The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials

A nanoscale continuum theory is established to directly incorporate interatomic potentials into a continuum analysis without any parameter fitting. The theory links interatomic potentials and atomic structure of a material to a constitutive model on the continuum level. The theory is applied to study the linear elastic modulus of a single-wall carbon nanotube. The Young's modulus predicted by this nanoscale continuum theory agrees well with prior experimental results and atomistic studies.     

Continuum Mechanics Modeling and Simulation of Carbon Nanotubes

The understanding of the mechanics of atomistic systems greatly benefits from continuum mechanics. One appealing approach aims at deductively constructing continuum theories starting from models of the interatomic interactions. This viewpoint has become extremely popular with the quasicontinuum method. The application of these ideas to carbon nanotubes presents a peculiarity with respect to usual crystalline materials: their structure relies on a two-dimensional curved lattice. This renders the cornerstone of crystal elasticity, the Cauchy–Born rule, insufficient to describe the effect of curvature. We discuss the application of a theory which corrects this deficiency to the mechanics of carbon nanotubes (CNTs). We review recent developments of this theory, which include the study of the convergence characteristics of the proposed continuum models to the parent atomistic models, as well as large scale simulations based on this theory. The latter have unveiled the complex nonlinear elastic response of thick multiwalled carbon nanotubes (MWCNTs), with an anomalous elastic regime following an almost absent harmonic range.     

An atomistic-based finite-deformation shell theory for single-wall carbon nanotubes

A finite-deformation shell theory is developed for single-wall carbon nanotubes (CNTs) based on the interatomic potential. The modified Born rule for Bravais multi-lattice is used to link the continuum strain energy density to the interatomic potential. The theory incorporates the effect of bending moment and curvature for a curved surface, and accurately accounts for the nonlinear, multi-body atomistic interactions as well as the CNT chirality. It avoids the amibiguous definition of nanotube thickness, and provides the constitutive relations among stress, moment, strain and curvature in terms of the interatomic potential.     

Non-local continuum modeling of carbon nanotubes: Physical interpretation of non-local kernels using atomistic simulations

The longitudinal, transverse and torsional wave dispersion curves in single walled carbon nanotubes (SWCNT) are used to estimate the non-local kernel for use in continuum elasticity models of nanotubes. The dispersion data for an armchair (10,10) SWCNT was obtained using lattice dynamics of SWNTs while accounting for the helical symmetry of the tubes. In our approach, the Fourier transformed kernel of non-local linear elastic theory is directly estimated by matching the atomistic data to the dispersion curves predicted from non-local 1D rod theory and axisymmetric shell theory. We found that gradient models incur significant errors in both the phase and group velocity when compared to the atomistic model. Complementing these studies, we have also performed detailed tests on the effect of length of the nanotube on the axial and shear moduli to gain a better physical insight on the nature of the true non-local kernel. We note that unlike the kernel from gradient theory, the numerically fitted kernel becomes negative at larger distances from the reference point. We postulate and confirm that the fitted kernel changes sign close to the inflection point of the interatomic potential. The numerically computed kernels obtained from this study will aid in the development of improved and efficient continuum models for predicting the mechanical response of CNTs.     

An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation

Carbon nanotubes (CNTs) display unique properties and have many potential applications. Prior theoretical studies on CNTs are based on atomistic models such as empirical potential molecular dynamics (MD), tight-binding methods, or first-principles calculations. Here we develop an atomistic-based continuum theory for CNTs. The interatomic potential is directly incorporated into the continuum analysis through constitutive models. Such an approach involves no additional parameter fitting beyond those introduced in the interatomic potential. The atomistic-based continuum theory is then applied to study fracture nucleation in CNTs by modelling it as a bifurcation problem. The results agree well with the MD simulations.     

An approach to multi-body interactions in a continuum-atomistic context: Application to analysis of tension instability in carbon nanotubes

The tensile strength of single-walled carbon nanotubes (CNT) is examined using a continuum-atomistic (CA) approach. The strength is identified with the onset of the CNT instability in tension. The focus of this study is on the effects of multi-body atomic interactions. Multiscale simulations of nanostructures usually make use of two- and/or three-body interatomic potentials. The three-body potentials describe the changes of angles between the adjacent bonds – bond bending. We propose an alternative and simple way to approximately account for the multi-body interactions. We preserve the pair structure of the potentials and consider the multi-body interaction by splitting the changing bond length into two terms. The first term corresponds to the self-similar deformation of the lattice, which does not lead to bond bending. The second term corresponds to the distortional deformation of the lattice, which does lead to bond bending. Such a split of the bond length is accomplished by means of the spherical–deviatoric decomposition of the Green strain tensor. After the split, the continuum-atomistic potential can be written as a function of two bond lengths corresponding to the bond stretching and bending independently. We apply an example exponential continuum-atomistic potential with the split bond length to the study of tension instability of the armchair and zigzag CNTs. The results of the study are compared with those obtained by Zhang et al. (2004. J. Mech. Phys. Solids 52, 977–998) who studied tension instability of carbon nanotubes by using the Tersoff–Brenner three-body potential, and with recent experimental results on the tensile failure of single walled carbon nanotubes.     

