  The properties of a ferroelectric, (001)-oriented, thin film clamped to a
substrate are investigated analytically and numerically. The emphasis is on the
tetragonal, polydomain, ferroelectric phase, using a three domain structure, as
is observed experimentally. The previously used, very restrictive set of
boundary conditions, arising from the domain walls, is relaxed, creating more
modes for energy relaxation. It is argued that this approach gives a more
realistic description of the clamped ferroelectric film.
  It is shown that for the ferroelectric oxides PbZr_(1-x)Ti_xO_3} the
tetragonal, polydomain phase is present over a wide range of substrate induced
strains for x_Ti>0.5, corresponding to the tetragonal side of the bulk phase
diagram. A polydomain, rhombohedral phase is present for x_Ti<0.5, at the bulk
rhombohedral side. Phase-temperature diagrams, and ferroelectric, dielectric
and piezoelectric properties, as well as lattice parameters, are calculated as
function of substrate induced strain and applied field. The analytical
formulation allows the decomposition of these properties into three different
causes: domain wall motion, field induced elastic effects and piezoelectric
effects. It is found that domain wall motion and polarization rotation of the
in-plane oriented domains under an applied field contribute most to the
properties, while the out-of-plane oriented domains hardly contribute.
