experimental Single crystals of Cu–14.3Al–4.1Ni (wt.%) were grown by seeded Bridgman technique in the form of a solid bar 18 mm in diameter and 200 mm in length. The bar axis was exactly aligned with [0 0 1] cubic austenite crystallographic direction. A tubular specimen (Fig. 1, length of the gauge part between two collars, l0 ¼ 37 mm, diameters, dint ¼ 5 mm, dext ¼ 7 mm) was machined from the single crystal. The specimen was given a standard thermal treatment by annealing at 900 °C for 2 h in an argon atmosphere and quenched into the ice water. Following that, the transformation temperatures were evaluated by DSC measurement (Fig. 2) as Ms ¼ 70 °C, Mf ¼ 78 °C, As ¼ 49 °C, Af ¼ 40 °C. The austenite crystal lattice orientation was evaluated using Laue X-ray diffraction measurement. The specimen was deformed at room temperature T ¼ 27 °C using a tension/torsion combined
Fig. 1. CuAlNi single crystal tube specimen, long axis orientation [0 0 1], gauge length l0 ¼ 37 mm, dint ¼ 5 mm, dext ¼ 7 mm.
Fig. 2. DSC record of the CuAlNi single crystal tube (small specimen cut from the gauge part after failure).
load testing machine Shimadzu Autograph AG10. A special combined load extensometer was attached to the collars (Fig. 1) bordering the gauge length of the specimen. The elongation l l0 (used to evaluate the axial strain e) and torsion angle a (used to evaluate the shear strain c), Eq. (1), were measured using this extensometer. The axial stress, r, and shear stress, s, were evaluated from axial force F and torque M, respectively, Eq. (2). e¼
l l0 ; l0
r¼
4F ð1 þ eÞ; 2 d2 Þ pðdext int
c¼
adext 2l0
ð1Þ
s¼
16Mdext 4 d4 Þ pðdext int
ð2Þ
When doing combined tension/torsion experiments, time evolution of one of the two variables describing axial (e; r) and torsional (c; s) loads, respectively, is always prescribed and the remaining two variables are measured as a specimen response. Depending on the choice of the two prescribed variables, different modes of the experiment control are possible. In the present work, at first, tensile tests were carried out as follows. Time evolution of the axial strain (de=dt ¼ 0:0001 s1 ) and rate of the shear stress (torque) (ds=dt ¼ 0 MPa/s) were prescribed (Fig. 3a) and axial stress and shear strain were measured. Next, torsion tests were performed with prescribed constant shear strain rate (dc=dt ¼ 0:0001 s1 ) and axial stress rate (dr=dt ¼ 0 MPa/s). The polished
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Fig. 3. Tensile test on CuAlNi single crystal tube at T ¼ 300 K. Tensile strain rate de=dt ¼ 0:0001 s1 while the tube was let free to twist (ds=dt ¼ 0 MPa/s). (a) Time dependence of variables e, c, r, s; (b) axial stress–strain, r–e, curve; (c) strain path, e–c, response in strain space; (d) strain path response in torsion angle–axial displacement coordinates.
surface of the tube was observed in situ during the tests by an attached optical microscope.
