碳纳米管力学行为的数值模拟
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摘要
纳米材料被誉为是21世纪的重要材料,并将构成未来智能社会的四大支柱之一。碳纳米管在纳米材料中最富有代表性,并且是性能最优异的材料之一。自从日本科学家首次发现碳纳米管,各国科学家对其进行了大量的研究。美、英、法、德、日及我国均相继成立了纳米材料研究中心,其重要的研究内容就是碳纳米管。1997年,单壁碳纳米管的研究成果与“克隆羊”和“火星探路者”一起,分别被评为当年的世界十大科学成就之一。碳纳米管具有优秀的力学、物理、化学及电学性质,在很多方面都有重要的应用。但是,目前在工程应用方面尚未取得完全的突破性进展。为此,研究人员必须对碳纳米管的结构及其性能进行全面的研究,这些研究具有重要的理论和工程应用价值。
本文围绕碳纳米管的力学行为,开展了包括建立模型及数值模拟方面的研究,具有一定的理论意义和实用价值。主要开展了分子结构力学方法、修正的分子结构力学方法和化学键单元方法在碳纳米管力学行为模拟方面的研究,具体内容如下。
1.分子结构力学方法在碳纳米管力学行为方面的数值模拟研究
分子结构力学方法是一种基于分子力学的数值计算方法。这种方法的特点是将原子间的共价键等效成宏观结构的梁,从而,将碳纳米管的微观结构等效为类似空间刚架的结构,然后用结构力学的方法求解。
迄今为止,学者们利用该方法开展的工作主要集中在理论上,但应用该方法对复杂的微纳米机电器件进行模拟时会出现一定困难。本文采用有限元方法求解结构力学问题,使用工程软件实现碳纳米管力学行为的有限元模拟,可以避免出现这种困难。数值模拟的结果表明,单壁碳纳米管的杨氏模量、剪切模量与其直径之间存在依赖现象,即碳纳米管的杨氏模量和剪切模量会随着管径的变化而变化。此外,对单壁碳纳米管系统,在不考虑阻尼时,振动基频与碳纳米管长度直径比之间也存在尺度依赖关系。最后,分子力学力场弹力常数对单壁碳纳米管力学行为的影响也在本部分中也做了详细的讨论。
2.修正的分子结构力学方法在碳纳米管力学行为方面的研究
经过仔细地研究分子力学方法在碳纳米管中的应用,本文作者所在研究组提出了一种基于分子力学的数值计算方法:修正的分子结构力学方法。在工作组前面工作的基础上,本文将该方法进一步拓展应用于碳纳米管的剪切模量及动态力学行为的分析。修正的分子结构力学方法不再是简单地将碳纳米管的原子结构类比成宏观的某些结构,而是用分子力学的力场势能函数表述碳纳米管系统的势能,这样的处理避免了简单的能量等效可能会产生的误差。本方法将系统能量按一定的方式离散,从能量最小原理出发,在小变形假设的基础上,直接建立系统方程。由于选择的力场包含了离面振动能量项,所以该方法有更广的适用范围及更好的精度。本部分系统地介绍了该方法从理论建模到程序实现的过程,在课题组前期工作基础上,系统地计算研究了单壁碳纳米管杨氏模量、剪切模量及振动基频与碳纳米管长度/直径之间的关系。
3.化学键单元方法在碳纳米管力学行为方面的研究
本文建立了一种基于分子力学的三维纳米尺度的有限元方法:化学键单元方法。化学键单元模拟了碳纳米管原子间碳-碳化学键的力学行为,单元的刚度矩阵通过联系分子力学与连续介质力学而得到。化学键单元的元素全部是分子力学力场弹力常数的函数。本方法避免了简单的能量等效方法带来的计算误差,运算方便,效率更高。应用工程有限元软件提供的扩展单元建立化学键单元,可以实现碳纳米管力学行为的数值模拟。单壁碳纳米管杨氏模量、剪切模量及振动固有频率与碳纳米管长度/直径之间的关系在本部分论文中做了详细的讨论。此外,论文中尝试应用该方法研究了简单的微纳米器件(纳米秤)的力学行为。最简单的纳米秤结构由一根悬臂的碳纳米管及附着于纳米管的微粒组成。本部分论文针对如何通过纳米秤系统共振频率的改变得到附加微粒质量的理论及数值模拟方法进行了讨论。关于单壁碳纳米管直径、长度以及微粒附着位置对纳米秤系统共振频率的影响及如何有效提高纳米秤敏感度的办法,也通过数值模拟开展了研究。最后,通过参数拟合的办法对单壁碳纳米管的有效厚度进行了研究,并采用非线性弹簧模拟了范德华力,对双壁碳纳米管的杨氏模量、剪切模量以及固定外壁-内壁自由的一阶往复振动进行了分析。
Nano-materials are considered to be one of the most important materials in the 21th century and they will be the fundamental structures of the intelligent society. The most representative material of nano-materials is carbon nanotube (CNT) which possesses excellent performances in many fields. Ever since Japanese scientist found CNT in 1991, lots of researches have been performed by the scholars all over the world. The research centers on nanomaterials have been found in many contries, including the USA, UK, France, Germany, Japan and China. The behavior and application of CNTs is a key task in these centers. The investigations of single-walled carbon nanotube (SWCNT) have been sellected as the "ten key" scientific achievements together with "Mars PathFinder" all over the world in 1997.
