II. Note on the present state of the theory of thin elastic shells

Abstract

In a paper read before the Royal Society in February, 1888, and published in ‘ Phil. Trans.,’ A, of that year, I advanced a theory of the mode of deformation that takes place when a thin shell is vibrating. The theory was founded on the form of the potential energy function, obtained by a method adapted from that of Kirchhoff for plates. It appears that, in case there are no surface-stresses on the faces of the shell, this function consists of two terms, of which one contains a certain function W 2 and the thickness 2 h as factors, and the other contains a function W 1 and h 3 as factors. The term W 3 depends entirely on quantities σ 1 , σ 2 , w , expressing the extension of the middle surface, while the form given for W 1 contained only quantities expressing the changes of curvature. Some previous theories proceeded as if W 1 alone occurred, and, in fact, this was the case with a paper by Lord Rayleigh in ‘Proceedings of the London Mathematical Society,’ vol. 13, 1882, on the “Infinitesimal Bending of Surfaces.of Revolution.”