On the effective elastic moduli of carbon nanotubes for nanocomposite structures

A critical review on the validity of different experimental and theoretical approaches to the mechanical properties of carbon nanotubes for advanced composite structures is presented. Most research has been recently conducted to study the properties of single-walled and multi-walled carbon nanotubes. Special attention has been paid to the measurement and modeling of tensile modulus, tensile strength, and torsional stiffness. Theoretical approaches such as molecular dynamic (MD) simulations, finite element analysis, and classical elastic shell theory were frequently used to analyze and interpret the mechanical features of carbon nanotubes. Due to the use of different fundamental assumptions and boundary conditions, inconsistent results were reported. MD simulation is a well-known technique that simulates accurately the chemical and physical properties of structures at atomic-scale level. However, it is limited by the time step, which is of the order of 10⁻¹⁵ s. The use of finite element modeling combined with MD simulation can further decrease the processing time for calculating the mechanical properties of nanotubes. Since the aspect ratio of nanotubes is very large, the elastic rod or beam models can be adequately used to simulate their overall mechanical deformation. Although many theoretical studies reported that the tensile modulus of multi-walled nanotubes may reach 1 TPa, this value, however, cannot be directly used to estimate the mechanical properties of multi-walled nanotube/polymer composites due to the discontinuous stress transfer inside the nanotubes.     

Evaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approach

The development of a finite element formulation that is appropriate for the computation of Young’s and Shear modulus of single walled carbon nanotubes (SWCNTs) is the purpose of this paper. The method utilizes the atomistic microstructure of the nanotubes. According to the three-dimensional atomic nanostructure of SWCNTs, nodes are defined at the atom locations. Appropriate spring-type elements interconnect these nodes to simulate properly interatomic interactions. This approach is implemented via the use of three-dimensional spring-like elements each node of which obeys to three translations and three rotations. In this way, molecular mechanics theory can be applied directly while the atomic bonds are modeled by using exclusively physical variables such as bond stretching, bond angle bending and torsional rotation resistance force constants. With the proposed method, the Young’s and shear modulus of numerous SWCNTs were determined. The effect of the nanotube radius and thickness on the mechanical behavior of SWCNTs was tested and demonstrated. The numerical results show good agreement with other corresponding values which are available in the literature.     

Investigation of chirality and diameter effects on the Young’s modulus of carbon nanotubes using non-linear potentials

The main goal of this research is to predict Young’s modulus of carbon nanotubes using a full non-linear finite element model. Spring elements are used to simulate molecular interactions in atomic structure of carbon nanotube. All interactions are simulated non-linearly. A parametric study is performed to investigate effects of chirality and diameter on the Young’s modulus of single walled carbon nanotubes. Unlike the results of presented linear finite element models, the results of current model imply on independency of Young’s modulus from chirality and diameter. Obtained results from this study are in a good agreement with experimental observations and published data.     

A progressive fracture model for carbon nanotubes

An atomistic-based progressive fracture model for simulating the mechanical performance of carbon nanotubes by taking into account initial topological and vacancy defects is proposed. The concept of the model is based on the assumption that carbon nanotubes, when loaded, behave like space-frame structures. The finite element method is used to analyze the nanotube structure and the modified Morse interatomic potential to simulate the non-linear force field of the C–C bonds. The model has been applied to defected single-walled zigzag, armchair and chiral nanotubes subjected to axial tension. The defects considered were: 10% weakening of a single bond and one missing atom at the middle of the nanotube. The predicted fracture evolution, failure stresses and failure strains of the nanotubes correlate very well with molecular mechanics simulations from the literature.     

Multiscale modeling of dynamic crack propagation

A multiscale approach is employed to investigate a center-cracked specimen with the purpose to redefine fracture toughness from the atomistic perspective and to simulate different modes of crack propagation. The specimen is divided into three regions: (1) far field, modeled by classical fracture mechanics, (2) near field, modeled by a multiscale field theory and analyzed by a generalized finite element method, and (3) crack tip atomic region, modeled by molecular dynamics (MD). The exact and analytical solution of the far field is utilized to specify boundary conditions at the interface between the far field and the near field. The interaction between the near field and the crack tip region is described by full-blown interatomic forces. In this work, crystals of perovskite (Barium Titanate) and rocksalt (Magnesia) have been studied. Fracture toughness is defined as a material property associated with instability of the MD simulation. Mode I, Mode II, and mixed mode fracture have been investigated and numerical results will be presented and discussed.     

