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Surface waves in piezothermoelastic transversely isotropic layer lying over piezothermoelastic transversely isotropic half-space

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Abstract

In this article, the propagation of Rayleigh surface waves in a piezothermoelastic transversely isotropic layer lying over a piezothermoelastic transversely isotropic half-space is investigated in the context of the Green–Naghdi model type III of hyperbolic thermoelasticity. The secular equation of Rayleigh surface waves is derived, and different cases are discussed. Phase velocity, attenuation coefficient and specific loss of surface waves are computed and presented graphically with respect to frequency, and a comparison of different wave characteristics for classical and generalized thermoelastic models is presented in the figures.

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Acknowledgements

The author is grateful to the reviewer for his valuable suggestion for the improvement of the paper.

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  1. Department of Mathematics, University of North Bengal, Darjeeling, 734013, India

    Siddhartha Biswas

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  1. Siddhartha Biswas
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Appendix

Appendix

$$\begin{aligned} {A}'= & {} {A}'_{14} {A}'_{17} -{A}'_{11} {A}'_{20} , \\ {B}'= & {} {A}'_{15} {A}'_{17} +{A}'_{14} {A}'_{18} -{A}'_{11} {A}'_{21} -{A}'_{12} {A}'_{20} , \\ {C}'= & {} {A}'_{16} {A}'_{17} +{A}'_{15} {A}'_{18} +{A}'_{14} {A}'_{19} -{A}'_{11} {A}'_{22} -{A}'_{12} {A}'_{20} -{A}'_{13} {A}'_{20} , \\ {E}'= & {} {A}'_{16} {A}'_{18} +{A}'_{15} {A}'_{19} -{A}'_{12} {A}'_{22} -{A}'_{13} {A}'_{21} , \\ {F}'= & {} {A}'_{16} {A}'_{19} -{A}'_{13} {A}'_{22}, \end{aligned}$$

where

$$\begin{aligned} {A}'_{11}= & {} b_{1} b_{9} \left( {b_{6} b_{14} +b_{9} } \right) , \\ {A}'_{12}= & {} b_{1} b_{9} \left( {b_{10} b_{14} -b_{9} b_{15} } \right) +\left( {b_{4} b_{9} -b_{5} b_{8} } \right) \left( {b_{9} +b_{6} b_{14} } \right) -\left( {b_{8} b_{14} +b_{9} b_{13} } \right) \left( {b_{2} b_{9} -b_{5} b_{6} } \right) , \\ {A}'_{13}= & {} \left( {b_{4} b_{9} -b_{5} b_{8} } \right) \left( {b_{10} b_{14} -b_{9} b_{15} } \right) +b_{5} b_{10} \left( {b_{8} b_{14} +b_{9} b_{13} } \right) , \\ {A}'_{14}= & {} b_{1} b_{9} \left( {b_{14} b_{7} -b_{9} b_{12} } \right) , \\ {A}'_{15}= & {} b_{1} b_{9} \left( {b_{9} b_{16} -b_{11} b_{14} } \right) +\left( {b_{4} b_{9} -b_{5} b_{8} } \right) \left( {b_{4} b_{7} -b_{9} b_{12} } \right) -\left( {b_{8} b_{14} +b_{9} b_{13} } \right) \left( {b_{3} b_{9} -b_{5} b_{7} } \right) , \\ {A}'_{16}= & {} \left( {b_{4} b_{9} -b_{5} b_{8} } \right) \left( {b_{9} b_{16} -b_{11} b_{14} } \right) -b_{5} b_{11} \left( {b_{8} b_{14} +b_{9} b_{13} } \right) , \\ {A}'_{17}= & {} b_{13} b_{17} \left( {b_{9} +b_{6} b_{14} } \right) -b_{17} \left( {b_{8} b_{14} +b_{9} b_{13} } \right) , \\ {A}'_{18}= & {} b_{13} b_{17} \left( {b_{10} b_{14} -b_{9} b_{15} } \right) +\left( {b_{13} b_{21} -b_{14} b_{20} } \right) \left( {b_{6} b_{14} +b_{9} } \right) \\&-\left( {b_{8} b_{14} +b_{9} b_{13} } \right) \left( {b_{21} -b_{15} b_{17} -b_{14} b_{18} } \right) , \\ {A}'_{19}= & {} \left( {b_{13} b_{21} -b_{14} b_{20} } \right) \left( {b_{10} b_{14} -b_{9} b_{15} } \right) +b_{15} b_{21} \left( {b_{8} b_{14} +b_{9} b_{13} } \right) , \\ {A}'_{20}= & {} b_{13} b_{17} \left( {b_{7} b_{14} -b_{9} b_{12} } \right) +b_{12} b_{17} \left( {b_{8} b_{14} +b_{9} b_{13} } \right) , \\ {A}'_{21}= & {} b_{13} b_{17} \left( {b_{9} b_{16} -b_{4} b_{14} } \right) +\left( {b_{13} b_{21} -b_{14} b_{20} } \right) \left( {b_{7} b_{14} -b_{9} b_{12} } \right) \\&+\left( {b_{8} b_{14} +b_{9} b_{13} } \right) \left( {b_{12} b_{21} +b_{14} b_{19} -b_{16} b_{17} } \right) , \\ {A}'_{22}= & {} \left( {b_{13} b_{21} -b_{14} b_{20} } \right) \left( {b_{9} b_{16} -b_{4} b_{14} } \right) -b_{16} b_{21} \left( {b_{8} b_{14} +b_{9} b_{13} } \right) . \end{aligned}$$

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Biswas, S. Surface waves in piezothermoelastic transversely isotropic layer lying over piezothermoelastic transversely isotropic half-space. Acta Mech 232, 373–387 (2021). https://doi.org/10.1007/s00707-020-02848-8

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