Nonlocal continuum models for carbon nanotubes subjected to static loading

Abstract

Static and buckling analyses of carbon nanotubes (CNTs) are carried out with newly developed nonlocal continuum models. Small-scale effects are explicitly derived for bending deformation solutions for CNTs subjected to general flexural loading first. Solutions via nonlocal continuum models are expressed by simple terms related to scale coefficients in addition to remaining terms via local continuum models in which the simplicity of the nonlocal continuum models is clearly observed. Discussions on various derivations of Young's modulus for CNTs from existing experimental work in the literature are provided, revealing the applicability of the nonlocal continuum models. In addition, a simple equation for the buckling load of CNTs with various general boundary conditions subject to axial loading via the nonlocal elastic beam model is explicitly derived for instability analysis. The results of this research provide benchmark solutions for the response of CNTs subject to general static loading, with small-scale effects modeled and revealed. Thus, the work has great potential in studying mechanical properties of CNTs of various sizes.

Variational Principles for Multiwalled Carbon Nanotubes Undergoing Vibrations Based on Nonlocal Timoshenko Beam Theory

“…, n [7][8][9][10]28]. The parameter η = e 0 a appears in the nonlocal theory of beams and helps define the small scale effects accurately where e 0 is a constant for adjusting the model by experimental results and a is an internal characteristic length [17][18][19][20][21][22][23][24][25][26].…”

“…Having (18)- (19), (20)- (21), and (22)-(23) with the same right-hand sides ensures that the variational principle can be derived for the present problem. From (17), it follows that…”

“…Beam theories capable of taking the small scale effects into account are based on the nonlocal theory of elasticity which was developed in early seventies [16,17]. The nonlocal theory was applied to the study of nanoscale Euler-Bernoulli and Timoshenko beams in a number of papers [18][19][20][21][22][23][24][25][26][27]. Variational formulations for various nonlocal beam models were given in [23].…”

A Nonlocal Elasticity Approach for the In-Plane Static Analysis of Nanoarches

“…Application of nonlocal elasticity for the formulation of nonlocal version of the Euler-Bernoulli beam model is initially proposed by Peddieson et al [2]. Since then, the nonlocal theory, including nano-beam, plate and shell models were successfully developed using nonlocal continuum mechanics and many researchers reported on bending, vibration, buckling and wave propagation of nonlocal nanostructures [3][4][5]. Most of these studies focused on straight beam formulation, however, it is known that these structures might not be perfectly straight [6].…”

A new displacement-based framework for non-local Timoshenko beams

“…For instance, Eringen's integral theory has been used by Peddieson et al [30], Wang and Shindo [31], Civalek and Demir [32] to derive non-local EB beam models, by Wang and Liew [33], Wang and et al [34] to derive non-local TM beam models, the latter in conjunction with the principle of virtual work. A non-local EB beam model has been presented by Challamel and Wang [35] based on a gradient elastic model and a non-local integral elastic model, where the constitutive relation is expressed by combining local and nonlocal curvatures.…”