We relate the micromechanics of vortex evolution to that of force chain buckling and, on this basis, formulate the conditions for strain localization in a continuum model of dense granular media. Using the traditional bifurcation analysis of shear bands, we show that kinematic vortex fields are in fact solutions to the boundary value problem satisfying null boundary conditions. To establish an empirical basis for our study, we first develop a method to identify the location of the core and boundary of each vortex from a given displacement field in two dimensions. We then employ this method to characterize the residual deformation field (i.e., the deviation of particle motions from the continuum deformation) in a physical experiment and a discrete element simulation of dense granular samples submitted to biaxial compression. Vortices in the failure regime are essentially confined to the shear band. Primary vortices, the clear majority, rotate in the same direction as the shear band; secondary vortices, so-called wakes, rotate in the opposite direction. Primary vortices align in spatial succession along the central axis of the band; wakes form next to the band boundaries, in between and beside two adjacent primary vortices. Force chain buckling, the governing mechanism for shear bands, is responsible for vortex formation in the failure regime. Vortex dynamics are consistent with stick-slip dynamics. From quiescent conditions of jamming or stick, vortical motions arise from force chain buckling and associated relative particle rotations and sliding; these in turn precipitate intermittent periods of unjamming or slip, evident in the attendant drops in stress ratio and bursts in both kinetic energy and local nonaffine deformation. A kinematic vortex field inside shear bands is proposed that is consistent with the equations of continuum mechanics and the underlying instability of force chain buckling: such a field is periodic with a repeating unit cell comprising a primary vortex at the center of the band, with two trailing wakes close next to the band boundaries.

SUMMARYRecent analysis of data from triaxial tests on sand and discrete element simulations indicate the final pattern of failure is encoded in grain motions during the nascent stages of loading. We study vortices that are evident from grain displacements at the start of loading and bear a direct mathematical connection to boundary conditions, uniform continuum strain and shear bands. Motions of three grains in mutual contact, that is, 3‐cycles, manifest vortices. In the initial stages of loading, 3‐cycles initiate a rotation around a region Ω* where the shear band ultimately develops. This bias sets a course in 3‐cycle evolution, determining where they will more likely collapse. A multiscale spatial analysis of 3‐cycle temporal evolution provides quantitative evidence that the most stable, persistent 3‐cycles degrade preferentially in Ω*, until essentially depleted when the shear band is fully formed. The transition towards a clustered distribution of persistent 3‐cycles occurs early in the loading history—and coincides with the persistent localisation of vortices in Ω*. In 3D samples, no evidence of spatial clustering in persistent 3‐cycle deaths is found in samples undergoing diffuse failure, while early clustering manifests in a sample that ultimately failed by strain localisation. This study not only delivered insights into the possible structural origins of vortices in dense granular systems but also a tool for the early detection of the mode of failure—localised versus diffuse—a sample will ultimately undergo. Copyright © 2014 John Wiley & Sons, Ltd.

The formation of shear bands is a key attribute of degradation and failure in soil, rocks, and many other forms of amorphous and crystalline materials. Previous studies of dense sand under triaxial compression and two-dimensional analogues from simulations have shown that the ultimate shear band pattern may be detected in the nascent stages of loading, well before the band's known nucleation point (i.e., around peak stress ratio), as reported in the published literature. Here we construct a network flow model of force transmission to identify the bottlenecks in the contact networks of dense granular media: triaxial compression of Caicos ooid and Ottawa sand and a discrete element simulation of simple shear. The bottlenecks localise in the nascent stages of loading -in the location where the persistent shear band ultimately forms. This corroborates recent findings on vortices that suggest localised failure is a progressive process of degradation, initiating early in the loading history at sites spanning the full extent, yet confined to a subregion, of the sample. Bottlenecks are governed by the local and global properties of the sample fabric and the grain kinematics. Grains with large rotations and/or contacts having minimal load-bearing capacities per se do not identify the bottlenecks early in the loading history.

Plastic deformation in a plane strain compression test of a dense sand specimen is studied using functional networks. Kinematical information for the deforming material is obtained using digital image correlation (DIC) and summarized by two types of complex network with different connectivity rules establishing links between the network nodes which represent the DIC observation sites. In the first, nodes are connected to a minimum fixed number of neighbors with similar kinematics such that the resulting network forms one connected component. In the second, nodes are connected to other nodes whose kinematical behavior lies within a fixed distance of each other in an observation space. The fixed radius is determined using optimization with a stopping criterion again with the resulting network forming one connected component. We find different network properties of each network provide useful information about plastic deformation and nonaffine kinematical processes emerging within the material. In particular, persistent shear bands and mesoscale structures within them (e.g. vortices) appear to be closely related to values of network properties including closeness centrality, clustering coefficients, k-cores and the boundaries of community structures determined using local modularity.

International audienceWe study the evolution of structure inside a deforming, cohesionless granular material undergoing failure in the absence of strain localisation – so-called diffuse failure. The spatio-temporal evolution of the basic building blocks for self-organisation (i.e. force chains and minimal contact cycles) reveals direct insights into the structural origins of failure. Irrespective of failure mode, self-organisation is governed by the cooperative behaviour of truss-like 3-cycles providing lateral support to column-like force chains. The 3-cycles, which are initially in scarce supply, form a minority subset of the minimal contact cycle bases. At large length-scales (i.e. sample size), these structures are randomly dispersed, and remain as such while their population progressively falls as loading proceeds. Bereft of redundant constraints from the 3-cycles, the force chains are initially just above the isostatic state, a condition that progressively worsens as the sample dilates. This diminishing capacity for redistribution of forces without incurring physical rearrangements of member particles renders the force chains highly prone to buckling. A multiscale analysis of the spatial patterns of force chain buckling reveals no clustering or localisation with respect to the macroscopic scale. Temporal patterns of birth-and-death of 3-cycles and 3-force chains provide unambiguous evidence that significant structural reorganisations among these building blocks drive rheological behaviour at all stages of the loading history. The near-total collapse of all structural building blocks and the spatially random distribution of force chain buckling and 3-cycles hint at a possible signature of diffuse failure