Investigation of the Effect of Pre-Stressed on Vibration Frequency of Rectangular Nanoplate Based on a Visco-Pasternak Foundation

Journal of Solid Mechanics

مقاله 8، دوره 6، شماره 1، خرداد 2014، صفحه 98-121   

نوع مقاله: Research Paper

نویسندگان

M Goodarzi^{* 1}؛ M Mohammadi¹؛ A Farajpour²؛ M Khooran³

¹Department of Mechanical Engineering, College of Engineering, Ahvaz Branch, Islamic Azad University

²Young Researches and Elites Club, North Tehran Branch, Islamic Azad University,

³Department of Mechanical Engineering, Shahid Chamran University of Ahvaz

چکیده

In the present work, the free vibration behavior of rectangular graphene sheet under shear in-plane load is studied. Nonlocal elasticity theory has been implemented to study the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) subjected to shear in-plane load. The SLGSs is embedded on a viscoelastic medium which is simulated as a Visco-Pasternak foundation. Using the principle of virtual work, the governing equations are derived for the rectangular nanoplates. Differential quadrature method (DQM) is employed and numerical solutions for the vibration frequency are obtained. The influence of surrounding elastic medium, material property, aspect ratio, nonlocal parameter, length of nanoplate and effect of boundary conditions on the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) is studied. Six boundary conditions are investigated. Numerical results show that the vibration frequencies of SLGSs are strongly dependent on the small scale coefficient and shear in-plane load. The present analysis results can be used for the design of the next generation of nanodevices that make use of the vibration properties of the graphene.

