6 - The ‘static-geometric analogy’

Summary

If the various shallow-shell equations from chapter 8 which govern the behaviour of the S-and B-surfaces are assembled together, some remarkable formal analogies between them become obvious. Analogies of this kind were first pointed out in the 1940s by Lur'e and Goldenveiser (see Lur'e, 1961; Goldenveiser, 1961, §30), and they are known collectively as the ‘static-geometric analogy’. They are peculiar to the theory of thin shells and have no counterpart in, e.g. the classical equations of three-dimensional elasticity.

These analogies emerge particularly clearly in the formulation of the equations of elastic shells in terms of the static and kinematic interaction of distinct ‘stretching’ and ‘bending’ surfaces. The following exposition follows closely that given by Calladine (1977b). Since it relates explicitly to shallowshell equations, for which in particular the coordinates are aligned with the directions of principal curvature, the discussion cannot be regarded as complete. In fact the analogy holds when the equations are set up in terms of the most general curvilinear coordinate system; but it is usually regarded as being restricted to shells with zero surface loading (Naghdi, 1972, p.613). As will be seen, the introduction of change of Gaussian curvature (g) as a kinematic variable makes possible the extension of the analogy to shells loaded by pressure (p); and indeed these two variables turn out to be analogous in the present context.