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Simulation of cubic to monoclinic-II transformations in a single crystal Cu–Al–Ni tube
International Journal of Plasticity 23 (2007) 161–182 www.elsevier.com/locate/ijplas
Simulation of cubic to monoclinic-II transformations in a single crystal Cu–Al–Ni tube Garrett J. Hall a, Sanjay Govindjee
b,d,*
, Petr Sˇittner c, V. Nova´k
c
a
Cal Poly, San Luis Obispo, Structural Engineering and Applied Mechanics, Department of Civil and Environmental Engineering, California Polytechnic State University San Luis Obispo, CA 93407, USA b University of California, Berkeley, Structural Engineering, Mechanics and Materials, Department of Civil and Environmental Engineering University of California, Berkeley, CA 94720, USA c Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, Prague 8, 182 21, Czech Republic d ETH Zurich, Center of Mechanics, Institute of Mechanical Systems, Tannenstrasse 3, 8092 Zurich, Switzerland Received 24 July 2005 Available online 22 August 2006
Abstract Motivated by recent unusual experimental results [Sˇittner, P., Hashimoto, K., Kato, M., Tokuda, M., 2003. Stress induced martensitic transformations in tension/torsion of CuAlNi single crystal tube. Scripta Materialia 48, 1153–1159] indicating a strong coupling between axial and torsional deformations in single crystal monoclinic-II shape memory alloy tubes, various computational aspects of numerically modelling the observed behavior in monoclinic-II materials are discussed. As a necessary first step, the shape strain computations in monoclinic-II systems are examined. It is found that ambiguities regarding the lattice parameters and proper unit cell in the existing literature have implications for the macroscopic constitutive theories of such alloys. This is demonstrated in the context of a detailed numerical investigation of the experimental problem of interest, wherein it is found that numerical simulations demonstrate proper quantitative response, a strong axial ! torsional coupling given a biasing initial condition, and the observed lack of torsional ! axial coupling. Implications of the findings are discussed with regard to both the further development of numerical models and future avenues for experimental research. 2006 Elsevier Ltd. All rights reserved. Keywords: Shape memory alloys; Phase transformation; Single crystal; Quasi-convexity
*
Corresponding author. Fax: +44 41 632 1145. E-mail addresses: [email protected] (G.J. Hall), [email protected] (S. Govindjee), [email protected] (P. Sˇittner), [email protected] (V. Nova´k). 0749-6419/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2006.04.004
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1. Introduction A recent numerical investigation by Hall and Govindjee (2003) of single crystal torsion tubes predicted the appearance of significant axial extensions under applied torsional loads (i.e. torsional ! axial coupling), and suggested the single crystal torsion tube as an ideal model problem for the study of solid–solid phase transformation energetics. The usefulness of this geometry/loading combination was also recognized by Thamburaja and Anand (2002). Since that time an experimental study by Sˇittner et al. (2003) has been performed on a Cu–Al–Ni (b01 ) specimen of similar geometry albeit with different boundary conditions (see also Nouailhas and Cailletaud (1995) for a related single crystal study, and Thamburaja and Anand (2001) and Li and Sun (2002) for related polycrystalline tube experiments). Among the significant findings in Sˇittner et al. (2003) was the observation of spontaneous axial ! torsional coupling in a single crystal undergoing a cubic ! monoclinic-II pseudoelastic phase transformation, and the description of a possible microstructure formation consistent with the observed deformation. Based on these observations, in the present paper we revisit the problem in order to compare the physical observations with those predicted by the lower bound model of Govindjee et al. (2003) and Hall and Govindjee (2002, 2003). Towards that end, in Section 2 we begin with an examination of shape strain computations in monoclinic-II alloys as required for input to the constitutive model. The two common descriptions of the unit cell which appear in the literature, 18R and 6M1, are compared in light of the implications for the resulting energetics of transformation. A detailed discussion concerning Cu–Al–Ni is provided as this is the material employed in the experimental work of interest. Here, it is found that the literature is somewhat ambiguous as to the correct lattice parameters, a fact which has both theoretical and numerical implications. With the shape strain computations in mind we then proceed in Section 3 to a numerical investigation of the problem of interest. Using an extension of a previously developed constitutive theory (Govindjee et al., 2003; Hall and Govindjee, 2002, 2003), various numerical approximations of the boundary value problem in the experiment are made, each demonstrating particular aspects of the numerical implementation of the model. Specifically, in Section 3, we drive the model in various combinations of tension/extension torsion/twist to provide insight into the coupled mode transformations in monoclinic-II materials. 2. Shape strain computations Macroscopic constitutive models for shape memory alloy materials which are derived from consideration of the underlying microstructure necessarily include a representation of the shape strain which accompanies phase transformation and/or reorientation. It is generally accepted that for most of the common shape memory alloys that the lattice parameters and unit cells associated with phase transformation are well characterized for a given alloy composition. However, in the case of monoclinic-II materials, there has been some debate regarding the proper characterization of the unit cell. In particular the computation of the shape strains in monoclinic-II ðb01 Þ systems is often based upon somewhat ambiguous and/or inconsistent statements concerning the geometry of the transformations strains. The apparently most consistent method proposed to date is
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due to Hane (1999) wherein he proposes to resuscitate the notion of the 6M1 lattice structure to replace the more common 18R system following the lead of Otsuka et al. (1993). In what follows, we report upon such computations and highlight some of the inconsistencies associated with the 18R unit cell method of computing the shape strains, and point out differences between some of the numerical values provided in the literature. These issues are further considered in the macroscopic numerical modeling which follows in Section 3. 2.1. Background The monoclinic-II system is a bit different from other martensitic transformation systems such as orthorhombic, tetragonal, or even monoclinic-I; see Pitteri and Zanzotto (1998) for a discussion of the two monoclinic systems. The monoclinic-II system is known to possess an austenite–martensite interface where the martensite is untwinned; see e.g. Saburi et al. (1976) for a discussion of a Cu–Zn–Ga alloy. For the computation of the transformation strains one needs to know the lattice correspondence between the cubic austenite and the monoclinic martensite. The traditional lattice correspondence employs an 18R lattice cell. However, the monoclinic angles reported in the literature do not permit a low energy austenite–martensite interface (a rank-one interface) for this correspondence. Thus it is conventional to introduce an ‘‘invariant lattice shear’’ to rectify this inconsistency. The amount of the ‘‘invariant lattice shear’’ is determined mathematically (not experimentally) so that a low energy interface can exist between austenite and untwinned martensite. More recently, Hane (1999), following Otsuka et al. (1993), has revisited the notion of a 6M1 lattice correspondence and shown that it more naturally produces transformation strains which are rank-one compatible with austenite; i.e. it produces transformation strains which are rank-one compatible with austenite without the need for a ‘‘invariant lattice shear’’. In this section, the pertinent equations for computing transformation strains from lattice parameters are reviewed and the numerical consequences are determined. 2.2. Transformation strains In what follows all tensor components are given in either the orthonormal cubic austenite basis, denoted by a subscript A, or in the orthonormal martensite basis which is aligned with [0 1 1]A, ½ 1 0 0A , ½0 1 1A , denoted by a subscript M. Once the deformation map, F, from the cubic structure to the monoclinic-II structure has been determined, the transformation stretches (squared) are determined as U2 = FTF. For the infinitesimal deformation theory the transformation strains are determined as U 1. For existence of an austenite to untwinned martensite interface with normal m and habit-plane displacement b one needs U2 to have one eigenvalue equal to unity, one below unity and one above unity (Ball and James, 1987, Prop. 4). 2.2.1. 18R To compute the transformation strains for the 18R monoclinic structure one starts from the cubic lattice parameter a0, the 18R parameters a, b, c and monoclinic angle b. In the martensite basis, the deformation gradient is given by
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2
1 0
6 F ¼ 40 1 0 0
3 2 pffiffiffi 2a=a0 7 6 05 4 0 1 M 0
g
0 b=a0 0
0 0
pffiffiffi 2c sinðbÞ=9a0
3 7 5 :
ð1Þ
M
If the description were fully consistent then g would be equal to cot(b). This however does not produce a transformation strain with the ‘‘observed’’ compatibility with an austenite phase. Thus, when reporting the transformation strains for such systems g is typically chosen so the resulting transformation strains are rank-one compatible with austenite. The transformation stretch is computed from the polar decomposition of F and is typically given in the austenite basis by executing a change of coordinates from the martensite basis to the austenite basis. 2.2.2. 6M1 To compute the transformation strains for the 6M1 monoclinic structure one starts from the cubic lattice parameter a0, the 18R parameters a, b, c and monoclinic angle b. First, one converts the 18R lattice parameter c and the monoclinic angle to the 6M1 lattice cell. The relations are given by Hane (1999) and James and Hane (2000) a 1 b1 ¼ b þ tan ð2Þ c sinðbÞ and c1 ¼
c sinðbÞ : 3 sinðb1 Þ
In the martensite basis, the deformation gradient is given by 3 2 3 2 pffiffiffi 1 0 g1 0 0 2a=a0 7 6 7 6 F ¼ 40 1 0 5 4 5 : 0 b=a0 0 pffiffiffi 0 0 1 M 0 0 2c1 sinðb1 Þ=3a0 M
ð3Þ
ð4Þ
The description here is fully consistent and g1 is equal to cot(b1). Further, rank-one compatibility to austenite from untwinned martensite is very nearly satisfied using reported monoclinic angles without the need for inconsistent lattice shears. 2.3. Application In this section we briefly present the application of the above transformations to several materials that have been studied in the literature. 2.3.1. Cu–Zn–Ga This alloy was studied by Saburi et al. (1976) at 20.4 at.% Zn and 12.5 at.% Ga. They ˚ , b = 5.33 A ˚ , c = 38.22 A ˚ and b ¼ 88:33. The report 18R lattice parameters of a = 4.40 A ˚ which was measured on a slightly different cubic lattice parameter is given as a0 = 5.86 A alloy (21.8 at.% Zn and 11.8 at.% Ga). These authors also report the shear value g = ±0.0775 which corresponds to a shear angle of 85.57. Note the inconsistency with the monoclinic angle – a point which is usually rationalized by appealing to the notion of stacking faulting. In Saburi and Nenno (1981) the value of g = 0.086, which corresponds
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to a shear angle of 85.08 and is termed the ideal 18R structure with no stacking faults; note, however, that this latter value is still inconsistent with the monoclinic angle. Using g = 0.0775 the resulting transformation stretch (from the 18R relations) is compatible with austenite to within the accuracy of the reported parameters. If one uses a consistent value for g then we find (from the 18R relations) the middle eigenvalue of U2 is 1.0390. However, if one computes the transformation strain using the 6M1 structure then one finds a stretch tensor that is nearly rank-one compatible with austenite; the middle eigenvalue of U2 is 0.9926. For the 6M1 cell this occurs without having to introduce any arbitrary lattice shears. 2.3.2. Cu–Zn–Al This alloy was studied by Pitteri and Zanzotto (1997) at 15 at.% Zn and 17 at.% Al. The ˚ , a = 4.553 A ˚ , b = 5.452 A ˚ , c = 38.977 A ˚ and reported lattice parameters were a0 = 5.996 A b = 87.5. Using the 18R structure and a consistent shear value gives a transformation such that the closest eigenvalue of U2 to unity is 1.0254. Again, if we compute the transformation stretch from the 6M1 structure we find a stretch value which is more nearly compatible with austenite. In this case the closest eigenvalue of U2 to unity is 1.0041. Slightly different results are found using the reported values in Bhattacharya and Kohn (1996) to generate a transformation stretch where the closest eigenvalue of U2 to unity is 1.0052. Here also it is clear that the 6M1 structure gives a superior result. 2.3.3. Cu–Al–Ni This alloy was studied in Otsuka et al. (1974) at 14.2 wt.% Al and 4.3 wt.% Ni and by Tokonami et al. (1979) at 14.0 wt.% Al and 4.2 wt.% Ni. The lattice parameters from these ˚ , b = 5.356 A ˚, two sources are different. From Otsuka et al. (1974) we have a = 4.382 A ˚ c = 38.00 A; other parameters are not reported. From Tokonami et al. (1979) we have ˚ , b = 5.330 A ˚ , c = 38.19 A ˚ , b = 89.0. a = 4.430 A Otsuka et al. (1974) lattice parameters: Since the lattice parameters from Otsuka et al. (1974) are incomplete we fill in the missing ones using those in Hane (1999). Using the 18R structure and a consistent shear value gives a transformation stretch which is nearly compatible with austenite; the closest eigenvalue of U2 to unity is 1.0426. If we compute the transformation stretch from the 6M1 structure we find a more nearly compatible stretch whose closest eigenvalue of U2 to unity is 0.9801. Meanwhile, the reported values in Bhattacharya and Kohn (1996) generate a 6M1 transformation stretch whose closest eigenvalue of U2 to unity is 0.9645. Tokonami et al. (1979) lattice parameters: Here the austenite lattice spacing was not ˚ . Using the 18R structure and a consistent shear value reported, so we assume a0 = 5.838 A gives a transformation stretch which is nearly compatible with austenite: 1.0536 is the closest eigenvalue of U2 to unity. Finally, if one computes the transformation stretch from the 6M1 structure we find again a more nearly compatible transformation stretch with austenite. In this case the closest eigenvalue of U2 to unity is 0.9932. 2.4. Discussion of shape strain computations It can be seen in all the cases that the transformation stretches generated from the 6M1 system come closer to being compatible with austenite over those generated from the 18R system when no arbitrary adjustments are made in the 18R computation. As shown by
