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Rayleigh wave at the surface of a prestressed elastic medium with arbitrary anisotropy

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Abstract

The propagation of harmonic plane waves is considered in a general anisotropic elastic medium, in the presence of prestress. A system of three homogeneous equations governs the existence of a Rayleigh wave at the stress-free plane boundary of the medium. This requires solving a complex secular equation for the implicit real phase velocity of the Rayleigh wave. For the presence of the radical, this irrational equation is never easy to solve by any standard analytical or numerical method. In this study, the system of three equations is transformed to get a real secular equation for the propagation of a Rayleigh wave, which can always be solved numerically for the real phase velocity of the Rayleigh wave. This phase velocity defines a complex slowness vector, which is used to calculate the polarisation of the Rayleigh wave at different depths in an elastic half-space. Effects of prestress and anisotropic symmetry on the phase velocity of the Rayleigh wave and associated particle motion are analysed by a numerical example.

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Authors and Affiliations

  1. Department of Mathematics, Kurukshetra University, Kurukshetra, 136 119, India

    M. D. Sharma

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  1. M. D. Sharma
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Appendix

Appendix

The elastic tensor \(B_{ijkl}\) represents the elastic behaviour of a general anisotropic prestressed medium. In two-suffixed notations, this fourth-order tensor is replaced by a six-order asymmetric square matrix \(\{b_{IJ}\}, I=i\delta _{ij}+(9-i-j),~ J=k\delta _{kl}+(9-k-l);~i,j,k,l=1,2,3.\) Considering the slowness vector \(\mathbf{p}=(p_1,p_2,q)\) as a row-matrix, the elements of the asymmetric square matrix \(\mathbf{Z}\) in (5) are expressed as follows:

$$\begin{aligned} Z_{11}= & {} \mathbf{p}{} \mathbf{A_{1}}{} \mathbf{p}',~~Z_{22}=\mathbf{p}{} \mathbf{A_{2}}\mathbf{p}',~~Z_{33}=\mathbf{p}{} \mathbf{A_{3}}{} \mathbf{p}',\nonumber \\ Z_{12}= & {} \mathbf{p}{} \mathbf{A_{4}}{} \mathbf{p}',~~Z_{13}=\mathbf{p}{} \mathbf{A_{5}}\mathbf{p}',~~Z_{21}=\mathbf{p}{} \mathbf{A_{6}}{} \mathbf{p}',\nonumber \\ Z_{23}= & {} \mathbf{p}{} \mathbf{A_{7}}{} \mathbf{p}',~~Z_{31}=\mathbf{p}{} \mathbf{A_{8}}\mathbf{p}',~~Z_{32}=\mathbf{p}{} \mathbf{A_{9}}{} \mathbf{p}' \end{aligned}$$
(19)

where \(\mathbf{p}'\) denotes the transpose of \(\mathbf{p}\). The \(3\times 3\) matrices \(\mathbf{A_{r}}, (\mathbf{r}=\mathbf{1},\mathbf{2}\ldots ,\mathbf{9}),\) are given by

$$\begin{aligned} \mathbf{A_{1}}= & {} \{b_{11},b_{16},b_{15};b_{61},b_{66},b_{65};b_{51},b_{56},b_{55}\}; ~~~\mathbf{A_{2}}=\{b_{66},b_{62},b_{64};b_{26},b_{22},b_{24};b_{46},b_{42},b_{44}\};\nonumber \\ \mathbf{A_{3}}= & {} \{b_{55},b_{54},b_{53};b_{45},b_{44},b_{43};b_{35},b_{34},b_{33}\}; ~~~\mathbf{A_{4}}=\{b_{16},b_{12},b_{14};b_{66},b_{62},b_{64};b_{56},b_{52},b_{54}\};\nonumber \\ \mathbf{A_{5}}= & {} \{b_{15},b_{14},b_{13};b_{65},b_{64},b_{63};b_{55},b_{54},b_{53}\}; ~~~\mathbf{A_{6}}=\{b_{61},b_{66},b_{65};b_{21},b_{26},b_{25};b_{41},b_{46},b_{45}\};\nonumber \\ \mathbf{A_{7}}= & {} \{b_{65},b_{64},b_{63};b_{25},b_{24},b_{23};b_{45},b_{44},b_{43}\}; ~~~\mathbf{A_{8}}=\{b_{51},b_{56},b_{55};b_{41},b_{46},b_{45};b_{31},b_{36},b_{35}\};\nonumber \\ ~~~\mathbf{A_{9}}= & {} \{b_{56},b_{52},b_{54};b_{46},b_{42},b_{44};b_{36},b_{32},b_{34}\}. \end{aligned}$$
(20)

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Sharma, M.D. Rayleigh wave at the surface of a prestressed elastic medium with arbitrary anisotropy. Acta Mech 230, 3041–3053 (2019). https://doi.org/10.1007/s00707-019-02455-2

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