Two-dimensional Eshelby’s problem for two imperfectly bonded piezoelectric half-planes

General solutions are derived to the two-dimensional Eshelby’s problem of an inclusion of arbitrary shape embedded in one of two imperfectly bonded anisotropic piezoelectric half-planes. The inclusion undergoes uniform eigenstrains and eigenelectric fields. In this work four different kinds of imperfect interface models with vanishing thickness are considered: (i) a compliant and weakly conducting interface, (ii) a stiff and highly conducting interface, (iii) a compliant and highly conducting interface, and (iv) a stiff and weakly conducting interface. Furthermore the obtained general solutions are illustrated in detail through an example of an elliptical inclusion near the imperfect interface. It is observed that the full-field expressions of the three analytic function vectors characterizing the electroelastic field in the two piezoelectric half-planes including the elliptical inclusion can be elegantly and concisely presented through the introduction of an integral function. We also present the tractions and normal electric displacement along a compliant and weakly conducting imperfect interface induced by the elliptical inclusion. It is found that the imperfection of the interface has no influence on the leading term in the far-field asymptotic expansion of the tractions and normal electric displacement along the compliant and weakly conducting interface induced by an arbitrary shaped inclusion. The far-field expansions of the analytic function vectors in the two imperfectly bonded half-planes for an arbitrary shaped inclusion are also derived. Some new identities and structures of the matrices N i and N i ( - 1 ) for anisotropic piezoelectric materials are obtained.