On the long-wave onset of dispersion of the surface-wave velocity in coated solids

A linear-dispersion coefficient K describing the slope of the Rayleigh-Love velocity branch at its long-wave onset is subjected to analysis. For the case when the sagittal plane is a symmetry plane for both the layer and the substrate (plane strain deformations), it is found that the sign of K is that of the difference between the beam velocities of the layer and substrate materials. A simple approximant of K is obtained for the case of isotropic layer and substrate. In the 3D phase space of isotropic layer and substrate material parameters, the surface K=0 is established, and the subspace where K is positive/negative for the Rayleigh velocity of a layer smaller/greater than that of a substrate (which is contrary to their commonly encountered correspondence) is identified. The consideration is extended to the case of a functionally graded layer on a homogeneous substrate. It is shown that the sign of K remains being given by the sign of a difference of layer and substrate beam velocities; at the same time, the impact of layer density on this sign can no longer be eliminated through passing to the density-normalized elastic moduli as for a homogeneous layer.