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Effect of phase-lags on the transient waves in an axisymmetric functionally graded viscothermoelastic spherical cavity in radial direction

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Abstract

This paper aims to present the analysis of transient wave characteristics in a functionally graded viscothermoelastic infinite medium with spherical cavity in the context of generalized thermoelasticity. Continued series solution is used to solve simultanious differential equations for evaluating the field variables. Convergence of the series solution is implemented and investigated that the series of functions are absolutely and uniformly convergent. The formal solution for the field variables are obtained analytically and represented graphically. The effect of grading index and different theories of generalized thermoelasticity are also shown graphically to examine the behavior of the variations of the field variables.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Maharaja Agrasen University, Baddi, Solan, HP, 174103, India

    Dinesh Kumar Sharma

  2. Department of Mathematics, Dinabandhu Mahavidyalaya, Bongaon, West Bengal, India

    Mitali Bachher

  3. Department of Applied Mathematics, University of Calcutta, Kolkata, 700009, India

    Nantu Sarkar

Authors
  1. Dinesh Kumar Sharma
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  2. Mitali Bachher
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  3. Nantu Sarkar
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Corresponding author

Correspondence to Dinesh Kumar Sharma.

Appendix

Appendix

$$ \left. \begin{array}{l} G_{11}^{k} (s_{j} ) = \frac{i\Omega }{{\tilde{\updelta }_{0}^{*} \left\{ {(s_{j} + k + 2)^{2} \, - n^{2} } \right\}}},\,\,\,G_{12}^{k} (s_{j} ) = \frac{{A^{*} \left( {s_{j} + k + 1 + b^{*} } \right)}}{{(s_{j} + k + 2)^{2} \, - n^{2} }}, \hfill \\ G_{21}^{k} (s_{j} ) = \frac{{ - B^{*} \left( {s_{j} + k + 2 + b^{*} } \right)}}{{(s_{j} + k + 2)^{2} - (a^{*} )^{2} }},\,\,\,G_{22}^{k} (s_{j} ) = \frac{{\Omega^{*} \Omega^{2} \tilde{\uptau }_{q} }}{{(s_{j} + k + 2)^{2} - (a^{*} )^{2} }} \hfill \\ \end{array} \right\}\,\,\,\,(k = 1,\,2,\,3, \ldots ). $$
(44)
$$ \left\{ \begin{aligned} D_{11}^{2k} (s_{j} ) = (G_{12}^{2k - 2} (s_{j} )D_{21}^{2k - 1} (s_{j} ) - G_{11}^{2k - 2} (s_{j} )D_{11}^{2k - 2} (s_{j} ))\,\, \hfill \\ D_{22}^{2k} (s_{j} ) = \,\,(G_{21}^{2k - 2} (s_{j} )D_{12}^{2k - 1} (s_{j} ) - G_{22}^{2k - 2} (s_{j} )D_{22}^{2k - 2} (s_{j} ))\, \hfill \\ D_{12}^{2k + 1} (s_{j} ) = \,( - \,\,G_{12}^{2k - 1} (s_{j} )D_{22}^{2k} (s_{j} ) + G_{11}^{2k - 1} (s_{j} )D_{12}^{2k - 1} (s_{j} )) \hfill \\ D_{21}^{2k + 1} (s_{j} ) = ( - \,\,G_{21}^{2k - 1} (s_{j} )D_{11}^{2k} (s_{j} ) + G_{22}^{2k - 1} (s_{j} )D_{21}^{2k - 1} (s_{j} ))\, \hfill \\ \end{aligned} \right.;\,\,\,\,k = 1,\,2,\,3\, \ldots , $$
(45)

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Sharma, D.K., Bachher, M. & Sarkar, N. Effect of phase-lags on the transient waves in an axisymmetric functionally graded viscothermoelastic spherical cavity in radial direction. Int. J. Dynam. Control 9, 424–437 (2021). https://doi.org/10.1007/s40435-020-00659-2

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