Edited by Andrea J. Liu, University of Pennsylvania, Philadelphia, PA, and approved January 3, 2019 (received for review August 15, 2018)
Structural hierarchy is ubiquitous in nature and an emerging trend in engineered materials. Despite their many virtues, hierarchical materials also add complexity and dramatically increase the opportunities for random errors in assembly. Nonetheless, highly hierarchical tissues have evolved many times in diverse lineages; this prevalence suggests a common source of mechanical robustness. In this work, we introduce a tractable, model hierarchical lattice with controllable attributes on each length scale. We find, contrary to intuition, that adding additional levels of structure actually reduces the relative variation in mechanical properties, despite an increase in assembly errors. This finding informs our understanding of the emergence of hierarchically structured biological materials, while also offering a practical heuristic in materials design.
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.
Author contributions: J.A.M. and P.J.Y. designed research; J.A.M. performed research; J.A.M. analyzed data; and J.A.M. and P.J.Y. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1813801116/-/DCSupplemental.
Published under the PNAS license.
