Desiccation Cracks and their Patterns: Formation and Modelling in Science and Nature (Statistical Physics of Fracture and Breakdown)

Bringing together basic ideas, classical theories, recent experimental and theoretical aspects, this book explains desiccation cracks from simple, easily-comprehensible cases to more complex, applied situations. The ideal team of authors, combining experimental and theoretical backgrounds, and with experience in both physical and earth sciences, discuss how the study of cracks can lead to the design of crack-resistant materials, as well as how cracks can be grown to generate patterned surfaces at the nano- and micro-scales. Important research and recent developments on tailoring desiccation cracks by different methods are covered, supported by straightforward, yet deep theoretical models. Intended for a broad readership spanning physics, materials science, and engineering to the geosciences, the book also includes additional reading especially for students engaged in pattern formation research. Cover 1 Contents 7 Preface 13 List of Symbols 15 Chapter 1 Introduction 21 1.1 Why Study Drying Mud? 22 1.2 Objectives and Organization of the Book 24 1.3 Approach and Scope 26 References 27 Chapter 2 Elasticity 29 2.1 On Springs 29 2.2 Deformation, Displacement and Strain 31 2.3 Transformations of Strains, Principal Strains and Volumetric Strain 35 2.4 Stress 37 2.5 Thermodynamics and the Work of Deformation 41 2.6 Linear Elasticity 43 2.7 Different Formulations of Linear Elasticity 46 2.8 Plane Elasticity 49 2.8.1 Plane Strain and Plane Stress 49 2.8.2 Airy Stress Function 51 2.9 Summary 52 2.10 Further Reading 53 References 53 Chapter 3 Fracture Mechanics 55 3.1 Griffith and Fracture Energy 55 3.2 Stress Concentration 60 3.3 Stress Intensity Factors 61 3.4 Fracture Toughness and the Relationship Between K and G 64 3.5 Summary of the Critical Conditions for Fracture 66 3.6 An Example: Thin-Film Fracture 67 3.7 Nonlinear and Dissipative Effects of Fracture 73 3.7.1 A Plastic Zone Model of Fracture 74 3.7.2 A Mesoscopic View - The Path-Independent J-integral 76 3.7.3 Dynamic Elasticity and Dynamic Fracture 78 3.8 Crack Path Selection 80 3.9 Summary and Further Reading 84 References 85 Chapter 4 Poroelasticity 89 4.1 Pressure and Stress in a Two-component System 90 4.1.1 Fick's Laws 93 4.1.2 Darcy's Law 94 4.1.3 Network and Total Stress 94 4.2 Linear Poroelasticity 95 4.2.1 Poroelastic Energy Density 97 4.2.2 Poroelastic Constitutive Relations 98 4.2.3 Different Formulations of Poroelasticity 102 4.3 Relationship Between Poroelasticity and Thermoelasticity 104 4.4 Worked Examples of Poroelastic Deformation 106 4.5 Poroelasticity and a Driving Force for Fracture 108 4.6 Summary and Further Reading 112 References 114 Chapter 5 Colloids and Clays 117 5.1 DLVO Theory 118 5.1.1 van der Waals Potential 118 5.1.2 Electrostatic Potential 121 5.1.3 DLVO Theory and its Limitations 126 5.2 Clays 130 5.3 Summary and Further Reading 134 References 135 Chapter 6 Desiccation 137 6.1 Surface Tension and Capillary Pressure 138 6.1.1 Contact Lines and Capillary Rise 140 6.2 Solidification Through Evaporation 142 6.2.1 Skin Formation 142 6.2.2 Crystals and Cages 144 6.2.3 Aggregation 146 6.3 Pore-Scale Processes 148 6.3.1 Structure of a Drying Soil 150 6.3.2 Dynamics of a Drying Soil 151 6.4 Continuum Models of Drying 155 6.4.1 Surface Drying 155 6.4.2 Internal Transport: Carman-Kozeny 159 6.5 Further Reading 160 References 161 Chapter 7 Patterns of Crack Networks in Homogeneous Media 165 7.1 Introduction 165 7.2 Experimental Observations 166 7.2.1 Sequential Fragmentation and Length Scale Selection 168 7.2.2 Scaling of Crack Width 172 7.2.3 Distribution of Angles Between Cracks 173 7.3 Directional Drying 174 7.4 Characterizing the Crack Pattern: 2D View 175 7.4.1 Scale Invariance in Crack Patterns: Self-Similar and Self-Affine Structures 175 7.4.1.1 Scale Invariant Crack Width Distribution 176 7.4.1.2 Fractal Dimension of the Crack Edge 177 7.4.1.3 Self-Affinity of the Fracture Surface 178 7.4.1.4 Fractal Fracture Mechanics 180 7.4.2 Topology and Connectivity of the Crack Network 181 7.4.2.1 Minkowski