Temperature and composition dependence of the elastic constants of ...

Temperature and Composition Dependence of the Elastic Constants of Ni3Al S.V. PRIKHODKO, J.D. CARNES, D.G. ISAAK, H. YANG, and A.J. ARDELL The stiffness constants, cij, of monocrystalline Ni3Al of three different compositions, 23.2, 24.0, and 25.0 at. pct Al, were measured over the temperature range from 300 to 1100 K using the rectangular parallelepiped resonance (RPR) method. The bulk modulus, as well as the shear modulus, Young’s modulus, and Poisson’s ratio for randomly oriented polycrystalline stoichiometric Ni3Al, were derived from the stiffness constants. The data indicate that c44 is essentially independent of composition, decreasing slightly with increasing temperature for all three alloys. The values of c11 and c12, however, decrease with increasing aluminum content, the difference being small at room temperature but becoming larger at higher temperatures. We find that c11 and c12 are not as sensitive to aluminum concentration as is implied by previous results. A comparison of different shear moduli of Ni3Al and the saturated Ni-Al solid solution in equilibrium with it indicates that the ordered phase is generally elastically stiffer than the solid solution over the range of temperatures at which coarsening of the Ni3Al precipitate has been heavily investigated.

I. INTRODUCTION

THE ordered alloy Ni3Al (L12 structure) is technically important, because it is the so-called g 8 phase that occurs as precipitates in nickel-base superalloys. It is generally believed that the differences between the elastic constants and lattice parameters of the matrix and precipitates strongly influence the morphology and spatial correlations of the g 8 precipitates, which in turn can affect the mechanical properties of the alloys. We have recently measured the lattice constants of the equilibrium matrix and g 8 phases in four binary nickel-base alloys to temperatures up to ,1000 K.[1] We have also measured the cij of a monocrystalline NiAl solid solution alloy containing 12.69 at. pct Al to 1300 K.[2] These data, in conjunction with previous measurements of the cij on solid solutions of other compositions,[3] provide a thorough characterization of the cij as functions of temperature and composition for the Ni-Al solid solution over nearly the entire range of its phase stability. What is missing from a complete picture of the elastic behavior of two-phase NiAl g/g 8 alloys are measurements of the cij of Ni3Al as a function of composition and temperature. The single-crystal elastic constants cij of Ni3Al have been measured by four groups of investigators.[4–7] The data of Kayser and Stassis[4] and Wallow et al.,[5] who measured the cij of Ni3Al of stoichiometric composition, are in good agreement, but the results reported by the other two investigators are not. For example, the room-temperature values of the cij measured by Ono and Stern[6] and Dickson et al.[7] vary from 15 to 40 pct lower than those of Kayser and Stassis. However, the Ni3Al phase in the alloy of Ono and Stern contained 24 pct Al, while that in the alloy of Dickson et al. contained only 22.5 pct Al. These differences in the values of the cij are rather large and suggest that the cij could S.V. PRIKHODKO, Spectroscopist, H. YANG, Undergraduate Student, and A.J. ARDELL, Professor, Department of Materials Science and Engineering, and J.D. CARNES, Graduate Student, and D.G. ISAAK, Associate Researcher, Institute of Geophysics and Planetary Physics, are with the University of California, Los Angeles, CA 90095-1595. Manuscript submitted January 28, 1999. METALLURGICAL AND MATERIALS TRANSACTIONS A

