A piezoelectric constitutive theory with rotation gradient effects

Some recent experiments evidenced that the piezoelectric coefficients of such piezoelectric materials as PZT and BiTiO₃ have an evident dependence on the grain size. With respect to the success of strain gradient theories in interpreting size effect phenomena of conventional (non-piezoelectric) solids and the dependence of piezoelectricity on rotation of polar clusters, a new piezoelectric theory with rotation gradient effects is formulated in the present paper to elucidate the size effect problems of piezoelectric solids. The constitutive relations of materials with different symmetries are specialized, and their corresponding independent material constants are discussed. For the typical 6 mm class of piezoelectric crystalline materials, a potential function method is presented for solving plane problems. The analytical solution of a thin piezoelectric film bonded on a rigid substrate illustrates the size-dependent prediction of the present theory.