Dispersion of Rayleigh waves in weakly anisotropic media with vertically-inhomogeneous initial stress

Herein we present a procedure by which a high-frequency asymptotic formula can be derived for dispersion relations of Rayleigh waves that propagate in various directions along the free surface of a vertically-inhomogeneous, prestressed, and generally anisotropic half-space. The procedure is based on three assumptions, namely: (i) the incremental elasticity tensor of the material half-space can be written as the sum of a homogeneous isotropic part CIso and a depth-dependent perturbative part A; (ii) at the free surface both the initial stress and A are small as compared with CIso; (iii) the mass density, the initial stress, and A are smooth functions of depth from the free surface. We derive formulas and Lyapunov-type equations that can iteratively deliver each term of an asymptotic expansion of the surface impedance matrix, which leads to the aforementioned high-frequency asymptotic formula for Rayleigh-wave dispersion. As illustration we consider a thick-plate sample of AA 7075-T651 aluminum alloy, which has one face treated by low plasticity burnishing that induced a (depth-dependent) prestress at and immediately beneath the treated surface. We model the sample as a prestressed, weakly-textured orthorhombic aggregate of cubic crystallites and work out explicitly, up to the third order, the dispersion relations that pertain to Rayleigh waves propagating in several directions along the treated face of the sample.