On two circular inclusions in harmonic problems

Authors:
E. Honein, T. Honein and G. Herrmann

Journal:
Quart. Appl. Math. **50** (1992), 479-499

MSC:
Primary 73C99; Secondary 73B27, 73K20, 73V35

DOI:
https://doi.org/10.1090/qam/1178429

MathSciNet review:
MR1178429

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Abstract: In this paper, we derive the solution for two circular cylindrical elastic inclusions perfectly bonded to an elastic matrix of infinite extent, under anti-plane deformation. The two inclusions have different radii and possess different elastic properties. The matrix is subjected to *arbitrary* loading. The solution is obtained, via iterations of Möbius transformations, as a rapidly convergent series with an explicit general term involving the complex potential of the corresponding homogeneous problem, i.e., when the inclusions are absent and the matrix material occupies the entire space and is subjected to the same loading. This procedure has been termed ``heterogenization."

**[1]**L. M. Milne-Thomson,*Hydrodynamical images*, Proc. Cambridge Philos. Soc.**36**(1940), 246–247. MR**0001693****[2]**L. M. Milne-Thomson,*Theoretical hydrodynamics*, 4th ed, The Macmillan Co., New York, 1960. MR**0112435****[3]**Wolfgang K. H. Panofsky and Melba Phillips,*Classical electricity and magnetism*, Second edition. Addison-Wesley Series in Physics, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1962. MR**0135824****[4]**L. D. Landau and E. M. Lifshitz,*Course of theoretical physics. Vol. 8*, Pergamon International Library of Science, Technology, Engineering and Social Studies, Pergamon Press, Oxford, 1984. Electrodynamics of continuous media; Translated from the second Russian edition by J. B. Sykes, J. S. Bell and M. J. Kearsley; Second Russian edition revised by Lifshits and L. P. Pitaevskiĭ. MR**766230****[5]**T. Honein and G. Herrmann,*The two-dimensional problem of thermo-elasticity for an infinite region bounded by a circular cavity*, Thermodynamical Couplings in Solids, H. D. Bui and Q. S. Nguyen, eds., Elsevier Science Publ. B. V. (North-Holland), Amsterdam, 1987**[6]**T. Honein and G. Herrmann,*The involution correspondence in plane elastostatics for regions bounded by a circle*, Trans. ASME J. Appl. Mech.**55**(1988), no. 3, 566–573. MR**962465**, https://doi.org/10.1115/1.3125831**[7]**T. Honein and G. Herrmann,*On bonded inclusions with circular or straight boundaries in plane elastostatics*, J. Appl. Mech.**57**, 850-856 (1990)**[8]**T. Honein and G. Herrmann,*A circular inclusion with slipping interface in plane elastostatics*, Micromechanics and Inhomogeneity, G. J. Weng, M. Taya, and H. Abé, eds. Springer-Verlag, 1990**[9]**Hans Schwerdtfeger,*Geometry of complex numbers*, Dover Publications, Inc., New York, 1979. Circle geometry, Moebius transformation, non-Euclidean geometry; A corrected reprinting of the 1962 edition; Dover Books on Advanced Mathematics. MR**620163****[10]**E. Smith,*The interaction between dislocations and inhomogeneities*-I, Internat. J. Engrg. Sci.**6**, 129-143 (1968)**[11]**Lin Wei-Wen, T. Honein, and G. Herrmann,*A novel method of stress analysis of elastic materials with damage zones*, Yielding, Damage, and Failure of Anisotropic Solids, EGFS, J. P. Boehler, ed., 1989, Mechanical Engineering Publ., London, 1988, 609-613**[12]**L. R. Ford,*Automorphic Functions*, McGraw-Hill, New York, 1929; 2nd ed., Chelsea, New York, 1951**[13]**J. G. Goree and H. B. Wilson Jr.,*Transverse shear loading in an elastic matrix containing two circular cylindrical inclusions*, J. Appl. Mech., June 1967, 511-513 (1967)**[14]**B. Budiansky and G. F. Carrier,*High shear stress in stiff-fiber composites*, J. Appl. Mech.**51**. 733-735 (1984)**[15]**Paul S. Steif,*Shear stress concentration between holes*, J. Appl. Mech.**56**, 719-721 (1989)**[16]**M. Abramowitz and I. Stegun, eds.,*Handbook of Mathematical Functions*, Dover, New York, 1965**[17]**G. P. Sendeckyj,*Screw dislocations near circular inclusions*, Phys. Status Solidi**3**, 529-535 (1970)**[18]**W. Burnside,*On functions determined from their discontinuities and a certain form of boundary condition*, Proc. London Math. Soc. (3)**22**, 346-358 (1891)**[19]**W. Burnside,*On a class of automorphic functions*, Proc. London Math. Soc. (3)**23**, 49-88 (1892)**[20]**Alain Drotz,*Détermination analytique de potentiels d’écoulements non visqueux, incompressibles et bidimensionnels autour d’obstacles circulaires*, J. Méc. Théor. Appl.**6**(1987), no. 1, 23–45 (French, with English summary). MR**886821**

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DOI:
https://doi.org/10.1090/qam/1178429

Article copyright:
© Copyright 1992
American Mathematical Society