Vibration Analysis of Orthotropic Triangular Nanoplates Using Nonlocal Elasticity Theory and Galerkin Method

Document Type: Research Paper


1 Department of Mechanical Engineering, Isfahan University of Technology

2 Department of Mechanical Engineering, University of Jiroft


In this article, classical plate theory (CPT) is reformulated using the nonlocal differential constitutive relations of Eringen to develop an equivalent continuum model for orthotropic triangular nanoplates. The equations of motion are derived and the Galerkin’s approach in conjunction with the area coordinates is used as a basis for the solution. Nonlocal theories are employed to bring out the effect of the small scale on natural frequencies of nano scaled plates. Effect of nonlocal parameter, lengths of the nanoplate, aspect ratio, mode number, material properties, boundary condition and in-plane loads on the natural frequencies are investigated. It is shown that the natural frequencies depend highly on the non-locality of the nanoplate, especially at the very small dimensions, higher mode numbers and stiffer edge condition.


[1] Iijima S., 1991, Helical microtubules of graphitic carbon, Nature 354: 56-58.
[2] Stankovich S., Dikin D.A., Dommett G. H. B., Kohlhaas K., Zimney E., Stach E., Piner R., Nguyen S., Ruoff R., 2006, Graphene-based composite materials, Nature 442: 282-286.
[3] Mylvaganam K., Zhang L., 2004, Important issues in a molecular dynamics simulation for characterizing the mechanical properties of carbon nanotubes, Carbon 42(10): 2025-2032.
[4] Sears A., Batra R. C., 2004, Macroscopic properties of carbon nanotubes from molecular-mechanics simulations, Physical Review B 69(23): 235406.
[5] Sohi N., Naghdabadi R., 2007, Torsional buckling of carbon nanopeapods, Carbon 45: 952-957.
[6] Popov V. N., Doren V. E. V., Balkanski M., 2000, Elastic properties of single-walled carbon nanotubes, Physical Review B 61: 3078-3084.
[7] Sun C., Liu K., 2008, Dynamic torsional buckling of a double-walled carbon nanotube embedded in an elastic medium, European Journal of Mechanics - A/Solids 27: 40-49.
[8] Behfar K., Seifi P., Naghdabadi R., Ghanbari J., 2006, An analytical approach to determination of bending modulus of a multi-layered Graphene sheet, Thin Solid Films 496(2): 475-480.
[9] Wong E. W., Sheehan P. E., Lieber C. M., 1997, Nanobeam mechanics: elasticity, strength, and toughness of nano rods and nano tubes, Science 277: 1971-1975.
[10] Liew K. M., He X. Q., Kitipornchai S., 2006, Predicting nanovibration of multi-layered Graphene sheets embedded in an elastic matrix, Acta Materialia 54: 4229-4236.
[11] Eringen C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703-4710.
[12] Eringen C., 2002, Nonlocal Continuum Field Theories, Springer, New York.
[13] Wang Q., Wang C. M., 2007, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modeling carbon nanotubes, Nanotechnology 18(7): 075702.
[14] Civalek Ö., Akgöz B., 2010, Free vibration analysis of microtubules as cytoskeleton components: nonlocal euler-bernoulli beam modeling, Journal of Scientia Iranica 17(5): 367-375.
[15] Civalek Ö., Çigdem D., 2011, Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Applied Mathematical Modeling 35: 2053-2067.
[16] Murmu T., Adhikari S., 2010, Nonlocal transverse vibration of double-nanobeam-systems, Journal of Applied Physics 108: 083514.
[17] Khademolhosseini F., Rajapakse R. K. N. D., Nojeh A., 2010, Torsional buckling of carbon nanotubes based on nonlocal elasticity shell models, Computational Materials Science 48: 736-742.
[18] Wang Q., Varadan V. K., 2006, Vibration of carbon nanotubes studied using nonlocal continuum mechanics, Smart Materials and Structures 15: 659-666.
[19] Reddy J. N., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45: 288-307.
[20] Luo X., Chung D. D. L., 2000, Vibration damping using flexible graphite, Carbon 38: 1510-1512.
[21] Zhang L., Huang H., 2006, Young’s moduli of ZnO nanoplates: Ab initio determinations, Applied Physics Letters 89: 183111.
[22] Freund L. B., Suresh S., 2003, Thin Film Materials, Cambridge University Press, Cambridge.
[23] Scarpa F., Adhikari S., Srikantha Phani A., 2009, Effective elastic mechanical properties of single layer Graphene sheets, Nanotechnology 20: 065709.
