Primary and secondary resonances of functionally graded graphene platelet-reinforced nanocomposite beams

Abstract

The present study investigated the nonlinear harmonic vibration of functionally graded multilayer graphene nanoplatelet (GPL)-reinforced nanocomposite beams on the basis of the third-order shear deformation theory. The GPL volume fraction shows a layer-wise change, while in each individual layer GPLs are uniformly dispersed in the matrix. The effective Young’s moduli of the GPL-reinforced nanocomposite (GPLRC) beams were estimated through the Halpin–Tsai micromechanics model. The mass densities as well as the effective Poisson’s ratios of the GPLRC beams were predicted by the rule of mixture. The nonlinear partial differential equations of motion were discretized by means of the Galerkin procedure. A parametric study was carried out by using the multiple scales method to examine the effects of GPL distribution pattern, weight fraction, geometry, and size on the nonlinear response of the primary, secondary, and combination resonances. Results show that an addition of a very low weight fraction of GPL nanofillers significantly reduces the primary, superharmonic, subharmonic, and combinational resonant responses of the beams. The square-shaped GPLs with fewer graphene layers are the most favorable reinforcements.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11302087), the Australian Research Council under Discovery Project scheme (DP140102132, DP160101978), and the Natural Science Foundation of Jiangsu Province (Grant No. BK20160486).

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Appendices

Appendix A

The matrix \(\mathbf G (\omega )\) for the C–C, C–H, and H–H beams in Eq. (30) is

$$\begin{aligned} \mathbf G (\omega )= & {} \left[ {{\begin{array}{llll} 0&{}\quad 1&{}\quad 1&{}\quad 1 \\ {\sin \lambda _1 }&{}\quad {\cos \lambda _1 }&{}\quad {e^{\lambda _2 }}&{}\quad {e^{-\lambda _2 }} \\ {\lambda _1 }&{}\quad 0&{}\quad {\lambda _2 }&{}\quad {-\lambda _2 } \\ {\lambda _1 \cos \lambda _1 }&{}\quad {-\lambda _1 \sin \lambda _1 }&{}\quad {\lambda _2 e^{\lambda _2 }}&{}\quad {-\lambda _2 e^{-\lambda _2 }} \\ \end{array} }} \right] ,\\ \mathbf G (\omega )= & {} \left[ {{\begin{array}{llll} 0&{}\quad 1&{}\quad 1&{}\quad 1 \\ {\sin \lambda _1 }&{}\quad {\cos \lambda _1 }&{}\quad {e^{\lambda _2 }}&{}\quad {e^{-\lambda _2 }} \\ {\lambda _1 }&{}\quad 0&{}\quad {\lambda _2 }&{}\quad {-\lambda _2 } \\ {-\lambda _1^2 \sin \lambda _1 }&{}\quad {-\lambda _1^2 \cos \lambda _1 }&{}\quad {\lambda _2^2 e^{\lambda _2 }}&{}\quad {\lambda _2^2 e^{-\lambda _2 }} \\ \end{array} }} \right] , \end{aligned}$$

and

$$\begin{aligned} \mathbf G (\omega )= & {} \left[ {{\begin{array}{llll} 0&{}\quad 1&{}\quad 1&{}\quad 1 \\ {\sin \lambda _1 }&{}\quad {\cos \lambda _1 }&{}\quad {e^{\lambda _2 }}&{}\quad {e^{-\lambda _2 }} \\ 0&{}\quad {-\lambda _1^2 }&{}\quad {\lambda _2^2 }&{}\quad {\lambda _2^2 } \\ {-\lambda _1^2 \sin \lambda _1 }&{}\quad {-\lambda _1^2 \cos \lambda _1 }&{}\quad {\lambda _2^2 e^{\lambda _2 }}&{}\quad {\lambda _2^2 e^{-\lambda _2 }} \\ \end{array} }} \right] , \end{aligned}$$

respectively.

Appendix B

\(\varGamma _{i1} , \varGamma _{i2} ,\ldots , \varGamma _{i10} \) in Eqs. (41a)–(41c) are

$$\begin{aligned} \varGamma _{i1}= & {} \gamma _{i111} , \quad \varGamma _{i2} =\gamma _{i222} , \quad \varGamma _{i3} =\gamma _{i333} , \\ \varGamma _{i4}= & {} \gamma _{i112} +\gamma _{i121} +\gamma _{i211} , \\ \varGamma _{i5}= & {} \gamma _{i122} +\gamma _{i212} +\gamma _{i221} ,\\ \varGamma _{i6}= & {} \gamma _{i113} +\gamma _{i131} +\gamma _{i311} , \\ \varGamma _{i7}= & {} \gamma _{i133} +\gamma _{i313} +\gamma _{i331} , \\ \varGamma _{i8}= & {} \gamma _{i223} +\gamma _{i232} +\gamma _{i322} ,\\ \varGamma _{i9}= & {} \gamma _{i233} +\gamma _{i323} +\gamma _{i332} , \\ \varGamma _{i10}= & {} \gamma _{i123} +\gamma _{i132} +\gamma _{i213} +\gamma _{i231} +\gamma _{i312} +\gamma _{i321} \end{aligned}$$

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Li, X., Song, M., Yang, J. et al. Primary and secondary resonances of functionally graded graphene platelet-reinforced nanocomposite beams. Nonlinear Dyn 95, 1807–1826 (2019). https://doi.org/10.1007/s11071-018-4660-9

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Keywords

  • Graphene nanoplatelets
  • Functionally graded materials
  • Primary resonance
  • Secondary resonance
  • Combination resonance