Analysis of elastic properties of carbon nanotube reinforced nanocomposites with pinhole defects

Due to their unique molecular structure, carbon nanotubes exhibit outstanding properties. They are regarded as ideal reinforcements of composites. In this paper, the effects of pinhole defects on mechanical properties are investigated for wavy carbon nanotubes based nanocomposites using 3-D Representative Volume Element with long carbon nanotubes. The carbon nanotubes are modeled as continuum hollow cylindrical shape elastic material with pinholes, having some curvature in its shape. These defects are considered on the single walled carbon nanotubes. The mechanical properties like Young’s modulus of elasticity are evaluated for various values of waviness index, as well as type and number of pinhole defects. The effects of interactions between both defects as well as their influence on the nanocomposites are studied under an axial loading condition. Numerical equations are used to extract the effective material properties for the different geometries of Representative Volume Elements with non-defective carbon nanotubes. The finite element method results obtained for non-defective carbon nanotubes are consistent with analytical results for cylindrical Representative Volume Elements, which validate the proposed model. It is observed that the presence of pinhole defects as well as waviness, can significantly reduces the effective reinforcement, when compared with nanotubes without pinhole defects and this reinforcement decreases with the increase of the number of pinhole defects.     

Multiscale simulation of nanostructures based on spatial secant model: a discrete hyperelastic approach

The main objective of this paper is to present a coarse-grained material model for the simulation of three-dimensional nanostructures. The developed model is motivated by the recent progress in establishing continuum models for nanomaterials and nanostructures. As there are conceptual differences between the continuum field defined in the classical sense and the nanomaterials consisting of discrete, space-filling atoms, existing continuum measures cannot be directly applied for mapping the nanostructures due to the discreteness at small length scale. In view of the fundamental difficulties associated with the direct application of the continuum approach, we introduce a unique discrete deformation measure called spatial secant and have developed a new hyperelastic model based on this measure. We show that the spatial secant-based model is consistently linked to the underlying atomistic model and provides a geometric exact mapping in the discrete sense. In addition, we outline the corresponding computational framework using the finite element and/or meshfree method. The implementation is within the context of finite deformation. Finally we illustrate the application of the model in studying the mechanics of low-dimensional carbon nanostructures such as carbon nanotubes (CNT). By comparing with full-scale molecular mechanics simulations, we show that the proposed coarse-grained model is robust in that it accurately captures the non-linear mechanical responses of the CNT structures.     

Multiscale Experimental Mechanics of Hierarchical Carbon‐Based Materials

Investigation of the mechanics of natural materials, such as spider silk, abalone shells, and bone, has provided great insight into the design of materials that can simultaneously achieve high specific strength and toughness. Research has shown that their emergent mechanical properties are owed in part to their specific self‐organization in hierarchical molecular structures, from nanoscale to macroscale, as well as their mixing and bonding. To apply these findings to manmade materials, researchers have devoted significant efforts in developing a fundamental understanding of multiscale mechanics of materials and its application to the design of novel materials with superior mechanical performance. These efforts included the utilization of some of the most promising carbon‐based nanomaterials, such as carbon nanotubes, carbon nanofibers, and graphene, together with a variety of matrix materials. At the core of these efforts lies the need to characterize material mechanical behavior across multiple length scales starting from nanoscale characterization of constituents and their interactions to emerging micro‐ and macroscale properties. In this report, progress made in experimental tools and methods currently used for material characterization across multiple length scales is reviewed, as well as a discussion of how they have impacted our current understanding of the mechanics of hierarchical carbon‐based materials. In addition, insight is provided into strategies for bridging experiments across length scales, which are essential in establishing a multiscale characterization approach. While the focus of this progress report is in experimental methods, their concerted use with theoretical‐computational approaches towards the establishment of a robust material by design methodology is also discussed, which can pave the way for the development of novel materials possessing unprecedented mechanical properties.     

Molecular mechanics in the context of the finite element method

In molecular mechanics, the formalism of the finite element method can be exploited in order to analyze the behavior of atomic structures in a computationally efficient way. Based on the atom‐related consideration of the atomic interactions, a direct correlation between the type of the underlying interatomic potential and the design of the related finite element is established. Each type of potential is represented by a specific finite element. A general formulation that unifies the various finite elements is proposed. Arbitrary diagonal‐ and cross‐terms dependent on bond length, valence angle, dihedral angle, improper dihedral angle and inversion angle can also be considered. The finite elements are formulated in a geometrically exact setting; the related formulas are stated in detail. The mesh generation can be performed using well‐known procedures typically used in molecular dynamics. Although adjacent elements overlap, a double counting of the element contributions (as a result of the assembly process) cannot occur a priori. As a consequence, the assembly process can be performed efficiently line by line. The presented formulation can easily be implemented in standard finite element codes; thus, already existing features (e.g. equation solver, visualization of the numerical results) can be employed. The formulation is applied to various interatomic potentials that are frequently used to describe the mechanical behavior of carbon nanotubes. The effectiveness and robustness of this method are demonstrated by means of several numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.     

