Contents

Introduction…………………………………………………....

9

Chapter 1.  Geometric and physical models of fractals……….

11

1.1. Geometric scaling…………………………………….

13

1.2. The Cantor set………………………………………..

14

1.3. The Koch curves……………………………………..

17

1.4. Fractal sets on plane and in space……………………

21

1.5. Modes of fractal clasters……………………………..

25

1.6. Fractal dimensions……………………………………

27

Chapter 2. The space fractal in nature, physical processes  and phenomena…………………………………...

32

2.1. The nature space fractals……………………………..

32

2.2. Fractal structures of diffusional growth.……………..

35

2.3. The Safman-Talor’s gidrodynamic instability and the viscous fingers structures……………………………..

44

2.4. The percolation structures……………………………

49

Chapter 3. The fractal time series and self-organized criticality............................................................................

56

3.1. The Brown’s motion………………………………….

56

3.2. Statistics of wave heights and Hurst’s law…………...

59

3.3. Self-organized criticality and flicker noise…………..

61

3.4. The empirical laws of seismoacoustics and SOC……

72

Chapter 4. The fractal time series and exponential laws in physics of toughness and plasticity of solids….......

76

4.1 Jerky flow of metals and alloys……………………….

76

4.2. Exponential laws in spectrum of acoustic emission during creep of ice crystal……………………….......

77

4.3. SOC in electromagnetic signal-precursor of fracture of ice…………………………………………………

80

4.4. SOC and hypothesis about universal mechanism of fracture of solids…………………………………….

92

Chapter 5. Morfology transitions between fractal and euclidean patterns of noneguilibriun growth…….

94

5.1. Morphology transition between fractal and enclidean forms of Luders band………………………………...

94

5.2.  Kinetic phase diagrams of fractal and enclidean patterns of non-equilibrium growth of ice in supercooled water…………………………………………

109

Control questions........................................................................

121

Literature……………………………………………………...

123