Nonaffine behavior of three-dimensional semiflexible polymer networks

Three-dimensional semiflexible polymer networks are the structural building blocks of various biological and structural materials. Previous studies have primarily used two-dimensional models for understanding the behavior of these networks. In this paper, we develop a three-dimensional nonaffinity measure capable of providing direct comparison with continuum level homogenized quantities, i.e., strain field. The proposed nonaffinity measure is capable of capturing possible anisotropic microstructures of the filamentous networks. This strain-based nonaffinity measure is used to probe the mechanical behavior at different length scales and investigate the effects of network mechanical and microstructural properties. Specifically, it is found that although all nonaffinity measure components have a power-law variation with the probing length scale, the degree of nonaffinity decreases with increasing the length scale of observation. Furthermore, the amount of nonaffinity is a function of network fiber density, bending stiffness of the constituent filaments, and the network architecture. Finally, it is found that the two power-law scaling regimes previously reported for two-dimensional systems do not appear in three-dimensional networks. Also, unlike two-dimensional models, the exponent of the power-law relation depends weakly on the density of the three-dimensional networks.

Figure 1

The normalized nonaffinity components ${\eta }_{jk}, j,k=1...3$ as a function of normalized probing length scale for networks with fiber number density $N=200$ and $l_b/L_0=0.001$ when subjected to uniaxial tensile strain ${\varepsilon }^{\mathrm{ap}}={\varepsilon }_{11}$.

Figure 2

The first component of the normalized nonaffinity components ${\eta }_{11}$ as a function of the normalized probing length scale for networks composed of fibers with different flexibility parameters $l_b/L_0$ and $N=200$ when subjected to uniaxial tensile strain ${\varepsilon }^{\mathrm{ap}}={\varepsilon }_{11}$.

Figure 3

The first component of the normalized nonaffinity components ${\eta }_{11}$ as a function of the normalized probing length scale for networks with $l_b/L_0=0.01$ and different fiber number densities $N$ when subjected to uniaxial tensile strain ${\varepsilon }^{\mathrm{ap}}={\varepsilon }_{11}$. The inset shows the variation in the exponent of the power-law scaling as a function of $N$.

Figure 4

The first component of nonaffinity ${\eta }_{11}$ as a function of the normalized probing length scale for two-dimensional and three-dimensional networks with $l_b/L_0=0.01$ and different fiber number densities $N$ when subjected to uniaxial tensile strain ${\varepsilon }^{\mathrm{ap}}={\varepsilon }_{11}$. It is seen that the scaling regime with a small exponent does not appear in three-dimensional networks. Moreover, at a given fiber number density, three-dimensional systems show significantly more nonaffine behavior than the respective two-dimensional networks.

Figure 5

The influence of filament flexibility $l_b/L_0$ on the ratio of nonaffine and affine total energy $U^{\mathrm{nonaffine}}/U^{\mathrm{affine}}$ of the three-dimensional networks with different fiber number densities.

Figure 6

The influence of fiber network microstructure on the first $\left({\eta }_{11}\right)$ and third $\left({\eta }_{33}\right)$ components of the normalized nonaffinity measure for networks with $l_b/L_0=0.01$ and $N=200$ when subjected to uniaxial tensile strain ${\varepsilon }^{\mathrm{ap}}={\varepsilon }_{11}$. Three types of network architecture are considered: random, semirandom, and preferentially oriented. The representative microstructure of (a) random networks and (b) preferentially oriented networks is shown. (c) As the networks become preferentially oriented the nonaffinity increases but the exponent of the power-law relation remains unaltered. The ratio of the first $\left({\eta }_{11}\right)$ and third $\left({\eta }_{33}\right)$ nonaffinity components is related to the degree of anisotropy as discussed in the text.