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.     

The Peierls stress in non-local elasticity

The effect of non-locality on the Peierls stress of a dislocation, predicted within the framework of the Peierls-Nabarro model, is investigated. Both the integral formulation of non-local elasticity and the gradient elasticity model are considered. A modification of the non-local kernel of the integral formulation is proposed and its effect on the dislocation core shape and size, and on the Peierls stress are discussed. The new kernel is longer ranged and physically meaningful, improving therefore upon the existing Gaussian-like non-locality kernels. As in the original Peierls-Nabarro model, lattice trapping cannot be captured in the purely continuum non-local formulation and therefore, a semi-discrete framework is used. The constitutive law of the elastic continuum and that of the glide plane are considered both local and non-local in separate models. The major effect is obtained upon rendering non-local the constitutive law of the continuum, while non-locality in the rebound force law of the glide plane has a marginal effect. The Peierls stress is seen to increase with increasing the intrinsic length scale of the non-local formulation, while the core size decreases accordingly. The solution becomes unstable at intrinsic length scales larger than a critical value. Modifications of the rebound force law entail significant changes in the core configuration and critical stress. The discussion provides insight into the issue of internal length scale selection in non-local elasticity models.     

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.     

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.     

A rod model for three dimensional deformations of single-walled carbon nanotubes

A continuum model for single-walled carbon nanotubes (SWCNT) is presented which is based on an extension to the special Cosserat theory of rods (Kumar and Mukherjee, 2011). The model allows deformation of a nanotube’s lateral surface in a one dimensional framework and hence is an efficient substitute to the commonly used two dimensional shell models for nanotubes. The model predicts a new coupling mode in chiral nanotubes – coupling between twist and cross-sectional shrinkage implying that the three deformation modes (extension, twist and cross-sectional shrinkage) are all coupled to each other. Atomistic simulations based on the density functional based tight binding method (DFTB) are performed on a (9, 6) SWCNT and the simulation data is used to estimate material parameters of this rod model. A peculiar behavior of the nanotube is observed when it is axially stretched – induced rotation of each cross-section is equal in magnitude but opposite to that of its two neighboring cross-sections. This is shown to be an effect of relative shift/inner-displacement between the two SWCNT sub-lattices.     

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.     

Discrete and non-local elastica

In this paper, the buckling and post-buckling behavior of an elastic lattice system referred to as the discrete elastica problem is investigated using an equivalent non-local continuum approach. The geometrically exact post-buckling analysis of the elastic chain, also called Hencky system, is first numerically solved using the shooting method. This discrete physical model is also mathematically equivalent to a finite difference formulation of the continuum elastica. Starting from the exact difference equations of the discrete problem, a continualization method is applied for approximating the difference operators by differential ones, in order to better characterize the discrete system by an enriched continuous one. It is shown that the new continuum associated with the discrete system exactly fits the discrete elastica post-buckling problem, where the non-locality is of Eringen׳s type (also called stress gradient non-local model). An asymptotic expansion is performed for both the discrete and the non-local continuum models, in order to approximate the post-buckling branches of the discrete system. Some numerical investigations show the efficiency of the non-local approach, especially for capturing the scale effects inherent to the cell size of the lattice model.     

Multiscale modeling of the dynamics of solids at finite temperature

We develop a general multiscale method for coupling atomistic and continuum simulations using the framework of the heterogeneous multiscale method (HMM). Both the atomistic and the continuum models are formulated in the form of conservation laws of mass, momentum and energy. A macroscale solver, here the finite volume scheme, is used everywhere on a macrogrid; whenever necessary the macroscale fluxes are computed using the microscale model, which is in turn constrained by the local macrostate of the system, e.g. the deformation gradient tensor, the mean velocity and the local temperature. We discuss how these constraints can be imposed in the form of boundary conditions. When isolated defects are present, we develop an additional strategy for defect tracking. This method naturally decouples the atomistic time scales from the continuum time scale. Applications to shock propagation, thermal expansion, phase boundary and twin boundary dynamics are presented.     

