碳纳米管的相关力学问题的研究
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摘要
自从Iijima发现碳纳米管以来，它已引起了众多科学家的关注和研究。已有的相关研究工作表明碳纳米管可能具有奇特的力学、电学和光学等特性，因此可望具有广阔的应用前景。然而，不论是实验还是理论工作，目前仍然处于探索阶段，且对于碳纳米管的这些特性的预测结果也仍存在着较大的分歧。因此，基于所发展的高阶Cauchy-Born广义连续体本构模型以及范德华力广义参变本构模型，本文分别研究了单壁碳纳米管的一系列相关力学特性、压缩变形机理以及平行或交叉排列的碳纳米管间的黏附行为。所取得的研究进展包括：
本文提出了一种基于高阶Cauchy-Born准则建立单壁碳纳米管本构模型的方法。通过引入高阶变形梯度，合理地修正了传统Cauchy-Born准则在描述纳米管变形几何关系时所存在的缺陷。利用原子间相互作用势以及能量等效原理，得到了基于广义连续介质模型的单壁碳纳米管的本构关系。由此得到的本构参数不仅与变形梯度张量F，而且与其梯度▽F相关，同时也包含了碳纳米管的微结构信息，因此是一种考虑微结构效应的广义连续介质模型。
基于所发展的高阶Cauchy-Born广义连续体本构模型，首先对石墨片和单壁碳纳米管的一系列无限小变形的力学特性(如：轴向和环向模量、泊松比、剪切模量)以及应变能进行了研究，同时也考察了手性、半径、势能参数以及内位移放松等因素对这些力学特性的影响。研究发现：当单壁碳纳米管的半径较小时，手性和半径对其力学特性的影响十分明显；而当管半径较大时，这种影响不再重要，以至于随着管半径的增大单壁碳纳米管的这些的力学特性最终趋于石墨片相应的结果。从数值结果上看，势能参数和内位移放松对所预测的单壁碳纳米管的力学特性有着很大的影响。进一步，我们也研究了简单的有限变形下单壁碳纳米管的应变能和切向模量的变化。结果表明，单壁碳纳米管的切向模量几乎线性地随着轴向拉伸而减小；且对于半径相近而手性不同的碳纳米管而言，其系统应变能几乎完全一致。继而，手性碳纳米管的轴向变形与其径向和扭转变形间的耦合机理的研究表明，最大的耦合效应发生于手性角为π／12的碳纳米管中。
从变形机理方面重新审视了石墨片和碳纳米管的弯曲刚度，并指出了部分文献中关于这两个概念使用和计算上的不恰当性。具体而言，石墨片和碳纳米管的弯曲刚度这两个概念本质上是不同的，前者指的是材料的内禀特性，而后者是结构特性。当采用经典连续体理论模拟碳纳米管时，等效的连续体应该被看作是带有微结构的广义连续介质，因此本文建议把碳纳米管的结构弯曲刚度处理为一个独立的结构刚度参数。进而，基于高阶Cauchy-Born广义本构模型给出了合理的计算石墨片和碳纳米管弯曲刚度的方法和结果。
基于高阶Cauchy-Born广义连续体本构模型，结合无网格方法进行了单壁碳纳米管的压缩屈曲变形的分析，从应变能的角度分析了屈曲前后碳纳米管的变形机理。所获得的与分子动力学模拟和原子有限元计算相一致的结果证明了目前所发展的模型的可行性和正确性。
本文也提出了处理非成键原子间范德华力的广义参变本构模型以及基于此进行碳纳米管结构力学行为数值模拟的数学规划算法。纳米管中原子间短程力作用采用分子结构力学模型来模拟，而作为长程作用的范德华力用杆单元来模拟，这类杆单元有着特殊的非线性本构关系。对于这种高非线性问题的处理，本文基于所建立的广义参变本构模型与参数二次规划算法求解。与一般的数值方法相比较，本文的方法不需要传统的那种冗长的、反复的迭代，并具有非常好的收敛性，因此为碳纳米管结构力学行为的有效预测提供了保障。基于此我们研究了平行和交叉排列的碳纳米管的稳定构型和接触黏附行为，并发现单壁纳米管间的范德华力直接影响着其最终的稳定构型。所获得的与文献相一致的数值结果也证明了目前所发展方法的正确性和有效性。
Since their discovery by Iijima, carbon nantoubes (CNTs) have attracted much attention and interest from scientists in various disciplines. Many studies show that CNTs possess possibly unique mechanical, electronical and optical properties which result in potential applications in industry. However, both of experimental and theoretical works for CNTs have been being in a groping stage, and there are still some significant discrepancies among the results of mechanical properties of CNTs. Thus, based on the higher-order Cauchy-Born constitutive model and the generalized parametric constitutive law for van der Waals force developed here, a series of mechanical properties, compression buckling and adhesion behaviors for single-walled carbon nanotubes (SWCNTs) are studied respectively in the present thesis. The following progresses are made:
Based on the higher order Cauchy-Born rule, a constitutive model of SWCNT is established. In the present model, by including the second order deformation gradient tensor in the kinematic description, the limitation of the standard Cauchy-Born rule for the modeling of nanoscale crystalline films can be alleviated with less computational efforts. Based on the established relationship between the atomic potential and the macroscopic continuum strain energy density, analytical expressions for the tangent modulus tensors are derived. Moreover, the interatomic potential and the atomic structure of CNT are incorporated into the proposed constitutive model in a consistent way. Therefore SWCNT can be viewed as a macroscopic generalized continuum with microstructure.
