  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.
