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Resonant ultrasound spectroscopy: applications, current status and limitations
Journal of Alloys and Compounds 310 (2000) 243–250
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www.elsevier.com / locate / jallcom
Resonant ultrasound spectroscopy: applications, current status and limitations q R.B. Schwarz*, J.F. Vuorinen
1
Materials Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Abstract Over the last 10 years, resonant ultrasound spectroscopy (RUS) has become a versatile laboratory technique for measuring second-order elastic constants and ultrasonic attenuation in solids. The technique is based on measuring the spectrum of mechanical resonances for a sample of known shape (usually a parallelepiped). This spectrum cannot be deconvoluted to deduce the elastic constants. Instead, an approximate spectrum is calculated from the known sample dimensions, its mass, and a set of ‘guessed’ elastic constants. A multidimensional minimization of the error between the measured and calculated spectra enables deducing all the elastic constants of the solid from a single frequency scan. Currently, the technique can be applied to crystals of orthorhombic symmetry (9 elastic constants) or higher using desktop computers and software developed for this purpose. 2000 Published by Elsevier Science S.A. Keywords: Ultrasonics; Acoustic properties; Amorphous materials
1. Introduction Traditionally, the elastic properties of solids have been derived from measurements of the phase velocity of plane acoustic waves. Techniques such as the pulse-echo method [1,2], although quite powerful, are sometimes difficult to implement because (a) the technique requires large single crystals and (b) a number of independent measurements, often on separate samples, are needed to fully characterize the elastic properties of the crystal. While the pulse-echo method can be easily applied to crystals of high symmetry (e.g. cubic), it becomes progressively difficult to use as the crystal symmetry decreases. This is because, to determine all the elastic constants, one is eventually forced to measure sound velocities along some non-principle crystal directions where the phase velocity and the group velocity are no longer colinear [3]. Additional difficulties with the pulse-echo technique are encountered when doing measurements at very low or very
q Invited paper presented at the ‘12th International Conference on Internal Friction and Ultrasonic Attenuation of Solids’, Buenos Aires, Argentina, July 18–23, 1999. *Corresponding author. E-mail address: [email protected] (R.B. Schwarz), [email protected] (J.F. Vuorinen). 1 Permanent address: Tampere University of Technology, Institute of Materials Science, Tampere SF-33 101, Finland.
high temperatures. At low temperatures one is faced with the problem of preparing an effective bond between the thin piezoelectric transducer and the sample [4], and at high temperatures one faces the lack of high-temperature piezoelectric materials. Most of these problems can be avoided when using the resonant ultrasonic spectroscopy (RUS) technique. The origins of RUS are traceable to geophysics problems. A great achievement of geophysics has been the use of the measured resonant vibration modes of the Earth as a whole, excited by a major earthquake, to find the elastic properties of the planet. To that end, various pioneering authors derived solutions to the equations of motion of an isotropic and homogeneous elastic sphere [5–7]. It was soon realized that the seismology data could not be explained in terms of a homogeneous sphere and that the Earth had to be modeled by a layered sphere, much akin an onion. Frazer and LeCraw [8] and Holland [9] were the first to apply the RUS method to materials-science problems. Frazer and LeCraw used a one-transducer experimental setup to measure the resonant frequencies of a homogeneous sphere. They then used the known analytical solution for the vibrations of a homogeneous sphere and a graphical method to deduce the elastic properties of the specimen. Geophysicists Soga and Anderson [10] improved the method of Frazer and LeCraw by developing a two-transducer setup, with the sample in-between, which is
0925-8388 / 00 / $ – see front matter 2000 Published by Elsevier Science S.A. PII: S0925-8388( 00 )00925-7
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still widely used today. A first success of the RUS method was the measurement of the elastic moduli of small spherical lunar rocks (|0.3 mm diameter) brought back by the Apollo missions. Clearly, the elastic properties of these small samples could not be studied by the conventional pulse-echo techniques. Surprisingly, the sound velocity in the lunar minerals were found to be extremely low, on the order of 1.3 km / s, rather than the expected value of about 6 km / s typical of earth rocks. This prompted Schreiber and Anderson [11] to joke that the moon was indeed made out of cheese, which has similarly low sound velocity. The discrepancy was later found to be due to the high density of microcracks in the lunar samples. Earth rocks may have similar cracks but these are usually filled with water, enabling the transmission of the acoustic wave across the crack. In 1971, H. Demarest, a student of O. Anderson, solved the problem of the resonant modes of a cubic sample of cubic symmetry, with the axis of the sample oriented along high-symmetry crystallographic directions [12]. Further advances were made by Ohno, who derived numerical methods to calculate the resonant modes of parallelepipedshaped samples having tetragonal, orthorhombic, and trigonal symmetry [13,14]. These solutions required the use of digital computers, which were becoming available by that time. Together, the papers of Demarest and Ohno cover nearly all the important aspects of RUS. Excellent reviews of the RUS technique have been written by Heyliger et al. [15], Maynard [16], Leisure and Willis [17], and Migliori and Sarrao [18].
2. Experimental
2.1. Calculation of the resonant frequencies The calculation of the resonant frequencies starts with the Lagrangian of the elastic body. A free-standing body can sustain non-attenuated vibrations at a series of resonant frequencies, vm . At these frequencies, the time-independent part of the Lagrangian,
E
L I 5 Cijkl ´ij (u)´kl (u) dV
(1)
V
is an extremum under the normalization condition
E u u d dV 5 1, i j ij
(2)
Legendre polynomials as the basis functions, noting that these functions lead to a rapid convergence in the numerical solution for a cube-shaped specimen. Ohno used the same basis functions. Visscher [19] used a power series expansion as the basis and showed that, although the convergence is slower, the numerical problem is quite solvable, especially with the current fast desk-top computers. The power series expansion is simpler to implement for samples of arbitrary shape. The numerical code developed by Visscher is now available in commercial programs [20] applicable to samples of cylindrical and parallelepiped geometry. The derivation of the elastic constant tensor by the RUS method is an indirect iterative procedure. Knowing the sample dimensions, density, and elastic constants, one can easily calculate the spectrum of resonant frequencies, as described above. The inverse problem of deducing the elastic constants from a measured spectrum of mechanical resonances, however, has no known solution. For the indirect method, a starting resonant frequency spectrum, f cal (n51,2, . . . ), is calculated using estimated values for n the elastic constants (from theory or from literature data for similar materials) and the known sample dimensions and density. The difference between the calculated and measured resonance frequency spectrum, f mea (n5 n 1,2, . . . ), is quantified by a figure-of-merit function, F5
O w (f n
n
cal n
2 f nmea )2 ,
(3)
where w n (n51, 2, . . . ) are weight coefficients reflecting the confidence on individual resonance measurements. Then, a minimization of the function F is sought by regressing the values of all the elastic constants using computer software developed for this process. In effect, such programs search for a global minimum of the function F in a multidimensional space of dimensions equal to the number of unknown elastic constants. The curvature of F for the various elastic constants is different and thus some elastic constants are determined with higher precision than others. The accuracy of the RUS method depends on the quality of the sample and on the number of resonances measured. Experience has shown that this number should be at least 5 to 10 times the number of independent elastic constants to be determined. Typically, the pure shear elastic constants are the easiest to determine, with errors of less than 0.1%. The off-diagonal elastic constants are determined with the least accuracy, typically about 2% [17,18].
