Three-phase parabolic inhomogeneities with internal uniform stresses in plane and anti-plane elasticity

• Autorzy: Wang X. , Schiavone P.

Identyfikatory

DOI

10.24423/aom.3371

• Języki publikacji: EN

Abstrakty

EN

We examine the in-plane and anti-plane stress states inside a parabolic inhomogeneity which is bonded to an infinite matrix through an intermediate coating. The interfaces of the three-phase parabolic inhomogeneity are two confocal parabolas. The corresponding boundary value problems are studied in the physical plane rather than in the image plane. A simple condition is found that ensures that the internal stress state inside the parabolic inhomogeneity is uniform and hydrostatic. Furthermore, this condition is independent of the elastic properties of the coating and the two geometric parameters of the composite: in fact, the condition depends only on the elastic constants of the inhomogeneity and the matrix and the ratio between the two remote principal stresses. Once this condition is met, the mean stress in the coating is constant and the hoop stress on the coating side is also uniform along the entire inhomogeneity-coating interface. The unconditional uniformity of stresses inside a three-phase parabolic inhomogeneity is achieved when the matrix is subjected to uniform remote anti-plane shear stresses. The internal uniform anti-plane shear stresses inside the inhomogeneity are independent of the shear modulus of the coating and the two geometric parameters of the composite.

Słowa kluczowe

EN

three-phase parabolic inhomogeneity; coating; internal uniform stresses; plane elasticity; anti-plane elasticity

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2020
• Tom: Vol. 72, nr 1
• Strony: 27--38
• Opis fizyczny: Bibliogr. 23 poz., rys.

Twórcy

autor

Wang X.

• School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China

autor

Schiavone P.

• Department of Mechanical Engineering, University of Alberta, 10-203 Donadeo Innovation Centre for Engineering, Edmonton, Alberta Canada T6G 1H9

Bibliografia

• 1. N.J. Hardiman, Elliptic elastic inclusion in an infinite plate, Quarterly Journal of Mechanics and Applied Mathematics, 7, 226–230, 1954.
• 2. J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion and related problems, Proceedings of the Royal Society of London, A 241, 376–396, 1957.
• 3. J.D. Eshelby, The elastic field outside an ellipsoidal inclusion, Proceedings of the Royal Society of London, A 252, 561–569, 1959.
• 4. J.D. Eshelby, Elastic inclusions and inhomogeneities, Progress in Solid Mechanics, Vol. II, 89–140, 1961.
• 5. G.P. Sendeckyj, Elastic inclusion problem in plane elastostatics, International Journal of Solids and Structures, 6, 1535–1543, 1970.
• 6. S.X. Gong, S.A. Meguid, A general treatment of the elastic field of an elliptic inhomogeneity under anti-plane shear, ASME Journal of Applied Mechanics, 59, S131–S135, 1992.
• 7. T.C.T. Ting, Anisotropic Elasticity-Theory and Applications, Oxford University Press, New York, 1996.
• 8. C.Q. Ru, P. Schiavone, On the elliptic inclusion in anti-plane shear, Mathematics and Mechanics of Solids, 1, 327–333, 1996.
• 9. Y.A. Antipov, P. Schiavone, On the uniformity of stresses inside an inhomogeneity of arbitrary shape, IMA Journal of Applied Mathematics, 68, 299–311, 2003.
• 10. L.P. Liu, Solution to the Eshelby conjectures, Proceedings of the Royal Society of London, A 464, 573–594, 2008.
• 11. H. Kang, E. Kim, G.W. Milton, Inclusion pairs satisfying Eshelby’s uniformity property, SIAM Journal on Applied Mathematics, 69, 577–595, 2008.
• 12. X. Wang, Uniform fields inside two non-elliptical inclusions, Mathematics and Mechanics of Solids, 17, 736–761, 2012.
• 13. M. Dai, C.F. Gao, C.Q. Ru, Uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformation, Proceedings of the Royal Society of London, A 471 (2177): 20140933, 2015.
• 14. X. Wang, P. Schiavone, Uniformity of stresses inside a parabolic inhomogeneity, Zeitschrift für angewandte Mathematik und Physik (in press).
• 15. C.Q. Ru, Three-phase elliptical inclusions with internal uniform hydrostatic stresses, Journal of the Mechanicsand Physics of Solids, 47, 259–273, 1999.
• 16. R.M. Christensen, K.H. Lo, Solutions for effective shear properties in three-phase sphere and cylinder models, Journal of the Mechanics and Physics of Solids, 27, 315–330, 1979.
• 17. H.A. Luo, G.J. Weng, On Eshelby’s S-tensor in a three-phase cylindrically concentric solid and the elastic moduli of fiber-reinforced composites, Mechanics of Materials, 8, 77–88, 1989.
• 18. C.Q. Ru, P. Schiavone, A. Mioduchowski, Uniformity of stresses within a three-phase elliptic inclusion in anti-plane shear, Journal of Elasticity, 52, 121–128, 1999.
• 19. G.W. Milton, S.K. Serkov, Neutral coated inclusions in conductivity and anti-plane elasticity, Proceedings of the Royal Society of London, A 457, 1973–1997, 2001.
• 20. P. Jarczyk, V. Mityushev, Neutral coated inclusions of finite conductivity, Proceedings of the Royal Society of London, A 468 (2140), 954–970, 2012.
• 21. Yu.V. Obnosov, A generalized Milne–Thomson theorem for the case of parabolic inclusion, Applied Mathematical Modelling, 33, 1970–1981, 2009.
• 22. J.R. Philip, Seepage shedding by parabolic capillary barriers and cavities, Water Resources Research, 34, 2827–2835, 1998.
• 23. N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff Ltd., Groningen, 1953.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).