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Hane (1999) with only very minor adjustments in the monoclinic angle the 6M1 system can be made to generate ideal compatibility with austenite whereas the 18R system requires larger changes to the monoclinic angle (or the introduction of an arbitrary stacking fault shear). The differences in the numerical values one finds in the computed shape strains from the 18R and 6M1 structure appear rather insignificant with differences generally in the third significant digit; see Appendix B. However, for models which are derived explicitly from microstructural considerations, the variations may be non-negligible. To see this, consider that the free energy of mixing (which arises in the transition from the microscale to the macroscale) critically depends upon rank-one connections between variants in the material (see Ball and James, 1987; Kohn, 1991; Govindjee et al., 2003 and the references therein). Implicitly, such models consider all possible twinning combinations, including austenite– martensite (a–m) interfaces, martenstite–martensite (m–m) interfaces, and various order Results independent of linear/nonlinear twinning calculation assumption.
1 2 3
1
2
3
4
5
6
7
8
9
10
11
12
NA
S
S
S
S
S
X
X
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S
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S
S
S
S
X
X
X
X
S
S
NA
S
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X
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S
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S
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X
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X
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S
S
S
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S
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S
NA
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NA
S
S
S
NA
S
S
NA
S
4 5 6 7 8 9 10 11
Otsuka 6M1 Tokonami 6M1 Otsuka/Tokonami 18R
12
NA
Fig. 1. Twinning chart for Monoclini-II Cu–Al–Ni. Martensite twin solutions (m–m) are indicated by ‘‘S’’; the variant ordering is listed in Appendix A. The connectivity map for (a–m–m) twinning is seen to be sensitive to small changes in the lattice parameters and unit cell interpretation. Note that although these interfaces are not observed experimentally, they play a role in the derivation of macroscopic free energy functions for shape memory alloys.
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austenite–martensite–martensite laminate type interfaces (a–m–m). Thus if the rank-one connections are changed by perturbations in the lattice parameters, so is the landscape of the derived free energy function. Using the Cu–Al–Ni shape strains computed above as an example, nonlinear and linear twinning calculations (Ball and James, 1987; Bhattacharya, 1993) may be performed in each case. As previously noted, austenite and the lattice correspondence variants of martensite are close to, though not perfectly, twin compatible. Fig. 1 summarizes the rank-one connections between martensite twin pairs (m–m) by the letter ‘‘S’’ and indicates their mutual compatibility with austenite (a–m–m) for 6M1 and 18R structures using the lattice parameters from Otsuka et al. (1974) and Tokonami et al. (1979). It is seen that the map of rank-one connections is sensitive to slight variations in the lattice parameters. Note that the results shown are independent of the choice of linear or non-linear twinning computation. Also note that although a–m–m interfaces have not been reported experimentally, they necessarily come into the derivation of the macroscopic free energy functions from microstructural considerations. 3. Numerical simulations 3.1. Background The simulations of this section are based on experiments reported in Sˇittner et al. (2003). A tubular specimen measuring 7 mm on the outer diameter, 5 mm on the inner diameter, and 37 mm along the gauge length was manufactured. The specimen head had a screw with a ground flat. It was fixed inside a hollow cylinder grip using an internal screw and a cross bar going through the holes in the grip was attached to the flat part which prohibited rotation of the specimen inside the grips. The material was a single crystal of Cu–14.3Al–4.1Ni [wt.%] with the [0 0 1] cubic austenite crystallographic direction in alignment with the tube axis. In the experimental program two sets of boundary conditions were investigated: (1) applied axial load while torsionally unrestrained, and (2) applied twist while axially unrestrained. The major features of the macroscopic mechanical response for case (1) as reported in Sˇittner et al. (2003) are provided in Fig. 2. An axial ! torsional coupling is clearly evident as the bar rotated 5 under an applied axial extension of 2.5 mm. In contrast, case (2) results were not reported as the experiments failed to display torsional ! axial coupling under the specific thermomechanical conditions of the test. Both experiments were conducted at room temperature (27 C) which, for the specific crystal used, is consistent with a pseudoelastic response for the reported transformation temperature Af ðc01 Þ ¼ 40 C. In Sˇittner et al. (2003) the observed mechanical responses were rationalized based upon the results of basic estimates using anisotropic elasticity of the austenite and the crystallography of cubic to monoclinic transformation in Cu–Al–Ni. Here we approach the issue from the view point of a detailed boundary value problem. Our method is implemented on top of an isotropic elasticity model but does take into account the centrally important issue of anisotropic transformation effects (Sˇittner and Nova´k, 2000). The reported geometry, material properties, and boundary conditions were used to construct various numerical approximations to the experiment. In what follows the model (3.2), the methodology (3.3), and the results (3.4) of the computations are presented. A discussion of the findings is contained in Section 4.
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Fig. 2. Experimental mechanical response of a single crystal shape memory alloy Cu–Al–Ni (b01 ) tube as reported in Sˇittner et al. (2003).
3.2. Constitutive theory The theoretical formulation employed in the computations is based on the work of Govindjee et al. (2003) and Hall and Govindjee (2002) which has been extended to apply to materials for which the energy of mixing is degenerate. Such is the case for cubic–monoclinic materials in general, and the monoclinic-II material studied here in particular. Thus it was necessary to append a secondary condition in the optimization for the variant fractions in order to ensure a unique solution. This was done by choosing the minimum norm solution among all valid solutions, i.e., a singular value decomposition approach. It is noted that this does not change the essential nature of the free energy in the model as a convex approximation. The basic material parameters used in the computations are listed in Table 1. To this is added the shape strains associated with each of the twelve martensitic variants, and the orientation of the parent cubic lattice. Unless indicated otherwise, the [0 0 1] austenite Table 1 Material properties Young’s modulus Poisson’s ratio Density Termal expansion Specific heat Latent heat Reference temperature
E m q a cv kt h0
20.6 GPa 0.25 6448.1 kg/m3 6.5 · 106 1/K 400.0 J/(kg K) 6600 J/kg 169 K
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cubic crystallographic direction was placed in alignment with the tube axis. Due to the assumption of elastic isotropy, the properties given in Table 1 are adjusted to be appropriate for loading in the [0 0 1] direction. Since microstructure formation is also of interest, the chosen ordering of the martensitic variants is provided in Appendix A. As discussed previously (Section 2) there have been conflicting accounts of the lattice structure, and hence, the values of the shape strains. In what follows we perform simulations based on an 18 R unit cell as reported in Bhattacharya and Kohn (1996), a 6M1 unit cell with lattice parameters per Tokonami et al. (1979), and a 6M1 unit cell with lattice parameters per Otsuka et al. (1974). Consistent with the discussion of Section 2 specific values for {a, b, d, } in Cu–Al–Ni near 14 at.% Al and 4 at.% Ni are given in Table 2. 3.3. Methodology Computations reported herein were performed using the finite element analysis package (FEAP) by Taylor (1998). The tube is modeled using standard displacement based eight node brick elements and small deformation theory. All computations were made at 294.26 K. Applied loading included combinations of axial displacements and end rotations using flat tube end boundary conditions consistent with the aforementioned experiment. A convergence study was conducted using various uniform mesh densities consisting of meshes ranging from approximately 4300 degrees of freedom up to 119,000 degrees of freedom. All of the results were qualitatively and quantitatively consistent (e.g. axial reaction variations of less than 2%). Based on this the mesh shown in Fig. 4 was used for all results reported. In order to be consistent with Sˇittner et al. (2003), when reporting shear stresses in line plots we back compute them from the resultant torques by employing the basic linear mechanics torsion solution. When reporting end rotations or displacements we average the values over the nodes on the end of the modeled tube. In summary, the computations are carried out in the standard fashion by satisfying the weak form of the momentum balance equations. The constitutive evaluations are made via full relaxation to the convexified free energy as fully described in Govindjee et al. (2003) and Hall and Govindjee (2002). 3.4. Simulation results The series of numerical simulations performed are broken into four categories based on the boundary and initial conditions employed. The first set of simulations, as reported in Section 3.4.1, apply both an axial extension path and an applied twist path similar to that recorded in the experiments of Sˇittner et al. (2003). The second set of simulations apply an
Table 2 Shape strain parameters
a b d
18R Bhattacharya and Kohn (1996)
6M1 Tokonami et al. (1979)
6M1 Otsuka et al. (1974)
0.0442 0.0822 0.0160 0.0600
0.0520 0.0867 0.0202 0.0516
0.0436 0.0822 0.0170 0.0508
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axial extension while enforcing torque free boundary conditions: this closely models the boundary conditions of the experiment. This second set of runs are reported in Section 3.4.2. The third set of simulations (see Section 3.4.3) are similar to the second, only with the addition of various initial conditions such as imperfections in the specimen, misalignment, and residual stresses. The final set of boundary conditions allowed for an applied twist angle with zero axial stress (see Section 3.4.4). The main points of the results are compared and discussed in Section 4. 3.4.1. Applied extension, applied twist To gain insight into the observed behavior, the first set of simulations sought to examine predictions made by the model along the specific displacement path taken by the experimental specimen. This is of interest since by applying the experimentally observed twist in addition to the axial extension, one can examine the torque response and the variant formation history of the model for a physically based motion. As described previously, one of the austenite cubic basis vectors was aligned with the tube axis; this will be referred to as perfect alignment. The experimentally recorded displacement and rotation path was applied to the end of the tube (as indicated in Fig. 3 lower left) and the mechanical response was computed assuming shape strains as per 18R of Table 2. The resulting response is given in Fig. 3. The essential feature is the low magnitude of the shear stress indicating a very slight torque is required to induce the twist under the applied axial extension. By simple elastic isotropy one would predict shear stresses in excess of 45 MPa at this level of rotation.
