Structural dependence of nonlinear elastic properties for carbon nanotubes using a continuum analysis

Based on molecular mechanics coupled with the atomistic-based continuum theory, a structural mechanics approach is presented to examine the nonlinear elastic properties of carbon nanotubes (CNTs) subjected to large axial deformations. According to molecular mechanics, the interaction force between atoms is modeled using the Morse potential. The nanoscale continuum theory is established to directly incorporate the Morse potential function into the constitutive model of CNTs. In this paper, we simulate and examine the influence of CNT structures on the stress–strain response. The linear elastic property of CNTs is independent of the helicity of the hexagonal carbon lattice along the tubes, while their nonlinear elastic behavior shows a larger chirality dependence. The present theoretical approach supplies a set of very simple formulas and is able to serve as a good approximation of the mechanical properties of CNTs. PACS 62.20.-x; 62.20.Dc; 62.25.+g     

Nonlinear structural mechanics based modeling of carbon nanotube deformation

A nonlinear structural mechanics based approach for modeling the structure and the deformation of single-wall and multiwall carbon nanotubes (CNTs) is presented. Individual tubes are modeled using shell finite elements, where a specific pairing of elastic properties and mechanical thickness of the tube wall is identified to enable successful modeling with shell theory. The effects of van der Waals forces are simulated with special interaction elements. This new CNT modeling approach is verified by comparison with molecular dynamics simulations and high-resolution micrographs available in the literature. The mechanics of wrinkling of multiwall CNTs are studied, demonstrating the role of the multiwalled shell structure and interwall van der Waals interactions in governing buckling and postbuckling behavior.     

Vibrational analysis of carbon nanotubes using molecular mechanics and artificial neural network

This paper presents the molecular mechanics based finite element modeling of carbon nanotubes (CNTs) and their applications as mass sensors. The beam element with elastic behavior is considered as the bond between the carbon atoms and its properties are obtained using equating continuum and molecular characteristics. The first five natural frequencies of CNTs in cantilever and doubly clamped boundary conditions (BCs) and their corresponding mode shapes are studied in detail. Furthermore, a multilayer perceptron neural network is used to predict the fundamental vibration frequencies of the CNTs with different diameters and lengths. In addition, variations of the natural frequencies of the CNTs with distorted cross sections are investigated. Moreover, the effects of some attached masses with various values on the first three natural frequencies of a considered CNT are studied here.     

Effects of Radius and Orientation of Single-Walled Carbon Nanotubes on Their Nonlinear Tensile Deformation Behaviour

By capturing the atomic information and reflecting the behaviour governed by a nonlinear potential function, an analytical molecular mechanics approach is applied to establish the constitutive relation for single-walled carbon nanotubes （SWCNTs）. The nonlinear tensile deformation curves of zigzag and armchair nanotubes with different radii are predicted, and the elastic properties of these SWCNTs are obtained. A conclusion is made that the nanotube radius has little effect on the mechanical behaviour of SWCNTs subject to simple tension, while the nanotube orientation has larger influence.     

Analysis of Single-Walled Carbon Nanotubes Using a Chemical Bond Element Model

提出了一种纳米尺度的有限元方法,碳纳米管中的碳-碳化学键被模拟为键单元．按照平衡关系,根据有限元理论,作用于每个碳原子上的作用力可以写成键单元的刚度矩阵与每个碳原子位移的乘积．在分子力学的基本假设下,键单元刚度矩阵的每个元素可以写为分子力学中力场常数的函数,这样建立起了宏观力学方法(有限元)与纳米尺度力学方法(分子力学)之间的联系．应用该方法模拟了扶椅型与锯齿型单壁碳纳米管的力学行为从而验证了该方法的有效性．分析结果说明单壁碳纳米管的弹性模量与管厚度的选取直接相关．此外,弹性模量对所选取的分子力学中的力场常数非常敏感,管的弹性模量显示出对半径的尺度依赖性,但是管长度对弹性模量的影响小到可以被忽略．     

Studying the buckling and vibration characteristics of single-walled zinc oxide nanotubes using a nanoscale finite element model

The free vibration and axial buckling of achiral zinc oxide nanotubes (ZnONTs) are studied in this paper based on a three-dimensional finite-element model in which bonds are modeled using beam elements and mass elements are placed at the joints of beams instead of atoms. To determine the mechanical properties of the nanotubes, a linkage is established between molecular mechanics and density functional theory. The fundamental frequency and critical buckling load of ZnONTs with different geometries, chiralities and boundary conditions are calculated. It is shown that zigzag nanotubes are more stable than armchair ones. Investigating the effect of aspect ratio on the critical force shows that longer nanotubes are less stable. Also, it is indicated that increasing the length of the nanotubes will result in decreasing the frequency. Moreover, as the aspect ratio increases, the effect of end conditions diminishes.     

