Two fundamental mechanisms govern the stiffening of cross-linked networks

Abstract

Biopolymer networks, such as those constituting the cytoskeleton of a cell or biological tissue, exhibit a nonlinear strain-stiffening behavior when subjected to large deformations. Interestingly, rheological experiments on various in vitro biopolymer networks have shown similar strain-stiffening trends regardless of the differences in their microstructure or constituents, suggesting a universal stiffening mechanism. In this article, we use computer simulations of a random network comprised of cross-linked biopolymer-like fibers to substantiate the notion that this universality lies in the existence of two fundamental stiffening mechanisms. After showing that the large strain response is accompanied by the development of a stress path, i.e., a percolating path of axially stressed fibers and cross-links, we demonstrate that the strain stiffening can be caused by two distinctly different mechanisms: 1) the pulling out of stress-path undulations; and 2) reorientation of the stress path. The former mechanism is bending-dominated and can be recognized by a power-law dependence with exponent 3/2 of the shear modulus on stress, whereas the latter mechanism is stretching-dominated and characterized by a power-law exponent 1/2. We demonstrate how material properties of the constituents, as well as the network microstructure, can affect the transition between the two stiffening mechanisms and, as such, control the dominant power-law scaling behavior.

Figure 1

The nonlinear behavior of biopolymer networks plotted as the shear modulus, G, versus stress, T, or the shear modulus normalized by the initial shear modulus, G/G₀, versus stress normalized by the critical stress at the onset of nonlinearity, T/T_c, for (A) F-actin/scruin networks (8); (B) F-actin/rigor-HMM networks with actin concentration 19 μM, mean F-actin length 21 μm, and various increasing molar HMM/actin concentration ratios, $R=c_{\text{HMM}}/c_{\text{a}}$ (arrow) (13); (C) F-actin/filamin networks for lower F-actin concentrations (15); and (D) vimentin networks (39). To see this figure in color, go online.

Figure 2

The cross-linked network model. (A) Example of a generated RVE with fibers shown in blue and cross-links in red. (B) A cross-link is a two-node element whose behavior is controlled by four independent spring constants. The cross-link is idealized by coupling the four spring constants to each other, $s=s_1=s_3=s_2/l_{\text{c}}^2=s_4/l_{\text{c}}^2$, where $l_{\text{c}}$ is the mean length of the fiber sections. The nondimensional system parameter $\tilde{s}=s{l}_{\text{c}}^3/\kappa$ used in our analysis, compares the cross-link stiffness to the fiber bending stiffness. To see this figure in color, go online.

Figure 3

The effect of cross-link number $\tilde{s}$ of the networks with constant connectivity ${\tilde{n}}_{\text{X}}\approx 0.34$, $l_{\text{c}}\approx 0.86\mu \text{m}$. (A) The ensemble-averaged small strain response as a function of cross-link number $\tilde{s}$. The initial shear modulus, G₀, is normalized by the initial network shear modulus in the rigidly cross-linked limit $G_0^{\infty }$. The standard deviation is represented by the gray region. (B and C) The ensemble-averaged shear modulus, $\tilde{G}=G/G_0$, as a function of strain, Γ, and normalized stress, $\tilde{\text{T}}=\text{T}/{\text{T}}_{\text{c}}$, respectively, for selected $\tilde{s}$ values, as indicated by the symbols in (A). The response of the RCL networks is shown in blue. Standard deviations (up to about twice the symbol size) are not shown. (D–G) View of the deformed network at stress states indicated by the letters in (B) and (C) for the RCL case at $\tilde{\text{T}}\approx 1$ (D), $\tilde{\text{T}}\approx 20$ (E), and $\tilde{\text{T}}\approx 400$ (F) and for $\tilde{s}={10}^4$ at $\tilde{\text{T}}\approx 500$ (G). Filaments under tension/compression are shown in blue/light blue and cross-links in red. The thickness of a constituent is taken to be linearly proportional to its axial force normalized by the maximum axial force in the network, to help identify stress paths. To see this figure in color, go online.

