Finite strain theories and comparisons with seismological data
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All the finite strain equations that we are aware of that are worth considering in connection with the interior of the Earth are given, with the assumptions on which they are based and corresponding relationships for incompressibility and its pressure derivatives in terms of density. In several cases, equations which have been presented as new or independent are shown to be particular examples of more general equations that are already familiar. Relationships for deriving finite strain equations from atomic potential functions or vice versa are given and, in particular it is pointed out that the Birch-Murnaghan formulation implies a sum of power law potentials with even powers. All the equations that survive simple plausibility tests are fitted to the lower mantle and outer core data for the PEM earth model. For this purpose the model data are extrapolated to zero temperature, using the Mie-Grüneisen equation to subtract the thermal pressure (at fixed density) and the pressure derivative of this equation to substract the thermal component of incompressibility. Fitting of finite strain equations to such zero temperature data is less ambiguous than fitting raw earth model data and leads immediately to estimates of the low temperature zero pressure parameters of earth materials. On this basis, using the best fitting equations and constraining core temperature to give an extrapolated incompressibilityK 0=1.6×1011Pa, compatible with a plausible iron alloy, the following numerical data are obtained: Core-mantle boundary temperature 3770 K Zero pressure, zero temperature densities: lower mantle 4190 kg m−3 outer core (solidified) 7500 kg m−3 Zero pressure, zero temperature incompressibility of the lower mantle 2.36×1011Pa
However, an inconsistency is apparent betweenP(ϱ) andK(ϱ) data, indicating that, even in the PEM model, in which the lower mantle is represented by a single set of parameters, it is not perfectly homogeneous with respect to composition and phase.
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