International Journal of Fracture

, Volume 167, Issue 1, pp 3–14 | Cite as

Closure of circular arc cracks under general loading: effects on stress intensity factors

  • Elizabeth Ritz
  • David D. Pollard
Original Paper


The two-dimensional circular arc crack solution of Muskhelishvili (Some basic problems of the mathematical theory of elasticity, P. Noordhoff Ltd, Groningen, Holland, 1953) has been used widely to study curved crack behavior in an infinite, homogeneous and isotropic elastic material. However, for certain orientations and magnitudes of the remotely applied loads, portions of the crack will close. Since the analytical solution is incorrect once the crack walls come into contact, the displacement discontinuity method is combined with a complementarity algorithm to solve this problem. This study uses stress intensity factors (SIFs) and displacement discontinuities along the crack to define when the analytical solution is not applicable and to better understand the mechanism that causes partial closure under various loading conditions, including uniaxial tension and pure shear. Closure is mainly due to material from the concave side of the crack moving toward the outer crack surface. Solutions that allow interpenetration of the crack tips yield non-zero mode I SIFs, while crack tip closure under proper contact boundary conditions produce mode I SIFs that are identically zero. Partial closure of a circular arc crack will alter both mode I and II SIFs at the crack tips, regardless of the positioning or length of the closed section along the crack. Friction on the crack surfaces in contact changes the total length and positioning of closure, as well as generally decreases the magnitude of opening along the portions of the crack that are not closed.


