THE LARGE DEFORMATION OF NONLINEARLY ELASTIC SHELLS IN AXISYMMETRICAL FLOWS

This paper treats the large deformation of closed nonlinearly elastic axisymmetric shells under an external pressure field generated by the steady, irrotational, axisymmetric flow of an incompressible, inviscid fluid. The flow is assumed to have a prescribed velocity U and pressure P at infinity. The deformation of the shells is described by a geometrically exact theory. The parameters U and P and the deformed shape of the shell uniquely determine the velocity field of the steady flow. The most difficult part of the analysis is to show that the velocity and pressure of the flow on the shell depend continuously and compactly on the function describing the shape. The pressure field on the shell is substituted into the equilibrium equations for the shell, yielding a system of ordinary functional-differential equations. These are converted into a fixed-point form, which is analyzed by a global implicit function theorem. The problem has technical difficulties that do not arise in problems with rigid obstacles.