Bifurcation of spherical and cylindrical shells at finite deformation.

Abstract

In this dissertation we investigate the possibility of bifurcations in the solution paths of symmetrically loaded spherical and cylindrical shells, both thick-walled and membrane shells of incompressible, isotropic materials capable of withstanding finite deformations elastically. At the points of interest it becomes possible for non-symmetric solutions to exist, for example, bulges in pressurized cylindrical shells. Several authors have considered various aspects of the problems considered here, usually from the viewpoint of stability, using either the equations of small deformations superposed on large, the incremental equations, as formulated by Green, Rivlin and Shield (1952), using the Cauchy stress tensor on current axes, or by using an energy method. We have found, however, that the incremental equations are more appropriately formulated using the conjugate stress-deformation pair, nominal stress and deformation gradient, referred to an arbitrary reference configuration. The main advantage of this formulation is that changes in geometry due to the incremental displacements do not have to be calculated explicitly. Using this formulation of the equations we then investigate spherical and cylindrical shells subjected to internal and external pressures and also rotating cylindrical membranes. In all cases it is possible to conclude the analysis while making no assumptions on the form of the strain-energy function, other than those relating to incompressibility and isotropy. This then allows an investigation of the relationship between the form of strain-energy function, which we express as a function of the principal stretches directly, rather than through invariant functions of the stretches, and the occurrence of the possible bifurcation modes.

Date of Award	1979
Original language	English
Awarding Institution	University of Bath