Atomistic-continuum and ab initio estimation of the elastic moduli of single-walled carbon nanotubes - Chandraseker, Karthick; Mukherjee, Subrata (Computational materials science, 2007)
This paper deals with the calculation of elastic moduli and stress–strain curves for single-walled carbon nanotubes (SWNTs) using a computationally efficient, atomistically enriched continuum analysis. This approach is adopted to estimate shear and Young’s moduli and obtain stress–strain curves for carbon nanotubes (CNTs) subject to coupled extension and twist deformations. This accounts for the effects of natural extension-twist coupling [K. Chandraseker, S. Mukherjee, ASME J. Appl. Mech. 73 (2) (2006) 315–326; K. Chandraseker, S. Mukherjee, Y.X. Mukherjee, Int. J. Solids Struct. 43 (2006) 7128–7144] on SWNT constitutive properties. The constitutive properties are evaluated by assuming a cylindrical reference configuration [Chandraseker and Mukherjee, 2006; Chandraseker et al., 2006] rather than a planar graphene sheet [P. Zhang, Y. Huang, P.H. Geubelle, P.A. Klein, K.C. Hwang, Int. J. Solids Struct. 39 (2002) 3893–3906; M. Arroyo, T. Belytschko, Phys. Rev. B 69 (2004) 115415] thereby allowing for the anisotropy and change in strain energy that results from the finite deformation required to roll up a graphene sheet into a nanotube [M. Arroyo, T. Belytschko, Phys. Rev. B 69 (2004) 115415]. The Tersoff–Brenner multi-body empirical interatomic potential for carbon [J. Tersoff, Phys. Rev. B 37 (1988) 6991–7000; D.W. Brenner, Phys. Rev. B 42 (1990) 9458–9471] is used to model the C–C bond energies in this work. This enables exact analytic evaluation of the derivatives of the strain energy density rather than a numerical approach. Consistent values obtained corresponding to these material properties indicate that they do not depend strongly on the chirality of the nanotube [R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998 [7]]. The relative magnitudes of the Young’s and shear moduli obtained from this approach fall within the well known range in classical elasticity theory in most cases, and the computed values for the moduli agree well with existing experimental results and atomistic studies that employ the same interatomic potential. Further, in the present work, the moduli are also evaluated using a more accurate, albeit computationally expensive, ab initio density-functional-theoretic (DFT) approach (see for e.g., [D. Sáncez-Portal, E. Artacho, J.M. Soler, A. Rubio, P. Ordejón, Phys. Rev. B 59 (1999) 12678; K.N. Kudin, G.E. Scuseria, B.I. Yakobson, Phys. Rev. B 64 (2001) 235406]). A comparison between these values and the ones from the atomistic-continuum analysis brings to notice some of the advantages and limitations of both these approaches.