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RETRACTED ARTICLE: Chaotic simulation of the multi-phase reinforced thermo-elastic disk using GDQM

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This article was retracted on 09 June 2023

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Abstract

In this research, a mathematical derivation is made to develop a nonlinear dynamic model for the nonlinear frequency and chaotic responses of the multi-scale hybrid nano-composite reinforced disk in the thermal environment and subject to a harmonic external load. Using Hamilton’s principle and the von Karman nonlinear theory, the nonlinear governing equation is derived. For developing an accurate solution approach, generalized differential quadrature method (GDQM) and perturbation approach (PA) are finally employed. Various geometrically parameters are taken into account to investigate the chaotic motion of the viscoelastic disk subject to harmonic excitation. The fundamental and golden results of this paper could be that in the lower value of the external harmonic force, different FG patterns do not have any effects on the motion response of the structure. But, for the higher value of external harmonic force and all FG patterns, the chaos motion could be seen and for the FG-X pattern, the chaosity is more significant than other patterns of the FG. As a practical designing tip, it is recommended to choose plates with lower thickness relative to the outer radius to achieve better vibration performance.

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Abbreviations

h, R 0, and R i :

Thickness, inner, outer radius of the disk, respectively

F and NCM:

Fiber and nanocomposite matrix, respectively

\(\rho ,\,\,E,\,\nu ,\,\alpha \,\,\,and\,\,\,G\) :

Density, Young’s module, Poisson’s ratio, thermal expansion and shear parameters, respectively

\(V_{\text{NCM}} ,{{ V}}_{F}\) :

Volume fractions of nanocomposite matrix and fiber, respectively

\(E^{\text{CNT}}\), \(t^{\text{CNT}}\), \(l^{\text{CNT}}\), \(d^{\text{CNT}}\), and \(V_{\text{CNT}}\) :

Young’s module, thickness, length, diameter, and volume fraction of carbon nanotubes, respectively.

\(V_{\text{CNT}}^{*} ,\,\,\,W_{\text{CNT}}\) :

Effective volume fraction and weight fraction of the CNTs, respectively

Nt, V CNT :

Layer number and volume fraction of CNTs

U, V, W :

Displacement fields of a disk

u, w and Øx :

Displacements of the mid-surface in R and Z directions and rotations of the transverse normal around θ direction, respectively

\(\varepsilon_{RR}\) and \(\varepsilon_{\theta \theta }\) :

Corresponding normal strains in \(R\) and \(\theta\) directions, respectively

\(\gamma_{RZ}\) :

Shear strain in the \(RZ\) plane

T, U, W :

Corresponding kinetic energy, strain energy of the system and the work done, respectively

K W, C, N T :

Winkler coefficient, damping parameter, and thermal resistance force, respectively.

q dynamic and F :

Dynamical force and force, respectively

I i :

Mass inertias

\(\sigma_{RR} \,\,and\,\,\sigma_{\theta \theta }\) :

Corresponding normal stress in R and \(\theta\) directions

\(\tau_{RZ}\) :

Shear stress in the RZ plane

\(Qij\), \({\bar{Q}}_{ij}\) and \(\theta\) :

Stiffness elements, stiffness elements related to orientation angle and the orientation angle, respectively

\(\omega_{L} ,\,\,\overline{\omega }_{L}\) :

Linear non-dimensional linear natural frequencies, respectively

\(\omega_{NL} ,\,\,\overline{\omega }_{NL}\) :

Nonlinear non-dimensional nonlinear natural frequencies, respectively

C, P 1, P 2 and \(\gamma\) :

Damping coefficient, linear part of the w, nonlinear part (order one) of the w and nonlinear part (order two) of the w, respectively

\(a\) :

Deflection which is dimensionless

\(\Omega ,\,\,\,\sigma \,\,and\,\,\varepsilon\) :

Excitation frequency, detuning parameter and perturbation parameter, respectively

T 0 and T 1 :

Excitation term

\(\overline{q}\) :

The weakness form of the external force

\(\overline{A}\,\,\text{and}\,\,A\) :

Unknown complex conjugate and complex functions, respectively.

\(\omega_{0}\) :

Primary resonance

\(\alpha \,\,\,\text{and}\,\,\,\beta\) :

Amplitude and phase, respectively

M :

Magnification factor

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Funding

The study was funded by National Natural Science Foundation of China (51675148), The Outstanding Young Teachers Fund of Hangzhou Dianzi University (GK160203201002/003), and National Natural Science Foundation of China (51805475).

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Correspondence to Mostafa Habibi or Abdelouahed Tounsi.

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This article has been retracted. Please see the retraction notice for more detail: https://doi.org/10.1007/s00366-023-01859-y