On the functional form of non-local elasticity kernels

The functional form of non-local elasticity kernels is studied within the context of the integral formalism. The study is limited to linear isotropic elasticity. The kernels are derived analytically based on the discrete structure of the material at the atomic scale. Atomistic simulations are used to validate the results. Materials in which the interatomic interactions are represented by pair, as well as embedded atom-type potentials are considered. The derived kernels have a range which extends up to the cut-off radius of the interatomic potential, are positive at the origin, and become negative approximately one atomic distance away, thus departing from the commonly assumed Gaussian functional form. The functional form of the potential and the radial distribution function of interacting neighbors about a representative atom fully define their shape. This new continuum model involves two material length scales that are both derived from atomistics for a Morse solid and for Al. Two applications are considered in closure. It is shown that in strained superlattices, the non-local model predicts maximum stresses that are much larger than those obtained within the local theory. This observation has implications for defect nucleation in these structures. Furthermore, the new non-local model improves upon the Gaussian one by predicting a more realistic wave dispersion relationship, with essentially zero group velocity at the boundary of the Brillouin zone.     

Numerical analyses for the atomistic-based shell theory of carbon nanotubes

A shell theory established from the interatomic potential for carbon nanotubes is compared with the atomistic simulations. This shell theory is implemented in the finite element program ABAQUS via its user-material subroutine UGENS for shells. The numerical results for the representative loadings of tension, torsion and bending agree well with the atomistic simulations, which provide direct validation of this atomistic-based shell theory for carbon nanotubes.     

Modifications to the Cauchy–Born rule: Applications in the deformation of single-walled carbon nanotubes

This paper presents a study of the Cauchy–Born (CB) rule as applied to the deformation analysis of single-walled carbon nanotubes (SWNTs) that are modeled as 2-dimensional manifolds. The C–C bond vectors in the SWNT are assumed to deform according to the local deformation gradient as per the CB rule or a modified version thereof. Aspects of the CB rule related to spatial inhomogeneity of the deformation gradient at the atomic scale are investigated in the context of a specific class of extension–twist deformation problems. Analytic expressions are derived for the deformed bond lengths using the standard CB rule as well as modified versions of the standard CB rule. Since the deformation map is conveniently prescribed in this work, it is possible to compare the performance of these deformation rules with the exact solution (i.e. the exact analytic expression for the deformed bond vectors) given directly by the deformation map. This approach provides insights into the CB rule and its possible modifications for use in more complicated deformations where an explicit deformation map is not available. Specifically, it is concluded that in the case of inhomogeneous deformations at the atomic scale for which the CB rule is only approximate (as demonstrated in Section 1 of this paper), the mean value theorem in calculus can be used as a guide to modify the CB rule and construct a more rigorous and accurate atomistic–continuum connection. The deformed bond lengths are used to formulate an enriched continuum hyperelastic strain energy density function based on interatomic potentials (the multi-body Tersoff–Brenner [Tersoff, J., 1988. New empirical approach for the structure and energy of covalent systems. Phys. Rev. B 37, 6991–7000; Brenner, D.W., 1990. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys. Rev. B 42, 9458–9471] empirical interatomic potential for carbon-carbon bonds is used in this work). The deformation map (and hence the deformation gradient, the bond vectors and the continuum strain energy density) contains certain parameters, some of which are imposed and others determined as a result of energy minimization in the standard variational formulation. Numerical results for kinematic coupling and binding energy per atom are presented in the case of imposed extension and twist deformations on representative chiral, zig-zag and armchair nanotubes using the CB rule and its modifications. These results are compared with the exact solution based on the deformation map which serves as a basis for evaluating the efficacy of these deformation rules. The ideas presented in this paper can also be directly extended to other lattices.     

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.     

Multiscale boundary conditions in crystalline solids: Theory and application to nanoindentation

This paper presents a systematic approach to treating the interfaces between the localized (fine grain) and peripheral (coarse grain) domains in atomic scale simulations of crystalline solids. Based on Fourier analysis of regular lattices structures, this approach allows elimination of unnecessary atomic degrees of freedom over the coarse grain, without involving an explicit continuum model for the latter. The mathematical formulation involves compact convolution operators that relate displacements of the interface atoms and the adjacent atoms on the coarse grain. These operators are defined by geometry of the lattice structure, and interatomic potentials. Application and performance are illustrated on quasistatic nanoindentation simulations with a crystalline gold substrate. Complete atomistic resolution on the coarse grain is alternatively employed to give the benchmark solutions. The results are found to match well for the multiscale and the full atomistic simulations.     

