  We study microstructure formation in two nonconvex singularly-perturbed
variational problems from materials science, one modeling austenite-martensite
interfaces in shape-memory alloys, the other one slip structures in the plastic
deformation of crystals. For both functionals we determine the scaling of the
optimal energy in terms of the parameters of the problem, leading to a
characterization of the mesoscopic phase diagram. Our results identify the
presence of a new phase, which is intermediate between the classical laminar
microstructures and branching patterns. The new phase, characterized by partial
branching, appears for both problems in the limit of small volume fraction,
that is, if one of the variants (or of the slip systems) dominates the picture
and the volume fraction of the other one is small.
