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Free vibration analysis of nanoplates with auxetic honeycomb core using a new third-order finite element method and nonlocal elasticity theory

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Abstract

This paper proposes a finite element method (FEM) for the free vibration analysis of sandwich nanoplates with an auxetic honeycomb core. The proposed method uses a third-order shear deformation theory accounting for both shear deformation and stretching effects without any need for shear correction factors. The size-dependent effect is solved using the nonlocal elasticity theory. The auxetic sandwich nanoplate with negative Poisson’s ratio is applied to achieve ultra-light features and high strength. The obtained numerical results by the proposed method are compared with other published works to demonstrate the accuracy and reliability. Moreover, the influence of the nonlocal factor, geometrics parameters, and material properties (especially the auxetic honeycomb parameters) on the free vibration behavior of sandwich nanoplates is also examined in the numerical examples.

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.02-2019.330.

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Authors and Affiliations

  1. Advanced Structural Engineering Laboratory, Faculty of Civil Engineering, Ho Chi Minh City Open University, Ho Chi Minh City, Vietnam

    Quoc-Hoa Pham & Phu-Cuong Nguyen

  2. Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi, Vietnam

    Trung Thanh Tran

  3. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Trung Nguyen-Thoi

  4. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Trung Nguyen-Thoi

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  1. Quoc-Hoa Pham
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  4. Trung Nguyen-Thoi
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Correspondence to Trung Nguyen-Thoi.

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Appendices

Appendix A

Lagrange’s polynomial function:

$$N_{1} = \frac{1}{4}\left( {1 - \chi } \right)\left( {1 - \zeta } \right);$$
$$N_{2} = \frac{1}{4}\left( {1 + \chi } \right)\left( {1 - \zeta } \right);$$
$$N_{3} = \frac{1}{4}\left( {1 + \chi } \right)\left( {1 + \zeta } \right);$$

\(N_{4} = \frac{1}{4}\left( {1 - \chi } \right)\left( {1 + \zeta } \right)\).

Appendix B

Hermite’s polynomial function:

$$H_{1} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 - \zeta } \right)\left( {2 - \chi - \zeta - \chi^{2} - \zeta^{2} } \right)$$
$$H_{2} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 - \zeta } \right)\left( {1 - \chi^{2} } \right)$$
$$H_{3} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 - \zeta } \right)\left( {1 - \zeta^{2} } \right)$$
$$H_{4} = \frac{1}{8}\left( {1 + \chi } \right)\left( {1 - \zeta } \right)\left( {2 + \chi - \zeta - \chi^{2} - \zeta^{2} } \right)$$
$$H_{5} = \frac{1}{8}\left( {1 + \chi } \right)\left( {1 - \zeta } \right)\left( {1 - \chi^{2} } \right)$$
$$H_{6} = \frac{1}{8}\left( {1 + \chi } \right)\left( {1 - \zeta } \right)\left( {1 - \zeta^{2} } \right)$$
$$H_{7} = \frac{1}{8}\left( {1 + \chi } \right)\left( {1 + \zeta } \right)\left( {2 + \chi + \zeta - \chi^{2} - \zeta^{2} } \right)$$
$$H_{8} = \frac{1}{8}\left( {1 + \chi } \right)\left( {1 + \zeta } \right)\left( {1 - \chi^{2} } \right)$$
$$H_{9} = \frac{1}{8}\left( {1 + \chi } \right)\left( {1 + \zeta } \right)\left( {1 - \zeta^{2} } \right)$$
$$H_{10} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 + \zeta } \right)\left( {2 - \chi + \zeta - \chi^{2} - \zeta^{2} } \right)$$
$$H_{11} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 + \zeta } \right)\left( {1 - \chi^{2} } \right)$$
$$H_{12} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 + \zeta } \right)\left( {1 - \zeta^{2} } \right)$$

with \(\chi ,\zeta\) are natural coordinates.

Appendix C

Geometrical parameters of sandwich nanoplates:

figure a

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Pham, QH., Nguyen, PC., Tran, T.T. et al. Free vibration analysis of nanoplates with auxetic honeycomb core using a new third-order finite element method and nonlocal elasticity theory. Engineering with Computers 39, 233–251 (2023). https://doi.org/10.1007/s00366-021-01531-3

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