Nonlinear boundary value problems for the annular membrane: A note on uniqueness of positive solutions

Abstract

The study of axisymmetric deformations of annular membranes under normal surface loads within the framework of the FSppl-Hencky small finite-deflection theory is continued after progress in this field has been made in the recent work of Grabmtiller and Weinitschke [5]. When a radial displacement is applied at the inner edge and a radial tension or a displacement at the outer edge, the mathematical question of uniqueness of tensile solutions of the resulting nonlinear boundary value problems has not been settled yet. In this paper, uniqueness is proved in a parameter range of the boundary conditions, where previous methods have failed.

Recent mathematical resuits in the nonlinear theory of flat and curved elastic membranes of revolution

“…The particular constellation of Problem (s, H) again leads to a void condition (N). In summary, we have [17,18]: Analogous existence results do not hold in the cases of Problems (h, S) and (h, H) for the F6ppl theory. A discussion of the crucial condition (N) surprisingly shows that rt-solutions are absent in an unbounded, simply connected subset of the respective parameter ranges (h, S) and (h, H).…”

“…Thus, a new existence proof for rt-solutions to Problem (s, S) is obtained because it is seen from (3.9) that g(z) := f(z) + q(z) solves the integral equation (3.8). In [17], the question of uniqueness of rt-solutions to each boundary value problem was fully resolved by a suitable application of Hopf's generalized maximum principle [36]. The above new integral equation method was also extended to Problems (s, H), (h, S) and (h, H).…”

Curved annular elastic membranes under partially vanishing surface load

“…Similarly, the flattening must be independent of Poissons's ratio and should not play any role in the theory of flat annular membranes. Flat aximembranes had been the subject of numerous studies, not only within a simplified version of Reissner's finite-rotation theory [8,11,12,16,22] but also within the so-called small-finite-deflection theory [8,9,10,13]. We refer to [23] for a comprehensive review.…”

Wrinkle-free solutions in the theory of curved circular membranes

“…Integral equation methods have first been applied to nonlinear membrane problems by Dickey [9] and later successfully been refined by adding concavity and monotonicity arguments. Equipped with these tools, the mathematical problems of existence and uniqueness of a stable membrane state were solved for flat circular and annular membranes not only within the small finite deflection theory [10,11,12] but also within a simplified version of the Reissner theory of finite rotations [13,14,15,16,17]. The results are summarized in [18].…”