Abstract
Disordered fibre networks are the basis of many man-made and natural materials, including structural components of living cells and tissue. The mechanical stability of such networks relies on the bending resistance of the fibres, in contrast to rubbers, which are governed by entropic stretching of polymer segments. Although it is known that fibre networks exhibit collective bending deformations, a fundamental understanding of such deformations and their effects on network mechanics has remained elusive. Here we introduce a lattice-based model of fibrous networks with variable connectivity to elucidate the roles of single-fibre elasticity and network structure. These networks exhibit both a low-connectivity rigidity threshold governed by fibre-bending elasticity and a high-connectivity threshold governed by fibre-stretching elasticity. Whereas the former determines the true onset of network rigidity, we show that the latter exhibits rich zero-temperature critical behaviour, including a crossover between various mechanical regimes along with diverging strain fluctuations and a concomitant diverging correlation length.
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References
Chawla, K. K. Fibrous Materials (Cambridge Univ. Press, 1998).
Kabla, A. & Mahadevan, L. Nonlinear mechanics of soft fibrous networks. J. R. Soc. Interface 4, 99–106 (2007).
Hough, L. A., Islam, M. F., Janmey, P. A. & Yodh, A. G. Viscoelasticity of single wall carbon nanotube suspensions. Phys. Rev. Lett. 93, 168102 (2004).
Hall, L. J. et al. Sign change of Poisson’s ratio for carbon nanotube sheets. Science 320, 504–507 (2008).
Bausch, A. R. & Kroy, K. A bottom-up approach to cell mechanics. Nature Phys. 2, 231–238 (2006).
Fletcher, D. A. & Mullins, R. D. Cell mechanics and the cytoskeleton. Nature 463, 485–492 (2010).
Kasza, K. E. et al. The cell as a material. Curr. Opin. Cell Biol. 19, 101–107 (2007).
Pedersen, J. A. & Swartz, M. A. Mechanobiology in the third dimension. Ann. Biomed. Eng. 33, 1469–1490 (2005).
Maxwell, J. C. On the calculation of the equilibrium and stiffness of frames. Phil. Mag. 27, 294–299 (1864).
Thorpe, M. F. Continuous deformations in random networks. J. Non-Cryst. Solids 57, 355–370 (1983).
Garboczi, E. J. & Thorpe, M. F. Effective-medium theory of percolation on central-force elastic networks. III. The superelastic problem. Phys. Rev. B 33, 3289–3294 (1986).
Wyart, M., Liang, H., Kabla, A. & Mahadevan, L. Elasticity of floppy and stiff random networks. Phys. Rev. Lett. 101, 215501 (2008).
Gardel, M. L. et al. Elastic behavior of cross-linked and bundled actin networks. Science 304, 1301–1305 (2004).
Storm, C., Pastore, J. J., MacKintosh, F. C., Lubensky, T. C. & Janmey, P. A. Nonlinear elasticity in biological gels. Nature 435, 191–194 (2005).
Chaudhuri, O., Parekh, S. H. & Fletcher, D. A. Reversible stress softening of actin networks. Nature 445, 295–298 (2007).
Lieleg, O., Claessens, M. M. A. E., Heussinger, C., Frey, E. & Bausch, A. R. Mechanics of bundled semiflexible polymer networks. Phys. Rev. Lett. 99, 088102 (2007).
Head, D. A., Levine, A. J. & MacKintosh, F. C. Deformation of cross-linked semiflexible polymer networks. Phys. Rev. Lett. 91, 108102 (2003).
Wilhelm, J. & Frey, E. Elasticity of stiff polymer networks. Phys. Rev. Lett. 91, 108103 (2003).
Onck, P. R., Koeman, T., van Dillen, T. & van der Giessen, E. Alternative explanation of stiffening in cross-linked semiflexible networks. Phys. Rev. Lett. 95, 178102 (2005).
Heussinger, C. & Frey, E. Floppy modes and nonaffine deformations in random fiber networks. Phys. Rev. Lett. 97, 105501 (2006).
Buxton, G. A. & Clarke, N. Bending to stretching transition in disordered networks. Phys. Rev. Lett. 98, 238103 (2007).
Huisman, E. M. & Lubensky, T. C. Internal stresses, normal modes, and nonaffinity in three-dimensional biopolymer networks. Phys. Rev. Lett. 106, 088301 (2011).
Jacobs, D. J. & Thorpe, M. F. Generic rigidity percolation in two dimensions. Phys. Rev. E 53, 3682–3693 (1996).
Feng, S. & Sen, P. N. Percolation on elastic networks: New exponent and threshold. Phys. Rev. Lett. 52, 216–219 (1984).
Feng, S., Sen, P. N., Halperin, B. I. & Lobb, C. J. Percolation on two-dimensional elastic networks with rotationally invariant bond-bending forces. Phys. Rev. B 30, 5386–5389 (1984).
