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Analysis of Waves at Boundary Surfaces at Distinct Media with Nonlocal Dual-Phase-Lag

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Abstract

This study discusses the behavior of plane waves in a double porous thermoelastic with nonlocal dual-phase-lag solid half-space \((M_{1} )\) in contact with inviscid liquid half-space \((M_{2} )\). The governing equations are expressed in two-dimensional form, and normal mode analysis is adopted to solve the problem for further investigation. It has been found that there exist four coupled longitudinal, one transverse wave in the medium \(M_{1}\), and one mechanical wave in the medium \(M_{2}\). These waves are under the influence of parameters of nonlocal dual-phase-lag and double porosity. Secular equations are determined by applying interfacial mechanical and thermal conditions. The compact form of wave characteristics like phase velocity, attenuation coefficient, penetrating depth, and specific loss is obtained. The component of displacement, temperature change, and volume fraction fields in the medium \(M_{1}\), along with normal velocity and acoustic pressure in the medium, \(M_{2}\) are obtained in closed form. Numerically simulated results are displayed in the form of graphs to depict the behavior of nonlocal and phase lag on basic characteristics of waves.

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Authors and Affiliations

  1. Indira Gandhi University Meerpur, Rewari, Haryana, India

    Rajesh Kumar, Vipin Gupta & M. S. Barak

  2. Himachal Pradesh University Regional Centre, Dharamshala, HP, India

    Vijayata Pathania

  3. Kurukshetra University, Kurukshetra, Haryana, India

    Rajneesh Kumar

Authors
  1. Rajesh Kumar
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  2. Vipin Gupta
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  3. Vijayata Pathania
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  4. Rajneesh Kumar
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  5. M. S. Barak
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Correspondence to M. S. Barak.

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Significant Statement: This model is simulated with many technical applications, including oil recovery, and geological carbon dioxide sequestration in the medium, like; ocean beds having double porosity, phase lags, and nonlocality effects which play a crucial role in this study.

