An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes

An analytical molecular structural mechanics model for the prediction of mechanical properties of defect-free carbon nanotubes is developed by incorporating the modified Morse potential with an analytical molecular structural model. The developed model is capable of predicting Young’s moduli, Poisson’s ratios and stress–strain relationships of carbon nanotubes under tension and torsion loading conditions. Results on the mechanical properties of single-walled carbon nanotubes show that Young’s moduli of carbon nanotubes are sensitive to the tube diameter and the helicity. Young’s moduli of both armchair and zigzag carbon nanotubes increase monotonically and approach Young’s modulus of graphite when the tube diameter is increased. The nonlinear stress–strain relationships for defect-free nanotubes have been predicted, which gives a good approximation on the ultimate strength and strain to failure of nanotubes. Armchair nanotubes exhibit higher tensile strength than zigzag nanotubes but their torsion strengths are identical based on the present study. The present theoretical investigation provides a very simple approach to predict the mechanical properties of carbon nanotubes.     

Young’s modulus prediction of hexagonal nanosheets and nanotubes based on dimensional analysis and atomistic simulations

The present work investigates Young’s modulus of hexagonal nanosheets and nanotubes based on dimensional analysis and molecular mechanics. Using second derivatives of the strain energy density revealed from molecular dynamics simulations at 0 K (i.e., molecular mechanics) with harmonic potentials for various combinations of force constants, Young’s modulus have been computed for single-walled armchair and zigzag nanotubes of different radii. This parametric study with the aid of dimensional analysis allows explicitly establishing Young’s modulus of (n, n) armchair and (n, 0) zigzag nanotubes as functions of the force constants, bond length and chiral index n. Proposed formulae are applied to estimate Young’s modulus of graphene, boron nitride, silicon carbide sheets and their nanotubes. The accuracy of the proposed formulae are verified and discussed with available data in the literature for these three sheets and their nanotubes.     

A structural mechanics approach for the analysis of carbon nanotubes

This paper presents a structural mechanics approach to modeling the deformation of carbon nanotubes. Fundamental to the proposed concept is the notion that a carbon nanotube is a geometrical frame-like structure and the primary bonds between two nearest-neighboring atoms act like load-bearing beam members, whereas an individual atom acts as the joint of the related load-bearing beam members. By establishing a linkage between structural mechanics and molecular mechanics, the sectional property parameters of these beam members are obtained. The accuracy and stability of the present method is verified by its application to graphite. Computations of the elastic deformation of single-walled carbon nanotubes reveal that the Young’s moduli of carbon nanotubes vary with the tube diameter and are affected by their helicity. With increasing tube diameter, the Young’s moduli of both armchair and zigzag carbon nanotubes increase monotonically and approach the Young’s modulus of graphite. These findings are in good agreement with the existing theoretical and experimental results.     

Atomistic-continuum modeling for mechanical properties of single-walled carbon nanotubes

The study attempts to explore the influences of the surface effect resulting in an initial relaxed unstrained deformation and the in-layer non-bonded van der Waals (vdW) atomistic interactions on the mechanical properties of single-walled carbon nanotubes (SWCNTs) using a proposed atomistic-continuum modeling (ACM) approach. The modeling approach incorporates atomistic modeling, by virtue of molecular dynamics (MD) simulation, for simulating the initial unstrained equilibrium state, and equivalent-continuum modeling (ECM), by way of finite element approximations (FEA), for modeling the subsequent static/dynamic behaviors.SWCNTs with various radius and two different chiralities, including zigzag and armchair type, are presented. To validate the proposed technique, the present results are compared with the literature data, including numerical and experimental values. Results show that the derived elastic moduli (1.2–1.4 TPa) when considering these two nanoeffects tend to be more consistent with the published experimental data. In specific, they can increase up to 17–23% Young’s modulus, 5–15% shear modulus, 6–11% natural frequencies and 10–30% critical buckling load of the SWCNTs, implying that without considering these two effects, the material behaviors of SWCNTs would be potentially underestimated.     

单壁碳纳米管的扭转力学特性研究

利用基于高阶Cauchy-Born准则所建立的单壁碳纳米管本构模型,针对不同手性的单壁碳纳米管的扭转力学特性进行了研究.研究发现结构呈现对称性的锯齿型和扶手型单壁碳纳米管具有完全对称的扭转特性,而结构不对称的手性型单壁碳纳米管具有正反相异的扭转特性.同时,针对一系列手性不同的单壁碳纳米管的扭转力学特性展开了详细的研究.研究的部分结果与采用其他方法得到的结果进行了对比,证实了所提出方法以及预测结果的有效性和可行性.     

