Modeling of elastic nonlinearities in ferroelectric materials including nonlinear losses: application to nonlinear resonance mode of relaxors single crystals

(1-x)Pb(Mg/sub 1/3/Nb/sub 2/3/)O/sub 3/-xPbTiO/sub 3/ (PMN-PT) and (1-x)Pb(Zn/sub 1/3/Nb/sub 2/3/)O/sub 3/-xPbTiO/sub 3/ (PZN-PT) single crystals are considered to behave like soft Pb(Zr,Ti)O/sub 3/ (PZT) ceramics because of their small mechanical quality factor Q/sub m/ and poor stability under external disturbances (Q/sub m/ > 500-1000 for hard PZT ceramic, and Q/sub m/ < 100 for soft PZT and PMN-PT and PZN-PT single crystals). At weak signal excitation of the first resonance mode, the displacement at the end of a lateral bar is proportional to the Q/sub 31/d/sub 31/ figure of merit that is very close to that found for hard PZT. Indeed the very large piezoelectric coefficient compensates the low Q/sub m/. But increasing alternating current (AC) field results in the appearance of strong nonlinearities through a shift of the resonance frequency and jumps phenomenon observed on increasing and decreasing frequency sweep. It is shown in this paper that these nonlinearities are due to the nonlinear elastic compliance that can be modeled by a third order development of the constitutive piezoelectric equations. Experiments on PMN-PT and PZN-PT single crystals are used for comparison with the model to show the viability of the approach. Both the frequency shift and jumps phenomenon are simulated with a very good agreement with experimental results. The importance is also shown of losses associated with the third order term responsible for the large decrease of the mechanical quality factor for high strain levels. Thus, the nonlinear losses are related to the hysteresis of domain wall motion when subjected to large displacements.