On the continuum modeling of carbon nanotubes

We have recently proposed a nanoscale continuum theory for carbon nanotubes. The theory links continuum analysis with atomistic modeling by incorporating interatomic potentials and atomic structures of carbon nanotubes directly into the constitutive law. Here we address two main issues involved in setting up the nanoscale continuum theory for carbon nanotubes, namely the multi-body interatomic potentials and the lack of centrosymmetry in the nanotube structure. We explain the key ideas behind these issues in establishing a nanoscale continuum theory in terms of interatomic potentials and atomic structures.     

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 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.     

Surface stress effect on the vibration and instability of nanoscale pipes conveying fluid based on a size-dependent Timoshenko beam model

Presented in this paper is a precise investigation of the effect of surface stress on the vibration characteristics and instability of fluid-conveying nanoscale pipes.To this end,the nanoscale pipe is modeled as a Timoshenko nanobeam.The equations of motion of the nanoscale pipe are obtained based on Hamilton's principle and the Gurtin-Murdoch continuum elasticity incorporating the surface stress effect.Afterwards,the generalized differential quadrature method is employed to discretize the governing equations and associated boundary conditions.To what extent important parameters such as the thickness,material and surface stress modulus,residual surface stress,surface density,and boundary conditions influence the natural frequency of nanoscale pipes and the critical velocity of fluid is discussed.     

Multiscale modeling of fracture of MgO: Sensitivity of interatomic potentials

Three different interatomic potentials, namely, B-G I Model, B-G II Model and L-C Model, are used in multiscale modeling and simulation of a center-cracked specimen made of magnesia subjected to monotonically increasing loading. The specimen is decomposed into a far field, a near field and a crack-tip region. The analytical solution in the far field from linear elastic fracture mechanics (LEFM) is utilized. The solution of the near field is based on a multiscale field theory. In the crack-tip region, molecular dynamics (MD) simulation is employed. These methodologies are integrated to simulate mixed mode fracture of magnesia (MgO). Three different interatomic potentials are examined and the interatomic potential and interatomic force between Mg-Mg, Mg-O and O-O are shown. The numerical results of crack propagation demonstrate that (1) crack closure is witnessed in B-G I Model but not in B-G II Model and L-C Model, (2) B-G II Model and L-C Model diverge in the early stage. The cause of instability and the remedy are also discussed.     

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.     

Couple stresses in crystalline solids: origins from plastic slip gradients, dislocation core distortions, and three-body interatomic potentials

In the presence of plastic slip gradients, compatibility requires gradients in elastic rotation and stretch tensors. In a crystal lattice the gradient in elastic rotation can be related to bond angle changes at cores of so-called geometrically necessary dislocations. The corresponding continuum strain energy density can be obtained from an interatomic potential that includes two- and three-body terms. The three-body terms induce restoring moments that lead to a couple stress tensor in the continuum limit. The resulting stress and couple stress jointly satisfy a balance law. Boundary conditions are obtained upon stress, couple stress, strain and strain gradient tensors. This higher-order continuum theory was formulated by Toupin (Arch. Ration. Mech. Anal. 11 (1962) 385). Toupin's theory has been extended in this work to incorporate constitutive relations for the stress and couple stress under multiplicative elastoplasticity. The higher-order continuum theory is exploited to solve a boundary value problem of relevance to single crystal and polycrystalline nano-devices. It is demonstrated that certain slip-dominated deformation mechanisms increase the compliance of nanostructures in bending-dominated situations. The significance of these ideas in the context of continuum plasticity models is also dwelt upon.     

TRANSVERSE VIBRATION OF A HANGING NONUNIFORM NANOSCALE TUBE BASED ON NONLOCAL ELASTICITY THEORY WITH SURFACE EFFECTS

The aim of this paper is to study the free transverse vibration of a hanging nonuni- form nanoscale tube. The analysis procedure is based on nonlocal elasticity theory with surface effects. The nonlocal elasticity theory states that the stress at a point is a function of strains at all points in the continuum. This theory becomes significant for small-length scale objects such as micro- and nanostructures. The effects of nonlocality, surface energy and axial force on the natural frequencies of the nanotube are investigated. In this study, analytical solutions are formulated for a clamped-free Euler-Bernoulli beam to study the free vibration of nanoscale tubes.     

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.     

