Buckling Analysis of a Double-Walled Carbon Nanotube Embedded in an Elastic Medium Using the Energy Method
Subject Areas : EngineeringA Ghorbanpour Arani 1 , M Shokravi 2 , M Mohammadimehr 3
1 - Department of Mechanical Engineering, University of Kashan
2 - Department of Mechanical Engineering, University of Kashan
3 - Department of Mechanical Engineering, University of Kashan
Keywords: Buckling, DWCNT, Elastic medium, Energy method,
Abstract :
The axially compressed buckling of a double-walled carbon nanotabe surrounded by an elastic medium using the energy and the Rayleigh-Ritz methods is investigated in this paper. In this research, based on the elastic shell models at nano scale, the effects of the van der Waals forces between the inner and the outer tubes, the small scale and the surrounding elastic medium on the critical buckling load are considered. Normal stresses at the outer tube medium interface are also included in the current analysis. An expression is derived relating the external pressure to the buckling mode number, from which the critical pressure can be obtained. It is seen from the results that the critical pressure is dependent on the outer radius to thickness ratio, the material parameters of the surrounding elastic medium such as Young’s modulus and Poisson’s ratio. Moreover, it is shown that the critical pressure descend very quickly with increasing the half axial wave numbers.
[1] Yakobson B.I., Brabec C.J., Bernholc J., 1996, Nanomechanics of Carbon Tubes: Instabilities beyond Linear Response, Physical Review Letters 76: 2511-2514.
[2] He X.Q., Kitipornchai S., Liew K.M., 2005, Buckling analysis of multi-walled carbon nanotubes: A continuum model accounting for van der Waals interaction, Journal of the Mechanics and Physics of Solids 53: 303-326.
[3] Ru C.Q., 2000, Effect of van der Waals forces on axial buckling of a double-walled carbon nanotube, Journal of Applied physics 87: 1712-1715.
[4] Han Q., Lu G., 2003, Torsional buckling of a DWCNT embedded in an elastic medium, European Journal of Mechanics A/Solids 22: 875-883.
[5] Donnell L.H., 1976, Beams, Plates, Shells, McGraw-Hill, New York.
[6] Ru C.Q., 2001, Axially compressed buckling of a double-walled carbon nanotube embedded in an elastic medium, Journal of the Mechanics and Physics of Solids 49: 1265-1279.
[7] Ranjbartoreh A.R., Ghorbanpour A., Soltani B., 2007, Double-walled carbon nanotube with surrounding elastic medium under axial pressure, Physica E 39: 230-239.
[8] Ghorbanpour Arani A., Rahmani R., Arefmanesh A., Golabi S., 2008, Buckling analysis of multi-walled carbon nanotubes under combined loading considering the effect of small length scale, Journal of Mechanical Science and Technology 22: 429-439.
[9] Ghorbanpour Arani A., Mohammadimehr M., Arefmanesh A., Ghasemi A., 2010, Transverse vibration of short carbon nanotube using cylindrical shell and beam models, Proc. IMechE Part C: Journal of Mechanical Engineering Science 224: doi: 10.1243/09544062JMES1659.
[10] Brush D.O., Almroth B.O., 1975, Buckling of Bars, Plates and Shells, Mc Graw-Hill, Singapore.
[11] Saito R., Matsuo R., Kimura T., Dresselhaus G., Dresselhaus M.S., 2001, Anomalous potential barrier of double-wall carbon nanotube, Chemical Physics Letters 348: 187-193.
[12] Girifalco L.A., Lad R.A., 1956, Energy of cohesion, compressibility, and the potential energy functions of the graphite system, Journal of Chemical Physics 25: 693-697.
[13] 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.
[14] Zhou X., Zhou J.J., Qu-Yang Z.C., 2000, Strain energy and Young’s modulus of single-wall carbon nanotubes calculated from electronic energy-band theory, Physical Review B 62: doi: 10.1103/PhysRevB.62.13692.
[15] Tu Z.C., Qu-Yang Z.C., 2002, Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young’s moduli dependent on larger number, Physical Review B 65: doi: 10.1103/PhysRevB.65.233407.
[16] Kudin K.N., Scuseria G.E., Yakobson B.I., 2001, C2F, BN and C nanoshell elasticity from abinitio computations, Physical Review B 64: doi: 10.1103/PhysRevB.64.235406.
[17] Fok S.L., 2002, Analysis of the buckling of long cylindrical shells imbedded in an elastic medium using the energy method, Journal of Strain Analysis for Engineering Design 37(5): 375-383.
[18] Zhang Y.Q., Liu G.R., Qiang H.F., Li G.Y., 2006, Investigation of buckling of buckling of double-walled carbon nanitubes embedded in an elastic medium using the energy method, International Journal of Mechanical Sciences 48: 53-61.