1st Edition

Fractals in Rock Mechanics

By Heping Xie Copyright 1993

    Important developments in the progress of the theory of rock mechanics during recent years are based on fractals and damage mechanics. The concept of fractals has proved to be a useful way of describing the statistics of naturally occurring geometrics. Natural objects, from mountains and coastlines to clouds and forests, are found to have boundaries best described as fractals. Fluid flow through jointed rock masses and clusterings of earthquakes are found to follow fractal patterns in time and space. Fracturing in rocks at all scales, from the microscale (microcracks) to the continental scale (megafaults), can lead to fractal structures. The process of diagenesis and pore geometry of sedimentary rock can be quantitatively described by fractals, etc.

    The book is mainly concerned with these developments, as related to fractal descriptions of fragmentations, damage and fracture of rocks, rock burst, joint roughness, rock porosity and permeability, rock grain growth, rock and soil particles, shear slips, fluid flow through jointed rocks, faults, earthquake clustering, and so on. The prime concerns of the book are to give a simple account of the basic concepts, methods of fractal geometry, and their applications to rock mechanics, geology, and seismology, and also to discuss damage mechanics of rocks and its application to mining engineering.

    The book can be used as a textbook for graduate students, by university teachers to prepare courses and seminars, and by active scientists who want to become familiar with a fascinating new field.

    1 Introduction.

    2 General Properties of Fractals
    2.1 Measure and dimension
    2.2 Fractals and fractal dimension
    2.3 How to measure fractal curves
    2.4 Classical fractals
    2.5 The Weierstrass function
    2.6 Self-similarity and self-affinity
    2.7 Measurements of fractal dimensions

    3 The Perimeter-Area Relation of Fractal Sets and Fractal Surfaces
    3.1 The perimeter-area relation for rains and clouds 
    3.2 The fractal dimension of rivers
    3.3 Fractal characters of fractured surfaces
    3.4 The perimeter-area relation of the Koch Island with a finite number of generations
    3.5 The degree of statistical self-similarity of fractured surfaces
    3.6 The fractal surfaces

    4 Random Fractals
    4.1 Random Cantor sets
    4.2 Random walk in one-dimension
    4.3 Fractal percolation

    5 Fractal Growth
    5.1 Dielectric breakdown model
    5.2 Viscous fingering
    5.3 DLA models
    5.4 Laplace fractals
    5.5 Laplace fractals and fracture of materials

    6 Multifractals
    6.1 The Cantor set with weight distribution
    6.2 The continuous spectrum of fractal dimensions
    6.3 Multifractal fracture

    7 Self-Inverse Fractals
    7.1 Standard geometric inversion
    7.2 Self-inverse sets, self-inversion fractals
    7.3 Ramified self-inverse fractals with symmetry
    7.4 Self-inverse fractal dust set
    7.5 Fractal oscualtion, fractal envelope
    7.6 The Appolonian model of liquid crystals

    8 Fuzzy Fractals
    8.1 Dimensions of fuzzy sets
    8.2 A fuzzy fractal set, its dimension of projection and section

    9 The Theory of Large Deformation and Its Application
    9.1 Nonlinear geometrical field theory of mechanics
    9.2 Principle of generalized dimension of Eddington
    9.3 Rigid body rotation, Gibbs' formula
    9.4 Local rotation, the concept of mean rotation angle
    9.5 Distribution function of rotation of line segments
    9.6 Local angular velocity, generalized Euler's kinematic formula for deforming body
    9.7 Nonlinear analysis for large deformation of underground openings in rock

    10 Damage Mechanics of Rock
    10.1 Damage mechanics of brittle materials
    10.2 Deformation and fracture of rock
    10.3 The experiment results of damage evolution of rock
    10.4 The analysis of damage mechanics of rock masses
    10.5 The analysis of non-smooth deformation of rock masses
    10.6 The analysis of damage mechanics of discontinuous rock masses
    10.7 Dynamic damage analysis in brittle rock

    11 Fractals and Fragmentation of Rock Materials
    11.1 Fractal distribution of fragmentation
    11.2 Renormalization group approach to fragmentation
    11.3 Fractal behaviors during the process of rock rupture

    12 Fractal Pores and Particles of Rocks and Soils
    12.1 Fractal models of porous media
    12.2 Fractal pores
    12.3 Fractal particles
    12.4 Fractal capillary tube model in soil water retention estimation

    13 Fractal Models of Rock Micro-Fractures
    13.1 A fractal model of intergranular brittle fracture
    13.2 A fractal model of transgranular brittle fracture
    13.3 A fractal model of the combined intergranular and transgranular fracture

    14 Fractal Analysis of Rock Damage and Fracture
    14.1 Fractal analysis of fracture surfaces of rock
    14.2 Scaling similiarities between fracture surface, energies and a structure parameter
    14.3 Fractal characters of microfracturing during creep of rock
    14.4 Fractal nature on the damage evolution of rock materials
    14.5 Formation of fractal cracks in a kinetic fractal model
    14.6 The fractal effect of irregularity of crack branching on the fracture toughness of rock materials
    14.7 Fractal effects on the crack extension rate
    14.8 Fractal phenomena of roof fall caused by excavation
    14.9 Fractal simulation of fracture networks in rock masses
    14.10 Fractal character and mechanism of rock bursts

    15 Fractal Description of the Roughness of Rock Joints
    15.1 Surface roughness, deformation and strength of rock joints
    15.2 Relationship between the JRC value and fractal dimension
    15.3 Fractal measurement of rock joints in the laboratory and field
    15.4 Broad bandwidth study of the topography of joint surfaces in rocks
    15.5 A fractal model of joint profiles
    15.6 Fractal geometry of the flow paths in jointed rocks

    16 Fractal Nature of the Cluster System
    16.1 Fractal geometry in the San Andreas fault system
    16.2 Fractal nature in seismology
    16.3 A fractal approach to the clustering of earthquakes
    16.4 A fractal model for crustal deformation
    16.5 Fractal topography

    Subject Index
    Author Index


    Xie Heping, a specialist in engineering mechanics, was born in Shuangfeng, Hunan Province in 1956. He graduated from the China University of Mining and Technology in 1982 and received his Ph. D. degree from the China University of Mining and Technology in 1987, where he became a professor in 1990. He was approved as doctoral candidate supervisor by the Academic Degree Committee of State Council. In 2001 he was elected academician of the Chinese Academy of Engineering. He serves as the president of Sichuan University, as member of the Subject Evaluation Team of the Academic Degree Committee of State Council, as vice president of the Chinese Society for Rock Mechanics and Engineering and as vice president of the China Coal Society.