13th International Conference on Fracture June 16–21, 2013, Beijing, China -10- case containing two identical elliptical inhomogenieties and it can be a benchmark for comparison when other numerical method is developed. 4. Conclusion We have successfully proposed a systematic method to solve an infinite plane containing elliptical inclusions under remote anti-plane shears and in-plane electric fields. Although a Jacobian term may appear in the degenerate kernel, boundary density and boundary contour integral by using the elliptical coordinates, it can be cancelled out in our formulation to preserve the orthogonal condition. Although the work containing two elliptical inhomogenieties is not available in the literature, the piezoelectricity with two circular inclusions is used as a limiting case to demonstrate the validity of present approach. Besides, the case containing two identical elliptical inhomogenieties was provided as a benchmark example. The developed program can be generally used for piezoelectricity problems containing elliptical inhomogenieties of arbitrary number, position, size and inclination angle. Acknowledgements This research was partially supported by the National Science Council in Taiwan through Grant NSC 101-2811-E-019-002. References [1] J.T. Chen, A.C. Wu, Null-field approach for piezoelectricity problems with arbitrary circular inclusions,” Eng Anal Bound Elem, 30 (2006) 971-993. [2] S.X. Gong, S.A. Meguid, A general treatment of the elastic field of an elliptical inhomogeneity under antiplane shear. J Appl Mech-ASME, 59 (1992) 131-135. [3] S.X. Gong, A unified treatment of the elastic elliptical inhomogeneity under antiplane shear, Arch Appl Mech, 65 (1995) 55-64. [4] H. Shen, P. Schiavone, C.Q. Ru, A. Mioduchowski, An elliptic inclusion with imperfect interface in anti-plane shear,” Int J Solids Struct, 37 (2000) 4557-4575. [5] N.A. Noda, T. Matsuo, Stress analysis of arbitrarily distributed elliptical inclusions under longitudinal shear loading. Int J Fract, 106 (2000) 81-93. [6] J. Lee, H.R. Kim, Volume integral equation method for multiple circular and elliptical inclusion problems in antiplane elastostatics. Composites Part: B, 43 (2012) 1224-1243. [7] Y.T. Lee, J.T. Chen, Null-field approach for the antiplane problem with elliptical holes and/or inclusions. Composites Part: B, 44 (2013) 283-294. [8] Y.E. Pak, Circular inclusion problem in antiplane piezoelectricity. Int J Solids Struct, 29 (1996) 2403-2419. [9] T. Honein, B.V. Honein, E. Honein, G. Herrmann, On the interaction of two piezoelectric fibers embedded in an intelligent material. J Intell Mater Syst Struct, 6 (1995) 229-236. [10] C.K. Chao, K.J. Chang, Interacting circular inclusions in antiplane piezoelectricy. Int J Solids Struct, 36 (1999) 3349-3373. [11] X. Wang, Y.P. Shen, On double circular inclusion problem in antiplane piezoelectricity. Int J Solids Struct, 38 (2001) 4439-4461. [12] J.L. Bleustein, New surface wave in piezoelectric materials, Appl Phys Lett, 46 (1968) 412-413. [13] S.A. Meguid, Z. Zhong, Electroelastic analysis of a piezoelectric elliptical inhomogeneity. Int J Solids Struct, 34, pp. 3401-3414, 1997. [14] Y.E. Pak, Elliptical inclusion problem in antiplane piezoelectricity: implications for fracture mechanics. Int. J. Solids Struct., Vol. 48 pp. 209-222, 2010.
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