Effective bending stiffness of carbon nanotubes

Ru, C. Q.

doi:10.1103/physrevb.62.9973

Cited by 385 publications

(204 citation statements)

References 22 publications

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“…Due to the aspect ratio and tube-like geometry of SWNT, many studies have been conducted concerning the buckling and bending response of nanotubes using both theoretical [12,15,[17][18][19][20] and experimental [21,22] approaches. In particular, Overney et al [17] conducted a computational study and calculated a bending parameter of a graphene sheet based on the vibrational modes of a nanotube.…”

Section: Bending Rigiditymentioning

confidence: 99%

“…It seems logical that these equations could describe the bending properties of graphene sheets and nanotubes due to geometric similarities to solid plates and/or beams. However, Ru [20] has pointed out that a discrepancy exists in the use of equation (1) when describing bending properties of carbon nanotubes. If the bending rigidity described by this equation is used in the bending analysis of carbon nanotubes, then an equivalent wall thickness must be used that is very small compared to the inter-planar spacing of graphene sheets.…”

Section: Bending Rigiditymentioning

confidence: 99%

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Equivalent-continuum modeling of nano-structured materials

Odegard

2002

Composites Science and Technology

577

302

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A method has been proposed for developing structure-property relationships of nano-structured materials. This method serves as a link between computational chemistry and solid mechanics by substituting discrete molecular structures with equivalent-continuum models. It has been shown that this substitution may be accomplished by equating the molecular potential energy of a nano-structured material with the strain energy of representative truss and continuum models. As important examples with direct application to the development and characterization of singlewalled carbon nanotubes and the design of nanotube-based structural devices, the modeling technique has been applied to two independent examples: the determination of the effectivecontinuum geometry and bending rigidity of a graphene sheet. A representative volume element of the chemical structure of graphene has been substituted with equivalent-truss and equivalentcontinuum models. As a result, an effective thickness of the continuum model has been determined. The determined effective thickness is significantly larger than the inter-planar spacing of graphite. The effective bending rigidity of the equivalent-continuum model of a graphene sheet was determined by equating the molecular potential energy of the molecular model of a graphene sheet subjected to cylindrical bending (to form a nanotube) with the strain energy of an equivalent-continuum plate subjected to cylindrical bending.

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Section: Bending Rigiditymentioning

confidence: 99%

Section: Bending Rigiditymentioning

confidence: 99%

Equivalent-continuum modeling of nano-structured materials

Odegard

2002

Composites Science and Technology

577

302

View full text Add to dashboard Cite

show abstract

“…Ru [12][13][14][15] has conducted extensive research on CNT by using the elastic shell model. His contribution lies in redefining some mechanical parameters' relationship for their applications in CNT modeling and analysis.…”

Section: Introductionmentioning

confidence: 99%

Buckling of double-walled carbon nanotubes modeled by solid shell elements

Wang

Zhang

et al. 2006

Journal of Applied Physics

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A solid shell element model is proposed for the elastic bifurcation buckling analysis of double-walled carbon nanotubes ͑DWCNTs͒ under axial compression. The solid shell element allows for the effect of transverse shear deformation which becomes significant in a stocky DWCNT with relatively small radius-to-thickness ratio. The van der Waals ͑vdW͒ interaction between the adjacent walls is simulated by linear springs. Using this solid shell element model, the critical buckling strains of DWCNTs with various boundary conditions are obtained and compared with molecular dynamics results and those obtained by other existing shell and beam models. The results obtained show that the solid shell element is able to model DWCNTs rather well, with the appropriate choice of Young's modulus, tube thickness, and spring constant for modeling the vdW forces.

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“…The continuum shell model could predicate all of the buckling patterns of the single-walled CNT that were displayed by their molecular-dynamics simulations. Therefore, Ru [10][11][12][13][14] applied the classical cylindrical shell theory to the buckling and vibration analyses of double-walled CNTs. He proposed a linear proportional relationship between the variation of the van der Waals ͑vdW͒ force and the normal deflection to model the vdW interaction, and obtained a simple relationship for the vdW interaction coefficient as c = ͑200 ergs/ cm 2 ͒ / ͑0.16d 2 ͒.…”

Section: Introductionmentioning

confidence: 99%

Buckling analysis of triple-walled carbon nanotubes embedded in an elastic matrix

Kitipornchai

Liew

2005

Journal of Applied Physics

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Explicit formulas are obtained to describe the buckling behavior of triple-walled carbon nanotubes (CNTs) that are embedded in an elastic matrix, with van der Waals (vdW) interaction taken into consideration. The investigation is based on the continuum shell theory in which the individual tube is treated as a cylindrical shell. The elastic matrix surrounding the outermost tube is modeled as a Pasternak foundation to account for not only the normal stress, but also the shear stress between the outermost tube and the surrounding matrix. Numerical analyses are carried out to estimate the critical buckling load of triple-walled CNTs, and the results indicate that the critical buckling loads approach a constant of around 0.26N∕m with the increase of the innermost radii regardless of whether or not the triple-walled CNT is embedded in an elastic matrix. The effects of vdW interactions before and after buckling on the critical buckling load are also examined for triple-walled CNTs with various innermost radii.

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Effective bending stiffness of carbon nanotubes

Cited by 385 publications

References 22 publications

Equivalent-continuum modeling of nano-structured materials

Equivalent-continuum modeling of nano-structured materials

Buckling of double-walled carbon nanotubes modeled by solid shell elements

Buckling analysis of triple-walled carbon nanotubes embedded in an elastic matrix

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