Stiff Polymers, Foams, and Fiber Networks

We study the elasticity of fibrous materials composed of generalized stiff polymers. It is shown that, in contrast to cellular foamlike structures, affine strain fields are generically unstable. Instead, a subtle interplay between the architecture of the network and the elastic properties of its building blocks leads to intriguing mechanical properties with intermediate asymptotic scaling regimes. We present exhaustive numerical studies based on a finite element method complemented by scaling arguments.

Figure 1

Illustration of the different architecture of (a) cellular and (b) fibrous materials. The foam in (a) is constructed by a Voronoi tessellation from the centers of the fibers in (b).

Figure 2

Shear modulus $G{l}^3/\kappa$ as function of persistence length $l_p/l$ for various densities $\rho l$ and $\alpha =1$. The second branch in the upper right corner ($\rho l\ge 120$) is obtained by suppressing the mechanical response (“$k_s\to \infty$”). The dashed line indicates the three regimes as obtained by Eq. (3).

Figure 3

Scaling function $g$ as a function of $\rho l_p$ for various values of $x=\rho l$. For comparison, we present also the scaling function of a random foam (dashed line).

Figure 4

Fraction of energy stored in the various segment lengths; the curves correspond to different persistence lengths at a density of $\rho l=80$, equivalent to $⟨l_s⟩/l\approx 2\times {10}^{-2}$.