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.     

Comparison of micromodels describing the elastic properties of diamond

A mechanical model of diatomic crystal lattice with force interaction between atoms and angular interaction between bonds taken into account is proposed. Some relations between the macroscopic moduli of elasticity and the microparameters of the longitudinal rigidity of interatomic bonds and of the angular interaction rigidity are obtained for crystals with diamond lattice. Comparison with experimental data and with other theories describing similar lattices is conducted by using two constants at the microlevel.     

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.     

Strongly nonlinear theory of nanostructure formation owing to elastic and nonelastic strains in crystalline solids

We develop an essentially nonlinear theory of elastic and nonelastic microstrains resulting in the formation of nanostructures. Using the model of mutually penetrating lattices, we generalize the well-known theory of acoustic and optical vibrations to the case of nonlinear interaction between sublattices. This permits treating the sublattice interaction forces as periodic (for example, sinusoidal) functions of the relative displacement of the sublattices. We obtain equations for the macroscopic and microscopic displacement fields containing two characteristic scales of the nanostructure. We find a number of their solutions describing the effects of decrease in the potential interatomic barriers in the external stress field and the formation of defects and domain nanostructure as a result of bifurcation transitions. We prove their stability.     

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.     

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.     

Pair vs many-body potentials: Influence on elastic and plastic behavior in nanoindentation of fcc metals

Molecular-dynamics simulation can give atomistic information on the processes occurring in nanoindentation experiments. In particular, the nucleation of dislocation loops, their growth, interaction and motion can be studied. We investigate how realistic the interatomic potentials underlying the simulations have to be in order to describe these complex processes. Specifically we investigate nanoindentation into a Cu single crystal. We compare simulations based on a realistic many-body interaction potential of the embedded-atom-method type with two simple pair potentials, a Lennard-Jones and a Morse potential. We find that qualitatively many aspects of nanoindentation are fairly well reproduced by the simple pair potentials: elastic regime, critical stress and indentation depth for yielding, dependence on the crystal orientation, and even the level of the hardness. The quantitative deficits of the pair potential predictions can be traced back: (i) to the fact that the pair potentials are unable in principle to model the elastic anisotropy of cubic crystals and (ii) as the major drawback of pair potentials we identify the gross underestimation of the stable stacking fault energy. As a consequence these potentials predict the formation of too large dislocation loops, the too rapid expansion of partials, too little cross slip and in consequence a severe overestimation of work hardening.     

An effective interaction potential model for the shape memory alloy AuCd

The unusual properties of shape memory alloys (SMAs) result from a lattice level martensitic transformation (MT) corresponding to an instability of the SMAs crystal structure. Currently, there exists a shortage of material models that can capture the details of lattice level MTs occurring in SMAs and that can be used for efficient computational investigations of the interaction between MTs and larger-scale features found in typical materials. These larger-scale features could include precipitates, dislocation networks, voids, and even cracks. In this article, one such model is developed for the SMA AuCd. The model is based on effective interaction potentials (EIPs). These are atomic interaction potentials that are explicit functions of temperature. In particular, the Morse pair potential is used and its adjustable coefficients are taken to be temperature dependent. An extensive exploration of the Morse pair potential is performed to identify an appropriate functional form for the temperature dependence of the potential parameters. A fitting procedure is developed for the EIPs that matches, at suitable temperatures, the stress-free equilibrium lattice parameters, instantaneous bulk moduli, cohesive energies, thermal expansion coefficients, and heat capacities of FCC Au, HCP Cd, and the B2 cubic austenite phase of the Au-47.5at%Cd alloy. The resulting model is explored using branch-following and bifurcation techniques. A hysteretic temperature-induced MT between the B2 cubic and B19 orthorhombic crystal structures is predicted. This is the behavior that is observed in the real material. In addition to reproducing the important properties mentioned above, the model predicts, to reasonable accuracy, the transformation strain tensor and captures the latent heat and thermal hysteresis to within an order of magnitude.     

Soliton dynamics and interaction in the Bose–Einstein condensates with harmonic trapping potential and time-varying interatomic interaction

Investigated in this paper is the quasi-one-dimensional Gross–Pitaevskii equation, which describes the dynamics of the Bose–Einstein condensates with the harmonic trapping potential and time-varying interatomic interaction. Via the Horita method and symbolic computation, analytic bright N-soliton solution is obtained. One-, two- and three-soliton solutions are analyzed graphically. Based on the limit analysis on the one- and two-soliton solutions, the modulation on the speed of the matter-wave bright solitons is realized. Via the parameters, the interaction between the matter-wave solitons are adjustable. Furthermore, an approach to construct the interference between the matter-wave solitons has been proposed. Finally, investigation on the three-soliton solution verifies our conclusions drawn from the one and two solitons. Our conclusions might be useful in the fields of the control on the matter-wave solitons, atom lasers, and atomic accelerators.     

