Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM
Introduction
After the invention of carbon nanotubes by Iijima [1], a new world of research is in progress for the proper and accurate analysis of nano-size structures. These nano-size structures exhibit superior physical, mechanical, chemical, electronic and electrical properties [2], [3], [4], [5]. These outstanding properties of CNTs lead to its usage in the emerging field of nanoelectronics, nanodevices, nanocomposites, etc.
Experiments at the nanoscale are extremely difficult and atomistic modeling remains prohibitively expensive for large-sized atomic system. Consequently continuum models continue to play an essential role in the study of carbon nanotubes. Thereby size-dependent continuum-based methods [6], [7], [8] are becoming popular in modeling small sized structures as it offers much faster solutions than molecular dynamics simulations for various engineering problems. Furthermore, size-dependent continuum mechanics is used because at small length scales the material microstructures (such as lattice spacing between individual atoms) become increasingly significant and their influence can no longer be ignored.
The most widely reported theory for analyzing small-scale structures is the nonlocal elasticity theory initiated by Eringen [9], [10]. In nonlocal elasticity theory the small-scale effects are captured by assuming that the stress at a point is a function not only of the strain at that point but also a function of the strains at all other points of the domain.
The importance of nonlocal elasticity theory motivated the scientific community to explore the behavior of micro/nano structures more accurately and easily. The feasibility of nonlocal continuum theory in the field of nanotechnology was first reported by Peddieson et al. [11]. Various works related to nonlocal elasticity theory are found in several references, e.g. [12], [13], [14], [15], [16], [17].
Recently, considerable attention has been turned on the mechanical behavior of single- and multi-walled carbon nanotubes embedded in polymer or metal matrix [18], [19], [20], [21], [22], [23], [24]. Due to axially compressive loads, buckling behavior of the carbon nanotubes combined with the influence of the surrounding elastic medium is of practical importance. Vibration and buckling analyses [25], [26], [27] of CNTs have shown the employment of Winkler-type elastic foundation for modeling continuous surrounding elastic medium. Recently Lee and Chang [28] have used Winkler-type model for vibration analysis of fluid-filled single-walled carbon nanotube (SWCNT) embedded in an elastic medium. Further, Murmu and Pradhan [29] carried out stability analysis of beam surrounded by elastic medium by using nonlocal beam theory and Winkler foundation model. However these works were based on the nonlocal Euler–Bernoulli (NL-EB) beam model.
The Winkler-type elastic foundation is approximated as a series of closely spaced, mutually independent, vertical linear elastic springs. The foundation modulus is represented by stiffness of the springs. However, this model is considered as a crude approximation of the true mechanical behavior of the elastic material. This is due to inability of the model to take into account the continuity or cohesion of the medium. The interaction between the springs is not taken into account in Winkler-like foundations. A more realistic and generalized representation of the elastic foundation can be accomplished by the way of a two-parameter foundation model. One such physical foundation model is the Pasternak-type foundation model [30]. This is also called as two-parameter foundation model. The first parameter of Pasternak foundation model represents the normal pressure while the second parameter accounts for the transverse shear stress due to interaction of shear deformation of the surrounding elastic medium.
Recently, Winkler and Pasternak foundation models were reported by Liew et al. [31] for the vibration analysis of nano-size graphene sheets embedded in polymer matrix. It is clear from literature survey that the use of Pasternak-type model for representing surrounding elastic medium are limited in literature.
In this paper, nonlocal elasticity theory has been implemented to investigate the stability response of single-walled carbon nanotubes embedded in an elastic medium. Both Winkler-type and Pasternak-type models are employed to simulate the interaction of the SWNTs with a surrounding elastic medium. The nonlocal small-scale coefficients get introduced into the nonlocal continuum theory through the constitutive relations. Also derived herein, for the first time, is the single nonlocal Timoshenko buckling equations for CNTS in elastic medium. A differential quadrature (DQ) approach is employed and numerical solutions for the critical buckling loads are obtained. Influences of nonlocal small-scale coefficient, Winkler modulus parameter, Pasternak shear modulus parameter and aspect ratio of the SWCNT on critical loads of the SWCNT are studied and discussed. It is hoped that the present analysis will be useful to researchers and engineers working on carbon nanotubes and CNT-based composites.
Section snippets
Nonlocal elasticity theory
The essence of the nonlocal elasticity theory is that the stress field at a reference point x in an elastic continuum depends not only on strain at that point but also on strains at all other points in the body. This is in accordance with the atomic theory of lattice dynamics and experimental observations on phonon dispersion. The scale effects are accounted for in the theory by considering internal size as a material parameter. The most general form of the constitutive relation for nonlocal
NL-TB model for SWCNT
Based on the Timoshenko beam theory, the displacement field at any point can be written aswhere x is the longitude coordinate, z the coordinate measured from the mid-plane of the beam and ψ the rotation of the cross-section. The terms u and w are the axial and transverse displacements, respectively, of the point (x, 0) on the mid-plane (i.e., z=0) of the beam. The nonzero strains according to Timoshenko beam theory are expressed as
For
Convergence of N-DQ approach
In this paper Eq. (27) is transformed to a DQ-analogous form for obtaining the critical buckling loads for a single-walled carbon nanotube that is embedded in an elastic medium (Fig. 1). The effective properties of SWCNT are taken as those of Reddy and Pang [33]. The Young's modulus E=1000 GPa, mass density ρ=2300 kg/m3, Poisson's ratio ν=0.19 and shear correction factor β=0.877 are considered in the analysis.
Based on Eq. (28) a computer code is developed in MATLAB. Convergence test is performed
Conclusions
Presented herein is the stability analysis of SWCNTs embedded in elastic medium based on Eringen's nonlocal elasticity theory and the Timoshenko beam theory. The nonlocal Timoshenko theory accounts for both the scale effect and the effect of transverse shear deformation, which becomes significant when dealing with CNTs that are short and stocky. Influence of the stiffness of the surrounding elastic medium on critical buckling loads of the CNTs is shown. Both Winkler-type and Pasternak-type
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