### Pure torsion of compressible non-linearly elastic circular cylinders

Debra A. Polignone, Cornelius O. Horgan
1991 Quarterly of Applied Mathematics
The large deformation torsion problem of an elastic circular cylinder, composed of homogeneous isotropic compressible nonlinearly elastic material and subjected to twisting moments at its ends, is described. The problem is formulated as a two-point boundary-value problem for a second-order nonlinear ordinary differential equation in the radial deformation field. The class of materials for which pure torsion (i.e., a deformation with zero radial displacement) is possible is described. Specific
more » ... scribed. Specific material models are used to illustrate the results. Introduction. The finite torsion of an elastic circular cylinder due to applied twisting moments at its ends was one of the classic problems solved by Rivlin [1,2] for isotropic incompressible nonlinearly elastic materials. The nonhomogeneous deformation found by Rivlin is sustainable, in the absence of body force, in all homogeneous isotropic incompressible materials. Thus it is a universal, or controllable, deformation. Furthermore, by virtue of the constraint of zero volume change, the deformation is that of pure torsion so that there is no extension in the radial direction and the cross-section of the cylinder remains circular. The situation for compressible materials is considerably more complicated. First, by virtue of Ericksen's theorem , the deformation cannot be universal. Second, there will, in general, be some radial extension. The torsion problem for a circular cylinder composed of a homogeneous isotropic compressible nonlinearly elastic material has been formulated by Green  (see also Green and Adkins [5, p. 67]) and properties of the solution discussed. The formulation leads to a nonlinear secondorder ordinary differential equation for the radial deformation r(R) (see Eq. (2.22) of the present paper). Here R and r denote radial components of a cylindrical polar coordinate system in the undeformed and deformed configurations respectively. The deformation r(R) depends on the form of the strain-energy density. An equivalent version of the ordinary differential equation may be written in terms of the principal stretches  (see Eq. (7.8) of the present paper). As remarked in [4, 6] , an analytic solution of the governing equation, even for very special strain-energy functions, is not readily obtained and so numerical approaches are called for. Other aspects of