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A method to derive homogeneous effective constitutive equations forperiodically layered elastic media is proposed. The crucial and novel ideaunderlying the procedure is that the coefficients of the dynamic effectivemedium can be associated with the matrix logarithm of the propagator over aunit period. The effective homogeneous equations are shown to have thestructure of a Willis material, characterized by anisotropic inertia andcoupling between momentum and strain, in addition to effective elasticconstants. Expressions are presented for the Willis material parameters whichare formally valid at any frequency and horizontal wavenumber as long as thematrix logarithm is well defined. The general theory is illustrated using theexample of scalar SH motion. Low frequency, long wavelength expansions of theeffective material parameters are also developed using a Magnus series andexplicit estimates for the rate of convergence are derived.

Effective Willis constitutive equations for periodically stratified anisotropic elastic media