Applied Mathematics and Mechanics

, Volume 36, Issue 8, pp 1033–1044

# Nonlinear vibration analysis of laminated composite Mindlin micro/nano-plates resting on orthotropic Pasternak medium using DQM

• A. G. Arani
• G. S. Jafari
Article

## Abstract

The nonlocal nonlinear vibration analysis of embedded laminated microplates resting on an elastic matrix as an orthotropic Pasternak medium is investigated. The small size effects of micro/nano-plate are considered based on the Eringen nonlocal theory. Based on the orthotropic Mindlin plate theory along with the von Kármán geometric nonlinearity and Hamilton’s principle, the governing equations are derived. The differential quadrature method (DQM) is applied for obtaining the nonlinear frequency of system. The effects of different parameters such as nonlocal parameters, elastic media, aspect ratios, and boundary conditions are considered on the nonlinear vibration of the micro-plate. Results show that considering elastic medium increases the nonlinear frequency of system. Furthermore, the effect of boundary conditions becomes lower at higher nonlocal parameters.

## Keywords

nonlinear vibration laminated micro-plate orthotropic Pasternak medium differential quadrature method

O322

74K20

## References

1. [1]
Akhavan, H., Hosseini, H. S., Rokni, D. T, H., Alibeigloo, A., and Vahabi, S. Exact solutions for rectangular Mindlin plates under in-plane loads resting on Pasternak elastic foundation, part I: buckling analysis. Computational Material Science, 44, 968–978 (2009)
2. [2]
Akhavan, H., Hosseini, H. S., Rokni, D. T. H., Alibeigloo, A., and Vahabi, S. Exact solutions for rectangular Mindlin plates under in-plane loads resting on Pasternak elastic foundation, part II: frequency analysis. Computational Material Science, 44, 951–961 (2009)
3. [3]
Morozov, N. F. and Tovstik, P. E. On modes of buckling for a plate on an elastic foundation. Mechanics of Solids, 45, 519–528 (2010)
4. [4]
Lu, H. X. and Li, J. Y. Vibration and stability of hybrid plate based on elasticity theory. Applied Mathematics andMechanics (English Edition), 30, 413–423 (2009)DOI 10.1007/s10483-009-0402-3
5. [5]
Reddy, J. N. A simple higher order theory for laminated composite plates. Journal of Applied Mechanics, 51, 745–752 (1984)
6. [6]
Ding, H. J., Chen, W. Q., and Xu, R. Q. On the bending, vibration and stability of laminated rectangular plates with transversely isotropic layers. Applied Mathematics and Mechanics (English Edition), 22, 17–24 (2001) DOI 10.1023/A:1015518832155
7. [7]
Ferreira, A. J. M., Roque, C. M. C., and Martins, P. A. L. S. Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method. Composite Part B, 34, 627–636 (2003)
8. [8]
Swaminathan, K. and Ragounadin, D. Analytical solutions using a higher-order refined theory for the static analysis of antisymmetric angle-ply composite and sandwich plates. Composite Structures, 64, 405–417 (2004)
9. [9]
Shen, H. S. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Composite Structures, 91, 9–19 (2009)
10. [10]
Heydari, M. M., Kolahchi, R., Heydari, M., and Abbasi, A. Exact solution for transverse bending analysis of embedded laminated Mindlin plate. Structural Engineering and Mechanics, 49, 661–672 (2014)
11. [11]
Eringen, A. C. On nonlocal elasticity. International Journal of Engineering Science, 10, 1–16 (1972)
12. [12]
Farahmand, H., Ahmadi, A. R., and Arabnejad, S. Thermal buckling analysis of rectangular microplates using higher continuity p-version finite element method. Thin Wall Structures, 49, 1584–1591 (2011)
13. [13]
Ahmadi, A. R., Farahmand, H., and Arabnejad, S. Buckling analysis of rectangular flexural microplates using higher continuity p-version finite-element method. International Journal of Multiscale Computational Engineering, 10, 249–259 (2012)
14. [14]
Ghorbanpour, A., A., Vossough, H., Kolahchi, R., and Mosallaie, B., A. A. Electro-thermo nonlocal nonlinear vibration in an embedded polymeric piezoelectric micro plate reinforced by DWBNNTs using DQM. Journal of Mechanical Science and Technology, 26, 3047–3057 (2012)
15. [15]
Ramezani, S. Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory. Nonlinear Dynamics, 73, 1399–1421 (2013)
16. [16]
Ghorbanpour, A. A., Jalilvand, A., and Kolahchi, R. Nonlinear strain gradient theory based vibration and instability of boron nitride micro-tubes conveying ferrofluid. International Journal of Applied Mechanics, 6, 1450060 (2014)Google Scholar
17. [17]
Ghorbanpour, A. A., Vossough, H., and Kolahchi, R. Nonlinear vibration and instability of a visco-Pasternak coupled double-DWBNNTs-reinforced microplate system conveying microflow. Mechanical Engineering Science (2015) DOI 10.1177/0954406215569587Google Scholar
18. [18]
Kutlu, A. and Omurtag, M. H. Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method. International Journal of Mechanical Science, 65, 64–74 (2012)
19. [19]
Phung-Van, P., De Lorenzis, L., Thai, C. H., Abdel-Wahab, M., and Xuan, H. N. Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements. Computational Materials Science, 96, 495–505 (2015)
20. [20]
Chakraverty, S. and Pradhan, K. K. Free vibration of functionally graded thin rectangular plates resting on winkler elastic foundation with general boundary conditions using Rayleigh-Ritz method. International Journal of Applied Mechanics, 6, 1450043 (2014)