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Uniform stresses inside a non-elliptical inhomogeneity and a nearby half-plane with locally wavy interface

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Abstract

We develop a two-parameter conformal mapping function for a doubly connected domain to solve the inverse problem in anti-plane and plane elasticity associated with a non-elliptical inhomogeneity with internal uniform stresses embedded in a half-plane bonded to another half-plane also with internal uniform stresses via a locally wavy interface. The internal uniform stresses are found to be independent of the specific shapes of the locally wavy and non-elliptical interfaces. The permissible range of the two parameters in the mapping function for a one-to-one mapping is obtained. Typical geometries corresponding to the three-phase composite are illustrated.

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References

  1. Antipov, Y.A.: Method of automorphic functions for an inverse problem of antiplane elasticity. Q. J. Mech. Appl. Math. 72(2), 213–234 (2019)

    Article  MathSciNet  Google Scholar 

  2. Bjorkman, G.S., Richards, R.: Harmonic holes-an inverse problem in elasticity. ASME J. Appl. Mech. 43, 414–418 (1976)

    Article  Google Scholar 

  3. Bjorkman, G.S., Richards, R.: Harmonic holes for nonconstant fields. ASME J. Appl. Mech. 46, 573–576 (1979)

    Article  Google Scholar 

  4. Cherepanov, G.P.: Inverse problem of the plane theory of elasticity. Prikl. Mat. Mekh. 38, 963–979 (1974)

    MathSciNet  Google Scholar 

  5. Dai, M., Gao, C.F., Ru, C.Q.: Uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformation. Proc. R. Soc. Lond. A 471(2177), 20140933 (2015)

    Article  Google Scholar 

  6. Gong, S.X., Meguid, S.A.: A general treatment of the elastic field of an elliptic inhomogeneity under anti-plane shear. ASME J. Appl. Mech. 59, S131–S135 (1992)

    Article  Google Scholar 

  7. Hardiman, N.J.: Elliptic elastic inclusion in an infinite plate. Q. J. Mech. Appl. Math. 7, 226–230 (1954)

    Article  MathSciNet  Google Scholar 

  8. Kang, H., Kim, E., Milton, G.W.: Inclusion pairs satisfying Eshelby’s uniformity property. SIAM J. Appl. Math. 69, 577–595 (2008)

    Article  MathSciNet  Google Scholar 

  9. Liu, L.P.: Solution to the Eshelby conjectures. Proc. R. Soc. Lond. A 464, 573–594 (2008)

    Article  MathSciNet  Google Scholar 

  10. Markenscoff, X.: Evolution of growth of interacting near-by micro-cavitie and macroscopic damage amplification. Math. Mech. Solids 19, 71–81 (2014)

    Article  MathSciNet  Google Scholar 

  11. Markenscoff, X., Dundurs, J.: Amplification of stresses in thin ligaments. Int. J. Solids Struct. 29, 1883–1888 (1992)

    Article  Google Scholar 

  12. Marshall, J.S.: On sets of multiple equally strong holes in an infinite elastic plate: parameterization and existence. SIAM J. Appl. Math. 79, 2288–2312 (2019)

    Article  MathSciNet  Google Scholar 

  13. Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordhoff Ltd., Groningen (1953)

    MATH  Google Scholar 

  14. Rice, J.R., Sih, G.C.: Plane problems of crack in dissimilar media. ASME J. Appl. Mech. 32, 418–423 (1965)

    Article  Google Scholar 

  15. Ru, C.Q.: Three-phase elliptical inclusions with internal uniform hydrostatic stresses. J. Mech. Phys. Solids 47, 259–273 (1999)

    Article  MathSciNet  Google Scholar 

  16. Ru, C.Q., Schiavone, P.: On the elliptic inclusion in anti-plane shear. Math. Mech. Solids 1, 327–333 (1996)

    Article  MathSciNet  Google Scholar 

  17. Sendeckyj, G.P.: Elastic inclusion problem in plane elastostatics. Int. J. Solids Struct. 6, 1535–1543 (1970)

    Article  Google Scholar 

  18. Ting, T.C.T.: Anisotropic Elasticity-Theory and Applications. Oxford University Press, New York (1996)

    Book  Google Scholar 

  19. Wang, X.: Uniform fields inside two non-elliptical inclusions. Math. Mech. Solids 17, 736–761 (2012)

    Article  MathSciNet  Google Scholar 

  20. Wang, X., Yang, P., Schiavone, P.: Uniform fields inside two interacting non-parabolic and non-elliptical inhomogeneities. Z. Angew. Math. Phys. 71(1), 25 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11272121) and through a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN – 2017 - 03716115112).

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Authors and Affiliations

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

    Xu Wang & Ping Yang

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

    Peter Schiavone

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  1. Xu Wang
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  2. Ping Yang
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  3. Peter Schiavone
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Correspondence to Xu Wang or Peter Schiavone.

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Wang, X., Yang, P. & Schiavone, P. Uniform stresses inside a non-elliptical inhomogeneity and a nearby half-plane with locally wavy interface. Z. Angew. Math. Phys. 71, 58 (2020). https://doi.org/10.1007/s00033-020-1281-1

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  • DOI: https://doi.org/10.1007/s00033-020-1281-1

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