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Modeling of Delayed Thermo Elastic Waves in a Polygonal Ring Reinforced with Graphene Platelets

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Abstract

The free vibration analysis of generalized thermo elastic waves in a clamped free polygonal ring incorporated with graphene platelets is addressed in the purview of delayed heat conduction equation. Homogeneous thermo elastic polygonal ring in isotropic plain strain form is considered for controlling equations. The distribution model of Halpin–Tsai is adopted for homogenization. The consistent formulation is derived by the nonlinear surface of the GPLRC ring via Fourier expansion collocation method. Numerical values are discussed for triangular, square, pentagonal and hexagonal GPLRC rings. To verify the analytical model, the field functions are presented in tabular and graphical forms with weight fraction and various patterns of GPLRC. This study may find the applications in design of thermal polygonal GPL ring structures.

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Authors and Affiliations

  1. Department of Mathematics, Karunya University, Coimbatore, 641 114, Tamil Nadu, India

    R. Selvamani

  2. Department of Applied Mathematics, University of Calcutta, Kolkata, West Bangal, 700009, India

    N. Sarkar

  3. Department of Mechanical Engineering, Imam Khomeini International University, Qazvin, Iran

    Farzad Ebrahami

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  1. R. Selvamani
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Appendix A

Appendix A

$$ \begin{aligned} a_{n}^{j} & = 2\left\{ {p(p - 1)J_{p} (\alpha_{i} ax) + (\alpha_{i} ax)J_{p + 1} (\alpha_{i} ax)} \right\}\cos 2(\theta - \gamma_{i} )\cos m\theta \\ & \quad - x^{2} \left\{ {(\alpha_{i} a)^{2} + \left[ {\overline{\lambda } + 2\cos^{2} (\theta - \gamma_{i} )} \right] + a_{i} } \right\}J_{p} \left( {\alpha_{i} ax} \right)\cos m\theta \\ & \quad + 2p\left\{ {(p - 1)J_{p} (\alpha_{i} ax) - (\alpha_{i} ax)J_{p + 1} (\alpha_{i} ax)} \right\}\sin m\theta \sin 2(\theta - \gamma_{i} ),\quad j = 1,2 \\ \end{aligned} $$
(A1)
$$ \begin{aligned} a_{n}^{3} & = - 2\left\{ {J_{p + 1} (\alpha_{3} ax)(\alpha_{3} ax) - J_{p} (\alpha_{3} ax)p\,(p - 1)} \right\}\cos m\,\theta \cos 2(\theta - \gamma_{i} ) \\ & \quad - 2\left\{ {J_{p} (\alpha_{3} ax)[(p - 1)p - (\alpha_{3} ax)^{2} ] + J_{p + 1} (\alpha_{3} ax)(\alpha_{3} ax)} \right\}\sin m\,\theta \sin 2(\theta - \gamma_{i} ) \\ \end{aligned} $$
(A2)
$$\begin{aligned} a_{n}^{4} & = 2\left\{ {p\,(p - 1)Y_{p} (\alpha_{4} ax)\cos m\,\theta - (\alpha_{4} ax)Y_{p + 1} (\alpha_{4} ax)\cos m\,\theta } \right\}\cos 2(\theta - \gamma_{i} ) \\ & \quad + 2\left\{ {Y_{p} (\alpha_{4} ax)\,[p(p - 1) - (\alpha_{4} ax)^{2} ]\sin m\,\theta + Y_{p + 1} (\alpha_{4} ax)(\alpha_{4} ax)\sin m\,\theta } \right\}\sin 2(\theta - \gamma_{i} ) \\ \end{aligned} $$
(A3)
