On finite displacements of curved annular elastic membranes without wrinkling

Abstract

Abstract. Axisymmetric deformations of curved annular elastic membranes subjected to vertical surface loads and radial edge loads or displacements are considered within Reissner's finite-rotation theory of thin shells of revolution, assuming linear stress-strain relations. The principal stresses in the membrane are determined by the solutions of a nonlinear second-order ODE, dependent on a geodesic variable. Analytical methods are used in order to determine the range of those boundary data for which the solutions of the differential equation are wrinkle-free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.1. Introduction. For many events, a linear theory of elastic membranes and thin shells of revolution is not applicable, for instance, if the applied load is not small enough. In these cases a nonlinear theory is required. Of particular interest are geometrically nonlinear theories where nonlinear strain-displacement relations but a linear stress-strain relation like Hooke's law are assumed.A membrane theory of this kind that is widely applied and that covers the special case of aximembranes, originates from E. Reissner's theory of thin shells of revolution

Curved annular elastic membranes under partially vanishing surface load

“…The totality of rm-solutions of problem (3.19), when specialized to t 0 = 0, has been thoroughly studied by Beck and Grabmüller [5]. In [5] a somewhat sharper condition than (Q) was needed forq(t) in order to guarantee the positivity of regular solutions:…”

“…The mathematical problems of existence and uniqueness of wrinkle-free solutions for curved annular membranes under several edge loads have been solved by A. Beck and H. Grabmüller [5] only recently imposing, however, some inappropriate restrictions upon the vertical surface load. Actually, the authors were unable to analyze the physically reasonable case when a vicinity of the inner membrane edge remains unloaded.…”

“…The study of stably equilibrated curved aximembranes under gravitational forces was possibly initiated by the doctoral thesis of A. Beck [2]. Extensions were derived by A. Beck and H. Grabmüller, not only for annular membranes [5] but also for circular configurations [4]. Some work on curved aximembranes was also done by J. Arango [1] who studied rotationally symmetric deformations under uniform normal pressure.…”

“…A discussion of the more restricted subset of those tensile solutions that admit of nonnegative circumferential stresses only, may be achieved by following the outlines of earlier work on this subject (cf. [5,8]) and will be omitted here. Displacement boundary conditions are most appropriately treated by utilizing a mapping argument which was first proposed by R. Pirner [16] and which requires the stress boundary value problem (the Dirichlet problem) to be solved.…”

“…Displacement boundary conditions are most appropriately treated by utilizing a mapping argument which was first proposed by R. Pirner [16] and which requires the stress boundary value problem (the Dirichlet problem) to be solved. Again we refer to previous work [5] for a thorough discussion. The objective now is to examine the Dirichlet problem for tensile solutions.…”