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Acta Materialia 59 (2011) 5924–5937 www.elsevier.com/locate/actamat
Neutron diffraction studies and multivariant simulations of shape memory alloys: Concurrent verification of texture development and mechanical response predictions Xiujie Gao a, Aaron Stebner b, Donald W. Brown c, L. Catherine Brinson b,d,⇑ a
General Motors Research and Development, Vehicle Development Research Laboratory, Warren, MI 48090, USA b Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA c Los Alamos National Laboratory, Mail Stop H805, Los Alamos, NM 87545, USA d Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA Received 19 October 2010; received in revised form 1 June 2011; accepted 1 June 2011 Available online 29 June 2011
Abstract A new methodology has been developed to compare texture development and macroscopic response predictions of micromechanical shape memory alloy models directly with diffraction data. Using these methods empirical neutron diffraction data from sequential, multiaxial, compressive loading schemes were compared with calculations from the simplified multivariant model. Through this process the ability of a multivariant model to predict both the texture development and mechanical response trends during the creation of complex stress states in martensitic polycrystalline NiTi was demonstrated for the first time. The result made it evident that a multivariant model is more completely validated through simultaneous verification of micro and macroscale predictions as opposed to only verifying macroscale predictions, as was the previous state of the art. The new methodology presented here provides the means to perform this multiscale verification for any multivariant model. It is also shown that in combining a multivariant model with diffraction techniques a new tool for examining the plausibility of variant growth and depletion mechanisms has been created. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Shape memory alloy; Texture evolution; Multivariant modeling; Micromechanical modeling; Diffraction
1. Introduction As the significance of texture evolution in shape memory behavior becomes more evident (as briefly reviewed in Stebner et al. [1]) and the number of shape memory alloy (SMA) compositions and applications that use them continue to increase, so does the desire for predictive models of shape memory behavior. Implementation of a phenomenological model that is robust enough to ⇑ Corresponding author at: Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA. Tel.: +1 847 467 2347. E-mail address:
[email protected] (L.C. Brinson).
replicate the entirety of macroscopic SMA responses for an arbitrary alloy system has proven to be a very challenging feat. Indeed, even formulating a macroscale model capable of replicating the entirety of a single NiTi composition’s behavior without intermediate recalibration and the pre-existence of an extensive thermomechanical database has not been achieved, although great advances towards this goal are being made [2–5]. Many of the behavioral phenomena that lead to shape memory qualities are based on changes in microstructure that differ from the mechanisms of traditional elastic– plastic metals, and explicit tracking of the microstructure is not accounted for in macroscopic, phenomenological formulations. In response to the desire for predictive
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.06.001
X. Gao et al. / Acta Materialia 59 (2011) 5924–5937
capabilities of shape memory-related phenomena driven by these non-traditional microstructural mechanisms micromechanical modeling of SMA has continued to thrive as an active area of research [6–23]. Evolving these models such that only a few fundamental experiments are needed to reliably predict a new material’s behavior would greatly reduce development times and costs, especially considering that many new generation SMA are composed of significant quantities of precious metals [24–26]. Critical to the performance of these models is their ability to accurately simulate martensite variant selection processes. Thus, in the ensuing article simulations are performed for isothermal (room temperature) deformation of martensite and the predictions of variant formation are compared with texture measurements observed in neutron diffraction experiments [1]. Because of their ability to explicitly capture relationships between micro and macroscale behaviors, micromechanics models may be used as a development tool in three unique ways. Microstructure optimization. 1. Microstructures may be numerically created: Through the subsequent simulation of thermomechanical loadings, behavior correlating with these microstructures become known. In this way many more microstructures may be evaluated than is practical through empirical observation alone, and finding an optimal and physically plausible microstructure for a desired behavior (e.g. transformation temperature, transformation strain, mechanical stability, etc.) becomes a numerical exercise. 2. Once a microstructure is known to be desirable, numerous thermo-mechanical paths thought to create the desired microstructure may be explored and evaluated. Thus the least expensive and most effective treatments may be evaluated. Multiscale modeling: From a hierarchical perspective, micromechanical models may be used to generate the material parameters and macroscopic responses needed to calibrate and/or benchmark phenomenological continuum formulations. Also, as computational resources continue to become more capable and accessible, the idea of an interactive micromechanical–phenomenological tool becomes more realistic. By automating the recalibration process needed by most system level models such a tool would greatly advance the ability to simulate SMA systems and components, especially those subjected to a variety of operating conditions over the course of their lifetimes. Physical insight: It is often difficult to empirically decouple and quantify the contribution of a known physical mechanism to a specific response, even with the great advances made in experimental techniques and instrumentation in recent years. Micromechanics modeling
