Positive Solutions of Annular Elastic Membrane Problems with Finite Rotations

Abstract

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.

On shells of revolution with the Love-Kirchhoff hypotheses

“…It is seen that, if the underlined term in (7.17) is dropped, the right hand side of (7.17) has the same monotonicity properties as in equation (7.1 In order to find solutions of (7.19), for v 0, and solutions to a corresponding integral equation for annular membranes for the full range of physically admissible parameters s, S, h, H, a novel theorem on the existence of positive solutions of integral equations of the type (7.19) due to Novak [33] is needed. Grabmiiller and Pirner [34,35] have succeeded in obtaining a complete existence and uniqueness theory of positive solutions of both circular and annular membranes based on equation (7.17), without the underlined term. The results differ markedly from those for the F6ppl membrane model.…”

“…While the differences are small for sufficiently small (P/Eh)' 3 , the range of existence of solutions for the F6ppl membrane may be entirely different from the existence range for the Reissner membrane. As shown in [34,35] the geometrically exact theory may permit only bounded solutions that are in some sense small, while the approximate theory has large solutions under the same loading conditions. Comparisons for circular plates have also been made in a number of papers and are summarized in [70,71], where references can be found.…”

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].…”

“…Concerning the displacement boundary value problem, we employ a mapping argument which originates from R. Pirner's diploma thesis [19] and which has been improved later on by H. Grabmiiller et al [14,15,16]. The idea of this method consists of mapping the set of parameters related to a stable state of the stress problem into the parameter set associated with the displacement problem.…”

On finite displacements of curved annular elastic membranes without wrinkling

“…The existence of solutions of the stress boundary-value problem is proved by an integral equation technique utilizing Schauder's fixed-point theorem, while positivity and uniqueness are obtained from a generalized maximum principle. Concerning the displacement boundaryvalue problems, a mapping argument is used, which was first proposed by R. Pirner [14] and which has been successfully applied later in a refined version in order to study stable equilibria of flat annular membranes, not only within a simplified version of Reissner's finite-rotation theory [7,10,11,21] but also within the so-called smallfinite-deflection theory [7,8,9,12]. We refer to [22] for a review.…”