Measurement of elastic constants of monoclinic nickel-titanium and ...
APPLIED PHYSICS LETTERS 102, 211908 (2013)
Measurement of elastic constants of monoclinic nickel-titanium and validation of first principles calculations A. P. Stebner,1,a) D. W. Brown,2 and L. C. Brinson1,3 1
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 3 Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA 2
(Received 19 April 2013; accepted 15 May 2013; published online 29 May 2013) Polycrystalline, monoclinic nickel-titanium specimens were subjected to tensile and compressive deformations while neutron diffraction spectra were recorded in situ. Using these data, orientationspecific and macroscopic Young’s moduli are determined from analysis of linear-elastic deformation exhibited by 13 unique orientations of monoclinic lattices and their relationships to each macroscopic stress and strain. Five of 13 elastic compliance constants are also identified: s11 ¼ 1.15, s15 ¼ 1.10, s22 ¼ 1.34, s33 ¼ 1.06, s35 ¼ 1.54, all 102 GPa1. Through these results, recent atomistic calculations of monoclinic nickel-titanium elastic constants are validated. C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4808040] V The martensite phase of shape-memory NiTi is comprised of B190 monoclinic structure.1 The elastic response of monoclinic NiTi has been a focus of recent research, as macroscopic-only assessments of the elastic response from stress-strain behaviors exhibited by this phase were not consistent with recent micromechanical observations and calculations.2–6 Toward quantification of the elastic anisotropy of this material, two sets of elastic constants have been calculated from first principles; one using pseudo-potentials6 (henceforth referred to as CWW ¼ SWW1 ), the other considering full electron interactions4 (CHKF ¼ SHKF1 ). Previous in situ neutron diffraction studies documented relative empiricalnumerical agreement of orientation-specific strain vs. macroscopic stress responses of CWW and polycrystalline, monoclinic NiTi,3 but the individual constants have not been validated against empirical observations for either set of calculations. One method to achieve first principles simulation validation would be to study the elastic response of a monoclinic NiTi single crystal subject to uniaxial deformation in multiple orientations near a temperature of 0 K. However, in the previous study of a monoclinic NiTi bi-crystal that was nearly a single crystal, elastic properties were not documented at any temperature.7 Also, commercially produced NiTi is polycrystalline, thus, it is desirable to validate C for use in polycrystalline models that account for elastic anisotropy (e.g., Refs. 3 and 9). In using C to study elasticinelastic strain partitioning exhibited by polycrystalline material, the choice of CWW vs. CHKF was shown to have significant impact on the resulting NiTi mechanics.5,8 Thus, the objective of this work was to empirically observe C from in situ neutron diffraction measurements and quantify the accuracy of CWW and CHKF applied to polycrystalline NiTi. Specimens with cylindrical gage sections of 5.10 mm 15.25 mm (tension) and 7.50 mm 15.00 mm (compression) were turned on a lathe from 10 mm diameter hot a)
Present address: Department of Mechanical Engineering, Colorado School of Mines, Golden, Colorado 80401, USA. Electronic mail: [email protected]
0003-6951/2013/102(21)/211908/5/$30.00
extruded rods. Complete details of this material and these experiments have been reported.5,8,10 Important to this report is that the material exhibits a stress-free martensite finish temperature Mf ¼ 4662 C (Ref. 3) and we observed the deformation of this material at room temperature (RT, 30 C). It has been demonstrated in other near-a Ti alloys that temperature does not dramatically affect the Young’s modulus (less than 5% difference between reported 0 and 300 K values),11 thus, cryogenic experiments are not critical to validate the first principles calculations given the current 30% disparity in the quantification of monoclinic NiTi elasticity at 295–300 K.5 The mean (std. dev.) virgin lattice parameters of this material measured for six different specimens using the ˚ , b ¼ 4.630(2) A ˚, HIPPO diffractometer12 are a ¼ 2.8931(9) A ˚ c ¼ 4.111(1) A, c ¼ 97.30(1) . Previous measurements of virgin B190 textures show that the initial texture of the specimens are very near random, indicated by a texture index of 1.15 and maximum pole densities not greater than 2 times random, with slight axisymmetric bias from hot extrusion.5,8,10 In the ensuing presentation and discussion of our data, Fit will be used as a superscript to denote a quantity or response that was measured from our data, while WW and HKF superscripts denote quantities and responses calculated using the respective sets of elastic constants. Subscripts will then identify whether a quantity refers to a specific hkl orientation, a macroscopic measure, or an ij direction in a Cartesian coordinate system. No subscript indicates a response modeled using the equations presented henceforth. The experiments reported in this work were performed on the SMARTS diffractometer at the Los Alamos Neutron Science Center. Mechanical deformations were applied to specimens using the horizontal hydraulic load frame with the loading axis oriented 45 relative to the incident neutron beam such that the detector banks recorded data for diffraction vectors within 11 alignment of parallel and perpendicular to the applied load.13 The white neutron beam was masked such that the center of the diffracting window was aligned with the center of the specimen gage sections using 5 5
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211908-2
Stebner, Brown, and Brinson
Appl. Phys. Lett. 102, 211908 (2013)
