On the SO(2) symmetric deformation of rotating rings with shear deformation

We study the SO(2) symmetric deformation of a circular ring, modeled using the beamshell theory of Libai and Simmonds. The equations of motion are based on general partial differential equations governing the elastodynamics of geometrically exact rings, which have been formulated by Dempsey (Proc. Roy. Sec. London A 452 (1996) 1927-1943). Thus, the formulation is valid for arbitrary pressure forcing and large deformations, although the results are tempered by a linear constitutive relation and an assumption of plane strain. With the assumption that the deformation retains SO(2) symmetry, the partial differential equations are reduced to a set of coupled ordinary differential equations. Within this restricted space of solutions, we study the existence and stability of relative equilibria and discuss the effects of constant hydrostatic pressure on the dynamical response. Specifically, interaction between inertial effects arising from rotational motion and the combined elastic and external pressure forces can produce unexpected behavior, including the existence of a non-trivial state which retains the symmetry, yet physically implies that material planes do not lie in the radial direction. Such a state is shown to affect the large-amplitude response of the system in a singular limit of the governing equations. (C) 1999 Elsevier Science Ltd. All rights reserved.