On the use of continuum mechanics to estimate the properties of nanotubes

doi:10.1016/S0038-1098(98)00626-7

Solid State Communications

Volume 110, Issue 4, 31 March 1999, Pages 227-230

https://doi.org/10.1016/S0038-1098(98)00626-7 Get rights and content

Abstract

In experimental and theoretical investigations of the properties of nanostructures, the equations of continuum beam theory are often used to interpret the mechanical response of nanotubes. In particular, Bernoulli–Euler beam bending theory is being utilized to infer the Young's Modulus. In this work, we examine the validity of such an approach using a simple elastic sheet model and show that at the nanotube scale the assumptions of continuum mechanics must be carefully respected in order to obtain reasonable results. Relations are derived for pure bending of nanotubes that show the explicit dependence of the “material properties” on system size when a continuum cross-section assumption is made. Two alternate approaches are proposed that provide a more reliable scheme for property extraction from experiments.

Introduction

Recently, experimental methods have allowed for the measurement of the structural motion of nanometer scale objects [1], [2], [3], [4]. In these works the notions of continuum mechanics have been used to infer that carbon nanotubes possess super-high Young's moduli. Other theoretical work using direct atomic simulation [5] has also reached similar conclusions. In [5], for instance, a comparison of bending stiffnesses between a C₂₀₀ single-wall nanotube and an Iridium beam of “similar dimensions” is presented. The bending stiffness of the carbon nanotubes was determined from a simulation of the atomic structure using the Keating potential; the bending stiffness of the Iridium beam was deduced using the continuum Bernoulli–Euler theory of beam bending. This type of comparison or data interpretation, however, must be done with due regard to the appropriate continuum hypothesis being employed—as has been previously acknowledged [4], [5]. In this paper, we show by using a highly idealized elastic sheet model for a nanotube [2], [6] that the reported super-high Young's moduli are direct consequences of the breakdown of the continuum cross-section hypothesis.

Section snippets

Young's Modulus

The Young's Modulus, E, of a material is defined as the ratio of the normal stress, σ=F/A, to the normal strain, ε=Δ/L, in a 1-dimensional tension test; see Fig. 1(a), where F is the applied force, A is the cross-sectional area of the specimen, Δ is the specimen elongation, and L is the specimen length. Thus, $E= σ ε = FL AΔ .$ The definition relies on the continuum hypothesis and is designed such that E is truly a material property. In other words, the definition of E is designed so that regardless of

Pure bending

Consider a beam of length L under pure bending (no transverse shear). The beam is made up of n tubular sheets of atoms that are spaced a distance s apart. The thickness of each sheet is t and the distance to the center of sheet j is denoted R_j. Thus the inner radius of the beam is R₁−t/2 and the outer radius is R_n+t/2. The mean radius of any sheet can be expressed as R_j=R₁+(j−1)(s+t); see Fig. 2. Assume further that L>10R_n so that the Bernoulli–Euler kinematic assumption is valid; i.e. the

Conclusions

In this report, we have derived approximate formulas for the determination of the break down of the continuum cross-section hypothesis. In particular we have shown that for nanobeams in bending, one needs a large number of atomic layers for the validity of the continuum cross-section assumption. Thus the interpretation of experimental data on nanostructures through the reporting of continuum engineering properties needs to be done with careful consideration of the implicit assumptions in

References (6)

M.M.J. Treacy, T.W. Ebbesen, J.M. Gibson, Nature 381, (1996)...
B.I. Yakobson, R.E. Smalley, Am. Sci. 85 (1997)...
N.G. Chopra, A. Zettl, Phys. Rev. B [submitted]...

There are more references available in the full text version of this article.

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