历年博士学位优秀论文精选-中英文(4)

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　　一维纳米结构的若干力学问题

　　作者姓名：王立峰

　　论文题目：一维纳米结构的若干力学问题

　　作者简介：王立峰，男，1977年12月出生，2002年4月师从于南京航空航天大学胡海岩教授，于2005年6月获博士学位。

　　中文摘要

　　碳纳米管、铜纳米线是构成未来纳米器件的重要元素，如何了解和描述其力学特性成为人们非常关注的科学问题。采用分子动力学方法研究这类纳尺度结构，只能得到个案结果，难以获得多个物理量之间的一般关系。连续介质力学方法能否适用于这类本质离散的纳尺度结构呢?若不适用，是否可以改造之?

　　本文以碳纳米管及铜纳米线等一维纳米结构为研究对象，采用连续介质力学分析与分子动力学计算相结合的方法，探索连续介质力学的有效性及失效时的改进，对碳纳米管中纵波与弯曲波的传播、单壁及多壁碳纳米管的屈曲、碳纳米管与刚性壁的碰撞、铜纳米线的表面效应引起的尺寸效应、铜纳米线的动力屈曲和纳尺度的温度等问题进行了研究，其主要创新及学术贡献如下：

　　1.用连续介质力学方法及基于

　　Tersoff-Brenner势的分子动力学方法对比研究了碳纳米管中弯曲波的传播及频散问题，主要考虑了转动惯量、剪切变形及非局部弹性所描述的微结构对碳纳米管中弯曲波频散的影响。建立了考虑转动惯量及剪切变形的非局部弹性梁动力学方程。基于考虑转动惯量及剪切变形的非局部弹性梁模型，Timoshenko梁模型及Euler梁模型，给出了单壁碳纳米管中弯曲波传播的频散关系。然后用分子动力学方法模拟了不同周期的弯曲波在碳纳米管中的传播。结果表明：Euler梁模型只在很小的波数范围内适用，Timoshenko梁模型能更好地给出单壁碳纳米管中弯曲波的频散关系;当波数非常大时，碳纳米管的微结构对波的传播将产生非常重要的影响，此时随着波数的增加相速度将会降低;考虑了转动惯量、剪切变形及非局部弹性的梁模型可以较好预测这时的频散关系。该研究结果发表在Physical

　　Review B上，被评价为“这是一篇非常有趣的论文，很有可能对碳纳米管动力学行为的研究产生重要的贡献和影响”。

　　2.用连续介质力学方法及基于Tersoff-Brenner势的分子动力学方法对比研究了碳纳米管中纵波的传播及频散问题，主要考虑了微结构对碳纳米管中纵波频散的影响。首先用分子动力学模拟了不同周期的纵波在碳纳米管中的传播。然后基于各种弹性杆模型、弹性壳模型、非局部弹性杆模型及非局部弹性壳模型得出了频散关系。结果表明：只有考虑非局部弹性的壳模型能很好地预测两支频散关系，微结构以及纵波与管壁径向运动的耦合会影响碳纳米管中高频纵波的频散。该研究结果发表在Nanotechnology上。

　　3.研究了碳纳米管的屈曲及后屈曲。先用基于 Tersoff-Brenner

　　势的分子动力学方法研究单壁碳纳米管在轴向载荷下的非线性后屈曲行为，结果表明：碳纳米管在后屈曲阶段近似为理想塑性弹簧，其屈曲过程是能量吸收的过程，碳纳米管可作为很好的吸能元件;连续介质力学可以较好地给出碳纳米管的屈曲点，但只能给出近似的后屈曲行为。该研究结果发表在Acta

　　Mechanica Solida

　　Sinica上。然后用分子动力学方法模拟了多壁碳纳米管在压缩、弯曲变形下力与变形的关系。通过与组成多壁碳纳米管的各单壁碳纳米管的比较分析，揭示了多壁碳纳米管层间Van

　　der Waals力对碳纳米管力学性质的影响。采用6-12形式的 Lennard-Jones势描述碳纳米管壁间Van der

　　Waals力。计算结果表明：多壁碳纳米管的比强度明显高于单壁碳纳米管;Van der

　　Waals力对杨氏模量影响不大，但对碳纳米管屈曲行为的影响却相当显著。该研究结果发表在固体力学学报上。

　　4.用分子动力学方法模拟碳纳米管与刚性壁的正碰撞过程，并与弹性动力学方法的分析结果进行对比。在分子动力学模拟中，采用Tersoff-Brenner势描述碳纳米管的原子间相互作用，用6-12形式的Lennard-Jones势描述碳纳米管与刚性壁间相互作用。结果表明：两种方法所得到的应力波传播速度吻合较好，应力波的传播过程是原子的动能和原子间势能的转化过程;与弹性动力学分析结果不同的是，在发生屈曲以前，碳纳米管与刚性壁的接触时间不仅与碳纳米管的长度近似成线性关系，还与管径及碰撞初速度有关;碰撞过程中，碳纳米管端部应力并非定值，但其平均值与弹性动力学计算结果相差不大。该研究结果发表在固体力学学报上。

