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Räisänen, Vilho I. and Herrmann, Hans J. 1998. Effect of texture on fracture of fibrous materials. Computer Methods in Applied Mechanics and Engineering, Vol. 161, Issue. 1-2, p. 103.
Åström, J. A. Mäkinen, J. P. Alava, M. J. and Timonen, J. 2000. Elasticity of Poissonian fiber networks. Physical Review E, Vol. 61, Issue. 5, p. 5550.
Niskanen, Kaarlo 2000. Cellulosic Pulps, Fibres and Materials. p. 249.
Wilhelm, Jan and Frey, Erwin 2003. Elasticity of Stiff Polymer Networks. Physical Review Letters, Vol. 91, Issue. 10,
Eichhorn, S.J. and Young, R.J. 2003. Deformation micromechanics of natural cellulose fibre networks and composites. Composites Science and Technology, Vol. 63, Issue. 9, p. 1225.
Wu, X.-F. and Dzenis, Y. A. 2005. Elasticity of planar fiber networks. Journal of Applied Physics, Vol. 98, Issue. 9, p. 093501.
Alava, Mikko and Niskanen, Kaarlo 2006. The physics of paper. Reports on Progress in Physics, Vol. 69, Issue. 3, p. 669.
Batchelor, Warren 2008. An analytical solution for the load distribution along a fibre in a nonwoven network. Mechanics of Materials, Vol. 40, Issue. 12, p. 975.
Sampson, W W 2009. Materials properties of paper as influenced by its fibrous architecture. International Materials Reviews, Vol. 54, Issue. 3, p. 134.
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Rawal, Amit Priyadarshi, Apurv Lomov, Stepan V. Verpoest, Ignaas and Vankerrebrouck, Jozef 2010. Tensile behaviour of thermally bonded nonwoven structures: model description. Journal of Materials Science, Vol. 45, Issue. 9, p. 2274.
Pradhan, Srutarshi Hansen, Alex and Chakrabarti, Bikas K. 2010. Failure processes in elastic fiber bundles. Reviews of Modern Physics, Vol. 82, Issue. 1, p. 499.
Chen, Fujia Porter, David and Vollrath, Fritz 2010. Silkworm cocoons inspire models for random fiber and particulate composites. Physical Review E, Vol. 82, Issue. 4,
Picu, R. C. 2011. Mechanics of random fiber networks—a review. Soft Matter, Vol. 7, Issue. 15, p. 6768.
Tsarouchas, D. and Markaki, A.E. 2011. Extraction of fibre network architecture by X-ray tomography and prediction of elastic properties using an affine analytical model. Acta Materialia, Vol. 59, Issue. 18, p. 6989.
Hatami-Marbini, H. Shahsavari, A. and Picu, R.C. 2013. Multiscale modeling of semiflexible random fibrous structures. Computer-Aided Design, Vol. 45, Issue. 1, p. 77.
Berkache, K. Deogekar, S. Goda, I. Picu, R.C. and Ganghoffer, J.-F. 2017. Construction of second gradient continuum models for random fibrous networks and analysis of size effects. Composite Structures, Vol. 181, Issue. , p. 347.
Chen, Naigeng and Silberstein, Meredith N. 2018. A micromechanics-based damage model for non-woven fiber networks. International Journal of Solids and Structures,
Gabbett, Cian Boland, Conor S. Harvey, Andrew Vega-Mayoral, Victor Young, Robert J. and Coleman, Jonathan N. 2018. The Effect of Network Formation on the Mechanical Properties of 1D:2D Nano:Nano Composites. Chemistry of Materials, Vol. 30, Issue. 15, p. 5245.
Chen, N. and Silberstein, M. N. 2018. Determination of Bond Strengths in Non-woven Fabrics: a Combined Experimental and Computational Approach. Experimental Mechanics, Vol. 58, Issue. 2, p. 343.
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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.
