Discussion: “Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes” (Wang, C. Y., Ru, C. Q., and Mioduchowski, A., 2004, ASME J. Appl. Mech., 71, pp. 622–631)

Abstract

I wish to point out that there are equations for the vibration ((1), pp. 259–261) and buckling (2) of elastically isotropic circular cylindrical shells that are as accurate as, but much simpler than, the so-called Exact Flügge Equations (Model III) that the authors use as their standard of comparison for the two sets of approximate equations they analyze, namely, the (simplified) Donnell Equations (Model I) and the Simplified Flügge Equations (Model II). (I use the adjective “so-called” because there is no set of two-dimensional shell equations that is “exact.”) On pp. 225–230 of (1) Niordson presents one possible derivation of the Morley-Koiter equations in terms of midsurface displacements in which the two equations of tangential equilibrium (or motion) are identical to the simplified Donnell equations—that is, the first two of the authors’ Flügge equations (3) with the coefficients of the small parameter (1−ν2)(D∕EhR2) set to zero—whereas the equation of normal equilibrium (or motion) may be obtained from the third Flügge equation by replacing the coefficient of (1−ν2)(D∕EhR2) in brackets by 2R2∇2w+w, where ∇2=∂2∕∂x2+R−2∂2∕∂θ2.