I describe some recent work aimed at generalizing Koiter's Koiter (1960, 1966) small-strain, finite-deformation model of elastic shells to the case of finite strains. The objective is to clearly understand the approximative nature of shell theory as a two-dimensional, small-thickness model of three-dimensional finite-elasticity theory. Most of the current research along these lines relies on the method of Gamma convergence (Friesecke et al. 2006), which is concerned with the limiting variational problem as thickness tends to zero, or, alternatively, on asymptotic analysis of the weak forms of the equilibrium equations (Ciarlet 2000, 2005). These methods have yielded rigorous derivations of membrane theory (Le Dret & Raoult 1996) and pure-bending theory (Friesecke et al (2006)) in the limit of small thickness. However, neither method has generated a model that accommodates simultaneous bending and stretching in a single framework. In contrast, Koiter's model, while limited to small strains, accommodates combined bending and stretching, and, although not a limit model in the sense of Gamma convergence or asymptotic analysis, has nevertheless been justified (Ciarlet 2005) through comparisons with solutions to the three-dimensional theory. Here I describe a systematic and straightforward approach to the problem of extending Koiter's model to finite strains. To ease the notation, and to exhibit the basic ideas in as simple a setting as possible, I confine attention to the case of plates. Certain formulae required in the extension to shells are discussed but not developed here in any detail.
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