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provides a means to investigate and verify mechanistic relations by decoupling the contribution of each mechanism to the overall behavior through calculation. The ability to simulate the interaction of multiple correspondence variants (CV) leads to the term “multivariant” modeling [12]. While the focus of this work is not to provide a thorough review of multivariant SMA models, it is relevant to briefly examine their evolution as it pertains to the ability to replicate complete textures of polycrystalline NiTi systems, and verification of those texture predictions. Early efforts focused predominantly on verifying macroscopic transformation strain prediction with some explanation of twin/slip systems; a few researchers calibrated initial textures from empirical data and subsequently verified macroscopic predictions of tension– compression asymmetry [27,28] and recovery strain dependence on texture [9]. More recently, fuller repertoires of multi-axial shape memory behavior have been captured for several alloy systems, including NiTi, but largely in situ texture evolution predictions remain unverified [6–15,17–19,21–23], with two exceptions. Nova´k and Sˇittner were the first to examine complete micro and macroscale predictions of a multivariant model during multiple step, thermo-mechanical loading processes [16,29]. However, their model formulation deviates from the majority of multivariant approaches, in that stress and strain fields within individual grains are not explicitly calculated [16]. Qiu et al. have simultaneously considered micro and macroscale predictions of a self-consistent model to examine the thermo-elastic behavior, hence elucidating thermal expansion coefficients [30] and anisotropic elastic constants [31] for polycrystalline NiTi. In its current state of development, though, this model lacks the ability to simulate mechanisms that give rise to shape memory and superelastic behaviors of NiTi systems. To date, to the best of the authors’ knowledge, a methodology to concurrently verify texture development and mechanical response calculations has not yet been presented for SMA multivariant models that predict macroscopic behavior based on intragranular crystallographic developments. This absence is significant in that the main benefits of such models, outlined above, rely on the ability to accurately replicate and interpret texture development as it relates to macroscopic thermo-mechanical behavior. A data set collected with the intent of being used for such a verification process was presented in Stebner et al. [1]. In that work polycrystalline NiTi parallelepiped specimens were sequentially compressed along their axial and transverse directions while in their martensitic room temperature state such that texture development during the creation of multiaxial stress states could be observed using the neutron diffraction instruments SMARTS [32] and HIPPO [33] at the Los Alamos Neutron Scattering Center (LANSCE). In this work methodologies for verifying in situ texture evolution predictions are developed. These
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are then applied through comparison of data from simulations performed using the simplified multivariant model (SMM) [18] and the aforementioned neutron diffraction experiments [1]. Through this process insights are gained that were not possible to achieve solely through macroscopic verification processes. Specifically, the SMM is found to correctly predict texture evolution as complex stress states are imparted upon martensitic NiTi specimens. Furthermore, the model achieves this agreement through prediction of growth and depletion of 24 (0.7206, 1, 1)[0 1 1] type-II twin habit plane variants (HPV). Thus for the first time, a multivariant model is verified in its ability to simulate the plausibility of variant systems and their evolutions that give rise to the macroscopic behaviors of martensitic, polycrystalline NiTi. Through discussion of these results the significance of concurrently verifying microscopic and macroscopic calculations of a mainstream multivariant model is evident. While in this work these methods are demonstrated using the SMM [18] and neutron diffraction experiments [1], the new methodology may be applied using any multivariant SMA model that calculates volume fraction evolutions of variants and any diffraction data set. 2. Simulation and verification methods 2.1. The simplified multivariant model The SMM [18] considers 24 HPV for a given twinning system and allows both transformation and reorientation inside a single crystal/grain. The fundamental system of equations solved for each single crystal/grain is: F d f_ n ¼ BðT T 0 Þ þ Rij enij þ kn k0 F C ;
if f_ n 6 0 ð1Þ
where superscript n represents the nth HPV, Fd is a constant during energy dissipation, fn is the volume fraction of the nth HPV, B is a linearized coefficient for the chemical free energy difference near the thermodynamic equilibrium temperature T0 of austenite and martensite, Rij is the average stress in a grain, enij is the transformation strain of the nth HPV, kn represents a boundary force applied to the nth HPV to keep its volume fraction within physical bounds (0 6 fn 6 1), k0 represents a boundary force applied to all HPV to keep the total volume fraction within physical bounds (0 6 R fn 6 1), and FC is the friction force or energy barrier if variant n is actively transforming. Note kn or k0 only appears when the volume fraction of the nth HPV or the total volume fraction is very close to 0 or 1. To obtain a polycrystalline response using the model the system of Eq. (1) is solved for each grain and then the self-consistent micromechanics method [34,35] is used to consider the interaction among the grains. Thus for a given change in temperature or applied macroscopic load the volume fractions of all 24 HPV and the corresponding transformation strain are able to be calculated for each grain and then assembled to provide the overall stress–strain