Phillips, J. C. Topology of covalent non-crystalline solids II: Medium-range order in chalcogenide alloys and A–Si(Ge). J. Non-Cryst. Solids 43, 37–77 (1981).
He, H. & Thorpe, M. F. Elastic properties of glasses. Phys. Rev. Lett. 54, 2107–2110 (1985).
Liu, A. J. & Nagel, S. R. Nonlinear dynamics: Jamming is not just cool any more. Nature 396, 21–22 (1998).
O’Hern, C. S., Silbert, L. E., Liu, A. J. & Nagel, S. R. Jamming at zero temperature and zero applied stress: The epitome of disorder. Phys. Rev. E 68, 011306 (2003).
Liu, A. J., Nagel, S. R., van Saarloos, W. & Wyart, M. in Dynamical Heterogeneities in Glasses, Colloids, and Granular Media (eds Berthier, L., Biroli, G., Bouchaud, J-P., Cipeletti, L. & van Saarloos, W.) (Oxford Univ.Press, 2010).
Schwartz, L. M., Feng, S., Thorpe, M. F. & Sen, P. N. Behavior of depleted elastic networks: Comparison of effective-medium and numerical calculations. Phys. Rev. B 32, 4607–4617 (1985).
Sahimi, M. & Arbabi, S. Mechanics of disordered solids. II. Percolation on elastic networks with bond-bending forces. Phys. Rev. B 47, 703–712 (1993).
Broedersz, C. P. & MacKintosh, F. C. Molecular motors stiffen non-affine semiflexible polymer networks. Soft Matter 7, 3186–3191 (2011).
Zabolitzky, J. G., Bergman, D. J. & Stauffer, D. Precision calculation of elasticity for percolation. J. Stat. Phys. 44, 211–223 (1986).
Arbabi, S. & Sahimi, M. Mechanics of disordered solids. I. Percolation on elastic networks with central forces. Phys. Rev. B 47, 695–702 (1993).
Das, M., MacKintosh, F. C. & Levine, A. J. Effective medium theory of semiflexible filamentous networks. Phys. Rev. Lett. 99, 038101 (2007).
Lax, M. Multiple scattering of waves. Rev. Mod. Phys. 23, 287–310 (1951).
Elliott, R. J., Krumhansl, J. A. & Leath, P. L. The theory and properties of randomly disordered crystals and related physical systems. Rev. Mod. Phys. 46, 465–543 (1974).
Soven, P. Contribution to the theory of disordered alloys. Phys. Rev. 178, 1136–1144 (1969).
Mao, X., Xu, N. & Lubensky, T. C. Soft modes and elasticity of nearly isostatic lattices: Randomness and dissipation. Phys. Rev. Lett. 104, 085504 (2010).
Feng, S., Thorpe, M. F. & Garboczi, E. Effective-medium theory of percolation on central-force elastic networks. Phys. Rev. B 31, 276–280 (1985).
Straley, J. Critical phenomena in resistor networks. J. Phys. C 9, 783–795 (1976).
Dykhne, A. M. Conductivity of a two-dimensional two-phase system. JETP 32, 63–65 (1971).
Efros, A. L. & Shklovskii, B. I. Critical behaviour of conductivity and dielectric constant near the metal–non-metal transition threshold. Phys. Status Solidi B 76, 475–485 (1976).
Heussinger, C. & Frey, E. Stiff polymers, foams, and fiber networks. Phys. Rev. Lett. 96, 017802 (2006).
DiDonna, B. A. & Lubensky, T. C. Nonaffine correlations in random elastic media. Phys. Rev. E 72, 066619 (2005).
Liu, J., Koenderink, G. H., Kasza, K. E., MacKintosh, F. C. & Weitz, D. A. Visualizing the strain field in semiflexible polymer networks: Strain fluctuations and nonlinear rheology of f-actin gels. Phys. Rev. Lett. 98, 198304 (2007).
Fisher, M. E. in Proc. School on Critical Phenomena, Stellenbosch, South Africa, 1982 Vol. 186 (ed. Hahne, F. J. W.) (Springer, 1983).
Chubynsky, M. V. & Thorpe, M. F. Algorithms for three-dimensional rigidity analysis and a first-order percolation transition. Phys. Rev. E 76, 041135 (2007).
Acknowledgements
This work was supported in part by NSF-DMR-0804900 (T.C.L. and X.M.) and in part by FOM/NWO (C.P.B. and F.C.M.). The authors thank M. Wyart, M. Das and L. Jawerth for useful discussions.
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C.P.B. and F.C.M. designed the simulation model, which was developed and executed by C.P.B.; X.M. and T.C.L. developed and executed the EMT. All authors contributed to the writing of the paper.
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Broedersz, C., Mao, X., Lubensky, T. et al. Criticality and isostaticity in fibre networks. Nature Phys 7, 983–988 (2011). https://doi.org/10.1038/nphys2127
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DOI: https://doi.org/10.1038/nphys2127
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