Abstract
This paper provides a closed-form solution for the Eshelby’s elliptic inclusion in antiplane elasticity. In the formulation, the prescribed eigenstarins are not only for the uniform distribution, but also for the linear form. After using the complex variable and the conformal mapping, the continuation condition for the traction and displacement along the interface in the physical plane can be reduced to a condition along the unit circle. The relevant complex potentials defined in the inclusion and the matrix can be separated from the continuation conditions of the traction and displacement along the interface. The expressions of the real strains and stresses in the inclusion from the assumed eigenstrains are presented. Results for the case of linear distribution of eigenstrain are first obtained in the paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Eshelby J.D.: The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A 241, 376–396 (1957)
Mura, T: Micromechanics of Defects in Solids. Martinus Nijhoff Publishers, Dordrecht (1982)
Nozaki H., Taya M.: Elastic fields in a polygon shaped inclusion with uniform eigenstrains. ASME J. Appl. Mech. 64, 495–502 (1997)
Nozaki H., Taya M.: Elastic fields in a polyhedral inclusion with uniform eigenstrains and related problems. ASME J. Appl. Mech. 68, 441–452 (2001)
Ru C.Q.: Analytic solution for Eshelby’s problem of an arbitrary shape in a plane or half-plane. ASME J. Appl. Mech. 66, 315–322 (1999)
Ru C.Q.: Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane. Acta Mech. 160, 219–234 (2003)
Wang M.Z., Xu B.X.: The arithmetic mean theorem of Eshelby tensor for a rotational symmetrical inclusion. J. Elast. 77, 12–23 (2005)
Zou W.N., He Q.C., Huang M.J.: Zheng QS Eshelby’s problem of non-elliptical inclusions. J. Mech. Phys. Solids 58, 346–372 (2010)
Wang X., Gao X.L.: On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite. Z. Angew. Math. Phys. 62, 1101–1116 (2011)
Gong S.X., Meguid S.A.: A general treatment of the elastic field of an elliptical inhomogeneity under antiplne shear. J. Appl. Mech. 45, s131–135 (1992)
Gong X.S.: A unified treatment of the elastic elliptical inclusion under antiplane shear. Arch. Appl. Mech. 65, 55–64 (1955)
Ru C.Q., Schiavone P.: On the elliptic inclusion in anti-plane shear, Math. Mech. Solids 1, 327–333 (1996)
Ru C.Q., Schiavone P.: A circular inclusion with circumferentially inhomogeneous interface in antiplane shear. Proc. R. Soc. Lond. A. 453, 2551–2572 (1997)
Xu B.X., Wang M.Z.: The arithmetic mean theorem for the N-fold rotational symmetrical inclusion in anti-plane elasticity. Acta Mech. 194, 233–242 (2007)
Gao X.L., Ma H.M.: Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem. Acta Mech. 223, 1067–1080 (2012)
Chen Y.Z., Hasebe N., Lee K.Y.: Multiple Crack Problems in Elasticity. WIT Press, Southampton (2003)
Muskhelishvili N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1953)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, Y.Z. Closed-form solution for Eshelby’s elliptic inclusion in antiplane elasticity using complex variable. Z. Angew. Math. Phys. 64, 1797–1805 (2013). https://doi.org/10.1007/s00033-013-0305-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-013-0305-5