We study the phenomenon of internal avalanching within the context of recently proposed ``Tetris'' lattice models for granular media. We define a recycling dynamics under which the system reaches a steady state which is self-structured, i.e. it shows a complex interplay between textured internal structures and critical avalanche behavior. Furthermore we develop a general mean-field theory for this class of systems and discuss possible scenarios for the breakdown of universality.
Categories: cond-mat.stat-mech