[24] Sakhaee-Pour A., 2009, Elastic buckling of single-layered Graphene sheet, Computational Materials Science 45: 266-270.
[25] Pradhan S. C., Phadikar J. K., 2009, Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration 325: 206-223.
[26] Pradhan S. C., Phadikar J. K.,2009, Small scale effect on vibration of embedded multilayered Graphene sheets based on nonlocal continuum models, Physics Letters A 37: 1062-1069.
[27] Phadikar J. K., Pradhan S. C., 2010, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Computational Materials Science 49: 492-499.
[28] Pradhan S. C., Kumar A., 2010, Vibration analysis of orthotropic Graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method, Computational Materials Science 50: 239-245.
[29] Aydogdu M., Tolga A., 2011, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E 43: 954-959.
[30] Jomehzadeh E., Saidi A. R., 2011, A study on large amplitude vibration of multilayered Graphene sheets, Computational Materials Science 50: 1043-1051.
[31] Jomehzadeh E., Saidi A. R., 2011, Decoupling the nonlocal elasticity equations for three dimension a vibration analysis of nano-plates, Journal of Composite Structures 93: 1015-1020.
[32] Salehipour H. , Nahvi H., Shahidi A.R., 2015, Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity, Physica E: Low-dimensional Systems and Nanostructures 66: 350-358.
[33] Ansari R. , Shahabodini A. , Faghih Shojaei M., 2016, Nonlocal three-dimensional theory of elasticity with application to free vibration of functionally graded nanoplates on elastic foundations, Physica E: Low-dimensional Systems and Nanostructures 76: 70-81.
[34] Kiani K., 2011, Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory, Journal of Sound and Vibration 330(20): 4896-4914.
[35] Kiani K., 2011, Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle, Part I: theoretical formulations, Physica E: Low-dimensional Systems and Nanostructure,s 44: 229-248.
[36] Kiani K., 2011, Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle, Part II: parametric studies, Physica E: Low-dimensional Systems and Nanostructures 44: 249-269.
[37] Kiani K., 2013, Vibrations of biaxially tensioned-embedded nanoplates for nanoparticle delivery, Indian Journal of Science and Technology 6(7): 4894-4902.
[38] Duan W. H., Wang C. M., 2007, Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology 18: 385704.
[39] Farajpour A., Mohammadi M., Shahidi A. R., Mahzoon M., 2011, Axisymmetric buckling of the circular Graphene sheets with the nonlocal continuum plate model, Physica E 43: 1820-1825.
[40] Babaie H., Shahidi A.R., 2011, Vibration of quadrilateral embedded multilayered Graphene sheets based on nonlocal continuum models using the Galerkin method, Acta Mechanica Sinica 27(6): 967-976.
[41] Malekzadeh P., Setoodeh A. R., Alibeygi Beni A., 2011, Small scale effect on the vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates, Journal of Composite Structures 93: 1631-1639.
[42] Anjomshoa A.,2013, Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory, Meccanica 48: 1337-1353.
[43] Shahidi R., Mahzoon M., Saadatpour M. M., Azhari M., 2005, Very large deformation analysis of plates and folded plates by finite strip method, Advances in Structural Engineering 8(6): 547-560.
[44] Liew K. M., Wang C. M., 1993, Pb-2 Rayleigh-Ritz method for general plate analysis, Engineering Structures 15(1): 55-60.
[45] Adali S., 2009, Variational principle for transversely vibrating multiwalled carbon nanotubes based on nonlocal Euler-Bernoulli beam model, Nano letters 9: 1737-1741.
[46] Reddy J. N., 1997, Mechanics of Laminated Composite Plates, Theory and Analysis, Chemical Rubber Company, Boca Raton, FL.
[47] Kim S., Dickinso S. M., 1990, The free flexural vibration of right triangular isotropic and orthotropic plates, Journal of Sound and Vibration 141(2): 291-311.
[48] Gorman J., 1985, Free vibration analysis of right triangular plates with combinations of clamped – simply supported conditions, Journal of Sound and Vibration 106(3): 419-431.
[49] Aghababaei R., Reddy J.N., 2009, Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates, Journal of Sound and Vibration 326: 277-289.