Stone-Wales拓扑缺陷对石墨烯拉伸力学性能的影响

采用Tersoff势对含Stone-Wales(SW)拓扑缺陷的单层石墨烯薄膜的单向拉伸力学性能进行了分子动力学模拟,分别研究了SW拓扑缺陷对扶手椅型和锯齿型石墨烯拉伸力学性能及变形机制的影响.研究结果表明,单个SW缺陷对两种手性石墨烯薄膜的杨氏模量几乎无影响,而对薄膜的强度、应变等力学性能和变形破坏机制的影响与手性有...     

Surface deformation-dependent mechanical properties of bending nanowires: an ab initio core-shell model

An ab initio core-shell model is proposed to evaluate the surface effect in bending nanowires, in which the elastic modulus depends on the surface relaxation and deformation induced by external loading. By using first-principles calculations based on the density functional theory(DFT), the surface and bulk properties are calculated for Ag, Pb, and Si nanowires. The obtained theoretical predictions of the effective Young’s modulus of nanowires agree well with the experimental data, which shows th...     

A molecular mechanics study on size-dependent elastic properties of single-walled boron nitride nanotubes

We present an analytical study for the elastic properties of single-walled boron nitride nanotubes via a molecular mechanics model. Closed-form expressions for Young's modulus, Poisson's ratio and surface shear modulus are derived as functions of the nanotube diameter. The results are helix angle sensitive and comparable to those from ab initio calculations. This work is a first effort to establish analytical model of molecular mechanics for composite nanotubes and reveals the dissimilarities between size-dependent elastic properties of carbon and boron nitride nanotubes.     

Analytical and numerical techniques to predict carbon nanotubes properties

In this paper, two different approaches for modeling the behaviour of carbon nanotubes are presented. The first method models carbon nanotubes as an inhomogeneous cylindrical network shell using the asymptotic homogenization method. Explicit formulae are derived representing Young’s and shear moduli of single-walled nanotubes in terms of pertinent material and geometric parameters. As an example, assuming certain values for these parameters, the Young’s modulus was found to be 1.71 TPa, while the shear modulus was 0.32 TPa. The second method is based on finite element models. The inter-atomic interactions due to covalent and non-covalent bonds are replaced by beam and spring elements, respectively, in the structural model. Correlations between classical molecular mechanics and structural mechanics are used to effectively model the physics governing the nanotubes. Finite element models are developed for single-, double- and multi-walled carbon nanotubes. The deformations from the finite element simulations are subsequently used to predict the elastic and shear moduli of the nanotubes. The variation of mechanical properties with tube diameter is investigated for both zig-zag and armchair configurations. Furthermore, the dependence of mechanical properties on the number of nanotubules in multi-walled structures is also examined. Based on the finite element model, the value for the elastic modulus varied from 0.9 to 1.05 TPa for single and 1.32 to 1.58 TPa for double/multi-walled nanotubes. The shear modulus was found to vary from 0.14 to 0.47 TPa for single-walled nanotubes and 0.37 to 0.62 for double/multi-walled nanotubes.     

Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model

An analytical model based on a molecular mechanics approach is presented to relate the elastic properties of a single-walled carbon nanotube to its atomic structure. We derive closed-form expressions for elastic modulus and Poisson's ratio as a function of the nanotube diameter. Properties at different length scales are directly connected via these expressions. The analytically calculated elastic properties for achiral nanotubes using force constants obtained from experimental data of graphite are compared to those based on tight binding numerical calculations. This study represents a preliminary effort to develop analytical methods of molecular mechanics for applications in nanostructure modeling.     

单壁碳纳米管屈曲的原子/连续介质混合模型

用数学和力学研究所，上海 200072)//力学学报.--2004,36(6).--744～748 提供了一种运用原子/连续介质混合(hybrid atomic/continuum,HAC)方法解决纳米力学问题的思路. 通过在连续介质力学模型中引入利用分子力学方法获得物性参数，建立了预测单壁碳纳米管临界屈曲参数的HAC模型. 结果表明, HAC模型具有与连续介质力学模型可比拟的简洁性, 同时可表征纳米管微观结构特征对屈曲参数的影响. 计算结果表明，Zigzag纳米管的抗屈曲性能优于Armchair纳米管. 基于Tersoff-Brenner作用势的分子动力学结果证实了这一结论.     

