The purpose of this research is to investigate the helical shear problem for a circular cylindrical tube composed of an isotropic hyperelastic incompressible material. Three specific constitutive models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy-Green tensor. For limiting values of a material parameter, all of these models reduce to the classical neo-Hookean form. The stress fields and displacements are characterized for each of these models. For two of the models considered, explicit closed-form analytic expressions are obtained. The results are compared with one another and with the predictions of the neo-Hookean model.
Get full access to this article
View all access options for this article.
References
1.
[1] Przybylo, P.A., and Arruda, E.M.: Experimental investigations and numerical modeling of incompresible elastomers during non-homogeneous deformations . Rubber Chemistry and Technology, 71, 730-749 (1999).
2.
[2] Boyce, M.C., and Arruda, E. M.: Constitutive models of rubber elasticity: a review . Rubber Chemistry and Technology, 73, 504-523 (2000).
3.
[3] Ogden, R.W.: Non-linear Elastic Deformations, Ellis Horwood, Chichester , 1984. Reprinted by Dover, New York, 1997.
4.
(Reprinted with minor modifications as Introduction to nonlinear elasticity, in Nonlinear Effects in Fluids and Solids, pp. 16-112, eds., M.M. Carroll & M.A. Hayes, Plenum Press, New York , 1996.
5.
[5] Erman, B., and Mark, J.E.: Structures and Properties of Rubberlike Networks, Oxford University Press, Oxford , 1997.
6.
[6] Horgan, C.O., and Saccomandi, G.: Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility . Journal of Elasticity, 56, 159-170 (1999).
7.
[7] Horgan, C.O., and Saccomandi, G.: Pure axial shear of isotropic incompressible nonlinearly elastic materials with limiting chain extensibility . Journal of Elasticity, 57, 307-319 (1999).
8.
[8] Horgan, C.O., and Saccomandi, G.: Pure azimuthal shear of isotropic, incompressible hyperelastic materials with limiting chain extensibility . International Journal of Nonlinear Mechanics, 36, 465-475 (2001).
9.
[9] Horgan, C.O., and Saccomandi, G.: Large deformations of a rotating solid cylinder for non-Gaussian isotropic, incompressible, hyperelastic materials . Journal of Applied Mechanics, 68, 115-117 (2001).
10.
[10] Horgan, C.O., and Saccomandi, G.: Constitutive modelling of rubber-like and biological materials with limiting chain extensibility . Mathematics and Mechanics of Solids, 7, 353-371 (2002).
11.
[11] Gent, A.N.: A new constitutive relation for rubber . Rubber Chemistry and Technology, 69, 59-61 (1996).
12.
[12] Gent, A.N.: Elastic instabilities of inflated rubber shells . Rubber Chemistry and Technology, 72, 263-268 (1999).
13.
[13] Arruda, E.M., and Boyce, M.C.: A three-dimensional constitutive model for the large deformation stretch behavior of rubber elastic materials . Journal of the Mechanics and Physics of Solids, 41, 389-412 (1993).
14.
[14] Kilian, H.G.: Equation of state of real networks . Polymer, 22, 209-217 (1981).
15.
[15] Knowles, J.K.: The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids . International Journal of Fracture, 13, 611-639 (1977).
16.
[16] Erman, B., and Mark, J.E.: Use of Fixman-Alben distribution function in the analysis of non-Gaussian rubber-like elasticity . Journal of Chemical Physics, 89, 3314-3316 (1988).
17.
[17] Rivlin, R.S.: Large elastic deformations of isotropic materials VI. Further results in the theory of torsion, shear and flexure . Philosophical Transactions of the Royal Society of London A, 242, 173-195 (1949).
18.
[18] Ogden, R.W., Chadwick, P., and Haddon, E.W.: Combined axial and torsional shear of a tube of incompressible isotropic elastic material . Quarterly Journal of Mechanics and Applied Mathematics, 26, 23-41 (1973).
19.
[19] Fosdick, R.L., and MacSithigh, G.: Helical shear of an elastic, circular tube with a non-convex stored energy . Archive for Rational Mechanics and Analysis, 84, 31-53 (1983).
20.
[20] Beatty, M.F., and Jiang, Q.: On compressible materials capable of sustaining axisymmetric shear deformations. Part 3: Helical shear of isotropic hyperelastic materials . Quarterly of Applied Mathematics, 57, 681-697 (1999).
21.
[21] Kirkinis, E., and Ogden, R.W.: On helical shear of a compressible elastic circular cylindrical tube . Quarterly Journal of Mechanics and Applied Mathematics, 56, 105-122 (2003).
