Dislocation microstructures and strain-gradient plasticity with one active slip plane

We study dislocation networks in the plane using the vectorial phase-field model in- troduced by Ortiz and coworkers, in the limit of small lattice spacing. We show that, in a scaling regime where the total length of the dislocations is large, the phase field model reduces to a simpler model of the strain-gradient type. The limiting model contains a term describing the three-dimensional elastic energy and a strain-gradient term describing the energy of the geometrically necessary dislocations, characterized by the tangential gra- dient of the slip. The energy density appearing in the strain-gradient term is determined by the solution of a cell problem, which depends on the line tension energy of disloca- tions. In the case of cubic crystals with isotropic elasticity our model shows that complex microstructures may form in which dislocations with different Burgers vector and or- ientation react with each other to reduce the total self-energy.