Appendices

Appendix A

$$ L_{0} = (d_{53} + d_{12} d_{59} )(d_{32} d_{43} - f_{3} ) $$
$$ \begin{gathered} L_{1} = \xi_{4}^{2} (d_{12} (f_{7} - d_{32} f_{12} ) + f_{4} + d_{32} f_{9} ) + d_{54} (f_{3} - d_{32} d_{43} ) + d_{53} (d_{34} + d_{32} (d_{44} + f_{2} f_{12} ) + d_{43} (d_{33} + \\ f_{1} f_{7} ) + d_{42} f_{3} - f_{2} f_{7} - f_{1} f_{3} f_{12} ) + d_{59} (d_{11} (d_{32} d_{43} - f_{3} ) + d_{12} (d_{32} d_{44} + d_{33} d_{43} + d_{42} f_{3} + d_{34} ) - f_{2} . \\ (f_{4} - d_{32} f_{9} ) + f_{1} (d_{43} f_{4} - f_{3} f_{9} )) + d_{56} (d_{43} f_{4} - f_{3} f_{9} - d_{12} (d_{43} f_{7} - f_{3} f_{12} )) + d_{12} d_{60} (f_{3} - d_{32} d_{43} ) \\ \end{gathered} $$
$$ \begin{gathered} L_{2} = \xi_{4}^{2} (d_{31} - d_{32} (d_{41} + d_{11} f_{12} ) + f_{7} (d_{11} - d_{12} d_{42} + f_{1} f_{9} ) - f_{12} (d_{12} d_{33} + f_{1} f_{4} ) + d_{33} f_{9} + d_{42} f_{4} ) + \\ d_{53} (d_{33} (d_{44} + f_{2} f_{12} ) + d_{34} (f_{1} f_{12} - d_{42} ) + f_{7} (d_{42} f_{2} + d_{44} f_{1} )) - d_{54} (d_{34} + d_{32} (d_{44} + f_{2} f_{12} ) + \\ d_{43} (d_{33} + f_{1} f_{7} ) - f_{3} (f_{1} f_{12} - d_{42} ) - f_{2} f_{7} ) + d_{55} (d_{32} d_{43} - f_{3} ) + d_{56} (d_{34} (f_{9} - d_{12} f_{12} ) - d_{43} (d_{31} \\ + d_{11} f_{7} ) + d_{44} (f_{4} - d_{12} f_{7} ) + f_{3} (d_{41} + d_{11} f_{12} ) + f_{2} (f_{4} f_{12} - f_{7} f_{9} )) + d_{57} (d_{43} (f_{7} d_{12} - f_{4} ) + f_{3} . \\ (f_{9} - d_{12} f_{12} )) + d_{58} (f_{4} - d_{32} f_{9} + d_{12} (d_{32} f_{12} - f_{7} )) + d_{59} (d_{11} (d_{34} + d_{32} d_{44} + d_{33} d_{43} + d_{42} f_{3} ) + \\ d_{31} (f_{2} - d_{43} f_{1} ) + d_{12} (d_{33} d_{44} - d_{34} d_{42} ) + d_{41} (f_{1} f_{3} - d_{32} f_{2} ) + f_{2} (d_{33} f_{9} + d_{42} f_{4} ) + f_{1} (d_{34} f_{9} + \\ d_{44} f_{4} )) + d_{60} (d_{11} (f_{3} - d_{32} d_{43} )) - d_{12} (d_{34} + d_{32} d_{44} + d_{33} d_{43} + d_{42} f_{3} ) + f_{2} (f_{4} - d_{32} f_{9} ) + \\ f_{1} (f_{3} f_{9} - d_{43} f_{4} ) \\ \end{gathered} $$
$$ \begin{gathered} L_{3} = \xi_{4}^{2} (d_{31} (f_{1} f_{12} - d_{42} ) - d_{11} (d_{33} f_{12} + d_{42} f_{7} ) - d_{41} (d_{33} + f_{1} f_{7} )) + d_{54} (d_{34} (d_{42} - f_{1} f_{12} ) - d_{33} (d_{44} \\ + f_{2} f_{12} ) - f_{7} (d_{42} f_{2} + d_{44} f_{1} )) + d_{55} (d_{34} + d_{32} (d_{44} + f_{2} f_{12} ) + d_{43} (d_{33} + f_{1} f_{7} ) + f_{3} (d_{42} - f_{1} f_{12} ) - \\ f_{2} f_{7} ) + d_{56} (f_{2} (d_{41} f_{7} - d_{31} f_{12} ) - d_{44} (d_{31} + d_{11} f_{7} ) - d_{34} (d_{41} + d_{11} f_{12} )) + d_{57} (d_{43} (d_{31} + d_{11} f_{7} ) + \\ d_{34} (d_{12} f_{12} - f_{9} ) + d_{44} (d_{12} f_{7} - f_{4} ) - f_{3} (d_{41} + d_{11} f_{12} ) + f_{2} (f_{7} f_{9} - f_{4} f_{12} )) + d_{58} (d_{11} (d_{32} f_{12} - f_{7} ) \\ - d_{31} + d_{32} d_{41} - d_{33} f_{9} - d_{42} f_{4} + d_{12} (d_{33} f_{12} + d_{42} f_{7} ) + f_{1} (f_{4} f_{12} - f_{7} f_{9} )) + d_{59} (d_{11} (d_{33} d_{44} - d_{34} . \\ d_{42} ) - d_{31} (d_{42} f_{2} + d_{44} f_{1} ) - d_{41} (d_{33} f_{2} + d_{34} f_{1} )) + d_{60} (d_{31} (d_{43} f_{1} - f_{2} ) - d_{11} (d_{34} + d_{32} d_{44} + d_{33} . \\ d_{43} + d_{42} f_{3} ) + d_{12} (d_{34} d_{42} - d_{33} d_{44} ) + f_{2} (d_{32} d_{41} - d_{33} f_{9} - d_{42} f_{4} ) - f_{1} (d_{34} f_{9} + d_{44} f_{4} - d_{41} f_{3} )) \\ \end{gathered} $$
$$ \begin{gathered} L_{4} = d_{55} (d_{33} (d_{44} + f_{2} f_{12} ) + d_{34} (f_{1} f_{12} - d_{42} ) + f_{7} (d_{42} f_{2} + d_{44} f_{1} )) + d_{57} (d_{31} (d_{44} + f_{2} f_{12} ) + d_{41} (d_{34} \\ - f_{2} f_{7} ) + d_{11} (d_{34} f_{12} + d_{44} f_{7} )) + d_{58} (d_{31} (d_{42} - f_{1} f_{12} ) + d_{41} (d_{33} + f_{1} f_{7} ) + d_{11} (d_{33} f_{12} + d_{42} f_{7} )) + \\ d_{60} (d_{11} (d_{34} d_{42} - d_{33} d_{44} ) + d_{31} (d_{42} f_{2} + d_{44} f_{1} ) + d_{41} (d_{33} f_{2} + d_{34} f_{1} )) \\ \end{gathered} $$