response. A description of the relevant algorithms can be found in the appendix of Gao and Brinson [18]. Note that this process solves the response of a material point in which grain size, variations in grain shape, grain position relative to other grains, precipitate structures, dislocation motion, and effects of other material imperfections are ignored. 2.2. Simulation details To run the SMM code discussed in Gao and Brinson [18] one needs to input the applied thermo-mechanical loading, the transformation temperatures, the elastic constants for both austenite and martensite, the rotation matrices for the austenite lattice of each grain with respect to the sample coordinates (the texture), the habit plane normal, the macroscopic shear direction and magnitude, the linearized coefficient (B) for the chemical free energy difference between austenite and martensite near the equilibrium temperature (T0), and the friction constant FC for energy dissipation. Values of these parameters used in the simulations are shown in Table 1. Note (column 2) that the assumption of an isotropic, equivalent austenite– martensite elastic response was made and elastic parameters were adopted from Otsuka and Wayman [36], in accordance with the state of understanding at the time of model development [8,13,17,19]. Recently, however, new light has been shed upon the elastic anisotropy of martensite and other researchers have implemented processes for modeling the evolution of this anisotropy [31,37,38]. In future model development these advancements will be analogously implemented. For 54.8 wt.% NiTi specimens, T0 = ½(Mf + Af) = 340 K, and the experiments were carried out at room temperature (298 K) [1]. The (0.7206, 1, 1)[0 1 1] type-II twin system of NiTi was employed to define the HPV, as it is commonly deemed the most prevalent within the material and has been used to describe martensitic NiTi behavior by other researchers [19,23,27,36]. The reader is referred to Gao and Brinson [18] for a further explanation of the simulation parameters and Gall and Sehitoglu [27], Liu et al. [39], Saburi and Nenno [40], and Jin and Weng [41] for further discussion of NiTi twinning systems. To initialize specimen texture in the simulations a preprocessing code assigns appropriate orientation angles for the austenite lattice in each grain relative to the sample coordinate system (see Fig. 2 for a schematic of the
Table 1 Parameters for SMM simulations. Theoretically calculated [46]
Estimated elastic constants [36]
Derived from the properties of 54.8 wt.% NiTi
^n ¼ ½0:2152; 0:4044; 0:8889 ^ ¼ ½0:7574; 0:4874; 0:4345 m g = 0.13078 Volume change = 0.34%
l = 32 GPa m = 0.3
B = 0.50 MPa K1 FC = 10.0 MPa
X. Gao et al. / Acta Materialia 59 (2011) 5924–5937
coordinate systems). Previously a perfect fiber symmetry had been adopted for the austenite in initializing simulations (i.e. [1 0 0]B2 coincident with the 1-direction of the sample and [0 1 0]B2 or [0 0 1]B2 randomly distributed about the 1-direction) [42]. However, preliminary work showed that the predicted textures were much sharper than those measured in neutron diffraction experiments, due both to actual small grain misalignment within the
2-direction 1-direction 3-direction Fig. 1. Oblique view of the specimen and its coordinates.
ys
yg1
xg12
2
g8 g12
yg8
HPV1
xg8 HPV7 HPV8
xs Fig. 2. Schematic showing a polycrystalline sample containing grains of different orientations. The degree of randomness of the grain orientations is defined through comparison of the grain coordinate system (xg#, yg#) with the specimen coordinates (xs, ys). During stress-free, temperatureinduced transformation, triangular, self-accommodating martensite morphology forms (composed of three HPVs such as 1, 7, and 8 shown in grain 8).
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specimens and the resolution of the experimental techniques [42]. To more realistically replicate the empirically observed self-accommodated martensite texture (Fig. 3) the initial B2 fiber texture was blurred by enforcing a Gaussian distribution of [1 0 0]B2 with a standard deviation of 11° about the 1-axis for the reported simulations (Fig. 5). Starting from this austenite texture the specimen is then cooled stress-free in the simulation and transformed into self-accommodated martensite (i.e. the 12 CV in each grain are of equal volume fraction), and the resulting texture is very near that which was empirically observed (Fig. 3) [1]. While direct measurement of the initial austenite texture was not made in the empirical data set, use of the [1 0 0]B2 fiber symmetry in these simulations was deduced from the observed initial martensite texture in the light of data on body-centered cubic metals [43] and NiTi ribbons [44], as documented in Stebner et al. [1]. Following calculation of the initial texture simulations of the experimental loading paths described in Stebner et al. [1] were performed using the SMM. Again, these paths consisted of compressing virgin specimens along their axial (1-) and transverse (2- and 3-) directions; transversely compressed specimens were subsequently compressed along their axial (1-) direction. The macroscopic responses of these simulations were directly calculated, and texture developments within the simulated specimens were transformed through the calculations described below such that they could be directly compared with the empirical results. 2.3. Texture calculation overview For the NiTi material under studyin its martensitic state each HPV is composed of two CV that are in a twinned relationship. Each CV represents a single crystallographic orientation and, in experiments, neutrons will diffract on certain planes of these single crystals. Therefore, by properly analyzing the output of a multivariant model, i.e. correlating each CV with the grain orientation relative
Fig. 3. Empirical and simulated virgin specimen pole figures. The specimen coordinate system for all pole figures throughout the paper is shown in the measured (1 0 0) plot.
