Advanced constitutive modeling of flexoelectric materials incorporating higher-order gradient effects: Towards the design and optimization of nanoscale devices

This work presents a comprehensive theoretical framework for flexoelectric materials by incorporating higher-order strain gradient and polarization gradient effects into the constitutive modeling. Using an extended strain gradient elasticity (SGE) approach, coupled with a generalized Toupin-like variational formulation, we derive governing equations, balance laws, and boundary conditions based on an enriched internal energy density function. Analytical solutions, expressed in terms of modified Bessel functions, provide key insights into the role of higher-order gradients in influencing displacement, polarization, and electric fields. The study highlights the critical impact of size effects on flexoelectric response, revealing that reducing material thickness enhances sensitivity and energy conversion efficiency. Furthermore, numerical simulations validate the theoretical model and demonstrate its applicability in the design of nanoscale flexoelectric sensors and energy harvesters. These findings establish a robust theoretical foundation for optimizing nanoscale electromechanical devices, with potential applications in biomedical sensors, structural health monitoring, and energy-efficient electronics.