Critical Buckling Load of Carbon Nanotube with Non-Local Timoshenko Beam Using the Differential Transform Method
In this paper, the Differential Transform Method (DTM) is employed to predict and to analysis the non-local critical buckling loads of carbon nanotubes with various end conditions and the non-local Timoshenko beam described by single differential equation. The equation differential of buckling of the nanobeams is derived via a non-local theory and the solution for non-local critical buckling loads is finding by the DTM. The DTM is introduced briefly. It can easily be applied to linear or nonlinear problems and it reduces the size of computational work. Inﬂuence of boundary conditions, the chirality of carbon nanotube and aspect ratio on non-local critical buckling loads are studied and discussed. Effects of nonlocal parameter, ratios L/d, the chirality of single-walled carbon nanotube, as well as the boundary conditions on buckling of CNT are investigated.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1317114Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 436
 Dresselhaus, M.S. and Avouris, P. “Carbon nanotubes: synthesis, structure, properties and application”, Top Appl Phys, 80, 1–11(2001),
 Ebrahimi, F., Reza, G. S., and Boreiry, M. “An investigation into the influence of thermal loading and surface effects on mechanical characteristics of nanotubes”, Structural Engineering and Mechanics, An Int'l Journal Vol. 57 No. 1(2016),
 Zidour, M., Hadji, L., Bouazza, M., Tounsi, A. and Adda Bedia, El A. “The mechanical properties of Zigzag carbon nanotube using the energy‒equivalent model”, Journal of Chemistry and Materials Research, Vol.3, 9–14 (2015),
 Hajnayeb, A. and Khadem, S.E. “An analytical study on the nonlinear vibration of a doublewalled carbon nanotube”, Structural Engineering and Mechanics, An Int'l Journal Vol. 54 No. 5(2015),
 S. Iijima, “Helical microtubules of graphitic carbon,” Nature, 354, 56–58 (1991).
 S. Iijima and T. Ichihashi, “Single-shell carbon nanotubes of 1 nm diameter,” Nature, 363, 603 (1993).
 Tagrara et al “On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams”, Steel and Composite Structures, Vol. 19, No. 5 1259-1277 (2015).
 Dai, H., Hafner, J.H., Rinzler, A.G., Colbert, D.T. and Smalley R.E “Nanotubes as nanoprobes in scanning probe microscopy”, Nature, 384, 147–50. (1996),
 Bouazza, M., Amara, K., Zidour, M., Tounsi, A., Adda Bedia El A. “Postbuckling analysis of nanobeams using trigonometric Shear deformation theory”, Applied Science Reports, 10(2), 112-121(2015).
 E. W. Wong, P. E. Sheehan, and C. M. Lieber, (1997). “Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotubes,” Science, 277, 1971-1975.
 S. Govindjee and J. L. Sackman, (1999). “On the use of continuum mechanics to estimate the properties of nanotubes,” Solid State Communications, 110, 227- 230.
 C. Q. Ru, (2001). “Axially compressed buckling of a doublewalled carbon nanotube embedded in an elastic medium,” J. Mech. Phys. Solids, 49, 1265-1279.
 C. Q. Ru, (2000). “Column buckling of multiwalled carbon nanotubes with interlayer radial displacements,” Physical Review B, 62, 16962-16967
 D. Qian, W. K. Liu, and R. S. Ruoff, (2001). “Mechanics of C60 in nanotubes,” J. Phys. Chem. B, 105, 10753-10758.
 Xie, G.Q., Han, X., Liu, G.R., Long, S.Y. “Effect of small size-scale on the radial buckling pressure of a simply supported multiwalled carbon nanotube”, Smart Mater. Struct., 15, 1143–1149 (2006).
 Arani, A. G., Cheraghbak, A. and Kolahchi, R. “Dynamic buckling of FGM viscoelastic nano-plates resting on orthotropic elastic medium based on sinusoidal shear deformation theory”, Structural Engineering and Mechanics, An Int'l Journal Vol. 60 No. 3(2016).
 Barati, M. R., and Shahverdi, H. “A four-variable plate theory for thermal vibration of embedded FG nanoplates under non-uniform temperature distributions with different boundary conditions”, Structural Engineering and Mechanics, An Int'l Journal Vol. 60 No. 4. (2016).
 Pradhan, S. C. and Phadikar, J. K “Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory”, Structural Engineering and Mechanics, An Int'l Journal Vol. 33 No. 2. (2009).
