Variational bounds for the effective moduli of heterogeneous piezoelectric solids

Variational bounds for the effective moduli of heterogeneous piezoelectric solids are developed by generalizing the Hashin-Shtrikman variational principles. Narrower bounds than Voigt-Reuss-type bounds are obtained by taking into account both the inclusion shape and the volume fraction. The proposed bounds for the effective electroelastic moduli are applicable to statistically homogeneous multiphase composites of any microgeometry and anisotropy and are self-consistent. A prescription for the calculation of the bounds is presented that takes advantage of existing, often closed-form expressions for the piezoelectric Eshelby tensor for ellipsoidal inclusions. Numerical results are presented and compared with measurements for four composite materials with different microstructures. The Hashin-Shtrikman-type bounds are much narrower than the Voigt-Reuss-type bounds. In many but not all cases they are sufficiently narrow to serve as good estimates of various elastic, dielectric and piezoelectric moduli, as assessed by comparison with measurements. Furthermore, the average of the Voigt- and Reuss-type bounds (which is often used for elastic polycrystals and composites) does not in general accurately describe the effective moduli of the heterogeneous solid either quantitatively or qualitatively.