BEM Circular arc crack Stress intensity factor Complementarity 



Boundary element method


Displacement discontinuity method


Stress intensity factors


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bhargava RR, Kumar R (1993) Dugdale model solution for an infinite plate with a circular arc crack. Eng Fract Mech 46: 265–273CrossRefGoogle Scholar
  2. Bobet A, Mutlu O (2005) Stress and displacement discontinuity element method for undrained analysis. Eng Fract Mech 72: 1411–1437CrossRefGoogle Scholar
  3. Bowie OL, Freese CE (1976) On the “Overlapping” Problem in crack analysis. Eng Fract Mech 8: 373–379CrossRefGoogle Scholar
  4. Broberg K (1999) Cracks and fracture. Academic Press, San DiegoGoogle Scholar
  5. Chao CK, Shen MH (1993) Thermal problem of curvilinear cracks in bonded dissimilar materials. J Appl Phys 73: 7129–7137CrossRefADSGoogle Scholar
  6. Chao R, Laws N (1992) Closure of an arc crack in an isotropic homogeneous material due to uniaxial loading. J Mech Appl Math 45: 629–640zbMATHCrossRefGoogle Scholar
  7. Cole JW, Milner DM, Spinks KD (2005) Calderas and caldera structures: a review. Earth Sci Rev 69: 1–26CrossRefADSGoogle Scholar
  8. Comninou M (1977) Interface crack with friction in contact zone. J Appl Mech Trans Asme 44: 780–781CrossRefGoogle Scholar
  9. Comninou M, Dundurs J (1978) Can two solids slide without slipping? Int J Solids Struct 14: 251–260zbMATHCrossRefMathSciNetGoogle Scholar
  10. Comninou M, Dundurs J (1979) On the frictional contact in crack analysis. Eng Fract Mech 12: 117–123CrossRefGoogle Scholar
  11. Cotterell B, Rice JR (1980) Slightly curved or kinked cracks. Int J Fracture 16: 155–169CrossRefGoogle Scholar
  12. Crouch SL (1976) Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution. Int J Numer Meth Eng 10: 301–343zbMATHCrossRefMathSciNetGoogle Scholar
  13. Crouch SL, Starfield AM (1983) Boundary element methods in solid mechanics. Allen & Unwin, LondonzbMATHGoogle Scholar
  14. De Bremaecker JC, Ferris MC (2004) Numerical models of shear fracture propagation. Eng Fract Mech 71: 2161–2178CrossRefGoogle Scholar
  15. De Bremaecker JC, Ferris MC, Ralph D (2000) Compressional fractures considered as contact problems and mixed complementarity problems. Eng Fract Mech 66: 287–303CrossRefGoogle Scholar
  16. Dirkse SP, Ferris MC (1995) The path solver: a non-monotone stabilization scheme for mixed complementarity problems. Optimization Meth Sofr 5: 123–156CrossRefGoogle Scholar
  17. Elvin N, Leung C (1999) A fast iterative boundary element method for solving closed crack problems. Eng Fract Mech 63: 631–648CrossRefGoogle Scholar
  18. England GL (1966) Steady-state stresses in concrete structures subjected to sustained temperatures and loads. I. Cases of uniaxial stress. Nucl Eng Des 3: 54–65CrossRefGoogle Scholar
  19. Erdogan F, Gupta GD (1972) On the numerical solution of singular integral equations. Quart Appl Math 29: 525–534zbMATHMathSciNetGoogle Scholar
  20. Ferris MC, Munson TS (1999) Interfaces to path 3.0: design, implementation and usage. Comput Optim Appl 12: 207–227zbMATHCrossRefMathSciNetGoogle Scholar
  21. Gao HJ, Chiu CH (1992) Slightly curved or kinked cracks in anisotropic elastic solids. Int J Solids Struct 29: 947–972zbMATHCrossRefGoogle Scholar
  22. Gercek H (2007) Poisson’s ratio values for rocks. Int J Rock Mech Mining Sci 44: 1–13CrossRefGoogle Scholar
  23. Khan D, Biswas K (2006) Circular arc crack under dynamic load: a generalized approach for energy release rate. Int J Fract 141: 27–35zbMATHCrossRefGoogle Scholar
  24. Lawn BR, Wilshaw TR (1975) Fracture of brittle solids. Cambridge University Press, New YorkGoogle Scholar
  25. Lee S, Ravichandran G (2003) Crack initiation in brittle solids under multiaxial compression. Eng Fract Mech 70: 1645–1658CrossRefGoogle Scholar
  26. Lorentzon M, Eriksson K (2000) A path independent integral for the crack extension force of the circular arc crack. Eng Fract Mech 66: 423–439CrossRefGoogle Scholar
  27. Mériaux C, Lister JR (2002) Calculation of dike trajectories from volcanic centers. J Geophys Res 107(B4):2077CrossRefGoogle Scholar
  28. Mijar AR, Arora JS (2000) Review of formulations for elastostatic frictional contact problems. Struct Multidisci Optim 20: 167–189CrossRefGoogle Scholar
  29. Muskhelishvili N (1953) Some basic problems of the mathematical theory of elasticity. P. Noordhoff Ltd, Groningen, HollandzbMATHGoogle Scholar
  30. Mutlu O, Pollard DD (2008) On the patterns of wing cracks along an outcrop scale flaw: a numerical modeling approach using complementarity. J Geophys Res 113: B06403CrossRefGoogle Scholar
  31. Olson JE (2007) Fracture aperture, length and pattern geometry development under biaxial loading: a numerical study with applications to natural, cross-jointed systems. Geol Soc Lond Special Publ 289: 123–142CrossRefGoogle Scholar
  32. Olson JE (1990) Fracture mechanics analysis of joints and veins. Dissertation: Stanford UniversityGoogle Scholar
  33. Perlman AB, Sih GC (1967) Elastostatic problems of curvilinear cracks in bonded dissimilar materials. Int J Eng Sci 5: 845–867zbMATHCrossRefGoogle Scholar
  34. Schultz RA (1988) Stress intensity factors for curved cracks obtained with the displacement discontinuity method. Int J Fracture 37: R31–R34CrossRefGoogle Scholar
  35. Shelton JW (1984) Listric normal faults; an illustrated summary. AAPG Bulletin 68: 801–815Google Scholar
  36. Shen D, Fan T (2004) Semi-inverse method for solving circular arc crack problems. Eng Fract Mech 71: 1705–1724CrossRefGoogle Scholar
  37. Tada H, Paris PC, Irwin GR (2000) The stress analysis of cracks handbook. American Society of Mechanical Engineers, New YorkCrossRefGoogle Scholar
  38. Thomas AL, Pollard DD (1993) The geometry of echelon fractures in rock—implications from laboratory and numerical experiments. J Struct Geology 15: 323–334CrossRefADSGoogle Scholar
  39. Tilbrook MT, Moon RJ, Hoffman M (2005) Curved crack propagation in homogeneous and graded materials. Fatigue Fract Eng Mater Struct 28: 939–950CrossRefGoogle Scholar
  40. Toya M (1974) A crack along the interface of a circular inclusion embedded in an infinite solid. J Mech Phys Solids 22: 325–348zbMATHCrossRefADSGoogle Scholar
  41. Tuckwell GW, Lonergan L, Jolly RJH (2003) The control of stress history and flaw distribution on the evolution of polygonal fracture networks. J Struct Geology 25: 1241–1250CrossRefADSGoogle Scholar
  42. Vijayakumar S, Curran JH (2007) Influence of boundary curvature on tangential stresses for the displacement discontinuity method. Eng Anal Bound Elem 31: 267–274zbMATHCrossRefGoogle Scholar
  43. Wen S, Li X (2000) Experimental study on young’s modulus of concrete. J Cent South Univ Technol 7: 43–45CrossRefGoogle Scholar
  44. Williams ML (1959) The stresses around a fault or crack in dissimilar media. Bull Seism Soc Am 49: 199–204Google Scholar
  45. Wu S, Bally AW, Cramez C (1990) Allochthonous salt, structure and stratigraphy of the north-eastern gulf of mexico. Part ii: Structure. Mar Petroleum Geology 7: 334–340CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Geological and Environmental SciencesStanford UniversityStanfordUSA

Personalised recommendations