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conditions and also are strongly influenced by the vdW coefficient used.
The performed investigations on the simulation of vibrations properties
classified by the method of modeling are presented in Tables 9.7-9.9. As
it can be seen 5-7, 5-8, and 5-9, the majority of investigations used con-
tinuum modeling and simply replaced a CNT with hollow thin cylinder
to study the vibrations of CNT. This modeling strategy cannot simulate
the real behavior of CNT, since the lattice structure is neglected. In other
word, these investigations simply studied the vibration behavior of a con-
tinuum cylinder with equivalent mechanical properties of CNT. Actually
nano-scale continuum modeling is preferred for investigating vibrations
of CNTs, since it was reported that natural frequencies of CNTs depend on
both chirality and boundary conditions.
9.6
SUMMARY AND CONCLUSION ON CNT SIMULATION
The CNTs modeling techniques can be classified into three main cate-
gories of atomistic modeling, continuum modeling and nano-scale con-
tinuum modeling. The atomistic modeling consists of MD, MC and Ab
initio methods. Both MD and MC methods are constructed on the basis of
second Newton's law. While MD method deals with deterministic equa-
tions, MC is a stochastic approach. Although both MD and MC depend
on potential, Ab initio is an accurate and potential free method relying on
solving Schrödinger equation. Atomistic modeling techniques are suffer-
ing from some shortcomings which can be summarized as: (I) inapplica-
bility of modeling large number of atoms (II) huge amount of computa-
tional tasks (III) complex formulations. Other atomistic methods such as
tight bonding molecular dynamic, local density, density functional theory,
Morse potential model and modified Morse potential model are also avail-
able which are in need of intensive calculations.
On the other hands, continuum modeling originated from continuum
mechanics are also applied to study mechanical behavior of CNTs. Com-
prising of analytical and numerical approaches, the validity of continuum
modeling has to be carefully observed wherein lattice structure of CNT
is replaced with a continuum medium. Numerical continuum modeling
is accomplished through finite element modeling using shell or curved
plate elements. The degree to which this strategy, that is, neglecting lat-
tice structure of CNT, will lead us to sufficiently accurate results is under
