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Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

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Abstract

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    Shojaa Ramezani

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  1. Shojaa Ramezani
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Correspondence to Shojaa Ramezani.

Appendix

Appendix

The ϒ I (I=1,2,…,9) parameters defined in Eqs. (84)–(86) are as follows:

(123)
(124)
(125)
(126)
(127)
(128)
(129)
(130)
(131)

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Ramezani, S. Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory. Nonlinear Dyn 73, 1399–1421 (2013). https://doi.org/10.1007/s11071-013-0872-1

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  • DOI: https://doi.org/10.1007/s11071-013-0872-1

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