CNT possesses excellent mechanical, physical, chemical and electrical properties and it has lots of potential application areas. However there are still many works to be done to get the breakthrough in the engineering applications. Many scientists heve devoted themselves to the investigations of CNT and their research works have been shown to have crucial theoretical and applied values in engineering applications.
This dissertation proposes a couple of new analysis methods and applies them in the simulation of the mechanical behaviors of CNTs. The research works are listed as below:
1. Simulation of the mechanical behaviors of SWCNT using molecular structural mechanics method
The molecular structural mechanics method bases on the molecular mechanics method. The inter-atoms covalent bonds are considered as macro-beams. Then the microscopic structures of SWCNTs are equivalently treated as rigid frames in macroscopic structures and solved using structural mechanics method.
By now, lots of works have been done by the researchers but they are mainly related to the theoretical analysis for simple problems. In this dissertation, the molecular structural mechanics method is combined with the finite element method and the simulation results of the Young's modulus, shear modulus and dynamic properties of SWCNT using commercial FE software are performed. The results show that the Young's modulus and shear modulus are sensitive to the diameters of SWCNT, i.e. they vary significantly with respect to the nanotube's diameter. Furthermore, for SWCNT, when the damping effect is not considered the fundamental frequency depends on the ratio of the nanotube's length and its diameter. In the end, the contributions of field constants in molecular mechanics method to the mechanical behaviors of SWCNT are discussed.
2. Simulation of the mechanical behaviors of SWCNTs using modified molecular structural mechanics method
A modified molecular structural mechanics method, based on molecular mechanics and similar to the finite element method, has been developed in the author's research group for the simulateion of the mechanical behaviors of SWCNTs. The atomic structures of SWCNTs are not simply treated as macro-structures but related to the deformed potential energy using the force-field energy functions. Under the small deformation assumption and the principle of minimum potential energy, the system equation can be established directly from the discretization of the system energy formulation. The inversion potential energy is considered in this formulation which improves the analysis accuracy and widens the analysis fields.
Based on the previous work of the author's research group, this dissertation systematically investigates the relationships of the Young's modulus and shear modulus with the diameter of SWCNT and the relationship of fundamental frequency with the ratio of the nanotube's length and its diameter.
3. Simulation of the mechanical behaviors of CNTs using the chemical bond element method
Also based upon molecular mechanics method, a kind of three dimensional (3-D) nano-scale finite element model, the chemical bond element model, is proposed for the simulations of the mechanical properties of SWCNTs. Chemical bonds between carbon atoms are modeled by the chemical bond elements. The constants of the sub-stiffness matrix are determined by using a linkage between the molecular mechanics and continuum mechanics. The entries in the sub-stiffness matrix of the chemical bonds elements are the functions of the force-field constants. The simulations of the mechanical behaviors of CNTs are carried out by means of commercial FE software, ANSYS, after the element stiffness sub-matrix is properly defined.
For SWCNT, the relationship of the Young's modulus and shear modulus with the nanotube's diameter and the relationship of fundamental frequency with the ratio of the nanotube's length and its diameter are investigated again by the new method. Furthermore, the simulation of the basic mechanical behaviors of the nanobalance is performed. The fundamental concepts and constitutions of nanobalance are presented and the principle using nanobalance to detect the mass of the attached particle is discussed. The theoretical and numerical solutions in calculating the mass of the attached particle are explored. The impacts of the length, diameter and the location of the attached particle to the resonant frequency of nanobalance are studied and some beneficial advices to enhance the sensitivities of nanobalance are given out. Besides, the effective thickness of SWCNT is proposed by parameter fitting method. In the end, the mechanical behaviors of double-walled carbon nanotube(DWCNT) are simulated where the Van der waals force between the inter-layer atoms is represented by a kind of non-linear spring. Investigations for DWCNT include its Young's modulus and shear modulus and a special vibration mode, i.e. the mode for inter-layer tube when the outer-layer atoms are fixed.
