The determination of the elastic properties of composite materials (multiphase aggregates, polycrystals, and porous or cracked solids) from the elastic properties of the components may be approached in several ways. The problem may be treated statistically, via scattering theory, through variational principles, or by the assumption of specific geometries for the material under consideration. Each of these methods is reviewed in turn. The widely used Voigt-Reuss-Hill average can be a poor approximation for both two-phase composites and polycrystals, and its replacement by the two Hashin-Shtrikman bounds is recommended. For pore-containing or crack-containing media, specific geometry models must be considered if useful results are to be obtained. If aggregate theory is used to estimate the moduli of individual components of a composite whose bulk properties are known, the shear moduli of the component phases must be matched (within a factor of 2 or 3) for the method to be useful. Results for nonlinear composites (which allow calculation of the pressure variation of aggregate moduli) have been obtained for only a few special cases.

The Elastic Properties of Composite Materials

The Velocity of Compressional Waves in Rocks to 10 Kilobars, Part 2

The elastic properties of candidate mantle phases are used to test the viability of olivine-rich (pyrolitic) and CaO + Al_2O_3-rich (eclogitic) assemblages for the mantle. High temperature adiabats for each phase of interest are constructed and compared to mantle seismic properties. Both pyrolitic and eclogitic assemblages satisfy the seismic properties between ∼ 200 and 400 km. Between 400 and 670 km depth an eclogitic assemblage yields a superior match to velocities and velocity gradients. The 400 km seismic discontinuity may represent a chemical boundary between pyrolite and picritic eclogite (“piclogite”) or phase transformations in the olivine + orthopyroxene components of a piclogitic assemblage containing about 16% olivine. High velocity gradients in the transition zone may be explained by the transformation of Ca-rich cpx to majorite garnet. Seismic properties at the top of the lower mantle are consistent with pyrolite, piclogite or perovskite, implying that the 670 km discontinuity may be a chemical boundary.

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Composition of the upper mantle: Geophysical tests of two petrological models

New data are presented for the room temperature, static compression of iron to 78 GPa with solid neon and argon as pressure-transmitting media. X ray diffraction studies have been performed on a geophysically relevant material, for the first time to such pressures under quasihydrostatic conditions, in a diamond anvil cell. The hydrostatic technique leads to increased precision in the measurement of high pressures and has placed closer constraints on the equation of state of e iron. From a linear least squares fit of a finite strain equation of state to the present data combined with earlier, low-pressure data for metastable e iron, the preferred values for the zero-pressure isothermal bulk modulus, K0, and first pressure derivative, K0′, are 192.7 (±9.0) GPa and 4.29 (±0.36), respectively. The zero-pressure volume for the e phase is 6.687 (±0.018) cm3/mol. On the basis of the pressure-volume curve calculated from fits of the finite strain equation of state, e iron appears to be less compressible under nonhydrostatic conditions, but the differences are within the error of the nonhydrostatic experiment. The results also confirm that the absence of a soft medium in static compression experiments with the diamond anvil cell results in an overestimate of the unit cell volume (measured with the incident X ray beam parallel to the load axis) for pressures calculated with the nonhydrostatic ruby calibration scale. It is found that for e iron, substantial compensation for this nonhydrostatic effect is implicit in the nonhydrostatic ruby pressure scale up to intermediate strains. The hydrostatic data and the e iron isotherm derived from shock wave experiments on iron samples are in very close agreement.

Static compression of iron to 78 GPa with rare gas solids as pressure-transmitting media

The single-crystal elastic moduli of stishovite have been determined experimentally from measured Brillouin scattering spectra. They are C11 = 4.53 ± 0.04, C33 = 7.76 ± 0.05; C44 = 2.52 ± 0.02, C66 = 3.02 ± 0.03; and C12 = 2.11 ± 0.05, C13 = 2.03 ± 0.04 in units of mbar. The estimated aggregate (Voigt-Reuss-Hill) adiabatic moduli are K = 3.16 ± 0.04 and μ = 2.20 ± .03 mbar. Previous values of the isothermal bulk modulus, obtained from hydrostatic compression experiments, are in agreement with the Reuss value of KT = 3.06 ± .04 mbar (as corrected to isothermal conditions). Taken together, these results suggest that the pressure derivative of the bulk modulus for stishovite is low relative to other rutile structure oxides.

The single‐crystal elastic moduli of stishovite

A new high‐pressure diamond anvil device is described which allows for x‐ray diffraction intensity measurements on single crystals up to hydrostatic pressures of about 100 kilobar. The present device fits into commercial x‐ray diffractometers such as precession cameras and four‐circle diffractometers. The pressure is determined by measuring the ruby R1‐line shift with a Fabry–Perot interferometer which is coupled to a microscope system. First measurements on Se carried out on a precession camera indicate that atomic position parameters and pressures can be determined with standard deviations of ±0.002 and ±0.7 kilobar.

R. Keller

Papers

Diamond anvil device for x‐ray diffraction on single crystals under pressures up to 100 kilobar