基于高阶Cauchy-Born准则的单壁碳纳米管本构模型

提出了一种基于高阶Cauchy—Born准则建立单壁碳纳米管本构模型的方法。通过引入高阶变形梯度,合理地修正了传统Cauchy—Born准则在描述纳米管变形几何关系时所存在的缺陷。利用原子间相互作用势以及能量等效原理,得到了基于广义连续介质模型的单壁碳纳米管的本构关系。由此得到的本构参数不仅与变形梯度张量F,而且与其梯度F相关,因此是一种广义连续介质模型。利用这样的本构模型,本文还对单壁碳纳米管的杨氏模量进行了预测,并与采用其他方法得到的结果进行了对比,从而证实了所提出方法的有效性。     

Quantum field induced strains in nanostructures and prospects for optical actuation

We develop the mechanics theory of a phenomenon in which strain is induced in nanoscale structures in the absence of applied stress, due solely to the presence of quantum mechanical confinement of charge carriers. The direct effect of strain on electronic structure has been widely studied in recent years, but the “reverse coupling” effect that we investigate, which is only appreciable in the smallest structures, has been largely ignored even though its effects are present in first principles atomistic calculations. We develop a simple effective mass approach that can be used to model this universal physical phenomenon allowing a transparent scheme to identify its occurrence. We relate quantum field induced strain to acoustic polarons and identify the presence of this effect in density functional theory calculations of strain and quantum confinement in free-standing Si and GaAs quantum dots. Finally, we discuss the use of this quantum confinement induced strain as a mechanism for universal optical actuation in nanowire structures in the context of recent experimental results on carbon nanotubes.     

Nanoscale finite element models for vibrations of single-walled carbon nanotubes: atomistic versus continuum

By the atomistic and continuum finite element models, the free vibration behavior of single-walled carbon nanotubes （SWCNTs） is studied. In the atomistic finite element model, the bonds and atoms are modeled by the beam and point mass elements, respectively. The molecular mechanics is linked to structural mechanics to determine the elastic properties of the mentioned beam elements. In the continuum finite element approach, by neglecting the discrete nature of the atomic structure of the nanotubes, they are modeled with shell elements. By both models, the natural frequencies of SWCNTs are computed, and the effects of the geometrical parameters, the atomic structure, and the boundary conditions are investigated. The accuracy of the utilized methods is verified in comparison with molecular dynamic simulations. The molecular structural model leads to more reliable results, especially for lower aspect ratios. The present analysis provides valuable information about application of continuum models in the investigation of the mechanical behaviors of nanotubes.     

Material and structural instabilities of single-wall carbon nanotubes

The nonlinear atomistic interactions usually involve softening behavior. Instability resulting directly from this softening are called the material instability, while those unrelated to this softening are called the structural instability. We use the finite-deformation shell theory based on the interatomic potential to show that the tension instability of single-wall carbon nanotubes is the material instability, while the compression and torsion instabilities are structural instability.     

An atomistic-continuum Cosserat rod model of carbon nanotubes

The focus of the present work is an atomistic-continuum model of single-walled carbon nanotubes (CNTs) based on an elastic rod theory which can exhibit geometric as well as material nonlinearity [Healey, T.J., 2002. Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7, 405-420]. In particular, the single-walled carbon nanotube (SWNT) is modeled as a one-dimensional elastic continuum with some finite thickness bounded by the lateral surface. Exploitation of certain symmetries in the underlying atomic structure leads to suitable representations of the continuum elastic strain energy density in terms of strain measures that capture extension, twist, bending, and shear deformations [Healey, T.J., 2002. Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7, 405-420]. Bridging between the atomic scale and the effective continuum is carried out by parameterization of the continuum elastic energy and determination of the parameters using unit cell atomistic simulations over a range of deformation magnitudes and types. Specifically, the proposed model takes into account (a) bending, (b) twist, (c) shear, (d) extension, (e) coupled extension and twist, and (f) coupled bending and shear deformations. The extracted parameters reveal benefits of accounting for important anisotropic and large-strain effects as improvements over employing traditional, linearly elastic, isotropic, small-strain, continuum models to simulate deformations of atomic systems such as SWNTs. It is envisioned that the proposed approach and the extracted model parameters can serve as a useful input to simulations of SWNT deformations using existing nonlinearly elastic continuum codes based, for example, on the finite element method (FEM).     

The influence of mechanical deformation on the electrical properties of single wall carbon nanotubes

Recent experimental studies and atomistic simulations have shown that carbon nanotubes (CNTs) display strong interplay between the mechanical deformation and electrical properties. We have developed a simple and accurate method to determine atom positions in a uniformly deformed CNT via a continuum analysis based on the interatomic potential. A shift vector is introduced to ensure the equilibrium of atoms. Such an approach, involving only three variables for the entire CNT, agrees very well with the molecular mechanics calculations. We then study the effect of mechanical deformation on the band gap change of single wall CNTs under tension, torsion, and combined tension/torsion via the k-space tight-binding method. Prior studies without this shift vector lead to significant overestimation of the band gap change. It is established that the conducting CNTs may easily become semi-conducting ones subject to mechanical deformation, but the semi-conducting CNTs never become conducting ones upon deformation.     

??????????????????о????

综述了连续介质方法在碳纳米管研究中的最新进展. 主要叙述梁、壳模型, 膜模型,多尺度方法,分子结构力学方法,非局部连续介质方法,以及无网格法的基本原理、 基本方法,及其最新进展,指出其局限性,并预测连续介质方法在碳纳米管研究的发展趋势 和方向.