V
and the extrema gives rv 2m . Here Cijkl is the elastic constant tensor and ´ij (u) is the position-dependent strain tensor. The solutions are found by expanding the displacements u in basis functions appropriate to the geometry of the body. By truncating the basis at a convenient limit N, the problem is reduced to that of diagonalizing a N 3N matrix (eigenvalue problem). Demarest chose normalized
2.2. Contact RUS measurements The most common method for detecting the mechanical resonant spectrum is illustrated in Fig. 1, where a small parallelepiped-shaped sample is lightly held between two piezoelectric transducers. One transducer is used to generate an elastic wave of constant amplitude and varying
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Fig. 3. Section of the resonant spectrum for a parallelepiped-shaped polycrystalline Ti–6Al–4V sample. Fig. 1. Schematic of the two-transducer resonant ultrasonic spectroscopy set up.
frequency, whereas the other is used to detect the resonances. The sample held at opposite corners, acts in effect as a high-Q, multi-frequency, narrow-band filter. To avoid electrical and mechanical cross talk between the two transducers, and to control the force applied to the sample, researchers have developed various transducer configurations [16,18]. The simple and inexpensive configuration shown in Fig. 2, used by Ledbetter et al. [21] and in our Laboratory, can be used to measure resonant frequencies in the range 10 kHz to 5 MHz, which covers the range of interest in most applications. In this configuration, two ‘pinducers’ [22] are held axially aligned on an aluminum block by means of four rubber ‘O’ rings. The ‘O’ rings avoid the acoustic cross talk at room temperature. Fig. 3 shows three of the many resonant modes for a poly-
Fig. 2. RUS two-transducer configuration based on ‘pinducers’.
crystalline sample of Ti–6Al–4V measured at room temperature using the fixture in Fig. 2. By replacing the rubber ‘O’ rings with Teflon sleeves, this configuration can be used down to cryogenic temperatures. The Curie temperature of the piezoelectric transducers and the stability of polymeric bonds limit the highest temperature at which the setup of Figs. 1 and 2 can be used. Darling et al. [23] using small lithium niobate transducers held by thin metallic membranes, developed a system useful to 750 K. Goto and Anderson [24] used alumina buffer rods to separate the transducers from the hot sample and extended the usable temperature range of the RUS technique up to 1820 K.
2.3. Non-contact RUS measurements It is possible to measure the resonant spectrum of specimens without a physical contact between sample and transducers, in what can be termed non-contact RUS. Such techniques are useful for characterizing samples that must be kept in controlled atmospheres. In the following we describe two non-contact RUS methods that rely on magnetic coupling. The first method is based on coating the sample with a thin film of a magnetostrictive material (e.g. nickel) whereas the second is based on the excitation of eddy currents in the presence of a constant magnetic field.
2.3.1. Magnetostrictive non-contact RUS method In the RUS method, the sample is usually a good mechanical resonator, with a figure of merit Q of at least 300. Because little energy is needed to maintain a high-Q sample at resonance, the resonances can be excited via a thin magnetostrictive film (,5 mm thick) attached to the sample, as shown schematically in Fig. 4. The coated sample is placed inside two coaxial solenoids, S 1 and S 2 , which act as driving and receiving transducers, respective-
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Fig. 4. Schematic of the non-contact RUS method based on the magnetostrictive excitation of resonances.
ly [25]. To enable the free mechanical resonances, the sample can be rested on loosely packed glass wool. An ac voltage of amplitude V1 and frequency v is applied to the excitation coil S 1 to generate a sinusoidal magnetic field H(v t) at the sample. The magnetostrictive film responds to the applied field generating a periodic stress, which, for the appropriate value of to, drives the sample into one of its mechanical resonances. The resonances are detected by solenoid S 2 . To decrease the direct magnetic coupling between S 1 and S 2 , the detection solenoid S 2 consists of two identical coils wound clockwise and counter-clockwise. The resonances are detected because the permeability of the magnetostrictive film decreases when the sample approaches a resonant frequency and the magnetostrictive film changes from a strain-free state (away from the resonance) to a stress-free state (at resonance). This technique allows RUS measurements between cryogenic temperatures and approximately 1000 K. The Curie temperature of the magnetostrictive material determines the upper temperature limit.
2.3.2. EMAT-based non-contact RUS methods Measurements of the elastic properties of solids via a direct electromagnetic transduction (EMAT) date to the pioneering work of Filimonov [26] and Nikiforengo [27], who used EMAT techniques to measure the velocity of propagation of plane waves in thin metal sheets. Johnson et al. [28,29] showed that EMAT could also be used to excite a free-standing spherical sample into any of its mechanical resonances, thus combining EMAT and RUS. To excite the resonances using EMAT, the conductive sample is placed inside a permanent magnetic field, H0 ¯1500 Oe, as shown schematically in Fig. 5 [30]. In addition, the sample is surrounded by a solenoid. Bursts of ac current at frequency v are sent to the solenoid, which induce in the sample surface ac-current bursts at frequency v. This ac current interacts with the field H0 generating alternating Lorentz forces near the sample surface. Mechanical resonances are excited at the appropriate values of v. The same solenoid is used to detect the resonances by measuring the induced signal during the time intervals in between successive
Fig. 5. Schematic of the non-contact EMAT / RUS method. (a) Sample surrounded by a solenoid is placed inside a magnetic field. (b) Schematic of the Lorentz forces generated on the sample surface [after Ref. [30]].
excitation current bursts. In addition, the temporal decay of the signal yields the internal losses in the sample. In another application of the EMAT / RUS combination, ‘trapped’ torsional resonant modes in aluminum bars have been used to develop high-resolution load transducers [31,32].
3. Selected examples of the RUS method
3.1. Measurements in amorphous metallic alloys Since the 1970’s, metallic glasses have been prepared by the rapid solidification of molten alloys at cooling rates on the order of 10 6 K s 21 . During the last five years, researchers have discovered methods to prepare some of these alloys in bulk form, at cooling rates of only 10 K s 21 [33,34]. The availability of bulk metallic glasses has enabled us to measure their elastic properties using RUS techniques [34,35]. An isotropic amorphous alloy has only two independent elastic constants. Fig. 6 shows the shear and bulk modulus of amorphous (Pd 0.625 Cu 0.375 ) 1002x Px alloys as a function of the phosphorous content, x [36]. These measurements were performed on cylindrical specimens of approximately 5 mm diameter and 6 mm long using a RUS setup similar to that shown in Fig. 1. Surprisingly, as the phosphorus content is varied, the elastic properties of these alloys go through extreme values. One possible interpretation of these results is in terms of varying degrees of short-range order in the asprepared metallic glasses, but this interpretation must be corroborated with independent atomistic structural measurements. In our present thinking, the Pd–Cu–P alloy with x519 has the lowest degree of short-range order and this is reflected as a minimum in the shear modulus and a maximum in the bulk modulus.
