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Exemplification in Exact and Approximate Secular Equation of Surface Wave Along Distinct Interfaces with Sliding Contact

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Abstract

Acomparison between the exact and approximate secular equation of Rayleigh waves in a Kelvin Voigt viscoelastic plate supported by an orthotropic continuum under initial stress and gravity is studied analytically as well as graphically. The effective boundary condition method is applied to obtain the approximate and exact formulas for the secular equation in a described model. Based on the obtained dispersion equation, the effects of sliding contact on the Rayleigh wave propagation is considered through numerical data. The obtained secular equation is totally explicit and it is a good tool for non destructively evaluating the adhesive bond between the layer and half space as well as their mechanical properties.

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Authors and Affiliations

  1. Department of Mathematics and Computing Indian Institute of Technology (Indian School of Mines), 826004, Dhanbad, India

    Shishir Gupta, Snehamoy Pramanik &  Smita

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  1. Shishir Gupta
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  2. Snehamoy Pramanik
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  3. Smita
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Correspondence to Snehamoy Pramanik.

Appendices

APPENDIX I

$$\begin{gathered} {{{\bar {\Delta }}}_{{02}}} = {{R}_{{11}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{14}}}} \right),\quad {{{\bar {\Delta }}}_{{002}}} = {{R}_{{11}}}\left[ {\frac{1}{6}\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{19}}}} \right){{X}_{{17}}} - i\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{14}}}} \right) + {{X}_{{18}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{25}}}} \right)} \right], \\ {{{\bar {\Delta }}}_{{03}}} = \frac{1}{6}\left[ {\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + 1} \right){{X}_{{16}}} - {{X}_{{22}}} - \left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{14}}}} \right)} \right],\quad {{{\bar {\Delta }}}_{{04}}} = \frac{{{{R}_{{11}}}}}{2}{{X}_{{20}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{14}}}} \right), \\ {{{\bar {\Delta }}}_{{06}}} = \frac{1}{3}\left[ {{{X}_{{29}}} - {{X}_{{26}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{14}}}} \right) - \left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{25}}}} \right)\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{14}}}} \right)} \right],\quad {{{\bar {\Delta }}}_{{07}}} = {{X}_{{15}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{25}}}} \right), \\ {{{\bar {\Delta }}}_{{08}}} = {{R}_{{11}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{25}}}} \right),\quad {{{\bar {\Delta }}}_{{09}}} = \left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{25}}}} \right){{X}_{{21}}} - i\left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{25}}}} \right) + \left( {\frac{{{{\zeta }^{2}}}}{{{{X}_{{32}}}}} + {{X}_{{14}}}} \right), \\ \end{gathered} $$
