We introduce a new phase field technique that incorporates the periodic nature of a crystal lattice by considering a free energy functional that is minimized by periodic density fields. This free energy naturally incorporates elastic and plastic deformations and multiple crystal orientations. The new phase field technique can be used to study a host of important phenomena in material processing that involve elastic and plastic effects in phase transformations. This novel phase field approach is used to study elastic and plastic deformation in nanocrystalline materials with a focus on the "reverse" Hall-Petch effect. In addition we apply the method to dendritic solidification in binary alloys and the role of dislocations in spinodal decomposition. / Thesis / Doctor of Philosophy (PhD)