Levy Type Solution for Nonlocal Thermo-Mechanical Vibration of Orthotropic Mono-Layer Graphene Sheet Embedded in an Elastic Medium

Document Type: Research Paper

Authors

1 Department of Engineering, Ahvaz Branch, Islamic Azad University

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

Abstract

In this paper, the effect of the temperature change on the vibration frequency of mono-layer graphene sheet embedded in an elastic medium is studied. Using the nonlocal elasticity theory, the governing equations are derived for single-layered graphene sheets. Using Levy and Navier solutions, analytical frequency equations for single-layered graphene sheets are obtained. Using Levy solution, the frequency equation and mode shapes orthotropic rectangular nanoplate are considered for three cases of boundary conditions. The obtained results are subsequently compared with valid result reported in the literature. The effects of the small scale, temperature change, different boundary conditions, Winkler and Pasternak foundations, material properties and aspect ratios on natural frequencies are investigated. It has been shown that the non-dimensional frequency decreases with increasing temperature change. It is seen from the figure that the influence of nonlocal effect increases with decreasing of the length of nanoplate and also all results at higher length converge to the local frequency. The present analysis results can be used for the design of the next generation of nanodevices that make use of the thermal vibration proper ties of the nanoplates.

Keywords


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

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

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

[4] 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.

[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, Shandong University Technology 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 of Applied Physics 54: 4703-4711.

[9] 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 (1): 23-27.

[10] 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:Low-Dimensional Systems and Nanostructures 43 (10): 1820-1825.

[11] 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 (5): 1605-1615.

[12] Farajpour A., Danesh M., Mohammadi M., 2011, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E: Low-Dimensional Systems and Nanostructures 44(3): 719-727.

[13] 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: Low-Dimensional Systems and Nanostructures 44(1): 135-140.

[14] 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: Engineering 45(1): 32-42.

[15] 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: Engineering 51: 121-129.

[16] 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.

[17] 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: Engineering 56: 629-637.

[18] 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.

[19] 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.

[20] 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 Material Science 52: 510-520.

[21] Mohammadi M., Ghayour M., Farajpour A., 2011, Analysis of free vibration sector plate based on elastic medium by using new version differential quadrature method, Journal of Solid Mechanics in Engineering 3(2): 47-56.

[22] Wang C.M., Duan W.H., 2008, Free vibration of nanorings/arches based on nonlocal elasticity, Journal of Applied Physics 104(1): 014303.

[23] Reddy J.N., Pang S.D., 2008, Nonlocal continuum theories of beams for the analysis of carbon nanotubes,Journal of Applied Physics 103(2): 023511.

[24] Murmu T., Pradhan S. C., 2009, Buckling analysis of single-walled carbon nanotubes embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E 41: 1232-1239.

[25] Wang L., 2009, Dynamical behaviors of double-walled carbon nanotubes conveying fluid accounting for the role of small length scale, Computational Material Science 45: 584-588.

[26] Xiaohu Y., Qiang H., 2007, Investigation of axially compressed buckling of a multi-walled carbon nanotube under temperature field, Composite Science Technology 67: 125-134.

[27] Sudak L.J., 2003, Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics, Journal of Applied Physics 94(11): 7281-7287.

[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 of Applied Physics 105(6): 064319.

[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] 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 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] Chen Y., Lee J.D., Eskandarian A., 2004, Atomistic viewpoint of the applicability of microcontinuum theories, International Journal of Solids Structures 41:2085-2097.

[37] Sakhaee-Pour A., Ahmadian M.T., Naghdabadi R., 2008, Vibrational analysis of single layered graphene sheets, Nanotechnology 19: 957–964.

[38] Pradhan S.C., Kumar A., 2010, Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method, Computational Material Science 50:239-245.

[39] 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.

[40] Zhang Y.Q., Liu X., Liu G.R., 2007, Thermal effect on transverse vibrations of double walled carbon nanotubes, Nanotechnology 18(44):445701.

[41] Benzair A., Tounsi A., Besseghier A., Heireche H., Moulay N., Boumia L., 2008, The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory, Journal of Applied Physics 41(22):225404.

[42] Lee H.L., Chang W.J., 2009, A closed-form solution for critical buckling temperature of a single-walled carbon nanotube, Physica E 41:1492–1494.

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