Free in-plane vibration of heterogeneous nanoplates using Ritz method


1 Assistant Professor, Faculty of Mechanical Engineerng, Urmia University of Technology, Urmia, Iran

2 M.Sc. Student, Faculty of Mechanical Engineerng, Urmia University of Technology, Urmia, Iran


In this paper, the Ritz method has been employed to analyze the free in-plane vibration of heterogeneous (non-uniform) rectangular nanoplates corresponding to Eringen’s nonlocal elasticity theory. The non-uniformity is taken into account using combinations of linear and quadratic forms in the thickness, material density and Young’s modulus. Two-dimensional boundary characteristic orthogonal polynomials are applied in the Ritz method in order to examine the nonlocal effect, aspect ratio, length of nanoplate and non-uniformity parameters on the vibrational behaviors of the nanoplate. Results are verified with the available published data and good agreements are observed. The outcomes confirm apparent dependency of in-plane frequency of nanoplate on the small scale effect, non-uniformity, aspect ratio and boundary conditions. For instance, frequency parameter decreases by increasing the nonlocal parameter in all vibration modes; the frequency parameters increase with length and aspect ratio of nanoplates. Furthermore, the effect of nonlocal parameters on the frequency parameter is more prominent at the higher aspect ratios. Finally, the effect of nonlocal parameter on the in-plane modes is also presented in this analysis.


  • Free in-plane vibration of non-uniform nanoplates is studied using the Ritz method.
  • Effects of small-scale, non-uniformity and aspect ratio are investigated.
  • The in-plane frequencies are found significantly affected by nonlocal parameters.
  • Frequencies increase with Young’s modulus and decrease with density of nanoplates.
  • The evolution of mode shapes with increasing nonlocal parameters is visualized.


Main Subjects

[1] S. Thomas, N. Kalarikkal, A. Manuel Stephan, B. Raneesh, A.K. Haghi, Advanced nanomaterials: Synthesis, properties, and applications, Apple Academic Press, 2014.

[2] T. Murmu, S. Adhikari, Nonlocal transverse vibration of double-nanobeam-systems, Journal of Applied Physics, 108 (2010) 083514.

[3] F. Baletto, R. Ferrando, Structural properties of nanoclusters: Energetic, thermodynamic, and kinetic effects, Reviews of Modern Physics, 77 (2005) 371-423.

[4] A.S. Afolabi, A.S. Abdulkareem, S.E. Iyuke, H.C. Van Zyl Pienaar, Continuous production of carbon nanotubes and diamond films by swirled floating catalyst chemical vapour deposition method, South African Journal of Science, 105 (2009) 278-281.

[5] K. Nagashio, T. Nishimura, K. Kita, A. Toriumi, Mobility variations in mono-and multi-layer graphene films, Applied Physics Express (APEX), 2 (2009) 025003.

[6] A.I. Gusev, A.A. Rempel, Nanocrystalline Materials, Cambridge International Science Publishing, Cambridge, U.K., 2004.

[7] X. Li, W. Liu, L. Sun, K.E. Aifantis, B. Yu, Y. Fan, Q. Feng, F. Cui, F. Watari, Effects of physicochemical properties of nanomaterials on their toxicity, Journal of Biomedical Materials Research Part A, 103 (2015) 2499-2507.

[8] I. Favero, S. Stapfner, D. Hunger, P. Paulitschke, J. Reichel, H. Lorenz, E.M. Weig, K. Karrai, Fluctuating nanomechanical system in a high finesse optical microcavity, Optics express, 17 (2009) 12813-12820.

[9] M. Poot, H.S.J. Van der Zant, Nanomechanical properties of few-layer graphene membranes, Applied Physics Letters, 92 (2008) 063111.

[10] P. Ball, Roll up for the revolution, Nature, 414 (2001) 142-144.

[11] R.H. Baughman, A.A. Zakhidov, W.A. De Heer, Carbon nanotubes: The route toward applications, Science, 297 (2002) 787-792.

[12] B.H. Bodily, C.T. Sun, Structural and equivalent continuum properties of single-walled carbon nanotubes, International Journal of Materials and Product Technology, 18 (2003) 381-397.

[13] C. Li, T.W. Chou, A structural mechanics approach for the analysis of carbon nanotubes, International Journal of Solids and Structures, 40 (2003) 2487-2499.

[14] R. Liu, L. Wang, Thermal vibration of a single-walled carbon nanotube predicted by semiquantum molecular dynamics, Physical Chemistry Chemical Physics, 17 (2015) 5194-5201.

[15] C. Li, T.W. Chou, Quantized molecular structural mechanics modeling for studying the specific heat of single-walled carbon nanotubes, Physical Review B, 71 (2005) 075409.

[16] T. Yumura, A density functional theory study of chemical functionalization of carbon nanotubes; Toward site selective functionalization, INTECH Open Access Publisher, 2011.

[17] S. Adali, Variational principles for nonlocal continuum model of orthotropic graphene sheets embedded in an elastic medium, Acta Mathematica Scientia, 32 (2012) 325-338.

