By Hari Singh Nalwa, Foreword by Richard E. Smalley
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Extra info for Encyclopedia of Nanoscience and Nanotechnology Volume 7
In the simplest terms, the shear modulus is the ratio of the applied stress to the strain of a material in the case where a deformation changes the shape of a body, but not its volume. The opposite case is true for the bulk modulus. It represents the stress/strain ratio for a deformation changing the volume of a body, but not its shape (hydrostatic). Both K and are deﬁned assuming a linear relationship between the applied stress and strain. The Young’s modulus for both SWNTs and MWNTs has been calculated, simulated, and measured in a variety of methods which are described in more detail below.
In such a case, the cantilever is inertially damped and, on the ultrasonic time scale, effectively rigid . Hence, the surface rigidity of materials with contact stiffness orders of magnitude higher than the cantilever spring constant can be quantitatively measured. To illustrate how surface nanomechanical rigidity can be extracted via UFM, Figure 5 shows a schematic of a prototypical force–displacement curve for a rigid nanoprobe tip in contact with a surface. Negative displacement (indentation) of the sample by the tip results in a strong repulsive force.
One of the earliest theoretical studies of the mechanical properties of SWNTs was carried out by Robertson et al. . 9 nm using two related many-body empirical potentials and local density functional (LDF) theory . The many-body potentials included atomic core–core repulsive terms and attractive terms related to the C valence electrons. It was found that the tensile or compressive strain energy per carbon atom of an SWNT relative to an unstrained tube varied as 1/R2 , where R is the tube radius.
Encyclopedia of Nanoscience and Nanotechnology Volume 7 by Hari Singh Nalwa, Foreword by Richard E. Smalley