A coupled mechanical‐charge/dipole molecular dynamics finite element method,with multi‐scale applications to the design of graphene nano‐devices

A new Molecular Dynamics Finite Element Method (MDFEM) with a coupled mechanical‐charge/dipole formulation is proposed. The equilibrium equations of Molecular Dynamics (MD) are embedded exactly within the computationally more favourable Finite Element Method (FEM). This MDFEM can readily implement any force field because the constitutive relations are explicitly uncoupled from the corresponding geometric element topologies. This formal uncoupling allows to differentiate between chemical‐constitutive, geometric and mixed‐mode instabilities. Different force fields, including bond‐order reactive and polarisable fluctuating charge–dipole potentials, are implemented exactly in both explicit and implicit dynamic commercial finite element code. The implicit formulation allows for larger length and time scales and more varied eigenvalue‐based solution strategies. The proposed multi‐physics and multi‐scale compatible MDFEM is shown to be equivalent to MD, as demonstrated by examples of fracture in carbon nanotubes (CNT), and electric charge distribution in graphene, but at a considerably reduced computational cost. The proposed MDFEM is shown to scale linearly, with concurrent continuum FEM multi‐scale couplings allowing for further computational savings. Moreover, novel conformational analyses of pillared graphene structures (PGS) are produced. The proposed model finds potential applications in the parametric topology and numerical design studies of nano‐structures for desired electro‐mechanical properties (e.g. stiffness, toughness and electric field induced vibrational/electron‐emission properties). Copyright © 2014 John Wiley & Sons, Ltd.     

Bending buckling behavior of perfect and defective single-walled carbon nanotubes via a structural mechanics model

A structural mechanics model is employed for the investigation of the bending buckling behavior of perfect and defective single-walled carbon nanotubes (SWCNTs). The effects of different types of defects (vacancies and Stone–Wales defects) at various locations on the critical bending buckling moments and curvatures are also studied for zigzag and armchair nanotubes with various aspect ratios (length/diameter). The locations of defects are along the length of the nanotube and around the circumference. Moreover, the results of this structural mechanics model are compared with a finite element model. The simple continuum model, especially, could be adopted to predict the critical buckling moments and curvatures of SWCNTs with large aspect ratio. Finally, the results of the present structural model are compared with those from molecular dynamics (MD) simulation, and there is good agreement between our model and the MD model.     

Numerical investigation of elastic mechanical properties of graphene structures

The computation of the elastic mechanical properties of graphene sheets, nanoribbons and graphite flakes using spring based finite element models is the aim of this paper. Interatomic bonded interactions as well as van der Waals forces between carbon atoms are simulated via the use of appropriate spring elements expressing corresponding potential energies provided by molecular theory. Each layer is idealized as a spring-like structure with carbon atoms represented by nodes while interatomic forces are simulated by translational and torsional springs with linear behavior. The non-bonded van der Waals interactions among atoms which are responsible for keeping the graphene layers together are simulated with the Lennard-Jones potential using appropriate spring elements. Numerical results concerning the Young’s modulus, shear modulus and Poisson’s ratio for graphene structures are derived in terms of their chilarity, width, length and number of layers. The numerical results from finite element simulations show good agreement with existing numerical values in the open literature.     

Probing the structure of DNA-carbon nanotube hybrids with molecular dynamics

DNA-carbon nanotube hybrids (DNA-CN) are novel nanoscale materials that consist of single-wall carbon nanotubes (SWCN) coated with a self-assembled monolayer of single-stranded DNA (ssDNA). Recent experiments on DNA-CN have shown that this material offers a remarkable set of technologically useful properties such as facilitation of SWCN sorting, chemical sensing, and detection of DNA hybridization. Despite the importance of DNA-CN, a detailed understanding of its microscopic structure and physical properties is lacking. To address this, we have performed classical all-atom molecular dynamics (MD) simulations exploring the self-assembly mechanisms, structure, and energetic properties of this nanomaterial. MD reveals that SWCN induces ssDNA to undergo a spontaneous conformational change that enables the hybrid to self-assemble via the pi-pi stacking interaction between ssDNA bases and SWCN sidewall. ssDNA is observed to spontaneously wrap about SWCN into compact right- or left-handed helices within a few nanoseconds. Helical wrapping is driven by electrostatic and torsional interactions within the sugar-phosphate backbone that result in ssDNA wrapping from the 3' end to the 5' end.     

