Surface Effects on Free Vibration Analysis of Nanobeams Using Nonlocal Elasticity: A Comparison Between Euler-Bernoulli and Timoshenko

Journal of Solid Mechanics

مقاله 7، دوره 5، شماره 3، آذر 2013، صفحه 290-304   

نوع مقاله: Research Paper

نویسندگان

Sh Hosseini – Hashemi^{* 1}؛ M Fakher²؛ R Nazemnezhad²

¹School of Mechanical Engineering, Iran University of Science and Technology--- Center of Excellence in Railway Transportation, Iran University of Science and Technology

²School of Mechanical Engineering, Iran University of Science and Technology

چکیده

In this paper, surface effects including surface elasticity, surface stress and surface density, on the free vibration analysis of Euler-Bernoulli and Timoshenko nanobeams are considered using nonlocal elasticity theory. To this end, the balance conditions between nanobeam bulk and its surfaces are considered to be satisfied assuming a linear variation for the component of the normal stress through the nanobeam thickness. The governing equations are obtained and solved for Silicon and Aluminum nanobeams with three different boundary conditions, i.e. Simply-Simply, Clamped-Simply and Clamped-Clamped. The results show that the influence of the surface effects on the natural frequencies of the Aluminum nanobeams follows the order CC

کلیدواژه‌ها

Surface effects؛ Nonlocal elasticity؛ Free vibration؛ Nanobeam؛ Euler-Bernoulli theory؛ Timoshenko theory

مراجع

 [1] He L., Lim C., Wu B., 2004, A continuum model for size-dependent deformation of elastic films of nano-scale thickness, International Journal of Solids and Structures 41(3):847-857.

[2] Eringen A.C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10(1):1-16.

[3] Eringen A.C., Edelen D., 1972, On nonlocal elasticity, International Journal of Engineering Science 10(3):233-248.

[4] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54(9): 4703-4710.

[5] Eringen A.C., 2002, Nonlocal Continuum Field Theories, Springer.

[6] Gurtin M., Murdoch A.I., 1975, A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis 57(4):291-323.

[7] Gurtin M., Weissmüller J., Larche F., 1998, A general theory of curved deformable interfaces in solids at equilibrium, Philosophical Magazine A 78(5):1093-1109.

[8] Asgharifard Sharabiani P., Haeri Yazdi M.R., 2013, Nonlinear free vibrations of functionally graded nanobeams with surface effects, Composites Part B: Engineering 45(1):581-586.

[9] Wang G., Feng X., Yu S., 2007, Surface buckling of a bending microbeam due to surface elasticity,Europhysics Letters 77(4):44002.

[10] Assadi A., Farshi B., 2011, Size-dependent longitudinal and transverse wave propagation in embedded nanotubes with consideration of surface effects, Acta Mechanica 222(1-2):27-39.

[11] Fu Y., Zhang J., Jiang Y., 2010, Influences of the surface energies on the nonlinear static and dynamic behaviors of nanobeams, Physica E: Low-Dimensional Systems and Nanostructures 42(9):2268-2273.

[12] Gheshlaghi B., Hasheminejad S.M., 2011, Surface effects on nonlinear free vibration of nanobeams, Composites Part B: Engineering 42(4):934-937.

[13] Guo J.-G., Zhao Y.P., 2007, The size-dependent bending elastic properties of nanobeams with surface effects, Nanotechnology 18(29):295701.

[14] Hosseini-Hashemi S., Nazemnezhad R., 2013, An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects, Composites Part B: Engineering 52:199-206.

[15] Liu C., Rajapakse R., 2010, Continuum models incorporating surface energy for static and dynamic response of nanoscale beams, Nanotechnology 9(4):422-431.

[16] Nazemnezhad R., Salimi M., Hosseini-Hashemi S., Sharabiani P.A., 2012, An analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density effects, Composites Part B: Engineering 43:2893-2897.

[17] Park H.S., 2012, Surface stress effects on the critical buckling strains of silicon nanowires, Computational Materials Science 51(1):396-401.

[18] Ren Q., Zhao Y.-P., 2004, Influence of surface stress on frequency of microcantilever-based biosensors, Microsystem Technologies 10(4):307-314.

[19] Song F., Huang G., Park H., Liu X., 2011, A continuum model for the mechanical behavior of nanowires including surface and surface-induced initial stresses, International Journal of Solids and Structures 48(14):2154-2163.

[20] Wang G.-F., Feng X.Q., 2009, Timoshenko beam model for buckling and vibration of nanowires with surface effects, Journal of Physics D: Applied Physics 42(15):155411.