A mechanics-of-materials model for predicting Young’s modulus of damaged woven fabric composites involving three damage modes

An analytical model for damaged woven fabric composites is developed using the theory of advanced mechanics of materials. The analysis is based on Castigliano’s second theorem and utilizes a damaged mosaic model laminate. Three damage modes (i.e., transverse yarn cracking, interface debonding, and sliding with friction at the interface) are considered. Only one independent interfacial parameter, the friction coefficient between warp and fill yarns, is introduced in the analysis. A closed-form formula is provided for estimating effective Young’s modulus of damaged woven laminates. A parametric study of some 192 sample cases of two different composite systems (i.e., glass fiber/epoxy and ceramic fiber/ceramic) is conducted to illustrate the application and significance of the newly derived analytical model. The numerical values of the effective Young’s modulus for the special case involving only transverse yarn cracking (the first damage mode) estimated by the present mechanics-of-materials model agree fairly well with those predicted by an elasticity-based model [Int. J. Solids Struct. 38 (2001) 855]. For the general case involving all three damage modes simultaneously, the present model reveals the complex nature of Young’s modulus reduction in a quantitative manner, which differs from existing models.     

STABILITY CHARACTERISTICS OF SINGLE-LAYERED ZINC OXIDE NANOSHEETS UNDER UNIAXIAL LOADING

This paper is aimed to propose a three-dimensional model which would be used for investigation on the mechanical behavior of single-layered zinc oxide nanosheets. To develop this model, molecular mechanics is coupled with the density functional theory. Simulating the hexagonal lattices of nanosheets as a hexagonal mechanical structure composed of structural beam elements, the buckling behavior of zinc oxide nanosheets is studied. Effects of different parameters on the stability of armchair and zigzag nanosheets are examined. It is shown that the buckling forces of zigzag nanosheets are slightly greater than those of armchair ones. However, with increasing size of nanosheets the effect of atomic structure on the stability of nanosheets diminishes.By studying the effect of end conditions on the buckling behavior of nanosheets, it is shown the stability of nanosheets is affected significantly by boundary conditions.     

Analytical solutions for elastic binary nanotubes of arbitrary chirality

Analytical solutions for the elastic properties of a variety of binary nanotubes with arbitrary chirality are obtained through the study of systematic molecular mechanics. This molecular mechanics model is first extended to chiral binary nanotubes by introducing an additional outof-plane inversion term into the so-called stick-spiral model,which results from the polar bonds and the buckling of binary graphitic crystals. The closed-form expressions for the longitudinal and circumferential Young's modulus and Poisson's ratio of chiral binary nanotubes are derived as functions of the tube diameter. The obtained inversion force constants are negative for all types of binary nanotubes, and the predicted tube stiffness is lower than that by the former stick-spiral model without consideration of the inversion term, reflecting the softening effect of the buckling on the elastic properties of binary nanotubes. The obtained properties are shown to be comparable to available density functional theory calculated results and to be chirality and size sensitive. The developed model and explicit solutions provide a systematic understanding of the mechanical performance of binary nanotubes consisting of Ⅲ–Ⅴ and Ⅱ–Ⅵ group elements.     

A study on the bending stiffness of single-walled carbon nanotubes and related issues

Single-walled carbon nanotubes (SWCNTs) are usually modeled as elastic tubes and their bending stiffness D is often related to their axial stretching modulus E (Young's modulus) as in mechanics of materials (i.e. D=EI where I is the moment of inertia of the tube). However, recent studies show that large discrepancies may exist when this relationship is used to predict Young's modulus of carbon nanotubes (CNTs) through bending dominated deformations. In the present paper, the bending stiffness of SWCNTs and some related issues are investigated by the combined use of the molecular-mechanics (M-M) model and the deformation mapping technique. Based on the analysis results, the contradictions mentioned above can be explained well. Furthermore, an analytical expression for the bending stiffness of SWCNTs is also presented. It shows that the bending stiffness of a SWCNT is approximately proportional to the cube of its radius which agrees well with the existing molecular dynamics simulation and continuum theory based results.     

Continuum models of multi-walled carbon nanotubes

Molecular mechanics (MM) simulations have been carried out to determine energetically favorable double-walled carbon nanotube (DWNT) structures, and analyze their infinitesimal extensional, torsional, radial expansion/contraction, and bending deformations. Loads are applied either to one wall or simultaneously to both walls of an open-ended DWNT. These results are compared against single-walled carbon nanotube (SWNT) results to determine differences and similarities between responses of SWNTs and DWNTs, and the validity of using SWNT results to predict the response of a DWNT. It is found that for small deformations such as simple tension and torsion, results for a DWNT can be derived from those for its constituent SWNTs within 3% error. Results of radial expansion/contraction of a SWNT are used to deduce an expression for the van der Waals force. Based on these results, a continuum model is proposed for a MWNT whose response to mechanical deformations computed using engineering theories is the same as that of the MWNT obtained via MM simulations. The continuum structure is comprised of concentric cylindrical tubes interconnected by truss elements. Young’s modulus, Poisson’s ratio, the thickness of each concentric tube, and the stiffness of the truss elements are given. The proposed continuum model is validated by studying bending and the onset of global buckling deformations of a DWNT and its proposed equivalent continuum structure. Carbon nanotubes can be replaced by their equivalent continuum structures when deriving mechanical properties of nanotube reinforced polymeric composites.     

