Variational bounds for the overall properties of piezoelectric composites

In this paper, variational bounds for the overall properties of periodic heterogeneous media with nonlinear and nonlocal piezoelectric constitutive relationships are obtained. First, elementary bounds are developed by extending to piezoelectric materials the well-known Voigt and Reuss bounds. Then, by generalizing the two Hashin-Shtrikman principles, eight new variational principles are derived and applied to obtain bounds. In fact, the variational principles developed are based on auxiliary electroelastic equilibrium problems, which can be solved by a transformation from the space domain to the Fourier domain. As an application, fibre-reinforced linear piezoelectric composites are considered, and expressions of upper and lower bounds for the overall properties of these composites are developed in closed form.