  Young's modulus determines the mechanical loads required to elastically
stretch a material, and also, the loads required to bend it, given that bending
stretches one surface while compressing the opposite one. Flexoelectric
materials have the additional property of becoming electrically polarized when
bent. While numerous studies have characterized this flexoelectric coupling,
its impact on the mechanical response, due to the energy cost of polarization
upon bending, is largely unexplored. This intriguing contribution of strain
gradient elasticity is expected to become visible at small length scales where
strain gradients are geometrically enhanced, especially in high permittivity
insulators. Here we present nano-mechanical measurements of freely suspended
SrTiO3 membrane drumheads. We observe a striking non-monotonic thickness
dependence of Young's modulus upon small deflections. Furthermore, the modulus
inferred from a predominantly bending deformation is three times larger than
that of a predominantly stretching deformation for membranes thinner than 20
nm. In this regime we extract a giant strain gradient elastic coupling of
~2.2e-6 N, which could be used in new operational regimes of
nano-electro-mechanics.
