Flexoelectricity in structured solids: Stacking and strain gradient modulation

Flexoelectricity, a coupling between strain gradients and electric polarization, has attracted significant interest due to its critical role in enhanced effects at small scales and its applicability across a diverse range of materials. Despite its universal presence across dielectric materials, designing flexoelectric structures for practical applications remains challenging due to the mathematical complexity of modeling higher-order electromechanical couplings. In this work, a modified mixed finite element model with a penalty-based constraint strategy is utilized to improve stability and enable accurate large-scale (over four million degrees of freedom) flexoelectric simulations. The framework is then applied to investigate how structural design can influence flexoelectric behavior. The analysis focuses on a pyramid-shaped unit cell because of the relatively high strain gradients that can be generated in a mechanically stable structure. To investigate performance enhancement of transducer structures, the effects of stacking and localized strain gradient modulation via embedded voids are investigated. Proper stacking of unit cell structures can enhance the transducer performance in an approximately linear fashion (i.e., doubling the number of unit cells doubles the output), however, mechanical boundary conditions must be carefully considered and designed. Furthermore, results reveal that stacking configurations can alter the distribution and magnitude of electric potential across the structure, and embedded voids introduce localized enhancements and tailored field profiles. These findings suggest that geometric design strategies—such as unit cell stacking and localized strain gradient modulation via embedded voids—offer a viable route for tuning the spatial characteristics of the flexoelectric response, enhancing performance in sensing applications.