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IssuesArchive of Issues2010-5pp.726-732

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I.P. Dobrovolskii, "An Inclusion Problem," Mech. Solids. 45 (5), 726-732 (2010)
Year 2010 Volume 45 Number 5 Pages 726-732
DOI 10.3103/S0025654410050080
Title An Inclusion Problem
Author(s) I.P. Dobrovolskii (Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, B.†Gruzinskaya 10, Moscow, 123995 Russia, dipedip@gmail.com)
Abstract An inclusion is a special region in a material, and this region experiences a transformation of the following nature. If the inclusion were free, then it would acquire a certain deformation with no stress arising in it; but since the inclusion is "pasted" into the material, this prevents free deformations and causes stresses arising in the inclusion itself and in the environment. Three systems of equations describing the problem are derived. For a space with a homogeneous isotropic matrix, an equivalent system of integral equations is obtained whose solution, for a homogeneous anisotropic ellipsoidal inclusion, is reduced to a system of linear algebraic equations. For the case where the moduli of elasticity in the inclusion and the homogeneous matrix coincide, an explicit solution for an inclusion of arbitrary shape is obtained.
Keywords equilibrium equations, homogeneous inclusion, volume potential
References
1.  J. D. Eshelby, Continual Theory of Dislocations (Inostr. Lit-ra, Moscow, 1963) [in Russian].
2.  R. V. Goldstein and E. I. Shifrin, "Integral Equations of the Problem on an Elastic Inclusion. Complete Analytical Solution of the Problem on an Elliptic Inclusion," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 50-76 (2004) [Mech. Solids (Engl. Transl.) 39 (1), 36-55 (2004)].
3.  R. V. Goldstein and E. I. Shifrin, "Stress State in an Elastic Space due to Phase Transformations in an Inclusion," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 48-64 (2005) [Mech. Solids (Engl. Transl.) 40 (5), 34-47 (2005)].
4.  V. I. Shevchenko, I. P. Dobrovolskii, and A. A. Lukk, "The Stress-Strain State of the Lithosphere in the Aegean Sector of the Mediterranean Mobile Belt," Fiz. Zemli, No. 12, 52-63 (2001) [Izv. Phys. Solid Earth (Engl. Transl.) 37 (12), 1015-1025 (2001)].
5.  I. P. Dobrovolsky and V. I. Shevchenko, "On the Origin of Subhorizontal Compressive Stresses and Deformations in a Mobile Belt," Fiz. Zemli, No. 2, 46-49 (2006) [Izv. Phys. Solid Earth (Engl. Transl.) 42 (2), 132-135 (2006)].
6.  V. A. Lomakin, Theory of Elasticity of Inhomogeneous Solids (Izd-vo MGU, Moscow, 1976) [in Russian].
7.  I. P. Dobrovolskii, "Problem on an Inhomogeneity in a Linearly Elastic Space or Half-Space," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 59-66 (2007) [Mech. Solids (Engl. Transl.) 42 (1), 50-56 (2007)].
8.  I. M. Lifshits and L. N. Rozentsveig, "On the Construction of the Green's Tensor for the Basic Equation of Elasticity for an Unbounded Elasto-Anisotropic Medium," Zh. Eksp. Teor. Fiz. 17 (9), 783-791 (1947).
9.  L. N. Sretenskii, Theory of Newtonian Potential (OGIZ-GOSTEKhIZDAT, Moscow-Leningrad, 1946) [in Russian].
10.  V. D. Kupradze (Editor) Three-Dimensional Problems of Mathematical Elasticity and Thermoelasticity (Nauka, Moscow, 1976) [in Russian].
11.  R. J. Asaro, "Somigliana Dislocations and Internal Stresses; with Application to Second Phase Hardening," Int. J. Engng Sci. 13 (3), 271-286 (1975).
Received 10 May 2007
Link to Fulltext http://www.springerlink.com/content/446176w32t0278v8/
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