Effective medium theory for elastic composites

SUMMARY AND CONCLUSIONS

I conclude that my preferred choice of effective medium theory (the CPA) satisfies all the known constraints on a viable theory: (a) it gives correct values and slopes for both large and small volume fractions of inclusions; (b) numerical evidence indicates that the results always satisfy the Hashin-Shtrikman bounds, the Beran-Molyneux-Miller bounds, and the McCoy-Silnutzer bounds; (c) the theory is also known [Berryman (1980b)] to reproduce Hill's exact result [Hill (1963)] for composites with uniform shear modulus -- which fact is a fairly simple exercise to check, so that the reader might find it instructive to carry this through.

The single-scatterer theory is designed to minimize multiple scattering effects while yielding formulas that are relatively easy to use. Nevertheless, the theory is not exact, and some potentially significant effects have been neglected. The neglected terms become more important for propagation of higher frequency elastic waves. But it is important to note that bounding methods and formulas are also much harder to implement rigorously for the frequency dependent (viscoelastic) case. This fact is surely one reason that the theory is seldom applied at significantly higher frequencies than typical seismic frequencies, or in regions of very much higher viscosity, and wave dissipation and dispersion. So, it is expected that, for small ranges of frequency -- and especially those that are pertinent to exploration seismology -- will naturally be included in the range of useful applications since the seismic band is fairly narrow. Then the viscoelastic effects can typically be treated without great additional difficulty. Some future efforts should nevertheless be directed towards extending this effective medium theory to scattering from clusters of inclusions at finite frequency -- thereby including within the expanded theory more of the important scattering effects discussed (but then specifically neglected, and therefore not treated in any detail) here.

Effective medium theory for elastic composites