Homogenization of Heterogeneous Elastic Materials with Applications to Seismic Anisotropy

Abstract:

The velocities of seismic waves passing through a complex Earth volume can be influenced by heterogeneities at length scales shorter than the seismic wavelength. As such, seismic wave propagation analyses can be performed by replacing the actual Earth volume by a homogeneous i.e., "effective", elastic medium. Homogenization refers to the process by which the elastic stiffness tensor of the effective medium is "averaged" from the elastic properties, orientations, modal proportions and spatial distributions of the finer heterogeneities.

When computing the homogenized properties of a heterogeneous material, the goal is to compute an effective or bulk elastic stiffness tensor that relates the average stresses to the average strains in the material. Tensor averaging schemes such as the Voigt and Reuss methods are based on certain simplifying assumptions. The Voigt method assumes spatially uniform strains while the Reuss method assumes spatially uniform stresses within the heterogeneous material. Although they are both physically unrealistic, they provide upper and lower bounds for the actual homogenized elastic stiffness tensor.

In order to more precisely determine the homogenized stiffness tensor, the stress and strain distributions must be computed by solving the three-dimensional equations of elasticity over the heterogeneous region. Asymptotic expansion homogenization (AEH) is one such structure-based approach for the comprehensive micromechanical analysis of heterogeneous materials. Unlike modal volume methods, the AEH method takes into account how geometrical orientation and alignment can increase elastic stiffness in certain directions. We use the AEH method in conjunction with finite element analysis to calculate the bulk elastic stiffnesses of heterogeneous materials. In our presentation, wave speeds computed using the AEH method are compared with those generated using stiffness tensors derived from commonly-used analytical estimates. The method is illustrated by comparing the homogenized properties and wave speeds with those presented by Backus [1962] for layered media, solutions that Voigt and Reuss methods cannot provide. We also investigate the influence of fold structures on seismic anisotropy in the crust.