　　5.研究了尺寸效应对铜纳米线杨氏模量的影响，以及铜纳米线的动力屈曲问题。首先根据基于内嵌原子势的分子动力学模拟结果，利用铜纳米线的非均匀性解释了尺寸效应对铜纳米线的杆模型和梁模型的等效杨氏模量的影响。和分子动力学的结果相对比，新的杆模型与以前的杆模型都符合得很好，但新的非均匀梁模型比以前的梁模型符合得更好。从对铜纳米线纵向振动和横向振动的固有频率的分子动力学模拟中同样得到纳米铜金属线的尺寸效应。该研究结果发表在International

　　Journal of computational

　　method上。然后用分子动力学方法和弹性动力学方法研究了铜纳米线的动力屈曲，模拟了不同载荷下铜纳米线的动力屈曲型态。用弹性动力学理论分析铜纳米线的动力屈曲，并与分子动力学模拟的结果进行了比较。分子动力学模拟得到的屈曲临界载荷略高于弹性动力学所给出的临界载荷。根据弹性动力学预测，应力波会对铜纳米线的动力屈曲产生显著影响。该研究结果发表在

　　International Journal of Nonlinear Sciences and Numerical Simulation上。

　　6.此外，初步探索了纳米尺度和温度有关的一些问题。研究了多壁碳纳米管间的热传导、碳纳米管的热膨胀、温度对纳米振荡器中阻尼的影响、纳米振荡器的转动问题以及基于碳纳米管的纳米振荡器的热激振动问题，讨论了分子动力学模拟过程中温度的定义。在研究中，为了避免人为控制温度对纳米结构整体动力学运动的影响，采用热传导方法对纳米结构加热，而不是通过直接对结构进行控制来研究温度问题。

　　基于上述研究发表的论文已被美国艺术与科学院、国家工程院院士T Belytschko教授(Meccanica 40: 455)、英国皇家学会院士 YW

　　Mai教授(J. Appl. Phys. 103, 074309)、美国复合材料学会主席RF Gibson教授 (Compos. Sci. Technol.

　　67:1)，BI Yakobson教授 (J. Mech. Phys. Solids,2008 Available online)、HJ Gao教授

　　(Appl. Phys. Lett. 93, 013106)、TW Chou教授(Phys. Rev. B 73, 245407)等著名学者引用

　　60余次。其中，发表在Phys. Rev. B上关于弯曲波频散研究的论文被美国复合材料学会主席RF Gibson教授在长篇综述中作为“特别重要”

　　的研究举例(Compos. Sci. Technol. 67:1);被英国曼彻斯特大学的QM Li等在 P. Roy. Soc. A Math. Phys.

　　(464:1941)上以近半页篇幅引述;被新加坡国立大学的CM Wang教授在多篇论文中列举为采用非弹性理论揭示纳尺度微结构效应的第一篇文献 (J. Phys.

　　D: Appl. Phys. 39: 3904; Int. J. Struct. Stabil. Dynam. 7:555; Nanotechnology

　　18: 105401);被H Askes等作为应变梯度理论应用于纳尺度动力学的注解(Int. J. Numer. Meth. Engng, 72:111)。在

　　Acta Mechanica Solida Sinica 上发表的论文所预测的碳纳米管在循环载荷下的迟滞现象，被HW Yap等发表在 Nano Letters

　　7: 1149上的实验结果所证实。

　　关键词： 分子动力学，碳纳米管，屈曲，碰撞，Van der Waals力，纵波，弯曲波，频散，应变梯度，温度

　　On Some Mechanics Problems in One Dimensional Nanostructure

　　Wang Lifeng

　　ABSTRACT

　　Carbon nanotubes and copper nanowires are so promising nano-components that

　　their mechanical properties have received more and more attention over the past

　　decade. In studying the nano-components, molecular dynamics simulations can only

　　predict the dynamics results case by case and can hardly reveal the relation

　　between multiple physical variables. On the other hand, it is still an open

　　problem whether the continuum mechanics is valid to those discrete

　　nano-components by nature. If not, is it possible to get any modified approach

　　to deal with nano-components?

