Rectangular Parallelepiped Resonance Research Articles

Elastic constants and anelastic properties of an anisotropic periclase sphere as determined by the resonant sphere technique

The resonant sphere technique (RST) is applied to measurements of elastic and anelastic properties of single‐crystal periclase (MgO) in order to demonstrate the potential value of this method as an experimental technique. The lowest 22 resonant frequencies and their attenuation properties were measured in the range 1.0 to 2.6 MHz. The oscillation modes were identified by comparing the observed resonant frequencies with computed ones. Using inversion theory, we find perturbations from an assumed set of elastic constants, so that the computed frequencies agree, in a least squares sense, with the measured ones. The final values of the elastic constants were obtained by adding the perturbations to the initial values of elastic constants. Furthermore, the internal friction coefficients were determined from the observed vibrational attenuation data by a least squares method. The values of elastic constants and internal friction coefficients are in good agreement with those determined by other methods. We find better accuracy, by up to 1 order of magnitude, than that found by using the rectangular parallelepiped resonance method. The RST can be an accurate method for measuring elastic and anelastic properties of small anisotropic materials. An applicability of the RST to the Earth's free oscillation is considered. Free oscillation of an elastically anisotropic sphere shows different frequency spectrum, depending on the degree of the elastic anisotropy. The frequency spectrum of an anisotropic sphere can be interpreted as the result of the split of degenerate modes of an isotropic sphere. Therefore the anomalous splitting of the Earth's free oscillation, which cannot be explained by the rotation and ellipticity of the Earth, may be attributed to the elastic anisotropy of the Earth.

Elastic constants of Ni3(Al,Mn) + B from 180 to 613 K

Intermetallic compounds have been the subject of increasingly intensive research in recent years, primarily because of their great potential for high temperature applications. Alloy development, especially improvement of inherently poor ductility of intermetallics has been the focus of the research work, but much effort has also been put to the determination of the mechanical properties of these alloys. For any material of scientific or practical interest, one of the fundamental properties to be determined is the elastic constants or elastic moduli of the material. In the characterization of materials, the elastic moduli are of special importance. They relate microscopically measurable physical properties directly to atomic level interactions in the crystal lattices or in the disordered arrangement of atoms in amorphous materials. For both scientific and technical purposes, it is also desirable to know how the elastic moduli of the material change with temperature. To extend the measuring range of rectangular parallelepiped resonance method to higher temperatures, the authors have developed a new non-contact measurement technique, based on magnetostrictive excitation and detection of the resonances in the sample. This technique is applicable form cryogenic temperatures up to about 950 C. In this work, the authors used this new measurement technique tomore » determine the elastic constants of Mn- and B-modified polycrystalline Ni[sub 3]Al intermetallic alloy in the temperature range 180--613 K. Both manganese and boron have been observed to improve the ductility of Ni[sub 3]Al.« less

Measurement of Elastic Moduli by Rectangular Parallelepiped Resonance Method II

The rectangular parallelepiped resonance method can be a very powerful method if universal procedure to obtain elastic moduli from measured resonance frequencies is established. The procedure proposed in this paper has the possibility to determine all elastic moduli of crystals with symmetry down to orthorhombic symmetry. However, we have only confirmed that the method gives correct elastic moduli of specimen with cubic symmetry without previous information. Although we need a great many of measured resonance frequencies from the lowest frequency to very high frequencies to determine all independent elastic moduli of specimen having lower symmetry than cubic symmetry, we can determine the shear moduli (C44, C55, C66) from much fewer of measured resonance frequencies.

Measurement of Elastic Modulus on Hydroxyapatite Ceramics Using the Rectangular Parallelepiped Resonance(RPR)Method

Measurement of Elastic Modulus on Hydroxyapatite Ceramics Using the Rectangular Parallelepiped Resonance(RPR)Method

Effects of cation disordering in a natural MgAl2O4 spinel observed by rectangular parallelepiped ultrasonic resonance and Raman measurements

Effects of cation disordering in a natural MgAl2O4 spinel observed by rectangular parallelepiped ultrasonic resonance and Raman measurements

Rectangular parallelepiped resonance--a technique of resonance ultrasound and its applications to the determination of elasticity at high temperatures.

The experimental work on resonance ultrasound, especially using the technique called rectangular parallelepiped resonance (RPR) is reviewed and summarized. The work on high-temperature (at temperatures beyond the Debye temperature) measurements of elastic constants is emphasized. Several applications of this high temperature work are described, including thermal equations of state, the determination of thermodynamic functions, and the determination of the bulk modulus at temperatures beyond experimental capability.

