Skip to main content
SpringerLink
Log in

A mode III arc-shaped crack with surface elasticity

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We study the effect of surface elasticity on an arc-shaped crack in a linearly elastic isotropic homogeneous material under antiplane shear deformation. The surface mechanics is incorporated by using a continuum-based surface/interface model of Gurtin and Murdoch. We obtain a complete solution by reducing the problem to two decoupled first-order Cauchy-type singular integro-differential equations. It is shown that different from the case of a straight crack, the stresses exhibit both the weak logarithmic and the strong square root singularities at the tips of the arc crack.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Antipov Y.A., Schiavone P.: Integro-differential equation for a finite crack in a strip with surface effects. Q. J. Mech. Appl. Math. 64, 87–106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen T., Dvorak G.J., Yu C.C.: Size-dependent elastic properties of unidirectional nano-composites with interface stresses. Acta Mech. 188, 39–54 (2007)

    Article  MATH  Google Scholar 

  3. Gurtin M.E., Murdoch A.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gurtin M.E., Murdoch A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

    Article  MATH  Google Scholar 

  5. Gurtin M.E., Weissmuller J., Larche F.: A general theory of curved deformable interface in solids at equilibrium. Philos. Mag. A 78, 1093–1109 (1998)

    Article  Google Scholar 

  6. Kim C.I., Ru C.Q., Schiavone P.: A clarification of the role of crack-tip conditions in linear elasticity with surface effects. Math. Mech. Solids 18, 59–66 (2013)

    Article  MathSciNet  Google Scholar 

  7. Kim C.I., Schiavone P., Ru C.Q.: The effects of surface elasticity on an elastic solid with mode-III crack: complete solution. ASME J. Appl. Mech. 77, 021011-1–021011-7 (2010a)

    Google Scholar 

  8. Kim C.I., Schiavone P., Ru C.Q.: Analysis of a mode III crack in the presence of surface elasticity and a prescribed non-uniform surface traction. Z. Angew. Math. Phys. 61, 555–564 (2010b)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kim C.I., Schiavone P., Ru C.Q.: Analysis of plane-strain crack problems (mode I and mode II) in the presence of surface elasticity. J. Elast. 104, 397–420 (2011a)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kim C.I., Schiavone P., Ru C.Q.: The effects of surface elasticity on mode-III interface crack. Arch. Mech. 63, 267–286 (2011b)

    MathSciNet  MATH  Google Scholar 

  11. Kim C.I., Schiavone P., Ru C.Q.: Effect of surface elasticity on an interface crack in plane deformations. Proc. R. Soc. Lond. A 467, 3530–3549 (2011c)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lengyel T.H., Schiavone P.: Displacement field in an elastic solid with mode-III crack and first-order surface effects. J. Mech. Mater. Struct. 7(8), 783–794 (2012)

    Article  Google Scholar 

  13. Markenscoff X., Dundurs J.: Annular inhomogeneities with eigenstrain and interphase modeling. J. Mech. Phys. Solids 64, 468–482 (2014)

    Article  MathSciNet  Google Scholar 

  14. Ru C.Q.: Simple geometrical explanation of Gurtin–Murdoch model of surface elasticity with clarification of its related versions. Sci. China 53, 536–544 (2010)

    MathSciNet  Google Scholar 

  15. Sigaeva, T., Schiavone, P.: The effect of surface stress on an interface crack in linearly elastic materials. Math. Mech. Solids (2014). doi:10.1177/1081286514534871

  16. Steigmann D.J., Ogden R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. Lond. A. 453, 853–877 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Walton J.R.: A note on fracture models incorporating surface elasticity. J. Elast. 109, 95–102 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zemlyanova, A.Y.: Curvilinear mode-I/mode-II interface fracture with a curvature-dependent surface tension on the boundary. arXiv preprint arXiv:1409.3501 (2014)

  19. Zemlyanova A.Y., Walton J.R.: Modeling of a curvilinear planar crack with a curvature-dependent surface tension. SIAM J. Appl. Math. 72, 1474–1492 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

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

Authors
  1. Xu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xu Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X. A mode III arc-shaped crack with surface elasticity. Z. Angew. Math. Phys. 66, 1987–2000 (2015). https://doi.org/10.1007/s00033-014-0482-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-014-0482-x

Mathematics Subject Classification

Keywords

Search

Navigation