The large deformation ofnonlinearly elastic rings in a two-dimensional compressible flow

The nonlinear nonlocal system of the equilibrium equations of an elastic ring under the action of an external two-dimensional uniformly subsonic potential barotropic steady-state gas flow is considered. The configurations of the elastic ring are identified by a pair of functions (zeta, psi). The simple curve zeta represents the shape of the ring and the real-valued function psi identifies the orientation of the material sections of the ring. The pressure field on the ring depends nonlocally on zeta, and on two parameters U and P which represent the pressure and the velocity at infinity. The system is shown to be equivalent to a fixed-point problem, which is then treated with continuation methods. It is shown that the solution branch ensuing from certain equilibrium states ((zeta0, psi0), 0, P0) in the solution-parameter space of ((zeta0, psi0), 0, P0) either approaches the boundary of the admissible ((zeta, psi), U, p)'s in a well-defined sense, or is unbounded, or is homotopically nontrivial in the sense that there exists a continuous map sigma from the branch to a two-dimensional sphere which is not homotopic in the sphere to a constant, while sigma restricted to the branch minus ((zeta0, psi0), 0, P0) is homotopic to a constant in the sphere. Furthermore, by fixing the pressure parameter at P0 and by considering the one-parameter problem in ((zeta, psi), U), the following holds. Every hyperplane in the solution-parameter space of the ((zeta, psi), U)'s which contains the equilibrium state ((zeta0, psi0), 0) and does not include a well-determined one-dimensional subspace intersects the solution branch above at a point different from ((zeta0, psi0), 0).