Large elastic deformations of isotropic materials. III. Some simple problems in cyclindrical polar co-ordinates

Abstract

Expressions for the components of strain and the incompressibility condition, for large deformations, are obtained in a cylindrical polar co-ordinate system. The stress-strain relations, equations of motion and boundary conditions for an incompressible, neo-Hookean material, in such a coordinate system, are also obtained and specialized to the case of cylindrical symmetry. These results are applied to the special cases of the simple torsion of a solid cylinder and of a hollow, cylindrical tube and to their combined simple extension and simple torsion. In the case of a solid cylinder, it is found that a state of simple torsion can be maintained by surface tractions applied to the ends of the cylinder only, and these consist of a torsional couple together with a compressive force. The necessary torsional couple is proportional to the amount of torsion and the compressive force to the square of the torsion. In the case of a hollow, cylindrical tube, it is again necessary to exert a torsional couple, proportional to the torsion, and a compressive force, proportional to the square of the torsion, on the plane ends, but it is also necessary to exert a normal surface traction, acting in a positive radial direction, on one or other of the curved surfaces of the tube and proportional to the square of the torsion.