mm2 boron-nitride slits for tension experiments, 6 6 mm2 for compression. The material exhibits an average, equiaxed parent grain size of 20 lm, and martensite crystallite sizes of 10 nm diameter 100 nm long,8,10 thus, the average response of billions of martensite crystallites was recorded within the diffracted volume. The positions of individual reflections of each of the recorded diffraction spectra were refined using the SMARTSWARE SMARTSSPF routine,14 which performs batch single peak refinements using the GSAS routine 15 RAWPLOT. Because of overlap of monoclinic peaks,3,5,10 it was necessary to fit some reflections in clusters to accurately distinguish background from the diffracted intensity of each reflection: ð002Þ=ð111), ð120Þ =ð012Þ, and the ð122Þ=ð200Þ reflections were fit in pairs while ð121Þ=ð120Þ=ð102Þ and ð 112Þ=ð121Þ=ð030Þ reflections were fit in triplets. The ð120Þ, ð102Þ, and ð200Þ reflections are not reported due to ambiguities in their refinements resulting from overlap with other orientations. Orientation specific strains (ehkl ) are calculated from these refinements according to ehkl ¼
0 dhkl dhkl ; 0 dhkl
(1)
where dhkl is the current planar d-spacing of a reflection and 0 dhkl is the d-spacing observed of the reflection in undeformed material. The responses of three unique specimens are studied in this work: one in compression, two in tension. Large mechanical deformations (618% strain) and inverse pole figure evolutions over the entirety of these deformations of these specimens have been reported;5,8,10 here, we only analyze the initial mechanical loading of each specimen (Fig. 1). Consistent with previous numbering,5 compression responses are reported for specimen 1 during strain-controlled loading (_e ¼ 5 104 ) from emacro ¼ 0.00% to 0.50% (20 data points). The specimen was unloaded from emacro ¼ 0.50%, and then reloaded to emacro ¼ 1.20% (9 additional data points). During diffraction measurements, specimens were held strain control (_e ¼ 0) for 35 min during each
FIG. 1. The absolute macroscopic stress-strain responses of the material in tension and compression. The compression specimen was unloaded after loading to e ¼ 0.5% and then reloaded. The macroscopic elastic responses of the mean and standard deviation of the micromechanically assessed macroscopic Young’s moduli listed in Table I are also shown.
diffraction measurement, and the stress relaxations observed in Fig. 1 occurred during the measurements, mostly within the first minute. ð012Þ, ð120Þ, and ð122Þ tension responses are reported for the 16 data points acquired in loading specimen 2 from emacro ¼ 0.00% to 2.00%, and the remaining tension data consists of 31 data points acquired for this same macroscopic deformation of specimen 3. The three reflections reported for specimen 2 were not able to be refined as well for specimen 3, which is why specimen 2 is used for these reflections even though there are not as many data points acquired in the regime of interest. Macro-mechanically, the material exhibits relative tension-compression symmetry during these initial deformations (Fig. 1), though symmetric microstructure evolution was not observed in inverse pole figure measurements.8,10 Average strains of unique orientations in the loading direction are plotted with respect to the macroscopic stress (rmacro ) in Fig. 2 and the macroscopic logarithmic strain (Lemacro ) in Fig. 3. In addition to the empirical measurements, expected elastic responses from SWW , SHKF are shown in both figures, noting that orientation-specific Young’s moduli (Ehkl ) may be calculated16 via Ehkl ¼ l41 s11 þ 2l21 l22 s12 þ 2l21 l23 s32 þ 2l31 l3 s15 þ l42 s22 þ 2l22 l23 s23 þ 2l1 l22 l3 s25 þ l43 s33 þ 2l1 l33 s35 þ l22 l23 s44 þ 2l1 l22 l3 s46 þ l21 l23 s55 þ l21 l22 s66 ;
(2)
where li are the direction cosines of a unit vector in the (hkl) direction with respect to a Cartesian coordinate system (i ¼ 1, 2, or 3 denotes the 1-, 2-, or 3-axis of the system, respectively). Knowing Ehkl , and assuming deformation of each orientation is initially elastic and that on average rmacro is initially uniformly distributed to the billions of observed crystallites, the following relations govern the initial Lemacro vs. ehkl and rmacro vs. ehkl responses: rmacro ¼ Ehkl ehkl ; Ehkl Lemacro ¼ ln 1 þ ehkl : Emacro
(3a) (3b)
HKF Calculations of EWW macro ¼ 111:9; Emacro ¼ 179:9 GPa as volume averaged from inverse pole figure measurements of each of the three undeformed specimens were previously reported,5,10 and these values were used to generate the responses of SWW , SHKF shown in Fig. 3 using Eq. (3b) and Ehkl calculated via Eq. (2) (Table I). The empirical data were also assessed to ascertain where each orientation ceases to experience predominantly linearelastic deformation. Measurements of ehkl are proportional to the average local stress experienced by each group of crystallites of a unique orientation, thus, when the relationship between ehkl and the macroscopic mechanical response (stress or strain) deviates from its initial linear proportion, inelastic deformation has begun to influence the (hkl) deformation. The number of data points and the macroscopic stress and strain bounds of linear-elastic deformation observed for each (hkl) are listed in Table I and indicated in Figs. 2 and 3. “Load 2” compression data were not used for any specimen, even if the unload data point after “Load 1” was contained
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211908-3
Stebner, Brown, and Brinson
Appl. Phys. Lett. 102, 211908 (2013)
FIG. 2. Strains (ehkl ) of 13 unique orientations of monoclinic crystallites are plotted vs. the macroscopic engineering stress (rmacro ). The elastic responses exhibited by the material in the regimes labeled “Elastic” are shown as EFit , and the elastic responses calculated from Refs. 4 and 6 using Eq. (2) are shown as EWW and EHKF , respectively. Error bars for ehkl are twice the statistical peak fitting error calculated by GSAS.