کلیدواژه‌ها

Vibration؛ Graphene sheet؛ Shear in-plane load؛ Visco-Pasternak foundation

مراجع

[1] Li X., Bhushan B., Takashima K., Baek C.W., Kim Y.K., 2003, Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nano indentation techniques, Ultramicroscopy 97:481-494.
[2] Fleck N. A., Muller G. M., Ashby M. F., Hutchinson J. W., 1994, Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia 42:475-487.
[3] Stolken J.S., Evans A.G., 1998, A microbend test method for measuring the plasticity length scale, Acta Materialia 46: 5109-5115.
[4] Chong A.C.M, Yang F., Lam D.C.C, Tong P., 2001, Torsion and bending of micron-scaled structures, Journal of Materials Research 16:1052-1058.
[5] Chowdhury R., Adhikari S., Wang C.W., Scarpa F., 2010, A molecular mechanics approach for the vibration of single walled carbon nanotubes, Computational Material Science 48:730-735.
[6] Behfar K., Naghdabadi R., 2005, Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium, Composite Science Technology 65:1159-1164.
[7] Sakhaee-pour A., Ahmadian M.T., Naghdabadi R., 2008, Vibrational analysis of single-layered graphene sheets, Nanotechnology 19(8):085702.
[8] Mohammadi M., Farajpour A., Goodarzi M., Dinari F., 2014, Thermo-mechanical vibration analysis of annular and circular graphene sheet embedded in an elastic medium, Latin American Journal of Solids & Structures 11(4): 659-682.
[9] Mindlin R. D., Tiersten H. F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11:415-448.
[10] Toupin R.A., 1962, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11:385-414.
[11] 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.
[12] 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.
[13] 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.
[14] Civalek Ö., Demir Ç., 2011, Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory, Applied Mathematical Modeling 35:2053-2067.
[15] Farajpour A., Danesh M., Mohammadi M., 2011, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E 44:719-727.
[16] Mohammadi M., Farajpour A., Goodarzi M., 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.
[17] 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.
[18] 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.
[19] Akgöz B., Civalek Ö., 2011, Application of strain gradient elasticity theory for buckling analysis of protein microtubules, Current Applied Physics 11:1133-1138.
[20] 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.
[21] 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.
[22] 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.
[23] Eringen A. C., 2002, Nonlocal Continuum Field Theories, Springer, New York.
[24] Aydogdu M., 2009, Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics, International Journal of Engineering Science 56:17-28.
[25] Aydogdu M., 2009, Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mechanics Research Communications 43:34-40.
[26] Narendar S., Gopalakrishnan S., 2009, Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes, Computational Materials Science 47:526-538.
[27] Wang C. M., Duan W. H., 2008, Free vibration of nanorings/arches based on nonlocal elasticity, Journal of Applied Physics 104:14303-14308.
[28] Moosavi H., Mohammadi M., Farajpour A., Shahidi S. H., 2011, Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory, Physica E 44:135-140.
[29] Murmu T., Pradhan S. C., 2009, Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E 41:1232-1239.
[30] Wang Y. Z., Li F. M., Kishimoto K., 2011,Thermal effects on vibration properties of double layered nanoplates at small scales, Composites Part B: Engineering 42:1311-1317.
[31] Reddy C.D., Rajendran S., Liew K. M., 2006, Equilibrium configuration and continuum elastic properties of finite sized graphene, Nanotechnology 17:864-870.
[32] Malekzadeh P., Setoodeh A. R, Alibeygi Beni A., 2011, Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium, Composite Structure 93: 2083-2089.
[33] Aksencer T., Aydogdu M., 2011, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E 43:954-959.
[34] Satish N., Narendar S., Gopalakrishnan S., 2012, Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics, Physica E 44:1950-1962.
[35] Prasanna T., Kumar J., Narendar S., Gopalakrishnan S., 2013, Thermal vibration analysis of monolayer graphene embedded in elastic medium based on nonlocal continuum mechanics, Composite Structures 100: 332-342.
[36] 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.
[37] 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.
[38] 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.
[39] 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.
[40] Ghorbanpour Arani A., Roudbari M.A., 2013, Nonlocal piezoelastic surface effect on the vibration of visco-Pasternak coupled boron nitride nanotube system under a moving nanoparticle, Thin Solid Films 542:232-241.
[41] 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.
[42] 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.
[43] Mohammadi M., Ghayour M., Farajpour A., 2011, Analysis of free vibration sector plate based on elastic medium by using new version of differential quadrature method, Journal of Solid Mechanics in Engineering 3:47-56.
[44] 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.
[45] Bert C. W, Malik M., 1996, Differential quadrature method in computational mechanics:a review, Applied Mechanic Review 49:1-27.
[46] Shu C., Richards Be., 1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier Stokes equations, International Journal for Numerical Methods in Fluids 15:791-798.
[47] 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(3):305-323.
[48] 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 4:128-143.
[49] Mohammadi M., Goodarzi M., Farajpour A., Ghayour M., 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.
[50] Romeo G., Frulla G., 1997, Post-buckling behaviour of graphite/epoxy stiffened panels with initial imperfections subjected to eccentric biaxial compression loading, International Journal of Non-Linear Mechanics 32:1017-1033.
[51] Saadatpour M. M., Azhari M., 1998, The Galerkin method for static analysis of simply supported plates of general shape, Computers and Structures 69:1-9.
[52] Babaei 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.
[53] 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(2):116-132.
[54] Bassilya S. F., Dickinson M., 1972, Buckling and lateral vibration of rectangular plates subject to in-plane loads a Ritz approach, Journal of Sound and Vibration 24:219-239.
[55] Cook I.T., Rockey K.C., 1963, Shear buckling of rectangular plates with mixed boundary conditions, Aeronautical Quarterly 14:349-356.
[56] Bijdiansky B., Connor R.W., 1948, Buckling Stress of Clamped Rectangular Flat Plate in Shear, Langley Memorial Aeronautical Laboratory, Langley Field, Virginia.
[57] Chen Y., Lee J.D., Eskandarian A., 2004, Atomistic viewpoint of the applicability of microcontinuum theories, International Journal of Solids and Structures 41:2085-2097.
[58] Wang L. F., Hu H.Y., 2005, Flexural wave propagation in single-walled carbon nanotubes, Physical Review B 71: 195412-195419.
[59] Duan W.H., Wang C. M., Zhang Y. Y., 2007, Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, Journal of Applied Physics 101: 024305.
[60] Wang Q., Wang C. M., 2007, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology 18: 075702.
[61] Shen L., Shen S. H., Zhang C. L., 2010, Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Computational Materials Science 48:680-685.