本文围绕碳纳米管的力学行为,开展了包括建立模型及数值模拟方面的研究,具有一定的理论意义和实用价值。主要开展了分子结构力学方法、修正的分子结构力学方法和化学键单元方法在碳纳米管力学行为模拟方面的研究,具体内容如下。
1.分子结构力学方法在碳纳米管力学行为方面的数值模拟研究
分子结构力学方法是一种基于分子力学的数值计算方法。这种方法的特点是将原子间的共价键等效成宏观结构的梁,从而,将碳纳米管的微观结构等效为类似空间刚架的结构,然后用结构力学的方法求解。
迄今为止,学者们利用该方法开展的工作主要集中在理论上,但应用该方法对复杂的微纳米机电器件进行模拟时会出现一定困难。本文采用有限元方法求解结构力学问题,使用工程软件实现碳纳米管力学行为的有限元模拟,可以避免出现这种困难。数值模拟的结果表明,单壁碳纳米管的杨氏模量、剪切模量与其直径之间存在依赖现象,即碳纳米管的杨氏模量和剪切模量会随着管径的变化而变化。此外,对单壁碳纳米管系统,在不考虑阻尼时,振动基频与碳纳米管长度直径比之间也存在尺度依赖关系。最后,分子力学力场弹力常数对单壁碳纳米管力学行为的影响也在本部分中也做了详细的讨论。
2.修正的分子结构力学方法在碳纳米管力学行为方面的研究
经过仔细地研究分子力学方法在碳纳米管中的应用,本文作者所在研究组提出了一种基于分子力学的数值计算方法:修正的分子结构力学方法。在工作组前面工作的基础上,本文将该方法进一步拓展应用于碳纳米管的剪切模量及动态力学行为的分析。修正的分子结构力学方法不再是简单地将碳纳米管的原子结构类比成宏观的某些结构,而是用分子力学的力场势能函数表述碳纳米管系统的势能,这样的处理避免了简单的能量等效可能会产生的误差。本方法将系统能量按一定的方式离散,从能量最小原理出发,在小变形假设的基础上,直接建立系统方程。由于选择的力场包含了离面振动能量项,所以该方法有更广的适用范围及更好的精度。本部分系统地介绍了该方法从理论建模到程序实现的过程,在课题组前期工作基础上,系统地计算研究了单壁碳纳米管杨氏模量、剪切模量及振动基频与碳纳米管长度/直径之间的关系。
3.化学键单元方法在碳纳米管力学行为方面的研究
本文建立了一种基于分子力学的三维纳米尺度的有限元方法:化学键单元方法。化学键单元模拟了碳纳米管原子间碳-碳化学键的力学行为,单元的刚度矩阵通过联系分子力学与连续介质力学而得到。化学键单元的元素全部是分子力学力场弹力常数的函数。本方法避免了简单的能量等效方法带来的计算误差,运算方便,效率更高。应用工程有限元软件提供的扩展单元建立化学键单元,可以实现碳纳米管力学行为的数值模拟。单壁碳纳米管杨氏模量、剪切模量及振动固有频率与碳纳米管长度/直径之间的关系在本部分论文中做了详细的讨论。此外,论文中尝试应用该方法研究了简单的微纳米器件(纳米秤)的力学行为。最简单的纳米秤结构由一根悬臂的碳纳米管及附着于纳米管的微粒组成。本部分论文针对如何通过纳米秤系统共振频率的改变得到附加微粒质量的理论及数值模拟方法进行了讨论。关于单壁碳纳米管直径、长度以及微粒附着位置对纳米秤系统共振频率的影响及如何有效提高纳米秤敏感度的办法,也通过数值模拟开展了研究。最后,通过参数拟合的办法对单壁碳纳米管的有效厚度进行了研究,并采用非线性弹簧模拟了范德华力,对双壁碳纳米管的杨氏模量、剪切模量以及固定外壁-内壁自由的一阶往复振动进行了分析。
Nano-materials are considered to be one of the most important materials in the 21th century and they will be the fundamental structures of the intelligent society. The most representative material of nano-materials is carbon nanotube (CNT) which possesses excellent performances in many fields. Ever since Japanese scientist found CNT in 1991, lots of researches have been performed by the scholars all over the world. The research centers on nanomaterials have been found in many contries, including the USA, UK, France, Germany, Japan and China. The behavior and application of CNTs is a key task in these centers. The investigations of single-walled carbon nanotube (SWCNT) have been sellected as the "ten key" scientific achievements together with "Mars PathFinder" all over the world in 1997.
