A General Theory of Shells and the Complementary Potentials

Abstract

This theory incorporates the attributes which are essential to the approximation of shells by finite elements. It is limited only by one assumption: Displacement is a linear function of distance along the normal to a reference surface. Deformation is decomposed into rotation and strain; the rotation carries elements of the reference surface to the same surface in any subsequent state. Transverse-shear deformations accommodate simple elements. The theory is couched in the potential P v and in the complementary potential P c ; these have the property, P v + P c = 0 for all admissible (equilibrated) states. The theory is also cast in the complementary functional P̄ c of stress and displacement, and the functional P̄ v of displacement, strain and stress; P̄ c and P̄ v are akin to the functionals of Hellinger-Reissner and Hu-Washizu. These alternate functionals provide the means to develop various hybrid elements.