Abstract
Several novel models, based on the nonlocal stress theory, on the lateral buckling of two- and three-dimensional (2D and 3D) ensembles of vertically aligned single-walled carbon nanotubes (SWCNTs) are developed. The existing van der Waals forces between the constitutive atoms of the neighboring SWCNTs are modeled by an elastic layer, and each individual SWCNT is modeled using the nonlocal Rayleigh beam theory. The governing equations for lateral buckling of both 2D and 3D SWCNTs ensembles due to axial compressive loads are derived. These are called discrete models since the lateral deformations of each SWCNT only express in terms of the longitudinal coordinate of the SWCNT. It will be revealed that the lateral deformation of SWCNTs can be also stated in terms of two and three coordinates for 2D and 3D ensembles of SWCNTs, respectively. The resulting models are called continuous models. Based on both discrete and continuous models, the critical axial buckling loads for both 2D and 3D ensembles of SWCNTs for a special case, when the outer SWCNTs are prohibited from any lateral movement, are obtained. A reasonably good agreement between the results of the discrete model and those of the continuous model is reported. Subsequently, the roles of small-scale parameter, intertube distance, and number of SWCNTs on the critical buckling loads are examined. The obtained results and the proposed models in the present work would be very helpful in design and fabrication of 2D and 3D groups of vertically aligned SWCNTs, particularly those whose main duties are to transfer securely the applied axial forces.
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References
Iijima S.: Helical microtubules of graphitic carbon. Nature 354, 56 (1991)
Saito R., Dresselhaus G., Dresselhaus M.: Physical Properties of Carbon Nanotubes. Imperial College Press, London (1998)
Thostenson E.T., Ren Z., Chou T.W.: Advances in the science and technology of carbon nanotubes and their composites: a review. Compos. Sci. Technol. 61, 1899–1912 (2001)
Coleman J.N., Khan U., Blau W.J., Gunko Y.K.: Small but strong: a review of the mechanical properties of carbon nanotube–polymer composites. Carbon 44, 1624–1652 (2006)
Ru C.Q.: Axially compressed buckling of a double-walled carbon nanotube embedded in an elastic medium. J. Mech. Phys. Solids 49, 1265–1279 (2001)
He X.Q., Kitipornchai S., Liew K.M.: Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. J. Mech. Phys. Solids 53, 303–326 (2005)
Xiaohu Y., Qiang H.: Investigation of axially compressed buckling of a multi-walled carbon nanotube under temperature field. Compos. Sci. Technol. 67, 125–134 (2007)
Yao X., Han Q.: The thermal effect on axially compressed buckling of a double-walled carbon nanotube. Euro. J. Mech. A/Solids 26, 298–312 (2007)
Yao X., Han Q., Xin H.: Bending buckling behaviors of single- and multi-walled carbon nanotubes. Comput. Mater. Sci. 43, 579–590 (2008)
Adali S.: Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory. Phys. Lett. A 372, 5701–5705 (2008)
Lee H.L., Chang W.J.: A closed-form solution for critical buckling temperature of a single-walled carbon nanotube. Phys. E 41, 1492–1494 (2009)
Yan Y., Wang W.Q., Zhang L.X.: Nonlocal effect on axially compressed buckling of triple-walled carbon nanotubes under temperature field. Appl. Math. Model. 34, 3422–3429 (2010)
Chan Y., Thamwattana N., Hill J.M.: Axial buckling of multi-walled carbon nanotubes and nanopeapods. Euro. J. Mech. A/Solids 30, 794–806 (2011)
Pradhan S.C., Reddy G.K.: Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM. Comput. Mater. Sci. 50, 1052–1056 (2011)
Eringen A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)
Eringen A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)
Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Eringen A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
Wang Q., Varadan V.K.: Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater. Struct. 15, 659–666 (2006)
