Criticality and isostaticity in fibre networks

Journal name:
Nature Physics
Year published:
Published online


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.

At a glance


  1. Fibre networks arranged on lattices in 2D and 3D.
    Figure 1: Fibre networks arranged on lattices in 2D and 3D.

    a,b, A small section of a sheared diluted triangular network near isostaticity with relatively stiff filaments (κ=101 in units of μ02; a) and floppy filaments (κ=10−5; b). The deviation of the local deformation from a uniform deformation is indicated by colour, where blue corresponds to a uniform or affine deformation and red corresponds to a highly non-affine deformation. c, An example of a small section of the diluted fcc network at p=0.7. To probe the mechanical properties of this network we shear the 111 plane (shown on top) along the direction of one of the bonds in this plane. d, Schematic representation of two crosslinked fibre segments indicating bond stretching (δij) between network nodes and angular deflection (Δθijk) along consecutive segments on the same fibre. The crosslinks themselves are freely hinged. e, Schematic representation of a deformed section of the network. The arrows indicate the deformation of nodes ui with respect to the undeformed reference state.

  2. Mechanics and non-affine strain fluctuations.
    Figure 2: Mechanics and non-affine strain fluctuations.

    a,b, The shear modulus, G, in units of μ/0d−1 as a function of the bond occupation probability, p, for a range of filament bending rigidities, κ, for a two-dimensional triangular lattice (a) and a three-dimensional fcc lattice (b). The numerial results for κ=0 are shown as dashed grey lines. The EMT calculations for a two-dimensional triangular lattice are shown as solid lines in a. c,d, The non-affinity measure, Γ, is shown as a function of p for various values of κ for a two-dimensional triangular lattice (c) and a three-dimensional fcc lattice (d). The values of κ in units of μ02 are 100 (green), 10−3 (cyan), 10−4 (red) and 10−6 (blue).

  3. Scaling analysis of the mechanics and anomalous elasticity.
    Figure 3: Scaling analysis of the mechanics and anomalous elasticity.

    a,b, Scaling of the shear modulus, G, in the vicinity of the isostatic point with the scaling form , with G in units of μ/0d−1 and the bending rigidity, κ, in units of μ02, for a diluted triangular lattice over a broad range of filament bending rigidities (κ in units of μ02: 10−1 black, 10−2 magenta, 10−3 cyan, 10−4 red, 10−5 purple and 10−6 blue) for the EMT calculations (a) and the simulations (b). The asymptotic form of the scaling function for low κ is shown as a dashed grey line in a. The scaling for the numerical data on a three-dimensional fcc lattice is shown as an inset in b. The EMT exponents are fCF=1, ϕ=2. In contrast, numerically we obtain fCF=1.4±0.1, ϕ=3.0±0.2 (2D) and fCF=1.6±0.2 ϕ=3.6±0.3 (3D). The scaling for the numerical data is carried out with respect to the isostatic point of the finite system pCF(W)=0.651 (2D, W=200) and pCF(W)=0.473 (3D, W=30). c, The shear modulus as a function of κ close to the isostatic point for a triangular lattice (p=0.643, blue circles) and an fcc lattice (p=0.47, red squares). At low κthere is a bending-dominated regime Gbend~κ; at intermediate κ there is a regime in which stretching and bending modes couple strongly with G~μ1−xκx, where x=0.50±0.01 (2D) and x0.40±0.01(3D). The EMT calculation for κ/μdouble greater thanp|ϕ is shown as a solid blue line.

  4. Finite-size scaling.
    Figure 4: Finite-size scaling.

    a, The non-affinity measure, Γ, for a two-dimensional triangular lattice at κ=0 for various system sizes, W. b,c, Finite-size scaling of Γaccording to the scaling form (b) and of the shear modulus with the scaling form (c). Here Δp=ppCF, where pCF=0.659±0.002 in the limit . The exponents we obtain are λCF/νCF=1.6±0.2, νCF=1.4±0.2 and fCF/νCF=0.9±0.1.

  5. Phase diagram.
    Figure 5: Phase diagram.

    The phase diagram for diluted fibre networks with connectivities that cover both the CF and bending thresholds. Above the bending-rigidity percolation point, zb, there are three distinct mechanical regimes: a stretching-dominated regime, with G~μ, a bending-dominated regime, with G~κ, and a regime in which bend and stretch modes couple, with G~μ1−xκx. Here xis related to the critical exponents, x=f/ϕ. We find here that x=0.50±0.01 (two-dimensional triangular lattice) and x=0.40±0.01 (three-dimensional fcc). The mechanical regimes are controlled by the CF isostatic point, zCF, which acts as a zero-temperature critical point.


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Author information


  1. Department of Physics and Astronomy, Vrije Universiteit, Amsterdam 1081 HV, The Netherlands

    • Chase P. Broedersz &
    • Frederick C. MacKintosh
  2. Lewis-Sigler Institute for Integrative Genomics and the Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

    • Chase P. Broedersz
  3. Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

    • Xiaoming Mao &
    • Tom C. Lubensky


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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