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|Title: ||Theoretical and numerical studies of single-walled carbon nanotubes based on the higher-order gradient continuum|
|Other Titles: ||Ji yu gao jie ti du lian xu de dan bi tan na mi guan de li lun he shu zhi yan jiu|
|Authors: ||Sun, Yuzhou (孫玉周)|
|Department: ||Department of Building and Construction|
|Degree: ||Doctor of Philosophy|
|Issue Date: ||2008|
|Publisher: ||City University of Hong Kong|
|Notes: ||CityU Call Number: TA455.C3 S86 2008|
xxii, 219 leaves : ill. 30 cm.
Thesis (Ph.D.)--City University of Hong Kong, 2008.
Includes bibliographical references (leaves -219)
|Abstract: ||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
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.|
|Online Catalog Link: ||http://lib.cityu.edu.hk/record=b2340563|
|Appears in Collections:||BC - Doctor of Philosophy |
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