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.     

Young’s modulus prediction of hexagonal nanosheets and nanotubes based on dimensional analysis and atomistic simulations

The present work investigates Young’s modulus of hexagonal nanosheets and nanotubes based on dimensional analysis and molecular mechanics. Using second derivatives of the strain energy density revealed from molecular dynamics simulations at 0 K (i.e., molecular mechanics) with harmonic potentials for various combinations of force constants, Young’s modulus have been computed for single-walled armchair and zigzag nanotubes of different radii. This parametric study with the aid of dimensional analysis allows explicitly establishing Young’s modulus of (n, n) armchair and (n, 0) zigzag nanotubes as functions of the force constants, bond length and chiral index n. Proposed formulae are applied to estimate Young’s modulus of graphene, boron nitride, silicon carbide sheets and their nanotubes. The accuracy of the proposed formulae are verified and discussed with available data in the literature for these three sheets and their nanotubes.     

An accurate molecular mechanics model for computation of size-dependent elastic properties of armchair and zigzag single-walled carbon nanotubes

In this paper, an analytical solution based on a molecular mechanics model is developed to evaluate the mechanical properties of armchair and zigzag single-walled carbon nanotubes (SWCNTs). Adopting the Perdew–Burke–Ernzerhof (PBE) exchange correlation, the density functional theory (DFT) calculations are performed within the generalized gradient approximation (GGA) to evaluate force constants used in the molecular mechanics model. After that, based on the principle of molecular mechanics, explicit expressions are proposed to obtain surface Young’s modulus, Poisson’s ratio and surface shear modulus of SWCNTs corresponding to both types of armchair and zigzag chiralities. Based on the DFT calculations, it is found that the flexural rigidity of graphene is independent of the type of chirality which indicates the isotropic characteristic of this material. Moreover, it is observed that for the all values of nanotube diameter, surface Young’s modulus for the armchair nanotube is more than that of zigzag nanotube. It is shown that the trend predicted by the present model is in good agreement with other models which confirms the validity as well as the accuracy of the present molecular mechanics model.     

六方氮化硼韧脆转变性能的理论研究

基于六方氮化硼原子间Tersoff势函数,本文建立了手性依赖的六方氮化硼“杆-簧”模型,得到了其在大变形下的非线性力学行为。结果表明六方氮化硼断裂模式从扶手椅型(Armchair,手性角为0o)到锯齿型(Zigzag,手性角为30o)由韧性断裂转变为脆性断裂。通过分析拉伸过程中键长键角变化情况,键角变化所起作用超过键长变化导致韧性断裂,而键长起主导作用时则为脆性断裂。探讨不同手性氮化硼非线性力学行为,获得两种断裂模式的分界点大约为手性角15o。相同条件下,石墨烯在拉伸过程中键长变化起支配作用,而氮化硼除了键长变化的影响之外,键角的变化对整个过程也有着至关重要的作用。最后,与分子动力学模拟结果对比显示本文的理论结果是可靠的。本研究对理解六方氮化硼优异的力学性能及其在微纳器件领域的应用提供了理论支撑。     

Vibration of two-dimensional hexagonal boron nitride

The dynamic behavior of two-dimensional nanostructures is important to the future application of nano devices. The vibrational behaviors of single-layered hexagonal boron nitride(h-BN) are studied by molecular dynamics simulation and continuum plate model. The bending stiffness and Poisson's ratios of h-BN along zigzag direction and armchair direction are calculated. H-BN is softer compared with graphene. The continuum plate model can predict the vibration of h-BN with four edge-clamped boundary conditions well. The electric fields in different directions have obvious influence on the vibration of h-BN. The natural frequency of h-BN changes linearly with the electric field intensity along the polarization direction. The natural frequency of h-BN decreases with the increase of electric field intensity along both positive and negative nonpolarization direction. While the natural frequency of h-BN increases with the increase of electric field intensity along both positive and negative transverse electric field.     