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Start Relative amount of minor in twin-pair, 0.27102 Number of stored rotation matrices for a grain: 2*24 (For all major & minor CVs in all HPVs) Number of stored plane normal for a grain: 7*2*24 (For all desired poles, major & minor CVs in all HPVs)
Lattice parameters of austenite and martensite Rotation matrices between major or minor CV lattice of all HPVs and austenite lattice (2*24)
Number of stored rotation matrices for a sample: 30*2*24 (For all gains, major & minor CVs in all HPVs) Number of stored plane normal for a sample: 30*7*2*24 (For all gains, desired poles, major & minor CVs in all HPVs)
Read in one step of simulation result (number of volume fractions of CV: 1440)
Number of desired poles (e.g., 7) Rotation matrices between all grains and sample (e.g., 30 grains)
Multivariant results (many loading steps)
Write out a texture file if this step is desired: (1440 lines) One line for one CV volume fraction: 3 Euler angles calculated from the stored rotation matrix & corresponding volume fraction
For each CV volume fraction (1440) Peak intensity: if after-rotated normal // diffraction vector, add it up If pole figure is desired for this step: If after-rotated normal // any grid point of pole figure, add it up
Write out pole figure files for all poles for desired step
Write out peak intensity file for all poles Fig. 4. Procedure to obtain peak intensities and pole figures from simulation results. The numbers used in the example described in Section 2.3 are shown to make the correlation between the descriptions and the numerical quantities clearer.
2
3
Fig. 5. The blurred fiber texture used to represent the initial austenite texture. The [1 0 0] axis of the 100 grains has a Gaussian distribution around the 1-direction of the specimen with a standard deviation of 11°. The [0 1 0] or [0 0 1] direction is randomly distributed around the rim as shown.
to the sample coordinate system, we can obtain texture development and compare results directly with the neutron
scattering data. A post-processing code was written to achieve this analysis starting with the output of the SMM calculations. Note that while the deviations induced by the monoclinic lattice of the martensite are considered, the few degrees of difference between the real direction correspondence and that derived from Bain distortion are ignored in the pole figure generating process. Using the known CV pair, their relative volume fractions for each HPV, and lattice correspondences [36,40,41] (discussed in more detail in Section 2.4), sequential rotation calculations were made to determine the evolving texture for the grid points in the pole figure of the desired set of planes. For example, to determine the (1 0 0) texture the normal of the (1 0 0) plane of the major or minor CV in each HPV was transformed, first with respect to the grain and then with respect to the specimen coordinates. If the transformed normal was aligned with a grid point of the pole figure the volume fraction of that CV was added to the intensity value of the (1 0 0) plane; otherwise no contribution from that CV was considered. This calculation was performed for every HPV volume fraction in all grains to find a total volume fraction contribution, which was then normalized with respect to random texture using the equation:
X. Gao et al. / Acta Materialia 59 (2011) 5924–5937
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Table 2 HPV number, the corresponding CV pairs, and the rotation matrices to transform a vector inside the CV lattice to a relative vector in the austenite lattice. HPV no.
CV pair
Rotation matrix
Here
[47]
[48]
Here
[47,48]
Major CV austenite
Minor CV austenite
1
1(+)
3(+)
5, 7
3, 4
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
2
1()
4(+)
7, 5
4, 3
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
3
10 (+)
6()
11, 10
6, 50
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
4
10 ()
50 (+)
10, 11
50 , 6
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
5
2(+)
60 (+)
12, 9
60 , 5
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
6
2()
5()
9, 12
5, 60
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
7
20 (+)
30 ()
6, 8
30 , 40
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
0, 0.7071, –0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
8
20 ()
40 ()
8, 6
40 , 30
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
9
3(+)
5(+)
9, 11
5, 6
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
10
3()
6(+)
11, 9
6, 5
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
11
30 (+)
2()
3, 2
2, 10
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
12
30 ()
10 (+)
2, 3
10 , 2
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
13
4(+)
20 (+)
4, 1
20 , 1
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
14
4()
1()
1, 4
1, 20
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
15
40 (+)
50 ()
10, 12
50 , 60
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
16
40 ()
60 ()
12, 10
60 , 50
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
0, 0.7071, 0.7071 0, 0.7071, 0.7071 1, 0.0000, 0.0000
17
5(+)
1(+)
1, 3
1, 2
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
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Table 2 (continued) HPV no.
