Based on the theory of invariants, from invariant polynomialconstitutive relations for piezoelectric materials which are either transversely isotropic or are of class mm2 are derived from the polynomial integrity basis functions. These constitutive relations are assumed to be smooth enough functions of their arguments to be expanded in terms of a Taylor series. These functions are expanded about the values their arguments take in the reference configuration and all terms up to the quadratic terms in the gradients of the mechanical displacement and electric potential are kept. The second-order theory so obtained is then specialized to the case of small deformations and weak electric fields, and the case of small deformations and relatively strong electric fields. The material parameters in the present theory are identified by relating them to those in the more conventional theories. 1995 Acoustical Society of America