Mechanic of Advanced and Smart Materials
Quarterly

Solution of out-of-Plane vibration of moderately thick rectangular nano-plate using nonlocal sinusoidal shear deformation theory

Document Type : Original Article

Authors

Peyman Yousefi 1
Mohammad Khodadadi 2

1 Assistant Professor, Department of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, Iran.

2 Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak,

https://doi.org/10.52547/masm.1.2.231
Abstract
In this paper, exact close form solution for out of plane free flexural vibration of moderately thick rectangular nano-plates are presented based on nonlocal sinusoidal shear deformation theory, with assumptions of the Levy's type boundary conditions, for the first time. The aim of this study is to evaluate the effect of small-scale parameters on the frequency parameters of the moderately thick rectangular nano-plates. To describe the effects of small-scale parameters on vibrations of rectangular nanoplates, the Eringen theory is used. the Levy's type boundary conditions is a combination of six different boundary conditions; specifically, two opposite edges are simply supported and any of the other two edges can be simply supported, clamped or free. Governing equations of motion and boundary conditions of the plate are derived by using the Hamilton’s principle. The present analytical solution can be obtained with any required accuracy and can be used as benchmark. Numerical results are presented to illustrate the effectiveness of the proposed method compared to other methods reported in the literature. Finally, the effect of boundary conditions, aspect ratios, small scale parameter and thickness ratios on nondimensional natural frequency parameters and frequency ratios are examined and discussed in detail.

Keywords

Nonlocal theory
Sinusoidal shear deformation
Out-of-plane vibration
[1] Leissa A.W. Vibration of plates, NASA, 1969.
[2] Leissa A.W. Recent Research in Plate Vibrations: classical theory. The Shock and Vibration Digest. 1977; 10(9):13-24.
[3] Leissa A.W. Recent research in plate vibrations: complicating effects. The Shock and Vibration Digest. 1977; 11(9):1973-1976.
[4] Leissa, A.W. The free vibration of rectangular plates. Journal of Sound and Vibration. 1973; 3(31):257-293.
[5] Mindlin R.D. Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. International Journal of Pressure Vessels and Piping. 1951; 15(67):220-231.
[6] Mindlin R.D, Schaknow A, Deresiewicz H. Flexural vibration of rectangular plates, Journal of Applied Mechanics. 1956; 1(23):430-433.
[7] Hosseini-Hashemi Sh, Rokni Damavandi Taher H, Akhavan H, Omidi M. Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory, Applied Mathematical Modelling.2010; 5(35):1276-1291.
[8] Khorshidi K. Effect of hydrostatic pressure and depth of fluid on the vibrating rectangular plates partially in contact with a fluid, Applied Mechanics and Materials. 2011; 110-116:927-935.
[9] Khorshidi K, Farhadi S. Free vibration analysis of a laminated composite rectangular plate in contact with a bounded fluid, Composite Structures. 2013; 45(104):176-186.
[10] Malekzadeh P, Fiouz A.R, Sobhrouyan M. Three-dimensional free vibration of functionally graded truncated conical shells subjected to thermal environment. International Journal of Pressure Vessels and Piping. 2012; 12(89):210-221.
[11] Dozio L. On the use of the Trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates. Thin-Walled Structures. 2011; 1(49):129-144.
[12] Rangoa R.F, Nallim L.G, Oller S. Formulation of enriched macro elements using trigonometric shear deformation theory for free vibration analysis of symmetric laminated composite plate assemblies. Composite Structures. 2015; 13(119):38-49.
[13] Mantari J, Soares C.G. A trigonometric plate theory with unknowns and stretching effect for advanced composite plates. Composite Structures. 2014; 5(107):396-405.
[14] Tounsi A, Houari M.S.A, Benyoucef S, Bedia E.A.A. A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aerospace Science and Technology. 2013;1(24):209-220.
[15] Sayyad A.S, Ghugal Y.M. Bending and free vibration analysis of thick isotropic plates by using exponential shear deformation theory. Applied and Computational mechanics. 2012; 2(100):290-299.
[16] Kharde S.B, Mahale A.K, Bhosale K.C, Thorat S.R. Flexural vibration of thick isotropic plates by using exponential shear deformation theory. International Journal of Emerging Technology and Advanced Engineering. 2013; 1(3):369-374.
[17] Yang Y, Zhang L, Lim C.W. Wave propagation in double-walled carbon nanotubes on a novel analytically nonlocal Timoshenko-beam model. Journal of Sound and Vibration. 2011; 330(8):1704-1717.
[18] Ke L.L, Wang Y.S. Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory. Physica E: Low-dimensional Systems and Nanostructures. 2011;43(5):1031-1039.
[19] Wang L. Wave propagation of fluid-conveying single-walled carbon nanotubes via gradient elasticity theory. Computational Materials Science. 2010; 4(49):761-766.
[20] Yang F, Chong A.C.M, Lam D.C.C, Tong P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures. 2002; 10(39):235-243.
[21] Ke L.L, Wang Y.S, Yang J, Kitipornchai S. Free vibration of size-dependent mindlin microplates based on the modified couple stress theory. Journal of Sound and Vibration. 2012; 1(333):94-106.
[22] Aksencer T, Aydogdu M. Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory. Physica E: Low-dimensional Systems and Nanostructures. 2011; 43(4):954-959.
[23] Aghababaei R, Reddy J. Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. Journal of Sound and Vibration. 2009; 326(1):277-289.
[24] Natarajan S, Chakraborty S, Thangavel M, Bordas S, Rabczuk T. Size-dependent free flexural vibration behavior of functionally graded nanoplates. Computational Materials Science. 2012; 30(65):74-80.
[25] Hosseini-Hashemi Sh, Zare M, Nazemnezhad R. An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity. Composite Structures. 2013; 2(100):290-299.
[26] Zhang P.Q, Lee H.P, Wang C.M, Reddy J.N. Non-local elastic plate theories. Proceedings of the Royal Society. 2007; 2088(463):32–40.
Mechanic of Advanced and Smart Materials
Volume 1, Issue 2
Winter 2022
Pages 231-246

  • Receive Date 06 January 2022
  • Revise Date 24 February 2022
  • Accept Date 15 February 2022

Home

Guide for Authors

Submit Manuscript

Contact Us