Appendix

Appendix

In Eqs. (32ac), Lij and Mij are expressed as follows:

$$\begin{gathered} \delta u_{o} : \hfill \\ L_{11} = A_{11} \frac{{\partial^{2} u}}{{\partial R^{2} }} - \frac{{A_{22} }}{{R^{2} }}u, \hfill \\ L_{12} = - D_{11} c_{1} \frac{{\partial^{3} w}}{{\partial R^{3} }} + \frac{{D_{22} c_{1} }}{{R^{2} }}\frac{\partial w}{{\partial R}} \hfill \\ L_{13} = B_{11} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - D_{11} c_{1} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - \frac{{B_{22} }}{{R^{2} }}\phi + \frac{{D_{22} c_{1} }}{{R^{2} }}\phi \hfill \\ M_{11} = I_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }},\,\,M_{12} = - I_{3} c_{1} \frac{{\partial^{3} w}}{{\partial R\partial t^{2} }},\,\,\,M_{13} = \left( {I_{1} - I_{3} c_{1} } \right)\frac{{\partial^{2} \phi }}{{\partial t^{2} }} \hfill \\ \end{gathered}$$
(i)
$$\begin{gathered} \delta w_{0} : \hfill \\ L_{21} = c_{1} D_{11} \frac{{\partial^{3} u}}{{\partial R^{3} }} - \frac{{c_{1} D_{22} }}{{R^{2} }}\frac{\partial u}{{\partial R}}, \hfill \\ L_{22} = - G_{11} c_{1}^{2} \frac{{\partial^{4} w}}{{\partial R^{4} }} + \frac{{G_{22} c_{1}^{2} }}{{R^{2} }}\frac{{\partial^{2} w}}{{\partial R^{2} }} + \left( {A_{55} - 3C_{55} c_{1} } \right)\frac{{\partial^{2} w}}{{\partial R^{2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\frac{{\partial^{2} w}}{{\partial R^{2} }} - N^{T} \frac{{\partial^{2} w}}{{\partial R^{2} }} \hfill \\ L_{23} = {\text{C}}\frac{\partial w}{{\partial t}},\,\,\,\,\,\,\,\,L_{24} = \frac{3}{2}{\text{A}}_{11} \frac{{\partial^{2} w}}{{\partial R^{2} }}\left( {\frac{\partial w}{{\partial R}}} \right)^{2} \hfill \\ L_{25} = c_{1} E_{11} \frac{{\partial^{3} \phi }}{{\partial R^{3} }} - G_{11} c_{1}^{2} \frac{{\partial^{3} \phi }}{{\partial R^{3} }} - \frac{{c_{1} }}{R}\frac{{E_{22} \partial \phi }}{R\partial R} - \frac{{c_{1} }}{R}\frac{{G_{22} c_{1} }}{R}\frac{\partial \phi }{{\partial R}} \hfill \\ \,\,\,\,\,\,\,\,\,\, + \left( {A_{55} - 3C_{55} c_{1} } \right)\frac{\partial \phi }{{\partial R}} - 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\frac{\partial \phi }{{\partial R}} \hfill \\ M_{21} = c_{1} I_{3} \frac{{\partial^{3} u}}{{\partial R\partial t^{2} }},\,\,M_{22} = I_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }} - c_{1}^{2} I_{6} \frac{{\partial^{4} w}}{{\partial R^{2} \partial t^{2} }}, \hfill \\ M_{23} = \left( {c_{1} I_{4} - c_{1}^{2} I_{6} } \right)\frac{{\partial^{3} \phi }}{{\partial R\partial t^{2} }}, \hfill \\ \, \hfill \\ \end{gathered}$$
(ii)
$$\begin{gathered} \delta \phi : \hfill \\ L_{31} = B_{11} \frac{{\partial^{2} u}}{{\partial R^{2} }} - c_{1} D_{11} \frac{{\partial^{2} u}}{{\partial R^{2} }} - \frac{{B_{22} }}{{R^{2} }}u + \frac{{D_{22} }}{{R^{2} }}u, \hfill \\ L_{32} = - E_{11} c_{1} \frac{{\partial^{3} w}}{{\partial R^{3} }} + G_{11} c_{1}^{2} \frac{{\partial^{3} w}}{{\partial R^{3} }} + \frac{{E_{22} c_{1} }}{{R^{2} }}\frac{\partial w}{{\partial R}} + \frac{{G_{22} c_{1}^{2} }}{{R^{2} }}\frac{\partial w}{{\partial R}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, - \left( {A_{55} - 3C_{55} c_{1} } \right)\frac{\partial w}{{\partial R}} + 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\frac{\partial w}{{\partial R}} \hfill \\ L_{33} = C_{11} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - E_{11} c_{1} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - c_{1} E_{11} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} + G_{11} c_{1}^{2} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\, - \frac{1}{{R^{2} }}\left\{ {C_{22} - E_{22} c_{1} } \right\}\phi + \frac{{c_{1} }}{{R^{2} }}\left\{ {E_{22} - G_{22} c_{1} } \right\}\phi \hfill \\ \,\,\,\,\,\,\, \, - \left( {A_{55} - 3C_{55} c_{1} } \right)\phi + 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\phi \hfill \\ M_{31} = \left( {I_{1} - c_{1} I_{3} } \right)\frac{{\partial^{2} u}}{{\partial t^{2} }},\,\,\,M_{32} = \left( {I_{6} c_{1}^{2} - I_{4} c_{1} } \right)\frac{{\partial^{3} w}}{{\partial R\partial t^{2} }}, \hfill \\ M_{33} = \left( {I_{6} c_{1}^{2} - 2c_{1} I_{4} + I_{2} } \right)\frac{{\partial^{2} \phi }}{{\partial t^{2} }}. \hfill \\ \end{gathered}$$
(iii)

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Al-Furjan, M.S.H., Habibi, M., rahimi, A. et al. RETRACTED ARTICLE: Chaotic simulation of the multi-phase reinforced thermo-elastic disk using GDQM. Engineering with Computers 38 (Suppl 1), 219–242 (2022). https://doi.org/10.1007/s00366-020-01144-2

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