Fractal Cracking of Concrete: Parameterization of Spatial Diffusion
Publication: Journal of Engineering Mechanics
Volume 125, Issue 6
Abstract
Concrete cracks on the tension face of beams in uniaxial bending are interpreted as a non-Fickian diffusive phenomenon resulting from a self-affine random fractal process. It is shown how a complete spatial description of the cracking geometry can be found from experimental data using both a (Hurst) scaling exponent and a diffusion-type coefficient. Once determined experimentally, these parameters are used to synthesize cracking patterns using fractional Brownian motion functions. In addition, it is shown how an effective Fokker-Planck diffusion equation can be used to describe the spatial geometry of the cracking.
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References
1.
Addison, P. S. (1996). “A method for modelling dispersion dynamics in coastal waters using fractional Brownian motion.”J. Hydr. Res., 34, 549–561.
2.
Addison, P. S. ( 1997). Fractals and chaos: An illustrated course. Institute of Physics Publishing, Bristol.
3.
Addison, P. S., Qu, B, Ndumu, A. S., and Pyrah, I. C. ( 1998). “A particle tracking model for non-Fickian subsurface diffusion.” Math. Geol., 30(6), 695–716.
4.
Addison, P. S., Qu, B., Nisbet, A., and Pender, G. ( 1997). “A non-Fickian particle-tracking diffusion model based on fractional Brownian motion.” Int. J. Numer. Methods in Fluids, 25, 1373–1384.
5.
Bazant, Z. P. ( 1997). “Scaling of quasibrittle fracture: Hypothesis of invasive and lacunar fractality, their critique and Weibull connection.” Int. J. Fracture, 83, 41–65.
6.
Bazant, Z. P., and Chen, E.-P. ( 1997). “Scaling of structural failure.” Appl. Mech. Rev., 50(10) 593–627.
7.
Bazant, Z. P., and Xi, Y. ( 1994). “Fracture of random quasibrittle materials: Markov process and Weibull-type models.” Structural safety and reliability, M. Shinozuka and J. T. P. Yao, eds., Balkema, Rotterdam, The Netherlands, 609–614.
8.
Borodich, F. M. ( 1997). “Some fractal models of fracture.” J. Mech. Phys. Solids, 45(2), 239–259.
9.
Bouchaud, J. P., Bouchaud, E., Lapasset, G., and Planès, J. ( 1993). “Models of fractal cracks.” Phys. Rev. Letters, 71(14), 2240–2243.
10.
Bunde, A., and Havlin, S., eds. ( 1994). Fractals in science. Springer, Berlin.
11.
Carpinteri A. ( 1994a). “Scaling laws and renormalization groups for strength and toughness of disordered materials.” Int. J. Solids Struct., 31(3), 291–302.
12.
Carpinteri, A. ( 1994b). “Fractal nature of material microstructure and size effects on apparent mechanical properties.” Mech. Mat., 18, 89–101.
13.
Carpenteri, A., and Chiaia, B. ( 1997). “Multifractal scaling laws in the breaking behaviour of disordered materials.” Chaos, solitons and Fractals, 8(2), 135–150.
14.
Carpinteri, A., and Ferro, G. ( 1994). “Size effect on tensile fracture properties: A unified explanation based on disorder and fractality of concrete microstructure.” Mat. and Struct., 27, 563–571.
15.
Chiaia, B., van Mier, J. G. M., and Vervuurt, A. ( 1998). “Crack growth mechanisms in four different concretes: Microscopic observations and fractal analysis.” Cement and Concrete Res., 28(1), 103–114.
16.
Chudnovsky, A., and Kunin, B. ( 1987). “A probabilistic model of brittle crack formation.” J. Appl. Phys., 62(10), 4124–4129.
17.
Hammad, A. M., and Issa, M. A. ( 1994). “Fractal dimension as a measure of roughness of concrete fracture trajectories.” Advanced Cement Based Mat., 1, 169–177.
18.
Herrmann, H. J. ( 1989). “Fractal deterministic cracks.” Physica D, 38, 192–197.
19.
Hurst, H. E. ( 1951). “Long term storage capacity of reservoirs.” Trans., ASCE, 116, 770–808.
20.
Issa, M. A., and Hammad, A. M. ( 1993). “Fractal characterisation of fracture surfaces in mortar.” Cement and Concrete Res., 23, 7–12.
21.
Issa, M. A., and Hammad, A. M. ( 1994). “Assessment and evaluation of fractal dimension of concrete fracture surface digitized images.” Cement and Concrete Res., 24(2), 325–334.
22.
Ji, X. H., Chan, S. Y. N., and Feng, N. Q. ( 1997). “A fractal model for simulating the formation of microcracks in the fracture process zone and a theoretical explanation of the size effect of the fracture energy of concrete.” Mag. of Concrete Res., 49(180), 253–258.
23.
Kreijger, P. C. ( 1984). “The skin of concrete composition and properties.” Mat. and Struct., Paris, 17(100), 275–283.
24.
Kunin, B., and Gorelik, M. ( 1991). “On representation of fracture profiles by fractional integrals of a Weiner process.” J. Appl. Phys., 70(12), 7651–7653.
25.
Mandelbrot, B. B. ( 1977). The fractal geometry of nature. W. H. Freeman and Company, San Francisco.
26.
Mandelbrot, B. B., and van Ness, J. W. ( 1968). “Fractional Brownian motions, fractional noises and applications.” SIAM Rev., 10, 422–437.
27.
Mandelbrot, B. B., and Wallis, J. R. ( 1969). “Computer experiments with fractional Gaussian noises. Part 3, Mathematical appendix.” Water Resources Res., 5, 260–267.
28.
Meakin, P. ( 1990). “Simple stochastic models for material failure.” Disorder and fracture, J. C. Charmet, ed., Plenum, New York, 119–140.
29.
Meakin, P. ( 1991). “Models for material failure and deformation.” Sci., 252, 226–234.
30.
Pande, C. S., Richards, L. R., and Smith, S. ( 1987). “Fractal characteristics of fractured surfaces.” J. Mat. Sci. Letters, 6, 295–297.
31.
Peitgen, H. O., and Saupe, D., eds. ( 1988). The science of fractal images. Springer, New York.
32.
Saouma, V. E., and Barton, C. C. (1994). “Fractals, fractures, and size effects in concrete.”J. Engrg. Mech., ASCE, 120(4), 835–854.
33.
Saouma, V. E., Barton, C. C., and Gamaleldin, N. A. ( 1990). “Fractal characterisation of fracture surfaces in concrete.” Engrg. Fracture Mech., 35, 47–53.
34.
Voss, R. F. ( 1985). “Random fractal forgeries.” Fundamental algorithms for computer graphics, Vol. F17, R. A. Earnshaw, ed., NATO ASI Series, Springer, Berlin.
35.
Wang, K. G., and Lung, C. W. ( 1990). “Long-time correlation effects and fractal Brownian motion.” Phys. Letters A, 151(3,4), 119–121.
36.
Xie, H. ( 1995). “Effects of fractal crack.” Theoretical and Appl. Fracture Mech., 23, 235–244.
37.
Yin, Z.-M. ( 1996). “New methods for simulation of fractional Brownian motion.” J. Computational Phys., 127, 66–72.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 125 • Issue 6 • June 1999
Pages: 622 - 629
History
Received: Jun 12, 1998
Published online: Jun 1, 1999
Published in print: Jun 1999
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