  In composite materials composed of soft polymer matrix and stiff,
high-aspect-ratio particles, the composite undergoes a transition in mechanical
strength when the inclusion phase surpasses a critical density. This phenomenon
(rheological or mechanical percolation) is well-known to occur in many
composites at a critical density that exceeds the conductivity percolation
threshold. Conductivity percolation occurs as a consequence of contact
percolation, which refers to the conducting particles' formation of a connected
component that spans the composite. Rheological percolation, however, has
evaded a complete theoretical explanation and predictive description. A natural
hypothesis is that rheological percolation arises due to rigidity percolation,
whereby a rigid component of inclusions spans the composite. We model
composites as random isotropic dispersions of soft-core rods, and study
rigidity percolation in such systems. Building on previous results for
two-dimensional systems, we develop an approximate algorithm that identifies
spanning rigid components through iteratively identifying and compressing
provably rigid motifs -- equivalently, decomposing giant rigid components into
rigid assemblies of successively smaller rigid components. We apply this
algorithm to random rod systems to estimate a rigidity percolation threshold
and explore its dependence on rod aspect ratio. We show that this transition
point, like the contact percolation transition point, scales inversely with the
average (aspect ratio-dependent) rod excluded volume. However, the scaling of
the rigidity percolation threshold, unlike the contact percolation scaling, is
valid for relatively low aspect ratio. Moreover, the critical rod contact
number is constant for aspect ratio above some relatively low value; and lies
below the prediction from Maxwell's isostatic condition.