CV pair
Rotation matrix
Here
[47]
[48]
Here
[47,48]
Major CV austenite
Minor CV austenite
18
5()
2(+)
3, 1
2, 1
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
19
50 (+)
4()
7, 6
4, 30
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
20
50 ()
30 (+)
6, 7
30 , 4
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
21
6(+)
40 (+)
8, 5
40 , 3
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
22
6()
3()
5, 8
3, 40
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
0, 0.7071, 0.7071 1, 0.0000, 0.0000 0, 0.7071, 0.7071
23
60 (+)
10 ()
2, 4
10 , 20
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
24
60 ()
20 ()
4, 2
20 , 10
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
1, 0.0000, 0.0000 0, 0.7071, 0.7071 0, 0.7071, 0.7071
ð2 volume fractionÞ=½ð1 cos hÞN grain
ð2Þ
where h is half of the cone angle and Ngrain is the number of grains. For X-ray or neutron scattering experiments the intensity for a pole is centro-symmetric independent of the symmetry of the crystal lattice. Thus the total volume fraction at a grid point is the average of the two centrosymmetric cones. A flow chart for the procedure is given in Fig. 4. As an example, considering Fig. 2, a hypothetical step in the simulations taken in which grain 8 reorients from its initial state of all HPV having equal volume fractions (fi = 1/24, i = 1 ... 24) to a state in which two dominant HPV each have volume fractions of 37%, six others have volume fractions of 4.2% and all the others are close to 0. After this step the contribution of grain 8 to the (1 0 0) texture is calculated using the dominant HPV with volume fractions of f1 = 0.37, f2 = 0.37, where HPV 1 is composed of CV 5 (major) and 7 (minor) and HPV 2 is composed of CV 7 (major) and 5 (minor) (see Table 2). To begin the calculation the following transformations of the (1 0 0) plane normal are performed: T 0 V major 100;g8;1 ¼ R8 H5 ½1 0 0
ð3Þ
T
ð4Þ
0 V minor 100;g8;1 ¼ R8 H7 ½1 0 0
where H0 and X0 are the transformation matrices which map a vector in the major or minor CV to a vector in the austenite (grain coordinate), respectively. R8 is a matrix that transforms a vector inside the coordinates of grain 8 to a vector in specimen coordinates. The transformed vectors
minor V major 100;g8;1 and V 100;g8;1 are located on the standard (1 0 0) pole figure grid (here 5° 5° increments are used) and their volume fractions (0.37 0.72898 for V major 100;g8;1 , 0.37 0.27102 for V minor ) are added to the volume fraction for that grid 100;g8;1 point (0.72898 is the relative amount of major in the major–minor twin pair [36], discussed in more detail in Section 2.4). To account for experimental resolution a cone angle from 5° to 20° can be used in assigning volume fractions to the pole figure grid. A cone angle of 5° or less assigns a contribution to only one grid location, while a larger cone angle can result in a contribution to several adjacent grid points. Similar calculations were performed to determine volume fraction distributions for the poles of interest. From the resulting information pole figures were plotted and thus are able to be correlated with experimental data from HIPPO. Peak intensity evolutions, or changes in multiples of random distribution (MRD), were also plotted as a function of applied strain and compared with the experimental data from SMARTS. Note that peak intensity is only a specific point in the pole figure, hence the more CV aligned to the direction of interest the higher the MRD.
2.4. Texture calculation parameters The necessary crystallographic details for calculating textures include the lattice parameters of austenite and martensite, the relative amount of minor in the major– minor twin pair, and the rotation matrices between major or minor CV lattice of all HPV and the austenite lattice.
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Since not all information is available for the alloy under consideration, Wechsler–Lieberman–Read (WLR) calculations [45] made by Matsumoto et al. [46] for another NiTi composition with similar lattice parameters were used. In ˚ , a = 2.889 A ˚ , b = 4.622 A ˚, their calculations a0 = 3.015 A ˚ c = 4.120 A, and c = 96.80°, which are very close to those ˚, measured for the 54.8 wt.% NiTi (a0 = 3.015 A ˚ ˚ ˚ a = 2.908 A, b = 4.677 A, c = 4.126 A, and c = 97.95° [1]). In their calculations Matsumoto et al. [46] input the lattice parameters, lattice correspondences between austenite and martensite, shear planes and shear directions to determine the relative amount of minor in the major–minor twin pair (0.27102), habit plane normal (0.2152, 0.4044, and 0.8889 are the three indices), macroscopic shear magnitude (0.13078) and direction (0.7574, 0.4874 and 0.4345 are the corresponding indices for the above habit plane normal), the CV pair comprising each HPV, and rotation matrices between austenite and all CV. In postprocessing the SMM data the orientation matrices used were derived from the simple lattice correspondences between the austenite and martensite phases, and are not the same as those obtained by full WLR calculation (leading to 3% error). Although their macroscopic shear direction differs slightly from that of Matsumoto et al., Jin and Weng performed very similar calculations and published a table of the correspondence between CV pair and their habit plane normals [41]. The HPV system and rotation matrices used in these simulations are listed in Table 2. Due to inconsistencies in the order in which lattice parameters are recorded, as well as numbering of the HPV and CV, the columns of Table 2 compare the conventions adopted in this work with those found elsewhere in the literature. The HPV numbering system (column 1, Table 2) lists, in sequential order, the HPV based on permutation of the signs and magnitudes of the three numbers composing the habit plane normal after Saburi and co-workers [47] (column 2, Table 2). This identification system differs from that of Miyazaki and colleagues [48] (column 3, Table 2), as they recorded their lattice parameters using b = 96.80° (c is designated the non-90° angle in this work). While the CV naming conventions of Saburi et al. [47] and Miyazaki et al. [48] agree relative to equivalent HPV (column 5, Table 2), in this work their system was transformed through n0 ! 2n and n ! 2n 1 to facilitate programming (e.g. 60 becomes 12 and 2 becomes 3; column 4, Table 2). The rotation matrices in columns six and seven come directly from the lattice correspondence.
