Statistical Physics of Fracture and Breakdown

Under extreme conditions the mechanical or electrical properties of solids tend to destabilize, leading to failure or breakdown. These instabilities often nucleate or spread from disorders in the structure of the solid. This book by two experts in the field investigates current techniques for modeling these failure and breakdown processes. It illustrates the basic modeling principles through a series of computer and laboratory simulations and 'table top' experiments. The book centers on three important case studies: electrical failures like fuse and dielectric breakdown; mechanical fractures; and earthquakes, which exhibit dynamic failure. The material will interest all graduate students and researchers studying disordered systems, whether their focus is the mechanical failure of solids, the electrical breakdown of conductors, or earthquake mechanics.

1 Introduction

1

(29)

1.1 Breakdown of disordered solids and dynamic frictional failures

1

(3)

1.2 A brief introduction to some theoretical ideas and models

4

(26)

1.2.1 A brief summary of percolation theory

5

(15)

1.2.2 Stress concentration and statistics of extremes

20

(7)

1.2.3 Self-organised criticality and sandpile models

27

(3)

2 Electrical breakdown in disordered solids

30

(50)

2.1 Introduction

30

(3)

2.2 The fuse problem

33

(28)

2.2.1 Qualitative analysis

33

(3)

2.2.2 Quantitative analysis: most probable failure current and distribution

36

(9)

2.2.3 Numerical simulations and experimental results

45

(3)

2.2.4 Other kinds of disorder: distribution of the failure threshold

48

(4)

2.2.5 The shortest path and the electromigration fuse model

52

(4)

2.2.6 Effects of temperature, AC fields and nonlinearity

56

(5)

2.3 The dielectric breakdown problem

61

(17)

2.3.1 Qualitative analysis

61

(1)

2.3.2 Duality in two dimensions

61

(3)

2.3.3 Lattice percolation

64

(3)

2.3.4 Continuum percolation

67

(1)

2.3.5 The shortest path

67

(1)

2.3.6 Dielectric breakdown with tunnelling bonds

68

(2)

2.3.7 Numerical simulations and experimental results

70

(8)

2.4 Conclusions

78

(2)

3 Fracture strength of disordered solids

80

(48)

3.1 Introduction

80

(2)

3.2 Fracture strength of a perfect solid containing a single crack

82

(9)

3.2.1 Stress concentration

84

(2)

3.2.2 Griffith's energy balance concept

86

(2)

3.2.3 Experimental and computer simulational verifications of Griffith's law

88

(3)

3.3 Fracture strength of brittle solids with small disorder and rough cracks

91

(4)

3.3.1 Griffith's law for fractal crack surfaces

92

(2)

3.3.2 Experimental observations

94

(1)

3.4 Fracture strength of strongly disordered solids

95

(11)

3.4.1 Estimates for the fracture exponent and comparisons with experiment

96

(9)

3.4.2 Brittleness and plastic yield

105

(1)

3.5 Fracture strength distribution

106

(7)

3.5.1 Extreme statistics and strength distribution

106

(3)

3.5.2 Comparisons with computer simulational and experimental results

109

(4)

3.6 Fracture strength scaling in systems with random breaking thresholds

113

(4)

3.6.1 Models with random spring constants

114

(1)

3.6.2 Models with random breaking strengths

114

(3)

3.7 Dynamics of fracture

117

(9)

3.7.1 Fracture propagation velocity

117

(2)

3.7.2 Large propagation velocity and morphology of fractured surfaces

119

(2)

3.7.3 Elastic precursor effects of complete fracture

121

(5)

3.8 Dynamic annealed impurity and self-organised criticality in fracture

126

(1)

3.9 Summary and conclusions

127

(1)

4 Earthquakes in model systems

128

(22)

4.1 Introduction

128

(2)

4.2 Burridge-Knopoff stick-slip model of earthquakes

130

(10)

4.2.1 Laboratory simulation model

130

(3)

4.2.2 Computer simulation model

133

(2)

4.2.3 Numerical studies and results

135

(5)

4.3 Self-organised criticality and cellular automata models of earthquakes

140

(3)

4.4 Earthquake fault patterns and percolation model of earthquakes

143

(2)

4.5 Precursors of self-organised criticality and earthquakes

145

(4)

4.5.1 Pulse response of the sandpile model

146

(2)

4.5.2 Response of the Burridge-Knopoff model to localised periodic pulses

148

(1)

4.6 Summary and conclusions

149

(1)

References

150

(9)

Index

159

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Statistical Physics of Fracture and Breakdown in Disordered Systems

Summary

Table of Contents

Supplemental Materials