Numerical simulation for bending modulus of carbon nanotubes and some explanations for experiment

Abstract The bending mechanical property of carbon nanotubes are numerically investigated in this paper. An advanced finite element analysis package, ABAQUS, is used to simulate the formation of rippling which is the appearance of wavelike distortion on the inner arc of the bent nanotubes, caused by the severe anisotropy of carbon nanotubes and a relatively large deformation. A non-linear bending moment–curvature relationship is obtained, which shows the tangential stiffness greatly decreases when rippling appears. This result can be used to explain the phenomenon and conclusion of the resonant experiment measuring the Young's modulus of carbon nanotubes, in which the Young's modulus calculated using linear theory is found to sharply decrease as the diameter increases [Science 283 (1999) 1513]. Here an analytical method is adopted to conduct a vibration analysis using a bi-linear bending constitution simplifying from the non-linear bending moment–curvature relationship, and the effective Young's modulus have been calculated for multi-walled carbon nanotubes of various sizes. The result carried out in the paper is similar to the measuring result is given by Poncharal et al. [Science 283 (1999) 1513].

[1]  Quanshui Zheng,et al.  Multiwalled carbon nanotubes as gigahertz oscillators. , 2002, Physical review letters.

[2]  W. D. Heer,et al.  Electrostatic deflections and electromechanical resonances of carbon nanotubes , 1999, Science.

[3]  T. Ebbesen,et al.  Exceptionally high Young's modulus observed for individual carbon nanotubes , 1996, Nature.

[4]  J. Z. Liu,et al.  Effect of a rippling mode on resonances of carbon nanotubes. , 2001, Physical review letters.

[5]  David Hui,et al.  The revolutionary creation of new advanced materials - Carbon nanotube composites , 2002 .

[6]  Herbert Shea,et al.  Carbon nanotubes: nanomechanics, manipulation, and electronic devices , 1999 .

[7]  S. Iijima.  Helical microtubules of graphitic carbon , 1991, Nature.

[8]  Erik Dujardin,et al.  Young's modulus of single-walled nanotubes , 1998 .

[9]  J. Bernholc,et al.  Nanomechanics of carbon tubes: Instabilities beyond linear response. , 1996, Physical review letters.

[10]  Honghao Wei.  ADVANCES IN NANOMECHANICS , 2002 .

[11]  Charles M. Lieber,et al.  Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nanorods and Nanotubes , 1997 .

[12]  M. Dresselhaus.  Carbon nanotubes , 1995 .

[13]  A. Maiti,et al.  Structural flexibility of carbon nanotubes , 1996 .

[14]  P. Avouris,et al.  Mechanical Properties of Carbon Nanotubes , 2001 .

[15]  Z. C. Tu,et al.  Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young's moduli dependent on layer number , 2001, cond-mat/0112454.

[16]  K. Lau.  Interfacial bonding characteristics of nanotube/polymer composites , 2003 .