Elastic shells

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(x¹,x²),

where r is a smooth vector-value function of two variables x¹, x². 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