Fractals and Mechanics of Fracture

Reference work entry


Classical mechanics including mechanics of fracture is often unsatisfactory when a solution predicts a singularity and the need arises to interpret the underlying physical meaning or lack thereof. A customary practice to deal with the singularity problem is to exclude a small region near the singular point, for which a different constitutive law – usually nonelastic – is postulated. This approach is adequate provided that the stress field outside the singular region is dominated by the elastic behavior. An alternative approach that successfully resolves problems involving singularities is the averaging process, also known as the quantization procedure – or – equivalently, discretization of the condition of the minimum of the potential energy of the system. In addition to the constitutive law, a certain “rule of decohesion” must be incorporated into the theory of fracture. An example of such a rule is the δCOD or the so-called “final stretch” criterion employed to describe the onset and the stable growth of a crack contained in a ductile solid. This criterion generalizes the well-known criteria of Griffith, Irwin–Orowan, Rice, and Wells.

Success of the novel approaches is particularly remarkable in the nanoscale domain, where the fractal geometry of cracks and the quantization rules need to be combined in order to describe adequately fracture processes at the lattice and/or atomistic level. Discrete cohesive crack representation with the fractal geometry incorporated into the mathematical model appears to produce most straightforward and useful results. Application of the Wnuk–Yavari correspondence principle relating the fractal and smooth blunt cracks demonstrates that even a minute amount of roughness of the crack surface is sufficient to cause a drop in the maximum stress measured at the tip of the crack from infinity to a well-defined finite value.

Early stages of fracture and the pre-fracture deformation states associated with a stable propagation of a subcritical crack in viscoelastic and/or ductile solids are described in some detail. The initial stable growth of crack manifests itself as a sequence of the local instability points, while the onset of catastrophic fracture corresponds to attainment of the global instability. The locus of these critical states supplants the Griffith result. Only in the limit of ideally brittle material behavior that both results, the present one and the classic one, coincide.

In the present review, introductory concepts of fractal and quantized fracture mechanics followed by the studies of delayed fracture in viscoelastic solids and the instabilities occurring in the process of ductile fracture are discussed.


