The effective piezoelectric properties of heterogeneous materials are evaluated in the context of periodic homogenization, whereby a variational formulation is developed, articulated with the extended Hill macrohomogeneity condition. The entire set of homogenized piezoelectric moduli is obtained as the volumetric averages of the microscopic properties of the individual constituents weighted by the displacement and polarization localization operators. This framework is extended in a second part of the paper to the computation of the flexoelectric effective properties, thereby accounting for higher gradient effects that may be induced by a strong contrast of properties of the composite constituents. The effective properties of inclusion-based composites are evaluated numerically as an illustration of the general homogenization theory and the respective effect of the volume fraction and relative tensile modulus of the reinforcement is assessed numerically.

在周期性均质化的背景下评估了异质材料的有效压电性能，从而开发了一种变型配方，并与扩展的Hill宏观均质性条件相吻合。整个均质化的压电模量集是由位移和极化定位算符加权的单个成分的微观性质的体积平均值。该框架在本文的第二部分中扩展到了柔电有效特性的计算，从而解决了可能由复合成分的性能强烈反差引起的更高的梯度效应。