Abstract
The purpose of this research is to investigate the basic issues that arise when generalized plane strain deformations are superimposed on anti-plane shear deformations in isotropic incompressible hyperelastic materials. Attention is confined to a subclass of such materials for which the strain-energy density depends only on the first invariant of the strain tensor. The governing equations of equilibrium are a coupled system of three nonlinear partial differential equations for three displacement fields. It is shown that, for general plane domains, this system decouples the plane and anti-plane displacements only for the case of a neo-Hookean material. Even in this case, the stress field involves coupling of both deformations. For generalized neo-Hookean materials, universal relations may be used in some situations to uncouple the governing equations. It is shown that some of the results are also valid for inhomogeneous materials and for elastodynamics.
Similar content being viewed by others
References
J.E. Adkins, A note on the finite plane-strain equations for isotropic incompressible materials. Proc. Cambridge Phil. Soc. 51 (1955) 363–367.
J.E. Adkins, A.E. Green and R.T. Shield, Finite plane strain. Phil. Trans. Roy. Soc. London 246 (1953) 181–213.
M.F. Beatty, Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues - with examples. Appl. Mech. Rev. 40 (1987) 1699–1733. Reprinted with minor modifications as "Introduction to nonlinear elasticity", In: M.M. Carroll and M.A. Hayes (eds), Nonlinear Effects in Fluids and Solids. Plenum Press, New York (1996) pp. 16–112.
R.L. Fosdick and B. Kao, Transverse deformations associated with rectilinear shear in elastic solids. J. Elasticity 8 (1978) 117–142.
R.L. Fosdick and J. Serrin, Rectilinear steady flow of simple fluids. Proc. Roy. Soc. London A 332 (1973) 311–333.
A.N. Gent, A new constitutive relation for rubber. Rubber Chemistry Technol. 69 (1996) 59–61.
A.E. Green and W. Zerna, Theoretical Elasticity. Oxford Univ. Press, Oxford (1963).
M.A. Hayes and K.R. Rajagopal, Inhomogeneous finite amplitude motions in a neo-Hookean solid. Proc. Roy. Ir. Acad. A 92 (1992) 137–147.
J.M. Hill, Partial solutions of finite elasticity-plane deformations. J. Appl. Math. Phys. 24 (1973) 401–408.
J.M. Hill, Partial solutions of finite elasticity-three dimensional deformations. J. Appl. Math. Phys. 24 (1973) 609–618.
J.M. Hill, The effect of precompression on the load-deflection relations of long rubber bush mountings. J. Appl. Polymer Sci. 19 (1975) 747–755.
J.M. Hill, A review of partial solutions of finite elasticity and their applications. Internat. J. Nonlinear Mech. 36 (2001) 447–463.
J.M. Hill, Exact integrals and solutions for finite deformations of the incompressible Varga elastic materials. In: Y.B. Fu and R.W. Ogden (eds), Nonlinear Elasticity: Theory and Applications. Cambridge Univ. Press, Cambridge (2001) pp. 160–200.
C.O. Horgan, Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Review 37 (1995) 53–81.
C.O. Horgan and G. Saccomandi, Antiplane shear deformations for non-Gaussian isotropic, incompressible hyperelastic materials. Proc. Roy. Soc. London A 457 (2001) 1999–2017.
C.O. Horgan and G. Saccomandi, Constitutive modelling of rubber-like and biological materials with limiting chain extensibility. Math. Mech. Solids 7 (2002) 353–371.
C.O. Horgan and G. Saccomandi, Finite thermoelasticity with limiting chain extensibility. J. Mech. Phys. Solids 51 (2003) 1127–1146.
C.O. Horgan and G. Saccomandi, Helical shear for hardening generalized neo-Hookean elastic materials. Math. Mech. Solids 8 (2003) 539–559.
J.K. Knowles, On finite anti-plane shear for incompressible elastic materials. J. Australian Math. Soc. B 19 (1976) 400–415.
J.K. Knowles, The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids. Internat. J. Fracture 13 (1977) 611–639.
R.W. Ogden, Non-linear Elastic Deformations. Ellis Horwood, Chichester (1984). Reprinted by Dover, New York (1997).
R.W.Ogden, Elements of the theory of finite elasticity. In: Y.B. Fu and R.W.Ogden (eds), Nonlinear Elasticity: Theory and Applications. Cambridge Univ. Press, Cambridge (2001) pp. 1–57.
G. Saccomandi, Universal results in finite elasticity. In: Y.B. Fu and R.W. Ogden (eds), Nonlinear Elasticity: Theory and Applications, Cambridge Univ. Press, Cambridge (2001) pp. 97–134.
G. Saccomandi, Universal solutions and relations in finite elasticity. In: M.A. Hayes and G. Saccomandi (eds), Topics in Finite Elasticity, CISM Lecture Notes 424. Springer, Wien/New York (2001) pp. 95–130.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Horgan, C.O., Saccomandi, G. Superposition of Generalized Plane Strain on Anti-Plane Shear Deformations in Isotropic Incompressible Hyperelastic Materials. Journal of Elasticity 73, 221–235 (2003). https://doi.org/10.1023/B:ELAS.0000029990.92029.a7
Issue Date:
DOI: https://doi.org/10.1023/B:ELAS.0000029990.92029.a7