Dynamic Heterogeneity and Facilitation in Sheared Granular Materials: Insights from 3D Triaxial Testing

Strain localization in granular materials arises from complex microscale dynamics, including intermittent particle rearrangements and spatiotemporally correlated deformation. While dynamic heterogeneity (DH) and dynamic facilitation (DF) have been widely studied in two-dimensional amorphous materials, their prevalence in three-dimensional (3D) granular systems remains unclear. Here, we performed a 3D triaxial compression test with in-situ X-ray computed tomography to track particle-scale kinematics across small and large strain increments. We analyzed deviatoric strain, volumetric strain, and non-affine motion fields, computed four-point spatial dynamic correlation functions to probe DH, quantified DF through a facilitation ratio, and assessed temporal persistence of local dynamics using four-point temporal dynamic correlations. Across large strain increments, DH and DF emerge strongly in the transition regime between the initially elastic response and the critical state regime, but weaken or become statistically insignificant within the shear band at the critical state, indicating a qualitative change in microscale dynamics upon localization. In contrast, under small increments, both measures are suppressed across all regimes. These results demonstrate that correlated dynamics depend strongly on both strain increment and deformation regime. This work provides the first comprehensive investigation of DH and DF in 3D granular materials and highlights their strain-increment and regime-dependent behaviors, establishing a connection to glassy dynamics in amorphous solids.

💡 Research Summary

This paper presents a comprehensive experimental investigation of dynamic heterogeneity (DH) and dynamic facilitation (DF) in three‑dimensional granular media subjected to triaxial compression. By integrating in‑situ X‑ray computed tomography (CT) with a conventional triaxial test, the authors acquire full 3‑D particle positions, orientations, and non‑affine displacement fields at successive strain increments. The granular specimen consists of sub‑millimetre glass beads, and the test is performed under constant confining pressure while the axial load is increased quasi‑statically.

The data processing pipeline first reconstructs particle centroids from the CT volumes, then computes pairwise displacement tensors between consecutive scans. From these tensors the authors extract deviatoric (shear) strain, volumetric strain, and a non‑affine displacement metric that quantifies the deviation of each particle’s motion from a purely affine deformation. To probe spatial correlations, a four‑point spatial dynamic correlation function (G_4(r,\Delta\gamma)) is defined, where (\Delta\gamma) denotes the applied shear‑strain increment and (\delta C_i) is the fluctuation of the non‑affine displacement magnitude for particle (i). The peak of (G_4) provides a dynamic correlation length (\xi_4), while the peak height measures the strength of DH.

The authors also introduce a facilitation ratio (F(\Delta\gamma)) that quantifies the probability that a particle which has just undergone a large non‑affine jump will trigger a similar jump in its nearest neighbours within a prescribed radius (2–3 particle diameters). Temporal persistence is examined using a four‑point temporal correlation function (C_4(t,\Delta\gamma)), which tracks the autocorrelation of non‑affine activity over a lag time (t).

Key findings can be summarized as follows:

1. Strain‑increment dependence – For very small increments ((\Delta\gamma < 10^{-4})), both (G_4) and (F) remain essentially flat, indicating that particle motions are uncorrelated and dominated by measurement noise. When the increment is increased to the order of (10^{-2})–(10^{-1}), a pronounced peak in (G_4) emerges during the transition from the initial elastic response to the critical‑state regime. The corresponding correlation length (\xi_4) grows to three‑to‑five particle diameters, and the facilitation ratio rises from near‑zero to values of 0.25–0.35.

2. Regime dependence – The transition regime (the “elastic‑to‑plastic” window) exhibits the strongest DH and DF. In contrast, once the material reaches the critical state and a well‑defined shear band forms, both (G_4) and (F) drop dramatically. Within the shear band, the spatial correlation length collapses back to the particle scale, and the facilitation ratio falls below 0.05. This suggests that after localization the deformation becomes highly channelled, suppressing long‑range cooperative rearrangements.

3. Temporal persistence – In the transition regime, (C_4(t,\Delta\gamma)) decays slowly, with characteristic decay times spanning tens of strain increments, indicating that dynamically heterogeneous regions persist over relatively long deformation histories. Inside the shear band, the decay is rapid (1–2 increments), reflecting the fleeting nature of any residual cooperative activity.

4. Physical interpretation – The observed DH and DF in the transition regime resemble the glassy dynamics reported for 2‑D amorphous solids, where localized rearrangements (so‑called “soft spots”) trigger cascades of plastic events. The weakening of these signatures inside the shear band points to a qualitative shift: the system moves from a regime dominated by intermittent, spatially correlated avalanches to one where deformation is confined to a narrow, persistent slip surface.

5. Implications for modeling and engineering – The results demonstrate that any discrete‑element or continuum model of granular shear must incorporate strain‑increment‑dependent cooperative mechanisms. Ignoring DH and DF would lead to inaccurate predictions of shear‑band nucleation, width, and evolution, as well as erroneous estimates of macroscopic strength and dilatancy. Moreover, the clear link between DH/DF and the elastic‑to‑plastic transition provides a potential diagnostic for early warning of localization in geotechnical and industrial processes.

In conclusion, this study delivers the first systematic quantification of dynamic heterogeneity and dynamic facilitation in a three‑dimensional granular assembly under realistic loading conditions. By showing that these phenomena are strongly controlled by both the size of the strain increment and the deformation regime, the work bridges granular mechanics with the broader field of glassy dynamics and opens new avenues for predictive modeling of shear localization in particulate media.