Table 1 Lattice parameters of the austenite b1 and martensite c01 , b01 phases transforming martensitically in CuAlNi [1] Phase
3. Results Fig. 3 shows a result of the tensile pseudoelastic test made up to 6% of axial strain. Looking at Fig. 3b, it is found that the tube deforms pseudoelastically with a narrow stress hysteresis. However, the martensitic transformation observed during stress free cooling of this crystal (DSC measurement, Fig. 2) exhibited a relatively wide temperature hysteresis of DH ¼ Af Ms ¼ 38 °C. While the former evidences the cubic to monoclinic b1 ! b01 martensitic transformation, the latter is characteristic for cubic to orthorhombic b1 ! c01 transformation in CuAlNi [1,9]. In fact, thermomechanically loaded CuAlNi single crystals [9] exhibit martensitic transformations mainly between three solid phases b1 , b01 and c01 depending on the composition [1], temperature [1], stress level [1], load axis orientation [9] and sense of loading [9,10]. Lattice parameters of these phases are given in Table 1. Which transition is going to take place in thermomechanical test under an imposed stress–temperature path can be understood with the help of a nonequilibrium r–T diagram [9,10] specific for each load axis orientation. Particularly for the [0 0 1] oriented crystals, the b1 ! b01 transformation is observed in tension at temperatures far above Ms , while the b1 ! c01 in tension close above Ms [1], in compression [10] and upon stress free cooling [1,9,10]. In the present case, the [0 0 1] oriented CuAlNi tube was deformed far above the Ms temperature in tension and the pseudoelastic response
b1
c01
b01
Structure
Cubic
Monoclinic
Lattice param. ) (A
a ¼ 5:83
Orthorhombic a ¼ 4:38, b ¼ 5:34, c ¼ 4:22
a ¼ 4:43, b ¼ 5:33, c ¼ 38:19, b ¼ 89
is hence safely due to the stress induced b1 ! b01 martensitic transformation. The experimental conditions as the test temperature, load axis orientation, stress level and sense of load are the same or similar as those in the earlier bar experiments [1,9], but the topology of the specimen (tube) and the type of the external constraint are different (the tube is free to twist). The strain path response recorded in the pseudoelastic test (Fig. 3c) is mainly of interest here. The stretched tube twists even if no torque is applied. Looking more closely, the strain response is well proportional in the elastic range, and only after the SIMT starts and proceeds, significant torsion strains appear and increase in the forward and decrease in the reverse loading branch. The strain path response is also shown in coordinates of torque angle–axial displacement (Fig. 3d) measured at the specimen collars. The tensile experiment was repeated several times starting with small maximum strains 2% and subsequently increasing them up to 10%. Surprisingly, during the SIMT (e ¼ 3%–6%), only faint traces inclined to the load axis were observed optically on the surface all over the tube circumference. Similar tensile tests
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with free torsional displacements were earlier made on CuAlZnMn polycrystalline tubes [2,3]. However, the curious nonproportional torsion strains never appeared. The Young modulus was evaluated from the elastic part of the stress–strain curve. It is surprisingly low (Eexp ¼ 20:6 GPa) compared to the elastic modulus measured earlier on polycrystalline CuAlZnMn tubes (E ffi 60 GPa) the elastic constants of which are comparable. Nevertheless, the experimental value is quite close to the value E100 ¼ 24:4 GPa calculated for uniaxial loading of the crystal along the [0 0 1] direction by Eq. (3) using experimentally determined elastic constants of the CuAlNi alloy (C11 ¼ 142:8 GPa, C12 ¼ 126:2 GPa, C44 ¼ 95:9 GPa) [12]. It is due to the strong elastic anisotropy of CuAlNi (2C44 =ðC11 C12 Þ ¼ 11:6) that the single crystal is softest in the [1 0 0] direction but very hard in the [1 1 1] direction (E111 ¼ 214 GPa). 1 C11 þ C12 ¼ Ehkl ðC11 C12 ÞðC11 þ 2C12 Þ 1 1 2 Ahkl ; C11 C12 2C44
Ahkl ¼
h 2 k 2 þ h 2 l2 þ k 2 l2 ðh2 þ k 2 þ l2 Þ2
ð3Þ
ð4Þ
Accordingly, we tried to study the response of the single crystal tube in the torsion tests with free axial displacements. However, the observed response was only elastic (G ¼ 83 GPa). The loading was reversed at the peak shear stress 400 MPa, since another specimen had broken in the previous torsion test at s ¼ 450 MPa before the SIMT could start. No traces of stress induced martensite plates were seen on the surface. This was surprising since the earlier studied CuAlZnMn polycrystalline tubes [2,3] transforming in tension at similar stresses (rtr ¼ 320 MPa) deformed elastically in torsion with low shear modulus (G ¼ 40 GPa) and started to transform at far lower shear stresses str ¼ 240 MPa. The tube finally fractured near one of the collars in a tensile test at stress r ¼ 360 MPa and maximum strain e ¼ 0:1 when we unsuccess-
fully tried to find the end of the transformation plateau.