100
200
300 0.03
axial stress
0.02 axial strain 0.01 shear strain 0
100 200 time (sec)
strain axial stress (MPa)
stress (MPa)
0 350 300 250 200 150 100 50 0
shear stress 0 300
0
5
0.01
0.02 axial strain
0.03
20
4
shear stress (MPa)
torsion angle (deg)
350 300 250 200 150 100 50 0
3 2 1 0
15 10 5 0
0
0.5 1 1.5 2 2.5 axial displacement (mm)
0
1
2 3 4 torsion angle (deg)
Fig. 3. Perfectly aligned, 18R, applied extension, applied twist mechanical response.
5
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The predicted volume average microstructure formation was dominated by only four martensitic variants. The spatial location of these variants were centered on the primary axes of austenite which coincides with the global axes used in the problem. This results in variants appearing on the tube faces as indicated in Table 3. With regard to variant formation, two items which stand out in the calculation are the following. First, only the variants with ea33 ¼ a þ appear. Second, the distribution of these four variants is consistent with the torsionally induced shear as is apparent from Table 3. The pattern of the distribution at the maximum load can be seen in Figs. 4 and 5 which indicate the distribution of variants 11 and 8 respectively. Symmetric patterns also exist on the opposite faces not shown, only with variants 12 and 7 appearing rather than 11 and 8. The above calculation was repeated using the shape strains consistent with Tokonami et al. (1979) 6M1 from Table 2. The results were similar, although it is noteworthy that the resultant torque was even lower (hence more accurate) than the previous run using the 18R parameters. This can be seen in the mechanical response shown in Fig. 6. Several runs of this type were also made with the axis of austenite shifted by 1 off the tube axis. It was found in every case that the amount of torque required to twist the tube would increase, indicating that the minimum was achieved with perfect alignment. Evidence of these observations are found in Figs. 7 and 8 which are for misaligned 18R and 6M1 runs respectively. In these runs it was also noted that the 90 periodicity of the maximum variant volume fraction (maxa{na}) around the tube was lost in favor of
Table 3 Variant distribution on the tube, where the global 3 axis is aligned with the tube and the global 1 and 2 axes coincide with austenite basis vectors Variant
Face/axis
eaij
7 8 11 12
2 +2 +1 1
a¼7 e13 ¼ þd a¼8 e13 ¼ d a¼11 e23 ¼ þd a¼12 ¼ d e23
Variant 11 Fraction 0.00E-00 3.32E-02 6.63E-02 9.95E-02 1.33E-01 1.66E-01 1.99E-01 2 1
Fig. 4. Variant a = 11 distribution on the +1 face for the loading indicated in Fig. 3.
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Variant 8 Fraction 0.00E-00 3.20E-02 6.40E-02 9.60E-02 1.28E-01 1.60E-01 1.92E-01 1 2
0 350 300 250 200 150 100 50 0 0
100
200
300 0.03
axial stress
0.02 axial strain 0.01 shear strain 100 200 time (sec)
strain axial stress (MPa)
stress (MPa)
Fig. 5. Variant a = 8 distribution on the +2 face for the loading indicated in Fig. 3.
shear stress 0 300
4 3 2 1 0 0
0
0.01 0.02 axial strain
0.03
20 shear stress (MPa)
torsion angle (deg)
5
350 300 250 200 150 100 50 0
0.5 1 1.5 2 axial displacement (m)
2.5
15 10 5 0
0
1
2 3 4 torsion angle (deg)
5
Fig. 6. Perfectly aligned, 6M1 (Tokonami et al., 1979), applied extension, applied twist mechanical response.
180 periodicity in which two of the faces (e.g. ±1 axes) had higher volume fractions than the other two (e.g. ±2 axes). 3.4.2. Applied extension, torque free The next category of simulations attempted to model the exact conditions of the experiment as axially fixed and torsionally free. The first set of results obtained was for a perfectly aligned 18R material. The mechanical response was similar to that of the previous runs with the exception that the model did not reproduce the physically observed twist.
G.J. Hall et al. / International Journal of Plasticity 23 (2007) 161–182
200
300 0.03
axial stress
axial strain
axial stress (MPa)
100
strain
stress (MPa)
0 350 300 250 200 150 100 50 0
0.02 0.01
shear strain 0
100 200 time (sec)
shear stress 0 300
350 300 250 200 150 100 50 0
0
0.01 0.02 axial strain
0.03
20
4
shear stress (MPa)
torsion angle (deg)
5
3 2 1 0
173
0
0.5 1 1.5 2 axial displacement (mm)
15 10 5 0
2.5
0
1 2 3 4 torsion angle (deg)
5
Fig. 7. Austenite 3 axis at a 1 rotation about [1 1 0], 18R shape strains, applied extension, applied twist mechanical response.
200
300
stress (MPa)
250 200
0.02 axial strain
150
0.01
100 50 0
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300
shear strains
strain
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axial stress (MPa)
0 350
shear stress 0 200 300
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0
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time (sec)
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shear stress (MPa)
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axial strain
0
0.5
1
1.5
2
axial displacement (mm)
2.5
15 10 5 0
0
1
2
3
4
5
torsion angle (deg)
Fig. 8. Austenite 3 axis at a 1 rotation about [0 1 0], 6M1 (Tokonami et al., 1979) shape strains, applied extension, applied twist mechanical response.
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Variant 12 Fraction 0.00E-00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E-00
Fig. 9. Variant a = 12 distribution for a perfectly aligned, 18R, applied axial extension, torque free loading run.
The reason for this is apparent from the distribution of volume fractions as seen in Fig. 9. Although only the expected variants a 2 {7, 8, 11, 12} appear, their pattern is perfectly diffuse radially around the tube. Thus, in the absence of some impetus, the model selects an equal distribution of the four variants as the (non-unique) minimal energy solution. In addition to the aforementioned torque free simulations, numerous runs were made with misalignments up to 1 between the tube axis and a major axis of austenite. It was found that the axial stress response was fairly insensitive to these perturbations, although the distribution of variants was at times drastically altered. Such behavior indicates that a slight perturbation of the system, either through geometry, alignment, thermal distribution, or some other factor may produce the desired twist numerically. The next section investigates the idea that initial or residual stresses in the system may be contributing to the observed behavior. 3.4.3. The effect of initial conditions Although the simulations of Section 3.4.1 correctly predicts the stress response under the applied displacement history, in Section 3.4.2 it was found that a pure applied axial extension did not spontaneously induce twisting. Working under the assumption that the particular variant formation observed requires some form of perturbation to form, both initial and residual stresses were examined as a potential source. Initial stress: As a first attempt, a small biasing initial stress distribution was placed in the tube. The distribution was specified by the function szh ¼ sozh R; sozh
ð5Þ sozh ,
is a constant and R is the radial position. By choosing a small value for this where has the effect of providing a small biasing (constant) torque. Although this construction is artificial, it allows one to investigate the perturbation required to induce the observed axial ! torsional coupling. Results from simulations conducted using initial stress values of szh 6 2.5 MPa produced the response provided in Fig. 10. The mechanical response is remarkably similar to the experimental result, although the predicted end rotation is higher than the observations. The martensitic variants and their spatial variation were similar to those seen
350 300 250 stress 200 150 100 50 0 0 50
175
0.06 0.04 strain
100
150 200 time (sec)
0.02 250
300
strain
stress (MPa)
G.J. Hall et al. / International Journal of Plasticity 23 (2007) 161–182
0 350
torsion angle (deg)
8 6 4 2 0
0
0.5
1 1.5 axial displacement (m)
2
2.5
Fig. 10. Results from simulation under applied axial displacement, unrestrained torsional boundary conditions, and small biasing initial stress.
in the simulations of Section 3.4.1. These results suggest that even a slight bias can induce the cooperative deformation mode suggested in Sˇittner et al. (2003) under the very specific conditions of the test. However, given the artificial nature of the bias introduced by the chosen initial stress, a residual stress was then considered as discussed next. Residual stress: A self equilibriating torque free initial residual stress field was implemented in the numerical simulation. The distribution of the stress followed i sizh szh Re szh ¼ R ; ð6Þ Ro Re Ro Re where Re ¼
3 R4o R4i : 4 R3o R3i
ð7Þ
Ro is the outer radius of the tube, Ri is the inner radius of the tube, and sizh is the residual shear stress at the inner surface of the tube. Setting sizh ¼ 40 MPa and again using boundary conditions from the experiment produced the results found in Fig. 11 for the 6M1 (Tokonami et al., 1979) material of Table 2. The qualitative response is very close to that observed experimentally, and by changing the strength of the residual stress field (sizh ), it was possible to quantitatively match the experimental data as well (the best case scenario is not shown here, as there is little justification for such a back-calculation). Of note is that again the simulations reproduce the observed cooperative deformation mode only when a favorable initial condition exists. The variants appearing in this computation are the same as in Section 3.4.1. However, their spatial distribution is quite different; see Fig. 12 for the distribution of variant a = 12. At issue here is the large degree of degeneracy in the model’s free energy of mixing for monoclinic materials. Such models are capable of faithfully resolving
G.J. Hall et al. / International Journal of Plasticity 23 (2007) 161–182
stress (MPa)
400
50
100
150
200
250
300
350 0.06
stress
300
0.04
200 strain
0.02
100 0
torsion angle (deg)
0
0
50
6 5 4 3 2 1 0 0
100
0.5
150 200 time (sec)
250
300 350
1 1.5 2 axial displacement (mm)
strain
176
0
2.5
Fig. 11. Applied axial extension simulation with torque-free residual shear stress. The material is consistent with Tokonami lattice parameters and the 6M1 unit cell.
