An investigation of the quasi-invariants of the static-geometric analogy for thin elastic shells

Abstract

The equations of the theory of thin elastic shells posses a characteristics symmetry which has enabled an intrinsic analogy to be established between the elastic constants and the static and geometric elements which appear in the formulation of the theory. This is the so-called static-geometric analogy which was formulated by Goldenveizer for isotrophic shells [1]. The present authors later generalized the static-geometric analogy to cover the cases of orthotropic [2] and anisotropic shells [3]. The static-geometric analogy enables us to write down equations of the theory of shells in complex form. The results obtained by Novozhilov in this field are well known [4]. In the case of cylindrical isotropic shells of arbitrary section, Novoshilov reduced the set of equations of the theory of shells to a single equation [4]. An analogous result obtained for a thin spherical shell [5] by Goldenveizer, who derived a set of equations in complex displacements for an isotropic shell. In [7] the static-geometric analogy is used to reduce the general equations for thin isotrophic shells to a single complex equation of the fourth order. With the aim of applying these methods to the case of orthotropic shells, the authors found that the factor 2h 2 E αE β 3(1−μ αμ β) enables the sets of equation of equilibrium and continuity to be combined to form a single set, and Hooke's equations to be reduced to a set of three linear non-differential equations in complex forces [8] It is well known that the static-geometric analogy enables each relation found to be duplicated. These results then suggest the formation of a single theory of thin shells in the complex domain which would reduce the number of unknowns and the order of the complex equations by half.