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

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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 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.

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Abbreviations

l :

Characteristic length of the couple stress theory or strain-gradient elasticity theory

\({{\bar{\eta }}}\) :

Dimensionless couple stress constant

h :

Thickness of the surface layer

k :

Wave number

\(\omega \) :

Circular frequency

c :

Phase velocity

\(u_{1}^{A}, u_{2}^{A}, u_{3}^{A}\) :

Displacement components in the surface layer

\(c_\mathrm{LA} \) :

Longitudinal wave velocity of the surface layer

\(c_\mathrm{TA}\) :

Shear wave velocity of the surface layer

\(\lambda _{A},\mu _{A} \) :

Lamé constants of the surface layer

\(\rho _{A} \) :

Density of the surface layer

\(u_{1}^{B}, u_{2}^{B}, u_{3}^{B}\) :

Displacement components in the half-space

\(c_\mathrm{LB} \) :

Longitudinal wave velocity of the half-space

\(c_\mathrm{TB} \) :

Shear wave velocity of the half-space

\(\lambda _{B}, \mu _{B} \) :

Lamé constants of the half-space

\(\rho _{B} \) :

Density of the half-space

\(p_{1}^{A}, p_{2}^{A}, p_{3}^{A}\) :

Force traction components of the surface layer

\(q_{2}^{A}\) :

Couple traction component of the surface layer

\(p_{1}^{B}, p_{2}^{B}, p_{3}^{B}\) :

Force traction components of the half-space

\(q_{2}^{B}\) :

Couple traction component of the half-space

\(u_{1}^{M}, u_{2}^{M}, u_{3}^{M}\) :

Displacement components of the middle plane of the plate

\(D_\mathrm{c} \) :

Classical flexural stiffness

\(D_\mathrm{l} \) :

Flexural stiffness due to the couple stresses

\({\lambda }'_{A} \) :

Lamé constant in the condition of plane stress

\(\sigma ^{r} \) :

Prestress in the plate

\(\delta _{ij} \) :

Kronecker delta

\(\sigma \) :

Residual surface tension

\(\lambda _{0},\mu _{0}\) :

Surface Lamé constants

\(\rho _{0}\) :

Surface density

\(f_{1}, f_{2}, f_{3} \) :

Body force components of the plate

\(c_{1}, c_{2}, c_{3} \) :

Body couple components of the plate

\(\sigma _{11}^{A}\) :

Normal stress of the surface layer in the \(x_{1}\) direction

\(M_{11}, N_{11} \) :

Stress resultants through the layer thickness

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Acknowledgements

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).

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  1. School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, 639798, Republic of Singapore

    Hui Fan & Jianmin Long

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Fan, H., Long, J. In-plane surface wave in a classical elastic half-space covered by a surface layer with microstructure. Acta Mech 231, 4463–4477 (2020). https://doi.org/10.1007/s00707-020-02769-6

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