Large deformation possible in every isotropic elastic membrane

Abstract

This paper is concerned with static solutions of finitely deformed elastic membranes regarded as thin shells. It deals with deformations that can be maintained, in the absence of body force, in every isotropic elastic membrane by the application of edge loads and/or uniform normal surface loads on the major surfaces of the thin shell-like body. The solutions, which are valid for both compressible and incompressible materials, are obtained with the use of a strain energy response function which depends on the metric tensor of the membrane in its deformed configuration. The main results are summarized by several theorems and their corollaries in accordance with three mutually exclusive cases for which the initial undeformed surface of the membrane (which may be a sector of a complete or closed surface) is, respectively, developable, spherical and a surface of variable Gaussian curvature satisfying certain differential criteria. The corresponding deformed surfaces are, respectively, a plane or a right circular cylinder, a sphere and a surface of constant mean curvature. These results are exhaustive in that they represent all finite deformation solutions possible in every isotropic elastic material characterized by the strain energy response mentioned above. Also discussed are some special cases of the general results and several families of solutions in terms of an alternative description which should be useful in application and which permit easy interpretations.