Viscoelastic-stiffness tensor of anisotropic media from oscillatory numerical experiments

A finely layered media behaves as an anisotropic medium when the dominat wavelengths are much larger than the layer thickness. If the constituent are anelastic, a generalization of Backus averaging predicts that the medium is effectively a transversely isotropic viscoelastic (TIV) medium. To test and validate the theory, we present a novel procedure to determine the complex and frequency-dependent stiffness components of a TIV medium. The methodology consists in performing numerical compressibility and shear harmonic tests on a representative sample of the material. These tests are described by a collection of non-coercive elliptic boundary-value problems formulated in the space-frequency domain, which are solved using a Galerkin finite-element procedure. Results on the existence and uniqueness of the continuous and discrete problems as well as optimal error estimates for the Galerkin finite-element method are derived. Numerical examples illustrates the implementation of the numerical oscillatory tests to determine the set of complex and frequency-dependent effective TIV coefficients and the associated phase velocities and quality factors for a periodic sequence of epoxy and glass thin layers. The results are compared to the analytical phase velocities and quality factors predicted by the Backus/Carcione theory.