Asymptotic construction of a dynamic shell theory: Finite-element-based approach

In the theory of shells, asymptotically correct dimensional reduction from the dynamic equations of three-dimensional (3-D) elasticity to the approximate two-dimensional (2-D) equations of shell theory is possible only within the long-wavelength approximation, just as in statics. However, as pointed out in the literature, such a procedure is not suitable for analysis of complicated high-frequency effects in the short-wavelength regime because displacements may change rapidly through the thickness. Therefore, in addition to the general derivation of 2-D equations within the long-wavelength regime, construction of a dynamic shell theory must involve a dimensional reduction procedure for the high-frequency behavior plus a separate and logically independent step for the short-wavelength regime. 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 procedure rigorously splits the 3-D problem into a linear one-dimensional (1-D) through-the-thickness analysis and a nonlinear 2-D shell analysis, taking into account both low- and high-frequency vibration within the long-wavelength regime. Clearly, this is not sufficient 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 3-D stress state in the short-wavelength regime with a 2-D theory. The main focus of this paper regards the through-thickness analysis, which is solved by a 1-D finite element approximation and which provides a means to calculate both inertia and stiffness coefficients for 2-D composite shell theory. This paper then represents the first attempt to develop a procedure based on finite elements for the through-thickness analysis associated with shell dynamics, which makes more the problem tractable in practice. Finally, numerical results are compared with published analytical solutions. The excellent agreement demonstrates the capabilities of the present model to analyze dynamic structural responses over a wide range of frequencies and wavelengths.