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Solution for a Crack Embedded in Multiply Confocally Elliptical Layers in Antiplane Elasticity

  • Y.-Z. Chen (a1)


This paper provides a general solution for a crack embedded in multiply confocally elliptical layers in antiplane elasticity. In the problem, the elastic medium is composed of an inclusion, many confocally elliptical layers and the infinite matrix with different elastic properties. In addition, the remote loading is applied at infinity. The complex variable method and the conformal mapping technique are used. On the mapping plane, the complex potentials for the inclusion and many layers are assumed in a particular form with two undetermined coefficients. The continuity conditions for the displacement and traction along the interface between two adjacent layers are formulated and studied. By enforcing those conditions along the interface, the exact relation between two sets of two undetermined coefficients in the complex potentials for j-th layer and j + 1-th layer can be evaluated. From the traction free condition along the crack faces, the correct form of the complex potential for the cracked inclusion is obtained. Finally, many numerical results are provided.


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1.Eshelby, J.D., “The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems,” Proceedings of the Royal Society A, 241, pp. 376396 (1957).
2.Mura, T., Micromechanics of Defects in Solids, Mar-tinus Nijhoff Publishers, Dordrecht (1987).
3.Zhou, K., Hoh, H.J., Wang, X., Keer, L.M., Pang, J.H.L. and Song, B., “A Review of Recent Works on Inclusions,” Mechanics of Materials, 60, pp. 144158 (2013).
4.Gong, S.X., “A Unified Treatment of the Elastic Elliptical Inclusion Under Antiplane Shear,” Archive of Applied Mechanics, 65, pp. 5564 (1995).
5.Ru, C.Q., and Schiavone, P., “On the Elliptic Inclusion in Anti-Plane Shear,” Mathematics and Mechanics of Solids, 1, pp. 327333 (1996)
6.Chao, C.K., and Young, C.W., “On the General Treatment of Multiple Inclusions in Antiplane Elas-tostatics,” International Journal of Solids and Structures, 35, pp. 35733593 (1998).
7.Ru, C.Q., Schiavone, P., and Mioduchowski, A., “Uniformity of Stresses Within a Three-Phase Elliptic Inclusion in Anti-Plane Shear,” Journal of Elasticity, 52, pp. 121128 (1999).
8.Shen, M.H., Chen, S.N., and Chen, F.M., “Anti-plane Study on Confocally Elliptical Inhomogeneity Problem Using an Alternating Technique,” Archive of Applied Mechanics, 75, pp. 302314 (2006).
9.Shen, M.H., Chen, S.N., and Chen, F.M., “Piezoelectric Study on Confocally Multicoated Elliptical Inclusion,” International Journal of Engineering Science, 43, pp. 12991312 (2005).
10.Chen, J.T., and Wu, A.C., “Null-Field Approach for the Multi-Inclusion Problem Under Antiplane Shears,” Journal of Applied Mechanics, 74, pp. 469487 (2007).
11.Chen, J.T., Shen, W.C., and Wu, A.C., “Null-Field Integral Equations for Stress Field Around Circular Holes Under Anti-Plane Shear,” Engineering Analysis with Boundary Elements, 30, pp.205217 (2006).
12.Chen, K.H., Chen, J.T., and Kuo, J.H., “Regularized Meshless Method for Antiplane Shear Problems,” International Journal for Numerical Methods in Engineering, 73, pp. 12511273 (2008).
13.Lee, Y.T., and Chen, J.T., “Null-Field Approach for the Antiplane Problem with Elliptical Holes And/Or Inclusions,” Journal of Composites B, 44, pp. 283294 (2013).
14.Ting, T.C.T., and Schiavone, P., “Uniform Antiplane Shear Stress Inside an Anisotropic Elastic Inclusion of Arbitrary Shape with Perfect or Imperfect Interface Bonding,” International Journal of Engineering Science, 48, pp. 6777 (2010).
15.Chen, Y.Z., “Closed form Solution for Eshelby's Elliptical Inclusion in Antiplane Elasticity Using Complex Variable,” Zeitschrift Fuer Angewandte Mathematik Und Physik, 69, pp. 17971805 (2013)
16.Chen, T., “A Confocally Multicoated Elliptical Inclusion Under Antiplane Shear: Some New Results,” Journal of Elasticity, 74, pp. 8797 (2004).
17.Chen, Y.Z., Hasebe, N., and Lee, K.Y., Multiple Crack Problems in Elasticity, WIT Press, Southampton (2003).
18.Muskhelishvili, N.I., Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Gro-ningen (1953).



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