Fractals in Rock Mechanics

Heping Xie

• Publisher: CRC Press
• Year: 2020
• ISBN:
  9781000150346
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Geology & Earth Sciences

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.

CHAPTER 1

Introduction

Geometry is a discipline in which the shapes of objects are studied. Classical geometry is used to investigate regular and smooth shapes (lines, curves, circles and so on). However, geometric shapes in classical geometry are approximate descriptions of objects in nature. For example, the rough surface of the earth is regarded as a smooth spherical or ellipsoid surface, and an upland is considered a trapezoidal slope (line) surface. Although these approximations do not prevent us from reaching conclusions that agree with practice, the need to obtain accurate descriptions of matter becomes critical to the development of science.

Mandelbrot (1975b, 1982) introduced a new geometry — fractal geometry — to describe irregular and nonsmooth phenomena in nature, and it appears that this new field of investigation will be more applicable to the study of natural phenomena than regular and smooth curves and surfaces of classical geometry.

The word ‘fractal’ is derived from the Latin fractus, meaning broken. The idea of ‘fractal’ originated during the mathematical crisis of 1875–1925 (Mandelbrot, 1982). At the beginning of 1870’s, most mathematicians believed that Euclidean geometry provided an accurate description of shape properties in both material and abstract space. These dimensions, as derived from Euclid (300 B.C.), stated that, for plane geometry:

(1) A point has no part;

(2) A line is a breadthless length;

(3) The extremities of lines are points;

(4) A surface has length and breadth only, and

(5) The extremities of surfaces are lines.

For spatial geometry, the assumptions were that:

(1) A solid has length, breadth and depth, and

(2) The extremity of a solid is a surface.

Meantime, most mathematicians believed that any continuous function must be differentiable.

In 1875, Weierstrass introduced a continuous but undifferentiated function in the form

where b is a real number larger than 1, and w is written either as w = b^H with Hϵ[0,1] or as w = b^D−2 with D∊[1, 2]. This function does not suit the original concept because ‘nowhere differentiable’ means that a curve does not have a tangent at all points. Thus the question arises as to whether the geometrical expression of this function is a curve.

The Weierstrass function upset the belief that continuous functions must be differentiable and led to a great deal of skepticism concerning the definition of a curve. For these reasons, Jordan redefined the concept of a curve as follows: A curve is the image of a continuous injection x = f(t), y = g(t); (t₀<t< t₁). If f(t₀) =f(t₁), g(t₀) = g(t₁), this image is called a Jordan closed curve.

Although Jordan’s definition of a curve eliminated some problems, it was too extensive, that is, only a continuum was required. In 1890, Peano found that the curve defined by Jordan would run over all points of a square; that is, the curve would result in continuous mapping of an interval on the whole of a square. This meant that the definition was still not satisfactory.

On the other hand, what did the length of a curve mean? The common formulation of the length of a curve was

where the existence of a derivative of y = f(x) was required.

Meantime, in 1904, Koch constructed a continuous, but nowhere differentiable closed curve which has an infinite perimeter, and by which a finite area is surrounded. Start with an equilateral triangle with side length 35. In the middle interval of its every side, a new equilateral triangle with the side length of 5 is made, and the base side of every new triangle is removed. Then, in the middle interval on each side 5 of this new figure, some newer equilateral triangles with the side length 5/3 are made again. Analogically, their base sides are all erased. After infinitely repeating the same procedure, Koch’s closed curve (Figure 2.6b) is constructed. Its perimeter length is successively 95, 125, 165,…, finally approaching oo, whereas the area surrounded by this curve becomes

which is an infinite series. It is seen that the ratio of this series is

, then

where A₀ is the initial area of the equilateral triangle with side length 35.

On the other hand, from set theory, Cantor proved an one-to-one correspondence between the points of a line and the points of a plane. Although this was not continuous correspondence from a line to a plane, it proved that dimension was not how many points existed in space, or the number of coordinates needed to fix a point in space. It indicated that the dimension of geometrical figures could change with one-to-one corresponding mapping, and the dimensional value could increase with single-valued and continuous mapping. This led to the lash of the concept of classical Euclidean dimension. Thus fractional dimension, similar dimension and fractal dimension were proposed successively. Finally a new geometry, fractal geometry, was born (Mandelbrot, 1975b, 1982, 1983). It is clear that fractal geometry is not a straight ‘application’ of 20th century mathematics, it is a new branch born belatedly from the crisis of mathematics (1875–1925).

The basic concept of fractal geometry is fractional dimension. Fractional dimension was proposed by Hausdorff (1919). But fractal geometry formed by generalization of this note is due to Β. Β. Mandelbrot. He proposed the idea of fractal geometry (early 1973) while holding a lecture in College de France. He believes that the concept of fractional dimension may be a powerful tool used to research physical phenomena, and fractal geometry can be applied to handle irregular shapes.