Variant 12 Fraction 0.00E-00 5.20E-02 1.04E-01 1.56E-01 2.08E-01 2.60E-01 3.12E-01 1 2 Fig. 12. Spatial distribution of the volume fractions for martensitic variant a = 12 under axial extension and unrestrained torsional motion. The simulation includes a torque free residual stress and employs the Tokonami lattice parameters with a 6M1 transformation description.
macroscopic measures of behavior but are sensitive to initial conditions and seemingly minor geometric features. What matters to the model is only that the spatial average of the transformation strain is correct. Returning to the discussion of Section 2, the effect of a small variations in the lattice parameters can be observed numerically. To demonstrate this, the same simulation was repeated using the 6M1 Otsuka et al. (1974) material from Table 2 (see also Section 2.3.3). The results are shown in Fig. 13 and although qualitatively similar they differ by approximately 30 MPa (out of 300 MPa total) in the stress response and just under 1 (out of 5 total) in the rotation – a rather large difference considering the small differences in the shape strains.
G.J. Hall et al. / International Journal of Plasticity 23 (2007) 161–182
50
100
150
200
250
300 350 0.06
stress
0.04
200 strain
0.02
100 0 0 6 5 4 3 2 1 0 0
50
100
0.5
150 200 time (sec)
250
1 1.5 2 axial displacement (mm)
300 350
strain
300
torsion angle (deg)
stress (MPa)
0 400
177
0
2.5
Fig. 13. Applied axial extension simulation with torque-free residual shear stress. The material is consistent with Otsuka lattice parameters and the 6M1 unit cell.
3.4.4. Applied twist, axially free The second set of boundary conditions explored experimentally in Sˇittner et al. (2003) consisted of an applied twist while allowing unrestrained axial motion. Under these conditions no transformation was observed for shear stresses approaching 400 MPa; an initial theoretical calculation by the aforementioned authors predicted transformation to begin at approximately 412 MPa. A numerical simulation of the problem was conducted using the 6M1 structure with parameters from Tokonami et al. (1979) and the material properties listed in Table 1; note that due to the isotropic elastic modulus assumption the associated shear modulus is drastically underestimated. Nevertheless the simulation predicted deviation from linearity in the range 450–500 MPa and with very little austenite–martensite transformation in the tube. However, by continuing the application of twist out to shear strains approaching 12% torsional ! axial coupling became increasingly apparent with a maximum axial displacement of (0.34 mm) at a physically unrealistic shear stress of over 600 MPa. 4. Concluding remarks Several issues regarding computations in monoclinic-II systems have been investigated. On the theoretical side, it was demonstrated that current ambiguities in the literature regarding measured lattice parameters are sufficiently broad to render nontrivial changes in the free energy landscape required for macroscopic modeling. Simultaneously, the debate between a 18R unit cell with stacking faults and a 6M1 unit cell has similar implications for macroscopic modeling. Most of the evidence we have examined supports the 6M1 viewpoint. On the numerical side, an investigation of the experimental observation of axial ! torsion coupling was conducted by way of finite element experiments. The following points summarize the main observations drawn from this part of the work.
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For the specific material and temperature state examined, coupling under applied twist, axially unrestrained motion occurs only at physically unrealistic levels. This is in agreement with the experiment, though it is noted that this mode of deformation could be observable in a suitably designed experiment. The computations produce near zero torque response when driven both axially and torsionally. The quantitative response seems quite reasonable, and the volume average variant distributions are consistent with what one would expect based on an examination of the shape strains listed in Section 3.2. Minimal shear stress is obtained in the perfectly aligned case which supports the notion, introduced in Sˇittner et al. (2003), of the special nature of conditions required for observation of a cooperative (coupled) deformation mode. In the axial extension, torque free simulations the axial stress response is not particularly sensitive to perturbations in the alignment of the austenite lattice. In contrast, the variant formation pattern is extremely sensitive to such perturbations. The model does not spontaneously exhibit coupling under applied extension, torsionally free conditions, perfectly aligned conditions. This case produces the expected four variants in equal volume fractions but in a diffuse pattern which results in zero twist. However, with the inclusion of a bias to seed the transformation, the numerics are in good agreement with experimental data. The final remark reveals an important point on the numerical modeling of unexpected coupled behaviors in shape memory alloys: to reproduce such spontaneous behavior, standard material parameters are insufficient and must be supplemented with the presently not well understood conditions which bias or seed the transformation. For example, theory predicts that spontaneous right-hand twist is equally likely with left hand twist or zero twist for that matter. This contradicts the observation of specimens which clearly exhibit a reproducible (in the experiments referenced) bias to twist rightward 4 times and leftward once under axial loading. The numerics suggest that either small torsional loads or residual stresses would explain the initiation of such spontaneous coupling, though further experimental investigation is required to clarify the issue. The point also highlights the need for the exercise of extreme care in the study of highly symmetric experimental modes of deformation. Acknowledgements P. Sittner would like to acknowledge support from Marie-Curie RTN Multimat (Contract No. MRTN-CT-2004-505226) and project A1048107 of Grant Agency of the Academy of Sciences of the Czech Republic. Appendix A. Variant ordering The constitutive theory is based on the lattice correspondence variants which arise in the transition from austenite to martensite. They enter into the computations through the associated shape or Bain strains. In the specific case of a cubic to monoclinic-II transition, there are twelve martensitic variants. To identify these variants, the following ordering has been adopted.
G.J. Hall et al. / International Journal of Plasticity 23 (2007) 161–182
2
aþ
6 ea¼1 ¼ 4 d 0 2 a 6 a¼3 e ¼4 d
0
aþ 6 ¼4 0 d
2
0 b 0
a 0 6 ea¼7 ¼ 4 0 b d 0 2 b 0 6 a¼9 e ¼ 40 aþ
ea¼11
0 2 b 6 ¼ 40 0
0
b
2
3
7 a 0 5; 0 b 3 d 0 7 a þ 0 5;
0
2
ea¼5
d
6 ea¼2 ¼ 4 d 0 2 a 6 a¼4 e ¼ 4 d
3
d 7 0 5; a 3 d 7 0 5;
aþ
2
ea¼6
0
aþ 6 ¼4 0 2
d
d a 0 d aþ 0
0
179
3
7 0 5; b 3 0 7 0 5; b
3 d 7 0 5; 0 a 3 0 d 7 b 0 5; 0 b
a 6 ea¼8 ¼ 4 0 aþ d 0 a þ 3 2 3 0 b 0 0 6 7 7 d 5; ea¼10 ¼ 4 0 a þ d 5; d a 0 d a 3 2 3 0 0 b 0 0 6 7 7 a d 5; ea¼12 ¼ 4 0 a d 5; d aþ 0 d a þ
Appendix B. Transformation stretches B.1. Application The transformation stretches discussed in Section 2 are included here for completeness; the ordering of the results follows that of the text. B.1.1. Cu–Zn–Ga Using data from Saburi et al. (1976) at 20.4 at.% Zn and 12.5 at.% Ga the 18R lattice ˚ , b = 5.33 A ˚ , c = 38.22 A ˚ and b ¼ 88:33 . The cubic lattice parameters are: a = 4.40 A ˚ parameter is given as a0 = 5.86 A which was measured on a slightly different alloy (21.8 at.% Zn and 11.8 at.% Ga). Using g = 0.0775 the resulting transformation stretch is given by 2 3 0:9096 0 0 6 7 U ¼4 0 1:0035 0:0172 5 : ðB:1Þ 0
0:0172 1:0843
A
If one uses a consistent value for g then we have 2 3 0:9096 0 0 6 7 U ¼4 0 1:0281 0:0185 5 : 0
0:0185 1:0584
A
ðB:2Þ
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If one computes the transformation strain using the 6 M1 structure then one finds 2 6 U¼4
0:9096
0
0 0
1:0888 0:0169
0
3
7 0:0169 5 : 0:9994 A
ðB:3Þ
B.1.2. Cu–Zn–Al This alloy was studied by Pitteri and Zanzotto (1997) at 15 at.% Zn and 17 at.% Al. The ˚ , a = 4.553 A ˚ , b = 5.452 A ˚ , c = 38.977 A ˚ and reported lattice parameters were a0 = 5.996 A b = 87.5. Using the 18R structure and a consistent shear value gives a transformation stretch of 2 3 0:9093 0 0 6 7 U¼4 0 1:0246 0:0262 5 : ðB:4Þ 0 0:0262 1:0703 A If we compute the transformation stretch from the 6M1 structure we find 2 3 0:9093 0 0 6 7 U¼4 0 1:0860 0:0254 5 : 0 0:0254
1:0097
ðB:5Þ
A
Slightly different results are found using the reported values in Bhattacharya and Kohn (1996) to generate a transformation stretch of 2 3 1:0866 0:0249 0 6 7 U ¼ 4 0:0249 1:0100 0 5 : ðB:6Þ 0 0 0:9093 A B.1.3. Cu–Al–Ni This alloy was studied in Otsuka et al. (1974) at 14.2 wt.% Al and 4.3 wt.% Ni and by Tokonami et al. (1979) at 14.0 wt.% Al and 4.2 wt.% Ni. The lattice parameters from these ˚ , b = 5.356 A ˚, two source are different. From Otsuka et al. (1974) we have a = 4.382 A ˚ c = 38.00 A; other parameters are not reported. From Tokonami et al. (1979) we have ˚ , b = 5.330 A ˚ , c = 38.19 A ˚ , b = 89.0. a = 4.430 A Otsuka et al. (1974) lattice parameters: Since the lattice parameters from Otsuka et al. (1974) are incomplete we fill the missing ones using those in Hane (1999). Using the 18R structure and a consistent shear value gives a transformation stretch of 2 3 0:9178 0 0 6 7 U¼4 0 1:0334 0:0194 5 : ðB:7Þ 0
0:0194
1:0516
A
If we compute the transformation stretch from the 6M1 structure we find a more nearly compatible stretch of
G.J. Hall et al. / International Journal of Plasticity 23 (2007) 161–182
2 6 U ¼4
0:9178 0 0
0
0
181
3
7 1:0945 0:0170 5 : 0:0170 0:9928 A
ðB:8Þ
The reported values in Bhattacharya and Kohn (1996) generate a transformation stretch of 2 3 1:1042 0:0160 0 6 7 U ¼ 4 0:0160 0:9842 0 5 : ðB:9Þ 0 0 0:9178 A ˚ as it Tokonami et al. (1979) lattice parameters: In this section we assume a0 = 5.838 A was not reported. Using the 18 R structure and a consistent shear value gives a transformation stretch of 2 3 0:9133 0 0 6 7 U ¼4 0 1:0417 0:0226 5 : ðB:10Þ 0
0:0226 1:0600
A
Finally, if one computes the transformation stretch from the 6M1 structure we find 2 3 0:9133 0 0 6 7 U ¼4 0 1:1036 0:0202 5 : ðB:11Þ 0 0:0202 1:0004 A References Ball, J., James, R., 1987. Fine phase mixtures as minimizers of energy. Archive for Rational Mechanics and Analysis 100, 13–52. Bhattacharya, K., 1993. Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Continuum Mechanics and Thermodynamics 5, 205–242. Bhattacharya, K., Kohn, R., 1996. Symmetry, texture, and the recoverable strain of shape memory alloys. Acta Materialia 44, 529–542. Govindjee, S., Mielke, A., Hall, G., 2003. The free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis. Journal of the Mechanics and Physics of Solids 51 (April), 763+. Hall, G., Govindjee, S., 2002. Application of a partially relaxed shape memory free energy function to estimate the phase diagram and predict global microstructure evolution. Journal of the Mechanics and Physics of Solids 50, 501–530. Hall, G., Govindjee, S., 2003. Application of the relaxed free energy of mixing to problems in shape memory alloy simulation. Journal of Intelligent Material Systems and Structures 13, 773–782. Hane, K., 1999. Bulk and thin film microstructures in untwinned Martensites. Journal of the Mechanics and Physics of Solids 47, 1917–1939. James, R., Hane, K., 2000. Martensitic transformations and shape-memory materials. Acta Materialia 48, 197– 222. Kohn, R., 1991. The relaxation of a double well energy. Continuum Mechanics and Thermodynamics 3, 193–236. Li, Z., Sun, Q., 2002. The initiation and growth of macroscopic martensite band in nano-grained NiTi microtube under tension. International Journal of Plasticity 18, 1481–1498. Nouailhas, D., Cailletaud, G., 1995. Tension-torsion behavior of a single-crystal superalloys: experiment and finite element analysis. International Journal of Plasticity 11, 451–470. Otsuka, K., Nakamura, T., Shimizu, K., 1974. Electron microscopy study of stress induced acicular b 0 1 martensite in Cu–Al–Ni alloy. Transactions of the Japan Institute of Metals 15, 200–210.
