Effect of Surface Energy on the Vibration Analysis of Rotating Nanobeam

Journal of Solid Mechanics

مقاله 5، دوره 7، شماره 3، آذر 2015، صفحه 299-311   

نوع مقاله: Research Paper

نویسندگان

M Safarabadi^{* 1}؛ M Mohammadi²؛ A Farajpour³؛ M Goodarzi²

¹School of Mechanical Engineering, College of Engineering, University of Tehran

²Department of Engineering, College of Mechanical Engineering, Ahvaz branch, Islamic Azad University

³Young Researches and Elites Club , North Tehran Branch, Islamic Azad University

چکیده

In this study, the free vibration behavior of rotating nanobeam is studied.  Surface effects on the vibration frequencies of nanobeam are considered. To incorporate surface effects, Gurtin–Murdoch model is proposed to satisfy the surface balance equations of the continuum surface elasticity. Differential quadrature method is employed and in order to establish the accuracy and applicability of the proposed model, the numerical results are presented to be compared with those available in the literature.  The effects of angular velocity, boundary conditions and surface elastic constants on the vibration characteristics are presented. Numerical results show that the softer boundary conditions cause an increase in the influence of the angular velocity on the nanobeam vibration frequencies.

کلیدواژه‌ها

Vibration؛ Rotating nanobeam؛ Differential quadrature method؛ Surface energy

مراجع

[1] Raighead H. G. C., 2000, Nanoelectromechanical systems, Science 290: 1532-1535.
[2] Ekinci K. L., Roukes M. L., 2005, Nanoelectromechanical systems, Review of Scientific Instruments 76: 1-12.
[3] Mindlin R. D., Tiersten H. F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11: 415-448.
[4] Toupin R. A., 1962, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11: 385-414.
[5] Akgöz B., Civalek Ö., 2013, Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory, Meccanica 48: 863-873.
[6] Akgöz B., Civalek Ö., 2011, Strain gradiant and modified couple stress models for buckling analysis of axially loaded micro-scales beam, International Journal of Engineering Science 49: 1268-1280.
[7] Civalek Ö., Demir C., Akgöz B., 2010, Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model, Mathematical and Computational Applications 15: 289-298.
[8] Civalek Ö., Demir Ç., 2011, Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory, Applied Mathematical Modeling 35: 2053-2067.
[9] Farajpour A., Danesh M., Mohammadi M., 2011, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E 44: 719-727.
[10] Mohammadi M., Goodarzi M., Ghayour M., Farajpour A., 2013, Influence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory, Composites: Part B 51: 121-129.
[11] Mohammadi M., Moradi A., Ghayour M., Farajpour A., 2014, Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures 11: 437- 458.
[12] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703- 4710.
[13] Eringen A.C., 2002, Nonlocal Continuum Field Theories, Springer, New York.
[14] 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: 847- 857.
[15] Ghorbanpour Arani A., Kolahchi R., Vossough H., 2012, Nonlocal wave propagation in an embedded DWBNNT conveying fluid via strain gradient, Physica B: Condensed Matter 407: 4281- 4286.
[16] Akgöz B., Civalek Ö., 2011, Application of strain gradient elasticity theory for buckling analysis of protein microtubules, Current Applied Physics 11: 1133-1138.
[17] Akgöz B., Civalek Ö., 2012, Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory, Archive of Applied Mechanics 82: 423- 443.
[18] Akgöz B., Civalek Ö., 2013, A size-dependent shear deformation beam model based on the strain gradient elasticity theory, International Journal of Engineering Science 70: 1-14.
[19] Gurtin M., Murdoch A. I., 1975, A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis 57: 291-323.
[20] Gurtin M., Weissmüller J., Larche F., 1998, A general theory of curved deformable interfaces in solids at equilibrium, Philosophical Magazine A 78: 1093-1109.
[21] Asgharifard Sharabiani P., Haeri Yazdi M.R., 2013, Nonlinear free vibrations of functionally graded nanobeams with surface effects, Composites Part B: Engineering 45: 581-586.
[22] Wang G., Feng X., Yu S., 2007, Surface buckling of a bending microbeam due to surface elasticity, Europhysics Letters 77: 44002.
[23] Assadi A., Farshi B., 2011, Size-dependent longitudinal and transverse wave propagation in embedded nanotubes with consideration of surface effects, Acta Mechanica 222: 27-39.
[24] 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: 2268 -2273.
[25] Alizada A. N., Sofiyev A. H., 2011, On the mechanics of deformation and stability of the beam with a nanocoating, Journal of Reinforced Plastics and Composites 30:1583-1595.
[26] Alizada A. N., Sofiyev A. H., Kuruoglu N., 2012, Stress analysis of a substrate coated by nanomaterials with vacancies subjected to uniform extension load, Acta Mechanica 223:1371-1383.
[27] Alizada A. N., Sofiyev A. H., 2011, Modified Young’s moduli of nanomaterials taking into account the scale effects and vacancies, Meccanica 46: 915- 920.
[28] Gheshlaghi B., Hasheminejad S. M., 2011, Surface effects on nonlinear free vibration of nanobeams, Composites Part B: Engineering 42: 934 -937.
[29] Guo J. G., Zhao Y.P., 2007, The size-dependent bending elastic properties of nanobeams with surface effects, Nanotechnology 18: 295701.
[30] 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.
[31] Liu C., Rajapakse R., 2010, Continuum models incorporating surface energy for static and dynamic response of nanoscale beams, Nanotechnology 9: 422- 431.
[32] 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.
