Structural hierarchy confers error tolerance in biological materials

Abstract

Structural hierarchy, in which materials possess distinct features on multiple length scales, is ubiquitous in nature. Diverse biological materials, such as bone, cellulose, and muscle, have as many as 10 hierarchical levels. Structural hierarchy confers many mechanical advantages, including improved toughness and economy of material. However, it also presents a problem: Each hierarchical level adds a new source of assembly errors and substantially increases the information required for proper assembly. This seems to conflict with the prevalence of naturally occurring hierarchical structures, suggesting that a common mechanical source of hierarchical robustness may exist. However, our ability to identify such a unifying phenomenon is limited by the lack of a general mechanical framework for structures exhibiting organization on disparate length scales. Here, we use simulations to substantiate a generalized model for the tensile stiffness of hierarchical filamentous networks with a nested, dilute triangular lattice structure. Following seminal work by Maxwell and others on criteria for stiff frames, we extend the concept of connectivity in network mechanics and find a similar dependence of material stiffness upon each hierarchical level. Using this model, we find that stiffness becomes less sensitive to errors in assembly with additional levels of hierarchy; although surprising, we show that this result is analytically predictable from first principles and thus potentially model independent. More broadly, this work helps account for the success of hierarchical, filamentous materials in biology and materials design and offers a heuristic for ensuring that desired material properties are achieved within the required tolerance.

Keywords: biophysics; evolution of biomaterials; network mechanics; soft matter; structural hierarchy.

Fig. 1.

(A) A dilute triangular lattice with one level of structure. Missing bonds are indicated with dashed lines. (B) A two-level triangular lattice, with each bond replaced by a smaller-scale, dilute triangular lattice. (C) An extension to three levels of structural hierarchy.

Fig. 2.

(A) Simulated stiffness plotted vs. bond portion for one length scale. (B) Heat map of simulated stiffness as a function of small and large bond portion for a two-level network. Stiffness is plotted in simulation units, as indicated in the key. (C) Heat map of simulated stiffness for a slice in bond portion space for three-level networks with full connectivity on the largest scale. Stiffness is plotted in simulation units, as indicated in the key. (D) Heat map of simulated stiffness for a slice in bond portion space for three-level networks with full connectivity on the smallest scale. Stiffness is plotted in simulation units, as indicated in the key. (E) Stiffness normalized by maximum stiffness for networks with one, two, and three levels of structure.

Fig. 3.

(A) Stiffness probability distribution functions (PDFs) estimated from histogram data for networks with one, two, and three levels of structural hierarchy. Points in 1D, 2D, and 3D bond portion with the same minute nominal stiffness were chosen, and Gaussian random variables were added to each bond portion. Note the spike in the PDF for the one-level network at zero stiffness. In this case, we consider a nominal stiffness of 0.025 and Gaussian random noise for each bond portion with zero mean and a SD of 0.005. Other cases are addressed in SI Appendix. (B) Relative error in stiffness vs. levels of hierarchy is plotted for $\overline{K}=0.001$, $\sigma =0.0001$. As additional levels of structural hierarchy are added, the relative error in the tensile stiffness decreases precipitously at first, and the effect saturates at a certain number of levels. Provided the assumptions leading to Eq. 9 hold, our analytical theory and numerical approach are in close agreement ($r^2\approx 1$). Here, the product of excess bond portions is 0.001, and the noise has amplitude 0.0001 on each scale. (C) The number of all small-scale bonds in a network, divided by the area enclosed by the outer perimeter of the network, is shown vs. number of hierarchical levels for stiffness values of 0.02, 0.03, and 0.04, in units of stretching modulus over small-scale bond length. Networks were chosen to have the same bond portion on all three levels. In each case, increasing hierarchy leads to markedly lower density of small-scale bonds, attesting to the ability of hierarchy to confer both robustness and efficiency.