An effective medium theory for three-dimensional elastic heterogeneities

Abstract

SUMMARY A second-order Born approximation is used to formulate 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 onepoint tensor variance 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’s 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 the singlepoint variance 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 second-order expressions fora1-Dstochasticlaminate.ComparisonswiththeexactBackustheoryshowthatthesecondorder approximation predicts the effective anisotropy for non-Gaussian media fairly well for relative rms fluctuations in the moduli smaller than about 30 per cent. A locally anisotropic model is formulated in which the local elastic properties have hexagonal symmetry, guided by a Gaussian random vector field that is transversely isotropic and specified by a horizontal-toverticalorientationratio ξ.Theself-consistenttheoryprovidesclosed-formexpressionsforthe dependence of the effective moduli on 0 <ξ <∞ and 0 <η< ∞. The effective-medium parametrizations described here appear to be suitable for incorporation into tomographic modelling.