Communicated by E. MeisterThe Foppl-Hencky theory of small finite deflections has been applied to the study of axisymmetric deformations of annular membranes by various authors. The mathematical problem of uniqueness of tensile solutions for the corresponding non-linear boundary value problems in the full range of physically meaningful boundary data was solved only recently.'* In this paper, a new integral equation technique of solution is developed which yields the existence of tensile solutions for a parameter range of the boundary data where previous investigations on this matter had not been successful. It is shown that tensile solutions no longer exist, when sufficiently large positive radial displacements are prescribed at the inner edge of the annulus.

Axisymmetric deformations of annular membranes subjected to normal surface loads and radial edge loads or displacements are considered within the F~Sppl nonlinear membrane theory. When the inner edge r = a is free of radial traction, the solution of the annular membrane problem is shown to reduce to the solution for the circular membrane (a = 0). For nonvanishing traction at r = a, the problem is reduced to a circular pseudo-membrane problem. For both cases, existence and uniqueness of tensile solutions of the annular membrane problem are proved, including a rigorous derivation of a stress concentration factor originally found by Schwerin by formal methods.

Following the study of axisymmetric deformations of annular membranes under a normal surface load on the basis of the Foppl-Hencky small-finite-deflection theory, which has been completed in recent works, the corresponding methods are here generalized as to apply to finite-deformation problems of the Simmonds-Libai simplified Reissner equations of finite rotations. The question of uniqueness of positive regular solutions is solved in general, and an integralequation technique is developed which yields existence and nonexistence results for an extensive range of the boundary data where previous investigations failed. In contrast to the Foppl theory, some of the boundary conditions are nonlinear, and hence an extension to the finite-rotation case is not straightforward. Moreover there are significant differences between the small-and the finite-rotation theory.

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