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Fig. 6. Shear and bulk moduli of amorphous (Pd 0.625 Cu 0.375 ) 1002x Px alloys as a function of the phosphorous content.
3.2. Elastic constants of composite materials Continuous-fiber (CF) reinforced composites can be tailored to achieve higher specific strength, stiffness, and creep resistance than found in monolithic materials. Microscopically, composites are elastically heterogeneous. Macroscopically, a composite specimen can be considered statistically homogeneous if the distribution of fibers within its volume is homogeneous and the separation between the fibers is much smaller than its overall dimensions. Predicting the deformation of a composite under arbitrary loading conditions requires a detailed knowledge of its average elastic properties. The simplest CF-composite is one for which the fibers are straight and homogeneously distributed, all aligned in one direction. The elastic prop-
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erties of this solid are homogeneous on a plane normal to the fiber direction. Such a transversely isotropic solid is characterized by only five independent elastic constants, similarly to a hexagonal-symmetry crystal. These five moduli describe the elastic properties of a homogeneous solid that is elastically equivalent to the heterogeneous composite [37]. Although the five elastic constants give a complete description of the stiffness of the composite, such description has not been generally adopted. The main difficulty has been measuring the five independent elastic constants. Engineers have often opted for a pragmatic, often incomplete, description of the elastic properties of composites in terms of a reduced number of engineering moduli, which are amenable to determination via static tests. The RUS technique alleviates this problem since the complete elastic constant tensor can be easily measured, as described next for continuous-fiber reinforced Al /Al 2 O 3 and Ti / SiC composites. The five independent elastic constants of an Al /Al 2 O 3 sample were measured at room temperature on parallelepiped-shaped specimens of 9310311 mm, using the simple apparatus shown in Fig. 2. These results are reported in Ref. [38]. Engineering applications of metal– matrix composites often require knowing the elastic properties of the material at off-axis loading conditions. Knowing the complete elastic constant tensor enables one to deduce the engineering moduli for any loading condition. As an example, Fig. 7 shows the Young’s modulus as a function of the angle b from the axial direction of the fibers. The figure also shows the values of moduli we calculated using rule-of-mixtures type equations and the known elastic properties of the matrix and fibers [38]. For increasing values of b, the measured Young’s modulus decreases rapidly from 230 GPa to about 137 GPa, reflecting the expected behavior for a soft aluminum matrix reinforced with stiff unidirectional continuous Al 2 O 3 fibers. It is remarkable that the stiffening effect of
Fig. 7. Young’s modulus E11 as a function of loading angle b for an aluminum matrix composite reinforced with unidirectional alumina fibers.
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the fibers has been almost completely lost for b 5458C. The five independent elastic constants of the Ti / SiC composite were measured from 10 to 293 K using a transducer setup similar to that in Fig. 2. These values were used to calculate the engineering moduli of the composite (solid symbols). These are shown in Fig. 8, together with the Young’s and shear moduli for polycrystalline titanium (open symbols) [39]. The elastic moduli of the two materials show similar temperature dependence. The main difference is that the E11 modulus for the composite is relatively insensitive to changing temperature, increasing by only 4% on cooling from 300 to 10 K. In comparison, the Young’s modulus of titanium increases by 10% over the same range. The difference may be attributed to the low temperature dependence of the Young’s modulus of the SiC fibers, for which we have not found values in the literature. Additional examples on the use of the RUS technique for the measurement of the elastic properties of composites can be found in the work of Ledbetter et al. [21].
3.3. Elastic constants measurements in very small samples One clear advantage of the RUS method over the conventional pulse-echo technique is its ability to measure the complete elastic constant tensor of very small samples.
Fig. 8. Engineering moduli of a unidirectional Ti–6Al–4V/ SiC composite (solid symbols) and Young’s and shear moduli of pure Titanium (open symbols) as a function of temperature.
In the following, we review some of the RUS measurements on small specimens. Over the last several years, there has been a growing interest in gallium nitride, a wide bandgap (3.4 eV) compound semiconductor. Due to the breakthrough in 1992 that made it possible to dope GaN to p-type, p–n junctions of GaN can now be fabricated. The ability to make GaN p–n junctions, as well as heterojunctions in the AlGaN / GaN / InGaN system, has opened the way for the development of blue light emitting diodes (which are already commercialized) and more recently, laser diodes. In addition, the wide bandgap and high melting point of GaN makes it, along with SiC, a promising material for high power and high temperature electronic devices. A major problem in the synthesis of GaN based devices is the lack of a lattice-matched substrate suitable for the epitaxial growth of defect-free thin films. The lattice mismatch of GaN with sapphire, the most commonly used substrate for optoelectronic GaN devices, is 213%, which causes large stresses in the epilayer. To decrease this misfit stress, a highly defective sacrificial buffer layer of AlN or GaN is used. The problem of determining the misfit stresses and stress distributions in the epilayer is critical for assessing the quality of the grown epilayers. To correctly estimate the misfit stress it is necessary to know accurately the elastic constants of GaN. Until recently, the elastic moduli of GaN were poorly known since GaN had only been available as thin films. A 0.285 mm thick wafer of GaN, grown at the Ioffee Physico-Technical Institue, Russia, became available in 1997, enabling us to measure its elastic properties. An oriented 0.2853232.3 mm 3 crystal, weighing about 8 mg, was cut from this wafer using a low-speed diamond-impregnated wire saw. We then used the RUS technique to measure the five second-order elastic constants of the sample. The results in units of GPa are: c 11 5 377, c 12 5 160, c 13 5 114, c 33 5 209, and c 44 5 81.4 [40]. These values are expected to be accurate to within a few percent. Some of the present elastic constants differ with previous measurements by more than 100%. The work of Spoor et al. [41] is another demonstration of the applicability of the RUS technique to high precision elastic constant measurements in small samples. Following the discovery of quasicrystals, researchers were interested in measuring their elastic constants. Of particular interest was determining whether AlCuLi quasicrystals were isotropic, as predicted by theory. However, good quality AlCuLi quasicrystals were only available as small specimens. Furthermore, these quasicrystals were brittle, and excessive pressure could have fractured the samples. An excessive pressure on the sample may also shift some of the resonant frequencies. Thus, the authors designed a special specimen holder in which the sample is placed between two 9 mm thick polyvinylidene fluoride films, which operate as piezoelectric transducers (see Fig. 1). With it, the authors were able to measure the resonant
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spectra of 70 and 280 mg AlCuLi quasicrystals, cut in parallelepiped shape. They used this data to demonstrate that AlCuLi quasicrystals are isotropic to within 0.07%, whereas the closely related cubic AlCuLi crystal is anisotropic. Following the discovery of superconductivity in ceramic perovskites, researchers became interested in measuring the elastic constants of these materials, since changes in elastic constants at the superconducting transition temperature may provide important clues as to the mechanism of superconductivity. Lei et al. [42] used the RUS method to measure the 9-independent elastic constants of a 1.443 1.1630.27 mm 3 orthorhombic single crystal of superconductor YBa 2 Cu 3 O 72d , the largest single crystal of this material available at that time. These measurements were taken as acoustic evidence against BSC-type superconductivity in YBa 2 Cu 3 O 72d [43].