4. Discussion There are two curious phenomena reported above to be discussed: (i) why the strain path response of the single crystal tube in the tension test is not proportional to the applied stress path i.e. why the torsion strains appear in the tensile test and (ii) why we failed to stress induce the martensitic transformation in the torsion test? In order to rationalize these phenomena, we shall consider the crystallography of the SIMT, elastic anisotropy and the effect of the constraints imposed by the specimen grips on the shape change of the single crystal. The crystallography of the stress induced b1 ! b01 transformation is well known [1]. There are 24 possible habit plane/shear direction systems with four habit planes clustered around the {0 1 1} poles leading to 24 martensite variant particles. The macroscopic stress interacts with the shape strain associated with these particles (the system with the largest Schmid factor for the shape strain is selected by the applied uniaxial stress [1]). Due to the symmetry arguments, eight transformation systems with habit planes denoted by filled and empty circles in Fig. 4 are equally preferred in the case of single crystal stressed in tension along the [0 0 1] direction. In the case of a bar specimen, one or two of these transformation systems become commonly active [1,11]. Corresponding b01 martensite thin plates (habit plane inclined 45° to the bar axis) are nucleated by the applied tensile stress, stretch to the specimen surface and thicken with increasing tensile stress till the whole central part of the specimen transforms into a b01 martensite single crystal. The transformation yields theoretically a tensile elongation of about 8.5% [1,9]. The shape strain, however, is not compatible with the constraint imposed by the axially aligned specimen grips and different martensite variants have to be activated near the specimen heads to accommodate this incompatibility. The single crystal tube can theoretically deform in the same manner as the bar. However, the
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Fig. 4. CuAlNi single crystal tube deformed in the tensile test with free torsional displacements: (a) crystal orientation in stereographic projection along the load axis [0 0 1] with denoted habit plane poles a–d (filled circles), habit plane traces (dashed great circles) and shear directions a1–d1 (crosses) of the four transformation systems involved in the proposed cooperative deformation mode; (b) sketch of the martensite plate traces observed in situ on the surface of the stretched tube.
propagation of the stress induced martensite bands through the tube has to follow the more complex tube topology and the tube, moreover, possessed an additional degree of freedom compared to the bar––it could twist. Besides the deformation mode characterized by one active set of martensite variants discussed above, there are other options for the [0 0 1] oriented CuAlNi tube involving possibly activity of more (equally stressed) transformation systems. For example, let us consider simultaneous activation of four habit plane/transformation shear systems a–d in figure 4a activated selectively in the four quadrants of the tube wall. Martensite particles with habit plane a will be induced near the place where the [0 1 0] crystal direction points out of the tube wall, b where the [ 1 0 0] direction points out, etc. The key point is that the shape strains corresponding to each activated martensite variant particle yield the same axial elongation but differently oriented shears in the (0 0 1) plane. These shears, when distributed in the tube as suggested in Fig. 4, may result in the apparently curious torsional displacements (Fig. 3c,d). The tube loaded by the applied stress in tension hence starts to twist with the onset of the SIMT. The sense of the twist depends on how the four activated set of martensite variants happen to be distributed in the four tube quadrants. Clearly, the high symmetry of the deformation geometry in the case of the [0 0 1] oriented load axis of the tube is essential for the activation of the proposed cooperative deformation mode. The experimental ob-
servation of the only 45° inclined fine traces all over the surface of the stretched tube (but no 90° lines) is thought to be a supporting evidence for it. In contrast to the tension, the deformation of the single crystal tube in torsion cannot be even theoretically achieved through the activation of martensite plates corresponding to the single martensite variant due to the topological reasons. A single crystallographic shear direction cannot support the imposed shear deformation homogeneously everywhere along the tube circumference. A cooperative deformation mode, however, can be possibly activated in the torsion test equally well as in tension, based upon the analogous considerations involving possibly any of the eight available transformation systems. The twisted tube shall hence theoretically extend or contract when the SIMT starts. In reality, however, this was not observed. The deformation geometry of the [0 0 1] oriented tube in torsion is, meanwhile, unsuitable for both elastic and SIMT deformation mechanisms. The single crystal is elastically stiff against the shears applied in torsion due to the large C44 elastic constant of CuAlNi (C44 is defined as the resistance to f0 0 1gh1 0 0i shears but for cubic crystals it is equal to the resistance against any ð0 0 1Þhh k 0i shears). The experimentally measured shear modulus of the tube in the torsion test (83 GPa) is lower than the theoretical C44 value (95.9 GPa) but still rather high. Since the SIMT in Cu-base alloys couples with the f0 1 1gh0 1 1i shears (elastic constant
