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Scaling of quasibrittle fracture: hypotheses of invasive and lacunar fractality, their critique and Weibull connection

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Abstract

Considerable progress has been achieved in fractal characterization of the properties of crack surfaces in quasibrittle materials such as concrete, rock, ice, ceramics and composites. Recently, fractality of cracks or microcracks was proposed as the explanation of the observed size effect on the nominal strength of structures. This explanation, though, has rested merely on intuitive analogy and geometric reasoning, and did not take into account the mechanics of crack propagation. In this paper, the energy-based asymptotic analysis of scaling presented in the preceding companion paper in this issue [1] is extended to the effect of fractality on scaling. First, attention is focused on the propagation of fractal crack curves (invasive fractals). The modifications of the scaling law caused by crack fractality are derived, both for quasibrittle failures after large stable crack growth and for failures at the initiation of a fractal crack in the boundary layer near the surface. Second, attention is focused on discrete fractal distribution of microcracks (lacunar fractals), which is shown to lead to an analogy with Weibull's statistical theory of size effect due to material strength randomness. The predictions ensuing from the fractal hypothesis, either invasive or lacunar, disagree with the experimentally confirmed asymptotic characteristics of the size effect in quasibrittle structures. It is also pointed out that considering the crack curve as a self-similar fractal conflicts with kinematics. This can be remedied by considering the crack to be an affine fractal. It is concluded that the fractal characteristics of either the fracture surface or the microcracking at the fracture front cannot have a significant influence on the law of scaling of failure loads, although they can affect the fracture characteristics.

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References

  1. Z.P. Bažant, Scaling of quasibrittle fracture: Asymptotic analysis. International Journal of Fracture 83 (1996) 19–40.

    Article  Google Scholar 

  2. B.B. Mandelbrot, D.E. Passoja and A. Paullay, Fractal character of fracture surfaces of metals. Nature 308 (1984) 721–722.

    Article  ADS  Google Scholar 

  3. S.R. Brown, A note on the description of surface roughness using fractal dimension. Geophysical Res. Letters 14(11) (1987) 1095–1098, and 15 (11) (1987) 286.

    ADS  Google Scholar 

  4. H. Xie, Th7e fractal effect of irregularity of crack branching on the fracture toughness of brittle materials. International Journal of Fracture 41 (1987) 267–274.

    Google Scholar 

  5. J.J. Mecholsky and T.J. Mackin, Fractal analysis of fracture in ocala chert. Journal of Materials Science and Letters 7 (1988) 1145–1147.

    Article  Google Scholar 

  6. R. Cahn, Fractal dimension and fracture. Nature 338 (Mar.) (1989) 201–202.

    Google Scholar 

  7. C.T. Chen and J. Runt, Fractal analysis of polystyrene fracture surfaces. Polymer Communications 30 (Nov.) (1989) 334–335.

    Google Scholar 

  8. E. Hornbogen, Fractals in microstructure of metals. International Materials Review 6 (1989) 277–296.

    ADS  Google Scholar 

  9. H. Xie, Studies on fractal models of microfractures of marble. Chinese Science Bulletin 34 (1989) 1292–1296.

    Google Scholar 

  10. G. Peng and D. Tian, The fractal nature of a fracture surface. Journal of Physics A: Mathematics and General 23 (1990) 3257–3261.

    Article  ADS  Google Scholar 

  11. V.C. Saouma, C. Barton and N. Gamal-el-Din, Fractal characterization of concrete crack surfaces. Engineering Fracture Mechanics 35(1) (1990).

  12. E. Bouchaud, G. Lapasset and J. Planes, Fractal dimension of fractured surfaces: a universal value? Europhysics Letters 13(1) (1990) 73–79.

    ADS  Google Scholar 

  13. T. Chelidze and Y. Gueguen, Evidence of fractal fracture. International Journal of Rock Mechanics and Mininig Sciences 27(3) (1990) 223–225.

    Article  Google Scholar 

  14. Q.Y. Long, L. Suquin and C.W. Lung, Studies of fractal dimension of a fracture surface formed by slow stable crack propagation. Journal of Physics 24(4) (1991).

  15. M.A. Issa, A.M. Hammad and A. Chudnovsky, Fracture surface characterization of concrete. Proc., 9th ASCE Conference on Engineering Mechanics, ASCE, N.Y. (1992).

    Google Scholar 

  16. K. Måløy, A. Hansen, E. Hinrichsen and S. Roux, Experimental measurement of the roughness of brittle cracks. Physical Review Letters 68(2) (1992) 213–215.

