In this paper we examine the combined extension and torsion of a compressible isotropic elastic cylinder of finite extent. The equilibrium equations are formulated in terms of the principal stretches and then applied to the special case of pure torsion superimposed on a uniform extension (an isochoric deformation). Explicit necessary and sufficient conditions on the strain-energy function for the material to support this deformation with vanishing traction on the lateral surfaces of the cylinder are obtained. Some strain-energy functions satisfying these conditions are considered, existing results are recovered as special cases and new results are obtained. We also point out how the strain-energy functions generated from the considered isochoric deformation considered (of a compressible material) can be used to generate energy functions and corresponding solutions for the incompressible theory.
Get full access to this article
View all access options for this article.
References
1.
[1] Rivlin, R.S.: Large elastic deformations of isotropic materials IV: further developments of the general theory. Phil. Trans. R. Soc. Lond. A241, 379-397 (1948).
2.
[2] Rivlin, R.S.: A note on the torsion of an incompressible highly elastic cylinder. Proc. Camb. Phil. Soc., 45, 485-487 (1948).
3.
[3] Rivlin, R.S.: Large elastic deformations of isotropic materials VI: further results in the theory of torsion, shear and flexure. Phil. Trans. R. Soc. Lond. A242, 173-195 (1949).
4.
[4] Rivlin, R.S.: Torsion of a rubber cylinder. J. Appl. Phys., 18, 444-449 (1947).
5.
[5] Rivlin, R.S. and Saunders, D.W.: Large elastic deformations of isotropic materials VII: experiments on the deformation of rubber. Phil. R. Soc. Lond. A243, 251-288 (1951).
6.
[6] Ericksen, J.L.: Deformations possible in every compressible isotropic, perfectly elastic material. J. Math. Phys., 34, 126-128 (1955).
7.
[7] Green, A.E. and Adkins, J.E.: Large Elastic Deformations, Clarendon Press, Oxford, 1970.
8.
[8] Green, A.E.: Finite elastic deformation of compressible isotropic bodies. Proc. R. Soc. Lond. A227, 271-278 (1955).
9.
[9] Faulkner, M.G. and Haddow, J.B.: Nearly isochoric finite torsion of a compressible isotropic elastic circular cylinder. Acta Mech., 13, 245-253 (1972).
10.
[10] Blatz, P.J. and Ko, W.L.: Application of finite elastic theory to the deformation of rubbery materials. Trans. Soc. Rheol., 6, 223-251 (1962).
11.
[11] Levinson, M.: Finite torsion of slightly compressible rubberlike circular cylinders. Int. J. Non-Linear Mech., 7, 445-463 (1972).
12.
[12] Currie, P.K. and Hayes, M.: On non-universal finite elastic deformations. In Proc. IUTAM Symp. on Finite Elasticity (D.E. Carlson and R.T. Shield eds.), Martinus Nijhoff Publishers, Dordrecht, 1981, pp. 143-150.
13.
[13] Beatty, M.F.: Topics in finite elasticity: hyperelasticity of rubber, elastomers and biological tissues—with examples. Appl. Mech. Rev., 40, 1699-1734 (1987).
14.
[14] Beatty, M.F.: Introduction to nonlinear elasticity. In Nonlinear Effects in Fluids and Solids (M.M Carroll and M.A. Hayes eds.), Plenum Press, New York, 1996, pp. 13-112.
15.
[15] Carroll, M.M.: Finite strain solutions in compressible isotropic elasticity. J. Elasticity, 20, 65-92 (1988).
16.
[16] Horgan, C.O. and Polignone, D.A.: A note on pure torsion of a circular cylinder for a compressible nonlinearly elastic material with nonconvex strain-energy. J. Elasticity, 37, 167-178 (1995).
17.
[17] Polignone, D.A. and Horgan, C.O.: Pure torsion of compressible nonlinearly elastic circular cylinders. Q. Appl. Math., 49, 591-607 (1991).
18.
[18] Jiang, X. and Ogden, R.W.: Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material. Int. J. Non-Linear Mech., 35, 361-369 (2000).
19.
[19] Jiang, X. and Ogden, R.W.: On azimuthal shear of a circular cylindrical tube of compressible elastic material. Q. J. Mech. Appl. Math., 51, 143-158 (1998).
20.
[20] Ogden, R.W.: Non-Linear Elastic Deformations, Dover Publications, New York, 1997.
21.
[21] Holzapfel, G.A., Gasser, C.T. and Ogden, R.W.: A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity, 61, 1-48 (2000).
22.
[22] Saccomandi, G.: Universal results in finite elasticity. In Nonlinear Elasticity: Theory and ApplicationsLMS Lecture Note Series 283 (Y.B. Fu and R.W. Ogden eds.), Cambridge University Press, Cambridge, 2001, pp. 97-134
23.
[23] Carroll, M.M. and Horgan, C.O.: Finite strain solutions for a compressible elastic solid. Q. Appl. Math., 48, 767-780 (1990).
24.
[24] Horgan, C.O.: Equilibrium solutions for compressible nonlinear elasticity. In Nonlinear Elasticity: Theory and ApplicationsLMS Lecture Note Series 283 (Y.B. Fu and R.W. Ogden eds.), Cambridge University Press, Cambridge, 2001, pp. 135-159.
25.
[25] Haughton, D.M.: Inflation and bifurcation of thick-walled compressible elastic spherical shells. IMA J. Appl. Math., 39, 259-272 (1987).
26.
[26] Haughton, D.M.: Circular shearing of compressible elastic cylinders. Q. J. Mech. Appl. Math., 46, 471-486 (1993).
27.
[27] Kirkinis, E. and Ogden, R.W.: On helical shear of an isotropic compressible elastic tube. Q. J. Mech. Appl. Math. in press (2002).
28.
[28] Beatty, M.F. and Jiang, Q.: On compressible materials capable of sustaining axisymmetric shear deformations. III. Helical shear of isotropic hyperelastic materials. Q. Appl. Math., 57, 681-697 (1999).