Numerical stabilities of loosely coupled methods for robust modeling of lightweight and flexible structures in incompressible and viscous flows

The growing interest to examine the hydroelastic dynamics and stabilities of lightweight and flexible materials requires robust and accurate fluid–structure interaction(FSI)models. Classically, partitioned fluid and structure solvers are easier to implement compared to monolithic methods;however, partitioned FSI models are vulnerable to numerical("virtual added mass") instabilities for cases when the solid to fluid density ratio is low and if the flow is incompressible.As a partitioned method, the loosely hybrid coupled(LHC)method, which was introduced and validated in Young et al.(Acta Mech. Sin. 28:1030–1041, 2012), has been successfully used to efficiently and stably model lightweight and flexible structures. The LHC method achieves its numerical stability by, in addition to the viscous fluid forces, embedding potential flow approximations of the fluid induced forces to transform the partitioned FSI model into a semi-implicit scheme. The objective of this work is to derive and validate the numerical stability boundary of the LHC. The results show that the stability boundary of the LHC is much wider than traditional loosely coupled methods for a variety of numerical integration schemes. The results also show that inclusion of an estimate of the fluid inertial forces is the most critical to ensure the numerical stability when solving for fluid–structure interaction problems involving cases with a solid to fluid-added mass ratio less than one.     

Interaction of molecules with the surface of a solid

The interaction of molecules and atoms with the surface of a solid is considered on the basis of classical mechanics. A two-dimensional square lattice with atoms arranged at the lattice points was taken as the model for describing a solid. It is assumed that only neighboring atoms interact in the solid, while the gas molecules interact with the atoms located in its surface layer.As a result of collisions with the surface, a gas molecule loses a part of its kinetic energy, and this process is characterized by the energy accommodation coefficient. In addition, another coefficient is introduced which takes account of that part of the energy of translational motion converted into energy of internal motion of the molecule (vibration and rotation). The possibility of the occurrence of inelastic losses and some special features of this phenomenon are illustrated by the interaction of a diatomic molecule with an isolated atom.The available experimental data on the interaction of particles of gas with the surface of a solid are essentially associated with the low-energy region (the temperature of the gas is less than or on the order of several hundreds of degrees). One of the objectives of this research was to find the distribution function of particles reflected from the surface; in particular, the hypothesis of diffuse-specular reflection is tested [1–3]. However, the few experimental results provide evidence of the effect of a large number of factors on the nature of the interaction, rather than make it possible to establish strict laws for the process.The theoretical investigations were conducted along the line of improving simple models-instead of modelling a solid by a one-dimensional chain of atoms [4,5], two and three-dimensional lattices are introduced [6–8]. It is noted in [6] that the interaction of gas particles with a one -dimensional chain differs from the interaction with a three-dimensional lattice, and this fact can lead to a considerable divergence in the values of the accommodation coefficient when the mass of the incident molecule is comparable with the mass of an atom of the solid. It also follows from [6] that if we describe the interaction between gas atoms and those of the solid by the Morse potential, then we can select the parameters of the potential so that the calculated data will agree with the experimental data. Moreover, good results are obtained if we make use of parameters of the potential determined on the basis of the combination rule [7].The interaction of atoms of gas with a three-dimensional lattice of finite size is considered in [8]. Forces with the Lennard-Jones potential act between gas atoms and atoms of the solid. The classical equations of motion of all particles were solved numerically on electronic computers.In these references, a gas particle (molecule or atom) is regarded as a whole; however, the problem of the influence of internal degrees of freedom on the coefficients of energy and momentum exchange between the gas particles and the solid is interesting. An attempt is made in this work to take this influence into account on the basis of classical mechanics.     

Discrete Homogenization in Graphene Sheet Modeling

Graphene sheets can be considered as lattices consisting of atoms and of interatomic bonds. Their bond lengths are smaller than one nanometer. Simple models describe their behavior by an energy that takes into account both the interatomic lengths and the angles between bonds. We make use of their periodic structure and we construct an equivalent macroscopic model by means of a discrete homogenization technique. Large three-dimensional deformations of graphene sheets are thus governed by a membrane model whose constitutive law is implicit. By linearizing around a prestressed configuration, we obtain linear membrane models that are valid for small displacements and whose constitutive laws are explicit. When restricting to two-dimensional deformations, we can linearize around a rest configuration and we provide explicit macroscopical mechanical constants expressed in terms of the interatomic tension and bending stiffnesses.     

