International Journal of Fracture

, Volume 137, Issue 1–4, pp 231–249 | Cite as

On Fractals and Size Effects

Article

Abstract

Following an extensive and critical review of fractals and size effects models, this paper seeks to generalize Bažant’s size effect law to fractal cohesive cracks. This is achieved through a Newtonian approach in which the cohesive and far field stress intensity factors of fractal cracks (derived by Yavari) are set equal. It will be shown that the fractal size effect law is a generalization of the one of Bažant (derived in Euclidian space). In light of the derived equation, the multi-fractal model of Carpinteri and the size effect law of Bažant are revisited. Finally, the paper will conclude with some general considerations pertaining to the so-called “New Kind of Science” developed by Wolfram, and its applicability to fracture mechanics.

Keywords

Cementitious material concrete fractals fracture mechanics size effect 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Balankin, A. 1997Physics of fracture and mechanics of self-affine cracksEngineering Fracture Mechanics57135203Google Scholar
  2. Barenblatt, G. 1962The mathematical theory of equilibrium crack in the brittle fractureAdvances in Applied Mechanics755125MathSciNetGoogle Scholar
  3. Bažant, Z., Cedolin, L. 1991Stability of StructuresOxford University PressOxfordGoogle Scholar
  4. Bažant, Z.P. 1976Instability, ductility and size effect in strain softening concreteJournal of Engneering Mechenics ASCE102331344Google Scholar
  5. Bažant, Z.P. 1984Size effect in blunt fracture: concrete, rock, metalJouranl of Engineering Mechanics, ASCE110518535Google Scholar
  6. Bažant, Z.P. 1997Scaling of quasibrittle fraacture: asymptotic analysisInternation Journal of Fracture Mechanics831940Google Scholar
  7. Bažant, Z.P. 2002Scaling of Structural StrengthHermes Penton ScienceLondonGoogle Scholar
  8. Bažant, Z.P. 2005Scaling theory for quasibrittle structural failureProceedings of National Academy of Sciences1011339713399Google Scholar
  9. Bažant, Z.P., Kazemi, M.T. 1990Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concreteInternational Journal of Fracture44111131Google Scholar
  10. Bažant, Z.P., Novák, D. 2000Energetic-statistical size effect in quasibrittle failure at crack inititation.ACI Materials Journal97381392Google Scholar
  11. Bažant, Z.P., Yavari, A. 2005Is the cause of size effect on structural strength fractal or energetic-statistical?Engineering Fracture Mechanics72131Google Scholar
  12. Carpinteri, A. 1994Scaling laws and renormlaization groups for strength and toughness disordered materialsInternational Journal of Solids and Structures31291302CrossRefMATHGoogle Scholar
  13. Carpinteri, A. and Chiaia, B. (1994). Multifractal scaling law for the fracture energy variation of concrete structures. In: W. F.H. (ed.): Proceedings of Second International Conference on Fracture Mechanics of Concrete Structures FraMCoS2. Zurich, Switzerland, pp. 581–596, Aedificato.Google Scholar
  14. Carpinteri, A., Chiaia, B., Cornetti, P. 2003On the mechanics of quasi-brittle materials with a fractal microstructureEngineering Fracture Mechanics7023212349Google Scholar
  15. Carpinteri, A., Chiaia, B., Ferro, G. 1995Size effects on nominal tensile strength of concrete strucutres: multifractality of material ligaments and dimensional transition from order to disorderMaterials and Structures28311317Google Scholar
  16. Cherepanov, G. 1979Mechanics of Brittle FractureMcGraw-HillNew YorkGoogle Scholar
  17. Cusatis, G., Bažant, Z., Cedolin, L. 2003Confinement-shear lattice model for concrete damage in tension and compression: I theoryASCE Journal of Engineering Mechanics12914391448Google Scholar
  18. Dugdale, D. 1960Yielding of steel sheets containing slitsJ. Mech. Phys. Sol.8100108CrossRefGoogle Scholar
  19. Einstein, A. 1905On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heatAnnalen der Physik17549560ADSMATHGoogle Scholar
  20. Engesser, F. 1895Über KnickfragenSchweizerische Bauzeitung262426(Sect. 8.1)Google Scholar
