Abstract :
In this paper, by systematically treating the integrals involved in the piezoelectric inclusion problem, we have obtained explicit results for the piezoelectric Eshelby tensors for a spheroidal inclusion aligned along the axis of the anisotropy in a transversely isotropic piezoelectric material. This problem was first treated by Dunn and Wienecke (Int. J. Solids Struct. 34 (27), 3571–3582) using a Greenʹs function approach, which closely follows Withersʹ approach (Phil. Mag. A 59 (4), 759–781) for an ellipsoidal inclusion problem in a transversely isotropic elastic medium. The same problem was recently treated by Michelitsch and Levin (Eur. Phys. J. B 14, 527–533), who also used a Greenʹs function approach. In this paper, we also obtain the piezoelectric Eshelby tensors for a spheroidal inclusion explicitly, but using a different approach. The method is a direct extension of a more unified approach, which has been recently developed by Mikata (Int. J. Engng. Sci. 38, 608–641), which is based on Deegʹs results (Ph.D. dissertation, Stanford University) on a piezoelectric inclusion problem. The main advantage of this method is that it is more straightforward and simpler than Dunn and Wienecke (1996), or Michelitsch and Levin (2000), and the results are a little bit more explicit than their solutions. The key step of this paper is an analytical evaluation of several integrals, which was made possible after a careful treatment of a certain bi-cubic equation.
