Preface to volume III

Abstract

This chapter discusses the mathematical foundations of the two-dimensional theory of shells. It provides, without any recourse to any a priori assumptions of a geometrical or mechanical nature, a mathematical justification of two-dimensional nonlinear and linear shell theories, by means of asymptotic methods, with the thickness as the small parameter. This approach leads to precise mathematical definitions of “membrane” and “flexural” shells, be they nonlinearly or linearly elastic. It automatically provides the limit two-dimensional energy in each case, together with the function space over which it should be minimized. This process highlights in particular the role played by two fundamental tensors, each associated with a displacement field of the middle surface, the change of metric and change of curvature tensors. Under fundamentally distinct sets of assumptions bearing on the geometry of the middle surface, on the boundary conditions, and on the order of magnitude of the applied forces, it is shown that in the linear case, the three-dimensional displacements, once properly scaled, converge towards a two-dimensional limit that satisfies either the linear two-dimensional equations of a membrane shell or the linear two-dimensional equations of a flexural shell.