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Influence of yielding base and rigid base on propagation of Rayleigh-type wave in a viscoelastic layer of Voigt type

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Abstract

The present study aims to study the propagation of Rayleigh-type wave in a layer, composed of isotropic viscoelastic material of Voigt type, with the effect of yielding base and rigid base in two distinct cases. With the aid of an analytical treatment, closed-form expressions of phase velocity and damped velocity for both the cases are deduced. As a special case of the problem it is found that obtained results are in good agreement with the established standard results existing in the literature. It is established through the study that volume-viscoelastic and shear-viscoelastic material parameter and yielding parameter have significant effect on phase and damped velocities of Rayleigh-type wave in both the cases. Numerical calculations and graphical illustration have been carried out for both the considered cases in the presence and the absence of viscoelasticity. A comparative study has been performed to analyse the effect of layer with yielding base, traction-free base and rigid base on the phase and damped velocities of Rayleigh-type wave.

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Acknowledgements

The authors convey their sincere thanks to Indian School of Mines, Dhanbad, for providing JRF to Ms. Shalini Saha and Mr. Kshitish Ch. Mistri and also facilitating us with its best facility for research. The authors sincerely acknowledge National Board of Higher Mathematics (NBHM) for providing financial support to carry out this research work through the project titled “Mathematical modelling of elastic wave propagation in highly anisotropic and heterogeneous media”.

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Authors and Affiliations

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

    S Saha, A Chattopadhyay, K C Mistri & A K Singh

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  1. S Saha
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  2. A Chattopadhyay
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  3. K C Mistri
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  4. A K Singh
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Correspondence to S Saha.

