A new derivation of the general equations of elastic shells

Abstract

Since the pioneering work of Love which resulted in the linear bending theory of shells known as Love's first approximation [1], the foundations of elastic shell theory, especially its constitutive equations, have received repeated attention and re-examination in the literature. In this connection, particular mention should be made of the approximate constitutive equations derived by Reissner [2], also known as Love's first approximation, those known as Flugge-Lur'e-Byrne equations and the approximate equations derived by Novoshilov as given in his book [3]. All of the foregoing approximate constitutive relations of the linear theory, obtained during the period 1888 to 1944, are developed under the Kirchhoff-Love hypothesis. An account of these, as well as other contributions some of which include the effect of transverse shear deformation, may be found in [4]. A systematic development of the theory of shells and the search for a satisfactory first approximation theory has had a revival of interest during the past few years. References to these developments are given in [4], [5] and [6]. The present derivation, given in [5] and [6], is carried out under the Kirchhoff-Love hypothesis. In particular, the entire boundary-value problem of shell theory is recast in terms of new variables for the strain measures as well as the stress and couple resultants. Particular attention is paid to an exact derivation of the constitutive equations and their first approximations which meet all invariance requirements. The natural boundary conditions for stress and couple resultants and all field equations consisting of compatibility, equilibrium and the constitutive equations involve only symmetric tensors The question of a satisfactory system of equations for bending of shells is briefly discussed.