Fractal models for the fragmentation of rocks and soils: a review

Abstract Fragmentation, the process of breaking apart into fragments, is caused by the propagation of multiple fractures at different length scales. Such fractures can be induced by dynamic crack growth during compressive/tensile loading or by stress waves during impact loading. Fragmentation of rocks occurs in resoonse to tectonic activity, percussive drilling, grinding and blasting. Soil fragmentation is the result of tillage and planting operations. Fractal theory, which deals with the scaling of hierarchical and irregur systems, offers new opportunities for modeling the fragmentation process. This paper reviews the literature on fractal models for the fragmentation of heterogeneous brittle earth materials. Fractal models are available for the fragmentation of: (1) classical aggregate; (2) aggregates with fractal pore space; and (3) aggregates with fractal surfaces. In each case, the aggregates are composed of building blocks of finite size. Structural failure is hierarchical in nature and takes place by multiple fracturing of the aggregated building blocks. The resulting number-size distribution of fragments depends on the probability of failure, P( 1 b i ) at each level in the hierarchy. Models for both scale-invariant and scale-dependent P( sol1 b i ) are reviewed. In the case of scale-invariant P( 1 b i ) , theory predict: D f = 3 + log [P( 1 b i )]/log[b] for classical aggregates; D f =D m + log [P( 1 b i )]/log[b] for aggregates with fractal pore space; and Df=Ds for aggregates with fractal surfaces. where b is a scaling factor and Df, Dm and Ds are the fragmentation, mass and surface fractal dimensions, respectively. The physical significance of these parameters is discussed, methods of estimating them are reviewed, and topics needing further research are identified.

[1]  H. Nagahama,et al.  Fractal fragment size distribution for brittle rocks , 1993 .

[2]  Leland Timothy Long,et al.  Statistical distribution of natural fractures and the possible physical generating mechanism , 1994 .

[3]  Mitsugu Matsushita,et al.  Fractal Viewpoint of Fracture and Accretion , 1985 .

[4]  G. P. Lawrence.  MEASUREMENT OF PORE SIZES IN FINE-TEXTURED SOILS: A REVIEW OF EXISTING TECHNIQUES , 1977 .

[5]  Gabor Korvin,et al.  Fractal models in the earth sciences , 1992 .

[6]  G. Menduni,et al.  Scaling properties of porous media with power-law particle size distributions , 1996 .

[7]  Hideki Takayasu,et al.  Fractals in the Physical Sciences , 1990 .

[8]  J. Li,et al.  Generation of fractal models for characterization of pulverized materials , 1993 .

[9]  M. Borkovec,et al.  ON PARTICLE-SIZE DISTRIBUTIONS IN SOILS , 1993 .

[10]  Shigeru Sano,et al.  Relationships between particle size and fracture energy or impact velocity required to fracture as estimated from single particle crushing , 1987 .

[11]  M. E. Kipp,et al.  Geometric statistics and dynamic fragmentation , 1985 .

[12]  F. Dullien.  Porous Media: Fluid Transport and Pore Structure , 1979 .

[13]  E. Perfect,et al.  Statistical Characterization of Dry Aggregate Strength Using Rupture Energy , 1994 .

[14]  Garrison Sposito,et al.  Fractal Fragmentation, Soil Porosity, and Soil Water Properties: I. Theory , 1991 .

[15]  E. Perfect,et al.  Comparison of functions for characterizing the dry aggregate size distribution of tilled soil , 1993 .

[16]  E. Perfect,et al.  Multifractal Model for Soil Aggregate Fragmentation , 1993 .

[17]  F. Bartoli,et al.  Structure and self‐similarity in silty and sandy soils: the fractal approach , 1991 .

[18]  D. Wolf,et al.  Refinement and re-evaluation of the drop-shatter soil fragmentation method☆ , 1984 .

[19]  Russell Reeve,et al.  A warning about standard errors when estimating the fractal dimension , 1992 .

[20]  Sidney Redner,et al.  Statistical Theory of Fragmentation , 1990 .

