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An Effective Medium Theory for Three-Dimensional Elastic Heterogeneities

Thomas H. Jordan

Published 2015, SCEC Contribution #2094

A second-order Born approximation is used to develop a self-consistent theory for the effective elastic parameters of stochastic media with ellipsoidal distributions of small-scale heterogeneity. The covariance of the stiffness tensor is represented as the product of a one-point tensor covariance and a two-point scalar correlation function with ellipsoidal symmetry, which separates the statistical properties of the local anisotropy from those of the geometric anisotropy. The spatial variations can then be rescaled to an isotropic distribution by a simple metric transformation; the spherical average of the strain Green function in the transformed space reduces to a constant Kneer tensor, and the second-order corrections to the effective elastic parameters are given by the contraction of the rescaled Kneer tensor against a single-point covariance of the stiffness tensor. Explicit results are derived for stochastic models in which the heterogeneity is transversely isotropic and its second moments are characterized by a horizontal-to-vertical aspect ratio η. If medium is locally isotropic, the expressions for the anisotropic effective moduli reduce in the limit η→∞ to Backus’s (1962) second-order expressions for a 1D stochastic laminate. Comparisons with the exact Backus theory show that the second-order approximation predicts the effective anisotropy for non-Gaussian media fairly well for relative root-mean-square fluctuations in the moduli smaller than about 30%. A locally anisotropic model is formulated in which the local elastic properties have hexagonal symmetry, guided by a Gaussian vector field that is transversely isotropic and specified by a horizontal-to-vertical orientation ratio ξ. The self-consistent theory provides closed-form expressions for the dependence of the effective moduli on 0≤ξ<∞ and 0≤η<∞. The effective-medium parameterizations described here are suitable for incorporation into tomographic modeling.

Jordan, T. H. (2015). An Effective Medium Theory for Three-Dimensional Elastic Heterogeneities. Geophysical Journal International, 203, 1343-1354.