within the macroscopic elastic bounds, as the unload event was inelastic with respect to Lemacro vs. ehkl for all observed orientations since the macroscopic strain did not fully recover (Fig 1). Using Eq. (3a), Ehkl for each reflection were determined from the rmacro vs. ehkl elastic responses. The resulting EFit responses according to Eq. (3a) are shown in Fig. 2 and Fit EFit hkl values are listed in Table I. The Ehkl were then used to fit Emacro to the Lemacro vs. ehkl elastic responses according to Eq. (3b) for each reflection, which are also reported in Fit Table I. The EFit hkl together with Emacro were used in Eq. (3b) Fit to generate the E responses shown in Fig. 3.
In our micromechanical analysis, only the assumption of uniform macroscopic stress distribution to monoclinic crystallites is needed; additional assumption of macroscopic strain distribution to crystallites is not required. Equation (3a) is substituted into the macroscopic Hooke’s law to derive Eq. (3b). While linear Lemacro vs. ehkl and rmacro vs. ehkl relations may be used to assess elastic deformation, linearity alone does not validate the self-consistent micromechanical assumption17 of initially uniform macroscopic stress distribution. To gage the “uniform” part of the assumption, an average Emacro ¼ 66.7 GPa and standard deviation of
FIG. 3. Strains of 13 unique orientations (ehkl ) of monoclinic crystallites are plotted vs. the macroscopic logarithmic strain (Lemacro ) together with the linear-elastic response of each orientation exhibited in the regimes labeled “Elastic,” denoted as EFit . The elastic responses calculated from Refs. 4 and 6 using Eqs. (2) and (3) are also shown as EWW and EHKF , respectively. Error bars for ehkl are twice the statistical peak fitting error calculated by GSAS.
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211908-4
Stebner, Brown, and Brinson
Appl. Phys. Lett. 102, 211908 (2013)
Fit TABLE I. Orientations ðhklÞ and the macroscopic modulus (EFit macro ) and orientation-specific Young’s moduli (Ehkl ) fit to their elastic data, which are contained max min max ; r ; ½e ; e ). Pearson’s linear correlation coefficient (q) for each EFit within the macroscopic stress and strain window (½rmin macro macro macro macro hkl and the number of WW HKF data points (n) used in the fits are also listed. Finally, Ehkl calculated from Refs. 4 and 6 (S , S ) according to Eq. (2) are listed together with their relative differences (Edif f Þ to EFit hkl .
ðhklÞ
EFit macro (GPa)
EFit hkl (GPa)
q
n
Lemin macro (%)
Lemax macro (%)
rmin macro (MPa)
rmax macro (MPa)
EWW hkl (GPa)
EWW dif f (%)
EHKF hkl (GPa)
EHKF dif f (%)
ð100Þ ð030Þ ð002Þ ð111Þ ð111Þ ð011Þ ð012Þ ð121Þ ð121Þ ð110Þ ð120Þ ð112Þ ð122Þ
89.3 62.6 55.6 71.3 81.9 57.8 55.4 88.9 61.9 65.2 50.8 75.3 50.9
94.4 101.4 74.4 90.1 99.2 108.9 108.1 92.3 43.9 32.3 36.6 152.8 50.9
0.906 0.889 0.874 0.943 0.705 0.943 1.000 0.960 0.927 0.966 0.997 0.774 0.994
7 17 6 13 12 22 2 13 11 6 5 11 5
0.070 0.090 0.060 0.250 0.200 0.200 0.000 0.250 0.050 0.060 0.020 0.150 0.020
0.000 0.059 0.000 0.000 0.000 0.079 0.110 0.000 0.039 0.000 0.219 0.000 0.219
52.1 60.7 47.3 116.6 101.2 101.2 0.8 116.6 42.4 47.3 25.4 84.9 25.4
5.0 35.5 5.0 5.0 5.0 47.3 61.0 5.0 22.4 5.0 105.2 5.0 105.2
128.2 146.2 111.1 87.6 83.3 177.3 137.3 83.6 52.2 41.2 33.8 175.2 115.3
35.8 44.2 49.3 2.8 16.0 62.8 27.0 9.4 18.9 27.6 7.7 14.7 126.5
138.7 177.0 148.1 179.6 140.6 199.9 170.6 189.3 174.1 146.6 142.2 210.3 207.7
46.9 74.6 99.1 99.3 41.7 83.6 57.8 105.1 296.6 353.9 288.5 37.6 308.1
13.6 GPa was calculated from the 13 fits (Table I). Even though some orientations immediately exhibit inelastic deformation in each tension (ð111Þ, ð111Þ, ð121Þ, ð110Þ, and ð 112Þ) and compression (ð012Þ, ð120Þ, and ð122Þ) (Figs. 2 and 3), in considering only elastic ðhklÞ responses, the predicted macroscopic response is in agreement with the initial macroscopic loading data (Fig. 1). As determination of EFit hkl is independent of EFit macro (but not vise-versa), the mean and standard deviation indicate that the empirical analyses are self-consistent to 20%. Using EFit hkl , 13 equations were written relating each Ehkl to monoclinic NiTi S according to Eq. (2) using a previously established Cartesian coordinate system for the monoclinic lattice.4–6,10 The rank of the resulting matrix of coefficients of sij was nine, which was insufficient to uniquely determine all 13 monoclinic sij from the empirical data. However, five sij were uniquely determined from these relationships, and they WW HKF in Table II. are listed (sFit ij ) and compared with sij , sij Note that because S was not fully determined from the experimental measurements, S1 ¼ C could not be calculated. The reported average Emacro ¼ 66.7 GPa from fitting these data is lower (30–50%) than recent in situ diffraction reports of Emacro for random textured monoclinic NiTi (109, 134 GPa),2,3 but consistent with ultrasonic measurement of Emacro and an earlier in situ diffraction report for monoclinic NiTi and NiTi-based alloys (60–85 GPa).18 The primary difference between our report and other recent neutron TABLE II. The five measured elastic compliance constants are listed (sFit ij ), as well as those calculated from Refs. 4 and 6 (SWW , SHKF ). Also given are relative differences, denoted with subscript “dif f .”