Based on the higher-order Cauchy-Born constitutive model, a seriers of infinitesimal mechanical properties and the strain energy for the graphite sheet and SWCNT are studied firstly with the consideration of the influences from tube chirality and radius, potential parameters and the inner displacement. The results show that infinitesimal properties of SWCNTs strongly depends on the tube radius as for smaller tubes, however, the dependence becomes very weak as for larger tubes and a plateau, which corresponds to the modulus of graphite predicted by the present method, is reached with increase of tube radius. It is also noted that potential parameters and the inner displacement have significant influences on mechanical properties of SWCNTs. Besides, the strain energy and the tangential elastic moduli for CNTs are also obtained under finite deformation. The results show that the tangential elasitic moduli decreases linearly with increasing the axial stretch, and the strain energy for SWCNTs with similar tube radius is consistent with each other. Then, the axial-deformation-induced circular and torsion response in chiral SWCNTs is analyzed systematically, and the maximum coupling response is obtained in SWCNT with chiral angle ofπ/12.
Some basic problems on the study of the bending stiffness of CNTs are analyzed based on their defomation mechanism. It is pointed out that the bending stiffness of a flat graphite sheet and that of CNTs are two different concepts. The former is an intrinsic material property while the later is a structural one. Since the smeared-out model of CNTs is a generalized continuum with microstructure, its effective bending stiffness should be regarded as an independent structural rigidity parameter which can not be determined simply by employing the classic formula in beam theory. Furthermore, based on the higher-order Cauchy-Born constitutive model, the reasonable computational methods and results for the bending stiffness of the graphite sheet and CNTs are given.
Based on the higher-order Cauchy-Born constitutive model, the compression deformation beyond the buckling point for SWCNTs is implemented numerically via the mesh-free method, and is analyzed based on the strain energy obtained. Results consistent with those based on molecular dynamic and atomic finite element methods confirm the feasibility and validity of the presented constitutive model.
A parametric variational principle for van der Waals force simulation between any two non-bonded atoms is also established together with the corresponding improved quadratic programming method for numerical simulation of mechanical behaviours of carbon nanotubes. Carbon-carbon covalent bond interaction in carbon nanotubes is modeled and computed based on molecular structural mechanics model. Van der Waals force is simulated by the network of bars with a special nonlinear mechanical constitutive law in the finite element analysis. In comparison with conventional numerical methods, the suggested method does not depend on displacement and stress iteration, but on the base exchanges in the solution of a standard quadratic programming problem. Thus, the model and method developed exhibit very good convergence behavior in computation and provide accurate predictions of the mechanical behaviours and displacement distributions in the nanotubes. Based on the present method, the equilibrium configuration and the adhesion behaviors for two parallel or cross SWCNTs are studied respectively. Numerical results demonstrate the validity and the efficiency of the proposed method.