High‐temperature elasticity of iron‐bearing olivines

The first high‐temperature data on the nine adiabatic elastic moduli for iron‐bearing olivine are reported. These measurements are on two single‐crystal specimens of natural olivine at ambient pressure and from room temperature to a maximum of 1500 K. The two specimens contain 8 and 9 modal percent fayalite, which required the oxygen fugacity be controlled at high temperature to preserve their chemical stability. The rectangular parallelepiped resonance apparatus was adapted to buffer the specimens from the atmosphere with a mixture of CO and CO2 gas. A small increase (∼1–2 GPa) in the adiabatic bulk modulus of each specimen, over that of end‐member forsterite, was found. The data are high quality to extreme temperatures, with good agreement found when comparing the temperature derivatives of the elastic moduli of the two specimens. Neither specimen exhibits measurable nonlinear temperature dependence in the computed isotropic bulk and shear moduli, which is in contrast to published forsterite data. The temperature derivatives of the isotropic bulk modulus KS are (−1.69, −1.80) × 10−2 GPa K−1 for the two olivine specimens, and the shear modulus G derivatives are (−1.38, −1.36) × 10−2 GPa K−1. These derivatives are only slightly larger in magnitude than |(∂KS/∂T)P| = 1.56 × 10−2 and |(∂G/∂T)P| = 1.30 × 10−2 GPa K−1 found previously for iron‐bearing olivine over a very small temperature range. There are also no significant differences between the temperature derivatives found here and the average derivatives of end‐member forsterite from data retrieved over a slightly larger temperature range. Several dimensionless parameters have been calculated from these results and are discussed in view of systematics which bear on high‐pressure phases in Earth's transition zone. One result from these systematics related to the seismic velocities in the Earth, and especially the shear wave velocities, is that an olivine content of less than 50% is implied at the 400‐km discontinuity if Earth's upper mantle is isochemical. Furthermore, the substitution of almost 10% iron for magnesium at the forsterite end of the olivine solid solution series has little effect on the dimensionless parameters or on the temperature derivatives of the elastic moduli at high temperature.

Elasticity of Ni-based L1 2-type intermetallic compounds

The three elastic stiffness constants of various Ni-based intermetallic Ni 3X (X = Mn, Fe, Al, Ga, Ge and Si) with L1 2 structure have been determined at room temperature by the rectangular parallelepiped resonance method. The elastic anisotrophy factor, A, of the compounds is generally large, ranging from 2.5 to 3.3 except for Ni 3Ge ( A = 1.71). The large anisotropy is believed to be responsible at least partly for the positive temperature dependence of the flow stress observed for these compounds. The elastic constants exhibit some correlation with the lattice constants. The minor element, X, can be regarded as exerting “chemical pressure” which causes the expansion of the Ni lattice, thus resulting in the decrease of the elastic constant.

Determination of elastic constants of single crystals with cubic symmetry by the rectangular parallelepiped resonance method

The rectangular parallelepiped resonance (RPR) method is known to be a powerful method for determining the elastic constants, but the amount of numerical calculation for data analysis is quite large if one adopts the conventional procedure. A new method of analysis is proposed, which is applicable to specimens of cubic crystals. In this method, the resonance frequencies are calculated and stored for various combinations of Poisson's ratio, ν and anisotropy factor, A. The measured spectrum of the resonance frequencies is compared in turn with the calculated spectra by a procedure of something like pattern comparison so as to determine roughly the values of the elastic constants. Final refinements can be done successfully by the interpolation of the stored data. As an example, the method is applied to the determination of the elastic constants of a Ni 3Ga single crystal, for which no previous data have been reported.

Rectangular parallellepiped resonance method for piezoelectric crystals and elastic constants of alpha-quartz

A theory is presented on the free osculations of a rectangular parallelepiped of piezoelectric crystal, by extending the theory of the rectangular parallelepiped resonance (RPR) method to determine elastic constants of crystals, as exemplified by an alpha-quartz specimen. The piezoelectric contribution to resonance frequencies was examined numerically on the specimen, and it was revealed that piezoelectricity causes approximately 5 kHz increase around 1 MHz. The resonance frequencies of the specimen were measured and inverted to elastic constants by least squares inversion. The inversion was by both the previous non-piezoelectric or elastic theory and by the present piezoelectric theory. The use of the non-piezoelectric theory resulted in an overestimate of 2σ or 0.6% in c11 and underestimate of σ or 6% in c12. These are the constants expected to be most affected by piezoelectricity. Errors are less than σ in the other constants. During measurement, it was found that the force applied to hold the specimen caused deviations from free oscillation and experimental errors of 5 kHz. The correction for this force is of some importance in RPR studies of piezoelectric crystals.

