Strain-driven criticality underlies nonlinear mechanics of fibrous networks

Networks with only central force interactions are floppy when their average connectivity is below an isostatic threshold. Although such networks are mechanically unstable, they can become rigid when strained. It was recently shown that the transition from floppy to rigid states as a function of simple shear strain is continuous, with hallmark signatures of criticality [Sharma et al., Nature Phys. 12, 584 (2016)]. The nonlinear mechanical response of collagen networks was shown to be quantitatively described within the framework of such mechanical critical phenomenon. Here, we provide a more quantitative characterization of critical behavior in subisostatic networks. Using finite-size scaling we demonstrate the divergence of strain fluctuations in the network at well-defined critical strain. We show that the characteristic strain corresponding to the onset of strain stiffening is distinct from but related to this critical strain in a way that depends on critical exponents. We confirm this prediction experimentally for collagen networks. Moreover, we find that the apparent critical exponents are largely independent of the spatial dimensionality. With subisostaticity as the only required condition, strain-driven criticality is expected to be a general feature of biologically relevant fibrous networks.

Figure 1

(a) Shear stiffness versus shear strain curves obtained from a phantom triangular lattice in 2D with $\langle z\rangle \simeq 3.4$. Different curves are obtained by varying the reduced bending rigidity $\tilde{\kappa },{10}^{-7}\left(▾\right),{10}^{-6}(▿),{10}^{-5}\left(▴\right),{10}^{-4}\left(▵\right),{10}^{-3}(•)$, and $\infty$ (thick red line). The onset strain for stiffening, ${\gamma }_0$, is shown as the blue dash-dotted line. The red dashed line shows the stiffness when $\tilde{\kappa }=0$. In absence of bending interactions, the stiffness remains zero for $\gamma \le {\gamma }_c$. The green dashed lines through the symbols show the predicted stiffness according to Eq. (9) with $f=0.8\pm 0.05$ and $\varphi =2.1\pm 0.2$. The top inset shows the strain, identified by the inflection point of the stiffness curves. In the limit of $\tilde{\kappa }\to 0$, the inflection point approaches the critical strain ${\gamma }_c$ of a central-force subisostatic network. As the inset shows, ${\gamma }_c$ is insensitive to the addition of finite fiber bending rigidity, up to $\tilde{\kappa }\lesssim {10}^{-3}$. The bottom inset shows $\mathcal{K}/\sigma$ versus $\sigma$ for different values of $\tilde{\kappa }$. The onset strain for stiffening, ${\gamma }_0$, obtained from the local minimum of $\mathcal{K}/\sigma$ versus $\sigma$, is indicated with a blue cross. (b) Experimentally obtained stiffness versus strain curve for a 1 mg/ml collagen network $(◯)$. Since $\mathcal{K}$ in simulations corresponds to $K/c$ in experiments, the experimentally obtained stiffness is normalized to the concentration $c$. The dashed line through the experimental data is fit according to Eq. (9) with the reduced bending rigidity $\tilde{\kappa }={10}^{-4}$ and the parameters $f=0.8,\varphi =2.3$ obtained from the collapse of stiffness curves obtained from simulations as explained in Sec. 3. The critical strain ${\gamma }_c=0.29$, marked with a red cross, is obtained as the inflection point of the stiffness curve. The onset strain for stiffening ${\gamma }_0$ is marked with a blue cross. The top inset shows the experimentally measured $K_{\max }$ versus concentration $c$ for collagen networks, where $K_{\max }$ is the maximum nonlinear modulus before the network ruptures. At large strains, when network stiffness is governed by stretching, $K_{\max }$ is expected to scale with the concentration $c$, shown as the black line. The lower inset shows $K/\sigma$ versus $\sigma$ obtained from the experimental data in (b). The minimum corresponds to the blue cross shown in (b).