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。     

An atomistic investigation of the kinetics of detwinning

This paper presents a “first principles” atomistic study of the dynamics of detwinning in a shape-memory alloy. In order to describe the macroscopic motion of twin boundaries, the continuum theory of twinning must be provided with a “kinetic relation”, i.e. a relation between the driving force and the propagation speed. This kinetic relation is a macroscopic characterization of the underlying atomistic processes. The goal of the present atomistic study is to provide the continuum theory with this kinetic relation by extracting the essential macroscopic features of the dynamics of the atoms. It also aims to elucidate the mechanism underlying the process of detwinning.The material studied is stoichiometric nickel-manganese, and interatomic interactions are described using three physically motivated Lennard-Jones potentials. The effect of temperature and shear stress on detwinning — specifically on the rate of transformation from one variant of martensite to the other — is examined using molecular dynamics. An explicit formula for this (kinetic) relation is obtained by fitting an analytic expression to the simulation results. The numerical experiments also verify that transverse ledge propagation is the mechanism underlying twin-boundary motion. All calculations are carried out in a two-dimensional setting.     

High-frequency antiplane wave propagation in ultra-thin films with nanostructures

Ultrasonic wave propagation is one of powerful and popular methods for measuring mechanical properties of solids even at nano scales. The extraction of material constants from the measured wave data may not be accurate and reliable when waves of short wavelengths are used. The objective of this paper is to study the high-frequency antiplane wave propagation in ultra-thin films at nanoscale. A developed continuum microstructure theory will be used to capture the effect of nanostructures in ultra-thin films. This continuum theory is developed from assumed displacement fields for nanostructures. Local kinematic variables are introduced to express these local displacements and are subjected to internal continuity conditions. The accuracy of the theory is verified by comparing the results with those of the lattice model for the antiplane problem in an infinite elastic medium. Specifically, dispersion curves and corresponding displacement fields for antiplane wave propagation in the ultra-thin films are studied. The inadequacy of the conventional continuum theory is discussed.     

凝聚态物质计算和模拟中使用的相互作用势

原子间相互作用势是凝聚态物质在原子尺度上进行计算机模拟的基础, 特别是用分 子动力学和Monte Carlo方法对凝聚态物质的性质和过程进行模拟时, 合适的原子 间相互作用势是得到有意义的结果的前提和条件. 可依据不同类型的相互作用如共 价键、离子键、金属键和Van der Waals力等构建不同类型的原子间相互作用势, 而且同一类型的相互作用也因所处理的性质或过程(如体积、表面、团簇、缺陷等) 不同所采用的形式也不相同, 这样就构建了大量的各种形式的原子间相互作用势. 本文对凝聚态的计算机模拟中常用的原子间相互作用势进行分类介绍和简要的评 述.     

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.     

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.     

Size effects in indentation response of thin films at the nanoscale: A molecular dynamics study

The indentation response of Ni thin films of thicknesses in the nanoscale was studied using molecular dynamics simulations with embedded atom method (EAM) interatomic potentials. A series of simulations were performed in films in the [1 1 1] orientation with thicknesses varying from 4 to 12.8 nm. The study included both single crystal films and films containing low angle grain boundaries perpendicular to the film surface. The simulation results for single crystal films show that as film thickness decreases larger forces are required for similar indentation depths but the contact stress necessary to emit the first dislocation under the indenter is nearly independent of film thickness. The low angle grain boundaries can act as dislocation sources under indentation. The mechanism of preferred dislocation emission from these boundaries operates at stresses that are lower as the film thickness increases and is not active for the thinnest films tested. These results are interpreted in terms of a simple model.     

Band structures of transverse waves in nanoscale multilayered phononic crystals with nonlocal interface imperfections by using the radial basis function method

A radial basis function collocation method based on the nonlocal elastic continuum theory is developed to com-pute the band structures of nanoscale multilayered phononic crystals. The effects of nonlocal imperfect interfaces on band structures of transverse waves propagating obliquely or verti-cally in the system are studied. The correctness of the present method is verified by comparing the numerical results with those obtained by applying the transfer matrix method in the case of nonlocal perfect interface. Furthermore, the influ-ences of the nanoscale size, the impedance ratio and the incident angle on the cut-off frequency and band structures are investigated and discussed in detail. Numerical results show that the nonlocal interface imperfections have signif-icant effects on the band structures in the macroscopic and microscopic scale.