[18] M.R. Karamooz Ravari, A.R. Shahidi, Axisymmetric buckling of the circular annular nanoplates using finite difference method, Meccanica, 48 (2013) 135-144.

[19] P. Lu, H.P. Lee, C. Lu, P.Q. Zhang, Dynamic properties of flexural beams using a nonlocal elasticity model, Journal of Applied Physics, 99 (2006) 073510.

[20] H.S. Shen, Nonlocal plate model for nonlinear analysis of thin films on elastic foundations in thermal environments, Composite Structures, 93 (2011) 1143-1152.

[21] C.Y. Wang, T. Murmu, S. Adhikari, Mechanisms of nonlocal effect on the vibration of nanoplates, Applied Physics Letters, 98 (2011) 153101.

[22] A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science, 10 (1972) 425-435.

[23] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54 (1983) 4703-4710.

[24] A.C. Eringen, Nonlocal continuum field theories, Springer Science & Business Media, 2002.

[25] T.P. Chang, Small scale effect on axial vibration of non-uniform and non-homogeneous nanorods, Computational Materials Science, 54 (2012) 23-27.

[26] Loya, J. López-Puente, R. Zaera, J. Fernández-Sáez, Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model, Journal of Applied Physics, 105 (2009) 044309.

[27] 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, 43 (2011) 954-959.

[28] A. Anjomshoa, Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory, Meccanica, 48 (2013) 1337-1353.

[29] Y.G. Hu, K.M. Liew, Q. Wang, X.Q. He, B.I. Yakobson, Nonlocal shell model for elastic wave propagation in single-and double-walled carbon nanotubes, Journal of the Mechanics and Physics of Solids, 56 (2008) 3475-3485.

[30] N.P. Bansal, J. Lamon, Ceramic matrix composites: Materials, modeling and technology, John Wiley & Sons, 2014.

[31] V. Yantchev, I. Katardjiev, Thin film Lamb wave resonators in frequency control and sensing applications: a review, Journal of Micromechanics and Microengineering, 23 (2013) 043001.

[32] T. Murmu, S.C. Pradhan, Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model, Physica E: Low-dimensional Systems and Nanostructures, 41 (2009) 1628-1633.

[33] J. Cumings, P.G. Collins, A. Zettl, Peeling and sharpening multiwall nanotubes, Nature, 406 (2000) 586.

[34] A.M. Brodsky, Control of phase transition dynamics in media with nanoscale nonuniformities by coherence loss spectroscopy, Journal of Optics, 12 (2010) 095702.

[35] S. Chakraverty, L. Behera, Free vibration of non-uniform nanobeams using Rayleigh–Ritz method, Physica E: Low-dimensional Systems and Nanostructures, 67 (2015) 38-46.

[36] A. Koochi, H.M. Sedighi, M. Abadyan, Modeling the size dependent pull-in instability of beam-type NEMS using strain gradient theory, Latin American Journal of Solids and Structures, 11 (2014) 1806-1829.

[37] X.J. Xu, Z.C. Deng, Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory, Applied Mathematics and Mechanics, 35 (2014) 1115-1128.

[38] S. Chakraverty, L. Behera, Free vibration of rectangular nanoplates using Rayleigh–Ritz method, Physica E: Low-dimensional Systems and Nanostructures, 56 (2014) 357-363.

[39] R.B. Bhat, Plate deflections using orthogonal polynomials, Journal of Engineering Mechanics, 111 (1985) 1301-1309.

[40] R.B. Bhat, Vibration of rectangular plates on point and line supports using characteristic orthogonal polynomials in the Rayleigh-Ritz method, Journal of sound and vibration, 149 (1991) 170-172.

[41] T.S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, Science Publisher, Inc., New York, 1978.

[42] S.M. Dickinson, A. Di Blasio, On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates, Journal of Sound and Vibration, 108 (1986) 51-62.

[43] W. Gautschi, G.H. Golub, G. Opfer, Applications and computation of orthogonal polynomials, ADVANCES IN, (1999) 251.

[44] B. Singh, S. Chakraverty, Boundary characteristic orthogonal polynomials in numerical approximation, Communications in Numerical Methods in Engineering, 10 (1994) 1027-1043.

[45] B. Singh, S. Chakraverty, Use of characteristic orthogonal polynomials in two dimensions for transverse vibration of elliptic and circular plates with variable thickness, Journal of Sound and Vibration, 173 (1994) 289-299.

[46] P. Lu, P.Q. Zhang, H.P. Lee, C.M. Wang, J.N. Reddy, Non-local elastic plate theories, in:  Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 2007, pp. 3225-3240.

[47] D.J. Gorman, Free in-plane vibration analysis of rectangular plates by the method of superposition, Journal of Sound and Vibration, 272 (2004) 831-851.

[48] L. Behera, S. Chakraverty, Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials, Applied Nanoscience, 4 (2014) 347-358.