  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 ten 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, stretching-stabilized,
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
mathematically similar dependence of material stiffness upon each hierarchical
level. Using this model, we find that the stiffness of such networks becomes
less sensitive to errors in assembly with additional levels of hierarchy;
though surprising, we show that this result is analytically predictable from
first principles, and thus likely 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.
