Effective intrinsic linear properties of laminar piezoelectric   composites and simple ferroelectric domain structures
  The effective properties of piezoelectric laminates have been analyzed, based on the calculation of internal fields and making use of a simple matrix manipulation method. The results are expressed in a compact notation which is convenient for numerical implementation and at the same time suitable for further analytical treatments. A detailed analysis of fully compatible ferroelectric domain structures shows, that the results for arbitrary piezoelectric laminates can be further simplified and specific property relationships for rank-1 laminates of tetragonal and rhombohedral crystals are derived. The method is finally applied to the analysis of various hierarchical domain structures. Detailed orientation relationships between the particular domains in some important domain pattern are given to make these structures accessible for the presented method. Some numerical results for tetragonal barium titanate illustrate the effects of different domain arrangements on the effective properties. 
Johannes R\"odel