Criticality enhances the reinforcement of disordered networks by rigid inclusions

TL;DR

This study reveals that rigid inclusions significantly enhance the stiffness of disordered elastic networks near the rigidity transition, with effects surpassing traditional composite theories due to emergent correlations.

Contribution

The paper introduces a combined theoretical and computational framework to analyze how rigid inclusions influence the mechanics of disordered networks near criticality, revealing anomalous stiffening effects.

Findings

Inclusion volume fraction strongly affects composite stiffness.

Enhanced stiffening exceeds classical composite predictions.

Scaling laws for shear modulus and fluctuations are confirmed.

Abstract

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 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 analogue 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 inter-particle 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 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.