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In composite materials composed of soft polymer matrix and stiff,high-aspect-ratio particles, the composite undergoes a transition in mechanicalstrength when the inclusion phase surpasses a critical density. This phenomenon(rheological or mechanical percolation) is well-known to occur in manycomposites at a critical density that exceeds the conductivity percolationthreshold. Conductivity percolation occurs as a consequence of contactpercolation, which refers to the conducting particles' formation of a connectedcomponent that spans the composite. Rheological percolation, however, hasevaded a complete theoretical explanation and predictive description. A naturalhypothesis is that rheological percolation arises due to rigidity percolation,whereby a rigid component of inclusions spans the composite. We modelcomposites as random isotropic dispersions of soft-core rods, and studyrigidity percolation in such systems. Building on previous results fortwo-dimensional systems, we develop an approximate algorithm that identifiesspanning rigid components through iteratively identifying and compressingprovably rigid motifs -- equivalently, decomposing giant rigid components intorigid assemblies of successively smaller rigid components. We apply thisalgorithm to random rod systems to estimate a rigidity percolation thresholdand explore its dependence on rod aspect ratio. We show that this transitionpoint, like the contact percolation transition point, scales inversely with theaverage (aspect ratio-dependent) rod excluded volume. However, the scaling ofthe rigidity percolation threshold, unlike the contact percolation scaling, isvalid for relatively low aspect ratio. Moreover, the critical rod contactnumber is constant for aspect ratio above some relatively low value; and liesbelow the prediction from Maxwell's isostatic condition.

Rigidity percolation in random 3D rod systems