Flexoelectricity shows promising applications for self-powered devices with its increased power density. This paper presents a second-order computational homogenization strategy for flexoelectric composite. The macro-micro scale transition, Hill-Mandel energy condition, periodic boundary conditions, and macroscopic constitutive tangents for the two-scale electromechanical coupling are investigated and considered in the homogenization formulation. The macrostructure and microstructure are discretized using $C^1$ triangular finite elements. The second-order multiscale solution scheme is implemented using ABAQUS with user subroutines. Finally, we present numerical examples including parametric analysis of a square plate with holes and the design of piezoelectric materials made of non-piezoelectric materials to demonstrate the numerical implementation and the size-dependent effects of flexoelectricity.

翻译：柔性电效应因其高功率密度在自供能器件中展现出广阔应用前景。本文提出了一种针对柔性电复合材料的二阶计算均匀化策略。在均匀化公式中，研究了宏微观尺度过渡、希尔-曼德尔能量条件、周期性边界条件以及双尺度机电耦合的宏观本构切线张量。采用$C^1$三角形有限单元分别对宏观结构和微观结构进行离散化。基于ABAQUS用户子程序实现了二阶多尺度求解方案。最后，通过含孔方板参数化分析及非压电材料制备压电材料的设计等数值算例，验证了数值实现方法的有效性并揭示了柔性电效应的尺寸依赖性特征。