Appendix B

$$ \begin{gathered} V_{q} = ((f_{3} f_{9} - d_{43} f_{4} + d_{12} (d_{43} f_{7} - f_{3} f_{12} ))m_{q}^{4} + (d_{43} (d_{31} + d_{11} f_{7} ) + d_{44} (d_{12} f_{7} - f_{4} ) - d_{41} f_{3} + \\ f_{12} (d_{12} d_{34} - d_{11} f_{3} - f_{2} f_{4} ) + f_{9} (f_{2} f_{7} - d_{34} ))m_{q}^{2} + d_{31} (d_{44} + f_{2} f_{12} ) + d_{11} (d_{34} f_{12} + d_{44} f_{7} ) \\ + d_{41} (d_{34} - f_{2} f_{7} ))/\sigma^{*} \\ \end{gathered} $$
$$ \begin{gathered} W_{q} = ((f_{4} - d_{12} f_{7} + d_{32} (d_{12} f_{12} - f_{9} ))m_{q}^{4} + (f_{12} (d_{11} d_{32} + d_{12} d_{33} + f_{1} f_{4} ) + f_{7} (d_{12} d_{42} - d_{11} ) - f_{9} (d_{33} \\ + f_{1} f_{7} ) + d_{32} d_{41} - d_{31} - d_{42} f_{4} )m_{q}^{2} + d_{31} (d_{42} - f_{1} f_{12} ) + d_{41} (d_{33} + f_{1} f_{7} ) + d_{11} (d_{33} f_{12} + \\ d_{42} f_{7} ))/\sigma^{*} \\ \end{gathered} $$
$$ \begin{gathered} S_{q} = ((f_{3} - d_{32} d_{43} )d_{12} m_{q}^{6} + (d_{11} (f_{3} - d_{32} d_{43} ) - d_{12} (d_{34} + d_{32} d_{44} + d_{33} d_{43} + d_{42} f_{3} ) + f_{2} (f_{4} - d_{32} f_{9} ) \\ + f_{1} (f_{3} f_{9} - d_{43} f_{4} ))m_{q}^{4} + (d_{12} (d_{34} d_{42} - d_{33} d_{44} ) - d_{11} (d_{32} d_{44} + d_{33} d_{43} + d_{34} - d_{42} f_{3} ) + d_{31} (d_{43} f_{1} \\ - f_{2} ) + d_{41} (d_{32} f_{2} - f_{1} f_{3} ) - f_{2} (d_{33} f_{9} + d_{42} f_{4} ) - f_{1} (d_{34} f_{9} + d_{44} f_{4} ))m_{q}^{2} + d_{11} (d_{34} d_{42} - d_{33} d_{44} ) + \\ d_{31} (d_{42} f_{2} + d_{44} f_{1} ) + d_{41} (d_{33} f_{2} + d_{34} f_{1} ))/\sigma^{*} \\ \end{gathered} $$
$$ \begin{gathered} \sigma^{*} = (f_{3} - d_{32} d_{43} )m_{q}^{4} + (f_{2} f_{7} - d_{34} - d_{43} (d_{33} + f_{1} f_{7} ) - d_{32} (d_{44} + f_{2} f_{12} ) + f_{3} (f_{1} f_{12} - d_{42} ))m_{q}^{2} + \\ d_{34} (d_{42} - f_{1} f_{12} ) - d_{33} (d_{44} + f_{2} f_{12} ) - f_{7} (d_{42} f_{2} + d_{44} f_{7} ) \\ \end{gathered} $$

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Kumar, R., Gupta, V., Pathania, V. et al. Analysis of Waves at Boundary Surfaces at Distinct Media with Nonlocal Dual-Phase-Lag. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 93, 573–585 (2023). https://doi.org/10.1007/s40010-023-00850-y

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