  Piezoelectricity is inherent only in noncentrosymmetric materials, but a
piezoelectric response can also be obtained in centrosymmetric crystals if
subjected to inhomogeneous deformation. This phenomenon, known as
flexoelectricity, affects the functional properties of insulators, particularly
thin films of high permittivity materials. We have measured
strain-gradient-induced polarization in single crystals of paraelectric
SrTiO$_3$ as a function of temperature and orientation down to and below the
105 K phase transition. Estimates were obtained for all the components of the
flexoelectric tensor, and calculations based on these indicate that local
polarization around defects in SrTiO$_3$ may exceed the largest ferroelectric
polarizations. A sign reversal of the flexoelectric response detected below the
phase transition suggests that the ferroelastic domain walls of SrTiO$_3$ may
be polar.
