  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.
