Skip to main content
SpringerLink
Log in

Finite deformations at the vertex of a bi-material wedge

  • Published:
International Journal of Fracture Aims and scope Submit manuscript

Abstract

The paper gives an asymptotic analysis of finite plane-strain deformations near the vertex of a bi-material wedge of arbitrary angle and subjected to the traction-free surface conditions. Each of two edge-bonded dissimilar wedges is assumed to be hyperelastic with the harmonic-type strain energy density introduced by John (1960). In contrast to the results of linearized elastostatics, the deformations and stresses around the vertex of an arbitrary bi-material wedge are found to be free of oscillatory singularities within the context of finite elastostatics. A simple relation is found between the material moduli and two wedge angles that classifies the higher-order singular field into two different types. The explicit conditions for the strict positivity of the Jacobian determinant in the vicinity of the vertex are given in terms of the known material and geometric parameters independently of the loading conditions. In particular, these conditions can be easily verified for several typical kinds of bi-material wedge. Finally, the expressions for the singular field near the wedge vertex are given and the main features of deformations and stresses are summarized.

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

  • Abeyaratne, R. (1983). Some finite elasticity problems involving crack-tips. Modelling Problems in Crack-tip Mechanics, (Edited by J.T. Pindera and B.R. Krasnowski), Martinus Nijhoff.

  • Bogy, D.B. (1971). Two edge-bonded elastic wedges of dissimilar materials and wedge angles under surface traction. ASME, Journal of Applied Mechanics 38, 377–386.

    Google Scholar 

  • Carroll, M.M. (1987). Finite strain solutions in compressible isotropic elasticity. Journal of Elasticity 19, 65–92.

    Google Scholar 

  • Chen, Y.Z. and Hasebe, Norio (1993). Singularity eigenvalue analysis of a crack along a wedge-shaped interface. ASME, Journal of Applied Mechanics 60, 781–783.

    Article  MATH  ADS  Google Scholar 

  • Dempsey, J.P. and Sinclair, G.B. (1981). On the singular behavior at the vertex of a bi-material wedge. Journal of Elasticity 11, 317–327.

    MATH  MathSciNet  Google Scholar 

  • England, A.E. (1965). A crack between dissimilar media. ASME, Journal of Applied Mechanics 32, 400–403.

    Google Scholar 

  • Geubelle, P. and Knauss, W.G. (1994). Finite strains at the tip of a crack in a sheet of hyperelastic material: I. Homogeneous case, II. Special bimaterial cases, III. General bimaterial case. Journal of Elasticity 35, 61–98, 99–138, 139–174.

    Article  MATH  Google Scholar 

  • Hein, V.L. and Erdogan, F. (1971). Stress singularities in a two-material wedge. International Journal of Fracture Mechanics 7, 317–330.

    Article  Google Scholar 

  • Herrmann, J.M. (1989). An asymptotic analysis of finite deformation near the tip of an interface-crack. Journal of Elasticity 21, 227–269.

    MATH  MathSciNet  ADS  Google Scholar 

  • John, F. (1960). Plane strain problems for a perfectly elastic material of harmonic type. Communications of Pure and Applied Mathematics XIII, 239–290.

    Google Scholar 

  • Knowles, J.K. and Sternberg, E. (1975). On the singularity induced by cartain mixed boundary conditions in linearized and nonlinear elastostatics. International Journal of Solids and Structures 11, 1173–1201.

    Article  MATH  MathSciNet  Google Scholar 

  • Knowles, J.K. and Sternberg, E. (1983). Large deformation near a tip of an interface-crack between two Neo-Hookean sheets. Journal of Elasticity 13, 257–293.

    Article  MATH  MathSciNet  Google Scholar 

  • Le, K.C. (1992). On the singular elastostatic field induced by a crack in a Hadamard material. Quarterly Journal of Mechanics and Applied Mathematics 45, 101–117.

    MATH  MathSciNet  Google Scholar 

  • Le, K.C. and Stumpf, H. (1993). The singular elastostatic field due to a crack in rubberlike materials. Journal of Elasticity 32, 183–222.

    Article  MATH  MathSciNet  Google Scholar 

  • Ogden, R.W. and Isherwood, D.A. (1978). Solution of some finite plane-strain problems for compressible elastic solids. Quarterly Journal of Mechanics and Applied Mathematics XXXI, 219–249.

    MathSciNet  Google Scholar 

  • Steigmann, D.J. and Pipkin, A.C. (1988). Stability of harmonic materials in plane strain. Quarterly of Applied Mathematics XLVI, 559–568.

    MathSciNet  Google Scholar 

  • Varley, E. and Cumberbatch, E. (1980). Finite deformation of elastic materials surrounding cylindrical holes. Journal of Elasticity 10, 341–405.

    Article  MATH  Google Scholar 

  • Williams, M.L. (1959). The stresses around a fault within the dissimilar media. Bulletin of the Seismological Society of America 49, 199–204.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2G8

    C.Q. Ru

Authors
  1. C.Q. Ru
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ru, C. Finite deformations at the vertex of a bi-material wedge. International Journal of Fracture 84, 325–358 (1997). https://doi.org/10.1023/A:1007357821832

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1007357821832

Keywords

Search

Navigation