Hu Y.G., Liew K.M., Wang Q.: Nonlocal elastic beam models for flexural wave propagation in double-walled carbon nanotubes. J. Appl. Phys. 106, 044301 (2009)
Kiani K.: A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect. Int. J. Mech. Sci. 52, 1343–1356 (2010)
Kiani K.: Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subjected to axial load using nonlocal shear deformable beam theories. Int. J. Mech. Sci. 68, 16–34 (2013)
Kiani, K.: Longitudinal, transverse, and torsional vibrations and stabilities of axially moving single-walled carbon nanotubes. Curr. Appl. Phys. 13, 1651–1660 (2013)
Kiani K., Mehri B.: Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. J. Sound Vib. 329, 2241–2264 (2010)
Kiani K.: Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part I: theoretical formulations. Acta Mech. 216, 165–195 (2011)
Kiani K.: Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part II: parametric study. Acta Mech. 216, 197–206 (2011)
Kiani K.: Longitudinal and transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effects. Phys. E 42, 2391–2401 (2010)
Wang L., Ni Q., Li M.: Buckling instability of double-wall carbon nanotubes conveying fluid. Comput. Mater. Sci. 44, 821–825 (2008)
Yan Y., Wang W.Q., Zhang L.X.: Dynamical behaviors of fluid-conveyed multi-walled carbon nanotubes. Appl. Math. Model. 33, 1430–1440 (2009)
Rasekh M., Khadem S.E.: Nonlinear vibration and stability analysis of axially loaded embedded carbon nanotubes conveying fluid. J. Phys. D Appl. Phys. 42, 135112 (2009)
Kiani K.: Vibration behavior of simply supported inclined single-walled carbon nanotubes conveying viscous fluidsflow using nonlocal Rayleigh beam model. Appl. Math. Model. 37, 1836–1850 (2013)
Chowdhury R., Adhikari S., Mitchell J.: Vibrating carbon nanotube based bio-sensors. Phys. E 42, 104–109 (2009)
Georgantzinos S.K., Anifantis N.K.: Carbon nanotube-based resonant nanomechanical sensors: a computational investigation of their behavior. Phys. E 42, 1795–1801 (2010)
Arash B., Wang Q., Varadan V.J.: Carbon nanotube-based sensors for detection of gas atoms. ASME J. Nanotechnol. Eng. Med. 2, 021010 (2011)
Kiani K., Ghaffari H., Mehri B.: Application of elastically supported single-walled carbon nanotubes for sensing arbitrarily attached nano-objects. Curr. Appl. Phys. 13, 107120 (2013)
Wang H., Dong K., Men F., Yan Y.J., Wang X.: Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix. Appl. Math. Model. 34, 878–889 (2010)
Wang X., Shen J.X., Liu Y., Shen G.G., Lu G.: Rigorous van der Waals effect on vibration characteristics of multi-walled carbon nanotubes under a transverse magnetic field. Appl. Math. Model. 36, 648–656 (2012)
Kiani K.: Transverse wave propagation in elastically confined single-walled carbon nanotubes subjected to longitudinal magnetic fields using nonlocal elasticity models. Phys. E 45, 86–96 (2012)
Arash B., Wang Q.: A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012)
Duan W.H., Wang C.M., Zhang Y.Y.: Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. J. Appl. Phys. 101, 024305 (2007)
Sundararaghavan V., Waas A.: Non-local continuum modeling of carbon nanotubes: physical interpretation of non-local kernels using atomistic simulations. J. Mech. Phys. Solids 59, 1191–1203 (2011)
Lennard-Jones J.E.: The determination of molecular fields: from the variation of the viscosity of a gas with temperature. Proc. R. Soc. Lond. Ser. A 106, 441–462 (1924)
Girifalco L.A., Lad R.A.: Energy of cohesion, compressibility and the potential energy function of graphite system. J. Chem. Phys. 25, 693–697 (1956)
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Kiani, K. Axial buckling analysis of vertically aligned ensembles of single-walled carbon nanotubes using nonlocal discrete and continuous models. Acta Mech 225, 3569–3589 (2014). https://doi.org/10.1007/s00707-014-1107-3
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DOI: https://doi.org/10.1007/s00707-014-1107-3