Nonlinear theory and stability of thick sandwich shells: PMM vol. 37, n≗3, 1973, pp. 532–543

Abstract

A derivation of the fundamental equations of a nonlinear theory of thick sandwich shells is given in tensor form. The shells are fabricated from alternating layers of different stiffness. It is considered that the hypotheses of the refined theory of shells of S.P. Timoshenko are valid for the hard layers, while the soft layers operate under transverse compression and shear. The change in metric during passage from one layer to another is taken into account. Diverse variants of the fundamental equations are presented, including equations for shells of an anisotropic couple-stress continual medium equivalent in an energy sense. The equations are linearized with respect to the membrane state for all the variants. The linearized equations are applied to stability problems of sandwich shells. The problem of the local stability of a cylindrical shell under axial compression is examined as an illustration. The change in the character of the buckling and the magnitude of the critical load is investigated as the relative stiffness of the layers changes. A study of the mechanical properties of composite materials bonded by high-strength layers, and the investigation of both the integrated and local effects upon deformation of the laminar media can be carried out on the basis of the theory developed in [1]. According to this theory, the behavior of the laminar media is described by a system of differential-difference equations permitting taking account of the discrete properties of the composite laminar medium. A development of this theory is given in [2] in application to sandwich shells. The equations obtained in [2] by using an assumption about the smallness of the displacements and strains can be used for thick shells whose thickness is commensurate with the minimum radius. The thickness of the layers of high stiffness should be small compared to this radius ( h ⪡ R). The assumption of smallness of the displacements should be discarded for the large class of problems associated with finite displacements, and nonlinear equations should be used. In deriving the nonlinear equations the assumption about the commensurability of displacements with the thickness of the hard layers ( u j ∼ h) and about the smallness of the displacements as compared with the minimal radius ( u j ⪡ R) is natural. Moreover, the influence of shears in the hard layers turns out to be essential in some problems. Nonlinear equations of the theory of thick sandwich shells of regular construction are derived below taking account of transverse shears in the layers of high striffness. As a particular case, the equations presented in [3–5], as well as the nonlinear equations for a single-layered shell [6], can be obtained from these equations. By applying the principle of continualization [7], nonlinear equations are obtained for anisotropic couple-stress media equivalent in an energy sense.