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Achieving a uniform stress field in a coated non-elliptical inhomogeneity in the presence of a mode III crack

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Abstract

We establish design criteria which guarantee uniformity of stresses inside a coated non-elliptical inhomogeneity influenced by the presence of a finite mode III crack in a matrix subjected to uniform remote anti-plane shear stresses. We employ a particular conformal mapping function containing an unknown real density function which is obtained via the numerical solution of an associated Cauchy singular integral equation with the aid of the Gauss–Chebyshev integration formula. Interestingly, in contrast to the (non-elliptical) shape of the coated inhomogeneity which is influenced solely by the presence of the nearby crack, the resulting internal uniform stress field remains unaffected by the crack.

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References

  1. Christensen, R.M., Lo, K.H.: Solutions for effective shear properties in three-phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315–330 (1979)

    Article  Google Scholar 

  2. Erdogan, F., Gupta, G.D.: On the numerical solution of singular integral equations. Q. Appl. Math. 29, 525–534 (1972)

    Article  MathSciNet  Google Scholar 

  3. Luo, H.A., Weng, G.J.: On Eshelby’s S-tensor in a three-phase cylindrically concentric solid and the elastic moduli of fibre-reinforced composites. Mech. Mater. 8, 77–88 (1989)

    Article  Google Scholar 

  4. Mikata, Y., Taya, M.: Stress field in and around a coated short fiber in infinite matrix subjected to uniaxial and biaxial loading. ASME J. Appl. Mech. 52, 19–24 (1985)

    Article  Google Scholar 

  5. Muskhelishvili, N.I.: Singular Integral Equations. P. Noordhoff Ltd., Groningen (1953)

    MATH  Google Scholar 

  6. Qiu, Y.P., Weng, G.J.: Elastic moduli of thickly coated particle and fiber-reinforced composites. J. Appl. Mech. ASME 58, 388–398 (1991)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Ru, C.Q., Schiavone, P., Mioduchowski, A.: Uniformity of stresses within a three-phase elliptic inclusion in anti-plane shear. J. Elast. 52, 121–128 (1999)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  10. Walpole, L.J.: A coated inclusion in an elastic medium. Math. Proc. Camb. Philos. Soc. 83, 495–506 (1978)

    Article  MathSciNet  Google Scholar 

  11. Wang, X., Chen, L., Schiavone, P.: Uniformity of stresses inside a non-elliptical inhomogeneity interacting with a mode III crack. Proc. Roy. Soc. London A (2018). https://doi.org/10.1098/rspa.2018.0304

    Article  Google Scholar 

  12. Wang, X., Gao, X.L.: On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite. Z. Angew. Math. Phys. 62, 1101–1116 (2011)

    Article  MathSciNet  Google Scholar 

  13. Xiao, Z.M., Chen, B.J.: A screw dislocation interacting with a coated fiber. Mech. Mater. 32, 485–494 (2000)

    Article  Google Scholar 

Download references

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 & Liang Chen

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

    Peter Schiavone

Authors
  1. Xu Wang
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  2. Liang Chen
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  3. Peter Schiavone
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Correspondence to Xu Wang or Peter Schiavone.

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Wang, X., Chen, L. & Schiavone, P. Achieving a uniform stress field in a coated non-elliptical inhomogeneity in the presence of a mode III crack. Z. Angew. Math. Phys. 69, 138 (2018). https://doi.org/10.1007/s00033-018-1032-8

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  • DOI: https://doi.org/10.1007/s00033-018-1032-8

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