Mechanical behaviour of carbon nanotubes under combined twisting–bending

The main objective of this paper is to investigate the mechanical behaviour (strength and stiffness) of carbon nanotubes (CNTs) under combinations of bending and twisting. In order to achieve this goal, molecular dynamics (MD) simulations of bended and twisted CNTs are performed. The LAMMPS code is used, the AIREBO potential is considered for CC bonds, the temperature is kept at 300 K and incremental bending and twisting rotations are imposed to the CNT. Two types of CNTs are analyzed, including zig-zag (8,0) and armchair (5,5) CNTs with similar radius and length. The CNTs are also analyzed for pure bending and pure twisting. The main results are shown in the form of diagrams of energy and moment against imposed rotations. Some relevant conclusions are drawn concerning the influence of loading (bending and twisting) on the stiffness, strength and failure of CNTs: namely, it is concluded that armchair CNTs possess higher strength and fracture toughness under twisting–bending loading than zigzag CNTs; additionally, it is found that both CNTs (armchair and zigzag) still support moderate-to-high bending levels without failure after being extremely twisted and torsionally buckled, even for twisting angles four times those corresponding to torsional buckling; finally, the results prove that CNTs, mostly armchair ones, exhibit very high twisting–bending stiffness and strength and can be used with confidence as torsional spring elements in nanoelectromechanical systems (NEMS).     

Buckling analysis of pristine and defected graphene

The buckling behavior of monolayer graphene (pristine and vacancy-defected) and bilayer graphene (pristine) loaded in the armchair direction was simulated for different boundary conditions using a truss FE model, representing the exact atomic lattice of graphene, and a FE model of an equivalent 2D plate. The critical buckling stress of pristine monolayer graphene was derived as a function of aspect ratio. The results from the two FE models coincide and are in very good agreement with established analytical solutions. With increasing the aspect ratio, the critical buckling stress of monolayer graphene decreases until the value of 2 from which the effect starts to diminish. Using the truss FE model, the effect of randomly dispersed vacancies on the critical buckling stress and buckling mode of monolayer graphene was studied. It was found that the critical buckling stress decreases dramatically with increasing the defect density: for a defect density of 10%, the critical buckling stress decreases by almost 50%. Moreover, the presence of defects was found to affect the highest buckling modes (above 3) even at low densities. Bilayer graphene has a totally different critical buckling stress than monolayer graphene due to the effect of van der Waals forces which depends on the applied boundary conditions.     

Stone-Wales拓扑缺陷对石墨烯拉伸力学性能的影响

采用Tersoff势对含Stone-Wales(SW)拓扑缺陷的单层石墨烯薄膜的单向拉伸力学性能进行了分子动力学模拟,分别研究了SW拓扑缺陷对扶手椅型和锯齿型石墨烯拉伸力学性能及变形机制的影响.研究结果表明,单个SW缺陷对两种手性石墨烯薄膜的杨氏模量几乎无影响,而对薄膜的强度、应变等力学性能和变形破坏机制的影响与手性有...     

Investigation of vacancy defects effects on the buckling behavior of SWCNTs via a structural mechanics approach

In this paper, the influence of various vacancy defects on the critical buckling loads and strains in carbon nanotubes under axial compression is investigated via a new structural model in ABAQUS software. The necessity of desirable conditions and expensive tests for experimental methods, in addition to the time expenditure required for atomic simulations, are the motivation for this work, which, in addition to yielding accurate results, avoids the obstacles of the previous methods. In fact, this model is a combination of other structural models designed to eliminate the deficiencies inherent in individual approaches. Because the present model is constructed in the CAE space of ABAQUS, there is no need to program for different loading and boundary conditions. A nonlinear connector is considered for modeling of stretching and torsional interactions, and a nonlinear spring is used for modeling of the angle variation interactions. A Morse potential is employed for stretching and bending potentials, and a periodic type of bond torsion is used for torsion interactions. The effect of different types of vacancy defects at various locations on the critical buckling loads and strains is studied for zigzag and armchair nanotubes with various aspect ratios (Length/Diameter). Comparison of our results with those of buckling of shells with cutouts indicates that vacancy defects in the carbon nanotubes can most likely be modeled as cutouts of the shells. Finally, results of the present structural model are compared with those from molecular dynamics (MD) simulation and show good agreement between our model and the MD model.     

An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes

An analytical molecular structural mechanics model for the prediction of mechanical properties of defect-free carbon nanotubes is developed by incorporating the modified Morse potential with an analytical molecular structural model. The developed model is capable of predicting Young’s moduli, Poisson’s ratios and stress–strain relationships of carbon nanotubes under tension and torsion loading conditions. Results on the mechanical properties of single-walled carbon nanotubes show that Young’s moduli of carbon nanotubes are sensitive to the tube diameter and the helicity. Young’s moduli of both armchair and zigzag carbon nanotubes increase monotonically and approach Young’s modulus of graphite when the tube diameter is increased. The nonlinear stress–strain relationships for defect-free nanotubes have been predicted, which gives a good approximation on the ultimate strength and strain to failure of nanotubes. Armchair nanotubes exhibit higher tensile strength than zigzag nanotubes but their torsion strengths are identical based on the present study. The present theoretical investigation provides a very simple approach to predict the mechanical properties of carbon nanotubes.     