be a very strong function of composition, as shown in Figure 1. In addition to the data in Figure 1, Franse et al. measured the Young’s modulus, E, of two polycrystalline alloys containing 24 and 25 pct Al and reported that E was much smaller in the alloy with the smaller aluminum concentration.[8] The apparent dependence of the elastic constants of Ni3Al with composition was a major factor prompting the present investigation. The possible dependence of the cij of Ni3Al on aluminum content, especially at compositions lower than 25 pct, is important because, in two-phase Ni-Al g/g 8 alloys, the Ni3Al phase contains only a little more than 23 pct Al.[9] Additionally, the previously measured cij[4,5] are reliable over the temperature range from 296.7 to 363 K, but the temperatures at which precipitation and coarsening have been measured are much higher, typically ranging from 700 to 1100 K. For this reason, our measurements were made at much higher temperatures than those used in previous investigations. II. EXPERIMENTAL PROCEDURES The rectangular parallelepiped resonance (RPR) method was used in the present investigation. The theoretical basis of the method is due to Demarest,[10] who derived the resonance frequencies of a vibrating specimen with cubic crystal symmetry and shape. Ohno[11] subsequently advanced the theory to include lower crystal symmetry and specimens in the shape of an arbitrary right-rectangular parallelepiped. Experimental details on the RPR method for measuring the temperature dependence of the elastic constants are provided by Sumino et al.,[12] and a modification of this technique, which extends the range of accessible temperatures to beyond 1700 K, was developed by Goto et al.[13] In the method of Goto et al., the sample is loosely held between two alumina rods at opposite corners; the rods are coupled to piezoelectric transducers using alumina rods. One transducer introduces a constant-amplitude signal of varying frequency in the radio frequency range, while the other detects the mechanical resonances as maxima in the transmitted signal. The spectrum of mechanical resonances can be calculated VOLUME 30A, SEPTEMBER 1999—2403

Fig. 1—Elastic constants of Ni3Al at room temperature as a function of Al concentration from previous work. Shown are the data of Kayser and Stassis (KS),[4] Wallow et al. (WNSN),[5] Ono and Stern (OS),[6] and Dickson et al. (DWC).[7]

knowing the elastic moduli, dimensions, and density of the specimen. There is no analytical method for solving the inverse problem of deducing the elastic moduli from the measured spectrum, which is why an indirect method is used. A starting set of resonant frequencies is calculated using estimated values of the cij (calculated theoretically or taken from the literature) together with the measured specimen dimensions and density. The difference between the calculated and measured resonant frequency spectrum is minimized, in a least-squares sense, by an iteration routine that provides a set of elastic constants that best fits the measured frequencies. Redundancy checks are made possible by observing more resonant modes than the number of independent cij of the specimen (three for cubic crystals such as Ni3Al). Ni3Al with three different compositions, 23.2, 24.0, and 25.0 at. pct Al, designated herein as alloys N1, N2, and N3, weighing about 40 g each, were arc-melted from pure nickel (99.99 pct) and aluminum (99.999 pct). Each alloy was remelted three times with negligible weight losses. Bulk samples containing large grains (more then 10 mm in “diameter”) were obtained by remelting the alloys at 1673 K in alumina crucibles in a vacuum furnace and very slowly cooling them through their melting ranges (1673 to 1623 K) over a period of 10 hours. Weight losses for each alloy varied between 0.5 and 0.7 pct. The final compositions were checked using energy-dispersive X-ray analysis and were consistent with the starting compositions, which should be regarded as nominal ones. After solidification, the samples were subjected to a homogenization anneal for 72 hours at 1473 K in a vacuum. The purpose of this annealing treatment was to enable diffusion to reduce the composition gradients within the dendritic regions of the crystals. Energy dispersive 2404—VOLUME 30A, SEPTEMBER 1999