Finite element simulation for estimating the mechanical properties of multi-walled carbon nanotubes

A finite element simulation technique for estimating the mechanical properties of multi-walled carbon nanotubes is developed. In the present modeling concept, individual carbon nanotube is simulated as a frame-like structure and the primary bonds between two nearest-neighboring atoms are treated as beam elements, the beam element properties are determined via the concept of energy equivalence between molecular dynamics and structural mechanics. As to the simulation of the interlayer van der Waals force which has intrinsic nonlinearity and complicated applying region, a simplifying method is proposed that the interlayer pressure caused by van der Waals force instead of the force itself is to be considered, and we make use of the linear part of the interlayer pressure near the equilibrium condition to avoid the nonlinearity in problem, then linear spring elements whose stiffness is determined by equivalent force concept can be utilized to simulate the interlayer van der Waals force such that significant modeling and computing effort is saved in performing the finite element analysis. Numerical examples for estimating the mechanical properties of nanotubes, such as axial and radial Young’s modulus, shear modulus, natural frequency, buckling load, etc., are presented to illustrate the accuracy of this simulation technique. By comparing to the results found in the literature and the possible analytical solutions, it shows that the obtained mechanical properties of nanotubes by the present method agree well with their comparable results. In addition, the relations between these mechanical properties and the nanotube size are also discussed.     

Tuning of nanotube mechanical resonances by electric field pulling

We show here that field emission (FE) can be used to directly observe the vibration resonances nu(R) of carbon nanotubes (CNTs) and that the tension created by the applied field allows the tuning of these resonances by up to a factor of 10. The resonances are observable by the changes they create in the FE pattern or the emitted FE current. The tuning is shown to be linear in voltage and to follow from the basic physics of stretched strings. The method allows one to study the mechanical properties of individual multiwall carbon nanotubes within an ensemble and follow their evolution as the CNTs are modified. The tuning and detection should be useful for nanometric resonant devices.     

Estimation of mechanical properties of nanomaterials using artificial intelligence methods

Computational modeling tools such as molecular dynamics (MD), ab initio, finite element modeling or continuum mechanics models have been extensively applied to study the properties of carbon nanotubes (CNTs) based on given input variables such as temperature, geometry and defects. Artificial intelligence techniques can be used to further complement the application of numerical methods in characterizing the properties of CNTs. In this paper, we have introduced the application of multi-gene genetic programming (MGGP) and support vector regression to formulate the mathematical relationship between the compressive strength of CNTs and input variables such as temperature and diameter. The predictions of compressive strength of CNTs made by these models are compared to those generated using MD simulations. The results indicate that MGGP method can be deployed as a powerful method for predicting the compressive strength of the carbon nanotubes.     

Nonlinear finite-element modeling of graphene and single- and multi-walled carbon nanotubes under axial tension

An effective finite-element (FE) approach for modeling the structure and the deformation of single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) is presented. An individual tube was modeled using a frame-like structure with beam elements. The effect of van der Waals forces, crucial in MWCNTs, was modeled by spring elements. The success of this new carbon nanotube (CNT) modeling approach was verified by comparing the simulation results for single- and multi-walled nanotubes and graphene with other experimental and computational results available in the literature. Simulations of final deformed configurations were in excellent agreement with the atomistic models for various deformations. The proposed approach successfully predicts the experimentally observed values for mechanical behavior of SWCNTs and MWCNTs. The results demonstrated that the proposed FE technique could provide a valuable tool for studying the mechanical behavior of different types of nanotubes, as well as their effectiveness as load-bearing entities in nanocomposite materials.     

