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A method for evaluating the piezoelectric response of unpoled ferroelectricceramics from elastic and dielectric measurements is proposed and tested onBaTiO$_3$. The method is based on the observation that the softening in aferroelectric phase with respect to the paraelectric phase is of piezoelectricorigin. The angular averages of the piezoelectric softening in unpoled ceramicsare calculated for ferroelectric phases of different symmetries. The expressionof the orientational average with the piezoelectric and dielectric constants ofsingle crystal tetragonal BaTiO$_3$ from the literature reproduces well thesoftening of the Young's modulus of unpoled ceramic BaTiO$_3$, after acorrection for the porosity. The agreement is good in the temperature regionsufficiently far from the Curie temperature and from the transition to theorthorhombic phase, where the effect of fluctuations should be negligible, butdeviations are found outside this region, and the reason for this is discussed.This validates the determination of the piezoelectric response by means ofpurely elastic measurements on unpoled samples. The method is indirect and, forquantitative assessments, requires the knowledge of the dielectric tensor. Onthe other hand, it does not require poling of the sample, and therefore isinsensitive to inaccuracies from incomplete poling, and can even be used withmaterials that cannot be poled, for example due to excessive electricalconductivity. While the proposed example of the Young's modulus of a ceramicprovides an orientational average of all the single crystal piezoelectricconstants, a Resonant Ultrasound Spectroscopy measurement of a single unpoledceramic sample through the ferroelectric transition can in principle measureall the piezoelectric constants, together with the elastic ones.

journal of applied physics

Quantitative evaluation of the piezoelectric response of unpoled  ferroelectric ceramics from elastic and dielectric measurements: tetragonal  BaTiO$_3$