本文提出了一种基于高阶Cauchy-Born准则建立单壁碳纳米管本构模型的方法。通过引入高阶变形梯度，合理地修正了传统Cauchy-Born准则在描述纳米管变形几何关系时所存在的缺陷。利用原子间相互作用势以及能量等效原理，得到了基于广义连续介质模型的单壁碳纳米管的本构关系。由此得到的本构参数不仅与变形梯度张量F，而且与其梯度▽F相关，同时也包含了碳纳米管的微结构信息，因此是一种考虑微结构效应的广义连续介质模型。
基于所发展的高阶Cauchy-Born广义连续体本构模型，首先对石墨片和单壁碳纳米管的一系列无限小变形的力学特性(如：轴向和环向模量、泊松比、剪切模量)以及应变能进行了研究，同时也考察了手性、半径、势能参数以及内位移放松等因素对这些力学特性的影响。研究发现：当单壁碳纳米管的半径较小时，手性和半径对其力学特性的影响十分明显；而当管半径较大时，这种影响不再重要，以至于随着管半径的增大单壁碳纳米管的这些的力学特性最终趋于石墨片相应的结果。从数值结果上看，势能参数和内位移放松对所预测的单壁碳纳米管的力学特性有着很大的影响。进一步，我们也研究了简单的有限变形下单壁碳纳米管的应变能和切向模量的变化。结果表明，单壁碳纳米管的切向模量几乎线性地随着轴向拉伸而减小；且对于半径相近而手性不同的碳纳米管而言，其系统应变能几乎完全一致。继而，手性碳纳米管的轴向变形与其径向和扭转变形间的耦合机理的研究表明，最大的耦合效应发生于手性角为π／12的碳纳米管中。
从变形机理方面重新审视了石墨片和碳纳米管的弯曲刚度，并指出了部分文献中关于这两个概念使用和计算上的不恰当性。具体而言，石墨片和碳纳米管的弯曲刚度这两个概念本质上是不同的，前者指的是材料的内禀特性，而后者是结构特性。当采用经典连续体理论模拟碳纳米管时，等效的连续体应该被看作是带有微结构的广义连续介质，因此本文建议把碳纳米管的结构弯曲刚度处理为一个独立的结构刚度参数。进而，基于高阶Cauchy-Born广义本构模型给出了合理的计算石墨片和碳纳米管弯曲刚度的方法和结果。
基于高阶Cauchy-Born广义连续体本构模型，结合无网格方法进行了单壁碳纳米管的压缩屈曲变形的分析，从应变能的角度分析了屈曲前后碳纳米管的变形机理。所获得的与分子动力学模拟和原子有限元计算相一致的结果证明了目前所发展的模型的可行性和正确性。
本文也提出了处理非成键原子间范德华力的广义参变本构模型以及基于此进行碳纳米管结构力学行为数值模拟的数学规划算法。纳米管中原子间短程力作用采用分子结构力学模型来模拟，而作为长程作用的范德华力用杆单元来模拟，这类杆单元有着特殊的非线性本构关系。对于这种高非线性问题的处理，本文基于所建立的广义参变本构模型与参数二次规划算法求解。与一般的数值方法相比较，本文的方法不需要传统的那种冗长的、反复的迭代，并具有非常好的收敛性，因此为碳纳米管结构力学行为的有效预测提供了保障。基于此我们研究了平行和交叉排列的碳纳米管的稳定构型和接触黏附行为，并发现单壁纳米管间的范德华力直接影响着其最终的稳定构型。所获得的与文献相一致的数值结果也证明了目前所发展方法的正确性和有效性。
Since their discovery by Iijima, carbon nantoubes (CNTs) have attracted much attention and interest from scientists in various disciplines. Many studies show that CNTs possess possibly unique mechanical, electronical and optical properties which result in potential applications in industry. However, both of experimental and theoretical works for CNTs have been being in a groping stage, and there are still some significant discrepancies among the results of mechanical properties of CNTs. Thus, based on the higher-order Cauchy-Born constitutive model and the generalized parametric constitutive law for van der Waals force developed here, a series of mechanical properties, compression buckling and adhesion behaviors for single-walled carbon nanotubes (SWCNTs) are studied respectively in the present thesis. The following progresses are made:
Based on the higher order Cauchy-Born rule, a constitutive model of SWCNT is established. In the present model, by including the second order deformation gradient tensor in the kinematic description, the limitation of the standard Cauchy-Born rule for the modeling of nanoscale crystalline films can be alleviated with less computational efforts. Based on the established relationship between the atomic potential and the macroscopic continuum strain energy density, analytical expressions for the tangent modulus tensors are derived. Moreover, the interatomic potential and the atomic structure of CNT are incorporated into the proposed constitutive model in a consistent way. Therefore SWCNT can be viewed as a macroscopic generalized continuum with microstructure.