X-ray chemical analysis indicated that the aluminum concentration increased as expected from the alloy having the lowest aluminum concentration (N1) to the alloy having the highest (N3), but no other measurements of concentration were performed. The microstructures of the alloys were checked using optical microscopy. Alloy N1 contained a small amount (up to 3 pct) of the Ni-Al solid solution phase, but no evidence was observed for the presence of this phase in alloys N2 and N3. For ease of interpretation of the data, the specimens used in the RPR technique are best shaped as right-rectangular parallelepipeds with faces normal to ^100&. One of the large grains in the annealed solidified buttons was chosen to be an RPR specimen and was oriented parallel to {100} using the back-reflection Laue method. The {100} face was then cut using a low-speed diamond wheel, after which the other faces were similarly oriented and cut. A final check of the orientation showed that deviation of each face from {100} did not exceed 1.5 deg. The faces of the specimens were mechanically abraded and metallographically polished to bring the dimensions of the samples to their final sizes, which are reported in Table I. The densities of the specimens were measured by the immersion method and also calculated using values of the lattice parameters at room temperature;[1,14] these are also shown in Table I. The small differences in the dimensions of the three sides of the parallelepipeds (Table I) were deliberately introduced to eliminate degeneracies in the resonant frequency spectra that occur when the lengths of each edge are identical. This also helps the measured resonant frequencies to be more readily identified as to their modal type. The measured densities were used for calculating the resonant spectra. The specimens were heated in a furnace with a Pt-13 pct Rh coil heating element. Temperature measurements were made using two Pt vs Pt-13 pct Rh thermocouples. Uncertainties in temperature due to measured gradients across the dimensions of the specimen are about 5 K at the highest temperatures attained. All testing was done in the protective atmosphere of argon; only minor oxidation of the samples was observed after heating. The measurements were repeated at selected temperatures, including room temperature before and after heating, and the results were found to be reproducible within experimental error (about 1.5 pct). This indicates that oxidation did not introduce errors into the measurements of the elastic constants. The PZT (PbZrO3-PbTiO3) transducers with a 1-MHz resonant frequency were attached to the ends of alumina buffer rods[15] and used to induce and detect the resonant frequencies. From initial estimates of the cij at room temperature, we theoretically calculated about 40 modal frequencies for each alloy. In reducing the data at each temperature, only modes that were reliably identified experimentally and did not cause mathematical instabilities in the recursion scheme were used for final calculation of the cij. For example, for alloy N1, 37 modes were used for all temperatures, whereas for alloy N2, 35 modes were used at 300 K, 34 modes from 400 to 700 K, and eight modes from 750 to 1100 K. For the stoichiometric alloy, eight modes were used over the entire temperature range up to 1100 K. In all cases, the dimensional changes associated with thermal expansion were included in reduction of the data. Numerical instabilities, most likely resulting from slight METALLURGICAL AND MATERIALS TRANSACTIONS A

Table I. Dimensions and Mass Densities of the Single-Crystal Specimens Used in the Measurements of the Elastic Constants Density (Mg/m3) Alloy

Pct Al

Edge Dimensions (mm)

Calculated

Measured

N1 N2 N3

23.2 24.0 25.0

4.616 6 0.002 3 4.633 6 0.003 3 4.662 6 0.002 4.734 6 0.005 3 4.779 6 0.005 3 4.821 6 0.001 4.927 6 0.003 3 4.965 6 0.005 3 5.021 6 0.007

7.5316 7.4831 7.4253

7.51 6 0.01 7.45 6 0.04 7.41 6 0.05

variations in either specimen geometry or concentration, necessitated the use of only eight modes during some calculations. Note that in each case the elastic constants are highly overdetermined, with 30 to 40 modes being used to calculate three constants; eight modes are sufficient, but having significantly more than that enables a more statistically significant calculation of the standard deviation. When each mode suggested a slightly different value for the calculated elastic constants, a mean value was taken. The standard deviation was calculated by considering the deviation of each contribution to the elastic constants. III. RESULTS AND DISCUSSION A. Data Analysis and Comparison with Previous Work Figure 2 illustrates the variation of c11, c12, and c44 for all three alloys with temperature; their values and standard deviations are reported in Table II. For all three alloys, the cij decrease with increasing temperature. The error limits at room temperature were less than 0.4 pct for all the cij for alloy N1, 1.2 pct for the alloy N2, and 0.7 pct for N3, but they increase at higher temperatures. The errors for alloy N3 were smaller than 0.9 pct over the entire temperature range but were larger than this for alloys N1 and N2, especially at high temperatures. The values of c44 for all three