3.4. Use of the RUS technique in quality control The application of mechanical resonances in the field of non-destructive evaluation is quite old. Early documented cases of British railroad engineers tapping the wheels of a train and using the sound to detect cracks is perhaps the first real use of resonant spectroscopy for non-destructive evaluation of parts [44]. As the RUS method is very sensitive to errors in the geometry, integrity, and homogeneity of the samples, it lends itself to high accuracy nondestructive evaluation of manufactured parts. Examples of RUS techniques to nondestructive evaluation of parts include cylindrical rolling equipment, ceramic oxygen sensors, ceramic substrates for integrated circuits, aircraft landing gears and wheels, and precision ball-bearings [45– 47]. Ledbetter et al. have studied the changes in the resonant spectrum of a steel bar caused by the introduction of a crack [48]. An illustrating example of non-destructive evaluation using a RUS technique is the testing of spheres for bearing applications. For a perfect sphere, the lowest resonant mode is 5-fold degenerate. If the sphere has cracks, roundness errors, texturing or / and internal strains, the five modes will no longer have the same resonant frequency and the resonance splits into 3 or 5 peaks. The splitting enables one to observe error in roundness of less than 1 part in a million [49]. The application of the RUS technique in non-destructive evaluation is not limited to small manufactured parts. A slightly modified RUS technique has been used to test the structural integrity of a 6-lane, 175-foot-long interstate highway bridge. The normal resonant modes of the bridge were recorded during normal traffic using a vehiclemounted microwave interferometer as transducer. Then, a small cut was made to the main I-beam of the bridge. Although this cut was too small to cause the bridge to fail, it was easily detected by the RUS method [47].
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4. Limitations and the future of the RUS techniques The current numerical codes for deducing the elastic constants from a measured resonant spectrum are applicable to samples of orthorhombic symmetry or higher (trigonal, tetragonal, hexagonal, and cubic). The accuracy of the RUS method is clearly dependent on the quality of the sample. The technique can be applied to parallelepipeds, cylinders, or spheres. Experience has shown that the best sample geometry for the RUS measurements is a parallelepiped. The reason is that some of the normal resonant models of cubes and cylinders are degenerate. Unless the sample shape is perfect, these modes will not overlap. The presence of closely spaced resonances makes mode identification difficult. In addition, the samples (cubes, parallelpipeds, or cylinders) must have their principal geometric directions carefully oriented along principal crystallographic axes. Clearly, this last restriction does not apply to isotropic materials. Researchers are expanding the limits of the RUS methods. In the coming years we will see the development of apparatus for the measuring the resonant modes of samples at very high temperatures. Certainly the use of buffer rods, as done by Goto and Anderson [24] is a viable solution, but the combination of RUS with the EMAT excitation of resonances may prove to be a simpler solution. Third-order elastic constants play an important role in solid state physics and in geophysics. They allow an evaluation of the anharmonicity in the interatomic potentials and thus describe the coupling between ultrasonic waves and lattice vibration waves. The third-order constants are also required for accurate equations of state used in the investigation of solid materials under pressure (e.g., planetary interiors). Of prime importance in these equations of state is the pressure dependence of the adiabatic bulk modulus. The extension of the RUS technique to measurements at elevated pressures has not received much attention, in part because there is no available theory to interpret the data. As discussed in Section 2.1, the analytical solutions for resonant modes derived by Ohno and Demarest assume that the surfaces of the vibrating specimens are free of external stresses. With the specimen under a hydrostatic pressure, this condition is no longer met. This change in the boundary conditions has an unknown effect on the resonant modes. Although data on third-order elastic constants can be derived from pressure derivatives of the second-order elastic constants [50], these changes cannot be directly deduced from RUS measurements made under hydrostatic pressures. The problem is that acoustic energy is lost by radiation into the pressurizing medium (usually a gas) and this lowers the figure of merit Q of the resonator, appearing as an additional change in the second-order elastic constants. Isaak et al. [51] used RUS to measure the Q of a fused silica parallelepiped excited into mechanical resonance while under compressed air, helium, or argon. As
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expected, the measured effects are larger for resonant modes that are mainly compressional than torsional. The pressure derivative of the shear modulus was found to depend linearly on the molecular mass of the gas. An extrapolation of the measured derivatives to zero molecular mass gave the sought-after value of (≠G / ≠P) T 5 23.42. Zhang et al. [52] derived simple models for the effects of pressure on other pressure derivatives, obtaining order-ofmagnitude agreement with the measurements. These results suggest to us that further insight into the effect of the pressurizing gas could be obtained by using CO 2 as a pressurizing gas. CO 2 becomes supercritical above a pressure of 73 Atm and a temperature of 318C. It is likely that the drastic decrease in its viscosity will lower the acoustic dissipation into the pressure medium sufficiently to make the gas-mass correction unnecessary. Then, the pressure derivatives of the complete second-order elastic constant tensor could be determined from two simple RUS measurements taken on the same sample at zero pressure and above the critical point of the gas.