Vibrational analysis of carbon nanotubes and graphene sheets using molecular structural mechanics approach

In this article, the vibrational properties of two kinds of single-layered graphene sheets and single-wall carbon nanotubes (SWCNT) are studied. The simulations are carried out for two types of zigzag carbon nanotubes (6,0), (12,0), armchair carbon nanotubes (4,4), (6,6) and zigzag and armchair graphene sheets with free-fixed and fixed–fixed end conditions.Fundamental frequency is determined by means of molecular structural mechanics approach. In this approach, carbon nanotubes (CNTs) and grapheme sheets are considered as space frames. By constructing equality between strain energies of each element in structural mechanics and potential energies of each bond, equivalent space frames can be achieved. Carbon atoms are considered as concentrated masses placed in beam joints (bond junctions).Results are presented as diagrams stating fundamental frequencies of nanotubes and graphene sheets with respect to aspect ratios. The results indicate that fundamental frequency decreases as aspect ratio increases. So it is preferred to use nanotubes and graphene sheets with lower aspect ratios for dynamic applications in order to prevent resonance and dynamic damage. Fundamental frequency of nanotubes is larger than that of graphene sheets. The results are in good agreement with results of previous researches.     

Determination of DNA-base orientation on carbon nanotubes through directional optical absorbance

We develop an approach for determining the orientation of DNA bases attached to carbon nanotubes (CNTs), by combining ab initio time-dependent density functional theory and optical spectroscopy measurements. The structures we find are in good agreement with the geometry of nucleosides on a (10,0) CNT obtained from molecular simulations using empirical force fields. The results shed light into the complex interactions of the DNA-CNT system, a candidate for ultrafast DNA sequencing through electronic probes.     

Mechanical properties of carbon nanotube by molecular dynamics simulation

The mechanical properties of single-walled carbon nanotube (SWCNT) are computed and simulated by using molecular dynamics (MD) in this paper. From the MD simulation for an armchair SWCNT whose diameter is 1.2 nm and length is 4.7 nm, we get that its Young modulus is 3.62 TPa, and tensile strength is 9.6 GPa. It is shown that the Young modulus and tensile strength of armchair SWCNTs are 12 order higher than those of ordinary metal materials. Therefore we can draw a conclusion that carbon nanotubes (CNT) belong to a particular material with excellent mechanical properties.     

Numerical simulation for bending modulus of carbon nanotubes and some explanations for experiment

The bending mechanical property of carbon nanotubes are numerically investigated in this paper. An advanced finite element analysis package, ABAQUS, is used to simulate the formation of rippling which is the appearance of wavelike distortion on the inner arc of the bent nanotubes, caused by the severe anisotropy of carbon nanotubes and a relatively large deformation. A non-linear bending moment–curvature relationship is obtained, which shows the tangential stiffness greatly decreases when rippling appears. This result can be used to explain the phenomenon and conclusion of the resonant experiment measuring the Young's modulus of carbon nanotubes, in which the Young's modulus calculated using linear theory is found to sharply decrease as the diameter increases [Science 283 (1999) 1513]. Here an analytical method is adopted to conduct a vibration analysis using a bi-linear bending constitution simplifying from the non-linear bending moment–curvature relationship, and the effective Young's modulus have been calculated for multi-walled carbon nanotubes of various sizes. The result carried out in the paper is similar to the measuring result is given by Poncharal et al. [Science 283 (1999) 1513].     

钢结构施工力学状态非线性分析方法

施工力学问题分析的难点在于结构时变、材料时变和边界时变的模拟。该文剖析了单元生死技术模拟施工过程的基本原理,从数学和物理上揭示了单元"杀死"、"漂移"和"激活"的机理。探讨了施工过程模拟的分步建模技术。对单元生死技术和分步建模技术的计算模型和精度进行了对比研究。基于经典梁单元,采用FORTRAN语言编制了这两种算法下的分析程序。算例分析表明:两种方法的理论及所编制程序正确,是施工力学问题求解的有效方法。     

由煤或焦炭制备纳米碳质材料的新进展

评述了以煤为碳源制备富勒烯、纳米碳管、竹节形碳管、铁嵌入的纳米碳棒和由碳包覆的金属纳米粒子等各种纳米材料。认为：等离子体电孤放电法是由煤制备各种纳米碳质材料最常用的方法，随电弧条件及电极性质的不同，所制备的纳米碳质材料可有各种不同形态及结构、由于煤是分子固体而石墨是晶格固体，两种碳源的反应机理有明显不同。在等离子体电弧加热时，煤分解并产生许多具有简单芳烃结构的分子，在纳米碳质材料的形成过程中，这些分子可能作为纳米碳质材料的结构单元，同时原煤中的矿物质在合成过程中也起着重要作用，因此煤本身的性质对纳米材料的制备极为重要。煤是成本低廉且储量最丰富的碳源，将是大规模工业化生产纳米碳质材料最好的碳源之一。