ij
2 sFit ij 10 1 (GPa )
102 sWW ij (GPa1)
sww ij dif f (%)
102 sHKF ij (GPa1)
sHKF dif f ij (%)
11 15 22 33 35
1.15 1.10 1.34 1.06 1.54
0.978 1.45 0.900 0.780 0.113
15.0 31.8 32.8 26.4 92.7
0.604 0.150 0.675 0.721 0.036
47.5 86.4 49.6 32.0 102.3
diffraction reports is that we did not assume that all monoclinic orientations follow Eq. (3a) while the macroscopic deformation appears linear-like (i.e., between 680 MPa (Ref. 2) or 6200 MPa (Ref. 3)), but rather individually investigated the micromechanical responses of each ðhklÞ for deviation from initial linear-elastic assumptions. Here, we also used the Lemacro vs. ehkl responses to fit Emacro instead of assembling Emacro solely from the rmacro vs. ehkl responses. In comparing predicted and observed magnitudes of Ehkl , visually (Fig. 2) and statistically (Table I) SWW responses are closer to the observed elastic responses than SHKF . For ð111Þ, ð121Þ, ð120Þ, and ð121Þ orientations, however, SHKF provide a better model of Ehkl =Emacro than SWW (Fig. 3). In considering the five measured sij (Table II), SWW are again in better relative agreement with the empirical observations than SHKF . These results indicate that SWW provide the best complete model of elastic anisotropy for monoclinic crystallites deforming within polycrystalline NiTi, since our empirical measurements were only conclusive in assessing five of the 13 independent constants. Furthermore, these results suggest that allowing relaxation of atomic positions and imposing stress controls (as was done by WW but not HKF)4 during first principles computations may result in more realistic predictions of elastic constants for polycrystalline materials. Both calculations of elastic constants, however, are stiffer than the observations by 15% or more, and by more than an order of magnitude in the 35-direction (Table II), where sHKF 35 is also of the opposite sign. This quantitative discrepancy indicates that there is still advancement desired toward establishing best-practice methodologies for computing elastic anisotropy of crystalline materials through atomistic simulations. Understanding how to achieve these advancements, however, is bounded by challenges toward fabrication and in situ observation of single crystals at the atomic scale and 0 K. We thank Thomas Sisneros and Bjørn Clausen of LANL for experimental assistance and Ron Noebe of NASA Glenn Research Center for providing the NiTi specimens. This
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211908-5
Stebner, Brown, and Brinson
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 under DOE Contract No. DE-AC52-06NA25396. A.S. acknowledges funding through fellowships from the Toshio Mura Endowment, Predictive Science and Engineering Design Cluster at Northwestern (PSED), Initiative for Sustainability and Energy at Northwestern (ISEN). A.S. and C.B. acknowledge the support of the Army Research Office, Grant No. W911NF-12-1-0013/P00002. 1
K. Otsuka, T. Sawamura, and K. Shimizu, Phys. Status Solidi A 5(2), 457 (1971); R. F. Hehemann and G. D. Sandrock, Scr. Metall. Mater. 5(9), 801 (1971). 2 S. Rajagopalan, A. L. Little, M. A. M. Bourke, and R. Vaidyanathan, Appl. Phys. Lett. 86(8), 081901 (2005). 3 S. Qiu, B. Clausen, S. A. Padula, R. D. Noebe, and R. Vaidyanathan, Acta Mater. 59(13), 5055 (2011). 4 N. Hatcher, O. Y. Kontsevoi, and A. J. Freeman, Phys. Rev. B 80(14), 144203 (2009). 5 A. P. Stebner, D. W. Brown, and L. C. Brinson, Acta Mater. 61, 1944 (2013). 6 M. F. X. Wagner and W. Windl, Acta Mater. 56(20), 6232 (2008). 7 Y. Kudoh, M. Tokonami, S. Miyazaki, and K. Otsuka, Acta Metall. Mater. 33(11), 2049 (1985).
