Criticality Enhances the Reinforcement of Disordered Networks by Rigid Inclusions

The mechanical properties of biological materials are spatially heterogeneous. Typical tissues are made up of a spanning fibrous extracellular matrix in which various inclusions, such as living cells, are embedded. While the influence of embedded inclusions on the stiffness of common elastic materials such as rubber has been studied for decades and can be understood in terms of the volume fraction and shape of inclusions, the same is not true for disordered filamentous and fibrous networks. Recent work has shown that, in isolation, such networks exhibit unusual viscoelastic behavior indicative of an underlying mechanical phase transition controlled by network connectivity and strain. How this behavior is modified when inclusions are present is unclear. Here, we present a theoretical and computational study of the influence of rigid inclusions on the mechanics of disordered elastic networks near the connectivity-controlled central-force rigidity transition. Combining scaling theory and coarse-grained simulations, we predict and confirm an anomalously strong dependence of the composite stiffness on inclusion volume fraction, beyond that seen in ordinary composites. This stiffening exceeds the well-established volume-fraction-dependent stiffening expected in conventional composites, e.g., as an elastic analog of the classic volume-fraction-dependent increase in the viscosity of liquids first identified by Einstein. We show that this enhancement is a consequence of the interplay between interparticle spacing and an emergent correlation length, leading to an effective finite-size scaling imposed by the presence of inclusions. We outline the expected scaling of the linear shear modulus and strain fluctuations with the inclusion volume fraction and network connectivity, confirm these predictions in simulations, and discuss potential experimental tests and implications for our predictions in real systems.