 Akbarov, S. D., Guliyev, H. H. and Yahnioglu, N. “Natural vibration of the three-layered solid sphere with middle layer made of FGM: three-dimensional approach”, Structural Engineering and Mechanics, An Int'l Journal Vol. 57 No. 2. (2016).
 Ait Yahia, S., Ait Atmane, H., Houari, M.S.A., Tounsi, A. “Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories”, Structural Engineering and Mechanics, 53(6), 1143 – 1165. (2015).
 Pairod, S. and Thanawut, W., “Vibration analysis of laminated plates with various boundary conditions using extended Kantorovich method”, Structural Engineering and Mechanics, An Int'l Journal Vol. 52 No. 1. (2014).
 T. Bensattalah, T. H. Daouadji, M. Zidour, A. Tounsi, E. A. Adda Bedia” investigation of thermal and chirality effects on vibration of single-walled carbon nanotubes embedded in a polymeric matrix using nonlocal elasticity theories” Mechanics of Composite Materials, Vol. 52, No. 4, (2016).
 M. Zidour, K. H. Benrahou, A. Semmah, M. Naceri, H. A. Belhadj, K. Bakhti, and A. Tounsi, “The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory,” Comput. Mater. Sci.,51, 252-260 (2012).
 Sudak, L. J. “Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics”, Journal of Applied Physics, 94, 7281 (2003).
 Sears, C., Batra, A. R. C. “Buckling of carbon nanotubes under axial compression”, Phys. Rev. B., 73, 085410 (2006).
 Semmah, A., tounsi, A., zidour, M., heireche H. and naceri M “Effect of the Chirality on Critical Buckling Temperature of Zigzag Single-walled Carbon Nanotubes Using the Nonlocal Continuum Theory”, Fullerenes, Nanotubes and Carbon Nanostructures, 23, 518–522. (2014).
 Ranjbartoreh, A. R., Wang, G. X., Ghorbanpour Arani, A., and Loghman, A. “Comparative consideration of axial stability of single- and double-walled carbon nanotube and its inner and outer tubes”, Physica E, 41, 202–208(2008).
 Kocaturk, T., and Doguscan, S. A. “Post-buckling analysis of Timoshenko beams with various boundary conditions under non-uniform thermal loading”, Structural Engineering and Mechanics, An Int'l Journal Vol. 40 No. 3(2011).
 Zhou, J.K. (1986), “differential transformation and its Aplication for Electrcial Circuits”, Huazhong University Pres,Wuhan, Chine,.
 Abdel-Halim Hassan, I. H., 2002a. Different applications for the differential transformation in the differential equations. Applied Mathematics and Computation 129: 183-201.
 Abdel-Halim Hassan, I.H., 2002b. On solving same eigenvalue problems by using a differential transformation. Applied Mathematics and Computation 127: 1-22.
 Ayaz, F., 2004. Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation 147: 547-567.
 Arikoglu, A. & Ozkol, I., 2006b. Solution of difference equations by using differential transform method. applied Mathematics and Computation 174: 1216-1228.
 Chen, C. K. and Ju, S. S. (2004), “Application of the differential transform method to a non-linear conservative system”, Applied Mathematics and Computation, 154, 431-441.
 Bao, W. X., Zhu, Ch.Ch. & Cui, W. Zh. 2004. Simulation of Young’s modulus of single-walled carbon nanotubes by molecular dynamics. Physica B. 352: 156–163.
 Liu, J. Z., Zheng, Q. S. & Jiang, Q. 2001. effect of a rippling mode on resonances of carbon nanotubes. Phys. Rev. Lett, 86: 4843.
 Tombler, T. W., Zhou, C. W., Alexseyev, L. et al. 2000. Reversible nanotube electro-mechanical characteristics under local probe manipulation. Nature, 405, 769.
 Tokio, Y., 1995. Recent development of carbon nanotube. Synth Met, 70: 1511-8.
 Wang, C. M., Reddy, J. N. & Lee, K. H. 2000. Shear Deformable Beams and Plates: Relationships with Classical Solutions. (Oxford: Elsevier).
 Eringen, A.C. 1983. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54: 4703–4710.
 Larbi Chaht, F., Kaci, A., Houari, M.S.A., Tounsi, A., Anwar Bég, O., & Mahmoud, S.R. 2015. Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect. Steel and Composite Structures, 18(2): 425 – 442.
 Zhao, Li., Zhu, J. & Xiao, D.W. 2016. Exact analysis of bi-directional functionally graded beams with arbitrary boundary conditions via the symplectic approach. Structural Engineering and Mechanics, An Int'l Journal 59 (1).