The talk will survey some recent results related by the common theme of
finding rank-one connections between energy wells in nonlinear elasticity models of crystals.
It will cover three cases (i) slip and twinning in Bravais lattices using the Ericksen
energy-well picture, (ii) compatibility of martensitic microstructures across polycrystal
grain boundaries and the Taylor set (joint with M. Galanopoulou), and (iii) remarkable
martensitic microstructures observed in a TiNbAl alloy, and their interpretation
in terms of T_N configurations (joint with T. Inamura & F. Della Porta).