Appl. Phys. Lett. 102, 211908 (2013) 8
A. P. Stebner, S. C. Vogel, R. D. Noebe, T. Sisneros, B. Clausen, D. W. Brown, A. Garg, and L. C. Brinson, “Micromechanical Elastic, Twinning, and Slip Strain Partitioning of Polycrystalline, Monoclinic NickelTitanium Large Uniaxial Deformations Measured via In Situ Neutron Diffraction,” J. Mech. Phys. Solids (In Revision). 9 S. Manchiraju, D. Gaydosh, O. Benafan, R. Noebe, R. Vaidyanathan, and P. M. Anderson, Acta Mater. 59(13), 5238 (2011). 10 A. P. Stebner, Ph.D. dissertation, Mechanical Engineering, Northwestern University, 2012. 11 E. R. Naimon, W. F. Weston, and H. M. Ledbetter, Cryogenics 14(5), 246 (1974). 12 H. R. Wenk, L. Lutterotti, and S. Vogel, Nucl. Instrum. Methods Phys. Res. A 515(3), 575 (2003). 13 M. A. M. Bourke, D. C. Dunand, and E. Ustundag, Appl. Phys. A: Mater. Sci. Process. 74, S1707 (1995). 14 B. Clausen, SMARTSware Manual (Los Alamos National Laboratory, 1998), LA-UR 04-6581. 15 A. C. Larson and R. B. VonDreele, GSAS (Los Alamos National Laboratory, 1986), LAUR 86-748; R. B. VonDreele, J. Appl. Crystallogr. 30, 517 (1997). 16 J. F. Nye, Physical Properties of Crystals, Their Representation by Tensors and Matrices (Clarendon Press, Oxford, 1957), p. 322. 17 E. Kroner, Acta Metall. Mater. 9(2), 155 (1961). 18 D. C. Dunand, D. Mari, M. A. M. Bourke, and J. A. Roberts, Metall. Mater. Trans. A 27(9), 2820 (1996); A. Stebner, S. Padula, R. Noebe, B. Lerch, and D. Quinn, J. Intell. Mater. Syst. Struct. 20(17), 2107 (2009); A. P. Stebner, M.S. thesis, Mechanical Engineering, University of Akron, 2007.
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Measurement of elastic constants of monoclinic nickel-titanium and validation of first principles calculations A. P. Stebner,1,a) D. W. Brown,2 and L. C. Brinson1,3 1
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 3 Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA 2
(Received 19 April 2013; accepted 15 May 2013; published online 29 May 2013) Polycrystalline, monoclinic nickel-titanium specimens were subjected to tensile and compressive deformations while neutron diffraction spectra were recorded in situ. Using these data, orientationspecific and macroscopic Young’s moduli are determined from analysis of linear-elastic deformation exhibited by 13 unique orientations of monoclinic lattices and their relationships to each macroscopic stress and strain. Five of 13 elastic compliance constants are also identified: s11 ¼ 1.15, s15 ¼ 1.10, s22 ¼ 1.34, s33 ¼ 1.06, s35 ¼ 1.54, all 102 GPa1. Through these results, recent atomistic calculations of monoclinic nickel-titanium elastic constants are validated. C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4808040] V The martensite phase of shape-memory NiTi is comprised of B190 monoclinic structure.1 The elastic response of monoclinic NiTi has been a focus of recent research, as macroscopic-only assessments of the elastic response from stress-strain behaviors exhibited by this phase were not consistent with recent micromechanical observations and calculations.2–6 Toward quantification of the elastic anisotropy of this material, two sets of elastic constants have been calculated from first principles; one using pseudo-potentials6 (henceforth referred to as CWW ¼ SWW1 ), the other considering full electron interactions4 (CHKF ¼ SHKF1 ). Previous in situ neutron diffraction studies documented relative empiricalnumerical agreement of orientation-specific strain vs. macroscopic stress responses of CWW and polycrystalline, monoclinic NiTi,3 but the individual constants have not been validated against empirical observations for either set of calculations. One method to achieve first principles simulation validation would be to study the elastic response of a monoclinic NiTi single crystal subject to uniaxial deformation in multiple orientations near a temperature of 0 K. However, in the previous study of a monoclinic NiTi bi-crystal that was nearly a single crystal, elastic properties were not documented at any temperature.7 Also, commercially produced NiTi is polycrystalline, thus, it is desirable to validate C for use in polycrystalline models that account for elastic anisotropy (e.g., Refs. 3 and 9). In using C to study elasticinelastic strain partitioning exhibited by polycrystalline material, the choice of CWW vs. CHKF was shown to have significant impact on the resulting NiTi mechanics.5,8 Thus, the objective of this work was to empirically observe C from in situ neutron diffraction measurements and quantify the accuracy of CWW and CHKF applied to polycrystalline NiTi. Specimens with cylindrical gage sections of 5.10 mm 15.25 mm (tension) and 7.50 mm 15.00 mm (compression) were turned on a lathe from 10 mm diameter hot a)
Present address: Department of Mechanical Engineering, Colorado School of Mines, Golden, Colorado 80401, USA. Electronic mail: [email protected]
0003-6951/2013/102(21)/211908/5/$30.00
