Structural hierarchy confers error tolerance in biological tissues

We present a novel approach to understand geometric-incompatibility-induced rigidity in under-constrained materials, including sub-isostatic 2D spring networks and 2D and 3D vertex models for dense biological tissues. We show that in all these models a geometric criterion, represented by a minimal length $\bar\ell_\mathrm{min}$, determines the onset of prestresses and rigidity. This allows us to predict not only the correct scalings for the elastic material properties, but al...

Disordered elastic networks provide a framework for describing a wide variety of physical systems, ranging from amorphous solids, through polymeric fibrous materials to confluent cell tissues. In many cases, such networks feature two widely separated rigidity scales and are nearly floppy, yet they undergo a dramatic stiffening transition when driven to sufficiently large strains. We present a complete scaling theory of the critical strain-stiffened state in terms of the small...

Many fibrous materials are modeled as elastic networks featuring a substantial separation between the stiffness scales that characterize different microscopic deformation modes of the network's constituents. This scale separation has been shown to give rise to emergent complexity in these systems' linear and nonlinear mechanical response. Here we study numerically a simple model featuring said stiffness scale-separation in two-dimensions and show that its mechanical response ...

In this letter a mathematical model to design nano-bio-inspired hierarchical materials is proposed. An optimization procedure is also presented. Simple formulas describing the dependence of strength, fracture toughness and stiffness on the considered size-scale are derived, taking into account the toughening biomechanisms. Furthermore, regarding nano-grained materials the optimal grain size is deduced: incidentally, it explains and quantitatively predicts the deviation from t...

The mechanical properties of biopolymer networks depend on their mean coordination number and stress state. Models based on spring networks have been shown to capture some properties of materials such as the actin cortex, but the effect of prestress combined with filament pruning, known to exist in the cortex, has not been studied. We show that in central-force spring networks below isostatic coordination that are rigidified by prestress, details of the pruning method signifi...

Designing strong and robust bio-inspired structures requires an understanding of how function arises from the architecture and geometry of materials found in nature. We draw from trabecular bone, a lightweight bone tissue that exhibits a complex, anisotropic microarchitecture, to generate networked structures using multi-objective topology optimization. Starting from an identical volume, we generate multiple different models by varying the objective weights for compliance, su...

Many biological tissues feature a heterogeneous network of fibers whose tensile and bending rigidity contribute substantially to these tissues' elastic properties. Rigidity percolation has emerged as a important paradigm for relating these filamentous tissues' mechanics to the concentrations of their constituents. Past studies have generally considered tuning of networks by spatially homogeneous variation in concentration, while ignoring structural correlation. We here introd...

Signatures of glassy dynamics have been identified experimentally for a rich variety of materials in which molecular networks provide rigidity. Here we present a theoretical framework to study the glassy behavior of both passive and active network materials. We construct a general microscopic network model that incorporates nonlinear elasticity of individual filaments and steric constraints due to crowding. Based on constructive analogies between structural glass forming liqu...

Reconstituted filamentous actin networks with myosin motor proteins form active gels, in which motor proteins generate forces that drive the network far from equilibrium. This motor activity can also strongly affect the network elasticity; experiments have shown a dramatic stiffening in in vitro networks with molecular motors. Here we study the effects of motor generated forces on the mechanics of simulated 2D networks of athermal stiff filaments. We show how heterogeneous in...

Mechanical metamaterials are periodic lattice structures with complex unit cell architectures that can achieve extraordinary mechanical properties beyond the capability of bulk materials. A new class of metamaterials is proposed, whose mechanical properties rely on deformation-induced transitions in nodal-topology by formation of internal self-contact. The universal nature of the principle presented, is demonstrated for tension, compression, shear and torsion. In particular, ...