Bounds on the effective elastic moduli of randomly oriented aggregates of hexagonal, trigonal, and tetragonal crystals are derived using the variational principles of Hashin and Shtrikman. The bounds are considerably narrower than the widely used Voigt and Reuss bounds. The Voigt-Reuss-Hill average lies within the Hashin-Shtrikman bounds in nearly all cases. Previous bounds of Peselnick and Meister are shown to be special cases of the present results.
Compressional velocity Vp to 8 kbar and 275°C in a homogeneous, 0.5% porosity, little‐altered harzburgite from the Antalya ophiolite complex in Turkey was measured along the three principal kinematic axes as defined from the preferred orientations of olivine and enstatite. The observed Vp anisotropy was 0.7 km/s for pressures P from 2 to 8 kbar and temperatures T from 25° to 275°C. The seismic anisotropy at 250°C and 5 kbar (15‐km depth) calculated from the single‐crystal elastic constants and the fabric data were within 3% of the observed velocities in the core directions. Measured values of the P and T derivatives of Vp for the three cores are 14 m/s kbar and −0.6 m/s °C, which are in reasonable agreement with calculated average values from single‐crystal data. These data result in a critical thermal gradient of 7.5°C/km. A model of oceanic crust and uppermost mantle (constructed from the structural data in this ophiolite complex) permits an estimation of the azimuthal seismic anisotropy in the uppermost mantle beneath this paleocrust under assumed conditions of 250°C and 15‐km depth. The result is Δ Vp = 0.3 km/s with Vp = 8.46 and 8.16 km/s normal and parallel to the ridge, respectively. The azimuthal shear velocity anisotropy is 0.18 km/s. Measurements of Vp are also presented for three lherzolite cores from Salt Lake Crater, Hawaii. The T derivatives of Vp for these cores agree with those obtained for the harzburgite; however, the P derivatives are consistently higher. The higher values of the P derivatives for the lherzolite sample are considered to result from its higher porosity.
Variational principles of anisotropic elasticity have been applied to aggregates of randomly oriented pure-phase polycrystals having hexagonal symmetry and trigonal symmetry. The bounds of the effective elastic moduli obtained in this way show a considerable improvement over the bounds obtained by means of the Voigt and Reuss assumptions. The Hill average is found to be in most cases a good approximation when compared to the bounds found from the variational method. The new bounds reduce in their limits to the Voigt and Reuss values.
Compressional (P) velocity anisotropy to 1.5 kbar at room temperature in a Iherzolite from the Lanzo massif in the Italian Alps was measured. These studies are relevant to the origin of the velocity anisotropy in the oceanic upper mantle. The results show the following. (1) The velocity anisotropy of the sample is about 7% and is independent of pressure to several kilobars. The anisotropy of the sample is referred to its in situ (field) orientation. (2) The seismic velocities were measured in three mutually orthogonal directions related to the field structures (foliation and lineation), and thus a means of estimating the velocity anisotropy in these peridotite massifs directly from the field structural data is provided. The field structures, in turn, are closely related to the fabrics and to the flow directions. (3) The anisotropy of the sample calculated from the fabric data is in good agreement with the measured anisotropy. The anisotropy of the Lanzo massif is calculated from several hundred field measurements of foliation and lineation referred to the measured anisotropy of the sample along those structural directions. (4) The results show that the laboratory measurements on the Iherzolite sample are similar to the values of density and seismic velocity of the Ivrea body shown in the geophysical model; however, the present results predict a seismic velocity anisptropy for the Lanzo massif, the maximum velocity trending NNE‐SSW and the minimum velocity trending WNW‐ESE. (5) These results support the interpretation of seismic anisotropy in the oceanic upper mantle essentially as given by Francis.
The internal friction in shear and modulus of rigidity of dry Solenhofen limestone has been investigated over a frequency range from 4 cps to 10 Mc/s at room temperature. The results found are: (1) The rigidity modulus is constant ( U = 2.64 × 1011 dynes/cm2) to within ±2 per cent over the total frequency range covered, provided that the samples have the same density. (2) The shear internal friction (as measured by the logarithmic decrement) in the cycle‐per‐second frequency range is about a factor of 5 lower than the internal friction in the megacycle frequency range; the logarithmic decrement at 4 cps = 3.4 × 10−3, the logarithmic decrement at 107 cps = 17 × 10−3. (3) The shear internal friction in the infrasonic frequency range increases by 18 per cent with the application of a 7.2‐kg/cm2 static axial tensile stress, but no large change in the internal friction occurs for axial compressive stresses of the same magnitude. (4) The shear internal friction is strain‐dependent even for strains as small as 10−6, a static axial tensile stress being superposed on the dynamic torsional stress.
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