$$\begin{gathered} {{{\bar {W}}}_{{1'}}} = - i{{{\bar {\xi }}}_{1}} + i{{{\bar {\xi }}}_{1}}{{{\bar {\gamma }}}_{1}}^{2} + {{{\bar {\gamma }}}_{1}},\quad {{{\bar {W}}}_{2}}' = - i{{{\bar {\xi }}}_{2}} + i{{{\bar {\xi }}}_{2}}\bar {\gamma }_{2}^{2} + {{{\bar {\gamma }}}_{2}},\quad {{{\bar {W}}}_{4}} = {{X}_{{30}}} + {{X}_{{31}}}\bar {\gamma }_{1}^{2} - 2i{{{\bar {\xi }}}_{1}}{{{\bar {\gamma }}}_{1}}, \\ {{{\bar {W}}}_{5}} = {{X}_{{30}}} + {{X}_{{31}}}{{{\bar {\gamma }}}_{2}}^{2} - 2i{{{\bar {\xi }}}_{2}}{{{\bar {\gamma }}}_{2}},\quad {{{\bar {W}}}_{6}} = i{{{\bar {\xi }}}_{1}} - {{{\bar {\gamma }}}_{1}},\quad {{{\bar {W}}}_{7}} = i{{{\bar {\xi }}}_{2}} - {{{\bar {\gamma }}}_{2}},\quad {{{\bar {F}}}_{0}} = {{X}_{{28}}}\left( {{{{\bar {W}}}_{{1'}}}{{{\bar {d}}}_{1}}{{{\bar {W}}}_{5}} - {{{\bar {W}}}_{{2'}}}{{{\bar {d}}}_{1}}{{{\bar {W}}}_{4}}} \right), \\ {{{\bar {F}}}_{1}} = X_{{28}}^{2}\left( {{{{\bar {W}}}_{{2'}}}{{{\bar {W}}}_{6}}{{{\bar {d}}}_{3}} - {{{\bar {W}}}_{{1'}}}{{{\bar {d}}}_{3}}{{{\bar {W}}}_{7}}} \right),\quad {{{\bar {F}}}_{2}} = {{X}_{{28}}}\left( {{{{\bar {W}}}_{{2'}}}{{{\bar {d}}}_{2}}{{{\bar {W}}}_{5}} - {{{\bar {W}}}_{{2'}}}{{{\bar {d}}}_{2}}{{{\bar {W}}}_{4}}} \right), \\ {{{\bar {F}}}_{3}} = X_{{28}}^{2}\left( {{{{\bar {W}}}_{{2'}}}{{{\bar {d}}}_{4}}{{{\bar {W}}}_{6}} - {{{\bar {W}}}_{{1'}}}{{{\bar {d}}}_{4}}{{{\bar {W}}}_{7}}} \right),\quad G = \frac{g}{{{{c}^{2}}k}}, \\ \end{gathered} $$
$$\begin{gathered} \zeta = \sqrt {\frac{{{{c}^{2}}}}{{\beta _{1}^{2}}}} ,\quad {{{\bar {\xi }}}_{i}} = \frac{{(\gamma _{1}^{2} - {{{\bar {a}}}^{2}})}}{{{{\zeta }^{2}}{{{\left( {\frac{{{{\beta }_{1}}}}{{{{\beta }_{2}}}}} \right)}}^{2}}G}}{{{\bar {\eta }}}_{1}}^{2},\quad {{\beta }_{2}} = \sqrt {\frac{{{{C}_{{44}}}}}{{{{\rho }_{2}}}}} ,\quad {{{\bar {\eta }}}_{1}} = \sqrt {\frac{{{{C}_{{11}}}}}{{{{C}_{{44}}}}} + 2{{I}_{2}}} , \\ {{{\bar {\eta }}}_{2}} = \sqrt {2\frac{{{{C}_{{33}}}}}{{{{C}_{{44}}}}} - \frac{{{{C}_{{11}}}}}{{{{C}_{{44}}}}} - \frac{{{{C}_{{13}}}}}{{{{C}_{{44}}}}} - 2{{I}_{2}}} , \\ {{I}_{2}} = \frac{{{{P}_{2}}}}{{{{C}_{{44}}}}},\quad {{I}_{1}} = \frac{{{{P}_{1}}}}{{2{{\mu }_{1}}}},\quad \bar {a} = \sqrt {{{\zeta }^{2}}{{{\left( {\frac{{{{\beta }_{1}}}}{{{{\beta }_{2}}}}} \right)}}^{2}}\left( {\frac{1}{{\frac{{{{C}_{{11}}}}}{{{{C}_{{44}}}}} + 2{{I}_{2}}}}} \right) - 1} , \quad \bar {b} = \sqrt {\frac{{2{{\zeta }^{2}}{{{\left( {\frac{{{{\beta }_{1}}}}{{{{\beta }_{2}}}}} \right)}}^{2}} - \frac{{{{C}_{{11}}}}}{{{{C}_{{44}}}}} + 2{{I}_{2}}}}{{2\frac{{{{C}_{{33}}}}}{{{{C}_{{44}}}}} - \frac{{\left( {{{C}_{{11}}} + {{C}_{{13}}}} \right)}}{{{{C}_{{44}}}}} - 2{{I}_{2}}}}} , \\ \end{gathered} $$
