Journal of Elasticity

, Volume 113, Issue 2, pp 167–177 | Cite as

The Anti-Plane Shear Problem in Nonlinear Elasticity Revisited

  • Edvige Pucci
  • Giuseppe Saccomandi


A classical problem in the framework of nonlinear elasticity theory is the characterization of the materials that may sustain a pure state of anti-plane shear in the absence of body forces. This problem has been solved by Knowles and by Hill in the framework of isotropic and incompressible elasticity in the seventies. Here we provide a simpler and shorter proof of these classical results. Moreover, we extend these results to nonlinear elastodynamics and we provide some new special solutions.


Anti-plane shear Compatibility problem Isotropic incompressible elasticity 

Mathematics Subject Classification (2010)

74B20 35N10 



The research is partially supported by PRIN-2009 project Matematica e meccanica dei sistemi biologici e dei tessuti molli and GNFM of Italian INDAM. We thank you Michel Destrade, Roger Fosdick, Jeremiah Murphy, Ray Ogden and two anonymous referees for providing constructive comments and help in improving the contents of this paper.


  1. 1.
    Adkins, J.E.: Some generalizations of the shear problem for isotropic incompressible materials. Proc. Camb. Philos. Soc. 50, 334–345 (1954) MathSciNetADSCrossRefMATHGoogle Scholar
  2. 2.
    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (1995) CrossRefMATHGoogle Scholar
  3. 3.
    Carroll, M.M.: Some results on finite amplitude elastic waves. Acta Mech. 3, 167–181 (1967) CrossRefGoogle Scholar
  4. 4.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics. Wiley, New York (1953) (volume 1 and 2) Google Scholar
  5. 5.
    Destrade, M., Ogden, R.W.: On the third- and fourth-order constants of incompressible isotropic elasticity. J. Acoust. Soc. Am. 128, 3334–3343 (2010) ADSCrossRefGoogle Scholar
  6. 6.
    Ericksen, J.L.: Overdetermination of the speed in rectilinear motion of non-Newtonian fluids. Q. Appl. Math. 14, 318–321 (1956) MathSciNetMATHGoogle Scholar
  7. 7.
    Fosdick, R.L., Serrin, J.: Rectilinear steady flow of simple fluids. Proc. R. Soc. Lond. A 332, 331–333 (1973) MathSciNetADSGoogle Scholar
  8. 8.
    Fosdick, R.L., Kao, B.G.: Transverse deformations associated with rectilinear shear in elastic solids. J. Elast. 8, 117–142 (1978) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Goursat, E.: Équations aux Dérivées Partielle du Second Ordre. Hermann, Paris (1896) Google Scholar
  10. 10.
    Hayes, M.A., Saccomandi, G.: Antiplane shear motions for viscoelastic Mooney-Rivlin materials. Q. J. Mech. Appl. Math. 57, 379–392 (2004) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hill, J.M.: Generalized shear deformations for isotropic incompressible hyperelastic materials. J. Aust. Math. Soc. Ser. B, Appl. Math 20, 129–141 (1978) CrossRefGoogle Scholar
  12. 12.
    Horgan, C.O.: Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Rev. 37, 53–81 (1995) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Horgan, C.O., Saccomandi, G.: Anti-plane shear deformations for non-Gaussian isotropic, incompressible hyperelastic materials. Proc. R. Soc. Lond. A 457, 1999–2017 (2001) MathSciNetADSCrossRefMATHGoogle Scholar
  14. 14.
    Huang, Y.N., Rajagopal, K.R.: On necessary and sufficient conditions for turbulent secondary flows in a straight tube. Math. Models Methods Appl. Sci. 5, 111–123 (1995) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Knowles, J.K.: On finite anti-plane shear for incompressible elastic materials. J. Aust. Math. Soc. Ser. B, Appl. Math 19, 400–415 (1976) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Knowles, J.K.: Universal states of finite anti-plane shear: Ericksen’s problem in miniature. Am. Math. Mon. 86, 109–113 (1979) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mayne, G.: Geometrical method in non-Newtonian fluid mechanics. Q. J. Mech. Appl. Math. 42, 239–247 (1989) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986) (first edition) (1993) (second edition) CrossRefMATHGoogle Scholar
  19. 19.
    Pucci, E., Saccomandi, G.: On the weak symmetry groups of partial differential equations. J. Math. Anal. Appl. 163, 588–598 (1992) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Seiler, W.: Involution—The Formal Theory of Differential Equations and Its Applications in Computer Algebra. Algorithms and Computation in Mathematics, vol. 24. Springer, Berlin (2010) MATHGoogle Scholar
  21. 21.
    Tsai, H., Rosakis, P.: On anisotropic compressible materials that can sustain elastodynamic anti-plane shear. J. Elast. 35, 213–222 (1994) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria IndustrialeUniversità degli Studi di PerugiaPerugiaItaly

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