Generalized multiscale homogenization approach to flexoelectric heterogeneous materials considering strain gradient contributions

Flexoelectricity, the electromechanical coupling induced by strain gradients, has gained increasing attention due to its relevance in the design of nanoscale devices, sensors, and energy-harvesting systems. Accurate prediction of effective properties in flexoelectric composites is crucial for guiding materials design; however, most existing studies consider only the flexoelectric tensor while neglecting the combined role of higher-order couplings. In this work, we present a generalized methodology based on the two-scale asymptotic homogenization method to evaluate the effective behavior of flexoelectric materials. The proposed framework explicitly incorporates the contributions of the flexoelectric tensor μ_ijkl, the strain-gradient elasticity tensor g_ijklmn, and the strain-gradient coupling tensor r_ijklm, allowing for a consistent treatment of higher-order interactions. As a particular case, explicit solutions are derived for stratified (multilayered) structures, which serve as a benchmark for more complex microstructures. Numerical examples illustrate the impact of microstructural symmetries, such as cubic, tetragonal, and isotropic arrangements, as well as the influence of the length-scale parameter and the constituent volume fractions on the effective tensors. The results demonstrate that second-order homogenization captures the interplay between microstructure, length scale, and electromechanical coupling, thereby providing a rigorous foundation for the design and optimization of advanced multifunctional materials.