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[21] H. Ledbetter, C. Fortunko, P. Heyliger, J Appl. Phys. 78 (1995) 1542. [22] Valpey Fisher, Hopkinton, Massachusetts 01748, USA. [23] Y. He, R.B. Schwarz, T. Darling, M. Hundley, S.H. Whang, Z.M. Wang, Mater. Sci. Eng. A239–A240 (1997) 157. [24] T. Goto, O.L. Anderson, Rev. Sci. Instrum. 59 (1988) 1405. [25] V.-T. Kuokkala, R.B. Schwarz, J. Sci. Instrum. 63 (1992) 3136. [26] S. Filimonov, B.A. Budenkov, N.A. Glukhov, Sov. J. Nondestr. Test. 7 (1971) 102. [27] Z.G. Nikiforengo, N.A. Glukhov, I.I. Averbukh, Sov. J. Nondestr. Test. 7 (1971) 427. [28] W.L. Johnson, S.J. Norton, F. Bendec, R. Pless, J. Acoust. Soc. Am. 91 (1992) 2637. [29] W.L. Johnson, B.A. Auld, G.A. Alers, J. Acoust. Soc. Am. 95 (1994) 1413. [30] W. Johnson, S.J. Norton, F. Bendec, R. Pless, J. Acoust. Soc. Am. 91 (1992) 2637. [31] W. Johnson, G.A. Alers, Rev. Sci. Instrum. 68 (1997) 102. [32] W.L. Johnson, G.A. Alers, and B.A. Auld, U. S. Patent No. 5 813 280 (1998) and No. 5 895 856. [33] A. Peker, W.L. Johnson, Appl. Phys. Lett. 63 (1993) 2342. [34] Y. He, T.D. Shen, R.B. Schwarz, Metall. Mater. Trans. 29A (1998) 1795. [35] Y. He, R.B. Schwarz, D. Mandrus, L. Jacobson, J. Non-Cryst. Solids 205–207 (1996) 602. [36] O. Jin, R.B. Schwarz, Unpublished Results, Los Alamos National Laboratory (1999). [37] R.M. Christensen, Mechanics of Composite Materials, John Wiley & Sons, New York, 1979. [38] J.E. Vuorinen, R.B. Schwarz, C. McCullough, J. Acous. Soc. Am. (in press, 1999). [39] G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties, MIT Press, 1971. [40] R.B. Schwarz, K. Khachaturyan, E.R. Weber, Appl. Phys. Lett. 70 (1997) 1122. [41] P.S. Spoor, J.D. Maynard, A.R. Kortan, Phys. Rew. Lett. 75 (1995) 3426. [42] M. Lei, J.L. Sarrao, W.M. Visscher, T.M. Bell, J.D. Thompson, A. Migliori, U.W. Welp, B.M. Veal, Phys. Rev. B 47 (1993) 6154. [43] M. Lei, A. Migliori, H. Ledbetter, Physica C 282–287 (1997) 1077. [44] A. Migliori, T.W. Darling, Ultrasonics 34 (1996) 473. [45] Using Resonant Inspection in Production Environment, Quasar International, Inc., Albuquerque, NM http: / / www.qusarintl.com / ndi). [46] A. Migliori, T.M. Bell, R.D. Dixon, R. Strong, Los Alamos National Laboratory Report LA-UR-93-225, 1993. [47] A. Migliori, T.W. Darling, Los Alamos National Laboratory Report LALP-95-9, 1995. [48] H. Ledbetter, P. Heyliger, K.-C. Pei, S. Kim, C. Fortunko, in: Review of Progress in Quantitative Nondestructive Evaluation, Vol. 14, Plenum, New York, 1995, pp. 2019–2025. [49] A. Migliori, R.D. Dixon, T.W. Darling, M. Lei, G. Rhodes, Los Alamos National Laboratory Report LA-UR-93-3193, 1993. [50] R.N. Thurston, K. Brugger, Phys. Rev. A 133 (1964) 1604. [51] D.G. Isaak, J.D. Carnes, O.L. Anderson, H. Oda, Acoust. Soc. Am. 104 (1998) 2200. [52] H. Zhang, R.S. Sorbello, J. Herro, J.R. Feller, D.E. Beck, M. Levy, Acoust. Soc. Am. 103 (1998) 2385.
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www.elsevier.com / locate / jallcom
Resonant ultrasound spectroscopy: applications, current status and limitations q R.B. Schwarz*, J.F. Vuorinen
1
Materials Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Abstract Over the last 10 years, resonant ultrasound spectroscopy (RUS) has become a versatile laboratory technique for measuring second-order elastic constants and ultrasonic attenuation in solids. The technique is based on measuring the spectrum of mechanical resonances for a sample of known shape (usually a parallelepiped). This spectrum cannot be deconvoluted to deduce the elastic constants. Instead, an approximate spectrum is calculated from the known sample dimensions, its mass, and a set of ‘guessed’ elastic constants. A multidimensional minimization of the error between the measured and calculated spectra enables deducing all the elastic constants of the solid from a single frequency scan. Currently, the technique can be applied to crystals of orthorhombic symmetry (9 elastic constants) or higher using desktop computers and software developed for this purpose. 2000 Published by Elsevier Science S.A. Keywords: Ultrasonics; Acoustic properties; Amorphous materials
1. Introduction Traditionally, the elastic properties of solids have been derived from measurements of the phase velocity of plane acoustic waves. Techniques such as the pulse-echo method [1,2], although quite powerful, are sometimes difficult to implement because (a) the technique requires large single crystals and (b) a number of independent measurements, often on separate samples, are needed to fully characterize the elastic properties of the crystal. While the pulse-echo method can be easily applied to crystals of high symmetry (e.g. cubic), it becomes progressively difficult to use as the crystal symmetry decreases. This is because, to determine all the elastic constants, one is eventually forced to measure sound velocities along some non-principle crystal directions where the phase velocity and the group velocity are no longer colinear [3]. Additional difficulties with the pulse-echo technique are encountered when doing measurements at very low or very
q Invited paper presented at the ‘12th International Conference on Internal Friction and Ultrasonic Attenuation of Solids’, Buenos Aires, Argentina, July 18–23, 1999. *Corresponding author. E-mail address: [email protected] (R.B. Schwarz), [email protected] (J.F. Vuorinen). 1 Permanent address: Tampere University of Technology, Institute of Materials Science, Tampere SF-33 101, Finland.
high temperatures. At low temperatures one is faced with the problem of preparing an effective bond between the thin piezoelectric transducer and the sample [4], and at high temperatures one faces the lack of high-temperature piezoelectric materials. Most of these problems can be avoided when using the resonant ultrasonic spectroscopy (RUS) technique. The origins of RUS are traceable to geophysics problems. A great achievement of geophysics has been the use of the measured resonant vibration modes of the Earth as a whole, excited by a major earthquake, to find the elastic properties of the planet. To that end, various pioneering authors derived solutions to the equations of motion of an isotropic and homogeneous elastic sphere [5–7]. It was soon realized that the seismology data could not be explained in terms of a homogeneous sphere and that the Earth had to be modeled by a layered sphere, much akin an onion. Frazer and LeCraw [8] and Holland [9] were the first to apply the RUS method to materials-science problems. Frazer and LeCraw used a one-transducer experimental setup to measure the resonant frequencies of a homogeneous sphere. They then used the known analytical solution for the vibrations of a homogeneous sphere and a graphical method to deduce the elastic properties of the specimen. Geophysicists Soga and Anderson [10] improved the method of Frazer and LeCraw by developing a two-transducer setup, with the sample in-between, which is
0925-8388 / 00 / $ – see front matter 2000 Published by Elsevier Science S.A. PII: S0925-8388( 00 )00925-7
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still widely used today. A first success of the RUS method was the measurement of the elastic moduli of small spherical lunar rocks (|0.3 mm diameter) brought back by the Apollo missions. Clearly, the elastic properties of these small samples could not be studied by the conventional pulse-echo techniques. Surprisingly, the sound velocity in the lunar minerals were found to be extremely low, on the order of 1.3 km / s, rather than the expected value of about 6 km / s typical of earth rocks. This prompted Schreiber and Anderson [11] to joke that the moon was indeed made out of cheese, which has similarly low sound velocity. The discrepancy was later found to be due to the high density of microcracks in the lunar samples. Earth rocks may have similar cracks but these are usually filled with water, enabling the transmission of the acoustic wave across the crack. In 1971, H. Demarest, a student of O. Anderson, solved the problem of the resonant modes of a cubic sample of cubic symmetry, with the axis of the sample oriented along high-symmetry crystallographic directions [12]. Further advances were made by Ohno, who derived numerical methods to calculate the resonant modes of parallelepipedshaped samples having tetragonal, orthorhombic, and trigonal symmetry [13,14]. These solutions required the use of digital computers, which were becoming available by that time. Together, the papers of Demarest and Ohno cover nearly all the important aspects of RUS. Excellent reviews of the RUS technique have been written by Heyliger et al. [15], Maynard [16], Leisure and Willis [17], and Migliori and Sarrao [18].
