Influence of nonlocal elasticity tensor and flexoelectricity in a rod: An asymptotic homogenization approach

This paper presents a methodology based on the asymptotic homogenization method (AHM) to model flexoelectric composites with nonlocal elasticity. The nonlocal elasticity tensor accounts for the long-range interactions between the strain gradient and the electric field, which affect the effective flexoelectric coefficients and the composite's overall response. The local problems, the general expression of the higher-order contributions in the asymptotic expansion, and the homogenized formulation of the equilibrium problem for a one-dimensional flexoelectric composite material are derived, and details of the AHM are given. Closed formulas for the effective flexoelectric coefficients with higher-order contributions of the asymptotic expansions are found, which is a novel contribution to the field. Finally, numerical examples are reported and discussed. Herein, the influence of different material constituents on the effective properties of various one-dimensional periodic composite materials is studied. The solutions to the homogenized problem for different material configurations are given.