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[0, 1, 1]M. The [0, 1, 1]M and [0 1 1]M directions are equivalent, hence they are collectively referred to as group B. 3.1. Macroscopic mechanical response Fig. 6 shows the empirical and calculated mechanical responses for deformation in the 1-direction for both a virgin specimen as well as specimens pre-deformed in the 2-direction. The overall character of the predictions in Fig. 6 is similar to the experimental curves in that the stress to initiate significant reorientation is lower, and wider reorientation plateaux are observed in the pre-strained samples. However, the model does not quantitatively reproduce the experimental flow curves when examining the apparent strain hardening response of the plateaux and the final strain levels, due to assumptions made in the model formulation. Primarily, to simplify the interaction energy calculation of the model, the interaction energy a variant must overcome to convert to another variant is assumed to be constant for all variants [18], while in reality these energy barriers are more complex. Interaction energy has been well defined for the martensite–austenite energy barriers [49], and current efforts are focused on developing
3. Results CV 1–4 share a common feature in that their [1 0 0]B2 direction is parallel to the [1 0 0]M direction. For ease of discussion this subset of CV is designated group A. Similarly, the [1 0 0]B2 direction of CV 5, 8, 10, and 11 are aligned with [0 1 1]M, while the [1 0 0]B2 direction of the remaining CV (6, 7, 9, and 12) are aligned with
Fig. 6. Simulated and empirical mechanical responses for loading in the 1direction of a virgin specimen (top) and specimens pre-deformed in the 2direction (bottom). The steps in the experimental curves are due to pauses in the loading sequences to collect neutron data. These tests were performed on martensitic NiTi specimens at room temperature.
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Table 3 Measured post-deformation strains and Poisson ratios. Loading
e11
e22
e33
m12 0.64 0.46
Axial, 1-direction
Experimental Simulated
4.15 4.13
2.66 1.88
1.83 1.91
Transverse, 2-direction
Experimental Simulated
1.08 1.39
3.26 3.27
1.98 1.75
Axial, 1-direction after transverse
Experimental Simulated
3.38 3.37
3.29 3.11
0.61 0.35
m21
0.33 0.42
m23
m13
Rmij
0.44 0.46
1.08 0.92
0.61 0.54
0.97 0.92
0.94 0.96 0.18 0.10
1.15 1.02
Bold emphasizes strains in the principle deformation direction. Specimen directions relative to the parallelepiped geometry are defined in Fig. 1.
Table 4 Volume fractions of grains belonging to CV groups A and B and the differences between empirically observed and simulated values. Loading
Group A
Group B
Experimental
Simulated
Difference
Experimental
Simulated
Difference
Virgin specimen 1-Direction straining 2-Direction (pre-)straining 1-Direction straining after pre-straining
0.29 0.65 0.18 0.66
0.34 0.83 0.10 0.78
0.05 0.18 0.08 0.12
0.69 0.30 0.75 0.26
0.66 0.17 0.90 0.22
0.03 0.13 0.15 0.04
ways to incorporate more accurate representations of variant–variant conversion energy barriers in the next generation multivariant model. It is also reasonable to credit some of this discrepancy to the isotropic treatment of the elastic response of the material [31,37,38]. Finally, using this self-consistent approach to obtain the polycrystalline response means that localized inter-granular and defectrelated microstructure phenomena, such as slip and the effects of precipitates/inclusions, are not modeled. However, for as-quenched NiTi loaded to moderate strain levels these mechanisms are not thought to play a significant role in the overall deformation response [50]. Also, in Stebner et al. [1] the empirically observed differences in initial responses due to the different starting textures were discussed. Here the model predicted identical initial responses for the as-cast and pre-strained specimens due to the use of isotropic elastic constants. To model the evolution of anisotropic elastic constants within the SMM an approach analogous to that of Qiu et al. [31] can be adopted, although it would have to be modified through known CV to HPV relationships to correspond to HPV moduli for this model, since HPV volume fractions are being calculated [18,31]. Table 3 shows both measured and simulated postdeformation directional strains and Poisson ratios resulting from the aforementioned loadings. Overall, good agreement is observed between the experimental and simulation results as the model captured the major trends, including the seemingly counter-intuitive values that were discussed in the data presentation, such as the effective Poisson ratio values approaching 1 and 0 during 1-direction compression of specimen 5. Again, these results are due to the dominance of inelastic deformation, namely twinning [1]. The exception to this observation is that the asymmetric
dilation empirically observed during 1-direction compression of a virgin specimen is not simulated. This result confirms the conclusion in Stebner et al. [1] that this asymmetry is not due to material crystallography. The summation of mij being close to unity for all simulations verifies that the predictions are physically plausible. 3.2. Virgin austenite and martensite texture Fig. 3 shows the observed and calculated (1 0 0), (0 1 0), (0 0 1), and (0 1 1) pole figures representing the initial texture of the material. Two major texture components are apparent, the (1 0 0) fiber parallel to the cylinder axis (1direction) and (0 1 1) parallel to the cylinder axis, associated with CV groups A and B, respectively. The (1 0 0) fiber component (CV group A) is associated with the (0 1 0), (0 0 1), and (0 1 1) density on the rim of the pole figures. The (0 1 1) component (CV group B) is associated with (1 0 0) density on the rim and (0 1 0) and (0 0 1) density spread in a continuous ring at H 45°. Overall the simulations capture the observed texture well, although the calculated textures are somewhat sharper and reach higher MRD values. To make a more quantitative comparison the observed and simulated orientation distribution figures have been integrated over a selected area of specimen space to find the population of CV groups A and B as defined above. The volume fraction of grains belonging to groups A and B in the virgin specimen as well as the difference between the simulated and observed texture components are seen in Table 4. CV groups A and B comprise one-third and two-thirds of the volume fraction, respectively, as would be expected in self-accommodated martensite, although there are small differences between the observed and predicted values.
