Dynamic Variational-Asymptotic Procedure for Laminated Composite Shells—Part II: High-Frequency Vibration Analysis

Abstract

Shell theories intended for low-frequency vibration analysis are frequently constructed from a generalization of the classical shell theory in which the normal displacement (to a first approximation) is constant through the thickness. Such theories are not suitable for the analysis of complicated high-frequency effects in which displacements may change rapidly along the thickness coordinate. Clearly, to derive by asymptotic methods, a shell theory suitable for high-frequency behavior requires a different set of assumptions regarding the small parameters associated with the characteristic wavelength and timescale. In Part I such assumptions were used to perform a rigorous dimensional reduction in the long-wavelength low-frequency vibration regime so as to construct an asymptotically correct energy functional to a first approximation. In Part II the derivation is extended to the long-wavelength high-frequency regime. However, for short-wavelength behavior, it becomes very difficult to represent the three-dimensional stress state exactly by any two-dimensional theory; and, at best, only a qualitative agreement can be expected. To rectify this difficult situation, a hyperbolic short-wave extrapolation is used. Unlike published shell theories for this regime, which are limited to homogeneous and isotropic shells, all the formulas derived herein are applicable to shells in which each layer is made of a monoclinic material.

An asymptotically exact theory of functionally graded piezoelectric shells

“…with I[f ] being defined as in the previous case. Finally, with D from (32) being plugged in (31), we obtain…”

“…The synthesis of these two approaches, called the variational-asymptotic method, first proposed by Berdichevsky [2] and developed further by Le [29], seems to avoid the disadvantages of both approaches described above and proved to be quite effective in constructing approximate equations for thin-walled structures. Note that this method has been applied, among others, to derive the 2-D theory of homogeneous piezoelectric shells by Le [24,27], the 2-D static theory of purely elastic sandwich plates and shells by Berdichevsky [4,3], the theory of smart beams by Roy et al [42], the theory of low-and high frequency vibration of laminate composite shells by Lee and Hodges [31,32], and just recently, the theory of smart sandwich shells by Le and Yi [30]. Note also the closely related method of gamma convergence used in homogenization of periodic and random microstructures [8] and dimension reduction of plate theories [15].…”

“…For the high-frequency vibrations of elastic and piezoelectric shells and rods where the kinetic energy density should be kept in the variational-asymptotic analysis see[5,6,23,25,26,28,29,32].…”

Nonlinear Free Vibration Analysis of Laminated Carbon/Epoxy Curved Panels

“…The nonlinear FEM steps utilised to compute the large amplitude free vibration frequency responses of the doubly curved laminated composite shell panels using the HSDT kinematics model by Singh and Panda 22 and Mahapatra 23 , et al including the effect of the environmental effect. Also, a large volume of the research articles on the dynamic behaviour of the laminated composite plate and shell structure reported using variational asymptotic method (VAM) [24][25][26][27][28][29][30][31] . Studies indicate that the VAM approach is capable of analysing the geometrical nonlinearity including geometries effect with adequate accuracy.…”