A wave-based computational method for free vibration and buckling analysis of rectangular Reddy nanoplates

Document Type : Research Paper

Authors

School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran.

Abstract

In this paper, the wave propagation method is combined with nonlocal elasticity theory to analyze the buckling and free vibration of rectangular Reddy nanoplate. Wave propagation is one of the powerful methods for analyzing the vibration and buckling of structures. It is assumed that the plate has two opposite edges simply supported while the other two edges may be simply supported or clamped. It is the first time that the wave propagation method is used for thick nanoplates. In this study, firstly the matrices of propagation and reflection are derived. Then, these matrices are combined to provide an exact method for obtaining the natural frequencies and critical buckling loads which can be useful for future studies. It is observed that obtained results of the wave propagation method are in good agreement with the obtained values by literature. At the end the obtained results are presented to evaluate the influence of different parameters such as nonlocal parameter, aspect ratio and thickness to length ratio of nanoplate.

Keywords

Main Subjects

[1]       R. Mindlin, H. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and analysis, Vol. 11, No. 1, pp. 415-448, 1962.
[2]       R. D. Mindlin, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, Vol. 16, No. 1, pp. 51-78, 1964.
[3]       A. C. Eringen, Theory of micromorphic materials with memory, International Journal of Engineering Science, Vol. 10, No. 7, pp. 623-641, 1972.
[4]       A. C. Eringen, 2002, Nonlocal continuum field theories, Springer Science & Business Media,
[5]       H. Makvandi, S. Moradi, D. Poorveis, K. H. Shirazi, A new approach for nonlinear vibration analysis of thin and moderately thick rectangular plates under inplane compressive load, Journal of Computational Applied Mechanics.
[6]       A. Zargaripoor, Bahrami, Arian, Nikkhah Bahrami, Mansour, Free vibration and buckling analysis of third-order shear deformation plate theory using exact wave propagation approach, Journal of Computational Applied Mechanics, January 2018, 2018.
[7]       R. Javidi, M. Moghimi Zand, K. Dastani, Dynamics of Nonlinear rectangular plates subjected to an orbiting mass based on shear deformation plate theory, Journal of Computational Applied Mechanics, 2017.
[8]       S. Pradhan, J. Phadikar, Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration, Vol. 325, No. 1-2, pp. 206-223, 2009.
[9]       T. Murmu, S. Pradhan, Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model, Physica E: Low-dimensional Systems and Nanostructures, Vol. 41, No. 8, pp. 1628-1633, 2009.
[10]     R. Aghababaei, J. Reddy, Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates, Journal of Sound and Vibration, Vol. 326, No. 1-2, pp. 277-289, 2009.
[11]     T. Aksencer, M. Aydogdu, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E: Low-dimensional Systems and Nanostructures, Vol. 43, No. 4, pp. 954-959, 2011.
[12]     P. Malekzadeh, M. Shojaee, Free vibration of nanoplates based on a nonlocal two-variable refined plate theory, Composite Structures, Vol. 95, pp. 443-452, 2013.
[13]     S. Chakraverty, L. Behera, Free vibration of rectangular nanoplates using Rayleigh–Ritz method, Physica E: Low-dimensional Systems and Nanostructures, Vol. 56, pp. 357-363, 2014.
[14]     P. Malekzadeh, M. Shojaee, A two-variable first-order shear deformation theory coupled with surface and nonlocal effects for free vibration of nanoplates, Journal of Vibration and Control, Vol. 21, No. 14, pp. 2755-2772, 2015.
[15]     S. Chakraverty, L. Behera, Small scale effect on the vibration of non-uniform nanoplates, Structural Engineering and Mechanics, Vol. 55, No. 3, pp. 495-510, 2015.
[16]     M. Panyatong, B. Chinnaboon, S. Chucheepsakul, Nonlocal second-order shear deformation plate theory for free vibration of nanoplates, Suranaree Journal of Science & Technology, Vol. 22, No. 4, 2015.
[17]     L. Behera, S. Chakraverty, Effect of scaling effect parameters on the vibration characteristics of nanoplates, Journal of Vibration and Control, Vol. 22, No. 10, pp. 2389-2399, 2016.
[18]     S. Faroughi, S. M. H. Goushegir, Free in-plane vibration of heterogeneous nanoplates using Ritz method, Journal of Theoretical and Applied Vibration and Acoustics, Vol. 2, No. 1, pp. 1-20, 2016.
[19]     M. H. Shokrani, A. R. Shahidi, Size-dependent free vibration analysis of rectangular nanoplates with the consideration of surface effects using finite difference method, Journal of applied and computational mechanics, Vol. 1, No. 3, pp. 122-133, 2015.
[20]     S. Sarrami-Foroushani, M. Azhari, Nonlocal buckling and vibration analysis of thick rectangular nanoplates using finite strip method based on refined plate theory, Acta Mechanica, Vol. 227, No. 3, pp. 721-742, 2016.
[21]     S. Hosseini-Hashemi, M. Kermajani, R. Nazemnezhad, An analytical study on the buckling and free vibration of rectangular nanoplates using nonlocal third-order shear deformation plate theory, European Journal of Mechanics-A/Solids, Vol. 51, pp. 29-43, 2015.
