Fractals in Rock Mechanics

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 Fractals2.1 Measure and dimension2.2 Fractals and fractal dimension2.3 How to measure fractal curves2.4 Classical fractals2.5 The Weierstrass function2.6 Self-similarity and self-affinity2.7 Measurements of fractal dimensions 3 The Perimeter-Area Relation of Fractal Sets and Fractal Surfaces3.1The perimeter-area relation for rains and clouds3.2 The fractal dimension of rivers3.3 Fractal characters of fractured surfaces3.4 The perimeter-area relation of the Koch Island with a finite number of generations3.5 The degree of statistical self-similarity of fractured surfaces3.6 The fractal surfaces 4 Random Fractals4.1 Random Cantor sets4.2 Random walk in one-dimension4.3 Fractal percolation 5 Fractal Growth5.1 Dielectric breakdown model5.2 Viscous fingering5.3 DLA models5.4 Laplace fractals 5.5 Laplace fractals and fracture of materials 6 Multifractals6.1 The Cantor set with weight distribution6.2 The continuous spectrum of fractal dimensions6.3 Multifractal fracture 7 Self-Inverse Fractals7.1 Standard geometric inversion7.2 Self-inverse sets, self-inversion fractals7.3 Ramified self-inverse fractals with symmetry7.4 Self-inverse fractal dust set7.5 Fractal oscualtion, fractal envelope7.6 The Appolonian model of liquid crystals 8 Fuzzy Fractals8.1 Dimensions of fuzzy sets8.2 A fuzzy fractal set, its dimension of projection and section 9 The Theory of Large Deformation and Its Application9.1 Nonlinear geometrical field theory of mechanics9.2 Principle of generalized dimension of Eddington9.3 Rigid body rotation, Gibbs'formula9.4 Local rotation, the concept of mean rotation angle9.5 Distribution function of rotation of line segments9.6 Local angular velocity, generalized Euler's kinematic formula for deforming body9.7 Nonlinear analysis for large deformation of underground openings in rock 10 Damage Mechanics of Rock10.1Damage mechanics of brittle materials10.2 Deformation and fracture of rock10.3 The experiment results of damage evolution of rock10.4 The analysis of damage mechanics of rock masses10.5 The analysis of non-smooth deformation of rock masses10.6 The analysis of damage mechanics of discontinuous rock masses10.7 Dynamic damage analysis in brittle rock 11 Fractals and Fragmentation of Rock Materials11.1 Fractal distribution of fragmentation11.2 Renormalization group approach to fragmentation11.3 Fractal behaviors during the process of rock rupture 12 Fractal Pores and Particles of Rocks and Soils12.1 Fractal models of porous media12.2 Fractal pores12.3 Fractal particles12.4 Fractal capillary tube model in soil water retention estimation 13 Fractal Models of Rock Micro-Fractures13.1 A fractal model of intergranular brittle fracture13.2 A fractal model of transgranular brittle fracture13.3 A fractal model of the combined intergranular and transgranular fracture 14 Fractal Analysis of Rock Damage and Fracture14.1Fractal analysis of fracture surfaces of rock14.2 Scaling similiarities between fracture surface, energies and a structure parameter14.3 Fractal characters of microfracturing during creep of rock14.4 Fractal nature on the damage evolution of rock materials14.5 Formation of fractal cracks in a kinetic fractal model14.6The fractal effect of irregularity of crack branching on the fracture toughness of rock materials14.7 Fractal effects on the crack extension rate14.8 Fractal phenomena of roof fall caused by excavation14.9 Fractal simulation of fracture networks in rock masses14.10 Fractal character and mechanism of rock bursts 15 Fractal Description of the Roughness of Rock Joints15.1 Surface roughness, deformation and strength of rock joints15.2 Relationship between the JRC value and fractal dimension15.3 Fractal measurement of rock joints in the laboratory and field15.4 Broad bandwidth study of the topography of joint surfaces in rocks15.5 A fractal model of joint profiles15.6 Fractal geometry of the flow paths in jointed rocks 16 Fractal Nature of the Cluster System16.1 Fractal geometry in the San Andreas fault system16.2 Fractal nature in seismology16.3 A fractal approach to the clustering of earthquakes16.4 A fractal model for crustal deformation16.5 Fractal topography ReferencesSubject IndexAuthor 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 electedacademician 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.