Journal of Engineering Mathematics

, Volume 95, Issue 1, pp 347–357 | Cite as

Azimuthal shear of doubly fibre-reinforced, non-linearly elastic cylindrical tubes

  • M. El Hamdaoui
  • J. Merodio


A doubly fibre-reinforced, non-linearly elastic tube subject to azimuthal shear is analysed. The materials involved are neo-Hookean models augmented with a function that accounts for the existence of two unidirectional reinforcements. This function endows the material with its anisotropic character and quantifies the effect associated with the two preferred material directions. The nature of the anisotropy considered has a particular influence on the shear response of the material, in contrast to previous analyses in which the effect of the anisotropy was taken to depend only on the stretch in the fibre directions. The inner boundary of the tube is fixed while the outer boundary is subject to a given shear traction. Non-smooth solutions arise inside the tube at a critical value of the shear stress, which can be obtained with the well-known Maxwell line analysis.


Discontinuous solutions Fibre-reinforced materials Finite deformations Maxwell line 



The authors acknowledge support from the Ministerio de Ciencia, Spain, under Project DPI2011-26167. Mustapha El Hamdaoui also thanks the Ministerio de Economía y Competitividad, Spain, for funding under Project DPI2008-03769 and Grant BES-2009-027812.


  1. 1.
    Rivlin RS (1949) Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Philos Trans R Soc Lond Ser A 242(845):173–195MATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Merodio J, Ogden RW (2005) Remarks on instabilities and ellipticity for a fiber-reinforced compressible nonlinearly elastic solid under plane deformation. Q Appl Math 63(2):325–333MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Kassianidis F, Ogden RW, Merodio J, Pence TJ (2007) Azimuthal shear of a transversely isotropic elastic solid. J Math Mech Solids 13(8):690–724MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gao DY, Ogden RW (2008) Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem. Z A M P 59(3):498–517MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Dorfmann A, Merodio J, Ogden RW (2010) Non-smooth solutions in the azimuthal shear of an anisotropic nonlinearly elastic material. J Eng Math 68(1):27–36MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    El Hamdaoui M, Merodio J, Ogden RW, Rodríguez J (2014) Finite elastic deformations of transversely isotropic circular cylindrical tubes. Int J Solids Struct 51(5):1188–1196CrossRefGoogle Scholar
  7. 7.
    Jiang X, Ogden RW (1998) On azimuthal shear of a circular cylindrical tube of compressible elastic material. Q J Mech Appl Math 51(1):143–158MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Jiang Q, Beatty M (2001) On compressible materials capable of sustaining axisymmetric shear deformations. Part 4: Helical shear of anisotropic hyperelastic materials. J Elast Phys Sci Solids 62(1):47–83MATHMathSciNetGoogle Scholar
  9. 9.
    Tsai H, Fan X (1999) Anti-plane shear deformations in compressible transversely isotropic materials. J Elast 54(1):73–88MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    O’Callaghan L, O’Reilly O, Zhornitskaya L (2011) On azimuthal shear waves in a transversely isotropic viscoelastic mixture: application to diffuse axonal injury. Math Mech Solids 16(6):625–636MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dagher MA, Soldatos KP (2011) On small azimuthal shear deformation of fibre-reinforced cylindrical tubes. J Mech Mater Struct 6(1–4):141–168CrossRefGoogle Scholar
  12. 12.
    Dagher MA, Soldatos KP (2013) Area-preserving azimuthal shear deformation of an incompressible isotropic hyper-elastic tube. J Eng Math 78(1):131–142CrossRefGoogle Scholar
  13. 13.
    Merodio J, Ogden RW (2006) The influence of the invariant \(I_8\) on the stress-deformation and ellipticity characteristics of doubly fiber-reinforced non-linearly elastic solids. Int J Non-linear Mech 41(4):556–563MATHCrossRefGoogle Scholar
  14. 14.
    James R (1979) Co-existent phases in the one-dimensional static theory of elastic bars. Arch Ration Mech Anal 72(2):99–140MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Continuum Mechanics and StructuresUniversidad Politécnica de MadridMadridSpain

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