2. Experimental
2.1. Calculation of the resonant frequencies The calculation of the resonant frequencies starts with the Lagrangian of the elastic body. A free-standing body can sustain non-attenuated vibrations at a series of resonant frequencies, vm . At these frequencies, the time-independent part of the Lagrangian,
E
L I 5 Cijkl ´ij (u)´kl (u) dV
(1)
V
is an extremum under the normalization condition
E u u d dV 5 1, i j ij
(2)
Legendre polynomials as the basis functions, noting that these functions lead to a rapid convergence in the numerical solution for a cube-shaped specimen. Ohno used the same basis functions. Visscher [19] used a power series expansion as the basis and showed that, although the convergence is slower, the numerical problem is quite solvable, especially with the current fast desk-top computers. The power series expansion is simpler to implement for samples of arbitrary shape. The numerical code developed by Visscher is now available in commercial programs [20] applicable to samples of cylindrical and parallelepiped geometry. The derivation of the elastic constant tensor by the RUS method is an indirect iterative procedure. Knowing the sample dimensions, density, and elastic constants, one can easily calculate the spectrum of resonant frequencies, as described above. The inverse problem of deducing the elastic constants from a measured spectrum of mechanical resonances, however, has no known solution. For the indirect method, a starting resonant frequency spectrum, f cal (n51,2, . . . ), is calculated using estimated values for n the elastic constants (from theory or from literature data for similar materials) and the known sample dimensions and density. The difference between the calculated and measured resonance frequency spectrum, f mea (n5 n 1,2, . . . ), is quantified by a figure-of-merit function, F5
O w (f n
n
cal n
2 f nmea )2 ,
(3)
where w n (n51, 2, . . . ) are weight coefficients reflecting the confidence on individual resonance measurements. Then, a minimization of the function F is sought by regressing the values of all the elastic constants using computer software developed for this process. In effect, such programs search for a global minimum of the function F in a multidimensional space of dimensions equal to the number of unknown elastic constants. The curvature of F for the various elastic constants is different and thus some elastic constants are determined with higher precision than others. The accuracy of the RUS method depends on the quality of the sample and on the number of resonances measured. Experience has shown that this number should be at least 5 to 10 times the number of independent elastic constants to be determined. Typically, the pure shear elastic constants are the easiest to determine, with errors of less than 0.1%. The off-diagonal elastic constants are determined with the least accuracy, typically about 2% [17,18].
V
and the extrema gives rv 2m . Here Cijkl is the elastic constant tensor and ´ij (u) is the position-dependent strain tensor. The solutions are found by expanding the displacements u in basis functions appropriate to the geometry of the body. By truncating the basis at a convenient limit N, the problem is reduced to that of diagonalizing a N 3N matrix (eigenvalue problem). Demarest chose normalized
2.2. Contact RUS measurements The most common method for detecting the mechanical resonant spectrum is illustrated in Fig. 1, where a small parallelepiped-shaped sample is lightly held between two piezoelectric transducers. One transducer is used to generate an elastic wave of constant amplitude and varying
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Fig. 3. Section of the resonant spectrum for a parallelepiped-shaped polycrystalline Ti–6Al–4V sample. Fig. 1. Schematic of the two-transducer resonant ultrasonic spectroscopy set up.
frequency, whereas the other is used to detect the resonances. The sample held at opposite corners, acts in effect as a high-Q, multi-frequency, narrow-band filter. To avoid electrical and mechanical cross talk between the two transducers, and to control the force applied to the sample, researchers have developed various transducer configurations [16,18]. The simple and inexpensive configuration shown in Fig. 2, used by Ledbetter et al. [21] and in our Laboratory, can be used to measure resonant frequencies in the range 10 kHz to 5 MHz, which covers the range of interest in most applications. In this configuration, two ‘pinducers’ [22] are held axially aligned on an aluminum block by means of four rubber ‘O’ rings. The ‘O’ rings avoid the acoustic cross talk at room temperature. Fig. 3 shows three of the many resonant modes for a poly-
Fig. 2. RUS two-transducer configuration based on ‘pinducers’.
crystalline sample of Ti–6Al–4V measured at room temperature using the fixture in Fig. 2. By replacing the rubber ‘O’ rings with Teflon sleeves, this configuration can be used down to cryogenic temperatures. The Curie temperature of the piezoelectric transducers and the stability of polymeric bonds limit the highest temperature at which the setup of Figs. 1 and 2 can be used. Darling et al. [23] using small lithium niobate transducers held by thin metallic membranes, developed a system useful to 750 K. Goto and Anderson [24] used alumina buffer rods to separate the transducers from the hot sample and extended the usable temperature range of the RUS technique up to 1820 K.
2.3. Non-contact RUS measurements It is possible to measure the resonant spectrum of specimens without a physical contact between sample and transducers, in what can be termed non-contact RUS. Such techniques are useful for characterizing samples that must be kept in controlled atmospheres. In the following we describe two non-contact RUS methods that rely on magnetic coupling. The first method is based on coating the sample with a thin film of a magnetostrictive material (e.g. nickel) whereas the second is based on the excitation of eddy currents in the presence of a constant magnetic field.
2.3.1. Magnetostrictive non-contact RUS method In the RUS method, the sample is usually a good mechanical resonator, with a figure of merit Q of at least 300. Because little energy is needed to maintain a high-Q sample at resonance, the resonances can be excited via a thin magnetostrictive film (,5 mm thick) attached to the sample, as shown schematically in Fig. 4. The coated sample is placed inside two coaxial solenoids, S 1 and S 2 , which act as driving and receiving transducers, respective-
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Fig. 4. Schematic of the non-contact RUS method based on the magnetostrictive excitation of resonances.
ly [25]. To enable the free mechanical resonances, the sample can be rested on loosely packed glass wool. An ac voltage of amplitude V1 and frequency v is applied to the excitation coil S 1 to generate a sinusoidal magnetic field H(v t) at the sample. The magnetostrictive film responds to the applied field generating a periodic stress, which, for the appropriate value of to, drives the sample into one of its mechanical resonances. The resonances are detected by solenoid S 2 . To decrease the direct magnetic coupling between S 1 and S 2 , the detection solenoid S 2 consists of two identical coils wound clockwise and counter-clockwise. The resonances are detected because the permeability of the magnetostrictive film decreases when the sample approaches a resonant frequency and the magnetostrictive film changes from a strain-free state (away from the resonance) to a stress-free state (at resonance). This technique allows RUS measurements between cryogenic temperatures and approximately 1000 K. The Curie temperature of the magnetostrictive material determines the upper temperature limit.
