  A method for evaluating the piezoelectric response of unpoled ferroelectric
ceramics from elastic and dielectric measurements is proposed and tested on
BaTiO$_3$. The method is based on the observation that the softening in a
ferroelectric phase with respect to the paraelectric phase is of piezoelectric
origin. The angular averages of the piezoelectric softening in unpoled ceramics
are calculated for ferroelectric phases of different symmetries. The expression
of the orientational average with the piezoelectric and dielectric constants of
single crystal tetragonal BaTiO$_3$ from the literature reproduces well the
softening of the Young's modulus of unpoled ceramic BaTiO$_3$, after a
correction for the porosity. The agreement is good in the temperature region
sufficiently far from the Curie temperature and from the transition to the
orthorhombic phase, where the effect of fluctuations should be negligible, but
deviations are found outside this region, and the reason for this is discussed.
This validates the determination of the piezoelectric response by means of
purely elastic measurements on unpoled samples. The method is indirect and, for
quantitative assessments, requires the knowledge of the dielectric tensor. On
the other hand, it does not require poling of the sample, and therefore is
insensitive to inaccuracies from incomplete poling, and can even be used with
materials that cannot be poled, for example due to excessive electrical
conductivity. While the proposed example of the Young's modulus of a ceramic
provides an orientational average of all the single crystal piezoelectric
constants, a Resonant Ultrasound Spectroscopy measurement of a single unpoled
ceramic sample through the ferroelectric transition can in principle measure
all the piezoelectric constants, together with the elastic ones.