The continuum elastic and atomistic viewpoints on the formation volume and strain energy of a point defect

We discuss the roles of continuum linear elasticity and atomistic calculations in determining the formation volume and the strain energy of formation of a point defect in a crystal. Our considerations bear special relevance to defect formation under stress. The elasticity treatment is based on the Green's function solution for a center of contraction or expansion in an anisotropic solid. It makes possible the precise definition of a formation volume tensor and leads to an extension of Eshelby's [Proc. R. Soc. London Ser. A 241 (1226), 376] result for the work done by an external stress during the transformation of a continuum inclusion. Parameters necessary for a complete continuum calculation of elastic fields around a point defect are obtained by comparing with an atomistic solution in the far field. However, an elasticity result makes it possible to test the validity of the formation volume that is obtained via atomistic calculations under various boundary conditions. It also yields the correction term for formation volume calculated under these boundary conditions. Using two types of boundary conditions commonly employed in atomistic calculations, a comparison is also made of the strain energies of formation predicted by continuum elasticity and atomistic calculations. The limitations of the continuum linear elastic treatment are revealed by comparing with atomistic calculations of the formation volume and strain energies of small crystals enclosing point defects.     

Construction of multi-dimensional isotropic kernels for nonlocal elasticity based on phonon dispersion data

Kernels for non-local elasticity are in general obtained from phonon dispersion relations. However, non-local elastic kernels are in the form of three-dimensional (3D) functions, whereas the dispersion relations are always in the form of one-dimensional (1D) frequency versus wave number curves corresponding to a particular wave direction. In this paper, an approach to build 2D and 3D kernels from 1D phonon dispersion data is presented. Our particular focus is on isotropic media where we show that kernels can be obtained using Fourier–Bessel transform, yielding axisymmetric kernel profiles in reciprocal and real spaces. These kernel functions are designed to satisfy the necessary requirements for stable wave propagation, uniformity of nonlocal stress and stress regularization. The proposed concept is demonstrated by developing some physically meaningful 2D and 3D kernels that will find useful applications in nonlocal mechanics. Relative merits of the kernels obtained via proposed methods are explored by fitting 1D kernels to dispersion data for Argon and using the kernel to understand the size effect in non local energy as seen from molecular simulations. A comparison of proposed kernels is made based on their predictions of stress field around a crack tip singularity.     

Study of dynamic stability of the double-walled carbon nanotube under axial loading embedded in an elastic medium by the energy method

In this paper, the dynamic stability of single- and double-walled carbon nanotubes (SWCNT and DWCNT) under dynamic axial loading is investigated using the continuum mechanics model and the minimum total energy method. The natural frequencies of the SWCNT and the critical dynamic axial load of the SWCNT and DWCNT are obtained using the Rayleigh-Ritz method. The effects of the elastic medium and the van der Waals forces between the two layers in the DWCNT are taken into account using the Winkler model and Lennard-Jones theory, respectively. The effect of the small length scale is also considered using the Eringen Model. The critical dynamic axial load is increased by inserting an inner carbon nanotube (CNT) into an isolated CNT embedded in an elastic medium.     

Non-local continuum model and its numerical simulation for localization problem

A non-local continuum model for strain-softening simply taking plastic strain or damage variable as a non-local variable is derived by using the additive decomposition principle of finite deformation gradient. At the same time, variational equations, their finite element formulations and numerical convoluted integration algorithm of the model in current configuration usually called co-moving coordinate system are given. Stability and convergence of the model are proven by means of the weak convergence theorem of general function and the convoluted integration theory. Mathematical and physical properties of the characteristic size for material or structure are accounted for within the context of a statistical weighted or kernel function, and way is investigated. Numerical simulation shows that this model is suitable for to analyzing deformation localization problems. The project supported by the National Natural Science Foundation of China (No. 19632030).     

Discontinuous bifurcation analysis in fracture energy-based gradient plasticity for concrete

Conditions for discontinuous bifurcation in limit states of selective non-local thermodynamically consistent gradient theory for quasi-brittle materials like concrete are evaluated by means of both geometrical and analytical procedures. This constitutive formulation includes two internal lengths, one related to the strain gradient field that considers the degradation of the continuum in the vicinity of the considered material point. The other characteristic length takes into account the material degradation in the form of energy release in the cracks during failure process evolution.The variation from ductile to brittle failure in quasi-brittle materials is accomplished by means of the pressure dependent formulation of both characteristic lengths as described by Vrech and Etse (2009).In this paper the formulation of the localization ellipse for constitutive theories based on gradient plasticity and fracture energy plasticity is proposed as well as the explicit solutions for brittle failure conditions in the form of discontinuous bifurcation. The geometrical, analytical and numerical analysis of discontinuous bifurcation condition in this paper are comparatively evaluated in different stress states and loading conditions.The included results illustrate the capabilities of the thermodynamically consistent selective non-local gradient constitutive theory to reproduce the transition from ductile to brittle and localized failure modes in the low confinement regime of concrete and quasi-brittle materials.     