Measured elastic moduli of single-crystal MgO up to 1800 K

Using the rectangular parallelepiped resonance method we measured the temperature dependence of the adiabatic elastic moduli of single-crystal MgO over the temperature range 300–1800 K. The high temperature limit of our measurements extends by 500 K the upper limit over which elasticity data on MgO are now available. Although our measured temperature dependence of C ij s are generally in good agreement with previous measurements over a more narrow range in temperature, we found that C 44 s decreases more rapidly with temperature, for T > 1000 K, than previous studies suggest. We also found that each of the slopes (ϱC 11 s /ϱT)p, (ϱKs/ϱT)p, and (C 44 s /ϱT)p become less negative with increasing temperature for T > 1400 K. From our measurements on elasticity we are able to confirm that the Gruneisen parameter at zero pressure is nearly constant with temperature up to 1800 K, with only a slight decrease above 1000 K. Utilizing our new data we present calculations showing the temperature dependence of thermodynamic parameters important in studies of earth's interior.

Elastic constants of corundum up to 1825 K

Elastic constants of single‐crystal corundum, α‐Al2O3, up to 1825 K (1.75 times the Debye temperature, θD) have been measured using a new device, the rectangular parallelepiped resonance method. This device uses a long, thin alumina buffer rod, which separates the transducer from the specimen and enables us to pick up the resonance vibration of the specimen at very high temperature. The elastic stiffness constants C11 and C33 show a large decrease, while C12 and C13 only slightly decrease with temperature. C44 decreases linearly with temperature, and C14 is almost constant in the measured range of temperature. The isotropic elastic parameters were calculated by means of Voigt, Reuss, and Hill averaging schemes, enabling us to analyze the high‐temperature elastic properties of corundum. The present elastic data also enable us to determine the thermodynamic parameters of corundum by combining them with the existing data on thermal expansivity and heat capacity. We find that (1) the evidence for anharmonicity in the thermal pressure is quite small, and the volume dependence of the thermal pressure diminishes above 1400 K; (2) the Grüneisen parameter decreases slightly with T at constant P above the Debye temperature; (3) the isothermal Anderson‐Grüneisen parameter approaches the value of the pressure derivative of the isothermal bulk modulus at high temperature; and (4) the parameter q = (∂ℓnγ/∂ℓnV)T also approaches the value 1 at high temperature.

Measurement of elastic constants of mantle-related minerals at temperatures up to 1800 K

The main purpose of this paper is to describe in some detail an experiment which allows the measurement of elastic constants of mantle minerals into the temperature range exceeding T θ = 1.5 , where θ is the Debye temperature. We need these experimental data to establish firmly the thermodynamic and mechanical behavior of solids well into the classical range of T, which is of especial interest to lower mantle geophysics. The second purpose is to describe two applications of the experiment, of special importance to geophysics: (1) the determination of υ p and υ s seismic velocities up to upper mantle temperatures, and the method of extrapolation to lower mantle temperatures; and (2) the calculation of critical parameters for use in mantle thermodynamics. The experiment is called rectangular parallelepiped resonance (RPR).

Sphere-resonance method to determine elastic constants of crystal

Theoretical basis is developed for the sphere-resonance method to determine elastic constants of a very small crystal. In this method, a sphere of crystal is formed, and frequencies of its free oscillations are measured by the resonance method. To calculate frequencies of the sphere, we use the Rayleigh–Ritz (variational) method as in the rectangular parallelepiped resonance method developed by Ohno [J. Phys. Earth 24, 355 (1976)]. It is necessary in the Rayleigh–Ritz method to evaluate matrix elements which are quadratic forms in the strain tensor with coupling by the elastic tensor. To do this the elastic tensor is transformed to spherical coordinates, and the contravariant canonical components of the elastic tensor are expanded by the generalized spherical harmonics YNml with nonzero coefficients only for l=0, 2, 4. For seven crystal systems, properties of the matrix elements are examined, and free oscillations are classified. We also obtain selection rules for excitation which are useful for the mode identification.

Elasticity and anharmonicity of potassium chloride at high temperature

High temperature elasticity of single crystal potassium chloride has been studied by the Rectangular Parallelepiped Resonance (RPR) method up to 870 K (⋍ 3.8 times the Debye temperature, ϑ). The elastic stiffness moduli, C 11 and C 44, decrease linearly with temperature while C 12 increases slightly with temperature. The RPR method is particularly suited to measurements of elasticity at high temperatures, since no glues (which decompose at high temperature) are used to connect the transducers to a specimen. As a consequence, the measured spectrum closely approximates the theoretical spectrum of a specimen freely suspended in space with no external contact. The present elasticity data permits the investigation of the thermodynamic properties of potassium chloride far above the Debye temperature when used together with the previous zero-pressure data on thermal expansivity and heat capacity. The equation of state of potassium chloride is virtually unaffected by anharmonicity, even at T/ϑ=3.8. One result is that the thermal pressure for KCl above the Debye temperature linearly increases with temperature. There is also small dependence on volume, in contrast to NaCl where there is no dependence on volume.

