Statistical models of fracture

Abstract

Disorder and long-range interactions are two of the key components that make material failure an interesting playfield for the application of statistical mechanics. The cornerstone in this respect has been lattice models of the fracture in which a network of elastic beams, bonds, or electrical fuses with random failure thresholds are subject to an increasing external load. These models describe on a qualitative level the failure processes of real, brittle, or quasi-brittle materials. This has been particularly important in solving the classical engineering problems of material strength: the size dependence of maximum stress and its sample-to-sample statistical fluctuations. At the same time, lattice models pose many new fundamental questions in statistical physics, such as the relation between fracture and phase transitions. Experimental results point out to the existence of an intriguing crackling noise in the acoustic emission and of self-affine fractals in the crack surface morphology. Recent advances in computer power have enabled considerable progress in the understanding of such models. Among these partly still controversial issues, are the scaling and size-effects in material strength and accumulated damage, the statistics of avalanches or bursts of microfailures, and the morphology of the crack surface. Here we present an overview of the results obtained with lattice models for fracture, highlighting the relations with statistical physics theories and more conventional fracture mechanics approaches. Contents PAGE 1. Introduction 351 2. Elements of fracture mechanics 354  2.1. Theory of linear elasticity 354  2.2. Cracks in elastic media 355  2.3. The role of disorder on material strength 357  2.4. Extreme statistics for independent cracks 359  2.5. Interacting cracks and damage mechanics 360  2.6. Fracture mechanics of rough cracks 363   2.6.1. Crack dynamics in a disordered environment: self-affinity and anomalous scaling 363   2.6.2. Crack roughness and fracture energy 366 3. Experimental background 368  3.1. Strength distributions and size-effects 368  3.2. Rough cracks 371  3.3. Acoustic emission and avalanches 379  3.4. Time-dependent fracture and plasticity 385 4. Statistical models of failure 386  4.1. Random fuse networks: brittle and plastic 386  4.2. Tensorial models 391  4.3. Discrete lattice versus finite element modeling of fracture 393  4.4. Dynamic effects 397   4.4.1. Annealed disorder and other thermal effects 397   4.4.2. Sound waves and viscoelasticity 398  4.5. Atomistic simulations 401 5. Statistical theories for fracture models 402  5.1. Fiber bundle models 402   5.1.1. Equal load sharing fiber bundle models 403   5.1.2. Local load sharing fiber bundle models 405   5.1.3. Generalizations of fiber bundle models 406  5.2. Statistical mechanics of cracks: fracture as a phase transition 408   5.2.1. Generalities on phase transitions 409   5.2.2. Disorder induced non-equilibrium phase transitions 411   5.2.3. Phase transitions in fracture models 413  5.3. Crack depinning 415  5.4. Percolation and fracture 417   5.4.1. Percolation scaling 417   5.4.2. Variations of the percolation problem 419   5.4.3. Strength of diluted lattices 420   5.4.4. Crack fronts and gradient percolation 422 6. Numerical simulations 424  6.1. The I–V characteristics and the damage variable 425  6.2. Damage distribution 430   6.2.1. Scaling of damage density 432   6.2.2. Damage localization 435   6.2.3. Crack clusters and damage correlations 437  6.3. Fracture strength 440   6.3.1. The fracture strength distribution 440   6.3.2. Size effects 443   6.3.3. Strength of notched specimens 445  6.4. Crack roughness 447  6.5. Avalanches 450 7. Discussion and outlook 454  7.1. Strength distribution and size-effects 455  7.2. Morphology of the fracture surface: roughness exponents 456  7.3. Crack dynamics: avalanches and acoustic emission 457  7.4. From discrete models to damage mechanics 458  7.5. Concluding remarks and perspectives 458 Acknowledgments 459 Appendix A: Algorithms 459 References 468