The Spontaneous Configuration of Ferroelectric-Ferroelastic Materials:Ⅱ ———Polycrystalline Materials

As the internal energy densityof ferroelectric-ferroelastic single cystals is the non-convexfunction of the transformation gradients and the electric displacements in the process of polarization, in natural conditions (i.e., without external forces and external electric fields), the stable spontaneous configuration of a ferroelectric-ferroelastic polycrystalline material consists of a coarse mixture of various ferroelectric domains. In this coarse mixture, the volume fractions of electric domains are determined by the continuity requirements of the displacement and the electric displacement on the grain boundaries. Because various kinds of twins can be formed in the material, the volume fraction of the domainswith the same spontaneous deformation gradient depends upon the orientations of the grain boundaries, the orientations of the twining crystal lattices, and the twiningmode. In addition, the volume fraction of ferroelectric domains also depends upon the shape of the twining regio