　　The objective of this dissertation is to check the validity of the

　　continuum mechanics, especially the continuum dynamics, for the nano-structures

　　of one dimension, such as the carbon nanotubes and copper nanowires, with help

　　of molecular dynamics simulations. The studies presented in the dissertation

　　include the dispersion of both flexural and longitudinal waves in carbon

　　nanotubes, the buckling of both single-walled and multi-walled carbon nanotubes,

　　the impact of carbon nanotubes with a rigid wall, the size effect on the

　　effective Young’s modulus of copper nanowires, the dynamic buckling of copper

　　nanowires and the effect of temperature in nano-scale problems. The results and

　　the main contributions of the dissertation are as following.

　　1. The flexural wave propagation in single-walled carbon nanotubes was

　　studied through the use of the continuum mechanics and the molecular dynamics

　　simulation based on the Terroff-Brenner potential. The study focuses on the wave

　　dispersion caused not only by the rotary inertia and the shear deformation in

　　the model of a traditional Timoshenko beam, but also by the non-local elasticity

　　characterizing the microstructure of a carbon nanotube in a wide frequency range

　　up to THz. For this purpose, the study starts with the dynamic equation of a

　　generalized Timoshenko beam made of the non-local elastic material, and then

　　gives the dispersion relations of the flexural wave in the non-local elastic

　　Timoshenko beam, the traditional Timoshenko beam and the Euler beam,

　　respectively. Afterwards, it presents the molecular dynamics simulations for the

　　flexural wave propagation in an armchair (5,5) and an armchair (10,10)

　　single-walled carbon nanotubes for a wide range of wave numbers. The simulation

　　results show that the Euler beam holds for describing the dispersion of flexural

　　waves in these two single-walled carbon nanotubes only when the wave number is

　　small. The Timoshenko beam provides a better prediction for the dispersion of

　　flexural waves in the two single-walled carbon nanotubes when the wave number

　　becomes a little bit large. Only the non-local elastic Timoshenko beam is able

　　to predict the decrease of phase velocity when the wave number is so large that

　　the microstructure of carbon nanotubes has a significant influence on the

　　flexural wave dispersion. The work has been published in Physical Review B. The

　　referee wrote: “This is an extremely interesting paper which would, most likely,

　　make a significant contribution and impact to the study of dynamic behavior of

　　CNTs”.

　　2. The study on the longitudinal wave propagation and dispersion in

　　single-walled carbon nanotubes was presented through the use of the continuum

　　mechanics and the molecular dynamics simulation based on the Terroff-Brenner

　　potential. The study focuses on the effects of non-local elasticity

　　characterizing the microstructure on the wave dispersion of single-walled carbon

　　nanotubes. The study begins with the numerical simulation of molecular dynamics

　　for the longitudinal wave of single-walled carbon nanotubes. Then, it presents

　　the wave dispersion relations based on the models of rods or shells, made of

　　either the elastic materials or the non-local elastic materials so as to

　　characterize the micro-structure, for the carbon nanotubes. Among them, only the

　　model of non-local elastic shell is able to get a good agreement with the

　　molecular dynamics in a wide frequency range up to THz. The study shows that

　　both the micro-structure and the coupling of longitudinal wave and radial motion

　　play an important role in the wave dispersion of carbon nanotubes. The work has

　　been published in Nanotechnology.

　　3. The buckling behavior of carbon nanotubes was revealed through the use

　　of molecular dynamics simulation based on the Tersoff-Brenner potential. The

　　dissertation presents the buckling and the postbuckling of single-walled carbon

　　nanotubes subjected to a cyclic axial compressive load and gives the bifurcation

　　behavior in buckling process simulated with very fine time steps. In the whole

　　cycle of nonlinear deformation, the carbon nanotubes exhibit the profound

　　hysteretic behavior and the energy absorption ability. The molecular dynamics

　　simulation indicates that the carbon nanotube behaves approximately as an ideal

　　plastic spring when the cyclic strain is applied within the same postbuckling

　　mode. In comparison, the theory of continuum mechanics gives a good prediction

　　for the critical buckling strength, but only provides a rough estimation for the

　　post-buckling behaviors. The work has been published in Acta Mechanica Solida

　　Sinica. It presents the molecular dynamics simulations for both single-walled

　　and multi-walled carbon nanotubes subjected to different external loads to

　　examine the influence of the van der Waals force on the property of multi-wall

　　carbon nanotubes. In this part, the van der Waals force between the walls is

　　approximated through the use of the ‘6-12’ Lennard-Jones potential. Although the

　　van der Waals force has no remarkable effect on the Young’s modulus, its

　　influence on the strength and buckling behavior of carbon nanotubes is

　　significant. The work has been published in Acta Mechanica Solida Sinica

　　(Chinses edition).

