Introduction

In this chapter, more advanced problems of finite deformation (geometric nonlinearity) of shells and plates are considered. Initially, Donnell's and Novozhilov's nonlinear theories for doubly curved shells with constant curvature are presented. Then, the classical theory for thin shells of arbitrary shape is presented, which makes use of the theory of surfaces. Composite, sandwich and innovative functionally graded materials are introduced in the next section. In order to deal with these special materials and with moderately thick shells, nonlinear shear deformation theories are introduced. These theories, formulated for shells, can easily be modified to be applied to laminated, sandwich and functionally graded plates by setting the surface curvature equal to zero. Finally, the effect of thermal stresses is addressed.

Doubly Curved Shells of Constant Curvature

A doubly curved shell with rectangular base is considered, as shown in Figure 2.1. A curvilinear coordinate system (O; x, y, z) having the origin O at one edge of the panel is assumed; the curvilinear coordinates are defined as x = ψ Rx and y = ϑ Ry, where ψ and θ are the angular coordinates and Rx and Ry are principal radii of curvature (constant); a and b are the curvilinear lengths of the edges and h is the shell thickness. The smallest radius of curvature at every point of the shell is larger than the greatest lengths measured along the middle surface of the shell.