Elasticity of Solids at High Pressures and Temperatures : Theory, Measurement, and Geophysical Application
<p>A theory for describing the elasticity of solids at simultaneous high pressures and high temperatures is developed by incorporating the fourth-order ahnarmonic theory of lattice dynamics into finite strain theory. The theory is applied to the analysis of a variety of data for MgO, SiO&...
| Summary: | <p>A theory for describing the elasticity of solids at
simultaneous high pressures and high temperatures is developed
by incorporating the fourth-order ahnarmonic theory of lattice
dynamics into finite strain theory. The theory is applied
to the analysis of a variety of data for MgO, SiO<sub>2</sub> and
NaCl, and the results for MgO and SiO<sub>2</sub> used as the basis of
a discussion of the constitution of the lower mantle. New
results are reported of measurements of elastic properties
of MgO shock-compressed to over 500 Kb.</p>
<p>The condition that finite strain equations be frame-
indifferent is shown to require that only strain tensors
belonging to a class of frame-indifferent strain tensors be
used in finite strain expansion. It is shown that the
generality of finite strain theory is not impaired by the
inclusion of an explicit theory of thermal effects. Explicit
equations for isotherms, isentropes and Hugoniots and for
the effective elastic moduli of materials of cubic symmetry
under hydrostatic stress are derived. The primary parameters
of these equations are related to the elastic moduli and
their pressure and temperature derivatives in an arbitrary
reference state using thermodynamic identities, some of which
are derived here.</p>
<p>Hugoniot data corresponding to different initial sample
densities of MgO, SiO<sub>2</sub> and NaCl and original ultrasonic data
of NaCl are used to test both the compressional and thermal
parts of the theory, and to refine the equations of state of
these materials. The frame-indifferent analogue, E, of the
usual "Eulerian" strain tensor, ε, is found to usually give
faster convergence of finite strain expansions than the
"Lagrangian" strain tensor, η. The effect of usinq different
strain measures on the values of parameters derived from
data is demonstrated, and the adverse effects of using inappropriately
derived parameters in extrapolation equations is demonstrated.
Thermal effects in Hugoniot data are reasonably well described,
but higher-order anharmonic effects appear to be required in the
theory in order to describe the high temperature ultrasonic and
thermal expansion data.</p>
<p>Measured velocities of rarefaction waves propagating
into shocked MgO are in accord with a two-stage longitudinal
(elastic)-hydrodynamic (plastic) decompression model, and
constrain the high-pressure elastic moduli of MgO.</p>
<p>The effects on the determination of the lower mantle constitution
of temperature, varying composition, the presence of phases denser
than oxides mixtures, and the presence of iron in the "low-spin"
electronic state are estimated, and a trade-off between many
of these factors demonstrated. Iron content could range between 6% and
15% by weight of FeO. Silica content could range from 33% to 50% or
more by weight. Phases a few percent denser than oxides mixtures
seem to be likely. The temperature is very indeterminable.</p> |
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