　　4. With help of molecular dynamics simulations the dissertation studied the

　　impact of carbon nanotubes with a rigid wall, where the interatomic interactions

　　are described by a Tersoff-Brenner potential and the interactions between the

　　atoms of carbon nanotubes and the rigid wall are approximated by the ‘6-12’

　　Lennard-Jones potential. The simulations show that the velocity of stress waves

　　predicted by molecular dynamics falls into the same range by the theory of

　　elasticity. Different from an elastic rod, the impact duration of a nanotube is

　　not only proportional to the length of nanotube, but also depends on the initial

　　impact velocity and the diameter of nanotube. Furthermore, the stress at the end

　　of nanotube is not a constant during impact. However, its mean value is close to

　　the result given by theory of elasticity. The work has been published in Acta

　　Mechanica Solida Sinica (Chinese Edition).

　　5. The molecular dynamics simulation on the basis of embedded atom

　　potential show the inhomogeneous property of the nanowires, and then establishes

　　the continuum model of either a rod or a beam to predict the size dependence of

　　the effective Young’s modulus. The comparison with molecular dynamics simulation

　　shows that the rod model here enables one to predict the effective Young’s

　　modulus as accurately as current models for the nanowires of different sizes of

　　cross sections under axial load. Furthermore, the beam model gives better

　　prediction than the current model for the nanowires subject to pure bending. The

　　size effect on the elastic property can also be observed from the longitudinal

　　and transverse natural vibration of the nanowires. In this case, the effective

　　Young’s modulus is nearly the same as that obtained through axial deformation

　　and pure bending, respectively. The work has been published in International

　　Journal of Computational Method. The dynamic buckling of a copper nanowire,

　　including the effects of stress wave was studied, with help of molecular

　　dynamics simulation and dynamic buckling theory in continuum mechanics. The

　　dynamic buckling of the copper nanowire under an axial step load is simulated.

　　As a comparison, the nanowire is modeled as an elastic rod subject to the axial

　　step load, and is analyzed on the basis of buckling theory. The molecular

　　dynamics simulation gives a slightly higher critical load than the buckling

　　theory. Furthermore, the buckling theory predicts that the stress wave has a

　　remarkable effect on the dynamic buckling of copper nanowires. The work has been

　　published in International Journal of Nonlinear Sciences and Numerical

　　Simulation.

　　6. As an exploration, the dissertation presents a preliminary study on the

　　temperature in nano-scale problems. To avoid the effect of temperature control

　　on the dynamic behavior of nano-structures in molecular dynamics simulations,

　　the conduction method is used to control the temperature of multi-walled carbon

　　nanotubes of concern. The dissertation gives the heat conduction and the

　　corresponding expansion of carbon nanotubes with an increase of temperature, the

　　effect of damping in nano-oscillators made of multi-walled carbon nanotubes，the

　　rotation and the intrinsic thermal vibration of nano-oscillators as well. The

　　dissertation terminates with a discussion about the definitions of temperature

　　in the molecular dynamics simulations for nano-oscillators.

　　The publications of the above studies have been cited over 60 times by many

　　leading scientists including Prof. T Belytschko (American for Academy of Arts

　　and Science, U. S. National Academy of Engineering) (Meccanica 40: 455), Prof.

　　YW Mai (Follow of Royal Society) (J. Appl. Phys. 103, 074309), Prof. RF Gibson

　　(President of American Society for Composites) (Compos. Sci. Technol.

　　67:1)，Prof. BI Yakobson (J. Mech. Phys. Solids, 2008 Available online), Prof. HJ

　　Gao (Appl. Phys. Lett. 93, 013106), Prof. TW Chou (Phys. Rev. B 73, 245407). For

　　example, Prof. RF Gibson et.al cited the work published in Phys. Rev. B as an

　　example of “particularly important” research in their review in Compos. Sci.

　　Technol. 67:1. Prof. QM Li et. al. cited the work as a long paragraph of over

　　400 words in P. Roy. Soc. A Math. Phys. 464: 1941. Prof. CM Wang from National

　　University of Singapore cited the work as the first reference of studing carbon

　　nanotubes by using non-local elastic models in his three papers, such as J.

　　Phys. D: Appl. Phys. 39: 3904, Int. J. Struct. Stabil. Dynam. 7:555 and

　　Nanotechnology 18: 105401. H Askes et. al. cited the work as a remark that the

　　strain gradient theory be used to nano-scale dynamics in Int. J. Numer. Meth.

　　Engng, 72:111. Furthermore, HW Yap reported t in Nano Letters 7:1149 that the

　　experimental results coincide with the large hysteresis between the loading and

　　unloading curves predicted in Acta Mechanica Solida Sinica.

　　Key words: Molecular dynamics，Carbon nanotube，Buckling，Impact，Van der waals

　　force，Longitudinal wave，Flexural wave, Wave dispersion，Strain

　　gradient，Temperature

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