Stress Intensity Factor Crack Length Linear Elastic Fracture Mechanic Fractal Geometry Fractal Crack 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. A.S. Balankin, Physics of fracture and mechanics of self-affine cracks. Eng. Fract. Mech. 57(2), 135–203 (1997)CrossRefGoogle Scholar
  2. F.M. Borodich, Fracture energy in a fractal crack propagating in concrete or rock. Doklady Russian Acad. Sci. 325, 1138–1141 (1992)Google Scholar
  3. F.M. Borodich, Some fractal models of fracture. J. Mech. Phys. Solids 45, 239–259 (1997)CrossRefMATHGoogle Scholar
  4. F.M. Borodich, Fractals and fractal scaling in fracture mechanics. Int. J. Fract. 95, 239–259 (1999)CrossRefGoogle Scholar
  5. A. Carpinteri, Scaling laws and renormalization groups for strength and toughness of disordered materials. Int. J. Solids Struct. 31, 291–302 (1994)CrossRefMATHGoogle Scholar
  6. A. Carpinteri, A. Spagnoli, A fractal analysis of the size effect on fatigue crack growth. Int. J. Fatigue 26, 125–133 (2004)CrossRefMATHGoogle Scholar
  7. A. Carpinteri, B. Chiaia, P. Cornetti, A scale invariant cohesive crack model for quasi-brittle materials. Eng. Fract. Mech. 69, 207–217 (2002)CrossRefGoogle Scholar
  8. G.P. Cherepanov, A.S. Balankin, V.S. Ivanova, Fractal fracture mechanics – a review. Eng. Fract. Mech. 51(6), 997–1033 (1995)CrossRefGoogle Scholar
  9. F.A. Field, A simple crack extension criterion for time-dependent spallation. J. Mech. Phys. Solids 19, 61 (1971); also in AMR, vol. 25 (1972), Rev. 2781CrossRefGoogle Scholar
  10. R.V. Goldstein, A.B. Mosolov, Fractal cracks. J. Appl. Math. Mech. 56, 563–571 (1992)CrossRefGoogle Scholar
  11. G.A.C. Graham, The correspondence principle of linear viscoelasticity theory for mixed boundary value problems involving time dependent boundary regions. Q. Appl. Math. 26, 167 (1968); also in AMR, vol. 22, Rev. 4036MATHGoogle Scholar
  12. A.A. Griffith, The phenomenon of rupture and flow in solids. Phil. Trans. Roy. Soc. Lond. A221, 163–398 (1921a)CrossRefGoogle Scholar
  13. J. Harrison, Numerical integration of vector fields over curves with zero area. Proc. Am. Math. Soc. 121, 715–723 (1994)CrossRefMATHGoogle Scholar
  14. J. Harrison, A. Norton, Geometric integration on fractal curves in the plane, research report. Indiana Univ. Math. J. 40, 567–594 (1991)MathSciNetCrossRefMATHGoogle Scholar
  15. C.E. Inglis, Stresses in a plate due to the presence of cracks and sharp corners. Trans. R. Inst. Naval Architects 60, 219 (1913)Google Scholar
  16. M. Ippolito, A. Mattoni, L. Colombo, Role of lattice discreteness on brittle fracture: Atomistic simulations versus analytical models. Phys. Rev. B 73, 104111 (2006). 6 pagesCrossRefGoogle Scholar
  17. G.R. Irwin, Handbuch der Physik, vol. 6 (Springer, Berlin, 1956), pp. 551–590Google Scholar
  18. H. Khezrzadeh, M.P. Wnuk, A. Yavari, Influence of material ductility and crack surface roughness on fracture instability. J. Phys. D Appl. Phys. 44, 395302 (2011) (22 pages)CrossRefGoogle Scholar
  19. W.G. Knauss, Stable and unstable crack growth in viscoelastic media. Trans. Soc. Rheol. 13, 291 (1969)CrossRefGoogle Scholar
  20. W.G. Knauss, Delayed failure. The Griffith problem for linearly viscoelastic materials. Int. J. Fract. 6, 7 (1970); also in AMR, vol. 24, Rev. 5923Google Scholar
  21. W.G. Knauss, The mechanics of polymer fracture. Appl. Mech. Rev. 26, 1–17 (1973)Google Scholar
  22. W.G. Knauss, H. Dietmann, Crack propagation under variable load histories in linearly viscoelastic solids. Int. J. Eng. Sci. 8, 643 (1970); also in AMR, vol. 24, Rev. 1097CrossRefGoogle Scholar
  23. W.G. Knauss, The time dependent fracture of viscoelastic materials, in Proceedings of the First International Conference on Fracture, vol. 2, ed. by M.L. Williams. p. 1139; also see the Ph.D. Thesis, California Institute of Technology 1963 (1965)Google Scholar
  24. B.V. Kostrov, L.V. Nikitin, Some general problems of mechanics of brittle fracture. Archiwum Mechaniki Stosowanej. (English version) 22, 749; also in AMR, vol. 25 (1972), Rev. 1987 (1970)Google Scholar
  25. B.B. Mandelbrot, D.E. Passoja, A.J. Paullay, Fractal character of fracture surfaces in metals. Nature 308, 721–722 (1984)CrossRefGoogle Scholar
  26. D. Mohanty, Experimental Study of Viscoelastic Properties and Fracture Characteristics in Polymers, M.S. Thesis at Department of Mechanical Engineering, South Dakota State University, Brookings, 1972Google Scholar
  27. A.B. Mosolov, Cracks with fractal surfaces. Doklady Akad. Nauk SSSR 319, 840–844 (1991)MathSciNetGoogle Scholar
  28. H.K. Mueller, Stress-intensity factor and crack opening for a linearly viscoelastic strip with a slowly propagating central crack. Int. J. Fract. 7, 129 (1971)CrossRefGoogle Scholar
  29. H.K. Mueller, W.G. Knauss, Crack propagation in a linearly viscoelastic strip. J. Appl. Mech. 38(Series E), 483 (1971a)CrossRefGoogle Scholar
  30. H.K. Mueller, W.G. Knauss, The fracture energy and some mechanical properties of a polyurethane elastomer. Trans. Soc. Rheol. 15, 217 (1971b)CrossRefGoogle Scholar
