Closed-form expressions for the effective moduli of heterogeneous piezoelectric materials

In this paper, analytical and semi-analytical expressions for the effective properties of transversely isotropic piezoelectric materials are derived based on the Mori–Tanaka model and Eshelby tensors using compact matrix formulations. Concise explicit relations are presented for the coupled and decoupled effective electroelastic moduli of piezoelectric materials with various inclusion types and shapes. In order to use these explicit relationships, some of the Eshelby tensor components are needed to be analytically or numerically computed. These closed-form relations are suitable for use in designing composite piezoelectric materials with optimized characteristics. Results of effective electroelastic moduli, obtained using the derived closed-form expressions, are presented for various types and shapes of piezoelectric inclusions and compared with self-consistent and incremental self-consistent numerical predictions.