Strain-stiffening critical exponents of fiber networks under uniaxial deformation

Abstract

Disordered fiber networks exhibit a floppy to rigid mechanical phase transition as a function of connectivity. Sub-isostatically connected networks can undergo this transition via straining. Critical exponents governing this transition have been estimated theoretically and by numerical simulations of various types of networks. In this study, we present improved results, achieved through a combination of refined numerical simulations, larger system sizes and incorporation of theoretical predictions for better post-simulation analysis. We also report the evolution of the critical strain and critical exponents as the network is sheared while being subjected to non-volume-preserving uniaxial deformations.

Figures (6)

• Figure 1: Sub-isostatic networks undergo a floppy to rigid transition at a critical strain $\gamma_c(z, \epsilon)$. Orange line shows the pure shear $\gamma_c(z)$ at which strain-stiffening occurs. Upon application of a uniaxial deformation $\epsilon$, the critical strain line shifts (red for extension and yellow for compression). Extending the network to $\epsilon=\epsilon_c$, the network is rigid even at zero shear (pink). For $z \le 2$, the network is not connected and can never become rigid. For $z \ge 2d$ (green), networks are always rigid.
• Figure 2: Phantomised 2D triangular lattices diluted to sub-isostatic connectivities and strained. (a) $z_c=4.0$ at zero strain. (b) $z=3.6$ under pure shear $\gamma$. (c) $z=3.2$ under uniaxial deformation $\epsilon= (L-L_0)/L_0$.
• Figure 3: Differential shear modulus $K$ vs shear strain $\gamma$ with varying reduced bending modulus $\tilde{\kappa}$. (a) Dotted line shows the critical strain $\gamma_c$, where the bending dominated regime with $K \sim \tilde{\kappa}$ crosses over into the stretching dominated regime. (b) Earlier onset of rigidity as connectivity approaches $z_c=4$. (c) Networks under deformation show behavior similar to networks with different $z$. Compression $(\epsilon<0)$ delays criticality (resembling lower $z$) while extension $(\epsilon>0)$ advances it (resembling higher $z$). (d) Finite size effects do not shift the critical strain significantly. Initial softening of the modulus (buckling) is more prominent in smaller systems.
• Figure 4: Scaling collapse due to Widom-scaling (Eq. \ref{['eq:widom']}) with $\lambda=3/2$. (a) Three branches represent the different regimes of the rigidity transition (Eq. \ref{['eq:Gdef']}). (b) Widom-scaling across different connectivities at zero deformation. (c) Widom-scaling under compression and extension for $z = 3.2$. (d) Widom-scaling across finite size systems.
• Figure 5: Near $\gamma_c$, non-affinity scales as $d\Gamma \sim |\Delta\gamma|^{-\lambda}$, in the limit $\kappa \to 0^+$. (a) $d\Gamma$ vs $\Delta \gamma = \gamma - \gamma_c$ for $\kappa=10^{-6}$ across systems sizes. Dotted line shows the slope $-3/2$. Inset: suppression of non-affine fluctuations due to finite size effects. Maximum of $\delta \Gamma$ scales as max$(\delta \Gamma) \sim W^{\lambda/\nu}$ with $\nu = 1.4 \pm 0.1$. (b) Finite-size scaling using $\lambda=3/2$ and $\nu = 1.4 \pm 0.1$, for networks with $z=3.2$ and $\kappa=10^{-6}$. (c) Finite-size scaling using $K-K_c$ for $\tilde{\kappa}=0$. For central force networks, the shear modulus $K$ exhibits a discontinous jump $K_c$ at $\gamma_c$.