In-plane surface wave in a classical elastic half-space covered by a surface layer with microstructure
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O R I G I NA L PA P E R
Hui Fan · Jianmin Long
In-plane surface wave in a classical elastic half-space covered by a surface layer with microstructure
Received: 21 January 2020 / Revised: 6 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract 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 straingradient 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.
List of symbols l η¯ h k ω c u 1A , u 2A , u 3A cLA cTA λA, μA ρA u 1B , u 2B , u 3B cLB cTB λB , μB
Characteristic length of the couple stress theory or strain-gradient elasticity theory Dimensionless couple stress constant Thickness of the surface layer Wave number Circular frequency Phase velocity Displacement components in the surface layer Longitudinal wave velocity of the surface layer Shear wave velocity of the surface layer Lamé constants of the surface layer Density of the surface layer Displacement components in the half-space Longitudinal wave velocity of the half-space Shear wave velocity of the half-space Lamé constants of the half-space
H. Fan · J. Long (B) School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Republic of Singapore E-mail: [email protected]
H. Fan, J. Long
ρB p1A , p2A , p3A q2A p1B , p2B , p3B q2B u 1M , u 2M , u 3M Dc Dl λA σr δi j σ λ0 , μ0 ρ0 f1, f2 , f3 c1 , c2 , c3 A σ11 M11 , N11
Density of the half-space Force traction components of the surface layer Couple traction component of the surface layer Force traction components of the half-space Couple traction component of the half-space Displacement components of the middle pla
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