Frequency Analysis of Embedded Orthotropic Circular and Elliptical Micro/Nano-Plates Using Nonlocal Variational Principle

Journal of Solid Mechanics

مقاله 2، دوره 7، شماره 1، خرداد 2015، صفحه 13-27   

نوع مقاله: Research Paper

نویسندگان

A Anjomshoa؛ A.R Shahidi؛ S.H Shahidi؛ H Nahvi^*

Department of Mechanical Engineering, Isfahan University of Technology

چکیده

In this paper, a continuum model based on the nonlocal elasticity theory is developed for vibration analysis of embedded orthotropic circular and elliptical micro/nano-plates. The nano-plate is bounded by a Pasternak foundation. Governing vibration equation of the nonlocal nano-plate is derived using Nonlocal Classical Plate Theory (NCPT). The weighted residual statement and the Galerkin method are applied to obtain a Quadratic Functional. The Ritz functions are used to form an assumed expression for transverse displacement which satisfies the kinematic boundary conditions. The Ritz functions eliminate the need for mesh generation and thus large degrees of freedom arising in discretization methods such as Finite Element Method (FEM). Effects of nonlocal parameter, lengths of nano-plate, aspect ratio, mode number, material properties and foundation parameters on the nano-plate natural frequencies are investigated. It is shown that the natural frequencies depend on the non-locality of the micro/nano-plate, especially at small dimensions.                                         

کلیدواژه‌ها

Nonlocal elasticity theory؛ Frequency Analysis؛ Elliptical nano-plate؛ Variational principle

مراجع

[1] Iijima S., 1991, Helical microtubules of graphitic carbon, Nature 354: 56-58.
[2] Stankovich S., Dikin D.A., Dommet G., Zimney E., Stach E., Piner R., Nguyen S., Ruoff R., Nahvi B.A., 2006, Graphene-based composite materials, Nature 442: 282-286.
[3] Mylvaganam K., Zhang L., 2004, Important issues in a molecular dynamics simulation for characterising 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 A.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 carbo 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, 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 nanorods and nanotubes, 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 A.C., 1972, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science 10(5): 425-435.
[12] Eringen A.C, 2002, Nonlocal Continuum Field Theories, Springer, New York.
[13] Wang C.M., Zhang Y.Y., He X.Q., 2007, Vibration of nonlocal Timoshenko beams, Nanotechnology 18(10):105401.
[14] Civalek Ö., Akgöz B., 2010, Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler-Bernoulli beam modeling, Scientia Iranica 17(5): 367-375.
[15] Civalek Ö., Çigdem D., 2011, Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Applied Mathematical Modelling 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 grapheme sheets, Nanotechnology 20(6):065709.
[24] Sakhaee-Pour A., 2009, Elastic properties of single-layered graphene sheet, Solid State Communications 149: 91-95.
[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] Aydogdu M., Tolga A., 2011, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E 43: 954-959.
[28] Jomehzadeh E., Saidi A.R., 2011, A study on large amplitude vibration of multilayered graphene sheets, Computational Materials Science 50: 1043-1051.
[29] Jomehzadeh E., Saidi A.R., 2011, Decoupling the nonlocal elasticity equations for three dimensional vibration analysis of nano-plates, Composite Structures 93: 1015-1020.
[30] Duan W.H., Wang C.M., 2007, Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology 18(38) :385704.
[31] Farajpour A., Mohammadi M., Shahidi A.R., Mahzoon M., 2011, Axisymmetric buckling of the circular grapheme sheets with the nonlocal continuum plate model, Physica E 43: 1820-1825.
[32] Babaie H., Shahidi A.R., 2011, Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method, Archive of Applied Mechanics 81: 1051-1062.
[33] Malekzadeh P., Setoodeh A.R., Alibeygi Beni A., 2011, Small scale effect on the vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates, Composite Structures 93: 1631-1639.
[34] Malekzadeh P., Setoodeh A.R., Alibeygi B. A., 2011, Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in elastic medium, Composite Structures 93: 2083-2089.
[35] Shahidi A.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.
[36] Liew K.M., Wang C.M., 1993, Pb-2 Rayleigh-Ritz method for general plate analysis, Engineering Structures 15(1): 55-60.
[37] Azhari M., Shahidi A.R., Saadatpour M.M., 2005, Local and post-local buckling of stepped and perforated thin plates, Applied Mathematical Modelling 29(7): 633-652.
[38] Ceribasi S., Altay G., 2009, Free vibration of super elliptical plates with constant and variable thickness by Ritz method, Journal of Sound and Vibration 319: 668-680.
[39] Adali S., 2009, Variational principle for transversely vibrating multiwalled carbon nanotubes based on nonlocal Euler-Bernoulli beam model, Nano Letters 9: 1737-1741.
[40] Adali S., 2011, Variational principle and natural boundary conditions for multilayered orthotropic grapheme sheets undergoing vibrations and based on nonlocal elasticity theory, Journal of Theoretical and Applied Mechanics 49(3): 621-639.
[41] Phadikar J.K, Pradhan S.C., 2010, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Computational Materials Science 49: S492-S499.
[42] Aksencer T., Aydogdu M., 2012, Forced transverse vibration of nanoplates using nonlocal elasticity, Physica E 44: 1752-1759.
[43] Akgoz B., Civalek O., 2013, Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory, Mechanica 48: 863-873.
[44] Hosseini-Hashemi S., Zare M., Nazemnezhad R., 2013, An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity, Composite Structures 100: 290-299.
[45] Mohammadi M., Goodarzi M., Ghayour M., Alivand S., 2012, Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial in-plane pre-load via nonlocal elasticity theory, Journal of Solid Mechanics 4(2): 128-143.
[46] Mohammadi M., Farajpour A., Goodarzi M., Heydarshenas R., 2013, Levy type solution for nonlocal thermo-mechanical vibration of orthotropic mono-layer graghene sheet embedded in an elastic medium, Journal of Solid Mechanics 5(2): 116-132.
[47] Mohammadi M., Goodarzi M., Ghayour M., Farajpour A., 2013, Influence of in-plane pre-load on the vibration frequency of circular grapheme sheet via nonlocal continuum theory, Composites Part B: Engineering 51: 121-129.
[48] Wang C.M, Wang L., 1994, Vibration and buckling of super elliptical plates, Journal of Sound and Vibration 171: 301-314.
[49] Lam K.Y., Liew K.M., Chow S.T., 1992, Use of two-dimensional orthogonal polynomials for vibration analysis of circular and elliptical plates, Journal of Sound and Vibration 154(2): 261-269.