Stretching the circular cylindrical elastic sheet
The theory of thin elastic sheets, neglecting bending, is applied to many thin surface problems. The classical theory for these surfaces is inadequate when the strains are small, the surface is unstressed before deformation, and boundary displacements are prescribed. The equations in this case are degenerate. Attempts to solve the exact equations for these elastic membranes have failed for small strains. The purpose of this paper is to determine the behavior of the solution to the exact equations for the elastic surface which is initially a circular cylinder and which is deformed by a uniform pull at the ends keeping the end radii fixed. A resolution of the small-strain behavior both analytically and numerically is of particular importance. It was found that the stresses were statically determinant and that the equations could be reduced to a single nonlinear ordinary differential equation with boundary values. This equation could be converted into a nonlinear integral equation. By using this equation it was possible to prove existence and uniqueness and determine the appropriate linear theory for small values of the strains. All results were verified by numerically solving the integral equation using Simpson's rule and the nonlinear Gauss-Seidel method. The truncation error was analyzed. The numerical results were also compared to the Bromberg-Stoker theory (a large displacement theory), and to another exact strain treatment due to Stoker. 1. Background. The membrane theory of shells is based on the assumption that bending forces can be neglected. When the membrane is assumed to be unstressed in its undeformed state, the classical linear theory of membranes is inapplicable. This theory results from linearizing the governing equations about a large prestressed state. A straightforward linearization of the governing equations yields a solution incapable of satisfying arbitrary boundary conditions. One resolution of this problem has been studied. Experiments by Ecke [9] confirmed earlier studies made by Foppl and Hencky which suggested that changes in the normal displacement could be of more significance than tangential changes. Foppl [10] studied this for symmetric deformations of plane circular membranes. In a now-classical paper in this area, Bromberg and Stoker [2] generalized Foppl's theory to arbitrary surfaces of revolution. The change expounded was to retain all nonlinear terms involving derivatives of the normal displacement. As a particular example, Bromberg and Stoker dealt with the spherical cap in some detail. They found that for small strains their equations became singular. This difficulty was overcome by developing an appropriate boundary-layer * Received December 26, 1979. Partial support provided by NSF Grant MPS 75-06366.