Table II. The Experimentally Measured Stiffness Constants, cij, and Standard Deviations in GPa from 300 to 1100 K Alloy

T(K)

N1 23.2 pct Al

300 400 500 600 700 800 900 1000 1100 300 400 500 600 700 800 900 1000 1100 300 400 500 600 700 800 900 1000 1100

N2 24.0 pct

N3 25.0 pct

c11 229.3 226.5 223.3 221.0 217.7 214.3 210.9 207.8 202.4 229.4 225.8 222.1 218.0 213.6 209.1 205.2 201.5 197.5 224.5 220.4 216.4 212.4 208.6 203.6 198.9 194.2 187.8

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

c12 0.71 1.29 2.61 3.78 4.29 4.91 5.48 6.25 6.23 1.97 2.60 3.25 3.84 1.80 2.48 3.47 5.23 7.90 0.97 0.97 0.99 0.96 0.97 1.02 1.00 1.05 1.06

154.0 153.6 152.7 152.7 151.7 150.6 149.5 148.8 145.8 154.2 152.8 151.3 149.4 147.2 143.6 141.2 139.9 138.4 148.6 146.8 145.1 143.2 141.6 138.9 136.2 133.8 129.5

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

c44 0.68 1.26 2.59 3.78 4.30 4.91 5.48 6.24 6.22 1.93 2.55 3.22 3.83 1.82 2.50 3.51 5.31 8.04 1.06 1.06 1.08 1.05 1.05 1.10 1.07 1.13 1.14

123.7 120.8 117.9 115.0 111.9 108.8 105.6 102.3 98.9 123.8 121.1 118.4 115.6 112.6 109.6 106.4 103.2 99.6 124.4 121.6 118.6 115.7 112.6 109.4 106.4 103.2 99.8

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

0.35 0.34 0.32 0.32 0.31 0.32 0.33 0.35 0.33 0.88 0.86 0.84 0.81 0.86 0.87 0.87 0.86 0.85 0.39 0.38 0.38 0.37 0.36 0.37 0.36 0.37 0.37

alloys are identical, within the limits of experimental error over the entire temperature range. The data in Figure 2 can be represented by the empirical equation. cij 5 k1 1 k2T 1 k3X 1 k4XT 1 k5T 2 1 k6X 2

[1]

where T is the temperature in Kelvin and X is the concentration of aluminum in at. pct. The values of the coefficients ki are presented in Table III. The solid curves in Figure 2 were calculated using Eq. [1], and it is evident that this equation represents the data very well. Table III. Values of the Coefficients in Equation [1] Coefficient

Fig. 2—The variation of c11, c12, and c44 with temperature for the three Ni3Al alloys investigated. METALLURGICAL AND MATERIALS TRANSACTIONS A

k1 k2 k3 k4 k5 k6

Units

c11

c12

c44

GPa 2527.401 2256.199 1.815992 GPa/K 0.120378 0.158156 20.02761 GPa/pct Al 64.23207 34.80893 10.55093 GPa/K ? pct Al 20.00618 20.00682 0.000124 GPa/K2 28.2 3 1026 28.8 3 1026 24.3 3 1026 GPa/(pct Al)2 21.3482 20.73615 20.21386 VOLUME 30A, SEPTEMBER 1999—2405