$$ \begin{aligned} a_{n}^{j} & = 2\left\{ {p(p - 1)Y_{p} (\alpha_{i} ax) + (\alpha_{i} ax)Y_{p + 1} (\alpha_{i} ax)} \right\}\cos 2(\theta - \gamma_{i} )\cos m\,\theta \\ & \quad - x^{2} \left\{ {(\alpha_{i} a)^{2} + \left[ {\overline{\lambda } + 2\cos^{2} (\theta - \gamma_{i} )} \right] + a_{i} } \right\}Y_{p} \left( {\alpha_{i} ax} \right)\cos m\,\theta \\ & \quad + 2p\left\{ {(p - 1)Y_{p} (\alpha_{i} ax) - (\alpha_{i} ax)Y_{p + 1} (\alpha_{i} ax)} \right\}\sin m\,\theta \sin 2(\theta - \gamma_{i} ),\quad j = 5,6 \\ \end{aligned} $$
(A4)
$$ \begin{aligned} b_{n}^{j} & = 2\left\{ {[p(p - 1) - (\alpha_{i} ax)^{2} ]J_{p} (\alpha_{i} ax) + (\alpha_{i} ax)J_{p + 1} (\alpha_{i} ax)} \right\}\cos m\,\theta \sin 2(\theta - \gamma_{i} ) \\ &\quad + 2p\left\{ {(\alpha_{i} ax)J_{p + 1} (\alpha_{i} ax) - (p - 1)J_{p} (\alpha_{i} ax)} \right\}\sin m\,\theta \cos 2(\theta - \gamma_{i} ),j = 1,2 \\ \end{aligned} $$
(A5)
$$ \begin{aligned} b_{n}^{3} & = - \left\{ {J_{p + 1} (\alpha_{3} ax)(\alpha_{3} a\,x)\cos m\,\theta - \,J_{p} (\alpha_{3} a\,x)(p - 1)\cos m\,\theta } \right\}\sin 2\theta - 2\gamma_{i} \\ & \quad - \left[ {2(\alpha_{3} ax)J_{p + 1} (\alpha_{3} ax)2p\,\,\sin m\,\theta \,\,\cos 2(\theta - \gamma_{i} )}\right.\\ & \quad \left.{ - \,[(\alpha_{3} ax)^{2} - 2p(p - 1)]2p\,\,\sin m\,\theta \,\,\cos 2(\theta - \gamma_{i} )J_{p} (\alpha_{3} ax)} \right] \\ \end{aligned} $$
(A6)
$$ \begin{aligned} b_{n}^{4} & = \left\{ {(p - 1)Y_{p} (\alpha_{4} ax) - (\alpha_{4} ax)Y_{p + 1} (\alpha_{4} ax)} \right\}\cos m\,\theta \,\,\,\sin 2(\theta - \gamma_{i} ) \\ & \quad - \left\{ {2(\alpha_{4} ax)Y_{p + 1} (\alpha_{4} ax) - \,[(\alpha_{4} ax)^{2} - 2p(p - 1)]Y_{p} (\alpha_{4} ax)} \right\}2p\,\,\sin m\,\theta \,\,\,\cos 2(\theta - \gamma_{i} ) \\ \end{aligned} $$
(A7)
$$ \begin{aligned} b_{n}^{j} & = 2\left\{ {[p(p - 1) - (\alpha_{i} ax)^{2} ]Y_{p} (\alpha_{i} ax)\cos m\,\theta + (\alpha_{i} ax)Y_{p + 1} (\alpha_{i} ax)\cos m\,\theta } \right\}\sin 2\theta - 2\gamma_{i} \\ & \quad + 2p\left\{ {\,(\alpha_{i} ax)Y_{p + 1} (\alpha_{i} ax)\sin m\,\theta - (p - 1)Y_{p} (\alpha_{i} ax)\sin m\,\theta } \right\}\cos 2(\theta - \gamma_{i} ),\,j = 5,\,6 \\ \end{aligned} $$
(A8)
$$ c_{n}^{j} = d_{i} \left\{ {p\cos \left( {\overline{n - 1} \theta + \gamma_{i} } \right)J_{p} \left( {\alpha_{i} ax} \right) - \left( {\alpha_{i} ax} \right)J_{p + 1} \left( {\alpha_{i} ax} \right)\cos (\theta - \gamma_{i} )\cos m\,\theta } \right\},\quad j = 1,2 $$
(A9)
$$ c_{n}^{3} = 0.0,\quad c_{n}^{4} = 0.0 $$
(A10)
$$ c_{n}^{j} = d_{i} \left\{ {p\cos \left( {\overline{n - 1} \theta + \gamma_{i} } \right)Y_{p} \left( {\alpha_{i} ax} \right) - \left( {\alpha_{i} ax} \right)Y_{p + 1} \left( {\alpha_{i} ax} \right)\cos (\theta - \gamma_{i} )\cos m\,\theta } \right\},\quad j = 5,6 $$
(A11)

The expressions \(\overline{a}_{n}^{j} ,b_{n}^{j} \& \overline{c}_{n}^{j}\) is got by interchanging \(\cos n\theta\) and \(\sin n\theta\) in the Eqs. (A1)–(A11).

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Selvamani, R., Sarkar, N. & Ebrahami, F. Modeling of Delayed Thermo Elastic Waves in a Polygonal Ring Reinforced with Graphene Platelets. Int. J. Appl. Comput. Math 8, 234 (2022). https://doi.org/10.1007/s40819-022-01435-w

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