Pure torsion of compressible non-linearly elastic circular cylinders

Abstract

Abstract. 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 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 [3], 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 [4] (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 [6] (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

Extension, inflation and torsion of a residually stressed circular cylindrical tube

“…Kirkinis and Ogden [9] derived conditions on the form of strain-energy function for which the isochoric deformation of pure torsion superimposed on a uniform extension can be supported with vanishing traction on the lateral surfaces of the cylinder, and pure torsion of a compressible isotropic elastic material was also examined in [10].…”

A note on the pure torsion of a circular cylinder for a compressible nonlinearly elastic material with nonconvex strain-energy

“…In Section 3, the constitutive model of interest, namely, the Blatz-Ko material, is briefly described. It is shown that the pure torsion solution of [1] is, in fact, the unique smooth radially symmetric solution for this material. The main results of the paper are contained in Section 4.…”

“…In particular, the analytic solution of a wide variety of problems in finite elastostatics for compressible materials has been possible for this material (see [8][9][10][11][12][13][14][15]1). Before proceeding in the next section to a discussion of implications of loss of ellipticity for pure torsion of the Blatz-Ko material (3.1), we show here that the pure torsion solution r(R) = R (3.3) obtained in [1,8,9] is, in fact, the unique smooth radially symmetric solution to the boundary value problem of concern. For the purpose of this proof, t it is convenient to take the cylinder radius to be unity (A = 1) and write the boundary value problem (see Eq.…”

“…We conclude this section by making some remarks concerning the stress distribution in the cylinder. For pure torsion of the Blatz-Ko material, it is shown in [1] that the stresses are given by T.=0, To0=0, T~o=0, T.z=0, (4.28) To~ = lazR, T~z = --~T2R 2. (4.29)…”

“…The radial deformation r(R) is governed by a second-order nonlinear ordinary differential equation (see Eq. (2.22) of[1]). The strain-energy density per unit undeformed volume for isotropic elastic compressible materials is given by W= W(I 1, 12, , 12, 13 are the principal invariants of the deformation tensor B = FF r, where F is the deformation gradient tensor.…”