  We propose a new metric called s-LID based on the concept of Local Intrinsic
Dimensionality to identify and quantify hierarchies of kinematic patterns in
heterogeneous media. s-LID measures how outlying a grain's motion is relative
to its s nearest neighbors in displacement state space. To demonstrate the
merits of s-LID over the conventional measure of strain, we apply it to data on
individual grain motions in a set of deforming granular materials. Several new
insights into the evolution of failure are uncovered. First, s-LID reveals a
hierarchy of concurrent deformation bands that prevails throughout loading
history. These structures vary not only in relative dominance but also spatial
and kinematic scales. Second, in the nascent stages of the pre-failure regime,
s-LID uncovers a set of system-spanning, criss-crossing bands: microbands for
small s and embryonic-shearbands at large s, with the former being dominant. At
the opposite extreme, in the failure regime, fully formed shearbands at large s
dominate over the microbands. The novel patterns uncovered from s-LID
contradict the common belief of a causal sequence where a subset of microbands
coalesce and/or grow to form shearbands. Instead, s-LID suggests that the
deformation of the sample in the lead-up to failure is governed by a complex
symbiosis among these different coexisting structures, which amplifies and
promotes the progressive dominance of the embryonic-shearbands over microbands.
Third, we probed this transition from the microband-dominated regime to the
shearband-dominated regime by systematically suppressing grain rotations. We
found particle rotation to be an essential enabler of the transition to the
shearband-dominated regime. When grain rotations are completely suppressed,
this transition is prevented: microbands and shearbands coexist in relative
parity.
