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nonshallow structure. Only more complex shell is capable of reproducing
the results of MD simulations.
Some parameters, such as wall thickness of CNTs are not well defined
in the continuum mechanics. For instance, value of 0.34 nm, which is in-
terplanar spacing between graphene sheets in graphite, is widely used for
tube thickness in many continuum models.
The finite element method works as the numerical methods for determin-
ing the energy minimizing displacement fields, while atomistic analysis is
used to determine the energy of a given configuration. This is in contrast
to normal finite element approaches, where the constitutive input is made
via phenomenological models. The method is successful in capturing the
structure and energetic of dislocations. Finite element modeling is directed
by using 3D beam element, which is as equivalent beam to construct the
CNT. The obtained results will be useful in realizing interactions between
the nanostructures and substrates and also designing composites systems.
9.4.1.3
NANO-SCALE CONTINUUM MODELING
Unlike to continuum modeling of CNTs where the entirely discrete struc-
ture of CNT is replaced with a continuum medium, nano-scale continuum
modeling provides a rationally acceptable compromise in the modeling
process by replacing C-C bond with a continuum element. In the other
hand, in nano-scale continuum modeling the molecular interactions be-
tween C-C bonds are captured using structural members whose properties
are obtained by atomistic modeling. Development of nano-scale continu-
um theories has stimulated more excitement by incorporating continuum
mechanics theories at the scale of nano. Nano-scale continuum modeling
is usually accomplished numerically in the form of finite element model-
ing. Different elements consisting of rod, truss, spring and beam are used
to simulate C-C bonds. The two common method of nano-scale continu-
um are
quasi-continuum
(QC) and
equivalent-continuum
methods, which
have been used in nano-scale continuum modeling.
The QC
method which presents a relationship between the deforma-
tions of a continuum with that of its crystal lattice uses the classical Cau-
chy-Born rule and representative atoms. The quasi-continuum method
mixes atomistic-continuum formulation and is based on a finite element
discretization of a continuum mechanics variation principle.