Buckling behavior of perfect and defective DWCNTs under axial, bending and torsional loadings via a structural mechanics approach

The buckling behavior of perfect and defective double-walled carbon nanotubes (DWCNTs) under axial compressive, torsional and bending loadings is investigated using a structural mechanics model. The effects of van der Waals (vdW) forces are further modeled using a nonlinear spring element. Critical buckling loads, critical buckling moments and the effects of vacancy defects were studied for armchair nanotubes with various aspect ratios. The results show that vacancy defects greatly reduce the critical buckling load of DWCNTs. The density of defects plays an important role in buckling of DWCNTs. The results of this numerical model are in good agreement with their comparable existing works.     

Axisymmetric compressive buckling of multi-walled carbon nanotubes under different boundary conditions

The paper studies the axisymmetric compressive buckling behavior of multi-walled carbon nanotubes (MWNTs) under different boundary conditions based on continuum mechanics model. A buckling condition is derived for determining the critical buckling load and associated buckling mode of MWNTs, and numerical results are worked out for MWNTs with different aspect ratios under fixed and simply supported boundary conditions. It is shown that the critical buckling load of MWNTs is insensitive to boundary conditions, except for nanotubes with smaller radii and very small aspect ratio. The associated buckling modes for different layers of MWNTs are in-phase, and the buckling displacement ratios for different layers are independent of the boundary conditions and the length of MWNTs. Moreover, for simply supported boundary conditions, the critical buckling load is compared with the corresponding one for axial compressive buckling, which indicates that the critical buckling load for axial compressive buckling can be well approximated by the corresponding one for axisymmetric compressive buckling. In particular, for axial compressive buckling of double-walled carbon nanotubes, an analytical expression is given for approximating the critical buckling load. The present investigation may be of some help in further understanding the mechanical properties of MWNTs.     

The coupled effects of mechanical deformation and electronic properties in carbon nanotubes

Coupled effects of mechanical and electronic behavior in single walled carbon nanotu besare investigated by using quantum mechanics and quantum molecular dynamics. It is found that external applied electric fields can cause charge polarization and significant geometric deformation in metallic and semi-metallic carbon nanotubes. The electric induced axial tension ratio can be up to 10% in the armchair tube and 8.5% in the zigzag tube. Pure external applied load has little effect on charge distribution，but indeed influences the energy gap. Tensile load leads to a narrower energy gap and compressive load increases the gap. When the CNT is tensioned under an external electric field, the effect of mechanical load on the electronic structures of the CNT becomes significant, and the applied electric field may reduce the critical mechanical tension load remarkably. Size effects are also discussed.     

The effects of chirality and boundary conditions on the mechanical properties of single-walled carbon nanotubes

The effects of chirality and boundary conditions on the elastic properties and buckling behavior of single-walled carbon nanotubes are investigated using atomistic simulations. The influences of the tube length and diameter are also included. It is found that the elastic properties of carbon nanotubes at small deformations are insensitive to the tube chirality and boundary conditions during compression. However, for large deformations occurred upon both compression and bending, the tube buckling behavior is shown to be very sensitive to both tube chirality and boundary conditions. Therefore, while the popular continuum thin shell model can be successfully applied to describe nanotube elastic properties at small deformation such as the Young’s modulus, it cannot correctly account for the buckling behavior. These results may allow better evaluation of nanotube mechanical properties via appropriate atomistic simulations.     

Stability of an elastic cytoskeletal tensegrity model

An elastic cytoskeletal tensegrity structure composed by six inextensible elastic struts and 24 elastic cables is considered. The model is studied, adopting delay convention for stability. Critical conditions for simple and compound instabilities are defined. Post-critical behavior is also described. Equilibrium states with buckling of the struts are also considered. It is revealed that critical Euler buckling load of the struts is a necessary but not a sufficient condition for the existence of bifurcated equilibrium states, caused by buckling of the struts.     