extruded rods. Complete details of this material and these experiments have been reported.5,8,10 Important to this report is that the material exhibits a stress-free martensite finish temperature Mf ¼ 4662 C (Ref. 3) and we observed the deformation of this material at room temperature (RT, 30 C). It has been demonstrated in other near-a Ti alloys that temperature does not dramatically affect the Young’s modulus (less than 5% difference between reported 0 and 300 K values),11 thus, cryogenic experiments are not critical to validate the first principles calculations given the current 30% disparity in the quantification of monoclinic NiTi elasticity at 295–300 K.5 The mean (std. dev.) virgin lattice parameters of this material measured for six different specimens using the ˚ , b ¼ 4.630(2) A ˚, HIPPO diffractometer12 are a ¼ 2.8931(9) A ˚ c ¼ 4.111(1) A, c ¼ 97.30(1) . Previous measurements of virgin B190 textures show that the initial texture of the specimens are very near random, indicated by a texture index of 1.15 and maximum pole densities not greater than 2 times random, with slight axisymmetric bias from hot extrusion.5,8,10 In the ensuing presentation and discussion of our data, Fit will be used as a superscript to denote a quantity or response that was measured from our data, while WW and HKF superscripts denote quantities and responses calculated using the respective sets of elastic constants. Subscripts will then identify whether a quantity refers to a specific hkl orientation, a macroscopic measure, or an ij direction in a Cartesian coordinate system. No subscript indicates a response modeled using the equations presented henceforth. The experiments reported in this work were performed on the SMARTS diffractometer at the Los Alamos Neutron Science Center. Mechanical deformations were applied to specimens using the horizontal hydraulic load frame with the loading axis oriented 45 relative to the incident neutron beam such that the detector banks recorded data for diffraction vectors within 11 alignment of parallel and perpendicular to the applied load.13 The white neutron beam was masked such that the center of the diffracting window was aligned with the center of the specimen gage sections using 5 5
102, 211908-1
C 2013 AIP Publishing LLC V
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211908-2
Stebner, Brown, and Brinson
Appl. Phys. Lett. 102, 211908 (2013)
mm2 boron-nitride slits for tension experiments, 6 6 mm2 for compression. The material exhibits an average, equiaxed parent grain size of 20 lm, and martensite crystallite sizes of 10 nm diameter 100 nm long,8,10 thus, the average response of billions of martensite crystallites was recorded within the diffracted volume. The positions of individual reflections of each of the recorded diffraction spectra were refined using the SMARTSWARE SMARTSSPF routine,14 which performs batch single peak refinements using the GSAS routine 15 RAWPLOT. Because of overlap of monoclinic peaks,3,5,10 it was necessary to fit some reflections in clusters to accurately distinguish background from the diffracted intensity of each reflection: ð002Þ=ð111), ð120Þ =ð012Þ, and the ð122Þ=ð200Þ reflections were fit in pairs while ð121Þ=ð120Þ=ð102Þ and ð 112Þ=ð121Þ=ð030Þ reflections were fit in triplets. The ð120Þ, ð102Þ, and ð200Þ reflections are not reported due to ambiguities in their refinements resulting from overlap with other orientations. Orientation specific strains (ehkl ) are calculated from these refinements according to ehkl ¼
0 dhkl dhkl ; 0 dhkl
(1)
where dhkl is the current planar d-spacing of a reflection and 0 dhkl is the d-spacing observed of the reflection in undeformed material. The responses of three unique specimens are studied in this work: one in compression, two in tension. Large mechanical deformations (618% strain) and inverse pole figure evolutions over the entirety of these deformations of these specimens have been reported;5,8,10 here, we only analyze the initial mechanical loading of each specimen (Fig. 1). Consistent with previous numbering,5 compression responses are reported for specimen 1 during strain-controlled loading (_e ¼ 5 104 ) from emacro ¼ 0.00% to 0.50% (20 data points). The specimen was unloaded from emacro ¼ 0.50%, and then reloaded to emacro ¼ 1.20% (9 additional data points). During diffraction measurements, specimens were held strain control (_e ¼ 0) for 35 min during each
FIG. 1. The absolute macroscopic stress-strain responses of the material in tension and compression. The compression specimen was unloaded after loading to e ¼ 0.5% and then reloaded. The macroscopic elastic responses of the mean and standard deviation of the micromechanically assessed macroscopic Young’s moduli listed in Table I are also shown.
diffraction measurement, and the stress relaxations observed in Fig. 1 occurred during the measurements, mostly within the first minute. ð012Þ, ð120Þ, and ð122Þ tension responses are reported for the 16 data points acquired in loading specimen 2 from emacro ¼ 0.00% to 2.00%, and the remaining tension data consists of 31 data points acquired for this same macroscopic deformation of specimen 3. The three reflections reported for specimen 2 were not able to be refined as well for specimen 3, which is why specimen 2 is used for these reflections even though there are not as many data points acquired in the regime of interest. Macro-mechanically, the material exhibits relative tension-compression symmetry during these initial deformations (Fig. 1), though symmetric microstructure evolution was not observed in inverse pole figure measurements.8,10 Average strains of unique orientations in the loading direction are plotted with respect to the macroscopic stress (rmacro ) in Fig. 2 and the macroscopic logarithmic strain (Lemacro ) in Fig. 3. In addition to the empirical measurements, expected elastic responses from SWW , SHKF are shown in both figures, noting that orientation-specific Young’s moduli (Ehkl ) may be calculated16 via Ehkl ¼ l41 s11 þ 2l21 l22 s12 þ 2l21 l23 s32 þ 2l31 l3 s15 þ l42 s22 þ 2l22 l23 s23 þ 2l1 l22 l3 s25 þ l43 s33 þ 2l1 l33 s35 þ l22 l23 s44 þ 2l1 l22 l3 s46 þ l21 l23 s55 þ l21 l22 s66 ;