$$\begin{gathered} {{\beta }_{{21}}} = \frac{{{{\beta }_{1}}}}{{{{\beta }_{2}}}},\quad {{{\bar {\gamma }}}_{1}} = - \frac{{\sqrt {{{{\bar {a}}}^{2}} + {{{\bar {b}}}^{2}} - \sqrt {{{{\bar {a}}}^{4}} - 2{{{\bar {b}}}^{2}}{{{\bar {a}}}^{2}} + {{{\bar {b}}}^{4}} + 4t_{{22}}^{2}} } }}{{\sqrt 2 }}, \\ {{{\bar {\gamma }}}_{2}} = - \frac{{\sqrt {{{{\bar {a}}}^{2}} + {{{\bar {b}}}^{2}} + \sqrt {{{{\bar {a}}}^{4}} - 2{{{\bar {b}}}^{2}}{{{\bar {a}}}^{2}} + {{{\bar {b}}}^{4}} + 4t_{{22}}^{2}} } }}{{\sqrt 2 }}, \\ \end{gathered} $$
$${{X}_{{11}}} = \beta _{{21}}^{2}\left( {\frac{1}{{\frac{{{{C}_{{11}}}}}{{{{C}_{{44}}}}} + 2{{I}_{2}}}}} \right),\quad {{X}_{{12}}} = \frac{{{{C}_{{11}}}}}{{{{C}_{{44}}}}} + \frac{{{{C}_{{13}}}}}{{{{C}_{{44}}}}} + 2{{I}_{2}},\quad {{X}_{{13}}} = 2\frac{{{{C}_{{33}}}}}{{{{C}_{{44}}}}} - \frac{{{{C}_{{11}}}}}{{{{C}_{{44}}}}} - \frac{{{{C}_{{13}}}}}{{{{C}_{{44}}}}} - 2{{I}_{2}},$$
$${{X}_{{14}}} = \frac{{R_{{22}}^{2}}}{{R_{{11}}^{2}}} - 1,\quad {{X}_{{15}}} = - \frac{1}{2}{{R}_{{11}}}\left( {\frac{{{{R}_{{22}}}}}{{{{R}_{{11}}}}} + {{R}_{{44}}}} \right),\quad {{X}_{{16}}} = \left( {\frac{{{{R}_{{66}}}}}{{{{R}_{{44}}}}} + \frac{{{{R}_{{11}}}}}{{{{R}_{{33}}}}}\frac{{{{R}_{{66}}}}}{{R_{{44}}^{2}}}} \right),$$
$${{X}_{{17}}} = \left( {\frac{{{{R}_{{66}}}}}{{{{R}_{{44}}}}} - \frac{{{{R}_{{22}}}}}{{{{R}_{{11}}}{{R}_{{44}}}}}} \right),\quad {{X}_{{18}}} = \frac{{{{R}_{{22}}}{{R}_{{66}}}}}{{{{R}_{{11}}}{{R}_{{44}}}}},\quad {{X}_{{19}}} = \frac{{R_{{22}}^{2}}}{{R_{{11}}^{2}}} - 1,\quad {{X}_{{20}}} = \frac{{{{R}_{{66}}}}}{{R_{{44}}^{2}}} - \frac{1}{{{{R}_{{44}}}}},$$
$${{X}_{{21}}} = \frac{{{{R}_{{22}}}}}{{{{R}_{{11}}}}} - {{R}_{{66}}},\quad {{X}_{{22}}} = \frac{{{{R}_{{66}}}{{R}_{{66}}}}}{{{{R}_{{44}}}}},\quad {{X}_{{23}}} = \beta _{{21}}^{2}G,\quad {{X}_{{24}}} = \frac{{\bar {\eta }_{1}^{2}}}{{\beta _{{21}}^{2}G}},\quad {{X}_{{25}}} = \frac{{{{I}_{1}}}}{{{{R}_{{11}}}}},$$
$${{X}_{{26}}} = \frac{{{{R}_{{11}}}}}{{{{R}_{{33}}}}},\quad {{X}_{{27}}} = \frac{1}{{{{{\bar {\eta }}}_{1}}{{{\bar {\eta }}}_{2}}}},\quad {{X}_{{28}}} = 2\frac{{{{\mu }_{1}}}}{{{{C}_{{44}}}}},\quad {{X}_{{29}}} = {{R}_{{66}}}{{R}_{{44}}},\quad {{X}_{{30}}} = \frac{{{{C}_{{13}}}}}{{{{C}_{{44}}}}},\quad {{X}_{{31}}} = \frac{{{{C}_{{33}}}}}{{{{C}_{{44}}}}},$$
$${{X}_{{32}}} = 2{{R}_{{11}}},\quad {{R}_{{44}}} = \left( {1 - \frac{{{{I}_{1}}}}{{(1 + i{{\delta }_{{{{\mu }_{1}}}}})}}} \right),\quad {{R}_{{66}}} = - \left( {\frac{{\frac{{{{\lambda }_{1}}}}{{2{{\mu }_{1}}}} + {{\delta }_{\lambda }}\frac{{{{\lambda }_{1}}}}{{2{{\mu }_{1}}}} + {{I}_{1}}}}{{\left( {1 + \frac{{{{\lambda }_{1}}}}{{2{{\mu }_{1}}}}} \right) + \frac{{{{\lambda }_{1}}}}{{2\mu }}{{\delta }_{\lambda }}}} + {{\delta }_{\mu }}} \right),$$

