Theoretical and numerical studies of single-walled carbon nanotubes based on the higher-order gradient continuum
Student thesis: Doctoral Thesis
Carbon nanotubes (CNTs) have attractive applicability prospects due to their outstanding physical, mechanical, and electrical properties. In addition to a large amount of experimental work, theoretical modeling plays an important role in capturing and understanding the delicate behavior of nanostructures. The development of efficient computational techniques is currently an ongoing and challenging process in the research domain of nanoscience and nanotechnology. The present research investigates the mechanical properties of CNTs in virtue of a fine continuum mechanics theory: the higher-order gradient continuum. With the derived hyper-elastic constitutive model, the structural and mechanical properties of single-walled carbon nanotubes (SWCNTs) are determined and analyzed in the theoretical scheme of the higher-order gradient continuum. The research then turns to the numerical simulation of the mechanical behavior of CNTs. Based on the established higher-order constitutive relationship, a mesh-free computational framework is developed to implement the numerical computation of SWCNTs for the first time. The mechanical response of SWCNTs under various loading methods is numerically modeled and investigated. The research finally explores the multiscale coupling of the developed mesh-free method with an atomic simulation. The essential idea of the higher-order gradient continuum is that the strain energy depends on both the first- and second-order deformation gradients. To derive the nanoscale continuum constitutive relationship, the deformation of microscale bond vectors is approximated with an extended Cauchy-Born rule, the higher-order Cauchy-Born rule, in which the effect of the second-order deformation gradient is involved. The continuum elastic potential is obtained by equating the deformation energy of a representative cell with that of an equivalent volume of the continuum. With the use of an interatomic potential, a nanoscale continuum constitutive model is thus established. While the second-order deformation gradient is considered, the approximation for microscale bond vectors is largely enhanced. Specifically, the second-order term describes the bending effect, and thus the derived constitutive model is more reasonable. A significant contribution of the present research is its investigation for the structural and mechanical properties of SWCNTs in the theoretical scheme of the higher-order gradient continuum. An undeformed SWCNT can be viewed as having been formed by rolling up a graphite sheet into a cylindrical shape. However, this rolling is not a grid transformation. With three geometrical parameters, the rolling process is appropriately written as a set of equations. The structural and elastic properties of SWCNTs are determined by minimizing the energy of a representative cell. The obtained structural properties are compared with those obtained by other researchers using an exact transforming map. Different types of CNTs are studied, and the dependence of their elastic properties on the chirality and radius of the tubes is investigated and analyzed. The established model can also be applied to the semi-analytical study of the mechanical response of SWCNTs under axisymmetrical loading conditions. The most important contribution of this thesis is the development of a mesh-free computational framework to implement the numerical simulation of SWCNTs based on the established higher-order constitutive relationship. Because the second-order deformation gradient is involved in the present theory, the interpolation of displacements generally requires C1 -continuity in the numerical simulation. In the finite element method (FEM), this requirement leads to great difficulty in establishing elements and constructing the interpolation functions. Mesh-free methods are newly developed computational techniques that have some distinct advantages over classical numerical methods. In particular, the mesh-free approximation possesses non-local properties and automatically satisfies the higher-order continuity requirement. This intrinsic non-local property leads to real rotation-free approximation, and displacements can be used as the only nodal freedoms. This advantage is employed in the present research, and the application of a mesh-free method in the study of CNTs is exploited. With the developed mesh-free method, the response of SWCNTs under the hydrostatic pressure is first simulated. For this case, the tube deforms uniformly along the axial direction, and the analysis can be simplified by taking off the axial freedom of degree. The structural transition occurs when the hydrostatic pressure reaches the critical value. The responses of SWCNTs under axial compression, torsion, and bending are simulated and studied, respectively. The computational precision and convergence of the mesh-free method are studied in comparison with the full atomic simulation, and the choice of the mesh-free parameter is discussed. As is well known, atomic simulation becomes difficult or impossible for large-scale problems. For homogeneous deformation, a smaller number of nodes can achieve a higher level of precision. This reveals that continuum simulation can overcome the difficulty in computational cost of atomic simulations. Mesh-free numerical simulations based on the standard Cauchy-Born rule are also carried out using the constitutive model. These computations reveal that simulation based on this rule cannot exhibit the true buckling patterns of SWCNTs, while the buckling deformation can be truly displayed with the present higher-order theory. This shows that the present theory can overcome the shortcomings of models based on the standard Cauchy-Born rule. Finally, the author investigates the application of the developed mesh-free method in multiscale analysis. The multiscale method is currently a popular technique for structural computation and analysis, and it takes advantage of both atomic simulation and the continuum method. The multiscale method uses atomic simulation for the localized region in which the discrete motion of atoms is important, and the continuum method for the remaining regions in which the deformation is homogeneous and smooth. It is a viable and efficient means of studying materials and systems across different scales. The key issue in the multiscale method is determining how to efficiently and smoothly bridge two scales. Most multiscale analyses overlook the importance of the rationality and feasibility of continuum models. The present study employs the bridging domain method to couple the developed mesh-free computational framework and an atomic simulation technique. Two multiscale computational examples are performed, and good computational efficiency is obtained. This shows that the established higher-order continuum theory and the developed mesh-free computational framework are reasonable, efficient, and accurate and surely have good applicability prospects in the future study of nanotubes and nanostructures.
- Nanostructured materials, Nanotubes, Carbon