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Otsuka, K., Ohba, T., Tokonami, M., Wayman, C., 1993. New description of long period stacking order structures of martensites in b-phase alloys. Scripta Metallurgica et Materialia 29, 1359–1364. Pitteri, M., Zanzotto, G., 1997. Electron microscopy of internally faulted Cu–Zn–Al martensite. Acta Metallurgica 25, 989–1000. Pitteri, M., Zanzotto, G., 1998. Generic and non-generic cubic-to-monoclinic transitions and their twins. Acta Materialia 46, 225–237. Saburi, T., Nenno, S., 1981. The shape memory effect and related phenomena. Proceedings of an International Conference on Solid Solid Phase Transformations, 1455–1479. Saburi, T., Nenno, S., Kato, S., Takata, K., 1976. Configurations of martensitic variants in Cu-Zn-Ga. Journal of the Less Common Metals 50, 223–236. Sˇittner, P., Nova´k, V., 2000. Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals. International Journal of Plasticity 16, 1243–1268. Sˇittner, P., Hashimoto, K., Kato, M., Tokuda, M., 2003. Stress induced martensitic transformations in tension/ torsion of CuAlNi single crystal tube. Scripta Materialia 48, 1153–1159. Taylor, R., November 1998. FEAP: A Finite Element Analysis Program. Available from:
Simulation of cubic to monoclinic-II transformations in a single crystal Cu–Al–Ni tube Garrett J. Hall a, Sanjay Govindjee
b,d,*
, Petr Sˇittner c, V. Nova´k
c
a
Cal Poly, San Luis Obispo, Structural Engineering and Applied Mechanics, Department of Civil and Environmental Engineering, California Polytechnic State University San Luis Obispo, CA 93407, USA b University of California, Berkeley, Structural Engineering, Mechanics and Materials, Department of Civil and Environmental Engineering University of California, Berkeley, CA 94720, USA c Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, Prague 8, 182 21, Czech Republic d ETH Zurich, Center of Mechanics, Institute of Mechanical Systems, Tannenstrasse 3, 8092 Zurich, Switzerland Received 24 July 2005 Available online 22 August 2006
Abstract Motivated by recent unusual experimental results [Sˇittner, P., Hashimoto, K., Kato, M., Tokuda, M., 2003. Stress induced martensitic transformations in tension/torsion of CuAlNi single crystal tube. Scripta Materialia 48, 1153–1159] indicating a strong coupling between axial and torsional deformations in single crystal monoclinic-II shape memory alloy tubes, various computational aspects of numerically modelling the observed behavior in monoclinic-II materials are discussed. As a necessary first step, the shape strain computations in monoclinic-II systems are examined. It is found that ambiguities regarding the lattice parameters and proper unit cell in the existing literature have implications for the macroscopic constitutive theories of such alloys. This is demonstrated in the context of a detailed numerical investigation of the experimental problem of interest, wherein it is found that numerical simulations demonstrate proper quantitative response, a strong axial ! torsional coupling given a biasing initial condition, and the observed lack of torsional ! axial coupling. Implications of the findings are discussed with regard to both the further development of numerical models and future avenues for experimental research. 2006 Elsevier Ltd. All rights reserved. Keywords: Shape memory alloys; Phase transformation; Single crystal; Quasi-convexity
*
Corresponding author. Fax: +44 41 632 1145. E-mail addresses: [email protected] (G.J. Hall), [email protected] (S. Govindjee), [email protected] (P. Sˇittner), [email protected] (V. Nova´k). 0749-6419/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2006.04.004
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1. Introduction A recent numerical investigation by Hall and Govindjee (2003) of single crystal torsion tubes predicted the appearance of significant axial extensions under applied torsional loads (i.e. torsional ! axial coupling), and suggested the single crystal torsion tube as an ideal model problem for the study of solid–solid phase transformation energetics. The usefulness of this geometry/loading combination was also recognized by Thamburaja and Anand (2002). Since that time an experimental study by Sˇittner et al. (2003) has been performed on a Cu–Al–Ni (b01 ) specimen of similar geometry albeit with different boundary conditions (see also Nouailhas and Cailletaud (1995) for a related single crystal study, and Thamburaja and Anand (2001) and Li and Sun (2002) for related polycrystalline tube experiments). Among the significant findings in Sˇittner et al. (2003) was the observation of spontaneous axial ! torsional coupling in a single crystal undergoing a cubic ! monoclinic-II pseudoelastic phase transformation, and the description of a possible microstructure formation consistent with the observed deformation. Based on these observations, in the present paper we revisit the problem in order to compare the physical observations with those predicted by the lower bound model of Govindjee et al. (2003) and Hall and Govindjee (2002, 2003). Towards that end, in Section 2 we begin with an examination of shape strain computations in monoclinic-II alloys as required for input to the constitutive model. The two common descriptions of the unit cell which appear in the literature, 18R and 6M1, are compared in light of the implications for the resulting energetics of transformation. A detailed discussion concerning Cu–Al–Ni is provided as this is the material employed in the experimental work of interest. Here, it is found that the literature is somewhat ambiguous as to the correct lattice parameters, a fact which has both theoretical and numerical implications. With the shape strain computations in mind we then proceed in Section 3 to a numerical investigation of the problem of interest. Using an extension of a previously developed constitutive theory (Govindjee et al., 2003; Hall and Govindjee, 2002, 2003), various numerical approximations of the boundary value problem in the experiment are made, each demonstrating particular aspects of the numerical implementation of the model. Specifically, in Section 3, we drive the model in various combinations of tension/extension torsion/twist to provide insight into the coupled mode transformations in monoclinic-II materials. 2. Shape strain computations Macroscopic constitutive models for shape memory alloy materials which are derived from consideration of the underlying microstructure necessarily include a representation of the shape strain which accompanies phase transformation and/or reorientation. It is generally accepted that for most of the common shape memory alloys that the lattice parameters and unit cells associated with phase transformation are well characterized for a given alloy composition. However, in the case of monoclinic-II materials, there has been some debate regarding the proper characterization of the unit cell. In particular the computation of the shape strains in monoclinic-II ðb01 Þ systems is often based upon somewhat ambiguous and/or inconsistent statements concerning the geometry of the transformations strains. The apparently most consistent method proposed to date is
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163
due to Hane (1999) wherein he proposes to resuscitate the notion of the 6M1 lattice structure to replace the more common 18R system following the lead of Otsuka et al. (1993). In what follows, we report upon such computations and highlight some of the inconsistencies associated with the 18R unit cell method of computing the shape strains, and point out differences between some of the numerical values provided in the literature. These issues are further considered in the macroscopic numerical modeling which follows in Section 3. 2.1. Background The monoclinic-II system is a bit different from other martensitic transformation systems such as orthorhombic, tetragonal, or even monoclinic-I; see Pitteri and Zanzotto (1998) for a discussion of the two monoclinic systems. The monoclinic-II system is known to possess an austenite–martensite interface where the martensite is untwinned; see e.g. Saburi et al. (1976) for a discussion of a Cu–Zn–Ga alloy. For the computation of the transformation strains one needs to know the lattice correspondence between the cubic austenite and the monoclinic martensite. The traditional lattice correspondence employs an 18R lattice cell. However, the monoclinic angles reported in the literature do not permit a low energy austenite–martensite interface (a rank-one interface) for this correspondence. Thus it is conventional to introduce an ‘‘invariant lattice shear’’ to rectify this inconsistency. The amount of the ‘‘invariant lattice shear’’ is determined mathematically (not experimentally) so that a low energy interface can exist between austenite and untwinned martensite. More recently, Hane (1999), following Otsuka et al. (1993), has revisited the notion of a 6M1 lattice correspondence and shown that it more naturally produces transformation strains which are rank-one compatible with austenite; i.e. it produces transformation strains which are rank-one compatible with austenite without the need for a ‘‘invariant lattice shear’’. In this section, the pertinent equations for computing transformation strains from lattice parameters are reviewed and the numerical consequences are determined. 2.2. Transformation strains In what follows all tensor components are given in either the orthonormal cubic austenite basis, denoted by a subscript A, or in the orthonormal martensite basis which is aligned with [0 1 1]A, ½ 1 0 0A , ½0 1 1A , denoted by a subscript M. Once the deformation map, F, from the cubic structure to the monoclinic-II structure has been determined, the transformation stretches (squared) are determined as U2 = FTF. For the infinitesimal deformation theory the transformation strains are determined as U 1. For existence of an austenite to untwinned martensite interface with normal m and habit-plane displacement b one needs U2 to have one eigenvalue equal to unity, one below unity and one above unity (Ball and James, 1987, Prop. 4). 2.2.1. 18R To compute the transformation strains for the 18R monoclinic structure one starts from the cubic lattice parameter a0, the 18R parameters a, b, c and monoclinic angle b. In the martensite basis, the deformation gradient is given by
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2
1 0
6 F ¼ 40 1 0 0
3 2 pffiffiffi 2a=a0 7 6 05 4 0 1 M 0
g
0 b=a0 0
0 0
pffiffiffi 2c sinðbÞ=9a0
3 7 5 :
ð1Þ
M