How do we understand the fractional dimension as opposed to an integer dimension for which we are well trained in classical geometry? An excellent example is a sponge model. A sponge cube looks three-dimensional in classical geometry, but it may be stressed in a plane because of a highly ordered structure of holes in it; thus, it also seems to be two-dimensional. This means that the integral dimensions such as 2 and 3 here only reflect apparent phenomena of objects. In fact, the dimension of the sponge cube may be in the range between 2 and 3 because of existence of a highly ordered structure of holes in it; that is, fractional dimension. This indicates that fractional dimension can reflect inherent properties of objects.

Since Mandelbrot’s books (1975b, 1977, 1982, 1983) have been published, the concept of fractals has caught the imagination of scientists in many fields, and papers concerning fractals in various contexts now appear almost daily. Mandelbrot has discussed trees, rivers, lungs, water levels, turbulence, economics, word frequencies and many, many other topics by using his fractal geometrical concepts. In his books, he purposefully has no introduction and no conclusion in order to stress his belief that as more work is done in this field, his ideas will reveal further insight into the geometry of nature (Feder, 1988).

Fractals proved to be more surprising is, spurred on by developments in computer technology, their wide applications in computer art and in the motion picture industry to generate realistic but synthetic landscapes. Fractal ‘planets’ rising over the horizon of its moon, mountains, valleys and islands have been shown as beautiful cover photographs in journals and magazines. These illustrations are based on algorithms of fractals (Peitgen & Richter, 1986; Peitgen & Saupe, 1988). They look natural. Voss has described some of ideas he used in the generation of his spectacular pictures (Voss, 1985a, b). Some of the computer graphics involved in the creation of the beautiful fractal objects are described in the books of Barnsley (1988), Stevens (1989, 1990), and Edgar (1990).

As a series of discipline of studies, the intellectual beauty of fractals has led to a revolution in statistical physics over the last decade. ‘Fractional dimension’ has been studied for over 70 years, although the term ‘fractal’ was not coined until 1975 by Mandelbrot. A large number of physicists became involved around 1980 (Wong, 1988). Fractals are ‘hybrids’ between familiar shapes and typically have noninteger dimensions. The mass of a ‘mass fractal’ is proportional to its radius to a power between 1 and 3; this power is 3 for a solid ball, 2 for a plane and 1 for a line. Similarly, the area of a ‘surface fractal’ scales with the radius of the object with a noninteger power. This property of fractals can be used with great effectiveness in analyzing physical problems. There exists enough introductory and advanced material on fractals, which are particularly well suited for thermodynamics, statistical mechanics and applied mathematics studies at any level. Some striking books and proceedings have mainly described in detail the ideas and applications of fractals of physics, such as: Fischer & Smith (1985), Feder (1988), Pietronero & Tosatti (1986), Jullien & Rotet (1987), Stanley & Ostrowsky (1986), Barnsley & Denko (1986), Banaver et al. (1987), Brien (1989), Vicsek (1989), Robert & Linda (1989), Scholz & Mandelbrot (1989), Falconer (1990), Herrmann & Roux (1990), and Cherbit (1991).

Fractal geometry is used mainly to study irregular shapes with self-similarity, irregular figures with self-inverse, the mapping with self-squaring, self-affine fractal sets and so on. But the dominant content of fractal geometry is fractals with self-similarity (linear fractal, in short).

Self-similarity means that any portion of the fractal that is magnified by an arbitrary factor looks the same as the original fractal; that is, scale invariance. The phenomenon with the behaviors of self-similarity and scale invariance has been observed in many fields, but especially in the field of rock mechanics in which we are very interested. It may be said that the complexity of million-year-old sedimentary rock is being unraveled by concepts of fractals (Wong, 1988). In the past, physicists paid little attention to rock, mainly because they were discouraged by its apparent complexity. They are well trained in working with idealized models, but when faced with a piece of rock, not only do they not know where to begin, but they may also question whether it is even possible to find interesting physics in a ‘dirty’ and uncontrolled system. In the last few years, however, physicists have gained a new insight into rock using such concepts as the random system of fractals, percolation and diffusion-limited growth, and damage evolution. Mechanical behavior of rock materials is so complicated that it may be constructed in a fractal dimensional space. In fact, an element of rock materials is very similar to a sponge cube with highly ordered structures of micropores. ...

Table of contents

• Cover
• Half Title
• Half Title One
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Introduction
• 2 General Properties of Fractals
• 3 The Perimeter-Area Relation of Fractal Sets and Fractal Surfaces
• 4 Random Fractals
• 5 Fractal Growth
• 6 Multifractals
• 7 Self-Inverse Fractals
• 8 Fuzzy Fractals
• 9 THE THEORY OF LARGE DEFORMATION AND ITS APPLICATION
• 10 Damage Mechanics of Rock
• 11 Fractals and Fragmentation of Rock Materials
• 12 Fractal Pores and Particles of Rocks and Soils
• 13 Fractal Models of Rock Micro-Fractures
• 14 Fractal Analysis of Rock Damage and Fracture
• 15 Fractal Description of the Roughness of Rock Joints
• 16 FRACTAL NATURE OF THE CLUSTER SYSTEMS
• References
• Subject Index
• Author Index