Hide All1.Cox, H. L., Br. J. Appl. Phys. 3, 72 (1952).2. For shear-lag models for discontinuous fiber composites, see Robinson, I. M. and Robinson, J. M., J. Mater. Sci. 29, 4663 (1994).3.Page, D. H. and Seth, R. S., Tappi J. 63, 113 (1980).4.Perkins, R. W., in Materials Interactions Relevant to the Pulp Paper and Wood Industries, edited by Caulfield, D. F., Passaretti, J. D., and Sobczynski, S. F. (Materials Research Society, Pittsburgh, PA, 1990), Vol. 197, p. 99, and references therein.5. See, e.g., Galiolis, C., Composite Sci. Technol. 48, 15 (1994) and references therein.6.Andrews, M. C., Day, R. J., Hu, X., and Young, R. J., J. Mater. Sci. Lett. 11, 124 (1992).7.Murat, M., Anholt, M., and Wagner, H. D., J. Mater. Res. 7, 3120 (1992).8.Monette, L., Anderson, M. P., and Grest, G. S., J. Appl. Phys. 75, 1155 (1994).9.Duxbury, P. M., Beale, P. D., and Moukarzel, C., Phys. Rev. B 51 (6), 3476–3488; C. Moukarzel and P. M. Duxbury, J. Appl. Phys. 76, 4086 (1994).10.Blumentritt, B. F., Vu, B. T., and Cooper, S. L., Polym. Eng. Sci. 15, 428 (1975).11.Karbhari, M. and Wilkins, D. M., Scripta Metall. Mater. 25, 707 (1991).12.Åström, J., Saarinen, S., Niskanen, K., and Kurkijärvi, J., J. Appl. Phys. 75, 2383 (1994).13. Cf, e.g., Sahimi, M., Physica A 186, 160 (1992).14.Hansen, A., in Statistical models for the fracture of disordered media, edited by Herrmann, H. J. and Roux, S. (North-Holland, Amsterdam, 1990), p. 115 ff.15. Version 4.9, Hibbitt, Karlsson & Sorensen Inc., 1080 Main St., Pawtucket, RI 02860.16. We note here in passing that strictly speaking in our simulations the elastic modulus is not equal to Young's modulus, because the Poisson contraction is prohibited. The relation between the two in 2D is given by [see, e.g., Landau, L. D. and Lifshitz, E. M., Theory of Elasticity (Pergamon Press Ltd., London, 1959), p. 52] where E 0 is the elastic modulus of a network with fixed y-coordinate of the edges perpendicular to external stress, E is the elastic modulus of a network with Poisson contraction, and ν − 0.33 is the Poisson contraction coefficient that can be obtained from the Cox model.17.Corte, H. and Kallmes, O. J. in The Formation and Structure of Paper, Transactions of the Symposium held at Oxford, edited by Bolam, J. (William Clowes and Sons, Ltd., London, 1962), pp. 13–52.18.Pike, G. E. and Seager, C. H., Phys. Rev. B 10, 1421 (1974).19.Jangmalm, A. and Östlund, S., Nordic Pulp and Paper Res. J. 10, 156 (1995).20.Heyden, S. and Gustafsson, P. J., A network model for application to cellulose fiber materials, conference paper in 7th Int. Conference on Mechanical Behavior of Materials, May 1995, The Netherlands.21.Hansen, A., in Statistical models for the fracture of disordered media, edited by Herrmann, H. J. and Roux, S. (Elsevier Science Publishers, North-Holland, 1990), p. 134.22.Hashin, Z. and Shtrikman, S., J. Mech. Phys. Solids 11, 335 (1963); see also Z. Hashin, J. Appl. Mech. 50, 481 (1983) and references therein and M. Deng and C. T. J. Dodson, Paper: An Engineered Stochastic Structure (Tappi Press, 1994), pp. 204–206 for an application to fiber networks.23.Grubb, D. T., Li, Z-F., and Phoenix, S. L., Comp. Sci. Technol. 59, 237 (1995).24. See, e.g., Niskanen, , in Products of Papermaking—Transactions of the Tenth Fundamental Research Symposium held at Oxford: September 1993, p. 685, Fig. 32.25.Hansen, A., in Statistical models for the fracture of disordered media, edited by Herrmann, H. J. and Roux, S. (Elsevier Science Publishers, North-Holland, 1990), p. 149 ff.26.Duxbury, P. M., Guyer, R. A., and Machta, J., Phys. Rev. B 51, 6711 (1995).
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