Generalized thermomechanics with dual internal variables

The formal structure of generalized continuum theories is recovered by means of the extension of canonical thermomechanics with dual weakly non-local internal variables. The canonical thermomechanics provides the best framework for such generalization. The Cosserat, micromorphic, and second gradient elasticity theory are considered as examples of the obtained formalization.     

基于Riesz势空间分数阶算子的非局部粘弹性力学元件

将幂函数引入Eringen非局部线粘弹性本构,导出Riesz势形式的应力-应变关系.利用该关系,构造非局部弹簧和非局部阻尼器两类元件;利用元件的串联和并联,建立非局部Kelvin和非局部Maxwell粘弹性模型,推导模型的松弛模量和蠕变柔量.进一步,给出非局部粘弹性模型在生物组织超声波耗散建模中的应用.     

Finite deformation continuum model for single-walled carbon nanotubes

A continuum-based model for computing strain energies and estimating Young’s modulus of single-walled carbon nanotubes (SWCNTs) is developed by using an energy equivalence-based multi-scale approach. A SWCNT is viewed as a continuum hollow cylinder formed by rolling up a flat graphite sheet that is treated as an isotropic continuum plate. Kinematic analysis is performed on the continuum level, with the Hencky (true) strain and the Cauchy (true) stress being employed to account for finite deformations. Based on the equivalence of the strain energy and the molecular potential energy, a formula for calculating Young’s modulus of SWCNTs is derived. This formula, containing both the molecular and continuum scale parameters, directly links macroscopic responses of nanotubes to their molecular structures. Sample numerical results show that the predictions by the new model compare favorably with those by several existing continuum and molecular dynamics models.     

Non-local finite element analysis of damped beams

Non-local viscoelastic beam models are used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local damping models the internal force of the non-local model is obtained as weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two-node beam elements. However, for non-local damping, nodes remote from the element do have an effect on the energy expressions, and hence on the damping matrix. The expressions of these direct and cross damping matrices may be obtained explicitly for some common spatial kernel functions and Euler–Bernoulli beam theory. Alternatively numerical integration may be applied to obtain solutions. Examples are given where the eigenvalues are compared to the exact solution for a pinned–pinned beam to demonstrate the convergence of the finite element method. The results for beams with other boundary conditions are used to demonstrate the versatility of the finite element technique.     

Lattice with long-range interaction of power-law type for fractional non-local elasticity

Lattice models with long-range interactions of power-law type are suggested as a new type of microscopic model for fractional non-local elasticity. Using the transform operation, we map the lattice equations into continuum equation with Riesz derivatives of non-integer orders. The continuum equations that are obtained from the lattice model describe fractional generalization of non-local elasticity models. Particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case.     

An atomistic and non-classical continuum field theoretic perspective of elastic interactions between defects (force dipoles) of various symmetries and application to graphene

Force multipoles are employed to represent various types of defects and physical phenomena in solids: point defects (interstitials, vacancies), surface steps and islands, proteins on biological membranes, inclusions, extended defects, and biological cell interactions among others. In the present work, we (i) as a prototype simple test case, conduct quantum mechanical calculations for mechanics of defects in graphene sheet and in parallel, (ii) formulate an enriched continuum elasticity theory of force dipoles of various anisotropies incorporating up to second gradients of strain fields (thus accounting for nonlocal dispersive effects) instead of the usual dispersion-less classical elasticity formulation that depends on just the strain (c.f. Peyla, P., Misbah, C., 2003. Elastic interaction between defects in thin and 2-D films. Eur. Phys. J. B. 33, 233-247). The fundamental Green's function is derived for the governing equations of second gradient elasticity and the elastic self and interaction energies between force dipoles are formulated for both the two-dimensional thin film and the three-dimensional case. While our continuum results asymptotically yield the same interaction energy law as Peyla and Misbah for large defect separations (∼1/rⁿ for defects with n-fold symmetry), the near-field interactions are qualitatively far more complex and free of singularities. Certain qualitative behavior of defect mechanics predicted by atomistic calculations are well captured by our enriched continuum models in contrast to classical elasticity calculations. For example, consistent with our atomistic calculations of defects in isotropic graphene, even two dilation centers show a finite interaction (as opposed to classical elasticity that predicts zero interaction). We explicitly find the physically consistent result that the self-energy of a defect is equivalent to half the interaction energy between two identical defects when they “merge” into each other. The atomistic, classical elastic and the enriched continuum predictions are thoroughly compared for two types of defects in graphene: Stone-Wales and divacancy.