[22]     D. Rong, J. Fan, C. Lim, X. Xu, Z. Zhou, A New Analytical Approach for Free Vibration, Buckling and Forced Vibration of Rectangular Nanoplates Based on Nonlocal Elasticity Theory, International Journal of Structural Stability and Dynamics, pp. 1850055, 2017.
[23]     A. Daneshmehr, A. Rajabpoor, A. Hadi, Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories, International Journal of Engineering Science, Vol. 95, pp. 23-35, 2015.
[24]     M. Hosseini, H. H. Gorgani, M. Shishesaz, A. Hadi, Size-Dependent Stress Analysis of Single-Wall Carbon Nanotube Based on Strain Gradient Theory, International Journal of Applied Mechanics, Vol. 9, No. 06, pp. 1750087, 2017.
[25]     M. Hosseini, M. Shishesaz, K. N. Tahan, A. Hadi, Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials, International Journal of Engineering Science, Vol. 109, pp. 29-53, 2016.
[26]     M. Z. Nejad, A. Rastgoo, A. Hadi, Effect of exponentially-varying properties on displacements and stresses in pressurized functionally graded thick spherical shells with using iterative technique, Journal of Solid Mechanics, Vol. 6, No. 4, pp. 366-377, 2014.
[27]     M. Z. Nejad, A. Hadi, Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Science, Vol. 106, pp. 1-9, 2016.
[28]     M. Z. Nejad, A. Hadi, Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Science, Vol. 105, pp. 1-11, 2016.
[29]     M. Z. Nejad, A. Hadi, A. Farajpour, Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials, Structural Engineering and Mechanics, Vol. 63, No. 2, pp. 161-169, 2017.
[30]     M. Z. Nejad, A. Hadi, A. Rastgoo, Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Science, Vol. 103, pp. 1-10, 2016.
[31]     M. Z. Nejad, A. Rastgoo, A. Hadi, Exact elasto-plastic analysis of rotating disks made of functionally graded materials, International Journal of Engineering Science, Vol. 85, pp. 47-57, 2014.
[32]     M. Shishesaz, M. Hosseini, K. N. Tahan, A. Hadi, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica, Vol. 228, No. 12, pp. 4141-4168, 2017.
[33]     A. Hadi, M. Z. Nejad, A. Rastgoo, M. Hosseini, Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory, Steel and Composite Structures, Vol. 26, No. 6, pp. 663-672, 2018.
[34]     M. Zamani Nejad, M. Jabbari, A. Hadi, A review of functionally graded thick cylindrical and conical shells, Journal of Computational Applied Mechanics, Vol. 48, No. 2, pp. 357-370, 2017.
[35]     A. Hadi, M. Z. Nejad, M. Hosseini, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, Vol. 128, pp. 12-23, 2018.
[36]     X. Zhang, G. Liu, K. Lam, Vibration analysis of thin cylindrical shells using wave propagation approach, Journal of sound and vibration, Vol. 239, No. 3, pp. 397-403, 2001.
[37]     C. Mei, B. Mace, Wave reflection and transmission in Timoshenko beams and wave analysis of Timoshenko beam structures, Journal of vibration and acoustics, Vol. 127, No. 4, pp. 382-394, 2005.
[38]     T. Natsuki, M. Endo, H. Tsuda, Vibration analysis of embedded carbon nanotubes using wave propagation approach, Journal of Applied Physics, Vol. 99, No. 3, pp. 034311, 2006.
[39]     L. Xuebin, Study on free vibration analysis of circular cylindrical shells using wave propagation, Journal of sound and vibration, Vol. 311, No. 3-5, pp. 667-682, 2008.
[40]     A. Bahrami, M. R. Ilkhani, M. N. Bahrami, Wave propagation technique for free vibration analysis of annular circular and sectorial membranes, Journal of Vibration and Control, Vol. 21, No. 9, pp. 1866-1872, 2015.
[41]     A. Bahrami, A. Teimourian, Nonlocal scale effects on buckling, vibration and wave reflection in nanobeams via wave propagation approach, Composite Structures, Vol. 134, pp. 1061-1075, 2015.
[42]     M. Ilkhani, A. Bahrami, S. Hosseini-Hashemi, Free vibrations of thin rectangular nano-plates using wave propagation approach, Applied Mathematical Modelling, Vol. 40, No. 2, pp. 1287-1299, 2016.
[43]     A. Bahrami, A. Teimourian, Study on the effect of small scale on the wave reflection in carbon nanotubes using nonlocal Timoshenko beam theory and wave propagation approach, Composites Part B: Engineering, Vol. 91, pp. 492-504, 2016.
[44]     A. Bahrami, A. Teimourian, Study on vibration, wave reflection and transmission in composite rectangular membranes using wave propagation approach, Meccanica, Vol. 52, No. 1-2, pp. 231-249, 2017.
[45]     A. Bahrami, A. Teimourian, Small scale effect on vibration and wave power reflection in circular annular nanoplates, Composites Part B: Engineering, Vol. 109, pp. 214-226, 2017.
[46]     A. Bahrami, Free vibration, wave power transmission and reflection in multi-cracked nanorods, Composites Part B: Engineering, Vol. 127, pp. 53-62, 2017.
[47]     J. N. Reddy, A simple higher-order theory for laminated composite plates, Journal of applied mechanics, Vol. 51, No. 4, pp. 745-752, 1984.
Volume 51, Issue 2
December 2020
Pages 253-274
  • Receive Date: 07 April 2018
  • Revise Date: 22 May 2018
  • Accept Date: 23 May 2018
  • First Publish Date: 01 December 2020