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Circular edge singularities for the Laplace equation and the elasticity system in 3-D domains

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Abstract

Asymptotics of solutions to the Laplace equation with Neumann or Dirichlet conditions in the vicinity of a circular singular edge in a three-dimensional domain are derived and provided in an explicit form. These asymptotic solutions are represented by a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the circular edge. We provide explicit formulas for a penny-shaped crack for an axisymmetric case as well as a case in which the loading is non-axisymmetric. Explicit formulas for other singular circular edges such as a circumferential crack, an external crack and a 3π/2 reentrant corner are also derived. The mathematical machinery developed in the framework of the Laplace operator is extended to derive the asymptotic solution (three-component displacement vector) for the elasticity system in the vicinity of a circular edge in a three-dimensional domain. As a particular case we present explicitly the series expansion for a traction free or clamped penny-shaped crack in an axisymmetric or a non-axisymmetric situation. The precise representation of the asymptotic series is required for constructing benchmark problems with analytical solutions against which numerical methods can be assessed, and to develop new extraction techniques for the edge flux/intensity functions which are of practical engineering importance in predicting crack propagation.

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References

  • Bach M, Nazarov SA, Wendland WL (1998) Propagation of a penny shaped crack under the Irwin criterion. In: Analysis, numerics and applications of differential and integral equations (Stuttgart 1996), Pitman Res Notes Math Ser, vol 379, pp 17–21. Longman, Harlow

  • Costabel M, Dauge M, Duduchava R (2003) Asymptotics without logarithmic terms for crack problems. Commun PDEs 28(5–6): 869–926

    Article  Google Scholar 

  • Costabel M, Dauge M, Yosibash Z (2004) A quasidual function method for extracting edge stress intensity functions. SIAM J Math Anal 35(5): 1177–1202

    Article  Google Scholar 

  • Dauge M (1988) Elliptic boundary value problems in corner domains—smoothness and asymptotics of solutions. Lecture notes in Mathematics 1341. Springer, Heidelberg

  • Kondratiev VA (1967) Boundary value problems for elliptic equations in domains with conical or angular points. Trans Moscow Math Soc 16: 227–313

    Google Scholar 

  • Leblond J, Torlai O (1992) The stress-field near the front of an arbitrarily shaped crack in a 3-dimensional elastic body. J Elast 29(2): 97–131

    Article  Google Scholar 

  • Leung A, Su R (1998) Eigenfunction expansion for penny-shaped and circumferential cracks. Int J Fract 89: 205–222

    Article  Google Scholar 

  • Omer N, Yosibash Z, Costabel M, Dauge M (2004) Edge flux intensity functions in polyhedral domains and their extraction by a quasidual function method. Int J Fract 129: 97–130

    Article  Google Scholar 

  • Sokolnikoff IS (1956) Mathematical theory of elasticity. McGraw-Hill, New York

    Google Scholar 

  • von Petersdorff T, Stephan E (1992) Singularities of the solution of the Laplacian in domains with circular edges. Appl Anal 45(1–4): 281–294

    Article  Google Scholar 

  • von Petersdorff T, Stephan EP (1990) Decompositions in edge and corner singularities for the solution of the Dirichlet problem of the Laplacian in a polyhedron. Math Nachr 149: 71–103

    Article  Google Scholar 

  • Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plates in extension. Trans ASME J Appl Mech 19: 526–528

    Google Scholar 

  • Yosibash Z, Omer N, Costabe M, Dauge M (2005) Edge stress intensity functions in polyhedral domains and their extraction by a quasidual function method. Int J Fract 136: 37–73

    Article  Google Scholar 

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Authors and Affiliations

  1. Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, 84105, Beer-Sheva, Israel

    Zohar Yosibash & Samuel Shannon

  2. IRMAR, University of Rennes 1, Rennes, France

    Monique Dauge & Martin Costabel

Authors
  1. Zohar Yosibash
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  2. Samuel Shannon
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  3. Monique Dauge
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  4. Martin Costabel
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Corresponding author

Correspondence to Zohar Yosibash.

Additional information

The first two authors gratefully acknowledge the partial support of this work by the Israel Science Foundation (grant No. 750/07).

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Yosibash, Z., Shannon, S., Dauge, M. et al. Circular edge singularities for the Laplace equation and the elasticity system in 3-D domains. Int J Fract 168, 31–52 (2011). https://doi.org/10.1007/s10704-010-9553-y

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  • DOI: https://doi.org/10.1007/s10704-010-9553-y

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