Figure 4

Simple shear of the box containing a percolating stress path comprised of fiber constituents (blue) and cross-links (red). The box at the critical strain, Γ_c, is shown in magenta. Subsequent shearing by γ is accomplished by extension of the stress path. To see this figure in color, go online.

Figure 5

The effect of connectivity, ${\tilde{n}}_{\text{X}}$. The ensemble-averaged responses $G\left(\text{Γ}\right)$(A), $\tilde{G}\left(\text{T}\right)$(B), and $\tilde{G}\left(\tilde{\text{T}}\right)$ (C) of networks for cross-link spring constants $s\to \infty$ (blue) and $s\approx 4\times {10}^{-5}\text{N}/\text{m}$ (green) and connectivity ${\tilde{n}}_{\text{X}}\approx 0.12$, $l_{\text{c}}\approx 1.2\mu \text{m}$ (circles), ${\tilde{n}}_{\text{X}}\approx 0.34$, $l_{\text{c}}\approx 0.86\mu \text{m}$ (triangles), and ${\tilde{n}}_{\text{X}}\approx 0.72$, $l_{\text{c}}\approx 0.34\mu \text{m}$ (squares). Standard deviations (up to about twice the symbol size) are not shown. To see this figure in color, go online.

Figure 6

(A) Normalized initial shear modulus, $G_0/G_{0\text{a}}$, of the RCL network with $l_{\text{c}}=0.86\mu \text{m}$ versus characteristic ratio $l_{\text{b}}/l_{\text{c}}$, where $G_{0\text{a}}$ is the initial shear modulus calculated for the RCL F-actin network with $\mu ={\mu }_{\text{a}}=4\times {10}^{-8}$ N and $\kappa ={\kappa }_{\text{a}}=6.75\times {10}^{-26}$ N/m. The $l_{\text{b}}/l_{\text{c}}$ for the RCL F-actin network is indicated by the star. Larger values of $l_{\text{b}}/l_{\text{c}}$ are obtained either for $\mu <{\mu }_{\text{a}}$ and constant $\kappa ={\kappa }_{\text{a}}$ (magenta solid symbols) or for $\kappa >{\kappa }_{\text{a}}$ and constant $\mu ={\mu }_{\text{a}}$ (blue open symbols). The straight lines indicate the scaling laws of open-cell foams for the two extreme limits 1. (B and C) The large strain response of the RCL network for various $l_{\text{b}}/l_{\text{c}}$ ratios shown in (A), where $G/G_0$ is the network shear modulus scaled by the initial modulus, $G_0$, and $\text{T}/{\text{T}}_{\text{c}}$ is the stress scaled by the stress value at the onset of nonlinearity, ${\text{T}}_{\text{c}}$. To see this figure in color, go online.

Figure 7

View through the deformed network realization showing load-bearing supportive frames ${\tilde{n}}_{\text{X}}\approx 0.72$ for the RCL limit in Fig. 5 (blue square) at $\text{T}\approx 30\text{Pa}$ (A), ${\tilde{n}}_{\text{X}}\approx 0.72$ for $s\approx 4e-5\text{N}/\text{m}$ in Fig. 5 (green square) at $\text{T}\approx 65\text{Pa}$ (B), ${\tilde{n}}_{\text{X}}\approx 0.34$ for $\tilde{s}=10$ in Fig. 3 (red diamond) at $\tilde{\text{T}}\approx 30$ (C). The axial stress map is obtained in the same way as in Fig. 3, D–F. To see this figure in color, go online.

Figure 8

Compilation of all curves of $G/G_0$ versus normalized stress, $\text{T}/{\text{T}}_{\text{c}}$, along with the corresponding values of ${\tilde{l}}_{\text{b}}/l_{\text{c}}$, for the cases in Fig. 3C (where cross-link stiffness is varied at constant connectivity) and Fig. 5C (at constant cross-link stiffness but varying connectivity). To see this figure in color, go online.