Temperature Effect on Vibration Analysis of Annular Graphene Sheet Embedded on Visco-Pasternak Foundati

Journal of Solid Mechanics

مقاله 8، دوره 5، شماره 3، آذر 2013، صفحه 305-323   

نوع مقاله: Research Paper

نویسندگان

M Mohammadi^{* 1}؛ A Farajpour²؛ M Goodarzi¹؛ H Mohammadi³

¹Department of Engineering, Ahvaz Branch, Islamic Azad University

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

³Department of Electrical Engineering, Shahid Chamran University of Ahvaz

چکیده

In this study, the vibration behavior of circular and annular graphene sheet embedded in a Visco-Pasternak foundation and coupled with temperature change and under in-plane pre-load is studied. The single-layered annular graphene sheet is coupled by an enclosing viscoelastic medium which is simulated as a Visco- Pasternak foundation. By using the nonlocal elasticity theory and classical plate theory, the governing equation is derived for single-layered graphene sheets (SLGSs). The closed-form solution for frequency vibration of circular graphene sheets has been obtained and nonlocal parameter, in-plane pre-load, the parameters of elastic medium and temperature change appears into arguments of Bessel functions. To verify the accuracy of the present results, the new version differential quadrature method (DQM) is also developed. Closed-form results are successfully verified with those of the DQM results. The results are subsequently compared with valid result reported in the literature. The effects of the small scale, pre-load, mode number, temperature change, elastic medium and boundary conditions on natural frequencies are investigated. The non-dimensional frequency decreases at high temperature case with increasing the temperature change for all boundary conditions. The effect of temperature change on the non-dimensional frequency vibration becomes the opposite at high temperature case in compression with the low temperature case. The present research work thus reveals that the nonlocal parameter, boundary conditions, temperature change and initial pre-load have significant effects on vibration response of the circular nanoplates. The present analysis results can be used for the design of the next generation of nanodevices that make use of the thermal vibration properties of the graphene.

کلیدواژه‌ها

Vibration؛ Annular graphene sheet؛ Temperature change؛ In-plane pre-load

مراجع

[1] Sorop T.G., Jongh L.J., 2007, Size-dependent anisotropic diamagnetic screening in superconducting nanowires, Physical Review B 75: 014510-014515.

[2] Iijima S., 1991, Helical microtubules of graphitic carbon, Nature 354: 56–58.

[3] Kong X.Y, Ding Y, Yang R, Wang Z.L., 2004, Single-crystal nanorings formed by epitaxial self-coiling of polar nanobelts, Science 303:1348-1351.

[4] Wong E.W., Sheehan P.E., Lieber C.M., 1997, Nanobeam mechanics:elasticity, strength and toughness of nanorods and nanotubes, Science 277:1971–1975.

[5] Zhou S.J., Li Z.Q., 2001, Metabolic response of platynota stultanapupae during and after extended exposure to elevated CO2 and reduced O2 atmospheres, Journal of insect physiology 31:401-409.

[6] Fleck N.A., Hutchinson J.W., 1997, Strain gradient plasticity, Applied Mechanics 33:295–361.

[7] Yang F., Chong A.C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids Structure 39:2731-2743.

[8] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal Applied Physics 54:4703-4711.

[9] Farajpour A., Danesh M., Mohammadi M., 2011, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E 44:719–727.

[10] 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.

[11] Aydogdu M., 2009, Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E 41:861-864.

[12] Babaei H., Shahidi A. R., 2010, Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method, Archive Applied Mechanics 81:1051–1062.

[13] Lu P., Lee H.P., Lu C., Zhang P.Q., 2006, Dynamic properties of flexural beams using a nonlocal elasticity model, Journal Applied Physics 99:073510.

[14] Duan W. H., Wang C. M., 2007, Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology 18:385704.

[15] 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(3): 437-458.

[16] 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 and Structures 11 (4): 659-683.

[17] 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.

[18] 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.

[19] 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.

[20] Wang Q., Wang C.M., 2007, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology 18(7):075702.

