Universal bounds for eective piezoelectric moduli

Rigorous upper and lower bounds for the e€ective moduli are obtained for heterogeneous piezoelectric materials, by generalizing the Hashin±Shtrikman variational principle to the coupled problems of piezoelectricity. The key in obtaining these bounds is the choice of the ®eld variables used in the variational principles, i.e., the strain tensor and the electric displacement vector or the stress tensor and the electric ®eld vector. Universal theorems, originally established for (uncoupled) mechanical and non-mechanical problems, are generalized for application to piezoelectricity problems, and the boundary conditions which provide upper and lower bounds for the average energy are identi®ed. These theorems lead to rigorous bounds for the e€ective piezoelectric moduli that are de®ned through the relation between the average ®eld quantities. Computable bounds are derived from Eshelby's tensors for the piezoelectricity problems. These tensors are obtained by applying the Fourier transform to the Green's functions of an unbounded homogeneous body. Ó 1998 Elsevier Science Ltd. All rights reserved.