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Fig. 7. Textures after compression to emax = 4.2%, eunload = 3.5% in the 1-direction.
3.3. Development of texture during deformation 3.3.1. Axial direction compression of a virgin specimen The observed and calculated textures of virgin specimens after 1-direction compression such that eunload ¼ 11 3:5% are shown in Fig. 7. The calculated pole figures suggest a moderately sharper texture than was empirically observed, indicated by the higher peak MRD. As noted in Stebner et al. [1], during compression the (1 0 0) fiber parallel to the straining direction (CV group A) strengthens considerably with a concomitant decrease in the (0 1 1) fiber (CV group B). The effect of the sharper textures is quantified in Table 4, where it is seen that the simulation over-predicted the volume fraction increase in CV group A by 0.13, as well as the decrease in the CV group B population by 0.10. Note that both the instrument used to measure the texture (HIPPO) and the analysis technique are not particularly well suited for such sharp textures (discussed in more detail in Stebner et al. [1]), as is manifest in
Fig. 8. Normalized peak intensity evolution as emax = 4.2% is applied in the 1-direction. The markers indicate the empirical data points and the lines replicate the simulation data.
the experimental volume fractions not summing to unity. Bearing this caveat in mind, these 10% differences can be described as good quantitative agreement between the predictions and observations. Qualitatively the ability of the model to replicate texture development is clearly evident. In Fig. 8 development of the (1 0 0) and (0 1 1) diffraction peak intensities, chosen because they represent the two primary texture components (CV groups A and B) of the material, are shown during 1-direction compressive straining. As in Stebner et al. [1], the intensities are normalized to an initial value of one for ease of comparison. The calculated normalized diffraction intensities extracted from the SMM data agree well, both with the measured intensity development as well as observations previously discussed. The over-prediction of reorientation by the model for both the (1 0 0) and (0 1 1) poles relative to the measurements is again observed. 3.3.2. Transverse direction compression of a virgin specimen Fig. 9 depicts the textures of the material after 2-direction compression such that eunload ¼ 4:3%. As for 22 1-direction compression, the model predicted depletion of (1 0 0) pole density in the 1-direction and a concurrent increase in (1 0 0) pole density in the 2-direction, but over-predicted the CV group population changes (Table 4), this time by 0.08 and 0.15 for CV groups A and B, respectively. It is also significant to note that the complicated features of the pole figures, discussed in detail in Stebner et al. [1] (see Appendix), are well captured by the SMM. The exception is the (0 0 1) rim pole density distribution, which was found to be not entirely correct in the experiments: rim populations of (0 1 0) and (0 0 1) poles should always be perpendicular to each other, yet in the empirical data it is seen that the (0 1 0) pole population at H = 90°, W = 90° and 270° is non-existent, while there are (0 0 1) poles at H = 90°, W = 0° and 180° (see Stebner et al. [1] for further details). However, here the simulated pole densities are consistently aligned with each other.
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Fig. 9. Textures after compression to eunload = 4.3% in the 2-direction.
Fig. 10. Textures after compression to emax = 6.4%, eunload = 5.3% in the 1-direction. Physical and numerical specimens were previously loaded to eunload = 4.3% in the 2-direction.
3.3.3. Axial direction compression of a transversely pre-loaded specimen Texture development after 1-direction deformation such that eunload ¼ 5:3% following the aforementioned 11 2-direction compression is depicted in Fig. 10. Again, the simulated pole figures well capture the main features of the observed texture evolution, (1 0 0) pole alignment with the load and (0 1 1) pole alignment perpendicular to the load, but over-predict the volume fraction changes by 0.20 and 0.19 for CV Groups A and B, respectively. While previously it was not clear if the disagreement was only due to the resolution of the experimental techniques, the increase in the amount of change as multiple deformation steps are applied is indicative that the simulations could be improved in future work. This issue is further discussed in Section 4. In situ development of the (1 0 0) and (0 1 1) normalized peak intensities are seen in Fig. 11. Similar to the 1-direction straining of virgin specimens, the SMM satisfactorily reproduces development of the diffracted peak intensity
with strain. This statement might seem surprising when looking at this figure, as visually it appears the model has missed the evolution of the (0 1 1) poles in the 2-direction and (1 0 0) poles in the 3-direction. However, in considering this plot in tandem with the corresponding pole figures (Figs. 9 and 10) it is apparent that these directional populations are initially very near zero MRD (Fig. 9), thus any slight deviation appears to be quite dramatic on the normalized scale, while in actuality they are not. 4. Discussion The dominant trends (growth and depletion of CV groups A and B, wider reorientation plateaux observed for the pre-textured specimens, etc.) were well captured by the SMM as complex stress states were created. This result verifies that the simulations correctly associate the growth and depletion of variants with the application of isothermal loadings for the chosen HPV system. However, the accumulation of empirical and numerical differences as
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Fig. 11. Normalized peak intensity evolution during 1-direction straining after deformation in the 2-direction. The markers indicate the empirical data points and the lines replicate the simulation data.
