An elastic half-plane with a fixed cycloid wavy boundary

Abstract

We present a rigorous analysis of the plane problem of an elastic half-plane with a fixed cycloid wavy boundary under biaxial tension using Muskhelishvili’s complex variable method. A closed-form expression for the pair of analytic functions characterizing the stresses and displacements in the elastic body is derived by means of analytic continuation. Furthermore, elementary expressions for the interfacial and hoop stresses along the cycloid interface and stress distributions along the right of an interface valley are obtained. For a cusped cycloid interface, the stresses exhibit the square root singularity at the cusp tips. The mode I stress intensity factor at a cusp tip is expediently extracted from the resulting analytic functions. The interfacial normal stress and hoop stress are constant along a cusped cycloid interface except at the isolated cusp tips.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    Akbari Mousavi, A.A., Al-Hassani, S.T.S.: Numerical and experimental studies of the mechanism of the wavy interface formations in explosive/impact welding. J. Mech. Phys. Solids 53(11), 2501–2528 (2005)

    Article  Google Scholar 

  2. 2.

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

    MathSciNet  Google Scholar 

  3. 3.

    Chiu, C.H., Gao, H.: Stress singularities along a cycloid rough surface. Int. J. Solids Struct. 30, 2983–3012 (1993)

    Article  Google Scholar 

  4. 4.

    Fan, H., Xiao, Z.M.: Dislocation interacting with a slightly wavy interface. Mech. Mater. 40(1), 2137–2161 (1996)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Gao, H.: Stress concentration at slightly undulating surfaces. J. Mech. Phys. Solids 39(4), 443–458 (1991)

    Article  Google Scholar 

  6. 6.

    Gao, H., Nix, W.D.: Surface roughening of heteroepitaxial thin films. Annu. Rev. Mater. Sci. 29, 173–209 (1999)

    Article  Google Scholar 

  7. 7.

    Jaramillo, D., et al.: On the transition from a waveless to a wavy interface in explosive welding. Mater. Sci. Eng. 91, 217–222 (1987)

    Article  Google Scholar 

  8. 8.

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

    MATH  Google Scholar 

  9. 9.

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

    MathSciNet  Article  Google Scholar 

  10. 10.

    Suo, Z.G.: Singularities interacting with interfaces and cracks. Int. J. Solids Struct. 25, 1133–1142 (1989)

    Article  Google Scholar 

  11. 11.

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

    Book  Google Scholar 

  12. 12.

    Wang, X., Schiavone, P.: Asymptotic elastic fields near an interface anticrack tip. Acta Mech. 230(12), 4385–4389 (2019)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Wang, X., Schiavone, P.: An edge dislocation near an anticrack in a confocal elliptical coating. Contin. Mech. Thermodyn. (2021). (in press)

  14. 14.

    Wang, Z.Y., Zhang, H.T., Chou, Y.T.: Characteristics of the elastic field of a rigid line inhomogeneity. ASME J. Appl. Mech. 52, 818–822 (1985)

    Article  Google Scholar 

  15. 15.

    Xiao, Z.M., Chen, B.J., Steele, C.R.: An edge dislocation interacting with a slightly wavy interface. Int. J. Eng. Sci. 40, 2137–2161 (2002)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Yang, L., Qu, J.: Fracture mechanics parameters for cracks on a slightly undulating interface. Int. J. Fract. 64(1), 79–91 (1993)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No: RGPIN – 2017 - 03716115112).

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Xu Wang or Peter Schiavone.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Schiavone, P. An elastic half-plane with a fixed cycloid wavy boundary. Z. Angew. Math. Phys. 72, 82 (2021). https://doi.org/10.1007/s00033-021-01519-5

Download citation

Keywords

  • Cycloid interface
  • Stress concentration
  • Stress intensity factor
  • Anticrack
  • Complex variable method
  • Conformal mapping
  • Analytic continuation

Mathematics Subject Classification

  • 30B40
  • 30C20
  • 30E25
  • 35Q74
  • 74-10
  • 74A60
  • 74B05
  • 74G70
  • 74M25
  • 74S70