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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Detailed investigation on single water molecule entering carbon nanotubes

The behavior of a water molecule entering carbon nanotubes (CNTs) is studied. The Lennard-Jones potential function together with the continuum approximation is used to obtain the van der Waals interaction between a single-walled CNT (SWCNT) and a single water molecule. Three orientations are chosen for the water molecule as the center of mass is on the axis of nanotube. Extensive studies on the variations of force, energy, and velocity distributions are performed by varying the nanotube radius and the orientations of the water molecule. The force and energy distributions are validated by those obtained from molecular dynamics (MD) simulations. The acceptance radius of the nanotube for sucking the water molecule inside is derived, in which the limit of the radius is specified so that the nanotube is favorable to absorb the water molecule. The velocities of a single water molecule entering CNTs are calculated and the maximum entrance and the interior velocity for different orientations are assigned and compared.     

Atomistic-based continuum representation of the effective properties of nano-reinforced epoxies

In this paper, an atomistic-based representative volume element (RVE) is developed to characterize the behavior of carbon nanotube (CNT) reinforced amorphous epoxies. The RVE consists of the carbon nanotube, the surrounding epoxy matrix, and the CNT/epoxy interface. An atomistic-based continuum representation is adopted throughout all the components of the RVE. By equating the associated strain energies under identical loading conditions, we were able to homogenize the RVE into a representative fiber. The homogenized RVE was then employed in a micromechanical analysis to predict the effective properties of the newly developed CNT-reinforced amorphous epoxy. Numerical examples show that the effect of volume fraction, orientation, and aspect ratio of the continuous fibres on the properties of the CNT-reinforced epoxy adhesives can be significant. These results have a direct bearing on the design and development of nano-tailored adhesives for use in structural adhesive bonds.     

Modelling temperature-dependent fracture nucleation of SWCNTs using atomistic-based continuum theory

The fracture behaviour of carbon nanotubes depends largely on temperature, defect distribution, and geometric features. In this paper, the effect of temperature upon fracture nucleation of single-walled carbon nanotubes (SWCNTs) is investigated using an atomistic-based continuum theory. The temperature effects are described in terms of a modified Cauchy–Born rule based on the assumptions that the deformation is sufficiently small and locally homogeneous. Furthermore, it is assumed that the atoms have the same local vibration mode at a given temperature. The first derivative of the free energy density, which is a function of both the deformation gradient and the temperature, enables the determination of the second Piola–Kirchhoff stress. In the present study, the fracture nucleation is modelled as a bifurcation of a homogeneously deformed nanotube at a critical strain. The model predictions show that the fracture strain decreases with increasing temperature, while the elastic stiffness remains largely unchanged.     

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.     

Static and dynamic analysis of two-dimensional graphite lattices

Plane problems of statics and dynamics of graphite lattice are considered in the linear approximation. Comparative analysis of two models of interatomic interaction is carried out. One of these models is based on pairwise moment interaction, and the other is the Brennermodel where the variation in the angles between the segments connecting the atom under study with three nearest neighbors is additionally taken into account. The lattice tensile and shear rigidity in two directions is studied by straightforward calculations. The propagation of harmonic tensile and shear waves it two directions is considered. In problems of both statics and wave propagation, the results are compared with similar results for the equivalent continuum. It turned out that in the problems of statics, the Brenner model (after averaging) leads to an isotropic momentless continuum, while the model with pair interaction lead to the moment Cosserat continuum. In problems of wave propagation, both of these models give the same qualitative results. The velocities of acoustic parallel extension-compression wave propagation in a lattice are close to the wave velocity in the continuum but do not coincide with it. The difference increases with decreasing wave length and depends on the wave propagation direction. In the case of shear wave propagation in a lattice, the velocity of acoustic shear wave propagation in the pair moment potential model significantly (in the leading terms) depends on the direction of its propagation. The optical short waves are discovered and some of their properties are described.     

Can a single-wall carbon nanotube be modeled as a thin shell?

Single-wall carbon nanotubes (SWCNT) have been frequently modeled as thin shells, but the shell thickness and Young's modulus reported in literatures display large scattering. The order of error to approximate SWCNTs as thin shells is studied in this paper via an atomistic-based finite-deformation shell theory, which avoids the shell thickness and Young's modulus, but links the tension and bending rigidities directly to the interatomic potential. The ratio of atomic spacing (Δ≈0.14 nm) to the radius of SWCNT, Δ/R, which ranges from zero (for graphene) to 40% [for a small (5,5) armchair SWCNT (R=0.35 nm)], is used to estimate the order of error. For the order of error O[(Δ/R)³], SWCNTs cannot be represented by a conventional thin shell because their constitutive relation involves the coupling between tension and curvature and between bending and strain. For the order of error O[(Δ/R)²], the tension and bending (shear and torsion) rigidities of SWCNTs can be represented by an elastic orthotropic thin shell, but the thickness and elastic modulus cannot. Only for the order of error O(Δ/R), a universal constant shell thickness can be defined and SWCNTs can be modeled as an elastic isotropic thin shell.     

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).     

An atomistically validated continuum model for strain relaxation and misfit dislocation formation

In this paper, molecular dynamics (MD) calculations have been used to examine the physics behind continuum models of misfit dislocation formation and to assess the limitations and consequences of approximations made within these models. Without compromising the physics of misfit dislocations below a surface, our MD calculations consider arrays of dislocation dipoles constituting a mirror imaged “surface”. This allows use of periodic boundary conditions to create a direct correspondence between atomistic and continuum representations of dislocations, which would be difficult to achieve with free surfaces. Additionally, by using long-time averages of system properties, we have essentially reduced the errors of atomistic simulations of large systems to “zero”. This enables us to deterministically compare atomistic and continuum calculations. Our work results in a robust approach that uses atomistic simulation to accurately calculate dislocation core radius and energy without the continuum boundary conditions typically assumed in the past, and the novel insight that continuum misfit dislocation models can be inaccurate when incorrect definitions of dislocation spacing and Burgers vector in lattice-mismatched systems are used. We show that when these insights are properly incorporated into the continuum model, the resulting energy density expression of the lattice-mismatched systems is essentially indistinguishable from the MD results.     

A semi-analytical approach for calculating the equilibrium structure and radial breathing mode frequency of single-walled carbon nanotubes

A semi-analytical model for determining the equi-librium configuration and the radial breathing mode (RBM) frequency of single-wall carbon nanotubes (CNTs) is pre-sented. By taking advantage of the symmetry characteristics, a CNT structure is represented by five independent vari-ables. A line search optimization procedure is employed to determine the equilibrium values of these variables by minimizing the potential energy. With the equilibrium con-figuration obtained, the semi-analytical model enables an efficient calculation of the RBM frequency of the CNTs. The radius and radial breathing mode frequency results obtained from the semi-analytical approach are compared with those from molecular dynamics (MD) and ab initio calculations. The results demonstrate that the semi-analytical approach offers an efficient and accurate way to determine the equilib-rium structure and radial breathing mode frequency of CNTs.