multiple loads were applied in Section 3.3.3 suggests that there is likely room for further improvement in the simulations, such as more accurate replication of variant to variant conversion energy barriers. In Stebner et al. [1] it was shown that (after the work of Dunand et al. [51,52]) the texture development observed in both the measured and calculated pole figures can be described by (1, 1, 1) type I twinning, but not by (0.7206, 1, 1)[0 1 1] type II twinning [1]. The 24 HPV system composed of twinned arrangements of CV described by the latter twin system was invoked in the simulations due to its prevalence in both micromechanical simulations and experiments of other researchers, as discussed in Section 2.2. This choice was made without reference to the experimental data analysis. Future work will examine whether using the (1, 1, 1) type I HPV system, as well as other known HPV systems [20] and variant structures (such as only CV), improves the agreement between the experimental results and calculations. Still, from these simulations it is significant to note that the effective average polycrystalline martensite texture evolution, which can be described by a twinning event in a (1, 1, 1) plane, was replicated by growth and depletion of volume fractions of (0.7206, 1, 1)[0 1 1] type II HPV. The simulated texture development agreed well in the absence of the ability of CV to coalesce about (1, 1, 1) twin planes within the plates. This result confirms that it is possible that the major deformation mechanism at these strain levels is HPV reorientation and not coalescence of
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the HPV to a single CV, which would require a type II twinning event of the chosen HPV system. To verify this suggestion, however, further calculations are required to determine the possibility of other martensite variant structures producing this effective twinning observation. The early results presented here, though, illustrate how coupling of multivariant modeling with diffraction allows the deconvolution of twinning mechanisms and their relationships to variant structures within diffraction data. Such an observation is not possible for a polycrystalline material through examination of diffraction data using the lens of a Bain strain model [53], previous state of the art in considering variant evolution [50–53]. Additionally, the current numerical implementation of the SMM uses a 24 HPV grouping of CV, which is a simplification of allowing reorientation of all 192 unique CV that may arise as martensites are formed and deformed within polycrystalline NiTi [20,40]. The existing code can be used to examine how accurately a HPV-based model can predict the micromechanical behavior of SMA by performing simulations using the known HPV systems of an alloy and using this verification process to compare the results achieved for different HPV systems. Beyond these studies, applying the developed methodologies to predictions made by extension of the SMM and other multivariant models can allow further decoupling of empirically observed phenomena. For instance, using the two-tier multivariant model [54] the individual roles HPV reorientation and coalescence of CV play in shape memory behavior may be examined, as well as their interactions. Using the model of Manchiraju and Anderson [23] the role of accommodation slip and grain to grain constraints may be considered. None of these understandings are possible, though, without first verifying both the microscopic and macroscopic predictions of such models. Although they were only applied to the SMM in this work, the model– diffraction coupling technique presented provides the means to carry out multiscale verification for any multivariant model. 5. Conclusion Verification methodologies were established that could now be applied to all multivariant models that explicitly predict the growth and depletion of variant volume fractions within SMA. Through their application to the SMM, for the first time it has been shown that a multivariant model can accurately describe both primary texture development and macroscopic response changes following complex multi-axial loading. It was also shown that the marriage of micromechanics and neutron diffraction through the developed methodologies provides a necessary means to conduct further investigations and decouple mechanistic contributions and relations in greater detail than was previously achievable, especially regarding intermediate variant structures that are known to exist in these alloys. Work is being done to more fully understand
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crystallographic behaviors within bulk polycrystalline NiTi specimens using these tools. These processes require that not only isothermal martensite deformation predictions of a few loading cycles be examined, but also the complete gambit of NiTi shape memory behavior throughout the desired lifetime of an alloy system. Thus these methodologies are being extended to tension–compression deformation of martensite as well as isostress thermo-mechanical loadings, and preliminary results have been presented [55,56]. Finally, the methodology presented here provides a means to validate the capabilities of any SMA multivariant model.
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Acknowledgements This work has benefited from the use of the Lujan Neutron Scattering Center at LANSCE, which is funded by the Office of Basic Energy Sciences of the Department of Energy. A.S. gratefully acknowledges funding from Telezygology Inc. A.S. and L.C.B. gratefully acknowledge funding from the Northwestern University Predictive Science and Engineering Design Cluster and Initiative for Sustainability and Energy at Northwestern (ISEN) programs, and the Boeing Company. X.G. and L.C.B. gratefully acknowledge funding from NASA Langley. The authors collectively thank Professor Raj Vaidyanathan at the University of Central Florida for many valued discussions and Dr. Carlos Tome´ for fielding questions and providing POLE7 [57] (used to create pole figures in preliminary analysis of the simulation data). The reviewer is also thanked for enhancing this article. Appendix A In Stebner et al. [1] it was noted that the (0 1 0) and (0 1 1) poles have predominantly reoriented from the 2 to the 3 direction, while the band of (0 0 1) pole density on the rim has reduced almost uniformly. This was then shown to be a result of the (0 1 1) fiber rotating such that the (1 0 0) poles in the 3 direction reorient to align with the compression, while the (0 1 0) and (0 0 1) poles initially positioned at H = 45° and W = 90° or 270° circumvolve to H = 45° and W = 0° or 180°. References
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