Numbers and Densities 185 7.4.2.2 Network Theory Approach: Mapping onto an Equivalent Network 187 7.4.3 Percolation 189 7.5 Instabilities: Spirals and Wavy Cracks, En Echelon/En Passant Cracks, Star Cracks, and Wing Cracks 193 7.5.1 En Echelon Cracks 194 7.5.2 En Passant Cracks 194 7.5.3 Spiral Cracks 195 7.5.4 Wavy Cracks 197 7.5.5 Star Bursts and More Patterns 198 7.6 Crack Dynamics and Branching Cracks 199 7.7 Transition Between Different Modes of Instability and Fracture 202 7.7.1 Dendrite to Fracture 202 7.7.2 Viscous Fingering to Fracture 204 7.7.3 Invasion Percolation to Fracture 205 7.8 Towards Three Dimensions: Geological Formations, Drying Soil and Peeling 208 7.8.1 Obreimoff's Experiment 208 7.8.2 Natural Mud Cracks in Quasi-2D 209 7.9 Simulation of Quasi-2D Patterns 210 7.9.1 2D Modelling of Fracture: The Fibre Bundle Model 211 7.9.2 Random Fuse Model 212 7.9.3 Spring Network Model 212 7.9.4 Other Models 216 7.10 Summary 217 7.11 Further Reading 217 References 218 Chapter 8 The Effects of Plasticity on Crack Formation 227 8.1 Introduction to Rheology 227 8.1.1 Elastic Material and Fluid 228 8.1.2 Linear Viscoelasticity 231 8.1.3 Bingham Model 234 8.2 Elastoplasticity for Slow Deformation Processes 236 8.2.1 Decomposition of Elastic and Plastic Deformation 236 8.2.2 Thermodynamics of Elastoplasticity 238 8.2.3 Yield Conditions and the Normality Law 239 8.2.4 Yield Conditions of Paste-Like Materials 242 8.3 Crack Propagation in a Layer of Wet Paste 243 8.3.1 Plumose Structure in Crack Surfaces 243 8.3.2 Microscopic Observation of Plastic Deformation 244 8.3.3 Measurements of the Speed of Crack Growth in a Uniform Paste Layer 247 8.4 Theoretical Approaches for Crack Velocities 250 8.4.1 Viscoelastic Effect on Crack Propagation: 1D Lattice Model of Rheological Elements 251 8.4.2 Competition of Global Plastic Relaxation and Crack Growth 253 8.5 Memory Effect of Paste Due to Its Plasticity 258 8.5.1 Memory of Vibration and Its Visualization as Desiccation Crack Pattern 259 8.5.1.1 Memory of Vibration and Lamellar Crack Pattern 259 8.5.1.2 Plasticity of Paste 261 8.5.1.3 Condition for the Memory Effect of Vibration: Experimental Results 263 8.5.2 Residual Tension Theory to Explain Memory Effect of Vibration 264 8.5.2.1 Quasi-linear Analysis 265 8.5.2.2 Governing Equations for Non-linear Analysis 269 8.5.2.3 Non-linear Analysis 271 8.5.2.4 Condition for the Memory Effect of Vibration: Theoretical Explanation 273 8.5.3 Position Control of Cracks by Memory Effect and Faraday Waves 274 8.5.4 Memory of Flow and a Role of Interaction Between Colloidal Particles 278 Further Reading 282 References 283 Chapter 9 Special Topics 287 9.1 Tailoring Crack Patterns 287 9.1.1 Effect of Electric Fields on Desiccation Cracks 288 9.1.1.1 Effects of a Direct Field (DC) 288 9.1.1.2 Effect of an Alternating Field (AC) 290 9.1.1.3 DC Field Effect in Drying Droplets 291 9.1.2 Effect of a Magnetic Field on Desiccation Cracks 294 9.1.3 Patterning Cracks Through Micro-Technology 296 9.2 Designing Crack-Resistant Materials and Composites 299 9.2.1 Composites of Soft and Hard Particles 304 9.2.1.1 Employing Heterogeneous Material 305 9.2.2 Crack Reduction with 'Liquid Particles' 308 9.3 Crack Patterns in Drying Droplets of Biofluids 310 9.3.1 Human Blood Droplets and Drying Dynamics 311 9.3.2 Effect of Relative Humidity on Drying Droplets 315 9.3.3 Substrate Effect on Drying Droplets of Blood 316 9.4 Evolving Crack Networks 317 9.4.1 Columnar Joints 318 9.4.2 Evolving Mud Cracks 324 9.4.3 Other Crack Patterns 328 9.5 Further Reading 330 References 331 Appendix A: A Primer on Vectors and Tensors 337 A.1 Tensor Notation 337 A.2 Tensor Multiplication 339 A.3 Tensor Transformations 341 A.4 Tensor Differentiation 343 Appendix B: Fractals: Self-Similar and Self-Affine Systems 347 B.1 Self-Similarity and Fractal Dimension 347 B.2 Self-Affine Systems 351 B.3 Further Reading 352 References 353 Appendix C: Formulation of Elastoplasticity Based on Dissipation Functions 355 References 356 Appendix D: Steady Propagating Solution of Langer Model 357 Appendix E: Stress Expression in Finite Deformation Theory 359 References 361 Index 363 EULA 370