Dynamic Variational-Asymptotic Procedure for Laminated Composite Shells—Part I: Low-Frequency Vibration Analysis

Abstract

An asymptotically correct dynamic shell theory, valid over a wide range of frequencies and wavelengths, is rigorously derived from an analytical point of view. The derivation provides insight and guidance for the numerical modeling of layered shells. This work is based on three essential theoretical foundations: (a) the concept of decomposition of the rotation tensor, which is to establish the dynamic three-dimensional elasticity problem in a compact and elegant intrinsic form for application to the complex geometry of shells; (b) the variational-asymptotic method, which is to perform a systematic and mathematical dimensional reduction in the long-wavelength regime for both low- and high-frequency vibration analysis; and (c) hyperbolic short-wavelength extrapolation, which is to achieve simple, accurate, and positive definite energy functionals for all wavelengths. Based on these, unlike most established shell theories that are limited to the long-wavelength low-frequency regime, the present theory describes in an asymptotically correct manner not only the low-frequency but also some of the first high-frequency branches of vibrations in the long-wave range. Moreover, it recovers the approximate three-dimensional stress state in both long- and short-wavelength ranges.