A general method is presented for the analytic solution of Eshelby’s problem concerned with a smooth inclusion of arbitrary shape embedded within the elastic half-plane of a bonded piezoelectric—elastic bimaterial. Since the well-known Stroh formulation, on which almost all existing works on Eshelby’s problem in piezoelectric media has been based, breaks down for isotropic elastic materials, the solutions available in the literature cannot be applied directly. Consequently, the method of solution for the current problem involves a combination of Stroh’s formulation for piezoelectric materials and the well-known Muskhelishvili method for isotropic elastic bodies. The general solution is derived in terms of an auxiliary function which can be constructed using conformal mappings that map the exterior of the inclusion onto the exterior of the unit circle. To illustrate the method, the case of a dilatational inclusion is examined and detailed results are shown for the mean stress within a thermal elliptic inclusion.
Hu, M.Stress from isolation trenches in silicone substrates. Journal of Applied Physics, 67, 1092-1101 (1990 ).
2.
Hu, S.M.Stress-related problems in silicone industry . Journal of Applied Physics, 70, R53-R80 (1991).
3.
Niwa, H., Yagi, H., Tsuchikava, H. and Kato, M.Stress distribution in an aluminum interconnect of very large scale integration . Journal of Applied Physics, 68, 328-333 ( 1990).
4.
Downes, J.R. , Faux, D.A. and O’Reilly , E.P.A simple method for calculating strain distribution in quantum wire dot structures . Journal of Applied Physics, 81, 6700-6702 ( 1997 ).
5.
Gosling, T.J. and Willis, J.R.Mechanical stability and electronic properties of buried strained quantum wire arrays. Journal of Applied Physics, 77, 5601-5610 (1995 ).
6.
Wang, X., Sudak, L.J. and Ru, C.Q.Elastic fields in two imperfectly bonded half-planes with a thermal inclusion of arbitrary shape. Zeitschrift für angewandte Mathematik und Physik, 58, 488-509 (2007).
7.
Seo, K. and Mura, T.The elastic fields in a half-space due to an ellipsoid inclusion with uniform dilatational eigenstrains. ASME Journal of Applied Mechanics , 46, 568-572 (1979).
8.
Yu, H.Y. and Sanday, S.C.Elastic field in jointed semi-infinite solids with an inclusion. Proceedings of the Royal Society London A, 434, 521-530 ( 1991).
9.
Ru, C.Q.Analytic solution for Eshelby’s problem of an inclusion of arbitrary shape in a plane or half-plane. ASME Journal of Applied Mechanics, 66, 315-322 (1999).
10.
Ru, C.Q.A two dimensional Eshebly problem for two bonded piezoelectric half-planes. Proceedings of the Royal Society London A, 457, 865-883 (2001).
11.
Ru, C.Q. , Schiavone, P. and Mioduchowski , A.Elastic fields in two jointed half-planes with an inclusion of arbitrary shape. Zeitschrift für angewandte Mathematik und Physik, 52, 18-32 ( 2001).
12.
Janas, V.F. and Safari, A.Overview of fine-scale piezoelectric ceramic/polymer composite processing. Journal of the American Ceramics Society , 78, 2945-2955 (1995 ).
13.
Choi, J.S. , Ashida, F. and Noda, N.Control of thermally induced elastic displacement of an isotropic structural plate bonded to a piezoelectric ceramic plate. Acta Mechanica, 122, 49-63 (1997).
14.
Li, Q. and Chen, Y.H.Solution for a semi-permeable interface crack in elastic dielectric/piezoelectric bimaterials. ASME Journal of Applied Mechanics, 75, 011010 (2008).
15.
Ou, Z.C. and Chen, Y.H.Interface crack problem in elastic dielectric/piezoelectric bimaterials. International Journal of Fracture, 130, 427-454 (2004).
16.
Ting, T.C.T.Anisotropic Elasticity. Theory and Applications. Oxford University Press, Oxford, 1996.
17.
Chen, T. and Lai, D.An exact correspondence between plane piezoelectricity and generalized plane strain in elasticity. Proceedings of the Royal Society London A, 453, 2689-2713 (1997).
18.
Ou, Z.C. and Chen, Y.H.General solution of the stress potential function in Lekhnitskii’s elastic theory for anisotropic and piezoelectric materials. Advanced Studies in Theoretical Physics , 1, 357-366 (2007).
19.
Ru, C.Q.A hybrid complex variable solution for piezoelectric/isotropic elastic interfacial cracks. International Journal of Fracture , 152, 169-178 (2008).
20.
Chen, Y.H. and Lu, T.J.Cracks and fracture in piezoelectric materials . Advances in Applied Mechanics, 39, 127-221 ( 2003).
21.
Chen, Y.H. and Lu, T.J.Recent developments and applications in invariant integrals. ASME Applied Mechanics Review, 56, 515-552 (2003).
22.
Li, Q. and Chen, Y.H.Solution of the semi-permeable interface crack in dissimilar piezoelectric materials. ASME Journal of Applied Mechanics, 74, 833-844 (2007).
23.
Chen, Y.H.Advances in conservation laws and energy release rates. Kluwer Academic, Dordrecht, 2002 .
24.
Stroh, A.N.Dislocation and cracks in anisotropic elasticity . Philosophical Magazine, 7, 625-646 ( 1958).
25.
Muskhelishvili, N.I.Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordhoff, Reinhold, 1963.
26.
Kantorovich, L.V. and Krylov, V.I.Approximate Method of Higher Analysis. Interscience, New York , 1958.
27.
Clements, D.L.A crack between dissimilar anisotropic media . International Journal of Engineering Science, 9, 257-265 (1971).
28.
Suo, Z., Kuo, C.M., Barnett, D.M. and Willis, J.R.Fracture mechanics for piezoelectric ceramics . Journal of the Mechanics and Physics of Solids, 40, 739-765 (1992).
29.
Ru, C.Q. and Schiavone, P.On the elliptical inclusion in anti-plane shear. Mathematics and Mechanics of Solids, 1, 327-333 (1996).
30.
Ru, C.Q.Exact solution for finite electrode layers embedded at the interface of two piezoelectric half-planes. Journal of the Mechanics and Physics of Solids, 48, 693-708 ( 2000).
