A two-dimensional lattice model for the formation and evolution of shearbands in granular media is proposed. Each lattice site is assigned a randomvariable which reflects the local density. At every time step, the strain islocalized along a single shear-band which is a spanning path on the latticechosen through an extremum condition. The dynamics consists of randomlychanging the `density' of the sites only along the shear band, and thenrepeating the procedure of locating the extremal path and changing it. Startingfrom an initially uncorrelated density field, it is found that this dynamicsleads to a slow compaction along with a non-trivial patterning of the system,with high density regions forming which shelter long-lived low-density valleys.Further, as a result of these large density fluctuations, the shear band whichwas initially equally likely to be found anywhere on the lattice, getsprogressively trapped for longer and longer periods of time. This state ishowever meta-stable, and the system continues to evolve slowly in a mannerreminiscent of glassy dynamics. Several quantities have been studiednumerically which support this picture and elucidate the unusual system-sizeeffects at play.
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