Does the shear-lag model apply to random fiber networks?
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Does the shear-lag model apply to random fiber networks? V. I. R¨ais¨anena) CSC– Center for Scientific Computing, P.O. Box 405, FIN-02101 Espoo, Finland
M. J. Alava Helsinki University of Technology, Laboratory of Physics, FIN-02150 Espoo, Finland
K. J. Niskanen KCL Paper Science Centre, P.O. Box 70, FIN-02151 Espoo, Finland
R. M. Nieminen CSC– Center for Scientific Computing, P.O. Box 405, FIN-02101 Espoo, Finland and Helsinki University of Technology, Laboratory of Physics, FIN-02150 Espoo, Finland (Received 15 April 1996; accepted 19 June 1997)
The shear-lag type model due to Cox (Br. J. Appl. Phys. 3, 72 (1952) is widely used to calculate the deformation properties of fibrous materials such as short fiber composites and random fiber networks. We compare the shear-lag stress transfer mechanism with numerical simulations at small, linearly elastic strains and conclude that the model does not apply to random fiber networks. Most of the axial stress is transferred directly from fiber to fiber rather than through intermediate shear-loaded segments as assumed in the Cox model. The implications for the elastic modulus and strength of random fiber networks are discussed.
I. INTRODUCTION
The classical Cox model1 of the elasticity of fibrous matter is based on an effective medium approximation for a fiber in equilibrium with an average surroundings (“cell approximation”).2 The model has been applied to composites, where the fibers are surrounded by a matrix, and to fiber networks, where the fibers are directly bonded together. We study the latter type of materials, of which paper and nonwovens are typical examples. The Cox model is widely used to describe paper properties.3,4 In this application, one considers a fiber attached to an effective homogeneous background consisting of perpendicular or nearly perpendicular fibers. The effective background accounts for the average elastic properties of randomly located and isotropically oriented background fibers. In the case of composites, experimental evidence shows convincingly that a single fiber in a homogeneous matrix responds to external loading according to the shear-lag mechanism.5,6 This holds for both the variation of stress along the fiber axis and the dependence of stress on the orientation angle of the fiber. However, recent computer simulations7–9 have shown that local stress fluctuations in disordered fiber composites cannot be ignored. Experimental information on the elasticity of random fiber composites is still inconclusive and
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Current address: ICA 1, University of Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart, Germany. J. Mater. Res., Vol. 12, No. 10, Oct 1997
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the validity of the shear-lag mechanism has not been verified.10,11 In the case of bonded fiber networks, where the fibers are in direct contact with each other, it is obvious that the effective medium shear-lag model cannot work at low density, close to the geometrical percolation thresho
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