## Abstract

A shell may be constructed geometrically by means of its middle surface and its thickness as follows. Let S be a two-dimensional smooth surface in the three-dimensional Euclidean space ɛ, bounded by a smooth closed curve ∂*S* (in the case of closed surfaces ∂*S* = ∅). The surface is described by a vector equation

**z**=**r**(*x*^{1},*x*^{2}),

where **r** is a smooth vector-value function of two variables *x*^{1}, *x*^{2}. At each point of the middle surface we restore the segment of length *h* in the direction perpendicular to the surface so that its centre lies on the surface. If the length *h* is sufficiently small, the segments do not intersect each other and fill some domain *B* ⊂ *ɛ* (see Figure 3.1). A linear elastic body occupying the domain *B* in its stress-free undeformed state is called an elastic shell, the surface **S** its middle surface, and *h* its thickness. A plate is the special case of the shell, whose middle surface is plane. The shell is said to be thin if *h* is much smaller than the characteristic sizes as well as the radius of curvature of the middle surface