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.     

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.     

三维双连续铜/石墨自润滑复合材料的构筑及其摩擦磨损性能研究

以聚氨酯海绵为三维连续网络结构模板,采用浸渍法在聚氨酯海绵骨架表面均匀涂敷石墨浆料构筑具有三维连续网络结构的石墨骨架,然后在石墨骨架中填充铜合金粉,经排胶-热压烧结工艺制备石墨相和金属铜呈三维双连续复合型结构的铜/石墨自润滑复合材料.研究考察了三维双连续复合结构对材料承载能力和抗冲击破坏能力的影响,并探究了材料在重载作用下的摩擦磨损行为.结果表明：通过三维双连续结构设计,能够有效改变石墨相的富集状态和分布形式,并借助连续金属铜基体的高承载作用,显著提升材料在重载作用下的减摩抗磨性能.在180 N载荷下与轴承钢相对摩擦时,块体663铜合金和均相铜/石墨复合材料均出现急剧磨损并与摩擦配副发生“卡咬”现象,其中块体663铜合金与配副由于“卡咬”严重而停止试验,均相铜/石墨复合材料的磨痕深度达1.38 mm.然而,具有三维双连续结构的铜/石墨复合材料的摩擦系数可保持约在0.12左右,磨痕深度为0.16 mm,展现出优异的长时间耐磨损性能,磨损率约为5.3×10-6 mm3/(N·m).同时,该结构设计能够大幅减少石墨相与金属铜间的弱界面数量,并有效利用连续石墨相对裂纹传播路径的“歧化”引导和金...     

一种模拟气液两相流的格子波尔兹曼改进模型

基于格子波尔兹曼自由能模型,提出了一种模拟黏性流场中大密度比气液两相流的改进模型. 为了提高模型的精度,在原始模型的基础上计入了邻近点间粒子数密度的传递速率控制,考虑了碰撞项的差分松弛;为了避免两相间大密度比造成的数值不稳定问题,分别采用六点和九点差分格式求解?和?2. 同时,与传统格子波尔兹曼方法不同,实现了由单步碰撞操作到两步操作的转化. 通过对无重力场中气泡的模拟及与已有模型的计算结果的对比分析,表明该模型具有更高的数值精度. 成功模拟了重力作用下,单个上浮气泡的形变和尾涡形成过程,以及水平和竖直方向上两个气泡的相互作用过程,并验证了其质量守恒和体积不可压缩性.      

Estimates of thermoelastic characteristics of composites reinforced by short anisotropic fibers

We construct a mathematical model describing thermomechanical interaction between composite structure elements (isotropic particles of the matrix and anisotropic short fibers) and the macroscopically isotropic elastic medium with desired thermoelastic characteristics. At the first stage of this model, the self-consistency method is used to obtain relations determining the elasticity moduli of the composite, and at the second stage, the model permits determining its linear thermal expansion coefficient. The dual variational statement of the linear thermoelasticity problem in an inhomogeneous solid permits obtaining two-sided estimates for the bulk elasticity modulus, shear modulus, and linear thermal expansion coefficient of the composite under study. The calculated dependencies presented in the paper permit predicting the thermoelastic characteristics of a composite reinforced by anisotropic short fibers (including those in the form of nanostructure elements).     

Macroscopic elastic properties of regular lattices

In this paper we study analytically the elastic properties of the 2-D and 3-D regular lattices consisting of bonded particles. The particle-scale stiffnesses are derived from the given macroscopic elastic constants (i.e. Young's modulus and Poisson's ratio). Firstly a bonded lattice model is presented. This model permits six kinds of relative motion and corresponding forces between each bonded particle pair. By comparing the strain energy distributions between the discrete lattices and the continuum, the explicit relationship between the microscopic and macroscopic elastic parameters can be obtained for the 2-D hexagonal lattice and the 3-D hexagonal close-packed and face-centered cubic structures. The results suggest that the normal stiffness is determined by Young's modulus and the particle size (in 3-D), and that the ratio of the shear to normal stiffness is related to Poisson's ratio. Rotational stiffness depends on the normal stiffness, shear stiffness and particle sizes. Numerical tests are carried out to validate the analytical results. The results in this paper have theoretical implications for the calibration of the spring stiffnesses in the Discrete Element Method.