Nonlinear multiple-scale solution of a cylindrical shell

Abstract

Abstract.A multiple-scale perturbation technique is used with a two-parameter expansion to study the asymptotic solution of Reissner's axisymmetric finite-deformation equations for a circular cylindrical shell with an edge-bending moment load. Beyond the assumptions of Reissner's differential equations, it is assumed that (1) the rotations of a shell element are finite but not excessively large, (2) thickness variations in the differential equations are of order one and (3) the boundary-layer behavior is of the linear bending type to a first approximation. An asymptotic solution is then found which is uniformly valid in that it contains boundary-layer effects and corrections for extending the analysis into the shell's interior. Upon considering certain limits, it is observed that the solution contains well-established linear and nonlinear approximations to the solution.1. Introduction. The analysis presented herein involves the use of the multiplescale method of perturbation [1,2] to solve the nonlinear deformation problem of a thin circular cylindrical shell with an edge-bending moment load. The multiple-scale method of solution has been used [3] to find the solution to the homogeneous linear differential equations governing variable-thickness circular cylindrical shells undergoing axisymmetric deformation. The present solution is based on approximations to Reissner's nonlinear finite-deformation equations [4] for an isotripic elastic shell and includes the effect of gradual changes in thickness.Heuristics leading to the solution of differential equations by the method of multiple scales have been advanced by Cochran [2] and by Cole and Kevorkian [5]. The analysis is based on the observation that the composite solution is dependent upon two coordinates, one of which is of order one in the boundary layer while the other is of order one in the outer region. The effect that each of these coordinates has on the governing differential equations is realized through the ingenious multiple-scale transformation which converts ordinary differential equations into partial differential equations. Arbitrary functions are found on the basis of "Lighthill's principle" of limiting the singularity of the solution. A brief discussion of this method of analysis is found in Van Dyke [1].A linear asymptotic solution for variable-thickness shells of revolution has been found by Hildebrand [6]. Hildebrand generates his solution by expanding the dependent variables in terms of a small geometric parameter. Also, special thickness variations have been considered [7] for linear shell problems.Some nonlinear effects of edge loading have been considered [8,9]. Of particular interest is Reissner's solution [9] using an asymptotic expansion in terms of load-depen-*