Vibration analysis of nanomechanical mass sensor using carbon nanotubes under axial tensile loads

Carbon nanotubes (CNTs) are nanomaterials with many potential applications due to their excellent mechanical and physical properties. In this paper, we proposed that CNTs with clamped boundary condition under axial tensile loads were considered as CNT-based resonators. Moreover, the resonant frequencies and frequency shifts of the CNTs with attached mass were investigated based on two theoretical methods, which are Euler–Bernoulli beam theory and Rayleigh’s energy method. Using the present methods, we analyzed and discussed the effects of the aspect ratio, the concentrated mass and the axial force on the resonant frequency of the CNTs. The results indicate that the length of CNTs could be easily changed and could provide higher sensitivity as nanomechanical mass sensor. Moreover, the resonant frequency shifts of the CNT resonator increase significantly with increasing tensile load acting on the CNTs.     

Mechanism of field electron emission from carbon nanotubes

Field electron emission (FE) is a quantum tunneling process in which electrons are injected from materials (usually metals) into a vacuum under the influence of an applied electric field. In order to obtain usable electron current, the conventional way is to increase the local field at the surface of an emitter. For a plane metal emitter with a typical work function of 5 eV, an applied field of over 1 000 V/μm is needed to obtain a significant current. The high working field (and/or the voltage between the electrodes) has been the bottleneck for many applications of the FE technique. Since the 1960s, enormous effort has been devoted to reduce the working macroscopic field (voltage). A widely adopted idea is to sharpen the emitters to get a large surface field enhancement. The materials of emitters should have good electronic conductivity, high melting points, good chemical inertness, and high mechanical stiffness. Carbon nanotubes (CNTs) are built with such needed properties. As a quasi-one-dimensional material, the CNT is expected to have a large surface field enhancement factor. The experiments have proved the excellent FE performance of CNTs. The turn-on field (the macroscopic field for obtaining a density of 10 μA/cm²) of CNT based emitters can be as low as 1 V/μm. However, this turn-on field is too good to be explained by conventional theory. There are other observations, such as the non-linear Fowler-Nordheim plot and multi-peaks field emission energy distribution spectra, indicating that the field enhancement is not the only story in the FE of CNTs. Since the discovery of CNTs, people have employed more serious quantum mechanical methods, including the electronic band theory, tight-binding theory, scattering theory and density function theory, to investigate FE of CNTs. A few theoretical models have been developed at the same time. The multi-walled carbon nanotubes (MWCNTs) should be assembled with a sharp metal needle of nano-scale radius, for which the FE mechanism is more or less clear. Although MWCNTs are more common in present FE applications, the single-walled carbon nanotubes (SWCNTs) are more interesting in the theoretical point of view since the SWCNTs have unique atomic structures and electronic properties. It would be very interesting if people can predict the behavior of the well-defined SWCNTs quantitatively (for MWCNTs, this is currently impossible). The FE as a tunneling process is sensitive to the apex-vacuum potential barrier of CNTs. On the other hand, the barrier could be significantly altered by the redistribution of excessive charges in the micrometer long SWCNTs, which have only one layer of carbon atoms. Therefore, the conventional theories based upon the hypothesis of fixed potential (work function) would not be valid in this quasi-one-dimensional system. In this review, we shall focus on the mechanism that would be responsible for the superior field emission characteristics of CNTs. We shall introduce a multi-scale simulation algorithm that deals with the entire carbon nanotube as well as the substrate as a whole. The simulation for (5, 5) capped SWCNTs with lengths in the order of micrometers is given as an example. The results show that the field dependence of the apex-vacuum electron potential barrier of a long carbon nanotube is a more pronounced effect, besides the local field enhancement phenomenon.     

Electron microscopy and in situ testing of mechanical deformation of carbon nanotubes

In this paper, the electron diffraction technique to determine the helicity and atomic structure of carbon nanotubes is reviewed, as well as different mechanical test methods, tensile test, bending test, compression test and vibration test of carbon nanotubes by in situ electron microscopy are summarized while the relationship between mechanical properties and structures revealed by experiments is addressed. Except for these, the electric current and electron beam irradiation effect and some other novel electron microscopy experiments are also incorporated.     