If the description were fully consistent then g would be equal to cot(b). This however does not produce a transformation strain with the ‘‘observed’’ compatibility with an austenite phase. Thus, when reporting the transformation strains for such systems g is typically chosen so the resulting transformation strains are rank-one compatible with austenite. The transformation stretch is computed from the polar decomposition of F and is typically given in the austenite basis by executing a change of coordinates from the martensite basis to the austenite basis. 2.2.2. 6M1 To compute the transformation strains for the 6M1 monoclinic structure one starts from the cubic lattice parameter a0, the 18R parameters a, b, c and monoclinic angle b. First, one converts the 18R lattice parameter c and the monoclinic angle to the 6M1 lattice cell. The relations are given by Hane (1999) and James and Hane (2000) a 1 b1 ¼ b þ tan ð2Þ c sinðbÞ and c1 ¼
c sinðbÞ : 3 sinðb1 Þ
In the martensite basis, the deformation gradient is given by 3 2 3 2 pffiffiffi 1 0 g1 0 0 2a=a0 7 6 7 6 F ¼ 40 1 0 5 4 5 : 0 b=a0 0 pffiffiffi 0 0 1 M 0 0 2c1 sinðb1 Þ=3a0 M
ð3Þ
ð4Þ
The description here is fully consistent and g1 is equal to cot(b1). Further, rank-one compatibility to austenite from untwinned martensite is very nearly satisfied using reported monoclinic angles without the need for inconsistent lattice shears. 2.3. Application In this section we briefly present the application of the above transformations to several materials that have been studied in the literature. 2.3.1. Cu–Zn–Ga This alloy was studied by Saburi et al. (1976) at 20.4 at.% Zn and 12.5 at.% Ga. They ˚ , b = 5.33 A ˚ , c = 38.22 A ˚ and b ¼ 88:33. The report 18R lattice parameters of a = 4.40 A ˚ which was measured on a slightly different cubic lattice parameter is given as a0 = 5.86 A alloy (21.8 at.% Zn and 11.8 at.% Ga). These authors also report the shear value g = ±0.0775 which corresponds to a shear angle of 85.57. Note the inconsistency with the monoclinic angle – a point which is usually rationalized by appealing to the notion of stacking faulting. In Saburi and Nenno (1981) the value of g = 0.086, which corresponds
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to a shear angle of 85.08 and is termed the ideal 18R structure with no stacking faults; note, however, that this latter value is still inconsistent with the monoclinic angle. Using g = 0.0775 the resulting transformation stretch (from the 18R relations) is compatible with austenite to within the accuracy of the reported parameters. If one uses a consistent value for g then we find (from the 18R relations) the middle eigenvalue of U2 is 1.0390. However, if one computes the transformation strain using the 6M1 structure then one finds a stretch tensor that is nearly rank-one compatible with austenite; the middle eigenvalue of U2 is 0.9926. For the 6M1 cell this occurs without having to introduce any arbitrary lattice shears. 2.3.2. Cu–Zn–Al This alloy was studied by Pitteri and Zanzotto (1997) at 15 at.% Zn and 17 at.% Al. The ˚ , a = 4.553 A ˚ , b = 5.452 A ˚ , c = 38.977 A ˚ and reported lattice parameters were a0 = 5.996 A b = 87.5. Using the 18R structure and a consistent shear value gives a transformation such that the closest eigenvalue of U2 to unity is 1.0254. Again, if we compute the transformation stretch from the 6M1 structure we find a stretch value which is more nearly compatible with austenite. In this case the closest eigenvalue of U2 to unity is 1.0041. Slightly different results are found using the reported values in Bhattacharya and Kohn (1996) to generate a transformation stretch where the closest eigenvalue of U2 to unity is 1.0052. Here also it is clear that the 6M1 structure gives a superior result. 2.3.3. Cu–Al–Ni This alloy was studied in Otsuka et al. (1974) at 14.2 wt.% Al and 4.3 wt.% Ni and by Tokonami et al. (1979) at 14.0 wt.% Al and 4.2 wt.% Ni. The lattice parameters from these ˚ , b = 5.356 A ˚, two sources are different. From Otsuka et al. (1974) we have a = 4.382 A ˚ c = 38.00 A; other parameters are not reported. From Tokonami et al. (1979) we have ˚ , b = 5.330 A ˚ , c = 38.19 A ˚ , b = 89.0. a = 4.430 A Otsuka et al. (1974) lattice parameters: Since the lattice parameters from Otsuka et al. (1974) are incomplete we fill in the missing ones using those in Hane (1999). Using the 18R structure and a consistent shear value gives a transformation stretch which is nearly compatible with austenite; the closest eigenvalue of U2 to unity is 1.0426. If we compute the transformation stretch from the 6M1 structure we find a more nearly compatible stretch whose closest eigenvalue of U2 to unity is 0.9801. Meanwhile, the reported values in Bhattacharya and Kohn (1996) generate a 6M1 transformation stretch whose closest eigenvalue of U2 to unity is 0.9645. Tokonami et al. (1979) lattice parameters: Here the austenite lattice spacing was not ˚ . Using the 18R structure and a consistent shear value reported, so we assume a0 = 5.838 A gives a transformation stretch which is nearly compatible with austenite: 1.0536 is the closest eigenvalue of U2 to unity. Finally, if one computes the transformation stretch from the 6M1 structure we find again a more nearly compatible transformation stretch with austenite. In this case the closest eigenvalue of U2 to unity is 0.9932. 2.4. Discussion of shape strain computations It can be seen in all the cases that the transformation stretches generated from the 6M1 system come closer to being compatible with austenite over those generated from the 18R system when no arbitrary adjustments are made in the 18R computation. As shown by
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Hane (1999) with only very minor adjustments in the monoclinic angle the 6M1 system can be made to generate ideal compatibility with austenite whereas the 18R system requires larger changes to the monoclinic angle (or the introduction of an arbitrary stacking fault shear). The differences in the numerical values one finds in the computed shape strains from the 18R and 6M1 structure appear rather insignificant with differences generally in the third significant digit; see Appendix B. However, for models which are derived explicitly from microstructural considerations, the variations may be non-negligible. To see this, consider that the free energy of mixing (which arises in the transition from the microscale to the macroscale) critically depends upon rank-one connections between variants in the material (see Ball and James, 1987; Kohn, 1991; Govindjee et al., 2003 and the references therein). Implicitly, such models consider all possible twinning combinations, including austenite– martensite (a–m) interfaces, martenstite–martensite (m–m) interfaces, and various order Results independent of linear/nonlinear twinning calculation assumption.
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Fig. 1. Twinning chart for Monoclini-II Cu–Al–Ni. Martensite twin solutions (m–m) are indicated by ‘‘S’’; the variant ordering is listed in Appendix A. The connectivity map for (a–m–m) twinning is seen to be sensitive to small changes in the lattice parameters and unit cell interpretation. Note that although these interfaces are not observed experimentally, they play a role in the derivation of macroscopic free energy functions for shape memory alloys.
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austenite–martensite–martensite laminate type interfaces (a–m–m). Thus if the rank-one connections are changed by perturbations in the lattice parameters, so is the landscape of the derived free energy function. Using the Cu–Al–Ni shape strains computed above as an example, nonlinear and linear twinning calculations (Ball and James, 1987; Bhattacharya, 1993) may be performed in each case. As previously noted, austenite and the lattice correspondence variants of martensite are close to, though not perfectly, twin compatible. Fig. 1 summarizes the rank-one connections between martensite twin pairs (m–m) by the letter ‘‘S’’ and indicates their mutual compatibility with austenite (a–m–m) for 6M1 and 18R structures using the lattice parameters from Otsuka et al. (1974) and Tokonami et al. (1979). It is seen that the map of rank-one connections is sensitive to slight variations in the lattice parameters. Note that the results shown are independent of the choice of linear or non-linear twinning computation. Also note that although a–m–m interfaces have not been reported experimentally, they necessarily come into the derivation of the macroscopic free energy functions from microstructural considerations. 3. Numerical simulations 3.1. Background The simulations of this section are based on experiments reported in Sˇittner et al. (2003). A tubular specimen measuring 7 mm on the outer diameter, 5 mm on the inner diameter, and 37 mm along the gauge length was manufactured. The specimen head had a screw with a ground flat. It was fixed inside a hollow cylinder grip using an internal screw and a cross bar going through the holes in the grip was attached to the flat part which prohibited rotation of the specimen inside the grips. The material was a single crystal of Cu–14.3Al–4.1Ni [wt.%] with the [0 0 1] cubic austenite crystallographic direction in alignment with the tube axis. In the experimental program two sets of boundary conditions were investigated: (1) applied axial load while torsionally unrestrained, and (2) applied twist while axially unrestrained. The major features of the macroscopic mechanical response for case (1) as reported in Sˇittner et al. (2003) are provided in Fig. 2. An axial ! torsional coupling is clearly evident as the bar rotated 5 under an applied axial extension of 2.5 mm. In contrast, case (2) results were not reported as the experiments failed to display torsional ! axial coupling under the specific thermomechanical conditions of the test. Both experiments were conducted at room temperature (27 C) which, for the specific crystal used, is consistent with a pseudoelastic response for the reported transformation temperature Af ðc01 Þ ¼ 40 C. In Sˇittner et al. (2003) the observed mechanical responses were rationalized based upon the results of basic estimates using anisotropic elasticity of the austenite and the crystallography of cubic to monoclinic transformation in Cu–Al–Ni. Here we approach the issue from the view point of a detailed boundary value problem. Our method is implemented on top of an isotropic elasticity model but does take into account the centrally important issue of anisotropic transformation effects (Sˇittner and Nova´k, 2000). The reported geometry, material properties, and boundary conditions were used to construct various numerical approximations to the experiment. In what follows the model (3.2), the methodology (3.3), and the results (3.4) of the computations are presented. A discussion of the findings is contained in Section 4.
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Fig. 2. Experimental mechanical response of a single crystal shape memory alloy Cu–Al–Ni (b01 ) tube as reported in Sˇittner et al. (2003).