2.3.2. EMAT-based non-contact RUS methods Measurements of the elastic properties of solids via a direct electromagnetic transduction (EMAT) date to the pioneering work of Filimonov [26] and Nikiforengo [27], who used EMAT techniques to measure the velocity of propagation of plane waves in thin metal sheets. Johnson et al. [28,29] showed that EMAT could also be used to excite a free-standing spherical sample into any of its mechanical resonances, thus combining EMAT and RUS. To excite the resonances using EMAT, the conductive sample is placed inside a permanent magnetic field, H0 ¯1500 Oe, as shown schematically in Fig. 5 [30]. In addition, the sample is surrounded by a solenoid. Bursts of ac current at frequency v are sent to the solenoid, which induce in the sample surface ac-current bursts at frequency v. This ac current interacts with the field H0 generating alternating Lorentz forces near the sample surface. Mechanical resonances are excited at the appropriate values of v. The same solenoid is used to detect the resonances by measuring the induced signal during the time intervals in between successive
Fig. 5. Schematic of the non-contact EMAT / RUS method. (a) Sample surrounded by a solenoid is placed inside a magnetic field. (b) Schematic of the Lorentz forces generated on the sample surface [after Ref. [30]].
excitation current bursts. In addition, the temporal decay of the signal yields the internal losses in the sample. In another application of the EMAT / RUS combination, ‘trapped’ torsional resonant modes in aluminum bars have been used to develop high-resolution load transducers [31,32].
3. Selected examples of the RUS method
3.1. Measurements in amorphous metallic alloys Since the 1970’s, metallic glasses have been prepared by the rapid solidification of molten alloys at cooling rates on the order of 10 6 K s 21 . During the last five years, researchers have discovered methods to prepare some of these alloys in bulk form, at cooling rates of only 10 K s 21 [33,34]. The availability of bulk metallic glasses has enabled us to measure their elastic properties using RUS techniques [34,35]. An isotropic amorphous alloy has only two independent elastic constants. Fig. 6 shows the shear and bulk modulus of amorphous (Pd 0.625 Cu 0.375 ) 1002x Px alloys as a function of the phosphorous content, x [36]. These measurements were performed on cylindrical specimens of approximately 5 mm diameter and 6 mm long using a RUS setup similar to that shown in Fig. 1. Surprisingly, as the phosphorus content is varied, the elastic properties of these alloys go through extreme values. One possible interpretation of these results is in terms of varying degrees of short-range order in the asprepared metallic glasses, but this interpretation must be corroborated with independent atomistic structural measurements. In our present thinking, the Pd–Cu–P alloy with x519 has the lowest degree of short-range order and this is reflected as a minimum in the shear modulus and a maximum in the bulk modulus.
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Fig. 6. Shear and bulk moduli of amorphous (Pd 0.625 Cu 0.375 ) 1002x Px alloys as a function of the phosphorous content.
3.2. Elastic constants of composite materials Continuous-fiber (CF) reinforced composites can be tailored to achieve higher specific strength, stiffness, and creep resistance than found in monolithic materials. Microscopically, composites are elastically heterogeneous. Macroscopically, a composite specimen can be considered statistically homogeneous if the distribution of fibers within its volume is homogeneous and the separation between the fibers is much smaller than its overall dimensions. Predicting the deformation of a composite under arbitrary loading conditions requires a detailed knowledge of its average elastic properties. The simplest CF-composite is one for which the fibers are straight and homogeneously distributed, all aligned in one direction. The elastic prop-
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erties of this solid are homogeneous on a plane normal to the fiber direction. Such a transversely isotropic solid is characterized by only five independent elastic constants, similarly to a hexagonal-symmetry crystal. These five moduli describe the elastic properties of a homogeneous solid that is elastically equivalent to the heterogeneous composite [37]. Although the five elastic constants give a complete description of the stiffness of the composite, such description has not been generally adopted. The main difficulty has been measuring the five independent elastic constants. Engineers have often opted for a pragmatic, often incomplete, description of the elastic properties of composites in terms of a reduced number of engineering moduli, which are amenable to determination via static tests. The RUS technique alleviates this problem since the complete elastic constant tensor can be easily measured, as described next for continuous-fiber reinforced Al /Al 2 O 3 and Ti / SiC composites. The five independent elastic constants of an Al /Al 2 O 3 sample were measured at room temperature on parallelepiped-shaped specimens of 9310311 mm, using the simple apparatus shown in Fig. 2. These results are reported in Ref. [38]. Engineering applications of metal– matrix composites often require knowing the elastic properties of the material at off-axis loading conditions. Knowing the complete elastic constant tensor enables one to deduce the engineering moduli for any loading condition. As an example, Fig. 7 shows the Young’s modulus as a function of the angle b from the axial direction of the fibers. The figure also shows the values of moduli we calculated using rule-of-mixtures type equations and the known elastic properties of the matrix and fibers [38]. For increasing values of b, the measured Young’s modulus decreases rapidly from 230 GPa to about 137 GPa, reflecting the expected behavior for a soft aluminum matrix reinforced with stiff unidirectional continuous Al 2 O 3 fibers. It is remarkable that the stiffening effect of
Fig. 7. Young’s modulus E11 as a function of loading angle b for an aluminum matrix composite reinforced with unidirectional alumina fibers.
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the fibers has been almost completely lost for b 5458C. The five independent elastic constants of the Ti / SiC composite were measured from 10 to 293 K using a transducer setup similar to that in Fig. 2. These values were used to calculate the engineering moduli of the composite (solid symbols). These are shown in Fig. 8, together with the Young’s and shear moduli for polycrystalline titanium (open symbols) [39]. The elastic moduli of the two materials show similar temperature dependence. The main difference is that the E11 modulus for the composite is relatively insensitive to changing temperature, increasing by only 4% on cooling from 300 to 10 K. In comparison, the Young’s modulus of titanium increases by 10% over the same range. The difference may be attributed to the low temperature dependence of the Young’s modulus of the SiC fibers, for which we have not found values in the literature. Additional examples on the use of the RUS technique for the measurement of the elastic properties of composites can be found in the work of Ledbetter et al. [21].
3.3. Elastic constants measurements in very small samples One clear advantage of the RUS method over the conventional pulse-echo technique is its ability to measure the complete elastic constant tensor of very small samples.
Fig. 8. Engineering moduli of a unidirectional Ti–6Al–4V/ SiC composite (solid symbols) and Young’s and shear moduli of pure Titanium (open symbols) as a function of temperature.
