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IssuesArchive of Issues2004-1pp.36-55

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R. V. Goldstein and E. I. Shifrin, "Integral equations of the problem of an elastic inclusion. Complete analytical solution of the problem of an elliptic inclusion," Mech. Solids. 39 (1), 36-55 (2004)
Year 2004 Volume 39 Number 1 Pages 36-55
Title Integral equations of the problem of an elastic inclusion. Complete analytical solution of the problem of an elliptic inclusion
Author(s) R. V. Goldstein (Moscow)
E. I. Shifrin (Moscow)
Abstract We consider the problem of an isotropic elastic space containing an isotropic elastic inclusion. Using a generalization of the Eshelby method, we construct new integral equations for the stress tensor components inside the inclusion. Detailed investigation is carried out in the case of the plane problem and an elliptic inclusion. In this case, explicit analytical expressions are obtained for the stress tensor components inside and outside the inclusion. Using the procedure of passing to the limit, we construct solutions for an elliptic hole and a rigid elliptic inclusion. In the case of circular inclusions, simplified formulas for stresses are obtained as a special case of the general formulas. For all special cases found in the literature in which analytical expressions of the stresses have been constructed, the solutions obtained in this paper are compared with those obtained previously.
References
1.  V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-Dimensional Mathematical Problems in Elasticity and Thermoelasticity [in Russian], Nauka, Moscow, 1976.
2.  V. Z. Parton and P. I. Perlin, Integral Equations of Elasticity [in Russian], Nauka, Moscow, 1977.
3.  N. J. Hardiman, " Elliptic elastic inclusion in an infinite elastic plane," Quart. J. Mech. Appl. Math., Vol. 7, pp. 226-230, 1954.
4.  J. Eshelby, "Determination of the elastic stress field caused by an ellipsoidal inclusion and related problems," in Continuous Dislocation Theory [Russian translation], pp. 103-139, Izd-vo Inostr. Lit-ry., Moscow, 1963.
5.  R. V. Goldstein and E. I. Shifrin, Plane Problem for the Stress State Caused by Phase Transitions in an Elliptic Region. Preprint No. 714 [in Russian], Institute for Problem in Mechanics, Russian Academy of Sciences, Moscow, 2003.
6.  N. I. Muskhelishvili, Some Basic Problems in Mathematical Elasticity [in Russian], Nauka, Moscow, 1966.
7.  L. I. Sedov, Continuum Mechanics [in Russian], Nauka, Moscow, 1976.
8.  S. P. Timoshenko and J. Goodier, Theory of Elasticity [Russian translation], Nauka, Moscow, 1979.
9.  H. Hahn,, Theory of Elasticity [Russian translation], Mir, Moscow, 1988.
10.  P. F. Papkovich, Theory of Elasticity, Gos. Izd-vo Oboron. Prom., Moscow, Leningrad, 1939.
11.  J. N. Goodier, "Concentration of stress around spherical and cylindrical inclusions and flaws," J. Appl. Mech., APM-55-7, pp. 39-44, 1933.
Received 08 October 2003
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