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."
- L. M. Milne-Thomson, Hydrodynamical images, Proc. Cambridge Philos. Soc. 36 (1940), 246–247. MR 1693, DOI https://doi.org/10.1017/s0305004100017242
- L. M. Milne-Thomson, Theoretical hydrodynamics, The Macmillan Co., New York, 1960. 4th ed. MR 0112435
- Wolfgang K. H. Panofsky and Melba Phillips, Classical electricity and magnetism, 2nd ed., Addison-Wesley Series in Physics, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1962. MR 0135824
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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
- 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, DOI https://doi.org/10.1115/1.3125831
T. Honein and G. Herrmann, On bonded inclusions with circular or straight boundaries in plane elastostatics, J. Appl. Mech. 57, 850–856 (1990)
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
- 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
E. Smith, The interaction between dislocations and inhomogeneities-I, Internat. J. Engrg. Sci. 6, 129–143 (1968)
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
L. R. Ford, Automorphic Functions, McGraw-Hill, New York, 1929; 2nd ed., Chelsea, New York, 1951
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)
B. Budiansky and G. F. Carrier, High shear stress in stiff-fiber composites, J. Appl. Mech. 51. 733–735 (1984)
Paul S. Steif, Shear stress concentration between holes, J. Appl. Mech. 56, 719–721 (1989)
M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1965
G. P. Sendeckyj, Screw dislocations near circular inclusions, Phys. Status Solidi 3, 529–535 (1970)
W. Burnside, On functions determined from their discontinuities and a certain form of boundary condition, Proc. London Math. Soc. (3) 22, 346–358 (1891)
W. Burnside, On a class of automorphic functions, Proc. London Math. Soc. (3) 23, 49–88 (1892)
- 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
L. M. Milne-Thomson, Hydrodynamical images, Proc. Cambridge Philos. Soc. 36, 246–247 (1940)
L. M. Milne-Thomson, Theoretical Hydrodynamics, Macmillan, London, 1960
W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed., Addison-Wesley, London, 1962
L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskiĭ, Electrodynamics of Continuous Media, 2nd ed., Pergamon, Oxford, 1984
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
T. Honein and G. Herrmann, The involution correspondence in plane elastostatics for regions bounded by a circle, J. Appl. Mech, 55, 566–573 (1988)
T. Honein and G. Herrmann, On bonded inclusions with circular or straight boundaries in plane elastostatics, J. Appl. Mech. 57, 850–856 (1990)
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
H. Schwerdtfeger, The Geometry of Complex Numbers, Dover, New York, 1979
E. Smith, The interaction between dislocations and inhomogeneities-I, Internat. J. Engrg. Sci. 6, 129–143 (1968)
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
L. R. Ford, Automorphic Functions, McGraw-Hill, New York, 1929; 2nd ed., Chelsea, New York, 1951
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)
B. Budiansky and G. F. Carrier, High shear stress in stiff-fiber composites, J. Appl. Mech. 51. 733–735 (1984)
Paul S. Steif, Shear stress concentration between holes, J. Appl. Mech. 56, 719–721 (1989)
M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1965
G. P. Sendeckyj, Screw dislocations near circular inclusions, Phys. Status Solidi 3, 529–535 (1970)
W. Burnside, On functions determined from their discontinuities and a certain form of boundary condition, Proc. London Math. Soc. (3) 22, 346–358 (1891)
W. Burnside, On a class of automorphic functions, Proc. London Math. Soc. (3) 23, 49–88 (1892)
Alain Drotz, Détermination analytique de potentiels d’écoulements non visqueux, incompressibles et bidimensionnels autour d’obstacles circulaires, J. Theoret. Appl. Mech. 6, 23–45 (1987)
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Article copyright:
© Copyright 1992
American Mathematical Society