Progressive fracture analysis of planar lattices and shape-morphing Kagome-structure

Lattice solids are being contemplated for use in prospective lightweight structural applications. Hence, the understanding of their mechanical behavior is essential. In this work, the tensile behavior of the triangular (T), hexagonal (H) and Kagome (K) planar lattices containing notches is explored using FE-based progressive failure analysis. As an elastic-brittle material is assumed for the struts, they fail in rupture. The effects of notch-length and relative density are examined. Computations show a dramatic decrease of both stiffness and strength of the lattices with increasing the notch-length. For the relative density, the opposite effect is ascertained. Fracture patterns are found to depend only on the topology of the lattices. T- and K-lattice show a similar and much stronger behavior than the H-lattice.Using the same numerical method, the bending behavior of the shape-morphing Kagome-structure (KS) is simulated in regard of the different materials used in the members of the structure. Failure analysis of the KS involves progression of yielding and initiation of buckling in the face-sheet and struts. It was found that the bending rigidity of the KS is governed by the stiffness of the material of the face-sheet while the material of the core determines whether yielding or buckling will initiate first.     

The effect of transverse shear deformation on the buckling of two-layer composite columns with interlayer slip

This paper presents an efficient mathematical model for studying the buckling behavior of geometrically perfect elastic two-layer composite columns with interlayer slip between the layers. The present analytical model is based on the linearized stability theory and is capable of predicting exact critical buckling loads. Based on the parametric analysis, the critical buckling loads are compared to those in the literature. It is shown that the discrepancy between the different methods can be up to approximately 22%. In addition, a combined and an individual effect of pre-buckling shortening and transverse shear deformation on the critical buckling loads is studied in detail. A comprehensive parametric analysis reveals that generally the effect of pre-buckling shortening can be neglected, while, on the other hand, the effect of transverse shear deformation can be significant. This effect can be up to 20% for timber composite columns, 40% for composite columns very flexible in shear (pyrolytic graphite), while for metal composite columns it is insignificant.     

Simulating the fluid forces and fluid-elastic instabilities of a clamped–clamped cylinder in turbulent axial flow

In this paper, the fluid forces and the dynamics of a flexible clamped–clamped cylinder in turbulent axial flow are computed numerically. In the presented numerical model, there is no need to tune parameters for each specific case or to obtain coefficients from experiments. The results are compared with the dynamics measured in experiments available in the literature. The specific case studied here consists of a silicone cylinder mounted in axial water flow. Computationally it is found that the cylinder loses stability first by buckling. The threshold for buckling is in quantitative agreement with experimental results and weakly nonlinear theory. At higher flow speed a fluttering motion is predicted, in agreement with experimental results. It is also shown that even a small misalignment between the flow and the structure can have a significant impact on the dynamical behavior. To provide insight in the results of these fluid–structure interaction simulations, forces are computed on rigid inclined and curved cylinders, showing the existence of two different flow regimes. Furthermore it is shown that the inlet turbulence state has a non-negligible effect on these forces and thus on the dynamics of the cylinder.     

热载荷作用下梯度多孔材料梁弯曲及过屈曲力学行为的研究

基于Bernoulli-Euler梁理论,引入物理中面解耦了复合材料结构的面内变形与横向弯曲特性,研究了梯度多孔材料矩形截面梁在热载荷作用下的弯曲及过屈曲力学行为．假设沿梁厚度方向材料的性质是连续变化的,利用能量法推导了矩形截面梁的控制微分方程和边界条件,并用打靶法对无量纲化的控制方程进行数值求解．利用计算得到的结果分析了材料的性质、热载荷、边界条件对矩形截面梁非线性力学行为的影响．结果表明,对称材料模型下,固支梁与简支梁均显示出了典型的分支屈曲行为特征,而其临界屈曲热载荷值均会随着孔隙率系数的增加而单调增加．非对称材料模型下,固支梁仍显示出分支屈曲行为特征,但其临界屈曲热载荷不再随着孔隙率系数的变化而单调变化;而对于两端简支梁,发生了弯曲变形,弯曲挠度随载荷的增大而增大．     

单壁碳纳米管的扭转力学特性研究

利用基于高阶Cauchy-Born准则所建立的单壁碳纳米管本构模型,针对不同手性的单壁碳纳米管的扭转力学特性进行了研究.研究发现结构呈现对称性的锯齿型和扶手型单壁碳纳米管具有完全对称的扭转特性,而结构不对称的手性型单壁碳纳米管具有正反相异的扭转特性.同时,针对一系列手性不同的单壁碳纳米管的扭转力学特性展开了详细的研究.研究的部分结果与采用其他方法得到的结果进行了对比,证实了所提出方法以及预测结果的有效性和可行性.