Force constants of BN,SiC, AlN and GaN sheets through discrete homogenization

The force constants related to the bond stretching and angular variation of boron nitride, silicon carbide, aluminium nitride and gallium nitride nanosheets are directly evaluated from ab-initio reference solutions of the Young’s modulus and the Poisson’s ratio. To this end, the analytical expressions of the elastic constants of a generic monolayer hexagonal diatomic sheet are derived, starting from its sticks-and-springs molecular mechanics model, through proper tools of the homogenization of periodic discrete media. Numerical benchmark assessments are given.     

Structure–property relations for balsa wood as a function of density: modelling approach

In order to find an alternative core material to balsa wood in composite sandwich structures, it is important to understand balsa’s elastic properties in relation to its complex microstructural organisation. In the present work, experimental data on the elastic constants and microstructural features of balsa wood were collected for different porosities (densities) and processed into structure–property relations. An inverse problem was solved to predict variation of the cell wall properties with density, such that the collected experimental structure–property relations were satisfied. The Young’s modulus of the cell wall material in the longitudinal direction was found to increase with balsa’s density, which is consistent with the knowledge that the cell wall material stiffens during tree maturation. The value reported in the literature falls in the middle of the predicted range. The proposed micromechanical model also accurately calculated elastic properties of balsa wood at the mesolevel including longitudinal, radial, and tangential directions. The model took into account the presence of ray cells. It was shown that the addition of 15 % of rays increased the radial Young’s modulus up to 4 times with only slight decrease in the longitudinal modulus.     

Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes

This paper investigates the transverse and torsional wave in single- and double-walled carbon nanotubes (SWCNTs and DWCNTs), focusing on the effect of carbon nanotube microstructure on wave dispersion. The SWCNTs and DWCNTs are modeled as nonlocal single and double elastic cylindrical shells. Molecular dynamics (MD) simulations indicate that the wave dispersion predicted by the nonlocal elastic cylindrical shell theory shows good agreement with that of the MD simulations in a wide frequency range up to the terahertz region. The nonlocal elastic shell theory provides a better prediction of the dispersion relationships than the classical shell theory when the wavenumber is large enough for the carbon nanotube microstructure to have a significant influence on the wave dispersion. The nonlocal shell models are required when the wavelengths are approximately less than 2.36×10⁻⁹ and 0.95×10⁻⁹ m for transverse wave in armchair (15,15) SWCNT and torsional wave in armchair (10,10) SWCNT, respectively. Moreover, an MD-based estimation of the scale coefficient e₀ for the nonlocal elastic cylindrical shell model is suggested. Due to the small-scale effects of SWCNTs and the interlayer van der Waals interaction of DWCNTs, the phase difference of the transverse wave in the inner and outer tube can be observed in MD simulations in wave propagation at high frequency. However, the van der Waals interaction has little effect on the phase difference of transverse wave.     

增强六手臂缺失支柱手性拉胀超材料力学性能理论研究

前期研究工作中, 基于有限元分析, 作者发展了一种在大变形范围内具有可调恒定负泊松比的新型增强六手臂缺失支柱手性拉胀超材料. 为了揭示微观结构?力学性能关系, 并进一步指导超材料目标参数设计, 本文在小变形框架下基于能量法建立了表征该拉胀材料等效泊松比和弹性模量的理论模型. 增强六手臂缺失支柱手性拉胀材料由“Z”型手臂元件组成. “Z”型手臂可以被假设为两端简支的欧拉?伯努利梁. 因此, 本文首先推导了两端受集中力和力偶的任意形状欧拉?伯努利梁的应变能. 然后, 考虑平衡条件和变形协调条件进一步给出了材料等效泊松比和弹性模量的理论表达式. 研究表明只有“Z”型梁的内外手臂比为2:1时, 理论表达式才有简洁的形式. 为了更好地利用所推导的理论表达, 基于理论推导, 本文开发了MATLAT图形用户界面 (GUI). 在GUI中输入可描述该超材料几何形状的独立几何参数, 即可直接获取其等效泊松比和弹性模量. 最后, 基于理论结果, 系统讨论了超材料微结构几何参数对其等效力学性能的影响, 并将理论解与有限元计算结果进行了对比. 结果表明, 可以通过调控微结构几何参数获取大范围的目标力学性能.