Shell Constitutive Equations
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Shell Constitutive Equations
2.1 Introduction A simple theory of plates was originally developed by Sophie Germain. The corrected version by Kirchhoff (1850) is widely used in the analysis of thin plates. Using Kirchhoff’s concept Love developed the complete two-dimensional theory of thin shells more than 100 years ago, and numerous. Contributions to this subject have been made since then. Any two-dimensional theory of shells approximates the real three-dimensional problem. Researchers have been seeking better approximations for the exact three-dimensional elasticity solutions for shells. In the last three decades, the developed refined two-dimensional linear theories of thin shells include important contributions Sanders (1959), Flugge (1960), and Niordson (1978). In these refined shell theories, the initial curvature effect is taken into consideration. Nevertheless, the deformation is based on the Love–Kirchhoff assumption, and the radial stress effect is neglected. We will refer to all the theories built on the Kirchhoff–Love assumption, as the classical theory. The refined theories by Sanders (1959), Flugge (1960), and Niordson (1978) provide very good results for the analysis of thin shells. The theory of Sanders-Koiter has been widely used in the finite element analysis of shells (Ashwell and Gallagher, 1976). Niordson (1971) showed, however, that Love’s strain energy expression has inherent errors of relative order [h/R + (h/L)2 ], where h is the thickness of the shell, R is the magnitude of the smallest principal radius of curvature, and L is a characteristic wave length of the deformation pattern of the middle surface. Consequently, when the refined theories of thin shells are applied to thick shells, with h/R not small compared to unity, the error can be quite large. Unlike the theory of thin shells, the comprehensive theory of thick shells, with not only transverse shear strains considered but also initial curvature and radial stresses, has received limited attention from researchers. Voyiadjis and Shi (1991) developed a refined shell theory for thick cylindrical shells that is very accurate and convenient for finite element analysis. We present here a refined shell theory for thick spherical shells, with the shell equations based on similar assumptions as those of Voyiadjis and Shi (1991). The work of Voyiadjis and Woelke (2004) can be considered a more general formulation of the Voyiadjis and Shi theory (1991).
G.Z. Voyiadjis, P. Woelke, Elasto-Plastic and Damage Analysis of Plates and Shells, C Springer-Verlag Berlin Heidelberg 2008
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2 Shell Constitutive Equations
In the following sections of this chapter we present detailed derivations of the shell constitutive equations. The general form of these expressions is complicated; they can, however, easily be reduced to commonly used shell equations, as shown below. At every stage of the formulation, we make references to the classical shell theory.
2.1.1 Thickness of the Shell Thick shells have a number of distinctly different features from t
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