APPENDIX II

$$\begin{gathered} {{{\bar {\bar {\Delta }}}}_{{02}}} = {{{\bar {R}}}_{{11}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{14}}}} \right), \\ {{{\bar {\bar {\Delta }}}}_{{002}}} = {{{\bar {R}}}_{{11}}}\left[ {\frac{1}{6}\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{19}}}} \right){{{\bar {X}}}_{{17}}} - i\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{14}}}} \right) + {{{\bar {X}}}_{{18}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{25}}}} \right)} \right], \\ {{{\bar {\bar {\Delta }}}}_{{03}}} = \frac{1}{6}\left[ {\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + 1} \right){{{\bar {X}}}_{{16}}} - {{{\bar {X}}}_{{22}}} - \left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{14}}}} \right)} \right],\quad {{{\bar {\bar {\Delta }}}}_{{04}}} = \frac{{{{{\bar {R}}}_{{11}}}}}{2}{{{\bar {X}}}_{{20}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{14}}}} \right), \\ {{{\bar {\bar {\Delta }}}}_{{06}}} = \frac{1}{3}\left[ {{{{\bar {X}}}_{{29}}} - {{{\bar {X}}}_{{26}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{14}}}} \right) - \left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{25}}}} \right)\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{14}}}} \right)} \right],\quad {{{\bar {\bar {\Delta }}}}_{{07}}} = {{{\bar {X}}}_{{15}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{25}}}} \right), \\ {{{\bar {\bar {\Delta }}}}_{{08}}} = {{{\bar {R}}}_{{11}}}\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{25}}}} \right),\quad {{{\bar {\bar {\Delta }}}}_{{09}}} = \left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{25}}}} \right){{{\bar {X}}}_{{21}}} - i\left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{25}}}} \right) + \left( {\frac{{{{\zeta }^{2}}}}{{{{{\bar {X}}}_{{32}}}}} + {{{\bar {X}}}_{{14}}}} \right), \\ \end{gathered} $$
