DETERMINATION OF ELASTIC MODULI OF NANOSTRUCTURES:THEORETICAL ESTIMATES AND EXPERIMENTAL TECHNIQUES

摘要

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 thenanoscale. 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 Lame 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 shel