International Journal of Fracture

, Volume 142, Issue 1–2, pp 103–117 | Cite as

Shielding and amplification of a penny-shape crack due to the presence of dislocations

Original Paper


The interaction between a penny-shape crack and a dislocation in crystalline materials is investigated within the framework of dislocation dynamics. The long-range and singular stress field resulting from the crack is determined by modeling the crack as continuous distribution of dislocation loops. This distribution is determined by satisfying the traction boundary condition at the crack face, resulting into a singular integral equation of the first kind that is solved numerically. This crack model is integrated with the dislocation dynamics simulation technique to yield the stress field of the combine system of crack and different types of dislocations situated at different positions in a three dimensional space. The integrated system is then used to investigate the dislocation behavior and its influence on the crack opening displacement and the characteristic of the stress field near the crack tip. It is shown that, depending on the relative position of the dislocation and its character, the dislocation may result in reduction in the stress amplitude at the crack tip and in some cases in closure of the crack tip. These analyses yield shielding and amplification zones near the crack providing an insight of the dislocation influence on the crack. The full dislocation dynamic analysis reveals the nature of the crack dislocation interaction and the manner in which the dislocation morphology changes as it is attracted to the crack surfaces, as well as the changes it causes to the crack profile.


Penny-shape crack Discrete dislocation dynamics Crack shielding and amplification 


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  1. Chen YZ, Lee KY (2001) Numerical solution of three-dimensional crack problem by using hypersingular integral equation. Comp Meth Appl Mech Engng 190(31):4019–4026MATHCrossRefGoogle Scholar
  2. Chen YZ, Lin XY, Peng ZQ (1997) Application of differential-integral equation to elliptical crack problem under shear load. Theor Appl Fract Mech 27(1):63–78CrossRefGoogle Scholar
  3. Dai DN, Nowell D, Hills DA (1983) Eigenstrain methods in three-dimensional crack problems: an alternative integration procedure. J Mech Phys Solids 41(6):1003–1017CrossRefGoogle Scholar
  4. Deshpande VS, Needleman A, der Giessen EV (2003) Discrete dislocation plasticity modeling of short cracks in single crystals. Acta Mater 51(1):1–15CrossRefGoogle Scholar
  5. Van der Giessen EV, Deshpande VS, Cleveringa HHM, Needleman A (2001) Discrete dislocation plasticity and crack tip fields in single crystals. J Mech Phys Solids 49(9):2133–2153CrossRefGoogle Scholar
  6. Gonhiem N, Sun LZ (1999) Fast-sum method for the elastic field of three-dimensional dislocation ensembles. Phys Rev B 60(1):28–140ADSGoogle Scholar
  7. Gonhiem N, Huang, J (2006) The elastic field of general shape 3D cracks. Phil Mag 86(27):4195–4212CrossRefADSGoogle Scholar
  8. Hadamard J (1953) Lectures on Gauchy’s problem in linear partial differential equations. Dover Publishers Inc, New YorkGoogle Scholar
  9. Hills DA, Kelly PA, Dai DN, Korsunsky AM (1996) Solution of crack problems. Kluwer Academic Publishers, DordrechtMATHGoogle Scholar
  10. Hirth JP, Lothe J (1982) Theory of dislocations. Krieger Publishers Company, MalabarGoogle Scholar
  11. Irwin GR (1962) Crack-extension force for a part-through crack in a plate. J Appl Mech 29:651–654Google Scholar
  12. Kassir MK, Sih G (eds) (1975) Mechanics of Fracture, vol 2. Nordholff Publishers, LeidenGoogle Scholar
  13. Kubin LP, Canova G, Condat M, et al. (1992) Dislocation microstructures and plastic flow: a 3D Simulation. In: Martin G, Kubin LP (eds) Solid State Phenomena (23 & 24):455–472Google Scholar
  14. Love AEH (1927) A treatise on the mathematical theory of elasticity. Cambridge University Press, CambridgeMATHGoogle Scholar
  15. Majumdar BS, Burns SJ (1981) Crack tip shielding: An elastic theory of dislocations and dislocation arrays near a sharp crack. Acta Metall 29(4):579–588CrossRefGoogle Scholar
  16. Mayrhofer K, Fischer FD (1992) Derivation of a new analytical solution for a general two-dimensional finite-part integral applicable in fracture mechanics. Int J Numer Meth Eng 33:1027–1047MATHCrossRefMathSciNetGoogle Scholar
  17. Murakami Y, Nemat-Nasser S (1983) Growth and stability of interacting surface flows of arbitrary shape. Engng Fract Mech 17(3):193–210CrossRefGoogle Scholar
  18. Needleman A (2000) Computational mechanics at the mesoscale. Acta Mater 48(1):105–124CrossRefMathSciNetGoogle Scholar
  19. Sneddon IN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. J. Wiley, New YorkMATHGoogle Scholar
  20. Theocharis PS, Ioakimidis NI (1979) A method of numerical solution of cauchy-type singular integral equations with generalized kernels and arbitrary complex singularities. J Comp Phys 30(3):309–323CrossRefGoogle Scholar
  21. Wang Q, Noda N, Honda M, Chen M (2001) Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method. Int J Fract 108:119–131CrossRefGoogle Scholar
  22. Zbib HM, Rhee M, Hirth JP (1998) On plastic deformation and the dynamics of 3D dislocations. Int J Mech Sci 40(2–3):113–127MATHCrossRefGoogle Scholar
  23. Zbib HM, Diaz de la Rubia T (2002) A multiscale model of plasticity. Int J Plasticity 18(9):1133–1163MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.School of Mechanical and Materials EngineeringWashington State UniversityPullmanUSA

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