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Wave propagation analysis of functionally graded nanocomposite plate reinforced with graphene platelets in presence of thermal excitation

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Abstract

This work is an attempt to examine the wave propagation of thermally loaded graphene platelets (GPL)s reinforced nanocomposite plates. Particularly, an analytical approach is made to determine the extent of wave dispersion in functionally graded nanocomposite plates reinforced with graphene platelets (GPL)s when thermal loading is added to the structure. Along with thermal loading, the plate structure is rested on an elastic foundation to derive a more realistic outcome. Variety of thermal loads with different temperature rises have been studies in the present research; these loads include: uniform temperature rise (UTR), sinusoidal temperature rise (STR), and linear temperature rise (LTR). Moreover, the nanocomposite (NC) plate is rested on a Winkler–Pasternak elastic foundation. GPL particles are dispersed in the polymer base along the thickness of the plate in four differing distribution patterns. Furthermore, Hamilton’s principle along with refined higher-order plate theory is utilized to derive the governing equations of the problem. Afterwards, these differential equations are solved in an analytical format to attain the wave frequency, as well as, the phase velocity of GPL-reinforced NC plate. Before, assessing the impact of numerous parameters on the wave propagation characteristics of GPL-reinforced NC plate, our analytical model had been validated with previous studies. Results of this study indicates that the wave dispersion of NC plates reinforced with GPLs is heavily depended on parameters such as GPL distribution pattern, thermal loading type, GPL weight fraction and so on.

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Authors and Affiliations

  1. Department of Mechanical engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran

    Farzad Ebrahimi & Hosein Ezzati

  2. Faculty of Mechanical Engineering, Zanjan University, Zanjan, Iran

    Mohammad Najafi

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  1. Farzad Ebrahimi
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Appendix

Appendix

$$\begin{array}{c}{k}_{11}=-{A}_{11}{\beta }_{1}^{2}-{A}_{66}{\beta }_{2}^{2}\\ {k}_{12}=\left({A}_{12}+{A}_{66}\right){\beta }_{1}{\beta }_{2}\\ {k}_{13}=i{B}_{11}{\beta }_{1}^{3}+i\left({B}_{12}+2{B}_{66}\right){\beta }_{2}^{2}{\beta }_{1}\\ {k}_{14}=i{B}_{11}^{s}{\beta }_{1}^{3}+i\left({B}_{12}^{s}+2{B}_{66}^{s}\right){\beta }_{2}^{2}{\beta }_{1}\end{array}$$
$$\begin{array}{c}{k}_{22}=-{A}_{22}{\beta }_{1}^{2}-{A}_{66}{\beta }_{2}^{2}\\ {k}_{23}=i{B}_{22}{\beta }_{2}^{3}+i\left({B}_{12}+2{B}_{66}\right){\beta }_{1}^{2}{\beta }_{2}\\ {k}_{24}={B}_{22}^{s}{\beta }_{2}^{3}+i\left({B}_{12}^{s}+2{B}_{66}^{s}\right){\beta }_{1}^{2}{\beta }_{2}\end{array}$$
$$\begin{array}{c}{k}_{33}=-{D}_{11}{\beta }_{1}^{4}-2\times \left({D}_{12}+2{D}_{66}\right){\beta }_{1}^{2}{\beta }_{2}^{2}-{D}_{22}{\beta }_{2}^{4}-{k}_{w}-\left({k}_{p}-{N}^{T}\right)\left({\beta }_{1}^{2}+{\beta }_{2}^{2}\right)\\ {k}_{34}=-{D}_{11}^{s}{\beta }_{1}^{4}-2\times \left({D}_{12}^{s}+2{D}_{66}^{s}\right){\beta }_{1}^{2}{\beta }_{2}^{2}-{D}_{22}^{s}{\beta }_{2}^{4}-{k}_{w}-\left({k}_{p}-{N}^{T}\right)\left({\beta }_{1}^{2}+{\beta }_{2}^{2}\right)\\ {k}_{44}=-{H}_{11}^{s}{\beta }_{1}^{4}-2\times \left({H}_{12}^{s}+2{H}_{66}^{s}\right){\beta }_{1}^{2}{\beta }_{2}^{2}-{H}_{22}^{s}{\beta }_{2}^{4}-{k}_{w}-\left({k}_{p}-{N}^{T}\right)\left({\beta }_{1}^{2}+{\beta }_{2}^{2}\right)-{A}_{55}^{s}{\beta }_{1}^{2}-{A}_{44}^{s}{\beta }_{2}^{2}\end{array}$$
$$\begin{array}{c}{m}_{11}={m}_{22}={I}_{0}\\ {m}_{12}=0\\ {m}_{13}=-{l}_{1}{\beta }_{1}i\end{array}$$
$$\begin{array}{c}{m}_{14}=-{J}_{1}{\beta }_{1}i\\ {m}_{23}=-{I}_{1}{\beta }_{2}i\\ {m}_{24}=-{J}_{1}{\beta }_{2}i\\ {m}_{33}={I}_{0}-{I}_{2}\left({\beta }_{1}^{2}+{\beta }_{2}^{2}\right)\\ {m}_{34}={I}_{0}-{J}_{2}\left({\beta }_{1}^{2}+{\beta }_{2}^{2}\right)\\ {m}_{44}={I}_{0}-{K}_{2}\left({\beta }_{1}^{2}+{\beta }_{2}^{2}\right)\end{array}$$

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Ebrahimi, F., Ezzati, H. & Najafi, M. Wave propagation analysis of functionally graded nanocomposite plate reinforced with graphene platelets in presence of thermal excitation. Acta Mech 235, 215–234 (2024). https://doi.org/10.1007/s00707-023-03728-7

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