(2)
where li are the direction cosines of a unit vector in the (hkl) direction with respect to a Cartesian coordinate system (i ¼ 1, 2, or 3 denotes the 1-, 2-, or 3-axis of the system, respectively). Knowing Ehkl , and assuming deformation of each orientation is initially elastic and that on average rmacro is initially uniformly distributed to the billions of observed crystallites, the following relations govern the initial Lemacro vs. ehkl and rmacro vs. ehkl responses: rmacro ¼ Ehkl ehkl ; Ehkl Lemacro ¼ ln 1 þ ehkl : Emacro
(3a) (3b)
HKF Calculations of EWW macro ¼ 111:9; Emacro ¼ 179:9 GPa as volume averaged from inverse pole figure measurements of each of the three undeformed specimens were previously reported,5,10 and these values were used to generate the responses of SWW , SHKF shown in Fig. 3 using Eq. (3b) and Ehkl calculated via Eq. (2) (Table I). The empirical data were also assessed to ascertain where each orientation ceases to experience predominantly linearelastic deformation. Measurements of ehkl are proportional to the average local stress experienced by each group of crystallites of a unique orientation, thus, when the relationship between ehkl and the macroscopic mechanical response (stress or strain) deviates from its initial linear proportion, inelastic deformation has begun to influence the (hkl) deformation. The number of data points and the macroscopic stress and strain bounds of linear-elastic deformation observed for each (hkl) are listed in Table I and indicated in Figs. 2 and 3. “Load 2” compression data were not used for any specimen, even if the unload data point after “Load 1” was contained
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211908-3
Stebner, Brown, and Brinson
Appl. Phys. Lett. 102, 211908 (2013)
FIG. 2. Strains (ehkl ) of 13 unique orientations of monoclinic crystallites are plotted vs. the macroscopic engineering stress (rmacro ). The elastic responses exhibited by the material in the regimes labeled “Elastic” are shown as EFit , and the elastic responses calculated from Refs. 4 and 6 using Eq. (2) are shown as EWW and EHKF , respectively. Error bars for ehkl are twice the statistical peak fitting error calculated by GSAS.
within the macroscopic elastic bounds, as the unload event was inelastic with respect to Lemacro vs. ehkl for all observed orientations since the macroscopic strain did not fully recover (Fig 1). Using Eq. (3a), Ehkl for each reflection were determined from the rmacro vs. ehkl elastic responses. The resulting EFit responses according to Eq. (3a) are shown in Fig. 2 and Fit EFit hkl values are listed in Table I. The Ehkl were then used to fit Emacro to the Lemacro vs. ehkl elastic responses according to Eq. (3b) for each reflection, which are also reported in Fit Table I. The EFit hkl together with Emacro were used in Eq. (3b) Fit to generate the E responses shown in Fig. 3.
In our micromechanical analysis, only the assumption of uniform macroscopic stress distribution to monoclinic crystallites is needed; additional assumption of macroscopic strain distribution to crystallites is not required. Equation (3a) is substituted into the macroscopic Hooke’s law to derive Eq. (3b). While linear Lemacro vs. ehkl and rmacro vs. ehkl relations may be used to assess elastic deformation, linearity alone does not validate the self-consistent micromechanical assumption17 of initially uniform macroscopic stress distribution. To gage the “uniform” part of the assumption, an average Emacro ¼ 66.7 GPa and standard deviation of
FIG. 3. Strains of 13 unique orientations (ehkl ) of monoclinic crystallites are plotted vs. the macroscopic logarithmic strain (Lemacro ) together with the linear-elastic response of each orientation exhibited in the regimes labeled “Elastic,” denoted as EFit . The elastic responses calculated from Refs. 4 and 6 using Eqs. (2) and (3) are also shown as EWW and EHKF , respectively. Error bars for ehkl are twice the statistical peak fitting error calculated by GSAS.
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211908-4
Stebner, Brown, and Brinson
Appl. Phys. Lett. 102, 211908 (2013)
Fit TABLE I. Orientations ðhklÞ and the macroscopic modulus (EFit macro ) and orientation-specific Young’s moduli (Ehkl ) fit to their elastic data, which are contained max min max ; r ; ½e ; e ). Pearson’s linear correlation coefficient (q) for each EFit within the macroscopic stress and strain window (½rmin macro macro macro macro hkl and the number of WW HKF data points (n) used in the fits are also listed. Finally, Ehkl calculated from Refs. 4 and 6 (S , S ) according to Eq. (2) are listed together with their relative differences (Edif f Þ to EFit hkl .
