Strain-controlled criticality governs the nonlinear mechanics of fibre networks

Abstract

Disordered fibrous networks are ubiquitous in nature as major structural components of living cells and tissues. The mechanical stability of networks generally depends on the degree of connectivity: only when the average number of connections between nodes exceeds the isostatic threshold are networks stable [1]. Upon increasing the connectivity through this point, such networks undergo a mechanical phase transition from a floppy to a rigid phase. However, even subisostatic networks become rigid when subjected to sufficiently large deformations. To study this strain-controlled transition, we perform a combination of computational modeling of fibre networks and experiments on networks of type I collagen fibers, which are crucial for the integrity of biological tissues. We show theoretically that the development of rigidity is characterized by a strain-controlled continuous phase transition with signatures of criticality. Our experiments demonstrate mechanical properties consistent with our model, including the predicted critical exponents. We show that the nonlinear mechanics of collagen networks can be quantitatively captured by the predictions of scaling theory for the straincontrolled critical behavior over a wide range of network concentrations and strains up to failure of the material.As shown by Maxwell, networks with only central-force interactions exhibit a transition from a floppy to rigid phase at the isostatic point, where the local coordination number, or connectivity z equals the threshold value of z = 2d in d dimensions [1]. At this point, the number of degrees of freedom is just balanced by the number of constraints, and the system is marginally stable to small deformations. The jamming transition [2][3][4][5][6] in granular materials and rigidity percolation [7][8][9][10] in disordered spring networks are examples of such a transition. An important feature of these systems is the order of the transition. Jamming exhibits signatures of both first-and second-order transitions, with discontinuous behaviour of the bulk modulus and continuous variation of the shear modulus [5,11,12]. For networks of springs or fibres, the transition from floppy to rigid is a continuous phase transition, in both bulk and shear moduli [2, 5, 7, 7, 8,14].Interestingly, the structural networks in biology almost always have connectivities below the central-force iso- static point. Networks such as the extracellular matrices of collagen that make up tissue are strictly sub-isostatic with respect to central forces: their typical connectivity is between 3 (local branching) and 4 (binary crosslinking), placing them below both 2D and 3D isostatic thresholds [17,18]. Such sub-isostatic networks can, nevertheless, become rigid as a result of other mechanical constraints, such as fibre bending [4, 7,19], or when subjected to external strain [21]. The threshold strain, at which the transition occurs, depends on the nature of applied deformation (shear or tensile) and on the average connectivity of the network, in particular, and othe...

Strain-driven criticality underlies nonlinear mechanics of fibrous networks

“…There are three regimes as a function of strain: a linear elastic regime at low strain, a regime of rapid stiffening above a threshold strain γ 0 , and a high-strain regime in which the strain dependence is weaker and the elastic modulus becomes independent ofκ. The initial, linear modulus scales linearly withκ, indicating a bend-dominated response, as has been reported in several prior computational studies [13,15,18,19,23,33,39]. The applicability of the computational model for athermal networks such as collagen depends on the following observations.…”

“…The most relevant experimental control variable is the total protein concentration c. For a given thickness of fibers, the volume fraction ϕ of a network scales linearly with c and using the above assumptions can be simply related to the reduced bending rigidityκ as ϕ ∼κ ∝ (a/ l 0 ) 2 [23,33,35,36], where a is the fiber thickness. It follows that K/ϕ (or K/c) in experiments can be directly compared with K(γ,κ) in simulations.…”

“…However, when the applied field is an external strain, the transition from floppy to rigid states occurs at a threshold strain that depends on the network structure, nature of the applied deformation, as well as the average connectivity [16]. We recently showed that sheared subisostatic networks exhibit a line of second-order transitions at a strain threshold γ c (z), for connectivities z well below the isostatic threshold [23].…”

“…In Ref. [23] it was shown that the critical exponents appear to depend on the average connectivity in the network. Here we present our findings on the evolution of critical exponents in more detail.…”

Topological Edge Floppy Modes in Disordered Fiber Networks

“…1b-d). Fiber networks are ubiquitous in nature, taking the form of cell cytoskeleton and extra-cellular matrix, and in manmade materials, taking the form of fiber hydrogels and aerogels, felt, etc., and exhibit fascinating physics [30][31][32][33][34][35][36][37][38][39][40][41]. Using both analytic theory and numerical simulation, we show that topological floppy edge modes exist in these disordered fiber networks, and their existence lead to strongly asymmetric mechanical properties at opposite ends of the fiber network.…”