In the following, we review some of the RUS measurements on small specimens. Over the last several years, there has been a growing interest in gallium nitride, a wide bandgap (3.4 eV) compound semiconductor. Due to the breakthrough in 1992 that made it possible to dope GaN to p-type, p–n junctions of GaN can now be fabricated. The ability to make GaN p–n junctions, as well as heterojunctions in the AlGaN / GaN / InGaN system, has opened the way for the development of blue light emitting diodes (which are already commercialized) and more recently, laser diodes. In addition, the wide bandgap and high melting point of GaN makes it, along with SiC, a promising material for high power and high temperature electronic devices. A major problem in the synthesis of GaN based devices is the lack of a lattice-matched substrate suitable for the epitaxial growth of defect-free thin films. The lattice mismatch of GaN with sapphire, the most commonly used substrate for optoelectronic GaN devices, is 213%, which causes large stresses in the epilayer. To decrease this misfit stress, a highly defective sacrificial buffer layer of AlN or GaN is used. The problem of determining the misfit stresses and stress distributions in the epilayer is critical for assessing the quality of the grown epilayers. To correctly estimate the misfit stress it is necessary to know accurately the elastic constants of GaN. Until recently, the elastic moduli of GaN were poorly known since GaN had only been available as thin films. A 0.285 mm thick wafer of GaN, grown at the Ioffee Physico-Technical Institue, Russia, became available in 1997, enabling us to measure its elastic properties. An oriented 0.2853232.3 mm 3 crystal, weighing about 8 mg, was cut from this wafer using a low-speed diamond-impregnated wire saw. We then used the RUS technique to measure the five second-order elastic constants of the sample. The results in units of GPa are: c 11 5 377, c 12 5 160, c 13 5 114, c 33 5 209, and c 44 5 81.4 [40]. These values are expected to be accurate to within a few percent. Some of the present elastic constants differ with previous measurements by more than 100%. The work of Spoor et al. [41] is another demonstration of the applicability of the RUS technique to high precision elastic constant measurements in small samples. Following the discovery of quasicrystals, researchers were interested in measuring their elastic constants. Of particular interest was determining whether AlCuLi quasicrystals were isotropic, as predicted by theory. However, good quality AlCuLi quasicrystals were only available as small specimens. Furthermore, these quasicrystals were brittle, and excessive pressure could have fractured the samples. An excessive pressure on the sample may also shift some of the resonant frequencies. Thus, the authors designed a special specimen holder in which the sample is placed between two 9 mm thick polyvinylidene fluoride films, which operate as piezoelectric transducers (see Fig. 1). With it, the authors were able to measure the resonant
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spectra of 70 and 280 mg AlCuLi quasicrystals, cut in parallelepiped shape. They used this data to demonstrate that AlCuLi quasicrystals are isotropic to within 0.07%, whereas the closely related cubic AlCuLi crystal is anisotropic. Following the discovery of superconductivity in ceramic perovskites, researchers became interested in measuring the elastic constants of these materials, since changes in elastic constants at the superconducting transition temperature may provide important clues as to the mechanism of superconductivity. Lei et al. [42] used the RUS method to measure the 9-independent elastic constants of a 1.443 1.1630.27 mm 3 orthorhombic single crystal of superconductor YBa 2 Cu 3 O 72d , the largest single crystal of this material available at that time. These measurements were taken as acoustic evidence against BSC-type superconductivity in YBa 2 Cu 3 O 72d [43].
3.4. Use of the RUS technique in quality control The application of mechanical resonances in the field of non-destructive evaluation is quite old. Early documented cases of British railroad engineers tapping the wheels of a train and using the sound to detect cracks is perhaps the first real use of resonant spectroscopy for non-destructive evaluation of parts [44]. As the RUS method is very sensitive to errors in the geometry, integrity, and homogeneity of the samples, it lends itself to high accuracy nondestructive evaluation of manufactured parts. Examples of RUS techniques to nondestructive evaluation of parts include cylindrical rolling equipment, ceramic oxygen sensors, ceramic substrates for integrated circuits, aircraft landing gears and wheels, and precision ball-bearings [45– 47]. Ledbetter et al. have studied the changes in the resonant spectrum of a steel bar caused by the introduction of a crack [48]. An illustrating example of non-destructive evaluation using a RUS technique is the testing of spheres for bearing applications. For a perfect sphere, the lowest resonant mode is 5-fold degenerate. If the sphere has cracks, roundness errors, texturing or / and internal strains, the five modes will no longer have the same resonant frequency and the resonance splits into 3 or 5 peaks. The splitting enables one to observe error in roundness of less than 1 part in a million [49]. The application of the RUS technique in non-destructive evaluation is not limited to small manufactured parts. A slightly modified RUS technique has been used to test the structural integrity of a 6-lane, 175-foot-long interstate highway bridge. The normal resonant modes of the bridge were recorded during normal traffic using a vehiclemounted microwave interferometer as transducer. Then, a small cut was made to the main I-beam of the bridge. Although this cut was too small to cause the bridge to fail, it was easily detected by the RUS method [47].
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4. Limitations and the future of the RUS techniques The current numerical codes for deducing the elastic constants from a measured resonant spectrum are applicable to samples of orthorhombic symmetry or higher (trigonal, tetragonal, hexagonal, and cubic). The accuracy of the RUS method is clearly dependent on the quality of the sample. The technique can be applied to parallelepipeds, cylinders, or spheres. Experience has shown that the best sample geometry for the RUS measurements is a parallelepiped. The reason is that some of the normal resonant models of cubes and cylinders are degenerate. Unless the sample shape is perfect, these modes will not overlap. The presence of closely spaced resonances makes mode identification difficult. In addition, the samples (cubes, parallelpipeds, or cylinders) must have their principal geometric directions carefully oriented along principal crystallographic axes. Clearly, this last restriction does not apply to isotropic materials. Researchers are expanding the limits of the RUS methods. In the coming years we will see the development of apparatus for the measuring the resonant modes of samples at very high temperatures. Certainly the use of buffer rods, as done by Goto and Anderson [24] is a viable solution, but the combination of RUS with the EMAT excitation of resonances may prove to be a simpler solution. Third-order elastic constants play an important role in solid state physics and in geophysics. They allow an evaluation of the anharmonicity in the interatomic potentials and thus describe the coupling between ultrasonic waves and lattice vibration waves. The third-order constants are also required for accurate equations of state used in the investigation of solid materials under pressure (e.g., planetary interiors). Of prime importance in these equations of state is the pressure dependence of the adiabatic bulk modulus. The extension of the RUS technique to measurements at elevated pressures has not received much attention, in part because there is no available theory to interpret the data. As discussed in Section 2.1, the analytical solutions for resonant modes derived by Ohno and Demarest assume that the surfaces of the vibrating specimens are free of external stresses. With the specimen under a hydrostatic pressure, this condition is no longer met. This change in the boundary conditions has an unknown effect on the resonant modes. Although data on third-order elastic constants can be derived from pressure derivatives of the second-order elastic constants [50], these changes cannot be directly deduced from RUS measurements made under hydrostatic pressures. The problem is that acoustic energy is lost by radiation into the pressurizing medium (usually a gas) and this lowers the figure of merit Q of the resonator, appearing as an additional change in the second-order elastic constants. Isaak et al. [51] used RUS to measure the Q of a fused silica parallelepiped excited into mechanical resonance while under compressed air, helium, or argon. As
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expected, the measured effects are larger for resonant modes that are mainly compressional than torsional. The pressure derivative of the shear modulus was found to depend linearly on the molecular mass of the gas. An extrapolation of the measured derivatives to zero molecular mass gave the sought-after value of (≠G / ≠P) T 5 23.42. Zhang et al. [52] derived simple models for the effects of pressure on other pressure derivatives, obtaining order-ofmagnitude agreement with the measurements. These results suggest to us that further insight into the effect of the pressurizing gas could be obtained by using CO 2 as a pressurizing gas. CO 2 becomes supercritical above a pressure of 73 Atm and a temperature of 318C. It is likely that the drastic decrease in its viscosity will lower the acoustic dissipation into the pressure medium sufficiently to make the gas-mass correction unnecessary. Then, the pressure derivatives of the complete second-order elastic constant tensor could be determined from two simple RUS measurements taken on the same sample at zero pressure and above the critical point of the gas.
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