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.
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