Based on the higher-order Cauchy-Born constitutive model, a seriers of infinitesimal mechanical properties and the strain energy for the graphite sheet and SWCNT are studied firstly with the consideration of the influences from tube chirality and radius, potential parameters and the inner displacement. The results show that infinitesimal properties of SWCNTs strongly depends on the tube radius as for smaller tubes, however, the dependence becomes very weak as for larger tubes and a plateau, which corresponds to the modulus of graphite predicted by the present method, is reached with increase of tube radius. It is also noted that potential parameters and the inner displacement have significant influences on mechanical properties of SWCNTs. Besides, the strain energy and the tangential elastic moduli for CNTs are also obtained under finite deformation. The results show that the tangential elasitic moduli decreases linearly with increasing the axial stretch, and the strain energy for SWCNTs with similar tube radius is consistent with each other. Then, the axial-deformation-induced circular and torsion response in chiral SWCNTs is analyzed systematically, and the maximum coupling response is obtained in SWCNT with chiral angle ofπ/12.
Some basic problems on the study of the bending stiffness of CNTs are analyzed based on their defomation mechanism. It is pointed out that the bending stiffness of a flat graphite sheet and that of CNTs are two different concepts. The former is an intrinsic material property while the later is a structural one. Since the smeared-out model of CNTs is a generalized continuum with microstructure, its effective bending stiffness should be regarded as an independent structural rigidity parameter which can not be determined simply by employing the classic formula in beam theory. Furthermore, based on the higher-order Cauchy-Born constitutive model, the reasonable computational methods and results for the bending stiffness of the graphite sheet and CNTs are given.
Based on the higher-order Cauchy-Born constitutive model, the compression deformation beyond the buckling point for SWCNTs is implemented numerically via the mesh-free method, and is analyzed based on the strain energy obtained. Results consistent with those based on molecular dynamic and atomic finite element methods confirm the feasibility and validity of the presented constitutive model.
A parametric variational principle for van der Waals force simulation between any two non-bonded atoms is also established together with the corresponding improved quadratic programming method for numerical simulation of mechanical behaviours of carbon nanotubes. Carbon-carbon covalent bond interaction in carbon nanotubes is modeled and computed based on molecular structural mechanics model. Van der Waals force is simulated by the network of bars with a special nonlinear mechanical constitutive law in the finite element analysis. In comparison with conventional numerical methods, the suggested method does not depend on displacement and stress iteration, but on the base exchanges in the solution of a standard quadratic programming problem. Thus, the model and method developed exhibit very good convergence behavior in computation and provide accurate predictions of the mechanical behaviours and displacement distributions in the nanotubes. Based on the present method, the equilibrium configuration and the adhesion behaviors for two parallel or cross SWCNTs are studied respectively. Numerical results demonstrate the validity and the efficiency of the proposed method.
引文
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