Effects of interfacial sliding on anti-plane waves in an elastic plate imperfectly attached to an elastic half-space

We study the anti-plane shear waves in a domain consisting of an elastic layer (plate) with a coating attached to an elastic half-space (substrate). We assume an imperfect contact between the layer and the half-space, allowing some sliding. We also assume some elastic bonds between the layer and the substrate. On the free top surface we apply the compatibility conditions within the Gurtin–Murdoch surface elasticity. We found two different solutions: (i) the transversely exponential–transversely exponential (TE–TE) regime with amplitudes decaying exponentially from the free top surface and the interface in both the plate and the half-space, and (ii) the transversely harmonic–transversely exponential (TH–TE) regime with harmonic wave behaviour in the transverse direction in the plate and exponential decay in the half-space. The TE regime of anti-plane waves in an elastic half-space with non-perfect contact is also considered as a special case. A detailed analysis of the derived dispersion relations reveals a crucial influence of the interface stiffness on the phase velocities of anti-plane waves. This effect consists in the decrease of the phase velocities when the interfacial bonds are weakened. The strongest effect of the interfacial sliding on the phase velocities was observed for the long-length waves belonging to the TE–TE regime. Based on the derived lower bounds for the wave numbers from which the TE–TE regime of anti-plane waves exists, we have developed the theoretical background and methodology for assessing the bond stiffness of thin plates imperfectly bonded to an elastic substrate.