[21] 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.

[22] 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.

[23] 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.

[24] Ke L. L., Yang J., Kitiporncahi S., Bradford M. A., Wang Y. S., 2013, Axisymmetric nonlinear free vibration of size-dependent functionally graded annular microplates, Composites: Part B 53:207-217.

[25] Rahmat Talabi M., Saidi A. R., 2013, An explicit exact analytical approach for free vibration of circular/annular functionally graded plates bonded to piezoelectric actuator/sensor layers based on Reddy’s plate theory, Applied Mathematical Modelling 37:7664-7684.

[26] Mohammadi M., Farajpour A., Moradi M., Ghayour M., 2013, Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites Part B 56:629-637.

[27] Civalek Ö., Akgöz B., 2013, Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix, Computational Materials Science 77:295-303.

[28] Murmu T., Pradhan S.C., 2009, Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory, Journal Applied Physics 105:064319-064327.

[29] Pradhan S.C., Phadikar J.K., 2009, Small scale effect on vibration of embedded multi layered graphene sheets based on nonlocal continuum models, Physics Letters A 373:1062–1069.

[30] Wang Y.Z., Li F.M., Kishimoto K., 2011, Thermal effects on vibration properties of doublelayered 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] Aksencer T., Aydogdu M., 2011, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E 43:954 -959.

[33] 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 Structures 93:2083–2089.

[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 Kumar T.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] Paul A., Tipler G. M., 2008, Physics for Scientists and Engineers, Worth Publishers, New York.

[39] Murray B.W., O'mara I., William C., Robert B., Lee P., 1990, Handbook of Semiconductor Silicon Technology, Park Ridge, New Jersey, Noyes Publications.

[40] Jiang H., Liu R., Huang Y., Hwang K.C., 2004, Thermal expansion of single wall carbon nanotube, Journal of Engineering of Material and Technology 126 (3):265-270.

[41] Alamusi Hu N., Jia B., Arai M., Yan C., Li J., Liu Y., Atobe S., Fukunaga H., 2012, Prediction of thermal expansion properties of carbon nanotubes using molecular dynamics simulations, Computational Materials Science 54:249-254.

[42] Yao X., Han Q., 2006, Buckling analysis of multiwalled carbon nanotubes under torsional load coupling with temperature change, Journal of Engineering Materials and Technology 128:419-427.

[43] Ghorbanpour Arani A., Shiravand A., Rahi M., Kolahchi M., 2012, Nonlocal vibration of coupled DLGS systems embedded on visco-pasternak foundation, Physica B 407:4123-4131.

[44] 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.

[45] Ghorbanpour Arani A., Roudbari M. A., Amir S., 2012, Nonlocal vibration of SWBNNT embedded in bundle of CNTs under a moving nanoparticle, Physica B 407:3646-3653.

[46] Wang X., Wang Y., 2004, Free vibration analyses of thin sector plates by the new version of differential quadrature method, Computer Methods in Applied Mechanics and Engineering 193:3957-3971.

[47] 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(2) :47-56.

[48] Wang C.M., Tan V.B.C, Zhang Y.Y., 2006, Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes, Journal of Sound and Vibration 294:1060-1072.

[49] Pradhan S. C., Kumar A., 2011, Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method., Composite Structures 93:774 -779.

[50] Shu C., 2000, Differential Quadrature and Its Application in Engineering, Great Britain Springer.

[51] Wang X., Wang Y., 2004, Re-analysis of free vibration of annular plates by the new version of differential quadrature method, Journal of Sound and Vibration 278:685-689.

[52] Karamooz Ravari M.R., Shahidi A.R., 2012, Axisymmetric buckling of the circular annular nanoplates using finite difference method, Meccanica 47:1-10.

[53] Murmu T., Pradhan S.C., 2009, Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elastici ty and Timoshe nko beam theory and using DQM, Physica E 41:1232 -1239.

[54] Liew K. M., He X. Q., Kitipornchai S., 2006, Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix, Acta Material 54:4229-4236.