Asymptotically Correct Dynamic Shell Theory: 1-D Through-Thickness Analysis

Abstract

In this paper, with the aid of small parameters that are inherent to the shell problem, a dimensional reduction yielding a shell theory distinct from established shell theories is carried out using the variational-asymptotic method. The dimensional reduction procedure rigorously splits the three-dimensional problem into a linear one-dimensional, throughthickness analysis and a nonlinear two-dimensional shell analysis, taking into account both low- and high-frequency vibration within the long-wavelength regime. Clearly, this is not sucient for the short-wavelength regime. Therefore, one more step is used in the present paper { called hyperbolic short-wavelength extrapolation { which enables the analysis to qualitatively describe a three-dimensional stress state in the short-wavelength regime with a two-dimensional theory. The main focus of this paper regards the through-thickness analysis, which is solved by a one-dimensional nite element approximation and which provides a means to calculate both inertia and stiness coecients for two-dimensional composite shell theory. This paper then represents the rst attempt to develop a procedure based on nite elements for the through-thickness analysis associated with shell dynamics, which makes more the problem tractable in practice.