Analytical and numerical studies for nanotubes and nanoplates based on a nonlocal elasticity approach

Abstract

Since the discovery of carbon nanotubes by Iijima in 1991, the properties of nanotubes raise great attention among researchers all around the world. Many works about the electronic, magnetic, chemical, thermal, mechanical properties of carbon nanotubes have been done by scientific researchers with different science background. Molecular dynamics and generalized continuum mechanics are popular methodologies for mechanical analysis of carbon nanotubes. Although molecular dynamics is an effective way to study the mechanical and other properties, it always requires extremely high computational expenses and it is applicable for only the nanosturcutures with a few numbers of atoms or molecular. Then, the generalized continuum mechanics methods were developed to capture the size effects of nanostructures based on the classic continuum mechanics. The modified couple stress elasticity, strain gradient elasticity and nonlocal elasticity, as the three main generalized continous theories, are becoming more widely used by researchers as possible substitutes to Molecular dynamics simulations. In late 1960s and early 1970s, Eringen developed the nonlocal elasticity theory. The theory states that the nonlocal stress at a point within a domain of interest is not only affected by the strain at that local point but also by strains at all other points in the entire domain in an integral manner. By using this nonlocal elasticity, some drawbacks in the classic elasticity theory can be avoided, and the size effect can be covered and explained for the nano/micro structures. Since the first applying the Eringen's nonlocal elasticity theory to analyze the mechanical response of carbon nanotubes by Peddieson in 2003, there are more than 180 papers published by far, which concerned with different derivatives nonlocal models for different nanostructures as nanorod, nanobeam ,nanotube, graphene, nanoplate, and different research topics about bending, buckling, free vibration and wave propagation. However, almost previous papers reach the same result that the rigidity predicted by nonlocal elasticities is lower than the one by classic theories. These results become very puzzled, for they are conflicted with the results by other research methodology as the couple stress theory and strain gradient theory, and they are not in accordance with the results of experiments under nanoscale. In 2007, Lim developed the exact nonlocal model based on the Eringen's nonlocal theory and variational principle. The so-called "exact" model makes the differences with the partial" model which was used widely in previous research papers by others. In Lim's model, the nonlocal stresses are solved from Eringen's nonlocal constitutive relation, and the nonlocal strain energy density is deduced in the forms of the classic strain and its deriatives. The higher-order governing equations and boundary conditions, both with infinite terms of strain derivatives, can be obtained by using the Hamilton's principle and variational principle. For applications of Lim's nonlocal model in this thesis, one dimensional nanostructures such as the carbon nanotube/nanobeam/nanorod and two-dimensional ones as the carbon annular circular nanoplate are considered. The buckling and free transversal and axial vibration problems for the carbon nanotubes and bending analysis by analytical and numerical methods for the annular circular nanoplate are presented in this thesis. Various examples reach the solution that the nanostructures' rigidity becomes increased with the nonlocal size effect, which is contrary with the previous research works based on the partial nonlocal model, but in accordance with that of other theories and some molecular dynamics simulation. Detailed discussions are also presented made in this thesis to better understand this new model. During the PhD peorid, the author made the main following contributions: 1) formula deduction for transversal bending, buckling and vibration analysis by using the variational principle and the Hamilton's principle; 2) studying the transversal buckling and vibration problems for carbon nanotubes and the axial vibration problem for nanorod by using the exact nonlocal model; 3) formula deduction and analyzing for transversal bending problem for annular circular nanoplate; 4) developing a FORTRAN programming to get the numerical solution for bending analysis of annular circular nanoplate, and this programming can be further revised for buckling and vibration analysis of nanoplate.

Date of Award	16 Jul 2012
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	C W LIM (Supervisor)