ðhklÞ
EFit macro (GPa)
EFit hkl (GPa)
q
n
Lemin macro (%)
Lemax macro (%)
rmin macro (MPa)
rmax macro (MPa)
EWW hkl (GPa)
EWW dif f (%)
EHKF hkl (GPa)
EHKF dif f (%)
ð100Þ ð030Þ ð002Þ ð111Þ ð111Þ ð011Þ ð012Þ ð121Þ ð121Þ ð110Þ ð120Þ ð112Þ ð122Þ
89.3 62.6 55.6 71.3 81.9 57.8 55.4 88.9 61.9 65.2 50.8 75.3 50.9
94.4 101.4 74.4 90.1 99.2 108.9 108.1 92.3 43.9 32.3 36.6 152.8 50.9
0.906 0.889 0.874 0.943 0.705 0.943 1.000 0.960 0.927 0.966 0.997 0.774 0.994
7 17 6 13 12 22 2 13 11 6 5 11 5
0.070 0.090 0.060 0.250 0.200 0.200 0.000 0.250 0.050 0.060 0.020 0.150 0.020
0.000 0.059 0.000 0.000 0.000 0.079 0.110 0.000 0.039 0.000 0.219 0.000 0.219
52.1 60.7 47.3 116.6 101.2 101.2 0.8 116.6 42.4 47.3 25.4 84.9 25.4
5.0 35.5 5.0 5.0 5.0 47.3 61.0 5.0 22.4 5.0 105.2 5.0 105.2
128.2 146.2 111.1 87.6 83.3 177.3 137.3 83.6 52.2 41.2 33.8 175.2 115.3
35.8 44.2 49.3 2.8 16.0 62.8 27.0 9.4 18.9 27.6 7.7 14.7 126.5
138.7 177.0 148.1 179.6 140.6 199.9 170.6 189.3 174.1 146.6 142.2 210.3 207.7
46.9 74.6 99.1 99.3 41.7 83.6 57.8 105.1 296.6 353.9 288.5 37.6 308.1
13.6 GPa was calculated from the 13 fits (Table I). Even though some orientations immediately exhibit inelastic deformation in each tension (ð111Þ, ð111Þ, ð121Þ, ð110Þ, and ð 112Þ) and compression (ð012Þ, ð120Þ, and ð122Þ) (Figs. 2 and 3), in considering only elastic ðhklÞ responses, the predicted macroscopic response is in agreement with the initial macroscopic loading data (Fig. 1). As determination of EFit hkl is independent of EFit macro (but not vise-versa), the mean and standard deviation indicate that the empirical analyses are self-consistent to 20%. Using EFit hkl , 13 equations were written relating each Ehkl to monoclinic NiTi S according to Eq. (2) using a previously established Cartesian coordinate system for the monoclinic lattice.4–6,10 The rank of the resulting matrix of coefficients of sij was nine, which was insufficient to uniquely determine all 13 monoclinic sij from the empirical data. However, five sij were uniquely determined from these relationships, and they WW HKF in Table II. are listed (sFit ij ) and compared with sij , sij Note that because S was not fully determined from the experimental measurements, S1 ¼ C could not be calculated. The reported average Emacro ¼ 66.7 GPa from fitting these data is lower (30–50%) than recent in situ diffraction reports of Emacro for random textured monoclinic NiTi (109, 134 GPa),2,3 but consistent with ultrasonic measurement of Emacro and an earlier in situ diffraction report for monoclinic NiTi and NiTi-based alloys (60–85 GPa).18 The primary difference between our report and other recent neutron TABLE II. The five measured elastic compliance constants are listed (sFit ij ), as well as those calculated from Refs. 4 and 6 (SWW , SHKF ). Also given are relative differences, denoted with subscript “dif f .”
ij
2 sFit ij 10 1 (GPa )
102 sWW ij (GPa1)
sww ij dif f (%)
102 sHKF ij (GPa1)
sHKF dif f ij (%)
11 15 22 33 35
1.15 1.10 1.34 1.06 1.54
0.978 1.45 0.900 0.780 0.113
15.0 31.8 32.8 26.4 92.7
0.604 0.150 0.675 0.721 0.036
47.5 86.4 49.6 32.0 102.3
diffraction reports is that we did not assume that all monoclinic orientations follow Eq. (3a) while the macroscopic deformation appears linear-like (i.e., between 680 MPa (Ref. 2) or 6200 MPa (Ref. 3)), but rather individually investigated the micromechanical responses of each ðhklÞ for deviation from initial linear-elastic assumptions. Here, we also used the Lemacro vs. ehkl responses to fit Emacro instead of assembling Emacro solely from the rmacro vs. ehkl responses. In comparing predicted and observed magnitudes of Ehkl , visually (Fig. 2) and statistically (Table I) SWW responses are closer to the observed elastic responses than SHKF . For ð111Þ, ð121Þ, ð120Þ, and ð121Þ orientations, however, SHKF provide a better model of Ehkl =Emacro than SWW (Fig. 3). In considering the five measured sij (Table II), SWW are again in better relative agreement with the empirical observations than SHKF . These results indicate that SWW provide the best complete model of elastic anisotropy for monoclinic crystallites deforming within polycrystalline NiTi, since our empirical measurements were only conclusive in assessing five of the 13 independent constants. Furthermore, these results suggest that allowing relaxation of atomic positions and imposing stress controls (as was done by WW but not HKF)4 during first principles computations may result in more realistic predictions of elastic constants for polycrystalline materials. Both calculations of elastic constants, however, are stiffer than the observations by 15% or more, and by more than an order of magnitude in the 35-direction (Table II), where sHKF 35 is also of the opposite sign. This quantitative discrepancy indicates that there is still advancement desired toward establishing best-practice methodologies for computing elastic anisotropy of crystalline materials through atomistic simulations. Understanding how to achieve these advancements, however, is bounded by challenges toward fabrication and in situ observation of single crystals at the atomic scale and 0 K. We thank Thomas Sisneros and Bjørn Clausen of LANL for experimental assistance and Ron Noebe of NASA Glenn Research Center for providing the NiTi specimens. This
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211908-5
Stebner, Brown, and Brinson
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 under DOE Contract No. DE-AC52-06NA25396. A.S. acknowledges funding through fellowships from the Toshio Mura Endowment, Predictive Science and Engineering Design Cluster at Northwestern (PSED), Initiative for Sustainability and Energy at Northwestern (ISEN). A.S. and C.B. acknowledge the support of the Army Research Office, Grant No. W911NF-12-1-0013/P00002. 1
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Appl. Phys. Lett. 102, 211908 (2013) 8
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