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C 0 ¼ ðC11 C12 Þ=2) and not with the shears in the (0 0 1) plane imposed in the torsion test (Fig. 4a), it will not be also easy to stress induce the b1 ! b01 transformation by applying torsion loads to the tube. The habit planes and the shear directions of the possibly activated transformation systems [1] are inclined either 45° or 90° (Fig. 4a) to the (0 0 1) plane. The crystallographic indexes of the shear direction imposed in torsion vary in the (0 0 1) zone along the tube circumference. Accordingly, the resolved shear stresses in the activated transformation systems vary too. In order to estimate the transformation start stress in the torsion test, we have calculated the critical shear stresses str needed to trigger the SIMT based on the value of the critical resolved shear stress srsts in the transformation system taken from the tensile test (srsts ¼ 0:48rtr ¼ 144 MPa). We may limit ourselves only to the transformation systems with habit planes inclined 45° marked in figure 4a. Considering that the shear direction in the (0 0 1) zone makes an angle u with the habit plane pole and k with the shear direction of the shape strain, the str can be calculated as str ¼ srsts =ðsin u cos kÞ. str changes periodically but stays in the range 300–473 MPa. Since the SIMT must proceed everywhere in the twisted tube in order to support the stable cooperative deformation mode in torsion, we obtain an approximate threshold for the transformation start shear stress in the torsion test str ¼ 473 MPa. Since this value was not reached in the torsion test, the cooperative deformation mode, even if geometrically possible, was not activated. This is, nevertheless, only a semiqualitative explanation. FEM modelling of the deformation of the CuAlNi single crystal tube via SIMT is in progress [13] to verify the idea of the cooperative deformation modes both in tension and torsion tests, as well as to calculate the stress–strain responses. It is interesting to see how the elastic and transformation anisotropies, affecting favorably the recoverable tensile deformation of the [0 0 1] oriented tube, apparently posed significant problems for its torsional deformation, even if the torsion mode is generally more effective to induce the shear stresses and hence the SIMT in torsion tests. Importance of this fact for torsion deformation of textured SMA polycrystal bars is evident.
In summary, the presented experimental result suggests that the SMA single crystal elements of more complex geometry may show very unusual nonproportional responses in mechanical or thermomechanical loads. The minimization of the deformation energy through the activation of specific cooperative deformation modes involving multiple martensitic variants yielding net macroscopic strains compatible with the external constraints is proposed to be the origin of the phenomena. Bhattacharya and James [14] studied theoretically constrained transformation of SMA single crystal films and predicted deformation modes for martensites that can be called cooperative in the sense introduced here. There might be many other cooperative deformation modes of SMA single crystals loaded both in austenite and martensite state, yet they are waiting to be explored. These deformation modes are not intrinsic properties of SMA single crystals (i.e. any CuAlNi single crystal tube will not generally show the nonproportional responses reported here) but originate from geometrical restrictions on the interfaces and shape changes involved in the SIMT or martensite twinning processes. The activity of a cooperative deformation mode of transforming SMA single crystal is a result of complex interplay between many distinct material and test parameters as elastic constants, crystallographic type of induced martensitic transformations, lattice parameters of austenite and martensite phases, load axis orientation and sense of loading, external constraint, etc. Since the modes will be activated only for particular sets of parameters describing the above conditions, modelling is clearly the most efficient way of looking for them. It may be reasonably expected that the SMA single crystals will deform via these cooperative deformation modes in a rather stable manner––i.e. for a large number of cycles without significant damage as usual for pseudoelasticity. Since it has recently become much easier to grow large single crystals of very good quality from various SMAs [15], there is even a reasonable prospect for development of engineering applications based on the peculiar deformation modes of SMA single crystals.
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5. Conclusions Pseudoelastic tests were carried out on the [0 0 1] oriented CuAlNi single crystal tube that was free to twist when loaded in tension and free to deform axially when loaded in torsion. In the tension test, the tube simultaneously extended and twisted when SIMT took place. In order to rationalize this curious nonproportional strain path response, a transformation mode involving cooperative activity of four habit plane/shear direction systems is proposed based upon the symmetry arguments and results of the in situ observations of the tube surface. No pseudoelasticity was observed in the torsion test, likely due to the very large transformation stress required for it.
Acknowledgements P. Sittner would like to express sincere thanks to the Japanese Education Ministry and Mie University for supporting his research stay in Japan in 2002 as well as he thanks to the members of the technology group at IP ASCR (head P. Lejcek) who had kindly grown and machined the single crystal tube specimens. The support of the Grant
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Agency of the ASCR (contract no. A1048107) is acknowledged.
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