  31. N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity (English translation) (Noordhoff, 1953)Google Scholar
  32. H. Neuber, Theory of Notch Stresses (Springer, Berlin, 1958)Google Scholar
  33. V.V. Novozhilov, On a necessary and sufficient criterion for brittle strength. J. Appl. Mech. USSR 33, 212–222 (1969)Google Scholar
  34. N. Pugno, R.S. Ruoff, Quantized fracture mechanics. Philos. Mag. 84(27), 2829–2845 (2004)CrossRefGoogle Scholar
  35. J.R. Rice, Mathematical analysis in the mechanics of fracture, in Fracture. An Advanced Treatise, ed. by H. Liebowitz, vol. II (Academic, New York, 1968)Google Scholar
  36. R.A. Schapery, A theory of crack growth in viscoelastic media. Int. J. Fract. 11, 141–159 (1973)CrossRefGoogle Scholar
  37. C.F. Shih, Relationship between the J-integral and crack opening displacement for stationary and growing cracks. J. Mech. Phys. Solids 29, 305–326 (1981)CrossRefMATHGoogle Scholar
  38. A. Spagnoli, Self-similarity and fractals in the Paris range of fatigue crack growth. Mech. Mater. 37, 519–529 (2005)CrossRefGoogle Scholar
  39. A.A. Wells, Application of fracture mechanics at and beyond general yielding. Br. J. Weld. 11, 563–570 (1961)Google Scholar
  40. H.M. Westergaard, Bearing pressure and cracks. J. Appl. Mech. 61(1939), A49–A53 (1939)Google Scholar
  41. M.L. Williams, On stress distribution at the base of a stationary crack. J. Appl. Mech. 24, 109–114 (1957)MathSciNetMATHGoogle Scholar
  42. M.L. Williams, The continuum interpretation for fracture and adhesion. J. Appl. Polym.Sci. 13, 29 (1969a)CrossRefGoogle Scholar
  43. M.L. Williams, The kinetic energy contribution to fracture propagation in a linearly viscoelastic material. Int. J. Fract. 4, 69 (1969b); also in AMR, vol. 22 (1969), Rev. 8521Google Scholar
  44. J.R. Willis, Crack propagation in viscoelastic media. J. Mech. Phys. Solids 15, 229 (1967); also in AMR, vol.22 (1969), Rev. 8625CrossRefMATHGoogle Scholar
  45. M.P. Wnuk, Energy Criterion for Initiation and Spread of Fracture in Viscoelastic Solids (Technical Report of the Engineer Experimental Station at SDSU, No.7, Brookings, 1968a)Google Scholar
  46. M.P. Wnuk, Nature of fracture in relation to the total potential energy. Brit. J. Appl. Phys. 1(Serious 2), 217 (1968b)Google Scholar
  47. M.P. Wnuk, Effects of time and plasticity on fracture. British J. Appl. Phys., Ser. 2 2, 1245 (1969)Google Scholar
  48. M.P. Wnuk, Prior-to-failure extension of flaws under monotonic and pulsating loadings, SDSU Technical Report No. 3, Engineering Experimental Station Bulletin at SDSU, Brookings (1971)Google Scholar
  49. M.P. Wnuk, Accelerating crack in a viscoelastic solid subject to subcritical stress intensity, in Proceedings of the International Conference on Dynamic Crack Propagation, Lehigh University, ed. by G.C. Sih (Noordhoff, Leyden, 1972), pp. 273–280Google Scholar
  50. M.P. Wnuk, Quasi-static extension of a tensile crack contained in a viscoelastic-plastic solid. J. Appl. Mech. 41, 234–242 (1974)CrossRefMATHGoogle Scholar
  51. M.P. Wnuk, R.D. Kriz, CDM model of damage accumulation in laminated composites. Int. J. Fract. 28, 121–138 (1985)CrossRefGoogle Scholar
  52. M.P. Wnuk, B. Omidvar, Effects of strain hardening on quasi-static fracture in elasto-plastic solid represented by modified yield strip model. Int. J. Fract. 84, 383–403 (1997)CrossRefGoogle Scholar
  53. M.P. Wnuk, A. Yavari, On estimating stress intensity factors and modulus of cohesion for fractal cracks. Eng. Fract. Mech 70, 1659–1674 (2003)CrossRefGoogle Scholar
  54. M.P. Wnuk, A. Yavari, A correspondence principle for fractal and classical cracks. Eng. Fract. Mech. 72, 2744–2757 (2005)CrossRefGoogle Scholar
  55. M.P. Wnuk, A. Yavari, Discrete fractal fracture mechanics. Eng. Fract. Mech. 75, 1127–1142 (2008)CrossRefGoogle Scholar
  56. M.P. Wnuk, A. Yavari, A discrete cohesive model for fractal cracks. Eng. Fract. Mech. 76, 548–559 (2009)CrossRefGoogle Scholar
  57. M.P. Wnuk, B. Omidvar, M. Choroszynski, Relationship between the CTOD and the J-integral for stationary and growing cracks. Closed form solutions. Int. J. Fract. 87(1998), 331–343 (1998)Google Scholar
  58. M.P. Wnuk, M. Alavi, A. Rouzbehani, Comparison of time dependent fracture in viscoelastic and ductile solids. Phys. Mesomech. 15(1–2), 13–25 (2012)CrossRefGoogle Scholar
  59. M.P. Wnuk, M. Alavi, A. Rouzbehani, A mathematical model of Panin’s pre-fracture zones and stability of subcritical cracks, in Physical Mesomechanics (Russian Academy of Sciences, Tomsk, 2013 in print)Google Scholar
  60. S.N. Zhurkov, Kinetic concept of the strength of solids. Int. J. Fract. 1, 311 (1965); also in Appl. Mech. Rev., vol. 20, 1967, Rev. 4080Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Civil Engineering and MechanicsCollege of Engineering and Applied Science, University of WisconsinMilwaukeeUSA

Personalised recommendations