Chapter 4 Dynamic Equations for an Isotropic Spherical Shell Using Power Series Method and Surface Differential Operators Reza Okhovat and Anders Bostr€om Abstract Dynamic equations for an isotropic spherical shell are derived by using a series expansion technique. The displacement field is split into a scalar (radial) part and a vector (tangential) part. Surface differential operators are introduced to decrease the length of the shell equations. The starting point is a power series expansion of the displacement components in the thickness coordinate relative to the mid-surface of the shell. By using the expansions of the displacement components, the three-dimensional elastodynamic equations yield a set of recursion relations among the expansion functions that can be used to eliminate all but the four of lowest order and to express higher order expansion functions in terms of these of lowest orders. Applying the boundary conditions on the surfaces of the spherical shell and eliminating all but the four lowest order expansion functions give the shell equations as a power series in the shell thickness. After lengthy manipulations, the final four shell equations are obtained in a more compact form which can be represented explicitly in terms of the surface differential operators. The method is believed to be asymptotically correct to any order. The eigenfrequencies are compared to exact three-dimensional theory and membrane theory. Keywords Spherical shell • Shell equations • Surface differential operators • Dynamic • Eigenfrequency • Power series 4.1 Introduction Shells are commonly used in many branches of engineering and the demands for shell structures are increasing. Therefore they have been investigated for a number of different types of shells and new methods are developed for designing and using shells in structures. Shells appear in many aspects like designing pressure vessels, fuselages of airplanes, boat and ship hulls, roof structures, bodies of cars, trains, and aeroplanes. A shell can be considered as a curved plate having small thickness compared to the other geometrical dimensions as well as to the wavelengths of importance. The most important superiority of shells in comparison to plates is that shell structures can provide high strength and low weight because of their membrane stiffness. Particularly, spherical shells appear in many structures and have many applications. Some dynamic shell theories have thus been developed for this case. All these theories seem to depend on more or less ad hoc kinematical assumptions and/or other approximations. The literature on shells is large. A comprehensive introduction to shell theory is given by Tuner [1] and shells with general shapes are studied by Heyman [2]. Basic equations for spherical and cylindrical shells are discussed by Niordson [3]. For the present purposes the most relevant references seem to be those of Shah et al. [4, 5] and Niordson [6]. Shah et al. [4] seem to be the first to give the exact three-dimensional solution for the eigenfrequencies of a spherical shell (of arbitrary thickness), drawing on earlier work by Morse and Feshbach [7]. In particular, governing equations for spherical and cylindrical shells are given by Soedel [8], and thin cylindrical shells are studied by Leissa [9]. R. Okhovat • A. Bostr€om Department of Applied Mechanics, Chalmers University of Technology, SE-41296 Gothenburg, Sweden e-mail: reza.okhovat@chalmers.se N. Sottos et al. (eds.), Experimental and Applied Mechanics, Volume 6: Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-06989-0_4, #The Society for Experimental Mechanics, Inc. 2015 29
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