In-plane surface wave in a classical elastic half-space covered by a surface layer with microstructure

Surface layers with microstructures are widely used in many engineering fields. The mechanical behavior of microstructures in solids can be described by gradient elasticity theories. [One of them is the couple stress theory (Mindlin and Tiersten in Arch. Ration. Mech. Anal. 11:415–448, 1962).] In the present paper, we study the in-plane surface wave propagating in a classical elastic half-space covered by a surface layer described by the couple stress theory. We firstly develop the full solution for the above configuration. Since our primary objective is to introduce the couple stress theory (or strain-gradient elasticity theory) into the surface elasticity model (Gurtin and Murdoch in Arch. Ration. Mech. Anal. 57:291–323, 1975), we are particularly interested in the case that the surface layer is very thin. Therefore, as our second step, by employing the Kirchhoff thin plate model, we establish the surface elasticity model considering couple stresses and derive the isotropic surface elasticity solution of the present problem. Thirdly, by employing the second-order strain-gradient model (Aifantis in Int. J. Eng. Sci. 30:1279–1299, 1992), we derive the dispersion equation of the surface wave for the case that the microstructure length scale is larger than the layer thickness. The last two solutions are compared with the full solution numerically for the lowest mode of the surface wave. It should be pointed out that the present study involves multi-field knowledge of surface waves, couple stress theory, and surface elasticity theory. Ministry of Education (MOE) The authors would like to thank the financial support from Singapore Ministry of Education Academic Research Fund Tier 1 (RG185/18). Jianmin Long also acknowledges the supports from the National Natural Science Foundation of China (11702081) and the Open Project of State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University (SV2017-KF-19).