Journal of Engineering Mathematics

, Volume 68, Issue 2, pp 179–196 | Cite as

Second-gradient plane deformations of ideal fibre-reinforced materials II: forming flows of fibre–resin systems when fibres resist bending



The continuum theory of ideal fibre-reinforced fluids, namely incompressible viscous fluids exhibiting some direction of inextensibility is extended to account for the fibre bending stiffness; namely a property that prevents discontinuity of the fibre slope under normal loading conditions. The principal kinematics of this new theoretical development is consistent with three-dimensional forming flows of fibre–resin systems though, for simplicity, formulation of relevant constitutive equations is confined within the framework of relevant plane flows. The theory adopts the macroscopic view that the resin matrix behaves as a viscous fluid but the resin and fibres form a homogeneous composite material. Consideration of the fibre bending resistance requires the inclusion of couple-stress and, hence, non-symmetric stress. The outlined theoretical developments are therefore relevant to polar-media behaviour; in this context, the anisotropic viscous fluids of interest become part of the material class of the so-called polar fluids. For plane flows of this type of fluids, a manner is also outlined in which the non-symmetric stress distributions sought can be determined by solving two simultaneous, first-order linear differential equations. Moreover, a relevant stress-resultants technique is adopted and extended appropriately to make possible complete determination of the kinematics dictating the creeping forming plane flow of the composite fluids of interest. Details of the mechanisms that capture fibre bending resistance are revealed and illustrated through a relatively simple example application. This considers and resolves the forming flow process of an ideal fibre-reinforced composite, moulded into a sharp corner under the action of an external line force.


Fibre bending stiffness Fibre-reinforced fluids Ideal fibre-reinforced materials Plane flows Polar fluids Viscous fluids 


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  1. 1.
    Hull BD, Rogers TG, Spencer AJM (1994) Theoretical analysis of forming flows of continuous fibre–resin systems. In: Advani SG (eds) Flow and rheology in polymer composites manufacturing. Elsevier, Amsterdam, pp 203–256Google Scholar
  2. 2.
    Pipkin AC, Rogers TG (1971) Plane deformations of incompressible fiber-reinforced materials. J Appl Mech Trans ASME 38: 634–640MATHGoogle Scholar
  3. 3.
    Spencer AJM (1972) Deformations of fibre-reinforced materials. Oxford University Press, LondonMATHGoogle Scholar
  4. 4.
    Rogers TG, O’Neil JM (1991) Theoretical analysis of forming flows of fibre-reinforced composites. Comp Manuf 2: 153–160CrossRefGoogle Scholar
  5. 5.
    Spencer AJM, Soldatos KP (2007) Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. Int J Non-linear Mech 42: 355–368CrossRefGoogle Scholar
  6. 6.
    Soldatos KP (2009) Finite plane strain of ideal fibre-reinforced materials—implications of a second gradient hyper-elasticity theory. J Eng Math 68(1). doi: 10.1007/s10665-009-9353-4
  7. 7.
    Soldatos KP (2009) Azimuthal shear deformation of an ideal fibre-reinforced tube according to a second gradient hyper-elasticity theory. In: Ambrosio J et al. (eds) Proceedings of the 7th EUROMECH solid mechanics conference, Sept 7–11. Lisbon, PortugalGoogle Scholar
  8. 8.
    Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In: Flügge S (eds) Encyclopedia of physics. Springer-Verlag, MoscowGoogle Scholar
  9. 9.
    Zheng Q-S (1994) Theory of representations for tensor functions. Appl Mech Rev 47: 554–587CrossRefGoogle Scholar
  10. 10.
    Korsgaard J (1990) On the representation of two-dimensional isotropic functions. Int J Eng Sci 28: 653–662MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rogers TG (1972) Finite deformations of strongly anisotropic materials. In: Hutton JF, Pearson JRA, Walters K (eds) Theoretical rheology. Applied Science Publishers, New York, pp 141–168Google Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Theoretical Mechanics, School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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