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

1993

J Eng Math

Self Cite

Rotationally symmetric deformations of a curved circular elastic membrane under a vertical surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations, the determination of the principal stresses in the membrane is shown to be equivalent to the solution of a single, second-order ODE, expressed in terms of a geodesic variable. Analytical techniques are applied in order to determine a limit curve of those boundary data, which subdivide the parameter range into complementary domains of existence and non-existence of tensile solutions. Finally, the more restricted subdomain of those boundary parameters is determined which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.

Wrinkle-free solutions in the theory of curved circular membranes

Beck

1993

J Eng Math

Self Cite

Curved annular elastic membranes under partially vanishing surface load

Arango¹,

Meisner

1998

Z. angew. Math. Phys.

The concern of this article is nonlinear elastic membrane theory with special emphasis on axisymmetric deformations of curved annular membranes subjected to a partially vanishing vertical surface load and with prescribed radial stresses on the edges. The nonlinear finiterotation theory of E. Reissner is employed, assuming linear stress-strain relations. This leads to consider a single second order ODE for the derivation of the principal stresses in the membrane. Analysis previously developed for a determination of that boundary data which corresponds to nonnegative radial and circumferential stress components is extended to the case, where parts of the membrane adjacent to the inner edge remain unloaded. The curved membrane is shown to flatten out on such parts which seems not yet to be observed.Mathematics Subject Classification (1991). 73C05, 73K15, 73H99, 34B15.

Existence theorems for some boundary value problems in the nonlinear theory of annular elastic membranes

Pirner

1992

Eur. J. Appl. Math

A simplified version of Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations reduces, when specialized to axi-symmetric deformations of annular membranes under a vertical surface load, to a nonlinear ordinary differential equation which is free of Poisson'ratio. Within this framework the questions of existence and non-existence of non-negative solutions of the associated stress and displacement boundary value problem are brought to a final answer. Progress in this direction was made in an earlier study (Grabmüller & Pirner 1987). In this paper a continuous monotone curve is constructed which effects a subdivision of the respective ranges of boundary data into complementary domains of existence and non-existence of strictly positive solutions