Granular vortices: Identification, characterization and conditions for the localization of deformation
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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, the 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.
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