Surface waves propagating in an elastic half-space considering both the effects of strain gradient and surface elasticity

The strain gradient elasticity theory and surface elasticity theory have been employed to describe the mechanical behavior of materials featuring microstructures within the interior and on the surface, respectively. In this paper, by using Hamilton’s principle, we established a combined model that takes into account both the effects of strain gradient and surface elasticity. Based on this combined model, we investigated the propagation of anti-plane and in-plane surface waves in an elastic half-space. For the anti-plane surface wave, we derived the dispersion equation of surface wave analytically. For the in-plane surface wave, we formulated the linear algebraic equations for the undetermined constants, with the elements of the coefficient matrix detailed in the appendix. We also obtained the range of values for the phase velocity of the anti-plane surface wave and the upper bound for the phase velocity of the in-plane surface wave. We examined the effects of strain gradient constants on the dispersion curves of both the anti-plane and in-plane surface waves. The results show that the dispersion behavior of surface waves becomes richer when both the effects of strain gradient and surface elasticity are considered compared to the case of considering only surface elasticity effect. In addition, we found that when the characteristic length associated with the kinetic energy is relatively large, anti-plane surface wave exists only at small wave numbers. When the wave number tends to zero, the phase velocity of the in-plane surface wave approaches that of the classical Rayleigh waves.