General helicoidal shells undergoing large, one-dimensional strains or large inextensional deformations

Abstract

An earlied paper showed that general helicoidal shells (which include shells of revolution and general cylindrical shells as limiting cases) admit arbitrarily large, one-dimensional strain fields. In the present paper, the associated rotation and stress function fields are found. Introduction of the Euler parameters from rigid body dynamics reduces the determination of the rotation field to a linear problem. The equilibrium and compatibility equations are shown to reduce to six coupled scalar equations involving three rotation functions, three stress functions, extensional strains, stress couples, and four constants, two measuring the gross axial displacement and twist of the shell, and two measuring the net axial force and torque. One field equation is a first integral of the compatibility equations; another, a first integral of the moment equilibrium equations. Reissner's equations for the pure bending of curved tubes and Wan's equations for the gross twisting and extension of right helicoidal shells fall out as special cases. Determination of the displacement field in large inextensional bending reduces to quadratures, generalizing Reissner's result for a slit shell of revolution.