  21. Feder, J. 1988FractalsPlenum PressNew YorkGoogle Scholar
  22. Galilei, G. (1638). Dialogues Concerning Two New Sciences. Dover Publications (1954) New York, NY. Originally published by Elzevir, The Neterlands, 1638.Google Scholar
  23. Griffith, A. 1921The phenomena of rupture and flow in solidsPhil. Trans. Roy. Soc. London A221163197ADSGoogle Scholar
  24. Hillerborg, A., Modéer, M., Petersson, P.E. 1976Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elementsCement and Concrete Research6773782CrossRefGoogle Scholar
  25. Hurst, H. 1951Long-term sorage capacity of reservoirsTransactions of the American Society of Civil Engineering116770808Google Scholar
  26. Inglis, C. 1913Stresses in a plate due to the presence of cracks and sharp cornersTrans. Inst. Naval Architects55219241Google Scholar
  27. Irwin, G. (1960). Plastic zone near a crack and fracture toughness. In: Proceedings of 7th Sagamore Conf. p. 63.Google Scholar
  28. Izquierdo-Encarnación, J. 2003Ars sine scientia nihil estConcrete International257Google Scholar
  29. Kelly, A. 1974Strong SolidsSecondOxford University PressOxfordGoogle Scholar
  30. Koch, H.v. 1904Sur une courbe continue sans tangente, obtenue par une construction géométrique elémentaireArchiv för Matemat., Astron. och Fys1681702MATHGoogle Scholar
  31. Mandelbrot, B. 1983The Fractal Geometry of NatureW.H. FreemanSan FranciscoGoogle Scholar
  32. Mandelbrot, B., Passoja, D., Paullay, A. 1984Fractal character of fracture surfaces of metalsNature308721722CrossRefGoogle Scholar
  33. Paris, P., Erdogan, F. 1963A critical analysis of crack propagation lawsJournal of Basic Engineering, ASME85528Google Scholar
  34. RILEM TC QFS2004Quasibrittle fracture scaling and size effectMaterials and Structures37547568CrossRefGoogle Scholar
  35. Saouma, V., Barton, C. 1994Fractals, fractures and size effects in concreteJournal of Engineering Mechanics of the American Society of Civil Engineers120835854Google Scholar
  36. Saouma, V., Natekar, D., Hansen, E. 2003Cohesive stresses and size effects in elasto-plastic and quasi-brittle materialsInternational Journal of Fracture119287298CrossRefGoogle Scholar
  37. Saouma, V.E., Barton, C., Gamal-El-Din, N. 1990Fractal characterization of cracked concrete surfacesEngineering Fracture Mechanics Journal354753Google Scholar
  38. Schertzer, D., Lovejoy, S. 1997Universal multifractals do exist!: comments on a statistical analysis of mesoscale rainfall as a random cascadeJournal of Applied Meteorology3612961303CrossRefADSGoogle Scholar
  39. Sierpinski, W. (1912). Sur une nouvelle courbe continue qui remplit toute une aire plane. Bull. l’Acad. des Sciences Cracovie A pp. 462–478.Google Scholar
  40. Turcotte, D. 1992Fractals and Chaos in Geology and GeophysicsCambridge University PressCambridgeGoogle Scholar
  41. Vashy, A. 1892Sur les Lois de similitude en physiqueAnnales Télégraphiques192528Google Scholar
  42. Weiss, J. 2001Self-affinity of fracture surfaces and implications on a possible size effect on fracture geometryInternational Journal of Fracture109365381CrossRefGoogle Scholar
  43. Weisstein, E. (2004). Rule 150. From MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/Rule150.html..
  44. Wnuk, M., Legat, J. 2002Work of fracture and cohesive stress distribution resulting from triaxiality dependent cohesive zone modelInternational Journal of Fracture1142946CrossRefGoogle Scholar
  45. Wnuk, M., Yavari, A. 2003On estimating stress intensity factors and modulus of cohesion for fractal cracksEngineering Fracture Mechanics7016591674CrossRefGoogle Scholar
  46. Wolfram, S. (2002). A New Kind of Science. Wolfram Media. (1197 pp.), Champaign, Ill.Google Scholar
  47. Yavari, A., Sarkani, S., Moyer, E. 2002The mechanics of self-similar and self-affine fractal cracksInternational Journal of Fracture114127CrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Department of Civil, Environmental and Architectural EngineeringUniversity of ColoradoBoulderUSA
  2. 2.Department of Structural EngineeringPolitecnico di MilanoItaly

Personalised recommendations