Effective-medium theory of a filamentous triangular lattice

We present an effective-medium theory that includes bending as well as stretching forces, and we use it to calculate the mechanical response of a diluted filamentous triangular lattice. In this lattice, bonds are central-force springs, and there are bending forces between neighboring bonds on the same filament. We investigate the diluted lattice in which each bond is present with a probability $p$. We find a rigidity threshold $p_{\mathrm{b}}$ which has the same value for all positive bending rigidity and a crossover characterizing bending, stretching, and bend-stretch coupled elastic regimes controlled by the central-force rigidity percolation point at $p_{\mathrm{CF}}\simeq 2/3$ of the lattice when fiber bending rigidity vanishes.

Figure 1

Phase diagram of the diluted filamentous triangular lattice showing the central-force and bending rigidity thresholds, respectively, at $p=p_{\mathrm{CF}}$ and $p=p_{\mathrm{b}}$, the bending-dominated regime at small $\kappa$ in the vicinity of $p_{\mathrm{b}}$, the crossover bend-stretch regime near $p_{\mathrm{CF}}$, and the stretching-dominated regime at large $\kappa$. (Adapted from Ref. [35].)

Figure 2

Schematic of a filament of length $5a$ divided into 5 segments of length $a$ (a) in the equilibrium configuration and (b) in a distorted configuration. Circles mark lattice nodes (located as positions $ia$), and crosses mark the centers of bonds located at positions $[i-(1/2\left)\right]a$. The different colors of the bonds indicate different values for the stretching and bending moduli. The angle of the bonds $i-1$ and $i$ are indicated in (b). In the limit of slow changes in ${\theta }_i$, the slope of the $h\left(s\right)\equiv u^{\perp }\left(s\right)$ is constant in bond $i$, and bond angle $i$ is the angle of the line connecting site $i-1$ with site $i$ for small ${\theta }_i$.

Figure 3

Filamentous triangular lattice with bonds randomly occupied with probability $p$. The unit vectors ${\mathbf{e}}_1$, ${\mathbf{e}}_2$, ${\mathbf{e}}_3$ are marked by “1, 2, 3,” the 3 stretch energy vectors ${\mathbf{B}}_n^s$ are marked by the 3 red single lines, and the 3 bending energy vectors ${\mathbf{B}}_n^b$ are marked by the 3 green double lines. The purple dashed double line marks the bending vector ${\mathbf{B}}_4^b$ if the origin is marked by 0.

Figure 4

Positions of sites ${\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4$, and interactions in the effective medium, including NN stretching term of rigidity ${\mu }_m$, NNN bending term of rigidity ${\kappa }_m$, and third neighbor effective coupling term of rigidity ${\lambda }_m$.

Figure 5

(a) Semilog plot for the EMT solution ${\mu }_m/\mu =G/G_0$ as a function of $p$ for $\mu =1$ and $\kappa =1,{10}^{-1},{10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5},{10}^{-6}$ from top to bottom, as indicated in the legend. (b) Linear plot of ${\mu }_m/\mu$ as a function of $p$, with parameters and color code the same as in (a). The red solid line indicates ${\mu }_m$ for the case of a central force triangular lattice ($\kappa =0$). Here and in Figs. 6 to 8, we have set $a=1$. The contents of (a) appeared in a different form in Ref. [35].

Figure 6

The EMT solution for ${\kappa }_m$ expressed in terms of the dimensionless combination ${\widehat{\kappa }}_m={\kappa }_m/\left(\mu a^2\right)$ (a) and ${\lambda }_m$ expressed in terms of the dimensionless combination ${\widehat{\lambda }}_m={\lambda }_m/\left(\mu a^2\right)$ (b). Parameters and color code are the same as in Fig. 5.

Figure 7

Asymptotic solution (dashed lines) and numerical solutions (data points) of ${\mu }_m/\mu =G/G_0$ near $p_{\mathrm{b}}$. Parameters and color code are the same as in Fig. 5.

Figure 8

(a), (b), and (c) show rescaled plots of the EMT solutions ${\mu }_m/\mu$, ${\kappa }_m$, and ${\lambda }_m$ using the scaling forms (3.39a) for $\mu =1$ and $\kappa ={10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5},{10}^{-6}$, with color code the same as in Fig. 5, and exponents taking the value as in Eq. (3.40a). The thin black lines represent the asymptotic forms of Eq. (3.41a) for small $\kappa$. The brown dash-dotted lines, the thick blue solid lines, and the purple dashed lines plot the functional form of ${\mu }_m$ obtained in the crossover, the stretching-dominated, and the bending-dominated regimes of Eq. (3.42), respectively. The contents of (a) appeared in a different form in Ref. [35].

Figure 9

Comparison between the shear modulus (normalized by the shear modulus at $p=1$, so it is equivalent to ${\mu }_m/\mu$) obtained from numerical simulations from Ref. [35] (data points), Das's EMT (solid lines), and our EMT (dashed lines). Different colors mark different values of $\kappa$ with the same color code as in Fig. 5, and from top to bottom the corresponding values of $\kappa$ are 1, ${10}^{-1},{10}^{-2},...,{10}^{-6}$.