The values of c11 and c12 decrease slightly with increasing aluminum concentration, the difference becoming more pronounced as the temperature increases. The difference between the cij for alloys N2 and N3 is almost twice as large as the experimental error over the entire temperature range, but the corresponding difference for alloys N1 and N2 is comparable to the experimental error. Despite this difference for alloys N1 and N2, the smooth variation of the cij with T for both alloys suggests that the difference observed is real. The present room-temperature data on alloy N3 are in excellent agreement with those of Kayser and Stassis[4] (the disagreement is 0.4, 0.3, and 0.6 pct for c11, c12, and c44, respectively) and Wallow et al.[5] (the disagreement here is somewhat larger, 1.9, 1.8, and 0.6 pct, respectively, but still quite small). Both groups of investigators reported errors in their measurements of about 6 0.25 pct. The room-temperature values of c11, c12, and c44 reported by Ono and Stern[6] differ from those of the present investigation by 13, 17, and 5 pct, respectively, while those reported by Dickson et al.[7] differ from ours by 26, 42, and 1.6 pct. A comparison of all the available room-temperature data on the stiffnesses of Ni-Al alloys, from 0 to 25 at. pct Al, is shown in Figure 3. The two-phase region of the Ni-Al phase diagram at temperatures from ,720 to 1075 K is indicated. As can be seen, the variation of the cij from pure nickel to Ni3Al does not exceed 15 pct; for c12 and c44, it is even smaller. The large disagreement with the data of Ono and Stern and Dickson et al. is difficult to explain, given the absence of

a strong dependence of the cij on aluminum concentration in Ni3Al. The fact that the specimens of Dickson et al. were not single-phase Ni3Al, but contained an unspecified amount of the Ni-Al solid solution, is unlikely to be the source of the large discrepancy between their data and all the others.

B. Estimates of the Polycrystalline Elastic Constants It is useful to estimate the polycrystalline elastic constants, assuming randomly oriented polycrystals, from the measured cij because these “isotropic” elastic constants are often of interest in engineering applications. To this end, the bulk modulus, B, was first calculated using the exact formula B5

c11 1 2c12 3

[2]

There are many options for estimating the shear modulus, G, and Young’s modulus, E, for comparison with previous measurements on “isotropic” polycrystalline Ni3Al. We have chosen to calculate an isotropic value of G using the mean of the upper Gu and lower Gl bounds of the Hashin–Shtrikman (HS)[17] values. The upper and lower bounds are given by the equations Gl 5 G110 1 3

F

G

F

G

5 2 4b 1 G100 2 G110

21

[3]

and 5 2 6b 2 Gu 5 G100 1 2 G110 2 G100

21

[4]

where G110 5

c11 2 c12 2

G100 5 c44

[5] [6]

b1 5

23 (B 1 2G110) 5G110 (3B 1 4G110)

[7]

b2 5

23 (B 1 2G100) 5G100 (3B 1 4G100)

[8]

and

Young’s modulus, E, and Poisson’s ratio, v, were then calculated from B and the mean value G 5 (Gl 1 Gu)/2 using the equations[16] 9BG 3B 1 G

[9]

3B 2 2G 2(3B 1 G)

[10]

E5 Fig. 3—Illustrating the variation of c11, c12, and c44 with Al concentration for the Ni-Al solid solution and Ni3Al phases. The shaded region on the left represents the approximate variation of the solubility limit of Al in Ni from ,720 to 1075 K. The uniformly shaded region labeled g 8 indicates the approximate range of homogeneity of the Ni3Al phase. The data are those of n Prikhodko et al.,[2] Pottebohm et al.,[3] ▫ Kayser and Stassis,[4] C Wallow et al.,[5] L Ono and Stern,[6] Dickson et al.,[7] and ● this investigation.

2406—VOLUME 30A, SEPTEMBER 1999

and v5

The dependencies of E, G, B, and v on T are reported in

METALLURGICAL AND MATERIALS TRANSACTIONS A

Table IV. Values of the Bulk Modulus, B, the Shear Modulus, G, Young’s Modulus, E, and Poisson’s Ratio, n, as Functions of Temperature for “Randomly Oriented Polycrystalline” Stoichiometric Ni3Al* T(K)

B

G

E

300 400 500 600 700 800 900 1000 1100

173.9 171.4 168.8 166.3 163.9 160.5 157.1 153.9 149.0

77.8 75.8 73.8 71.8 69.8 67.6 65.7 63.6 61.4

203.1 198.2 193.3 188.4 183.3 177.9 173.0 167.7 161.9

n 0.305 0.307 0.309 0.311 0.314 0.315 0.317 0.318 0.319

*All the moduli are in GPa.

