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 pressure dependence of the thermal expansion coefficient, α, previously reported as (∂lnα/∂lnV)T = 5.5 ± 0.5 by Chopelas and Boehler [1989] is refined, using systematics in the volume dependence of (∂T/∂P)s measured for a large number of materials at high pressures and high temperatures. Since (∂ln (∂T/∂P)s/∂(V/V0))T is found to be constant and material independent over a very large compression range, (∂lnα/∂lnV)T is proportional to the compression, V/V0. We find α decreases by a factor of 5 for MgO throughout the mantle, reaching a value of 1.0 · 10−5 K−1 at its bottom. Densities of perovskite (PV) and magnesiowustite (MW) are calculated for lower mantle conditions using our new α(P, T), a room temperature finite strain equation, and recent data on the Mg-Fe partitioning in the PV-MW system. Both minerals have nearly identical densities to those of PREM throughout the entire lower mantle, which allows variable PV:MW ratios. A lower mantle made entirely of PV with a molar ratio of Mg:Fe of 88:12 would be about 0.11 g/cm3 or 2.5% denser than this mixture, but this density would just be within the uncertainty in PREM. A change in chemistry at 660 km depth to a PV mantle requires a thermal boundary which would improve the match in the densities between PV and PREM. These density agreements therefore preclude evaluation of a mineralogical model for the lower mantle using density comparisons. Recent measurements on melting of Fe, FeO, and FeS, however, suggest temperatures at the core-mantle boundary below 3500 K, which tends to favor a geotherm without a large thermal boundary at 660 km depth.

Thermal expansivity in the lower mantle

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.

Ivan C. Getting

Papers

GRÜNEISEN PARAMETER OF NaCl AT HIGH COMPRESSIONS