In elastic dielectrics, piezoelectricity is the response of polarization to applied mechanical strain,
and vice versa. Piezoelectric coupling is controlled by a third-rank tensor and is allowed only in materials
that are non-centrosymmetric. Flexoelectricity, however, is the generation of electric polarization by the
application of a non-uniform mechanical strain field, i.e. a strain gradient, and is expected to be pronounced
at submicron thickness levels, especially at the nano-scale. Flexoelectricity is controlled by a fourth-rank
tensor and is therefore allowed in materials of any symmetry. As a gradient effect, flexoelectricity is size
dependent, while piezoelectric coupling has no size dependence. Any ordinary piezoelectric cantilever
model developed for devices above micron-level thickness has to be modified for nano-scale piezoelectric
devices since the effect of flexoelectric coupling will change the electroelastic dynamics at such small
scales. In this work, we establish and explore a complete analytical framework by accounting for both the
piezoelectric and flexoelectric effects. The focus is placed on the development of governing electroelastodynamic
piezoelectric-flexoelectric equations for the problems of energy harvesting, sensing, and
actuation. The coupled governing equations are analyzed to obtain the frequency response. The coupling
coefficient for the bimorph configuration is identified and its size dependence is explored.
Adriane G. Moura and Alper Erturk, "Size effects in piezoelectric cantilevers at submicron thickness levels due to flexoelectricity," Proc. SPIE 10164, Active and Passive Smart Structures and Integrated Systems 2017, 1016405 (Presented at SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring: March 26, 2017; Published: 11 April 2017); https://doi.org/10.1117/12.2260315.
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