$$\begin{gathered} {{{\bar {\bar {W}}}}_{{1'}}} = - i{{{\bar {\bar {\xi }}}}_{1}} + i{{{\bar {\bar {\xi }}}}_{1}}\bar {\bar {\gamma }}_{1}^{2} + {{{\bar {\bar {\gamma }}}}_{1}},\quad {{{\bar {\bar {W}}}}_{2}}' = - i{{{\bar {\bar {\xi }}}}_{2}} + i{{{\bar {\bar {\xi }}}}_{2}}\bar {\bar {\gamma }}_{2}^{2} + {{{\bar {\bar {\gamma }}}}_{2}},\quad {{{\bar {\bar {W}}}}_{4}} = {{{\bar {X}}}_{{30}}} + {{{\bar {X}}}_{{31}}}\bar {\bar {\gamma }}_{1}^{2} - 2i{{{\bar {\bar {\xi }}}}_{1}}{{{\bar {\bar {\gamma }}}}_{1}}, \\ {{{\bar {\bar {W}}}}_{5}} = {{{\bar {X}}}_{{30}}} + {{{\bar {X}}}_{{31}}}\bar {\bar {\gamma }}_{2}^{2} - 2i{{{\bar {\bar {\xi }}}}_{2}}{{{\bar {\bar {\gamma }}}}_{2}},\quad {{{\bar {\bar {W}}}}_{6}} = i{{{\bar {\bar {\xi }}}}_{1}} - {{{\bar {\bar {\gamma }}}}_{1}},\quad {{{\bar {\bar {W}}}}_{7}} = i{{{\bar {\bar {\xi }}}}_{2}} - {{{\bar {\bar {\gamma }}}}_{2}},\quad {{{\bar {\bar {F}}}}_{0}} = {{{\bar {X}}}_{{28}}}({{{\bar {\bar {W}}}}_{{1'}}}{{{\bar {\bar {d}}}}_{1}}{{{\bar {\bar {W}}}}_{5}} - {{{\bar {\bar {W}}}}_{{2'}}}{{{\bar {\bar {d}}}}_{1}}{{{\bar {\bar {W}}}}_{4}}), \\ {{{\bar {\bar {F}}}}_{1}} = \bar {X}_{{28}}^{2}({{{\bar {\bar {W}}}}_{{2'}}}{{{\bar {\bar {W}}}}_{6}}{{{\bar {\bar {d}}}}_{3}} - {{{\bar {\bar {W}}}}_{{1'}}}{{{\bar {\bar {d}}}}_{3}}{{{\bar {\bar {W}}}}_{7}}),\quad {{{\bar {\bar {F}}}}_{2}} = {{{\bar {X}}}_{{28}}}({{{\bar {\bar {W}}}}_{{2'}}}{{{\bar {\bar {d}}}}_{2}}{{{\bar {\bar {W}}}}_{5}} - {{{\bar {\bar {W}}}}_{{2'}}}{{{\bar {\bar {d}}}}_{2}}{{{\bar {\bar {W}}}}_{4}}), \\ {{{\bar {\bar {F}}}}_{3}} = {{{\bar {X}}}_{{28}}}^{2}({{{\bar {\bar {W}}}}_{{2'}}}{{{\bar {\bar {d}}}}_{4}}{{{\bar {\bar {W}}}}_{6}} - {{{\bar {\bar {W}}}}_{{1'}}}{{{\bar {\bar {d}}}}_{4}}{{{\bar {\bar {W}}}}_{7}}), \\ \zeta = \sqrt {\frac{{{{c}^{2}}}}{{\bar {\beta }_{1}^{2}}}} ,\quad {{{\bar {\bar {\xi }}}}_{i}} = \frac{{(\bar {\bar {\gamma }}_{1}^{2} - {{{\bar {\bar {a}}}}^{2}})}}{{{{\zeta }^{2}}{{{\left( {\frac{{{{{\bar {\beta }}}_{1}}}}{{{{{\bar {\beta }}}_{2}}}}} \right)}}^{2}}}}{{{\bar {\bar {\eta }}}}_{1}}^{2},\quad {{{\bar {\beta }}}_{2}} = \sqrt {\frac{\mu }{\rho }} ,\quad {{{\bar {\bar {\eta }}}}_{1}} = \sqrt {\frac{{\lambda + 2\mu }}{\mu }} ,\quad {{{\bar {\bar {\eta }}}}_{2}} = \sqrt {2\frac{{\lambda + 2\mu }}{\mu } - \frac{{\lambda + 2\mu }}{\mu } - \frac{\lambda }{\mu }} , \\ \bar {\bar {a}} = \sqrt {{{\zeta }^{2}}{{{\left( {\frac{{{{{\bar {\beta }}}_{1}}}}{{{{{\bar {\beta }}}_{2}}}}} \right)}}^{2}}\left( {\frac{1}{{\frac{{\lambda + 2\mu }}{\mu }}}} \right) - 1} ,\quad \bar {\bar {b}} = \sqrt {\frac{{2{{\zeta }^{2}}{{{\left( {\frac{{{{\beta }_{1}}}}{{{{\beta }_{2}}}}} \right)}}^{2}} - \frac{{\lambda + 2\mu }}{\mu }}}{{2\frac{{\lambda + 2\mu }}{\mu } - \frac{{\left( {2\lambda + 4\mu } \right)}}{\mu }}},} \\ \end{gathered} $$
$$\begin{gathered} {{\beta }_{{21}}} = \frac{{{{{\bar {\beta }}}_{1}}}}{{{{{\bar {\beta }}}_{2}}}},\quad {{{\bar {\bar {\gamma }}}}_{1}} = - \frac{{\sqrt {{{{\bar {\bar {a}}}}^{2}} + {{{\bar {\bar {b}}}}^{2}} - \sqrt {{{{\bar {\bar {a}}}}^{4}} - 2{{{\bar {\bar {b}}}}^{2}}{{{\bar {\bar {a}}}}^{2}} + {{{\bar {\bar {b}}}}^{4}} + 4\bar {t}_{{22}}^{2}} } }}{{\sqrt 2 }}, \\ {{{\bar {\bar {\gamma }}}}_{2}} = - \frac{{\sqrt {{{{\bar {\bar {a}}}}^{2}} + {{{\bar {\bar {b}}}}^{2}} + \sqrt {{{{\bar {\bar {a}}}}^{4}} - 2{{{\bar {\bar {b}}}}^{2}}{{{\bar {\bar {a}}}}^{2}} + {{{\bar {\bar {b}}}}^{4}} + 4\bar {t}_{{22}}^{2}} } }}{{\sqrt 2 }}, \\ \end{gathered} $$
$${{\bar {X}}_{{11}}} = \beta _{{21}}^{2}\left( {\frac{1}{{\frac{{\lambda + 2\mu }}{\mu }}}} \right),\quad {{\bar {X}}_{{12}}} = \frac{{\lambda + 2\mu }}{\mu } + \frac{\lambda }{\mu },\quad {{\bar {X}}_{{13}}} = 2\frac{{\lambda + 2\mu }}{\mu } - \frac{{\lambda + 2\mu }}{\mu } - \frac{\lambda }{\mu },$$
$${{\bar {X}}_{{14}}} = \frac{{\bar {R}_{{22}}^{2}}}{{\bar {R}_{{11}}^{2}}} - 1,\quad {{\bar {X}}_{{15}}} = - \frac{1}{2}{{\bar {R}}_{{11}}}\left( {\frac{{{{{\bar {R}}}_{{22}}}}}{{{{{\bar {R}}}_{{11}}}}} + {{{\bar {R}}}_{{44}}}} \right),\quad {{\bar {X}}_{{16}}} = \left( {\frac{{{{{\bar {R}}}_{{66}}}}}{{{{{\bar {R}}}_{{44}}}}} + \frac{{{{{\bar {R}}}_{{11}}}}}{{{{{\bar {R}}}_{{33}}}}}\frac{{{{{\bar {R}}}_{{66}}}}}{{\bar {R}_{{44}}^{2}}}} \right),$$