Appendix A

Appendix A

$$ \begin{aligned} S_{1} & = \left( {A_{1} - A_{2} } \right)\left\{ {\mu \left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) - 2p\mu^{\prime}E_{2} F_{2} } \right\} \\ & \quad - \,\left( {B_{1} - B_{2} } \right)\left\{ {p\mu^{\prime}\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) + 2E_{2} F_{2} \mu } \right\}, \\ \end{aligned} $$
$$ \begin{aligned} S_{2} & = \left( {B_{1} - B_{2} } \right)\left\{ {\mu \left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) - 2p\mu^{\prime}E_{2} F_{2} } \right\} \\ & \quad + \,\left( {A_{1} - A_{2} } \right)\left\{ {p\mu^{\prime}\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) + 2E_{2} F_{2} \mu } \right\}, \\ \end{aligned} $$
$$ \begin{aligned} S_{3} & = E_{1} C_{1} - F_{1} D_{1} - M_{1} C_{2} + N_{1} D_{2} , \\ S_{4} & = F_{1} C_{1} - N_{1} C_{2} - M_{1} D_{2} + D_{1} E_{1} , \\ \end{aligned} $$
$$ \begin{aligned} S_{5} & = 2\xi \left\{ {E_{1} \left( {A_{1} - A_{2} } \right) - F_{1} \left( {B_{1} - B_{2} } \right)} \right\}, \\ S_{6} & = 2\xi \left\{ {E_{1} \left( {B_{1} - B_{2} } \right) + F_{1} \left( {A_{1} - A_{2} } \right)} \right\}, \\ \end{aligned} $$
$$ \begin{aligned} S_{7} & = \mu \left\{ {\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)C_{1} - 2D_{1} E_{2} F_{2} } \right\} \\ & \quad - \,p\mu^{\prime}\left\{ {2C_{1} E_{2} F_{2} + D_{1} \left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)} \right\} \\ & \quad - \,M_{2} C_{2} + N_{2} D_{2} , \\ \end{aligned} $$
$$ \begin{aligned} S_{8} & = p\mu^{\prime}\left\{ {\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)C_{1} - 2D_{1} E_{2} F_{2} } \right\} \\ & \quad + \,\mu \left\{ {2C_{1} E_{2} F_{2} + D_{1} \left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)} \right\} \\ & \quad - \,N_{2} C_{2} - D_{2} M_{2} , \\ \end{aligned} $$
$$ \begin{aligned} S_{9} & = E_{1} A_{1} - F_{1} B_{1} - A_{2} M_{3} + B_{2} N_{3} , \\ S_{10} & = F_{1} A_{1} - B_{2} M_{3} - A_{2} N_{3} + B_{1} E_{1} , \\ \end{aligned} $$
$$ \begin{aligned} S_{11} & = 2\xi \left( {E_{1} C_{1} - D_{1} F_{1} } \right) - M_{4} C_{2} + N_{4} D_{2} , \\ S_{12} & = 2\xi \left( {F_{1} C_{1} + D_{1} E_{1} } \right) - N_{4} C_{2} - M_{4} D_{2} , \\ \end{aligned} $$
$$ \begin{aligned} M_{1} & = \frac{{E_{2} \left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) + 2E_{2} F_{2}^{2} }}{{2\left( {E_{2}^{2} + F_{2}^{2} } \right)}}, \\ N_{1} & = \frac{{2E_{2}^{2} F_{2} - F_{2} \left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)}}{{2\left( {E_{2}^{2} + F_{2}^{2} } \right)}}, \\ \end{aligned} $$
$$ M_{2} = 4\xi^{2} \frac{{\left( {E_{1} E_{2} - F_{1} F_{2} } \right).\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) + 2E_{2} F_{2} \left( {F_{1} E_{2} + F_{2} E_{1} } \right)}}{{\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)^{2} + 4E_{2}^{2} F_{2}^{2} }}, $$
$$ N_{2} = 4\xi^{2} \frac{{\left( {F_{1} E_{2} + F_{2} E_{1} } \right)\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) - 2E_{2} F_{2} \left( {E_{1} E_{2} - F_{1} F_{2} } \right)}}{{\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)^{2} + 4E_{2}^{2} F_{2}^{2} }}, $$
$$ \begin{aligned} M_{3} & = 2\xi^{2} \frac{{E_{1} \left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) + 2E_{2} F_{2} F_{1} }}{{\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)^{2} + 4E_{2}^{2} F_{2}^{2} }}, \\ N_{3} & = 2\xi^{2} \frac{{F_{1} \left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) - 2E_{2} F_{2} E_{1} }}{{\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)^{2} + 4E_{2}^{2} F_{2}^{2} }}, \\ \end{aligned} $$