Figure 2

(a) Schematic diagram of the phase behavior of disordered fibrous networks. The curve ${\gamma }_c\left(z\right)$ is the boundary between floppy and rigid states. (b) ${\gamma }_c$ versus average connectivity for phantom triangular networks in 2D and FCC lattice-based 3D networks. The critical strain decreases with increasing connectivity and approaches zero at the isostatic threshold $\langle z\rangle =2d$. The shaded region spans connectivities in the range $3.0\text{–}3.6$. The two open symbols correspond to ${\gamma }_c$ of collagen networks prepared at 4 mg/ml for two different polymerization temperatures. The symbols show average of three samples and error bars represent standard deviations. Per sample at least 100 junctions were measured to determine $\langle z\rangle$.

Figure 3

Shear stiffness $\mathcal{K}$ versus $\mathrm{\Delta }\gamma =\gamma -{\gamma }_c$ for different $\tilde{\kappa }$ obtained from simulations on a phantom triangular network in 2D with $\langle z\rangle \simeq 3.4$. In the limit of $\tilde{\kappa }\to 0$, the stiffness $K$ increases as a power-law in $\mathrm{\Delta }\gamma$ with the critical exponent $f$, which for the given network is $\simeq 0.77$.

Figure 4

(a) Collapse of shear stiffness versus shear strain curves of Fig. 1 according to Eq. (3). Simulation data from 3D network with same connectivity as in 2D of $\langle z\rangle \simeq 3.4$ collapse with the same critical exponents $f=0.8$ and $\varphi =2.1$.

Figure 5

Divergent fluctuations at the critical strain. (a) Average bending angle $\langle {\theta }^2\rangle$ obtained from simulations on a phantom triangular network in 2D with $\langle z\rangle \simeq 3.4$ for different values of $\tilde{\kappa }$ (see legend). The network size is $W^2={250}^2$. The thick black line indicates the expected small-strain ${\gamma }^2$ scaling. $\langle {\theta }^2\rangle$ increases monotonically with $\gamma$. The shaded region is approximately in the range ${\gamma }_c\text{–}{\gamma }_0$. In this range, the rate of increase of $\langle {\theta }^2\rangle$ is strongly dependent on $\tilde{\kappa }$. (b) ${\mathrm{\Gamma }}_{\theta }\left(\gamma \right)$ obtained as the derivative of data in (a) with respect to $\gamma$. In the limit of $\tilde{\kappa }\to 0,{\mathrm{\Gamma }}_{\theta }$ diverges at $\gamma ={\gamma }_c$. (c) ${\mathrm{\Gamma }}_{\theta }$ versus $\gamma$ for different system sizes (see legend). The bending rigidity is $\tilde{\kappa }={10}^{-7}$. (d) Collapse of data in (c) according to Eq. (5) with $\lambda =0.6\pm 0.1$ and $\nu =2.0\pm 0.1$.

Figure 6

Nonaffine displacements in a 2D phantom triangular network with $\langle z\rangle \simeq 3.4$ and $\tilde{\kappa }={10}^{-6}$ are shown as the network is deformed through the critical strain ${\gamma }_c$. The arrows indicate the deviation of a node from the imposed deformation. The magnitude of the vectorial displacements is largest at the critical strain. The color bar on the right indicates the elastic energy in bending (green) or stretching (red) form. The network size is $W=20l_0$.

Figure 7

The onset strain for stiffening scales as ${\gamma }_0\sim {\gamma }_c^{(\varphi -f)}$. The critical exponents are $\varphi =2.1$ and $f=0.8$. The experimental data are taken from collagen networks prepared at temperatures, $T={30}^{\circ }\mathrm{C} (\circ )$ and ${37}^{\circ }\mathrm{C} (\square )$. This scaling is a direct consequence of the measured $c^2$ scaling of the shear stress at ${\gamma }_0$ as shown in the inset.

Figure 8

Shear stiffness versus shear strain curves collapsed according to the Eq. (3) for phantom triangular networks in 2D prepared at different connectivities (see legend). The red and the blue data sets have been shifted by a decade up and down, respectively, for better visualization. The exponent $f$ changes significantly with $\langle z\rangle$. With $\varphi$ showing practically no dependence on the connectivity, the ratio $f/\varphi$ increases with the connectivity as shown in the inset.