Topological constraints suppress shear localization in granular chain ensembles

Entangled granular systems exhibit mechanical rigidity and resistance to deformation, reminiscent of cohesive materials, due to their reduced degrees of freedom and contact friction. A quantitative understanding of how classical granular phenomena, such as shear localization and plastic flow, appear in such geometrically cohesive systems remains unknown. Here, we investigate this using granular chain ensembles subjected to direct shear tests. Our experiments reveal that chains longer than four beads exhibit pronounced shear hardening, which is nearly independent of the applied normal stress and is accompanied by the complete suppression of shear localization. The volume dilation of the long chain ensembles also does not vanish in the steady state. We complement this phenomenology, which is distinct from that of typical frictional granular ensembles, with DEM simulations. The simulations reveal that tensile forces are generated due to particles being locally jammed, characterized by a high non-covalent coordination number. Consequently, this leads to a deformation that shows a very diffuse region of localization and enhanced shear hardening. Overall, our study highlights that granular chains provide a systematic route to map how connectivity constraints impact flow properties and mechanical rigidity.

💡 Research Summary

This paper investigates how topological constraints introduced by linking granular particles into chains affect shear deformation, strain hardening, and shear localization. The authors prepared ensembles of spherical beads (diameter = 2 mm) connected into chains of varying lengths (N = 1, 4, 8, 10, 12, 24, 48) and subjected them to direct shear tests (DST) under normal stresses ranging from 30 kPa to 100 kPa. Simultaneously, they performed discrete element method (DEM) simulations that faithfully reproduced the experimental conditions, modeling each chain as a series of “I‑shaped” link contacts between beads together with standard frictional bead‑bead contacts.

Key macroscopic findings are as follows. For single beads (N = 1), the shear stress ratio η = τ/σ_n initially rises but quickly reaches a peak and then declines with increasing shear strain, displaying the classic strain‑softening behavior of cohesionless granular media. Dilatancy follows the same trend, peaking before decreasing toward a critical state where volume change vanishes. In stark contrast, ensembles with chains longer than four beads exhibit pronounced strain hardening: η continuously increases with shear strain and only plateaus for the shortest multi‑bead case (N = 4). Moreover, dilatancy does not diminish; it keeps growing even at large strains, indicating that the material never reaches a conventional critical state. This hardening is essentially independent of the applied normal stress, although higher σ_n slightly reduces the absolute value of η for a given strain.

Shear localization was quantified using particle image velocimetry (PIV) to obtain the horizontal velocity gradient ∂Vx/∂z across the sample height. Regions where the gradient exceeded 30 % of its maximum were defined as the “shear zone.” For N = 1, the shear zone is narrow, resembling a classic shear band only a few bead diameters thick. As chain length increases, the shear zone widens dramatically, eventually spanning the entire deformation region for the longest chains (N = 24, 48). Thus, topological constraints suppress the formation of narrow, localized shear bands and promote a diffuse deformation field.

DEM simulations provide microscopic insight into this transition. Two contact families are distinguished: (i) link contacts that can carry both compressive and tensile forces, and (ii) frictional bead‑bead contacts that are purely compressive. During shear, the non‑covalent coordination number Znc (the number of frictional contacts per bead, excluding link contacts) rises for beads that become locally jammed. High Znc beads experience reduced degrees of freedom and act as “anchors” that transmit tensile forces through the chain links. The simulations reveal that tensile forces develop primarily near the initiation point of the deformation region and radiate outward, while compressive forces form a dense, system‑spanning network. Average tensile link forces increase sharply between N = 8 and N = 12 (≈ 20 % jump) and then saturate for longer chains, indicating a threshold chain length beyond which a continuous tensile backbone can sustain the applied shear.

The authors also examined particle rotations. For cohesionless beads (N = 1) and short chains (N = 4), cumulative rotations are symmetric about the deformation centre. In long‑chain ensembles, rotations concentrate near the deformation initiation point, reflecting an asymmetric redistribution of deformation energy. Stress tensors computed separately from frictional contacts and from link contacts show that vertical normal stress σzz varies along the deformation region, with peaks correlating with high Znc zones, confirming that locally jammed beads bear additional load.

Overall, the study demonstrates that introducing one‑dimensional connectivity constraints transforms granular media from a strain‑softening, shear‑band‑forming material into a strain‑hardening, shear‑diffuse system. The mechanism hinges on the emergence of tensile forces in the chain links, which are amplified by locally jammed beads possessing high non‑covalent coordination numbers. This “geometric cohesion” arising from entanglement and knotting provides a tunable route to control macroscopic rheology.

The implications are broad. Granular chains can serve as a model platform for designing metamaterials with programmable stiffness and shear resistance, for constructing earthquake‑resistant geostructures where shear localization must be avoided, and for optimizing 3‑D printing processes that rely on granular flow. By simply varying chain length, engineers can tailor the balance between stiffness, dilatancy, and localization, offering a versatile tool for advanced material design.