Open AccessNonlinear problems of equilibrium for axisymmetric membranes
Author(s) -
A. I. Bedov,
R. D. Vagapov,
А. И. Габитов,
A. S. Salov
Publication year - 2021
Publication title -
iop conference series. materials science and engineering
Language(s) - English
Resource type - Journals
eISSN - 1757-899X
pISSN - 1757-8981
DOI - 10.1088/1757-899x/1164/1/012009
Subject(s) - isotropy , orthotropic material , nonlinear system , mathematics , dimensionless quantity , boundary value problem , mathematical analysis , membrane , rotational symmetry , von mises yield criterion , series (stratigraphy) , mechanics , geometry , physics , thermodynamics , chemistry , finite element method , optics , biochemistry , quantum mechanics , paleontology , biology
The problems for equilibrium of circular and annular membranes made of elastic-orthotropic and elastoplastic isotropic materials are considered in this paper in a geometrically and physically nonlinear formulation meeting the Föppl-Karman assumptions. The outer contour is considered to be stationary, while the inner one being free, or reinforced by either an elastic ring or a fixed disk. The load is assumed to be uniformly distributed over the membrane surface. The results of calculations of circular and annular membranes with large deflections in comparison with thickness, deformations and squares of angular displacement to be comparable with each other but small in comparison with the unit are presented in the first part thereof. The resulting boundary value problems are reduced to the Cauchy problem to be numerically integrated by the shooting method under the iterative Steffensen algorithm. For circular membrane structures an analytical solution as a power series similar to the Prescott series for a circular membrane is assumed either. The results of numerical solutions of problems are presented as dimensionless characteristic functions. Analysis of this information will enable to solve the problem of the support, being optimal in a certain sense when the membrane is close to be equal in strength. Problems of elastoplastic equilibrium of isotropic membranes are presented in the second part thereof. The idealized Prandtl diagram is used as the main characteristic of the mechanical properties of the membrane material. The Tresca-Sant-Venant and Huber-Mises-Hencky conditions were used as criteria for the transition of a material from an elastic state to a plastic one. Solutions of physically nonlinear equilibrium problems for axisymmetric membranes are also presented as dimensionless characteristic functions.