    Article  ADS  Google Scholar 

  17. A.B. Mosolov and F.M. Borodich, Fractal fracture of brittle bodies under compression (in Russian). Doklady Akademii Nauk 324(3) (1992) 546–549.

    MathSciNet  Google Scholar 

  18. F. Borodich, Fracture energy of fractal crack, propagation in concrete and rock (in Russian). Doklady Akademii Nauk 325(6) (1992) 1138–1141.

    Google Scholar 

  19. D.A. Lange, H.M. Jennings and S.P. Shah, Relationship between fracture surface roughness and fracture behavior of cement paste and mortar. Journal of American Ceramic Society 76(3) (1993) 589–597. J. Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials. Cambridge University Press, Cambridge, U.K. (1990).

    Article  Google Scholar 

  20. H. Xie, Fractals in Rock Mechanics. Balkema, Rotterdam (1993).

    Google Scholar 

  21. A. Carpinteri, B. Chiaia and G. Ferro, Multifractal scaling law for the nominal strength variation of concrete structures. In Size Effect in Concrete Structures (Proc., Japan Concrete Institute International Workshop, held in Sendai, Japan, 1993), ed. by M. Mihashi, H. Okamura and Z.P. Bažant, E. and F.N. Spon, London-New York (1994) 193–206.

    Google Scholar 

  22. H. Xie, D.J. Sanderson and D.C.P. Peacock, A fractal model and energy dissipation for en echelon fractures. Engineering Fracture Mechanics 48(5) (1994) 665–662.

    ADS  Google Scholar 

  23. V.C. Saouma and C.C. Barton, Fractals, fracture and size effect in concrete. ASCE Journal of Engineering Mechanics 120(4) (1994) 835–854.

    Article  Google Scholar 

  24. A. Carpinteri, Fractal nature of material microstructure and size effects on apparent mechanical properties. Mechanics of Materials 18 (1994) 89–101.

    Article  Google Scholar 

  25. A. Carpinteri, Scaling laws and renormalization groups for strength and toughness of disordered materials. International Journal of Solids and Structures 31 (1994) 291–302.

    Article  MATH  Google Scholar 

  26. A. Carpinteri and G. Ferro, Size effect on tensile fracture properties: a unified explanation based on disorder and fractality of concrete microstructure. Materials and Structures 27 (1994) 563–571.

    Google Scholar 

  27. A. Carpinteri, B. Chiaia and G. Ferro, Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder. Materials and Structures 28(7) (1995), 311–317.

    Google Scholar 

  28. A. Carpinteri, B. Chiaia and G. Ferro, Multifractal scaling for the nominal strength variation of concrete structures. In Size Effect in Concrete Structures, (Proc. of JCI Intern. Workshop, held in Sendai), ed. by H. Mihashi, H. Okamura and Z.P. Bažant, E. and F.N. Spon, London, 193–206.

  29. A. Carpinteri, G. Ferro and S. Intervenizzi. In Fracture Mechanics of Concrete Structures (Proceedings of FraMCoS-2, held at E.T.H., Zürich), ed. by F.H. Wittmann, Aedificatio Publishers, Freiburg, Germany (1995) 557–570.

    Google Scholar 

  30. A. Carpinteri and B. Chiaia. In Fracture Mechanics of Concrete Structures (Proceedings of FraMCoS-2, held at E.T.H., Zürich), ed. by F.H. Wittmann, Aedificatio Publishers, Freiburg (1995) 581–596.

    Google Scholar 

  31. N.-Q. Feng, X.-H. Ji, Q.-F. Zhuang and J.-T. Ding, A fractal study of the size effect of concrete fracture energy. In Fracture Mechanics of Concrete Structures, Vol. 1 (Proc., 2nd Int. Conf. on Fracture Mech. of Concrete Structures (FraMCoS-2), held at ETH, Zürich), ed. by F.H. Wittmann, Aedificatio Publishers, Freiburg, Germany (1995) 597–606.

    Google Scholar 

  32. M.C. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw Hill, New York (1978) (Chapters 9–11).

    MATH  Google Scholar 

  33. G.I. Barenblatt, Similarity, Self-Similarity and Intermediate Asymptotics. Consultants Bureau, New York, N.Y. (1979).

    MATH  Google Scholar 

  34. Z.P. Bažant, Is Size Effect Caused by Fractal Nature of Crack Surfaces?, Report No. 94-10/402i, Department of Civil Engineering, Northwestern University, Evanston, Illinois (1994).