[33] 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: 3555-3563.
[34] Eltaher M. A., Mahmoud F. F., Assie A. E., Meletis E. I., 2013, Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams, Applied Mathematics and Computation 224: 760-774.
[35] Pradhan S. C., Murmu T., 2010, Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever, Physica E 42: 1944 -1949.
[36] Assadi A., Farshi B., 2011, Size dependent stability analysis of circular ultrathin films in elastic medium with consideration of surface energies, Physica E 43: 1111-1117.
[37] Assadi A., Farshi B., 2011, Size dependent vibration of curved nanobeams and rings including surface energies, Physica E 43: 975-978.
[38] Farajpour A., Dehghany M., Shahidi A. R., 2013, Surface and nonlocal effects on the axisymmetric buckling of circular graphene sheets in thermal environment, Composites Part B: Engineering 50: 333-343.
[39] Asemi S. R., Farajpour A., 2014, Decoupling the nonlocal elasticity equations for thermo-mechanical vibration of circular graphene sheets including surface effects, Physica E 60: 80-90.
[40] Farajpour A., Rastgoo A., Mohammadi M., 2014, Surface effects on the mechanical characteristics of microtubule networks in living cells, Mechanics Research Communications 57: 18-26.
[41] Shenoy V. B., 2005, Atomistic calculations of elastic properties of metallic fcc crystal surfaces, Physical Review B 71: 094104-094115.
[42] Lu P., He L., Lee H., Lu C., 2006, Thin plate theory including surface effects, International Journal of Solids and Structures 43: 4631- 4647.
[43] Rao S. S., 2007, Vibration of Continuous Systems, John Wiley & Sons.
[44] Gurtin M. E., Markenscoff X., Thurston R. N., 1976, Effect of surface stress on the natural frequency of thin crystals, Applied Physics Letters 29: 529-530.
[45] Danesh M., Farajpour A., Mohammadi M., 2012, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications 39: 23- 27.
[46] 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.
[47] Moosavi H., Mohammadi M., Farajpour A., Shahidi A. R., 2011, Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory, Physica E 44: 135-140.
[48] Mohammadi M., Goodarzi A., 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: 128-143.
[49] Mohammadi M., Farajpour A., Goodarzi M., Dinari F., 2014, Thermo-mechanical vibration of annular and circular graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures 11: 659- 682.
[50] Shu C., 2000, Differential Quadrature and its Application in Engineering, Springer, Great Britain.
[51] Civalek Ö., 2004, Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns, Engineering Structures 26: 171-186.
[52] Civalek Ö., 2007, Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method, International Journal of Mechanical Sciences 49: 752-765.
[53] Civalek Ö., 2007, Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods, Applied Mathematical Modelling 31: 606 -624.
[54] Civalek Ö., 2007, Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach, Journal of Computational and Applied Mathematics 205: 251-271.
[55] Mohammadi M., Farajpour A., Moradi A., Ghayour M., 2014, Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites: Part B 56: 629- 637.
[56] Mohammadi M., Farajpour A., Goodarzi M., Shehni Nezhad Pour H., 2014, Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium, Computational Materials Science 82: 510 -520.
[57] Mohammadi M., Ghayour M., Farajpour A., 2013, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites: Part B 45: 32- 42.
[58] Farajpour A., Shahidi A. R., Mohammadi M., Mahzoon M., 2012, Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics, Composite Structures 94: 1605-1615.
[59] Asemi S. R., Farajpour A., Asemi H. R., Mohammadi M., 2014, Influence of initial stress on the vibration of double-piezoelectricnanoplate systems with various boundary conditions using DQM, Physica E 63: 169 -179.
[60] Asemi S. R., Farajpour A., Mohammadi M., 2014, Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory, Composite Structures 116: 703-712.
[61] Asemi S. R., Mohammadi M., Farajpour A., 2014, A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory, Latin American Journal of Solids and Structures 11: 1541-1564.
[62] Mohammadi M., Farajpour A., Goodarzi M., Heydarshenas R., 2013, Levy type solution for nonlocal thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Journal of Solid Mechanics 5: 116-132.
[63] Mohammadi M., Farajpour A., Goodarzi M., Mohammadi H., 2013, Temperature effect on vibration analysis of annular graphene sheet embedded on visco-pasternak foundation, Journal of Solid Mechanics 5: 305-323.
[64] Goodarzi M., Mohammadi M., Farajpour A., Khooran M., 2014, Investigation of the effect of pre-stressed on vibration frequency of rectangular nanoplate based on a visco pasternak foundation, Journal of Solid Mechanics 6: 98-121.
[65] Bert C. W, Malik M., 1996, Differential quadrature method in computational mechanics, Applied Mechanic Review 49: 1-27.
[66] Shu C., Richards B.E., 1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier Stokes equations, International Journal for Numerical Methods in Fluids 15: 791-798.
[67] Miller R. E., Shenoy V. B., 2000, Size dependent elastic properties of structural elements, Nanotechnology 11: 139-147.
[68] Daw M. S., Baskes M. I., 1984, Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals, Physical Review B 29: 6443-6453.
[69] Hosseini-Hashemi S.h., Fakher M., Nazemnezhad R., 2013, Surface effects on free vibration analysis of nanobeams using nonlocal elasticity: a comparison between euler-bernoulli and timoshenko, Journal of Solid Mechanics 5: 290-304.