Fig. 5—Illustrating the variation of the differences between the shear modulus of the g 8 and g phases with temperature. Three different estimates of the shear modulus are shown (see text), normalized to that of the g solid solution (ss), which in all cases has its equilibrium composition. The increase in solubility of the g phase with increasing temperature is responsible for the nonmonotonic behavior of DG/Gss.

between our calculated values of E in Figure 4 and those measured by Davies and Stoloff does not arise from the particular choice of the method of calculation used to obtain E from the cij. We note that the values of E measured by Franse et al.,[8] extrapolated to room temperature, are significantly smaller than the values seen in Figure 4. C. Implications for Spatial Correlations of g 8 Precipitates during Coarsening Fig. 4—The variation of Young’s modulus, E, with temperature. The data of Davies and Stoloff[18] are shown for comparison.

Table IV, and the previously reported data on E vs T of Davies and Stoloff[18] are compared with our calculated values of E in Figure 4. As can be seen, our calculated values of E depend on T in nearly the same manner as those of Davies and Stoloff, though their values are smaller by about 12 pct over the entire range of T. The discrepancy is possibly due to preferred orientation of the crystals in their specimen, but this is speculative because Davies and Stoloff provided no information on the texture of their material.* It is quite *It is perhaps worth pointing out that the values of E reported by Davies and Stoloff exceed, by a considerable margin, the values of Young’s modulus for tensile deformation parallel to ^100&, (c11 2 c12)(c11 1 2c12)/(c11 1 c12) ' 106 GPa at 300 K. They are therefore consistent with physical expectation.

possible that the method we have used to calculate E does not provide a representative value for randomly oriented polycrystalline Ni3Al. In this context, however, it is interesting to point out that we also calculated the upper and lower bounds for G using the formulas of Voigt and Reuss (VR).[19] The average values of G from the VR bounds are almost identical to those of the HS bounds; hence, the discrepancy


Many years ago Ardell and Nicholson[20] postulated that the tendency of g 8 precipitates to align along ^100& during coarsening could be attributed to elastic interactions stemming from the fact that G for the Ni3Al phase was smaller than that of the matrix phase. This postulate was consistent with a calculation of the elastic interaction energy due to Eshelby[20] and with the best data available at the time. It is interesting to revisit this hypothesis in light of the present data. Accordingly, the shear modulus mismatch between the g 8 phase in equilibrium with the Ni-Al solid solution of approximate equilibrium composition was calculated over the temperature range 650 to 1050 K, because nearly all the prior investigations of aging have been done on alloys aged in this range of temperatures.[21] In the calculation of differences in shear moduli, we regard both phases to have their thermodynamic equilibrium compositions. In the case of the g 8 phase, this is 23.13 at. pct Al,[9] which is nearly constant from 650 to 1050 K. The composition of the saturated solid solution increases with increasing temperature and in principle changes with the overall composition of a two-phase alloy because of the complications due to coherent equilibrium.[22,23] To simplify this issue, we have used a recently published equilibrium solubility curve[24] and have calculated G using the mean values of the HS bounds, as well as considering G100 and