    Google Scholar 

  35. Z.P. Bažant, Can scaling of structural failure be explained by fractal nature of cohesive fracture? Appendix to a paper by Bažant and Li in Size-Scale Effects in the Failure Mechanisms of Materials and Structures (Proc., IUTAM Symposium, held at Politecnico di Torino, Italy, Oct. 1994), ed. by A. Carpinteri, E. and F.N. Spon, London, 284–289 (1996).

  36. Z.P. Bažant, Scaling theories for quasibrittle fracture: Recent advances and new directions, in Fracture Mechanics of Concrete Structures (Proc., FraMCoS-2, held at E.T.H., Zürich) (1995) 515–534.

  37. Z.P. Bažant, Scaling of quasibrittle fracture and the fractal question, ASME Journal of Materials and Technology 117 (1995), 361–367 (Materials Division Special 75th Anniversary Issue).

    Article  Google Scholar 

  38. A.C. Palmer and J.O. Sanderson, Fractal crushing of ice and brittle solids. Proceedings of the Royal Society London 433 (1991) 469–477.

    MATH  ADS  Google Scholar 

  39. S.U. Bhat, Modeling of size effect in ice mechanics using fractal concepts. Journal of Offshore Mechanics and Arctic Engineering 112 (1990) 370–376.

    Google Scholar 

  40. Z.P. Bažant and L. Cedolin, Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories, Oxford University Press, New York (1991).

    MATH  Google Scholar 

  41. Z.P. Bažant, Fracture in concrete and reinforced concrete. In Mechanics of Geomaterials: Rocks, Concretes, Soils (Preprints, IUTAM Prager Symposium held at Northwestern University, ed. by Z.P. Bažant, Evanston, Illinois), (1983) 281–317.

  42. Z.P. Bažant, Size effect in blunt fracture: Concrete, rock, metal. ASCE Journal of Engineering Mechanics 110 (1984) 518–535.

    Article  Google Scholar 

  43. ACI Committee 446, Fracture Mechanics (Z.P. Bažant, Chairman and Princ. Author) (1992). Fracture mechanics of concrete: Concepts, models and determination of material properties. Special publication, ACI 446, IR-91, American Concrete Institute, Detroit, MI., 1991 (146 pp.); reprinted in Fracture Mechanics of Concrete Structures, ed. by Z.P. Bažant, Elsevier, London, 1–140.

    Google Scholar 

  44. Z.P. Bažant, J. Ožbolt and R. Eligehausen. Fracture size effect: review of evidence for concrete structures. Journal of Structural Engineering ASCE 120(8) (1994) 2377–2398.

    Article  Google Scholar 

  45. Z.P. Bažant and M.T. Kazemi, Size effect on diagonal shear failure of beams without stirrups. ACI Structural Journal 88 (1991) 268–276.

    Google Scholar 

  46. P. Marti, Size effect in double-punch tests on concrete cylinders. ACI Materials Journal 86,No. 6 (1989) 597–601.

    Google Scholar 

  47. W. Weibull, Phenomenon of rupture in solids. Ingenioersvetenskaps Akad. Handl. 153 (1939) 1–55.

    Google Scholar 

  48. Z.P. Bažant, Y. Xi and S.G. Reid, Statistical size effect in quasi-brittle structures: I. Is Weibull theory applicable? ASCE Journal of Engineering Mechanics 117(11) (1991) 2609–2622.

    Google Scholar 

  49. Z.P. Bažant and Y. Xi, Statistical size effect in quasi-brittle structures: II. Nonlocal theory. ASCE Journal of Engineering Mechanics 117(11) (1991) 2623–2640.

    Google Scholar 

  50. P.E. Petersson, Crack growth and development of fracture zones in plain concrete and similar materials (Report TVBM-1006), Division of Building Materials, Lund Institute of Technology, Lund, Sweden (1991).

    Google Scholar 

  51. T. Hasegawa, T. Shioya and T. Okada, Size effect on splitting tensile strength of concrete. Proc. 7th Conference of Japan Concrete Institute (1985) 305–312.

  52. H. Xie and D.J. Sanderson, Fractal effect of crack propagation on dynamic stress intensity factors and crack velocities. International Journal of Fracture 74 (1995) 29–42.

    Article  Google Scholar 

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  1. Civil Engineering and Materials Science, Northwestern University, Evanston, Illinois, 60208, USA

    ZdĚnk P. BaŽant

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BaŽant, Z.P. Scaling of quasibrittle fracture: hypotheses of invasive and lacunar fractality, their critique and Weibull connection. International Journal of Fracture 83, 41–65 (1997). https://doi.org/10.1023/A:1007335506684

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