  We develop a nonlinear, three-dimensional phase field model for crystal
plasticity which accounts for the infinite and discrete symmetry group G of the
underlying periodic lattice. This generates a complex energy landscape with
countably-many G-related wells in strain space, whereon the material evolves by
energy minimization under the loading through spontaneous slip processes
inducing the creation and motion of dislocations without the need of auxiliary
hypotheses. Multiple slips may be activated simultaneously, in domains
separated by a priori unknown free boundaries. The wells visited by the strain
at each position and time, are tracked by the evolution of a G-valued discrete
plastic map, whose non-compatible discontinuities identify lattice
dislocations. The main effects in the plasticity of crystalline materials at
microscopic scales emerge in this framework, including the long-range elastic
fields of possibly interacting dislocations, lattice friction, hardening,
band-like vs. complex spatial distributions of dislocations. The main results
concern the scale-free intermittency of the flow, with power-law exponents for
the slip avalanche statistics which are significantly affected by the symmetry
and the compatibility properties of the activated fundamental shears.