Effects of carbon nanotube structures on mechanical properties

This paper describes a structural mechanics approach to modelling the mechanical properties of carbon nanotubes (CNTs). Based on a model of truss structures linked by inter-atomic potentials, a closed-form elastic solution is obtained to predict the mechanical properties of single-walled carbon nanotubes (SWNTs). Moreover, the elastic modulus of multi-walled carbon nanotubes (MWNTs) is also predicted for a group of the above mentioned SWNTs with uniform interval spacing. Following the structural mechanics approach, the elastic modulus, Poissons ratio, and the deformation behaviors of SWNTs were investigated as a function of the nanotube size and structure. Poissons ratio of SWNTs shows a chirality dependence, while the elastic modulus is insensitive to the chirality. The disposition of the strain energy of bonds shows quite a difference between the zigzag and armchair tubes subjected to axial loading. A zigzag tube is predicted to have a lower elongation property than an armchair tube. PACS 62.20-x; 62.20.Dc; 62.25+g     

Vibrations of carbon nanotube-reinforced composites

This work deals with a study of the vibrational properties of carbon nanotube-reinforced composites by employing an equivalent continuum model based on the Eshelby-Mori-Tanaka approach. The theory allows the calculation of the effective constitutive law of the elastic isotropic medium (matrix) with dispersed elastic inhomogeneities (carbon nanotubes). The devised computational approach is shown to yield predictions in good agreement with the experimentally obtained elastic moduli of composites reinforced with uniformly aligned single-walled carbon nanotubes (CNTs). The primary contribution of the present work deals with the global elastic modal properties of nano-structured composite plates. The investigated composite plates are made of a purely isotropic elastic hosting matrix of three different types (epoxy, rubber, and concrete) with embedded single-walled CNTs. The computations are carried out via a finite element (FE) discretization of the composite plates. The effects of the CNT alignment and volume fraction are studied in depth to assess how the modal properties are influenced both globally and locally. As a major outcome, the lowest natural frequencies of CNT-reinforced rubber composites are shown to increase up to 500 percent.     

Effect of Carbon Nanotubes on the Mechanical Properties of Polypropylene/Wood Flour Composites: Reinforcement Mechanism

Maleic anhydride grafted polypropylene (PP-g-MA) was employed as the compatibilizer and carbon nanotubes (CNTs) or hydroxylated CNTs as reinforcements for polypropylene/wood flour composites. The results showed that when the PP-g-MA loading level was 10 wt%, the bending strength, tensile strength, Izod notched impact strength, and elongation at break of PP-wood composites were enhanced by 85% (66.3 MPa), 93% (33.7 MPa), 5.8% (2.01 kJ/m²), and 64% (23%), respectively, relative to the uncompatibilized composites. The introduction of pristine CNTs only improved slightly the overall mechanical properties of the compatibilized composites due to poor interfacial compatibility. Unlike CNTs, incorporating hydroxylated CNTs (CNT-OH) could significantly improve all of the mechanical properties; for instance, at 0.5 wt% CNT-OH loading, the flexural strength and tensile strength reached 68.5 MPa, and 40.4 MPa about 6.6% higher than that for the composites with the same CNT loading. Furthermore, CNT-OH also remarkably enhanced the storage modulus. Contact angle and morphology observations indicated that the increases in mechanical properties could be attributed to the improvements of interfacial interactions and adhesions of CNTs with the matrix and fillers.     

Structural and vibrational properties of deformedcarbon nanotubes

The fantastic variation of the physical properties of carbon nanotubes (CNTs) and their bundles under mechanical strain and hydrostatic pressure makes them promising materials for fabricating nanoscale electromechanical coupling devices or transducers. In this paper, we review the recent progress in this field, with much emphasis on our first-principles numerical studies on the structural and vibrational properties of the deformed CNTs under uniaxial and torsional strains, and hydrostatic pressure. The nonresonant Raman spectra of the deformed CNTs are also introduced, which are calculated by the first-principles calculations and the empirical bond polarizability model.     

Investigating interfacial contact configuration and behavior of single-walled carbon nanotube-based nanodevice with atomistic simulations

Carbon nanotubes (CNTs), including single-walled carbon nanotubes (SWNTs) and multi-walled carbon nanotubes (MWNTs), are considered to be the promising candidates for next-generation interconnects with excellent physical and chemical properties ranging from ultrahigh mechanical strength, to electrical properties, to thermal conductivity, to optical properties, etc. To further study the interfacial contact configurations of SWNT-based nanodevice with a 13.56-Å diameter, the corresponding simulations are carried out with the molecular dynamic method. The nanotube collapses dramatically into the surface with the complete collapse on the Au/Ag/graphite electrode surface and slight distortion on the Si/SiO₂ substrate surface, respectively. The related dominant mechanism is studied and explained. Meanwhile, the interfacial contact configuration and behavior, depended on other factors, are also analyzed in this article.