CNT possesses excellent mechanical, physical, chemical and electrical properties and it has lots of potential application areas. However there are still many works to be done to get the breakthrough in the engineering applications. Many scientists heve devoted themselves to the investigations of CNT and their research works have been shown to have crucial theoretical and applied values in engineering applications.
This dissertation proposes a couple of new analysis methods and applies them in the simulation of the mechanical behaviors of CNTs. The research works are listed as below:
1. Simulation of the mechanical behaviors of SWCNT using molecular structural mechanics method
The molecular structural mechanics method bases on the molecular mechanics method. The inter-atoms covalent bonds are considered as macro-beams. Then the microscopic structures of SWCNTs are equivalently treated as rigid frames in macroscopic structures and solved using structural mechanics method.
By now, lots of works have been done by the researchers but they are mainly related to the theoretical analysis for simple problems. In this dissertation, the molecular structural mechanics method is combined with the finite element method and the simulation results of the Young's modulus, shear modulus and dynamic properties of SWCNT using commercial FE software are performed. The results show that the Young's modulus and shear modulus are sensitive to the diameters of SWCNT, i.e. they vary significantly with respect to the nanotube's diameter. Furthermore, for SWCNT, when the damping effect is not considered the fundamental frequency depends on the ratio of the nanotube's length and its diameter. In the end, the contributions of field constants in molecular mechanics method to the mechanical behaviors of SWCNT are discussed.
2. Simulation of the mechanical behaviors of SWCNTs using modified molecular structural mechanics method
A modified molecular structural mechanics method, based on molecular mechanics and similar to the finite element method, has been developed in the author's research group for the simulateion of the mechanical behaviors of SWCNTs. The atomic structures of SWCNTs are not simply treated as macro-structures but related to the deformed potential energy using the force-field energy functions. Under the small deformation assumption and the principle of minimum potential energy, the system equation can be established directly from the discretization of the system energy formulation. The inversion potential energy is considered in this formulation which improves the analysis accuracy and widens the analysis fields.
Based on the previous work of the author's research group, this dissertation systematically investigates the relationships of the Young's modulus and shear modulus with the diameter of SWCNT and the relationship of fundamental frequency with the ratio of the nanotube's length and its diameter.
3. Simulation of the mechanical behaviors of CNTs using the chemical bond element method
Also based upon molecular mechanics method, a kind of three dimensional (3-D) nano-scale finite element model, the chemical bond element model, is proposed for the simulations of the mechanical properties of SWCNTs. Chemical bonds between carbon atoms are modeled by the chemical bond elements. The constants of the sub-stiffness matrix are determined by using a linkage between the molecular mechanics and continuum mechanics. The entries in the sub-stiffness matrix of the chemical bonds elements are the functions of the force-field constants. The simulations of the mechanical behaviors of CNTs are carried out by means of commercial FE software, ANSYS, after the element stiffness sub-matrix is properly defined.
For SWCNT, the relationship of the Young's modulus and shear modulus with the nanotube's diameter and the relationship of fundamental frequency with the ratio of the nanotube's length and its diameter are investigated again by the new method. Furthermore, the simulation of the basic mechanical behaviors of the nanobalance is performed. The fundamental concepts and constitutions of nanobalance are presented and the principle using nanobalance to detect the mass of the attached particle is discussed. The theoretical and numerical solutions in calculating the mass of the attached particle are explored. The impacts of the length, diameter and the location of the attached particle to the resonant frequency of nanobalance are studied and some beneficial advices to enhance the sensitivities of nanobalance are given out. Besides, the effective thickness of SWCNT is proposed by parameter fitting method. In the end, the mechanical behaviors of double-walled carbon nanotube(DWCNT) are simulated where the Van der waals force between the inter-layer atoms is represented by a kind of non-linear spring. Investigations for DWCNT include its Young's modulus and shear modulus and a special vibration mode, i.e. the mode for inter-layer tube when the outer-layer atoms are fixed.
引文
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