[21]  P. Hallett,et al.  Identification of pre-existing cracks on soil fracture surfaces using dye , 1995 .

[22]  E. L. Skidmore,et al.  A soil-aggregate crushing-energy meter , 1983 .

[23]  H. Herrmann,et al.  Statistical models for the fracture of disordered media. North‐Holland, 1990, 353 p., ISBN 0444 88551x (hardbound) US $ 92.25, 0444 885501 (paperback) US $ 41.00 , 1990 .

[24]  L. N. Mielke,et al.  Fractal description of soil fragmentation for various tillage methods and crop sequences. , 1993 .

[25]  Donald L. Turcotte,et al.  Fractals and fragmentation , 1986 .

[26]  G. Menduni,et al.  Self-similar heterogeneity in granular porous media at the representative elementary volume scale , 1993 .

[27]  A. H. Thompson,et al.  Fractals in Rock Physics , 1991 .

[28]  E. Perfect,et al.  Applications of fractals in soil and tillage research: a review , 1995 .

[29]  Y. Yortsos,et al.  Application of Fractal Geometry to the Study of Networks of Fractures and Their Pressure Transient , 1995 .

[30]  Feodor M. Borodich,et al.  Some fractal models of fracture , 1997 .

[31]  R. Irani,et al.  Particle Size: Measurement, Interpretation, and Application , 1963 .

[32]  M. T. Hanson,et al.  A rational source of plane fractals and its application to fragmentation analysis of thin plates , 1996 .

[33]  Heping Xie,et al.  Fractals in Rock Mechanics , 2020 .

[34]  T. J. Marshall,et al.  Stability of structural aggregates of dry soil , 1950 .

[35]  Joe M. Bradford,et al.  Applications of a Laser Scanner to Quantify Soil Microtopography , 1992 .

[36]  D. Turcotte.  Fractals in geology and geophysics , 2009, Encyclopedia of Complexity and Systems Science.

[37]  Charles G. Sammis,et al.  An automaton for fractal patterns of fragmentation , 1991, Nature.

[38]  James P. M. Syvitski,et al.  Principles, methods, and application of particle size analysis: Frontmatter , 1991 .

[39]  E. Perfect,et al.  Linear and Nonlinear Estimates of Fractal Dimension for Soil Aggregate Fragmentation , 1995 .

[40]  John W. Crawford,et al.  On the relation between number-size distributions and the fractal dimension of aggregates , 1993 .

[41]  E. Perfect,et al.  Unbiased estimation of the fractal dimension of soil aggregate size distributions , 1994 .

[42]  Garrison Sposito,et al.  Fractal Fragmentation, Soil Porosity, and Soil Water Properties: II. Applications , 1991 .

[43]  Nicolas Rivier,et al.  Size‐distribution in sudden breakage by the use of entropy maximization , 1988 .

[44]  E. Perfect,et al.  Fractal Characterization of Soil Aggregation and Fragmentation as Influenced by Tillage Treatment , 1997 .

[45]  John W. Crawford,et al.  Quantification of fungal morphology, gaseous transport and microbial dynamics in soil: an integrated framework utilising fractal geometry , 1993 .

[46]  B. H. Kaye.  A random walk through fractal dimensions , 1989 .

[47]  E. Guyon,et al.  Disorder and fracture , 1990 .

[48]  E. Perfect,et al.  Fractal Dimensions of Soil Aggregate‐size Distributions Calculated by Number and Mass , 1992 .

[49]  Fred Kofi Boadu,et al.  The fractal character of fracture spacing and RQD , 1994 .

[50]  P Meakin,et al.  Models for Material Failure and Deformation , 1991, Science.

[51]  P. Barrett.  The shape of rock particles, a critical review , 1980 .

[52]  Stefan Sokołowski,et al.  A Modified Number‐based Method for Estimating Fragmentation Fractal Dimensions of Soils , 1996 .

[53]  A. Dexter,et al.  An analysis of the fragmentation of remoulded soils with regard to self-mulching behaviour , 1995 .

[54]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .