ABSTRACT
Textbooks on the theory of shells define their object of interest as thin, possibly curved, bodies with two dimensions much larger than a third one – the thickness. As a rule, when it comes to a more rigorous mathematical description, a shell is represented as a set of material points in space with position vectors p determined by the conditions of the form
or some equivalent form. In the above setting r(ϑα) is a position vector of some reference surface, sayM, (usually the middle surface) with the field of unit normals n(ϑα). Later this surface, endowed with mechanical properties, serves as the representative of the shell proper. If this surface has boundary parametrized by, say λ, then the surface p(λ, ζ) = r(λ) + ζn is called the lateral surface of the shell. It goes invariably without saying thatM cuts the shell, so that h+ > 0 and h− < 0. Then the surface p(ϑα, h+) is called the upper face of the shell and p(ϑα, h−) is its lower face and the number h = h+ − h− is taken to measure the thickness of the shell. At this point most authors assume that both h+ and h−, and thereby h, are constant over M; those that admit variability of thickness impose all the same some restrictions on the form of the function h = h(ϑα).
