Finite Rotations and Finite Elements of Thin Elastic Shells

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The proposed investigation includes theoretical and computational studies. The underlying concept was presented earlier by the author. The idea stems from the geometrical nature of the deformations which often occur in thin shells, plates, arches and beams; these deformations are usually characterized by small strains but often accompanied by finite rotations. These features have far-reaching consequences in the contemporary procedures for approximating such structures by the method of finite elements. Since the strains are small, relative rotations are small within elements. Consequently, one can describe the deformations of the individual element by linear equations and, if the material is hookean, the entire description of the element is given by linear equations. Then the occurrence of finite rotations and the associated nonlinearities can be introduced in the formation of the global matrix. The foregoing observations suggest practical advantages in the application and formulation of the models which are used to describe the thin structures and to predict their response. However, many questions remain unanswered, questions about the accuracy of such formulation, applicability of convergence criteria, and practical advantages and disadvantages of such procedures. The purpose of the proposal is to seek the answers by scrutinizing the theoretical bases, the alternative functionals, stationary and/or extremal criteria. Recent formulations of the complementary functionals provide means to that end. Computational programs for numerical and symbolic investigations are other means for exploring and for testing the practicality of the results.