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

Subject Areas : Engineering

A Anjomshoa ¹ (Department of Mechanical Engineering, Isfahan University of Technology)
A.R Shahidi ² (Department of Mechanical Engineering, Isfahan University of Technology)
S.H Shahidi ³ (Department of Mechanical Engineering, Isfahan University of Technology)
H Nahvi ⁴ (Department of Mechanical Engineering, Isfahan University of Technology)

Received: 2014-10-17 Accepted : 2014-12-26 Published : 2015-03-30

Keywords:

References:

[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.