High temperature elasticity of sodium chloride

High temperature elasticity of single crystal sodium chloride has been studied by the Rectangular Parallelepiped Resonance (RPR) method up to 770 K (≅ 2.5 times the Debye temperature). The elastic stiffness constants, C 11 and C 44, decrease linearly with temperature while C 12 is almost independent of temperature and shows a maximum around 500 K. The RPR method is particularly suited to measurements of elasticity at high temperature, since no glues are used to connect the transducers to a specimen, and as a consequence the measured spectrum closely approximates the theoretical spectrum of a specimen suspended freely with no external contact in space. The present elasticity data permits the investigation of the thermodynamic properties of sodium chloride far above the Debye temperature when used together with the previous zero pressure data on thermal expansivity and heat capacity. Anharmonicity affecting the equation of state of sodium chloride is discussed in detail. One result is that the parameters in Decker's equation of state for NaCI are verified. Another result is that the thermal pressure for NaCI linearly increases with temperature, and is independent of volume, for temperatures above the Debye temperature.

Determination of elastic constants of trigonal crystals by the rectangular parallelepiped resonance method

We present a theoretical method to calculate the resonance frequencies of a rectangular parallelepiped of a trigonal ( D 3, C 3v, D 3d crystal, and apply it to determine the elastic constants of Al 2O 3 by the rectangular parallelepiped resonance (RPR) method. We demonstrate complete agreement between observed and calculated resonance spectra up to the thirty-ninth mode. The determined elastic constants of Al 2O 3 agree well with previous data. This study offers a new tool for the determination of elastic constants of a trigonal crystal.

Elasticity and thermal expansion of a natural garnet up to 1,000K.

High temperature mechanical properties of a natural pyrope garnet have been investigated up to 1, 000K. The elastic moduli were measured by the Rectangular Parallelepiped Resonance (RPR) method. The frequencies of the lower 16 modes smoothly decrease with temperature from 300 to 1, 000K, and so do the elastic moduli.Thermal expansion of the specimen was measured by a dilatometric method. The least squares calculation based on an equation kY2-Y+E(θE, T)/Q=0(Y is relative volume expansion, and E(θE, T) is the Debye function) yields the constants Q=16.5MJ/mol, θE=698K, and k=1.50. From these constants, the Gruneisen's parameter γ0 (=Bo Vo/Q) and the pressure derivative of bulk modulus ∂B/∂P (=2k+1) at 0K are 1.2 and 4.0, respectively.By using these data, bulk moduli and other physical properties of this garnet are estimated up to higher temperatures (Table 2).

The elastic constants of single crystal Fe1-xO,MnO and CoO,and the elasticity of stoichiometric magnesiowuestite.

Elastic constants of single-crystalline Fe1-xO (x=0.08), MnO and CoO have been measured by the rectangular parallelepiped resonance (RPR) method at temperatures between 30°C and Néel point TN (TN≅-87°C for Fe0.92O, ≅ -150°C for MnO and ≅16°C for CoO). In Fe0.92O and MnO, the shear constant C44 (or Cs for CoO) shows a drastic change at and around the Néel temperature, while the bulk modulus Ks and another shear constant Cs (or C44 for CoO) continue to show a linear temperature dependence even in the vicinity of the Néel point. The anomalous temperature variation of C44 in Fe0.92O persists above room temperatures. By using several empirical rules, the elastic properties of stoichiometric wüstite and magnesiowüstite are also evaluated. The elastic constants (Mbar) and their temperature coefficients (kbar/deg) at room temperature are:

The elastic constants of Mn2SiO4, Fe2SiO4 and Co2SiO4, and the elastic properties of olivine group minerals at high temperature.

Elastic constants of single-crystal Mn2SiO4, Fe2SiO4 and Co2SiO4 olivine are measured by the rectangular parallelepiped resonance (RPR) method at temperatures between 20° and 400°C. Elastic constants Cij (Mbar) and their temperature derivatives ∂Cij/∂T (kbar/deg) are: The isotropic properties, density ρ, adiabatic bulkmodulus K3, rigidityμ, the Poisson's ratio σ, and their temperature derivatives calculated by the Voigt-Reuss-Hill scheme are: By combining the present results with the previous data on a magnesium rich olivine, the effect of cation substitution on the elastic properties of olivine group minerals is discussed and the elastic constants of olivine at high temperature in the earth's mantle are clarified.