3.2. Constitutive theory The theoretical formulation employed in the computations is based on the work of Govindjee et al. (2003) and Hall and Govindjee (2002) which has been extended to apply to materials for which the energy of mixing is degenerate. Such is the case for cubic–monoclinic materials in general, and the monoclinic-II material studied here in particular. Thus it was necessary to append a secondary condition in the optimization for the variant fractions in order to ensure a unique solution. This was done by choosing the minimum norm solution among all valid solutions, i.e., a singular value decomposition approach. It is noted that this does not change the essential nature of the free energy in the model as a convex approximation. The basic material parameters used in the computations are listed in Table 1. To this is added the shape strains associated with each of the twelve martensitic variants, and the orientation of the parent cubic lattice. Unless indicated otherwise, the [0 0 1] austenite Table 1 Material properties Young’s modulus Poisson’s ratio Density Termal expansion Specific heat Latent heat Reference temperature
E m q a cv kt h0
20.6 GPa 0.25 6448.1 kg/m3 6.5 · 106 1/K 400.0 J/(kg K) 6600 J/kg 169 K
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cubic crystallographic direction was placed in alignment with the tube axis. Due to the assumption of elastic isotropy, the properties given in Table 1 are adjusted to be appropriate for loading in the [0 0 1] direction. Since microstructure formation is also of interest, the chosen ordering of the martensitic variants is provided in Appendix A. As discussed previously (Section 2) there have been conflicting accounts of the lattice structure, and hence, the values of the shape strains. In what follows we perform simulations based on an 18 R unit cell as reported in Bhattacharya and Kohn (1996), a 6M1 unit cell with lattice parameters per Tokonami et al. (1979), and a 6M1 unit cell with lattice parameters per Otsuka et al. (1974). Consistent with the discussion of Section 2 specific values for {a, b, d, } in Cu–Al–Ni near 14 at.% Al and 4 at.% Ni are given in Table 2. 3.3. Methodology Computations reported herein were performed using the finite element analysis package (FEAP) by Taylor (1998). The tube is modeled using standard displacement based eight node brick elements and small deformation theory. All computations were made at 294.26 K. Applied loading included combinations of axial displacements and end rotations using flat tube end boundary conditions consistent with the aforementioned experiment. A convergence study was conducted using various uniform mesh densities consisting of meshes ranging from approximately 4300 degrees of freedom up to 119,000 degrees of freedom. All of the results were qualitatively and quantitatively consistent (e.g. axial reaction variations of less than 2%). Based on this the mesh shown in Fig. 4 was used for all results reported. In order to be consistent with Sˇittner et al. (2003), when reporting shear stresses in line plots we back compute them from the resultant torques by employing the basic linear mechanics torsion solution. When reporting end rotations or displacements we average the values over the nodes on the end of the modeled tube. In summary, the computations are carried out in the standard fashion by satisfying the weak form of the momentum balance equations. The constitutive evaluations are made via full relaxation to the convexified free energy as fully described in Govindjee et al. (2003) and Hall and Govindjee (2002). 3.4. Simulation results The series of numerical simulations performed are broken into four categories based on the boundary and initial conditions employed. The first set of simulations, as reported in Section 3.4.1, apply both an axial extension path and an applied twist path similar to that recorded in the experiments of Sˇittner et al. (2003). The second set of simulations apply an
Table 2 Shape strain parameters
a b d
18R Bhattacharya and Kohn (1996)
6M1 Tokonami et al. (1979)
6M1 Otsuka et al. (1974)
0.0442 0.0822 0.0160 0.0600
0.0520 0.0867 0.0202 0.0516
0.0436 0.0822 0.0170 0.0508
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axial extension while enforcing torque free boundary conditions: this closely models the boundary conditions of the experiment. This second set of runs are reported in Section 3.4.2. The third set of simulations (see Section 3.4.3) are similar to the second, only with the addition of various initial conditions such as imperfections in the specimen, misalignment, and residual stresses. The final set of boundary conditions allowed for an applied twist angle with zero axial stress (see Section 3.4.4). The main points of the results are compared and discussed in Section 4. 3.4.1. Applied extension, applied twist To gain insight into the observed behavior, the first set of simulations sought to examine predictions made by the model along the specific displacement path taken by the experimental specimen. This is of interest since by applying the experimentally observed twist in addition to the axial extension, one can examine the torque response and the variant formation history of the model for a physically based motion. As described previously, one of the austenite cubic basis vectors was aligned with the tube axis; this will be referred to as perfect alignment. The experimentally recorded displacement and rotation path was applied to the end of the tube (as indicated in Fig. 3 lower left) and the mechanical response was computed assuming shape strains as per 18R of Table 2. The resulting response is given in Fig. 3. The essential feature is the low magnitude of the shear stress indicating a very slight torque is required to induce the twist under the applied axial extension. By simple elastic isotropy one would predict shear stresses in excess of 45 MPa at this level of rotation.
100
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300 0.03
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0.02 axial strain 0.01 shear strain 0
100 200 time (sec)
strain axial stress (MPa)
stress (MPa)
0 350 300 250 200 150 100 50 0
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0
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4
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350 300 250 200 150 100 50 0
3 2 1 0
15 10 5 0
0
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0
1
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Fig. 3. Perfectly aligned, 18R, applied extension, applied twist mechanical response.
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The predicted volume average microstructure formation was dominated by only four martensitic variants. The spatial location of these variants were centered on the primary axes of austenite which coincides with the global axes used in the problem. This results in variants appearing on the tube faces as indicated in Table 3. With regard to variant formation, two items which stand out in the calculation are the following. First, only the variants with ea33 ¼ a þ appear. Second, the distribution of these four variants is consistent with the torsionally induced shear as is apparent from Table 3. The pattern of the distribution at the maximum load can be seen in Figs. 4 and 5 which indicate the distribution of variants 11 and 8 respectively. Symmetric patterns also exist on the opposite faces not shown, only with variants 12 and 7 appearing rather than 11 and 8. The above calculation was repeated using the shape strains consistent with Tokonami et al. (1979) 6M1 from Table 2. The results were similar, although it is noteworthy that the resultant torque was even lower (hence more accurate) than the previous run using the 18R parameters. This can be seen in the mechanical response shown in Fig. 6. Several runs of this type were also made with the axis of austenite shifted by 1 off the tube axis. It was found in every case that the amount of torque required to twist the tube would increase, indicating that the minimum was achieved with perfect alignment. Evidence of these observations are found in Figs. 7 and 8 which are for misaligned 18R and 6M1 runs respectively. In these runs it was also noted that the 90 periodicity of the maximum variant volume fraction (maxa{na}) around the tube was lost in favor of
Table 3 Variant distribution on the tube, where the global 3 axis is aligned with the tube and the global 1 and 2 axes coincide with austenite basis vectors Variant
Face/axis
eaij
7 8 11 12
2 +2 +1 1
a¼7 e13 ¼ þd a¼8 e13 ¼ d a¼11 e23 ¼ þd a¼12 ¼ d e23
Variant 11 Fraction 0.00E-00 3.32E-02 6.63E-02 9.95E-02 1.33E-01 1.66E-01 1.99E-01 2 1
Fig. 4. Variant a = 11 distribution on the +1 face for the loading indicated in Fig. 3.
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Variant 8 Fraction 0.00E-00 3.20E-02 6.40E-02 9.60E-02 1.28E-01 1.60E-01 1.92E-01 1 2
0 350 300 250 200 150 100 50 0 0
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200
300 0.03
axial stress
0.02 axial strain 0.01 shear strain 100 200 time (sec)
strain axial stress (MPa)
stress (MPa)
Fig. 5. Variant a = 8 distribution on the +2 face for the loading indicated in Fig. 3.
shear stress 0 300
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0
0.01 0.02 axial strain
0.03
20 shear stress (MPa)
torsion angle (deg)
5
350 300 250 200 150 100 50 0
0.5 1 1.5 2 axial displacement (m)
2.5
15 10 5 0
0
1
2 3 4 torsion angle (deg)
5
Fig. 6. Perfectly aligned, 6M1 (Tokonami et al., 1979), applied extension, applied twist mechanical response.
180 periodicity in which two of the faces (e.g. ±1 axes) had higher volume fractions than the other two (e.g. ±2 axes). 3.4.2. Applied extension, torque free The next category of simulations attempted to model the exact conditions of the experiment as axially fixed and torsionally free. The first set of results obtained was for a perfectly aligned 18R material. The mechanical response was similar to that of the previous runs with the exception that the model did not reproduce the physically observed twist.
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axial stress (MPa)
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5
3 2 1 0
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15 10 5 0
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5
Fig. 7. Austenite 3 axis at a 1 rotation about [1 1 0], 18R shape strains, applied extension, applied twist mechanical response.
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Fig. 8. Austenite 3 axis at a 1 rotation about [0 1 0], 6M1 (Tokonami et al., 1979) shape strains, applied extension, applied twist mechanical response.
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Variant 12 Fraction 0.00E-00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E-00
Fig. 9. Variant a = 12 distribution for a perfectly aligned, 18R, applied axial extension, torque free loading run.
The reason for this is apparent from the distribution of volume fractions as seen in Fig. 9. Although only the expected variants a 2 {7, 8, 11, 12} appear, their pattern is perfectly diffuse radially around the tube. Thus, in the absence of some impetus, the model selects an equal distribution of the four variants as the (non-unique) minimal energy solution. In addition to the aforementioned torque free simulations, numerous runs were made with misalignments up to 1 between the tube axis and a major axis of austenite. It was found that the axial stress response was fairly insensitive to these perturbations, although the distribution of variants was at times drastically altered. Such behavior indicates that a slight perturbation of the system, either through geometry, alignment, thermal distribution, or some other factor may produce the desired twist numerically. The next section investigates the idea that initial or residual stresses in the system may be contributing to the observed behavior. 3.4.3. The effect of initial conditions Although the simulations of Section 3.4.1 correctly predicts the stress response under the applied displacement history, in Section 3.4.2 it was found that a pure applied axial extension did not spontaneously induce twisting. Working under the assumption that the particular variant formation observed requires some form of perturbation to form, both initial and residual stresses were examined as a potential source. Initial stress: As a first attempt, a small biasing initial stress distribution was placed in the tube. The distribution was specified by the function szh ¼ sozh R; sozh
ð5Þ sozh ,
is a constant and R is the radial position. By choosing a small value for this where has the effect of providing a small biasing (constant) torque. Although this construction is artificial, it allows one to investigate the perturbation required to induce the observed axial ! torsional coupling. Results from simulations conducted using initial stress values of szh 6 2.5 MPa produced the response provided in Fig. 10. The mechanical response is remarkably similar to the experimental result, although the predicted end rotation is higher than the observations. The martensitic variants and their spatial variation were similar to those seen
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0 350
torsion angle (deg)
8 6 4 2 0
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2
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Fig. 10. Results from simulation under applied axial displacement, unrestrained torsional boundary conditions, and small biasing initial stress.
in the simulations of Section 3.4.1. These results suggest that even a slight bias can induce the cooperative deformation mode suggested in Sˇittner et al. (2003) under the very specific conditions of the test. However, given the artificial nature of the bias introduced by the chosen initial stress, a residual stress was then considered as discussed next. Residual stress: A self equilibriating torque free initial residual stress field was implemented in the numerical simulation. The distribution of the stress followed i sizh szh Re szh ¼ R ; ð6Þ Ro Re Ro Re where Re ¼
3 R4o R4i : 4 R3o R3i
ð7Þ
Ro is the outer radius of the tube, Ri is the inner radius of the tube, and sizh is the residual shear stress at the inner surface of the tube. Setting sizh ¼ 40 MPa and again using boundary conditions from the experiment produced the results found in Fig. 11 for the 6M1 (Tokonami et al., 1979) material of Table 2. The qualitative response is very close to that observed experimentally, and by changing the strength of the residual stress field (sizh ), it was possible to quantitatively match the experimental data as well (the best case scenario is not shown here, as there is little justification for such a back-calculation). Of note is that again the simulations reproduce the observed cooperative deformation mode only when a favorable initial condition exists. The variants appearing in this computation are the same as in Section 3.4.1. However, their spatial distribution is quite different; see Fig. 12 for the distribution of variant a = 12. At issue here is the large degree of degeneracy in the model’s free energy of mixing for monoclinic materials. Such models are capable of faithfully resolving
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stress (MPa)
400
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stress
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200 strain
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torsion angle (deg)
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100
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150 200 time (sec)
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strain
176
0
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Fig. 11. Applied axial extension simulation with torque-free residual shear stress. The material is consistent with Tokonami lattice parameters and the 6M1 unit cell.