VOLUME 30A, SEPTEMBER 1999—2407

G110, Eqs. [7] and [8], which represent shears on {100}^010& and {110}^110&, respectively. Values of cij for the Ni-Al solid solution were calculated from the empirical equation published by Prikhodko et al.[2] The results are plotted on Figure 5 in the form DG/Gss, where DG 5 Gg 8 2 Gss, and the subscripts represent the solid solution (ss) and g 8 phases. It is evident that over the temperature range 850 to 1000 K, DG/Gss . 0 for all three shear moduli. Evidently, in the temperature range where precipitation and coarsening have been quite thoroughly investigated, the shear modulus of the g 8 phase is larger than that of the matrix. This finding clearly contradicts the long-held belief that the alignment of the g 8 particles in aged Ni-Al alloys is due to the fact that they are elastically soft compared to the matrix phase. The combined roles of lattice mismatch and elastic constant mismatch obviously need to be re-examined for the situation in which both phases are elastically anisotropic; the original calculations of Eshelby[20] assumed elastic isotropy. ACKNOWLEDGMENTS Three of the authors (SVP, HY, and AJA) are grateful to the Department of Energy for its financial support of this research under Grant No. DE-FG03-96ER45573. JDC and DGI were supported by the Office of Naval Research and also thank the NSF for support under Grant No. EAR 9614654. REFERENCES 1. A.B. Kamara, A.J. Ardell, and C.N.J. Wagner: Metall. Mater. Trans. A., 1996, vol. 27A, pp. 2888-96. 2. S.V. Prikhodko, J.D. Carnes, D.G. Isaak, and A.J. Ardell: Scripta

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Mater., 1998, vol. 38, pp. 67-72. 3. N. Pottebohm, G. Neite, and E. Nembach: Mater. Sci. Eng., 1983, vol. 60, pp. 189-94. 4. F.X. Kayser and C. Stassis: Phys. Status Solidi (a), 1981, vol. 64, pp. 335-42. 5. F. Wallow, G. Neite, W. Schroer, and E. Nembach: Phys. Status Solidi (a), 1987, vol. 99, pp. 483-90. 6. K. Ono and R. Stern: Trans. AIME, 1969, vol. 245, pp. 171-72. 7. R.W. Dickson, J.B. Wachtman, Jr., and S.M. Copley: J. Appl. Phys., 1969, vol. 40, pp. 2276-79. 8. J.J.M. Franse, M. Rosena, and E.P. Wohlfarth: Physica, 1977, vol. 40, pp. 317-18. 9. A. Taylor and R.W. Floyd: J. Inst. Met., 1952–53, vol. 81, pp. 25-32. 10. H.H. Demarest: J. Acoust. Soc. Am., 1971, vol. 49, pp. 768-75. 11. I. Ohno: J. Phys. Earth, 1976, vol. 24, pp. 355-79. 12. Y. Sumino, O. Nishizawa, T. Goto, I. Ohno, and M. Ozima: J. Phys. Earth, 1977, vol. 25, pp. 377-92. 13. T. Goto, S. Yamamoto, I. Ohno, and O.L. Anderson: J. Geophys. Res., 1989, vol. 94, pp. 7588-7602. 14. A.J. Bradley and A. Taylor: Proc. R. Soc., 1937, vol. A159, pp. 56-72. 15. T. Goto and O.L. Anderson: Rev. Sci. Instrum., 1988, vol. 59, pp. 1405-08. 16. E. Schreiber, O.L. Anderson, and N. Soga: Elastic Constants and Their Measurements, McGraw-Hill, New York, NY, 1973. 17. Z. Hashin and S. Shtrikman: J. Mech. Phys. Solids, 1962, vol. 10, pp. 343-52. 18. R.G. Davies and N.S. Stoloff : Trans. AIME, 1965, vol. 233, pp. 714-19. 19. G. Simmons and H. Wang: Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook, The MIT Press, Cambridge, MA, 1971. 20. A.J. Ardell, R.B. Nicholson, and J.D. Eshelby: Acta Metall., 1966, vol. 14, pp. 1295-1309. 21. A.J. Ardell: in Phase Transformations ’87, G. Lorimer, ed., The Institute of Metals, London, 1988, pp. 485-94. 22. F. Li and A.J. Ardell: Scripta Metall., 1997, vol. 37, pp. 1123-28. 23. F. Li and A.J. Ardell: J. Phase Equil., 1998, vol. 19, pp. 334-39. 24. A.J. Ardell: in Experimental Methods of Phase Diagram Determination, J.E. Morral, R.S. Schiffman, and S.M. Merchant, eds., TMS, Warrendale, PA, 1994, pp. 57-66.