$$ M_{4} = \frac{{E_{2} \left( {\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)^{2} - 4E_{2}^{2} F_{2}^{2} } \right) + 4\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)E_{2} F_{2}^{2} }}{{2\xi^{2} \left( {E_{2}^{2} + F_{2}^{2} } \right)}}, $$
$$ N_{4} = \frac{{F_{2} \left( {\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)^{2} - 4E_{2}^{2} F_{2}^{2} } \right) + 4\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)E_{2}^{2} F_{2} }}{{2\xi^{2} \left( {E_{2}^{2} + F_{2}^{2} } \right)}}, $$
$$ \begin{aligned} A_{1} & = \text{coshE}_{1} H\text{cosF}_{1} H,\quad B_{1} = \text{sinhE}_{1} H\text{sinF}_{1} H, \\ A_{2} & = \text{coshE}_{2} H\text{cosF}_{2} H,\quad B_{2} = \text{sinhE}_{2} H\text{sinF}_{2} H, \\ \end{aligned} $$
$$ \begin{aligned} C_{1} & = \text{sinhE}_{1} H\text{cosF}_{1} H,\quad D_{1} = \text{coshE}_{1} H\text{sinF}_{1} H, \\ C_{2} & = \text{sinhE}_{2} H\text{cosF}_{2} H,\quad D_{2} = \text{coshE}_{2} H\text{sinF}_{2} H, \\ \end{aligned} $$
$$ E_{1} = (a_{1}^{2} + b_{1}^{2} )^{{\frac{1}{4}}} \cos \left( {\frac{{\theta_{1} }}{2}} \right),\quad F_{1} = (a_{1}^{2} + b_{1}^{2} )^{{\frac{1}{4}}} \sin \left( {\frac{{\theta_{1} }}{2}} \right), $$
$$ E_{2} = (a_{2}^{2} + b_{2}^{2} )^{{\frac{1}{4}}} \cos \left( {\frac{{\theta_{2} }}{2}} \right),\quad F_{2} = (a_{2}^{2} + b_{2}^{2} )^{{\frac{1}{4}}} \sin \left( {\frac{{\theta_{2} }}{2}} \right), $$
$$ a_{1} = \xi^{2} - \frac{{\rho p^{2} \left( {\lambda + 2\mu } \right)}}{{(\lambda + 2\mu )^{2} + p^{2} (\lambda^{\prime} + 2\mu^{\prime})^{2} }},\quad a_{2} = \xi^{2} - \frac{{\rho p^{2} \mu }}{{\mu^{2} + p^{2} \mu^{\prime 2} }}, $$
$$ \begin{aligned} b_{1} & = \frac{{\rho p^{3} \left( {\lambda^{\prime } + 2\mu^{\prime } } \right)}}{{(\lambda + 2\mu )^{2} + p^{2} (\lambda^{\prime} + 2\mu^{\prime})^{2} }},\quad b_{2} = \frac{{\rho p^{3} \mu^{\prime}}}{{\mu^{2} + p^{2} \mu^{\prime 2} }}, \\ \theta_{1} & = \tan^{ - 1} \left( {\frac{{b_{1} }}{{a_{1} }}} \right),\quad \theta_{2} = \tan^{ - 1} \left( {\frac{{b_{2} }}{{a_{2} }}} \right). \\ \end{aligned} $$
$$ \begin{aligned} R_{1} & = P_{1} A_{1} - Q_{1} B_{1} - A_{2} \left( {P_{1}^{2} - Q_{1}^{2} } \right) + 2B_{2} P_{1} Q_{1} , \\ R_{2} & = Q_{1} A_{1} + P_{1} B_{1} - B_{2} \left( {P_{1}^{2} - Q_{1}^{2} } \right) - 2A_{2} P_{1} Q_{1} , \\ R_{3} & = P_{1} A_{1} - Q_{1} B_{1} - A_{2} ,R_{4} = Q_{1} A_{1} + P_{1} B_{1} - B_{2} , \\ R_{5} & = P_{2} C_{1} - Q_{2} D_{1} - C_{2} ,R_{6} = P_{2} D_{1} + Q_{2} C_{1} - D_{2} , \\ R_{7} & = P_{2} C_{1} - Q_{2} D_{1} - C_{2} \left( {P_{2}^{2} - Q_{2}^{2} } \right) + 2P_{2} Q_{2} C_{2} , \\ R_{8} & = P_{2} D_{1} + Q_{2} C_{1} - D_{2} \left( {P_{2}^{2} - Q_{2}^{2} } \right) - 2P_{2} Q_{2} C_{2} , \\ \end{aligned} $$
$$ \begin{aligned} P_{1} & = \frac{{\xi^{2} + E_{2}^{2} - F_{2}^{2} }}{{\xi^{2} }}, \\ P_{2} & = \left\{ {\frac{{2\left( {E_{1} E_{2} - F_{1} F_{2} } \right)\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right) + 4E_{2} F_{2} \left( {E_{2} F_{1} + E_{1} F_{2} } \right)}}{{\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)^{2} + 4E_{2}^{2} F_{2}^{2} }}} \right\}, \\ \end{aligned} $$
$$ \begin{aligned} Q_{1} & = \frac{{E_{2} F_{2} }}{{\xi^{2} }}, \\ Q_{2} & = \left\{ {\frac{{4E_{2} F_{2} \left( {E_{1} E_{2} - F_{1} F_{2} } \right) + 2\left( {E_{2} F_{1} + E_{1} F_{2} } \right)\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)}}{{\left( {\xi^{2} + E_{2}^{2} - F_{2}^{2} } \right)^{2} + 4E_{2}^{2} F_{2}^{2} }}} \right\}. \\ \end{aligned} $$

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Saha, S., Chattopadhyay, A., Mistri, K.C. et al. Influence of yielding base and rigid base on propagation of Rayleigh-type wave in a viscoelastic layer of Voigt type. Sādhanā 42, 1459–1471 (2017). https://doi.org/10.1007/s12046-017-0690-0

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