Mechanics and Finite Elements of Shells

This article begins with a brief review of the foundations: The classical theory of Love is described with attention to the underlying hypothesis and consequent limitations. A more general theory is described which removes the constraints of Love; the inclusion of transverse strains admits simpler finite elements, accommodates the thick shell via layers and even a transition to the three-dimensional approximation. The concept of the finite element is reviewed in the context of the discrete approximation of shells. Specific attention is given to those problems which are peculiar to shells: The predominant roles of flexural and extensional deformations, the lesser role of transverse shear, can lead to excessive stiffness (“locking”). Origins and procedures are described to circumvent these problems. The review is intended to bridge some chasms between the mechanics of the continua and the discrete models of finite elements. As such, the emphasis is upon those mechanical attributes of shells and elements which play key roles in forming practical models. Since the limitations of space, time and the author’s knowledge, preclude a full expose, the review includes only commentaries on some topics, such as inelasticity, nonlinearity and instability. Citations include original sources and some recent works which provide entree to contemporary developments.