[21] Wang L., Hu H., 2005, Flexural wave propagation in single-walled carbon nanotubes, Physical Review B 71(19): 195412-195419.

[22] Wang Q., 2005, Wave propagation in carbon nanotubes via nonlocal continuum mechanics, Journal of Applied physics 98(12):124301-124306.

[23] Ece M., Aydogdu M., 2007, Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes, Acta Mechanica 190(1-4):185-195.

[24] Lim C., Li C., Yu J.l., 2010, Free vibration of pre-tensioned nanobeams based on nonlocal stress theory, Journal of Zhejiang University Science A 11(1):34-42.

[25] Maachou M., Zidour M., Baghdadi H., Ziane N., Tounsi A., 2011, A nonlocal levinson beam model for free vibration analysis of zigzag single-walled carbon nanotubes including thermal effects, Solid State Communications 151(20): 1467-1471.

[26] Mohammadi B., Ghannadpour S., 2011, Energy approach vibration analysis of nonlocal timoshenko beam theory, Procedia Engineering 10:1766-1771.

[27] Reddy J., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45(2):288-307.

[28] Reddy J., 2010, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, International Journal of Engineering Science 48(11):1507-1518.

[29] Wang C., Zhang Y., He X., 2007, Vibration of nonlocal timoshenko beams, Nanotechnology 18(10):105401.

[30] Xu M., 2006, Free transverse vibrations of nano-to-micron scale beams, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 462(2074):2977-2995.

[31] Bedroud M., Hosseini-Hashemi S., Nazemnezhad R., 2013, Buckling of circular/annular Mindlin nanoplates via nonlocal elasticity, Acta Mechanica 224(11):2663-2676.

[32] Bedroud M., Hosseini-Hashemi S., Nazemnezhad R., 2013, Axisymmetric/asymmetric buckling of circular/annular nanoplates via nonlocal elasticity, Modares Mechanical Engineering 13(5):144-152.

[33] Hosseini-Hashemi S., Bedroud M., Nazemnezhad R., 2013, An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity, Composite Structures 103:108-118.

[34] 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.

[35] Lee H.-L., Chang W.-J., 2010, Surface and small-scale effects on vibration analysis of a nonuniform nanocantilever beam, Physica E: Low-Dimensional Systems and Nanostructures 43(1):466-469.

[36] Wang K., Wang B., 2011, Vibration of nanoscale plates with surface energy via nonlocal elasticity, Physica E: Low-Dimensional Systems and Nanostructures 44(2):448-453.

[37] Lei X.w., Natsuki T., Shi J.x., Ni Q.q., 2012, Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal timoshenko beam model, Composites Part B: Engineering 43(1):64-69.

[38] Gheshlaghi B., Hasheminejad S.M., 2012, Vibration analysis of piezoelectric nanowires with surface and small scale effects, Current Applied Physics 12(4):1096-1099.

[39] Wang K., Wang B., 2012, The electromechanical coupling behavior of piezoelectric nanowires: Surface and small-scale effects, Europhysics Letters 97(6):66005.

[40] Malekzadeh P., Shojaee M., 2013, Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams, Composites Part B: Engineering 52:82-94.

[41] Mahmoud F., Eltaher M., Alshorbagy A., Meletis E., 2012, Static analysis of nanobeams including surface effects by nonlocal finite element, Journal of Mechanical Science and Technology 26(11):3555-3563.

[42] Gurtin M.E., Ian Murdoch A., 1978, Surface stress in solids, International Journal of Solids and Structures 14(6):431-440.

[43] Shenoy V.B., 2005, Atomistic calculations of elastic properties of metallic fcc crystal surfaces, Physical Review B 71(9):094104-094115.

[44] Gurtin M.E., Murdoch A.I., 1975, A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis 57(4):291-323.

[45] Lu P., He L., Lee H., Lu C., 2006, Thin plate theory including surface effects, International Journal of Solids and Structures 43(16):4631-4647.

[46] Rao S.S., 2007, Vibration of Continuous Systems, John Wiley & Sons.

[47] Timoshenko S.P., Goodier J., 2011, Theory of elasticity, International Journal of Bulk Solids Storage in Silos 1(4): 567-567.

[48] Ogata S., Li J., Yip S., 2002, Ideal pure shear strength of aluminum and copper, Science 298(5594):807-811.

[49] Zhu R., Pan E., Chung P.W., Cai X., Liew K.M., Buldum A., 2006, Atomistic calculation of elastic moduli in strained silicon, Semiconductor Science and Technology 21(7):906.

[50] Miller R.E., Shenoy V.B., 2000, Size-dependent elastic properties of nanosized structural elements, Nanotechnology 11(3):139.