$${{\bar {X}}_{{17}}} = \left( {\frac{{{{{\bar {R}}}_{{66}}}}}{{{{{\bar {R}}}_{{44}}}}} - \frac{{{{{\bar {R}}}_{{22}}}}}{{{{{\bar {R}}}_{{11}}}{{{\bar {R}}}_{{44}}}}}} \right),\quad {{\bar {X}}_{{18}}} = \frac{{{{{\bar {R}}}_{{22}}}{{{\bar {R}}}_{{66}}}}}{{{{{\bar {R}}}_{{11}}}{{{\bar {R}}}_{{44}}}}},\quad {{\bar {X}}_{{19}}} = \frac{{\bar {R}_{{22}}^{2}}}{{\bar {R}_{{11}}^{2}}} - 1, \quad {{\bar {X}}_{{20}}} = \frac{{{{{\bar {R}}}_{{66}}}}}{{\bar {R}_{{44}}^{2}}} - \frac{1}{{{{{\bar {R}}}_{{44}}}}},$$
$$\begin{gathered} {{{\bar {X}}}_{{21}}} = \frac{{{{{\bar {R}}}_{{22}}}}}{{{{{\bar {R}}}_{{11}}}}} - {{{\bar {R}}}_{{66}}},\quad {{{\bar {X}}}_{{22}}} = \frac{{{{{\bar {R}}}_{{66}}}{{{\bar {R}}}_{{66}}}}}{{{{{\bar {R}}}_{{44}}}}},\quad {{{\bar {X}}}_{{23}}} = \beta _{{21}}^{2},\quad {{{\bar {X}}}_{{24}}} = \frac{{\bar {\bar {\eta }}_{1}^{2}}}{{\beta _{{21}}^{2}}},\quad {{{\bar {X}}}_{{25}}} = 0, \\ {{{\bar {X}}}_{{26}}} = \frac{{{{{\bar {R}}}_{{11}}}}}{{{{{\bar {R}}}_{{33}}}}},\quad {{{\bar {X}}}_{{27}}} = \frac{1}{{{{{\bar {\bar {\eta }}}}_{1}}{{{\bar {\bar {\eta }}}}_{2}}}},\quad {{{\bar {X}}}_{{28}}} = 2,\quad {{{\bar {X}}}_{{29}}} = {{{\bar {R}}}_{{66}}}{{{\bar {R}}}_{{44}}},\quad {{{\bar {X}}}_{{30}}} = \frac{\lambda }{\mu },\quad {{{\bar {X}}}_{{31}}} = \frac{{\lambda + 2\mu }}{\mu }, \\ {{{\bar {X}}}_{{32}}} = 2{{R}_{{11}}},\quad {{{\bar {R}}}_{{44}}} = 1,\quad {{{\bar {R}}}_{{66}}} = - \left( {\frac{{\frac{\lambda }{{2\mu }}}}{{\left( {1 + \frac{\lambda }{{2\mu }}} \right)}}} \right), \\ \end{gathered} $$
$$\begin{array}{*{20}{l}} {{{{\bar {R}}}_{{11}}} = \left( {\left( {1 + \frac{{\bar {\lambda }}}{{2\bar {\mu }}}} \right)} \right),\quad {{{\bar {R}}}_{{22}}} = \frac{{\bar {\lambda }}}{{2\bar {\mu }}},\quad {{{\bar {R}}}_{{33}}} = 0.5,\quad {{{\bar {\beta }}}_{1}} = \sqrt {\frac{{\bar {\mu }}}{{\bar {\rho }}}} } \end{array}.$$

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Gupta, S., Pramanik, S. & Smita Exemplification in Exact and Approximate Secular Equation of Surface Wave Along Distinct Interfaces with Sliding Contact. Mech. Solids 56, 819–837 (2021). https://doi.org/10.3103/S0025654421050101

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