Variant 12 Fraction 0.00E-00 5.20E-02 1.04E-01 1.56E-01 2.08E-01 2.60E-01 3.12E-01 1 2 Fig. 12. Spatial distribution of the volume fractions for martensitic variant a = 12 under axial extension and unrestrained torsional motion. The simulation includes a torque free residual stress and employs the Tokonami lattice parameters with a 6M1 transformation description.
macroscopic measures of behavior but are sensitive to initial conditions and seemingly minor geometric features. What matters to the model is only that the spatial average of the transformation strain is correct. Returning to the discussion of Section 2, the effect of a small variations in the lattice parameters can be observed numerically. To demonstrate this, the same simulation was repeated using the 6M1 Otsuka et al. (1974) material from Table 2 (see also Section 2.3.3). The results are shown in Fig. 13 and although qualitatively similar they differ by approximately 30 MPa (out of 300 MPa total) in the stress response and just under 1 (out of 5 total) in the rotation – a rather large difference considering the small differences in the shape strains.
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50
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300 350 0.06
stress
0.04
200 strain
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50
100
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150 200 time (sec)
250
1 1.5 2 axial displacement (mm)
300 350
strain
300
torsion angle (deg)
stress (MPa)
0 400
177
0
2.5
Fig. 13. Applied axial extension simulation with torque-free residual shear stress. The material is consistent with Otsuka lattice parameters and the 6M1 unit cell.
3.4.4. Applied twist, axially free The second set of boundary conditions explored experimentally in Sˇittner et al. (2003) consisted of an applied twist while allowing unrestrained axial motion. Under these conditions no transformation was observed for shear stresses approaching 400 MPa; an initial theoretical calculation by the aforementioned authors predicted transformation to begin at approximately 412 MPa. A numerical simulation of the problem was conducted using the 6M1 structure with parameters from Tokonami et al. (1979) and the material properties listed in Table 1; note that due to the isotropic elastic modulus assumption the associated shear modulus is drastically underestimated. Nevertheless the simulation predicted deviation from linearity in the range 450–500 MPa and with very little austenite–martensite transformation in the tube. However, by continuing the application of twist out to shear strains approaching 12% torsional ! axial coupling became increasingly apparent with a maximum axial displacement of (0.34 mm) at a physically unrealistic shear stress of over 600 MPa. 4. Concluding remarks Several issues regarding computations in monoclinic-II systems have been investigated. On the theoretical side, it was demonstrated that current ambiguities in the literature regarding measured lattice parameters are sufficiently broad to render nontrivial changes in the free energy landscape required for macroscopic modeling. Simultaneously, the debate between a 18R unit cell with stacking faults and a 6M1 unit cell has similar implications for macroscopic modeling. Most of the evidence we have examined supports the 6M1 viewpoint. On the numerical side, an investigation of the experimental observation of axial ! torsion coupling was conducted by way of finite element experiments. The following points summarize the main observations drawn from this part of the work.
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For the specific material and temperature state examined, coupling under applied twist, axially unrestrained motion occurs only at physically unrealistic levels. This is in agreement with the experiment, though it is noted that this mode of deformation could be observable in a suitably designed experiment. The computations produce near zero torque response when driven both axially and torsionally. The quantitative response seems quite reasonable, and the volume average variant distributions are consistent with what one would expect based on an examination of the shape strains listed in Section 3.2. Minimal shear stress is obtained in the perfectly aligned case which supports the notion, introduced in Sˇittner et al. (2003), of the special nature of conditions required for observation of a cooperative (coupled) deformation mode. In the axial extension, torque free simulations the axial stress response is not particularly sensitive to perturbations in the alignment of the austenite lattice. In contrast, the variant formation pattern is extremely sensitive to such perturbations. The model does not spontaneously exhibit coupling under applied extension, torsionally free conditions, perfectly aligned conditions. This case produces the expected four variants in equal volume fractions but in a diffuse pattern which results in zero twist. However, with the inclusion of a bias to seed the transformation, the numerics are in good agreement with experimental data. The final remark reveals an important point on the numerical modeling of unexpected coupled behaviors in shape memory alloys: to reproduce such spontaneous behavior, standard material parameters are insufficient and must be supplemented with the presently not well understood conditions which bias or seed the transformation. For example, theory predicts that spontaneous right-hand twist is equally likely with left hand twist or zero twist for that matter. This contradicts the observation of specimens which clearly exhibit a reproducible (in the experiments referenced) bias to twist rightward 4 times and leftward once under axial loading. The numerics suggest that either small torsional loads or residual stresses would explain the initiation of such spontaneous coupling, though further experimental investigation is required to clarify the issue. The point also highlights the need for the exercise of extreme care in the study of highly symmetric experimental modes of deformation. Acknowledgements P. Sittner would like to acknowledge support from Marie-Curie RTN Multimat (Contract No. MRTN-CT-2004-505226) and project A1048107 of Grant Agency of the Academy of Sciences of the Czech Republic. Appendix A. Variant ordering The constitutive theory is based on the lattice correspondence variants which arise in the transition from austenite to martensite. They enter into the computations through the associated shape or Bain strains. In the specific case of a cubic to monoclinic-II transition, there are twelve martensitic variants. To identify these variants, the following ordering has been adopted.
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2
aþ
6 ea¼1 ¼ 4 d 0 2 a 6 a¼3 e ¼4 d
0
aþ 6 ¼4 0 d
2
0 b 0
a 0 6 ea¼7 ¼ 4 0 b d 0 2 b 0 6 a¼9 e ¼ 40 aþ
ea¼11
0 2 b 6 ¼ 40 0
0
b
2
3
7 a 0 5; 0 b 3 d 0 7 a þ 0 5;
0
2
ea¼5
d
6 ea¼2 ¼ 4 d 0 2 a 6 a¼4 e ¼ 4 d
3
d 7 0 5; a 3 d 7 0 5;
aþ
2
ea¼6
0
aþ 6 ¼4 0 2
d
d a 0 d aþ 0
0
179
3
7 0 5; b 3 0 7 0 5; b
3 d 7 0 5; 0 a 3 0 d 7 b 0 5; 0 b
a 6 ea¼8 ¼ 4 0 aþ d 0 a þ 3 2 3 0 b 0 0 6 7 7 d 5; ea¼10 ¼ 4 0 a þ d 5; d a 0 d a 3 2 3 0 0 b 0 0 6 7 7 a d 5; ea¼12 ¼ 4 0 a d 5; d aþ 0 d a þ
Appendix B. Transformation stretches B.1. Application The transformation stretches discussed in Section 2 are included here for completeness; the ordering of the results follows that of the text. B.1.1. Cu–Zn–Ga Using data from Saburi et al. (1976) at 20.4 at.% Zn and 12.5 at.% Ga the 18R lattice ˚ , b = 5.33 A ˚ , c = 38.22 A ˚ and b ¼ 88:33 . The cubic lattice parameters are: a = 4.40 A ˚ parameter is given as a0 = 5.86 A which was measured on a slightly different alloy (21.8 at.% Zn and 11.8 at.% Ga). Using g = 0.0775 the resulting transformation stretch is given by 2 3 0:9096 0 0 6 7 U ¼4 0 1:0035 0:0172 5 : ðB:1Þ 0
0:0172 1:0843
A
If one uses a consistent value for g then we have 2 3 0:9096 0 0 6 7 U ¼4 0 1:0281 0:0185 5 : 0
0:0185 1:0584
A
ðB:2Þ
180
G.J. Hall et al. / International Journal of Plasticity 23 (2007) 161–182
If one computes the transformation strain using the 6 M1 structure then one finds 2 6 U¼4
0:9096
0
0 0
1:0888 0:0169
0
3
7 0:0169 5 : 0:9994 A
ðB:3Þ
B.1.2. Cu–Zn–Al This alloy was studied by Pitteri and Zanzotto (1997) at 15 at.% Zn and 17 at.% Al. The ˚ , a = 4.553 A ˚ , b = 5.452 A ˚ , c = 38.977 A ˚ and reported lattice parameters were a0 = 5.996 A b = 87.5. Using the 18R structure and a consistent shear value gives a transformation stretch of 2 3 0:9093 0 0 6 7 U¼4 0 1:0246 0:0262 5 : ðB:4Þ 0 0:0262 1:0703 A If we compute the transformation stretch from the 6M1 structure we find 2 3 0:9093 0 0 6 7 U¼4 0 1:0860 0:0254 5 : 0 0:0254
1:0097
ðB:5Þ
A
Slightly different results are found using the reported values in Bhattacharya and Kohn (1996) to generate a transformation stretch of 2 3 1:0866 0:0249 0 6 7 U ¼ 4 0:0249 1:0100 0 5 : ðB:6Þ 0 0 0:9093 A B.1.3. Cu–Al–Ni This alloy was studied in Otsuka et al. (1974) at 14.2 wt.% Al and 4.3 wt.% Ni and by Tokonami et al. (1979) at 14.0 wt.% Al and 4.2 wt.% Ni. The lattice parameters from these ˚ , b = 5.356 A ˚, two source are different. From Otsuka et al. (1974) we have a = 4.382 A ˚ c = 38.00 A; other parameters are not reported. From Tokonami et al. (1979) we have ˚ , b = 5.330 A ˚ , c = 38.19 A ˚ , b = 89.0. a = 4.430 A Otsuka et al. (1974) lattice parameters: Since the lattice parameters from Otsuka et al. (1974) are incomplete we fill the missing ones using those in Hane (1999). Using the 18R structure and a consistent shear value gives a transformation stretch of 2 3 0:9178 0 0 6 7 U¼4 0 1:0334 0:0194 5 : ðB:7Þ 0
0:0194
1:0516
A
If we compute the transformation stretch from the 6M1 structure we find a more nearly compatible stretch of
G.J. Hall et al. / International Journal of Plasticity 23 (2007) 161–182
2 6 U ¼4
0:9178 0 0
0
0
181
3
7 1:0945 0:0170 5 : 0:0170 0:9928 A
ðB:8Þ
The reported values in Bhattacharya and Kohn (1996) generate a transformation stretch of 2 3 1:1042 0:0160 0 6 7 U ¼ 4 0:0160 0:9842 0 5 : ðB:9Þ 0 0 0:9178 A ˚ as it Tokonami et al. (1979) lattice parameters: In this section we assume a0 = 5.838 A was not reported. Using the 18 R structure and a consistent shear value gives a transformation stretch of 2 3 0:9133 0 0 6 7 U ¼4 0 1:0417 0:0226 5 : ðB:10Þ 0
0:0226 1:0600
A
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