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Rayleigh waves in a nonlocal thermoelastic layer lying over a nonlocal thermoelastic half-space

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Abstract

The paper deals with Rayleigh wave propagation in a nonlocal thermoelastic layer, and the layer is lying over a nonlocal thermoelastic half-space. The problem is treated in the context of Eringen’s nonlocal thermoelasticity and Green–Naghdi model type III of hyperbolic thermoelasticity. The frequency equation of Rayleigh waves is derived, and different cases are also discussed. The effect of the nonlocal parameter on phase velocity, attenuation coefficient, specific loss, and penetration depth is presented graphically.

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

    Siddhartha Biswas

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Appendix

Appendix

$$\begin{aligned} \alpha _{1}= & {} \delta _{1} \sin \eta _{1} H+\cos \eta _{1} H-\frac{e_{1} }{e_{3} }\left( {\delta _{3} \sin \eta _{3} H+\cos \eta _{3} H} \right) ,\\ \alpha _{2}= & {} \delta _{2} \sin \eta _{2} H+\cos \eta _{2} H-\frac{e_{2} }{e_{3} }\left( {\delta _{3} \sin \eta _{3} H+\cos \eta _{3} H} \right) ,\\ \gamma _{1}= & {} c_{1} \cos \eta _{1} H+d_{1} \delta _{1} \sin \eta _{1} H-\frac{e_{1} }{e_{3} }\left( {c_{3} \cos \eta _{3} H+d_{3} \delta _{3} \sin \eta _{3} H} \right) ,\\ \gamma _{2}= & {} c_{2} \cos \eta _{2} H+d_{2} \delta _{2} \sin \eta _{2} H-\frac{e_{2} }{e_{3} }\left( {c_{3} \cos \eta _{3} H+d_{3} \delta _{3} \sin \eta _{3} H} \right) ,\\ l_{1}= & {} \left( {ikc_{1} +\delta _{1} \eta _{1} } \right) \cos \eta _{1} H+\left( {ikd_{1} \delta _{1} -\eta _{1} } \right) \sin \eta _{1} H\\&-\frac{e_{1} }{e_{3} }\left[ {\left( {ikc_{3} +\delta _{3} \eta _{3} } \right) \cos \eta _{3} H+\left( {ikd_{3} \delta _{3} -\eta _{3} } \right) \sin \eta _{3} H} \right] ,\\ l_{2}= & {} \left( {ikc_{2} +\delta _{2} \eta _{2} } \right) \cos \eta _{2} H+\left( {ikd_{2} \delta _{2} -\eta _{2} } \right) \sin \eta _{2} H\\&-\frac{e_{2} }{e_{3} }\left[ {\left( {ikc_{3} +\delta _{3} \eta _{3} } \right) \cos \eta _{3} H+\left( {ikd_{3} \delta _{3} -\eta _{3} } \right) \sin \eta _{3} H} \right] ,\\ l_{3}= & {} iky_{1} -k_{1} ,\\ l_{4}= & {} iky_{2} -k_{2} ,\\ l_{5}= & {} iky_{3} -k_{3} ,\\ p_{1}= & {} \left( {ikm_{5} -e_{1} +d_{1} \eta _{1} \delta _{1} } \right) \cos \eta _{1} H+\left( {-c_{1} \eta _{1} +ikm_{5} \delta _{1} -f_{1} \delta _{1} } \right) \sin \eta _{1} H\\&-\frac{e_{1} }{e_{3} }\left[ {\begin{array}{l} \left( {ikm_{5} -e_{3} -d_{3} \delta _{3} \eta _{3} } \right) \cos \eta _{3} H \\ +\left( {-c_{3} \eta _{3} +ikm_{5} \delta _{3} -f_{3} \delta _{3} } \right) \sin \eta _{3} H \\ \end{array}} \right] ,\\ p_{2}= & {} \left( {ikm_{5} -e_{2} +d_{2} \eta _{2} \delta _{2} } \right) \cos \eta _{2} H+\left( {-c_{2} \eta _{2} +ikm_{5} \delta _{2} -f_{2} \delta _{2} } \right) \sin \eta _{2} H\\&-\frac{e_{2} }{e_{3} }\left[ {\begin{array}{l} \left( {ikm_{5} -e_{3} -d_{3} \delta _{3} \eta _{3} } \right) \cos \eta _{3} H \\ +\left( {-c_{3} \eta _{3} +ikm_{5} \delta _{3} -f_{3} \delta _{3} } \right) \sin \eta _{3} H \\ \end{array}} \right] ,\\ p_{3}= & {} ikr_{5} -k_{1} y_{1} -z_{1} ,\\ p_{4}= & {} ikr_{5} -k_{2} y_{2} -z_{2} ,\\ p_{5}= & {} ikr_{5} -k_{3} y_{3} -z_{3} ,\\ q_{1}= & {} e_{1} \cos \eta _{1} H+f_{1} \delta _{1} \sin \eta _{1} H-\frac{e_{1} }{e_{3} }\left( {e_{3} \cos \eta _{3} H+f_{3} \delta _{3} \sin \eta _{3} H} \right) ,\\ q_{2}= & {} e_{2} \cos \eta _{2} H+f_{2} \delta _{2} \sin \eta _{2} H-\frac{e_{2} }{e_{3} }\left( {e_{3} \cos \eta _{3} H+f_{3} \delta _{3} \sin \eta _{3} H} \right) ,\\ P_{1}= & {} \gamma _{2} \left\{ {-l_{3} \left( {p_{4} z_{3} -z_{2} p_{5} } \right) +l_{4} \left( {p_{3} z_{3} -z_{1} p_{5} } \right) +l_{5} \left( {p_{3} z_{2} -z_{1} p_{4} } \right) } \right\} \\&+y_{1} \left\{ {l_{2} \left( {z_{3} p_{4} -z_{2} p_{5} } \right) +l_{4} \left( {-p_{2} z_{3} +q_{2} p_{5} } \right) -l_{5} \left( {-p_{2} z_{2} +p_{4} q_{2} } \right) } \right\} \\&-y_{2} \left\{ {l_{2} \left( {z_{3} p_{3} -z_{1} p_{5} } \right) +l_{3} \left( {-p_{2} z_{3} +p_{5} q_{2} } \right) -l_{5} \left( {-p_{2} z_{1} +q_{2} p_{3} } \right) } \right\} \\&+y_{3} \left\{ {l_{2} \left( {p_{3} z_{2} -p_{4} z_{1} } \right) +l_{3} \left( {-p_{2} z_{2} +p_{4} q_{2} } \right) -l_{4} \left( {-p_{2} z_{1} +p_{3} q_{2} } \right) } \right\} ,\\ P_{2}= & {} \gamma _{1} \left\{ {-l_{3} \left( {p_{4} z_{3} -z_{2} p_{5} } \right) +l_{4} \left( {p_{3} z_{3} -z_{1} p_{5} } \right) +l_{5} \left( {p_{3} z_{2} -z_{1} p_{4} } \right) } \right\} \\&+y_{1} \left\{ {l_{1} \left( {z_{3} p_{4} -z_{2} p_{5} } \right) +l_{4} \left( {-p_{1} z_{3} +q_{1} p_{5} } \right) -l_{5} \left( {-p_{1} z_{2} +p_{4} q_{1} } \right) } \right\} \\&-y_{2} \left\{ {l_{1} \left( {z_{3} p_{3} -z_{1} p_{5} } \right) +l_{3} \left( {-p_{1} z_{3} +p_{5} q_{1} } \right) -l_{5} \left( {-p_{1} z_{1} +q_{1} p_{3} } \right) } \right\} \\&+y_{3} \left\{ {l_{1} \left( {p_{3} z_{2} -p_{4} z_{1} } \right) +l_{3} \left( {-p_{1} z_{2} +p_{4} q_{1} } \right) -l_{4} \left( {-p_{1} z_{1} +p_{3} q_{1} } \right) } \right\} , \\ P_{3}= & {} \gamma _{1} \left\{ {l_{2} \left( {p_{4} z_{3} -z_{2} p_{5} } \right) +l_{4} \left( {-p_{2} z_{3} +q_{2} p_{5} } \right) -l_{5} \left( {-p_{2} z_{2} +q_{2} p_{4} } \right) } \right\} \\&-\gamma _{2} \left\{ {l_{1} \left( {z_{3} p_{4} -z_{2} p_{5} } \right) +l_{4} \left( {-p_{1} z_{3} +q_{1} p_{5} } \right) -l_{5} \left( {-p_{1} z_{2} +p_{4} q_{1} } \right) } \right\} \\&-y_{2} \left\{ {l_{1} \left( {z_{3} p_{2} +q_{2} p_{5} } \right) -l_{2} \left( {-p_{1} z_{3} +p_{5} q_{1} } \right) -l_{5} \left( {p_{1} q_{2} -q_{1} p_{2} } \right) } \right\} \\&+y_{3} \left\{ {l_{1} \left( {-p_{2} z_{2} +p_{4} q_{2} } \right) -l_{2} \left( {-p_{1} z_{2} +p_{4} q_{1} } \right) -l_{4} \left( {p_{1} q_{2} -p_{2} q_{1} } \right) } \right\} , \\ P_{4}= & {} \gamma _{1} \left\{ {l_{2} \left( {p_{3} z_{3} -z_{1} p_{5} } \right) +l_{3} \left( {-p_{2} z_{3} +q_{2} p_{5} } \right) -l_{5} \left( {-p_{2} z_{1} +q_{2} p_{3} } \right) } \right\} \\&-\gamma _{2} \left\{ {l_{1} \left( {z_{3} p_{3} -z_{1} p_{5} } \right) +l_{3} \left( {-p_{1} z_{3} +q_{1} p_{5} } \right) -l_{5} \left( {-p_{1} z_{1} +p_{3} q_{1} } \right) } \right\} \\&-y_{1} \left\{ {l_{1} \left( {-z_{3} p_{2} +q_{2} p_{5} } \right) -l_{2} \left( {-p_{1} z_{3} +p_{5} q_{1} } \right) -l_{5} \left( {p_{1} q_{2} -q_{1} p_{2} } \right) } \right\} \\&+y_{3} \left\{ {l_{1} \left( {-p_{2} z_{1} +p_{3} q_{2} } \right) -l_{2} \left( {-p_{1} z_{1} +p_{3} q_{1} } \right) -l_{3} \left( {p_{1} q_{2} -p_{2} q_{1} } \right) } \right\} , \\ P_{5}= & {} \gamma _{1} \left\{ {l_{2} \left( {p_{3} z-p_{4} z_{1} } \right) +l_{3} \left( {-p_{2} z_{2} +q_{2} p_{4} } \right) -l_{4} \left( {-p_{2} z_{1} +q_{2} p_{3} } \right) } \right\} \\&-\gamma _{2} \left\{ {l_{1} \left( {z_{3} p_{2} -p_{4} z_{1} } \right) +l_{3} \left( {-p_{1} z_{2} +q_{1} p_{4} } \right) -l_{4} \left( {-p_{1} z_{1} +p_{3} q_{1} } \right) } \right\} \\&-y_{1} \left\{ {l_{1} \left( {-z_{2} p_{2} +q_{2} p_{4} } \right) -l_{2} \left( {-p_{1} z_{2} +p_{4} q_{1} } \right) -l_{4} \left( {p_{1} q_{2} -q_{1} p_{2} } \right) } \right\} \\&+y_{2} \left\{ {l_{1} \left( {-p_{2} z_{1} +p_{3} q_{2} } \right) -l_{2} \left( {-p_{1} z_{1} +p_{3} q_{1} } \right) -l_{3} \left( {p_{1} q_{2} -p_{2} q_{1} } \right) } \right\} . \end{aligned}$$

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Biswas, S. Rayleigh waves in a nonlocal thermoelastic layer lying over a nonlocal thermoelastic half-space. Acta Mech 231, 4129–4144 (2020). https://doi.org/10.1007/s00707-020-02751-2

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