Micromechanical model for protein materials: From macromolecules to macroscopic fibers

We propose a model for the mechanical behavior of protein materials. Based on a limited number of experimental macromolecular parameters (persistence and contour length) we obtain the macroscopic behavior of keratin fibers (human, cow, and rabbit hair), taking into account the damage and residual stretches effects that are fundamental in many functions of life. We also show the capability of our approach to describe the main dissipation and permanent strain effects observed in the more complex spider silk fibers. The comparison between our results and the data obtained experimentally from cyclic tests demonstrates that our model is robust and is able to reproduce with a remarkable accuracy the experimental behavior of all protein materials we tested.

Figure 1

Multiscale behavior of protein materials like human hair under stretching. When a hair is stretched, the $\alpha$-helix structures undergo a coiled-coil transition to an almost straight polypeptide chain in the form of $\beta$ sheets.

Figure 2

(a) Scheme of second structure unfolding during an AFM molecule stretching and (b) corresponding force-elongation curve (experimental behavior reproduced from Ref. [34]). The sawtooth shape is due to unfolding of an increasing number of domains with (c) reporting the corresponding total energy minimization approach (see Ref. [11]). (d) Continuum limit approximation with constant unfolding force plateaux.

Figure 3

(a) Cartoon showing the deformation induced chain unfolding, where the total stretch $\lambda$ is given by the product of the elastic stretch ${\lambda }_e$ and the permanent stretch ${\lambda }_n$. (b) Micro-macro multiscale network.

Figure 4

Cyclic tests on human (a), rabbit (b), and cow (c) hair at room temperature and at a strain rate of $0.1{\mathrm{s}}^{-1}$ (experimental behavior-continuous lines, theoretical model-dashed lines). For the theoretical model we used in (a) for human hair $l_p=0.4\mathrm{nm}$, $T=300\mathrm{K}$, $q=Q/l_c=45k_{B}T/\mathrm{nm}$, $\mu =44\mathrm{MPa}$, $p=-130\mathrm{MPa}$, $N^{\text{fib}}={10}^{18}$; in (b) for rabbit hair $l_p=0.45\mathrm{nm}$; $T=300\mathrm{K}$, $q=Q/l_c=180k_{B}T/\mathrm{nm}$, $\mu =280\mathrm{MPa}$, $p=-375\mathrm{MPa}$, $N^{\text{fib}}={10}^{18}$; in (c) for cow hair $l_p=0.33\mathrm{nm}$; $T=300\mathrm{K}$, $q=Q/l_c=45k_{B}T/\mathrm{nm}$, $\mu =200\mathrm{MPa}$, $p=-130\mathrm{MPa}$, $N^{\text{fib}}={10}^{18}$.

Figure 5

The dragline silk sample underwent loading and unloading cycles tested through a nanotensile machine (Agilent UTM 150) at room temperature and at a strain rate of $0.1{\mathrm{s}}^{-1}$ (experimental behavior-continuous lines, theoretical model-dashed lines). Here we assumed $l_p=0.78\mathrm{nm}$, $l_c=25\mathrm{nm}$, $N^{\text{fib}}=4.75\times {10}^{18}{\mathrm{m}}^{-2}$, $Q=550k_{B}T$, $T=300\mathrm{K}$, $\mu =670\mathrm{MPa}$, $p=-400\mathrm{MPa}$.