 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| E. A. Ivanova, N. F. Morozov, B. N. Semenov, and A. D. Firsova, "Determination of elastic moduli of nanostructures: theoretical estimates and experimental techniques," Mech. Solids. 40 (4), 60-68 (2005) |
| Year |
2005 |
Volume |
40 |
Number |
4 |
Pages |
60-68 |
| Title |
Determination of elastic moduli of nanostructures: theoretical estimates and experimental techniques |
| Author(s) |
E. A. Ivanova (St. Petersburg)
N. F. Morozov (St. Petersburg)
B. N. Semenov (St. Petersburg)
A. D. Firsova (St. Petersburg) |
| Abstract |
In recent years, research activities in the field of fabricating and
studying nanodimensional tubes have been abundant [1-5]. It is important
to analyze not only electronic and optical [6] but also mechanical
properties of nanostructures. Nanotubes can undergo large deformations
without losing elastic properties [3]. Therefore, stress-strain analysis of
nanotubes is usually based on elastic shell theory [7]. Moreover, the
elastic moduli are determined from discrete models that include only the
force interaction between the atoms forming a tube. However, the existence
of single-walled nanotubes [4, 5] suggests that the moment interaction
between the atoms should also be taken into account. Otherwise the atomic
layer forming a nanotube would have zero bending stiffness and a
single-walled nanotube would be unstable.
In the first part of the present paper, using the discrete model [8, 9] of
a monocrystal as an example, we devise a technique for determining the
bending stiffness of nanodimensional structures with regard to the moment
interaction at the nanoscale. We obtain corrections allowing for the moment
interaction and permitting one to describe the mechanical properties of
single-walled nanostructures. The main obstacle encountered in attempts to
use moment theories in applied problems is the essential lack so far of
established techniques for determining moment elastic moduli
experimentally. The difficulty is that the moment interactions in the
material are so weak that it is virtually impossible to observe their
manifestations in macroscopic experiments. At the nanoscale, the
contribution of moment interactions is larger and proves to be rather
significant for nanostructures containing several atomic layers. Therefore,
experiments with nanodimensional structures are a promising way to devise
techniques for determining elastic moduli in moment theories. To find the
Lamé moment coefficients [10], one can use, say, experiments in which the
bending stiffness of nanoobjects consisting of a single atomic layer is
determined. As a rule, the elastic moduli of thin macroscopic shells are
determined on the basis of experiments with plates. Nanoplates exist only
in a stressed state, being attached to substrate. When separated from the
substrate, the plates roll up and, in a stress-free state, become shells of
various configurations. Thus to determine elastic moduli of nanostructures
one needs a technique based on experiments with shells. As common
nanoobjects like nanotubes and fullerenes undergo an arbitrary deformation,
the material is subject to bending and tension simultaneously. Therefore,
all directly measurable quantities (e.g., fundamental frequencies) depend
in a complicated way on both bending and tensile stiffness. There exist
cylindrical shell vibrations at which the material is subject to bending
alone. However, it is very difficult to observe such vibrations of
nanodimensional objects, since the cylinder axis remains rectilinear and
the cross-section shape does not vary along the axis. In recent years,
along with nanotubes and fullerenes, nanoobjects of more complicated
configuration [11-14] have been fabricated. From the viewpoint of possible
experimental determination of bending stiffness, nanodimensional helical
coils are of special interest [11, 13]. This is due to the fact that for
arbitrary deformation of helical shells the material is primarily subject
to bending, so that one can neglect extension effects when interpreting
experimental data; the fundamental vibration modes of a helical shell are
much easier to observe than those of a cylindrical shell, related to pure
bending of the material. The last assertion is illustrated in Fig. 1, where
the first four vibration modes of a helical shell are shown.
In the second part of the paper, we analyze the dynamics of helical shells
[15]; this analysis can serve as a theoretical foundation for the
experimental determination of the bending stiffness of nanodimensional
shells. |
| References |
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| 15. | Yu. A. Ustinov, The Saint-Venant Problem for Pseudocylinders [in
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| 16. | E. A. Ivanova, A. M. Krivtsov, N. F. Morozov, and A. D. Firsova,
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| 17. | E. A. Ivanova, A. M. Krivtsov, N. F. Morozov, and A. D. Firsova,
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| 18. | P. A. Zhilin, "The main equations of nonclassical elastic shell
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|
| Received |
04 April 2005 |
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