Hashin-Shtrikman Bounds for Piezoelectric Composite Materials with Periodic Distributed Inclusions

The prediction of effective moduli of piezoelectric composites is one of the most important topics in composite materials. Early studies on the effective moduli of piezoelectric composites were mainly about ellipsoidal inclusions. This paper proposed the variational bounds for piezoelectric composite with cubic inclusions periodically distributed, which is an extension of Mejak's work on the elastic composites with periodic microstructure. This methodology is available for the prediction of the effective modulus bounds of the piezoelectric composites with inclusions of different geometries if the Hill tensors of the inclusions are available. The variational bounds for piezoelectric composites with cubic and spherical inclusions are respectively evaluated. The results show that the effective modulus bounds of the two piezoelectric composites is quite similar with each other. The maximum relative error is about 5%.