Stefano Zapperi

Intermittent dislocation flow in viscoplastic deformation.

The viscoplastic deformation (creep) of crystalline materials under constant stress involves the motion of a large number of interacting dislocations. Analytical methods and sophisticated ‘dislocation dynamics’ simulations have proved very effective in the study of dislocation patterning, and have led to macroscopic constitutive laws of plastic deformation. Yet, a statistical analysis of the dynamics of an assembly of interacting dislocations has not hitherto been performed. Here we report acoustic emission measurements on stressed ice single crystals, the results of which indicate that dislocations move in a scale-free intermittent fashion. This result is confirmed by numerical simulations of a model of interacting dislocations that successfully reproduces the main features of the experiment. We find that dislocations generate a slowly evolving configuration landscape which coexists with rapid collective rearrangements. These rearrangements involve a comparatively small fraction of the dislocations and lead to an intermittent behaviour of the net plastic response. This basic dynamical picture appears to be a generic feature in the deformation of many other materials. Moreover, it should provide a framework for discussing fundamental aspects of plasticity that goes beyond standard mean-field approaches that see plastic deformation as a smooth laminar flow.

Statistical models of fracture

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

Life-support system benefits from noise

Mechanical ventilators are used to provide life support for patients with respiratory failure. But over the long term, these machines can damage the lungs, causing them to collapse and the partial pressure of oxygen in the arteries to drop to abnormally low values. In conventional mechanical ventilation, the respiratory rate and volume of air inspired per breath are fixed, although during natural breathing these parameters vary appreciably. A computer-controlled ventilator has now been introduced that can use noise to mimic this variability. We describe a conceptual model of lung injury in which the partial pressure of arterial oxygen is improved significantly by computer-controlled rather than conventional mechanical ventilation, in agreement with recent experimental data.

Colloquium : Modeling friction: from nanoscale to mesoscale

The physics of sliding friction is gaining impulse from nanoscale and mesoscale experiments, simulations, and theoretical modeling. This Colloquium reviews some recent developments in modeling and in atomistic simulation of friction, covering open-ended directions, unconventional nanofrictional systems, and unsolved problems.

Paths to self-organized criticality

We present a pedagogical introduction to self-organized criticality (SOC), unraveling its connections with nonequilibrium phase transitions. There are several paths from a conventional critical point to SOC. They begin with an absorbing-state phase transition (directed percolation is a familiar example), and impose supervision or driving on the system; two commonly used methods are extremal dynamics, and driving at a rate approaching zero. We illustrate this in sandpiles, where SOC is a consequence of slow driving in a system exhibiting an absorbing-state phase transition with a conserved density. Other paths to SOC, in driven interfaces, the Bak-Sneppen model, and self- organized directed percolation, are also examined. We review the status of experimental realizations of SOC in light of these observations.

Dynamics of a ferromagnetic domain wall: Avalanches, depinning transition, and the Barkhausen effect

We study the dynamics of a ferromagnetic domain wall driven by an external magnetic field through a disordered medium. The avalanchelike motion of the domain walls between pinned configurations produces a noise known as the Barkhausen effect. We discuss experimental results on soft ferromagnetic materials, with reference to the domain structure and the sample geometry, and report Barkhausen noise measurements on Fe21Co64B15 amorphous alloy. We construct an equation of motion for a flexible domain wall, which displays a depinning transition as the field is increased. The long-range dipolar interactions are shown to set the upper critical dimension to dc53, which implies that mean-field exponents~with possible logarithmic correction! are expected to describe the Barkhausen effect. We introduce a mean-field infinite-range model and show that it is equivalent to a previously introduced single-degree-of-freedom model, known to reproduce several experimental results. We numerically simulate the equation in d53, confirming the theoretical predictions. We compute the avalanche distributions as a function of the field driving rate and the intensity of the demagnetizing field. The scaling exponents change linearly with the driving rate, while the cutoff of the distribution is determined by the demagnetizing field, in remarkable agreement with experiments.@S0163-1829~98!08833-X#

Self-Organized Branching Processes: Mean-Field Theory for Avalanches

We discuss mean-field theories for self-organized criticality and the connection with the general theory of branching processes. We point out that the nature of the self-organization is not addressed properly by the previously proposed mean-field theories. We introduce a new mean-field model that explicitly takes the boundary conditions into account; in this way, the local dynamical rules are coupled to a global equation that drives the control parameter to its critical value. We study the model numerically, and analytically we compute the avalanche distributions.

Plasticity and avalanche behaviour in microfracturing phenomena

Inhomogeneous materials, such as plaster or concrete, subjected to an external elastic stress display sudden movements owing to the formation and propagation of microfractures. Studies of acoustic emission from these systems reveal power-law behaviour. Similar behaviour in damage propagation has also been seen in acoustic emission resulting from volcanic activity and hydrogen precipitation in niobium. It has been suggested that the underlying fracture dynamics in these systems might display self-organized criticality, implying that long-ranged correlations between fracture events lead to a scale-free cascade of ‘avalanches’. A hierarchy of avalanche events is also observed in a wide range of other systems, such as the dynamics of random magnets and high-temperature superconductors in magnetic fields, lung inflation and seismic behaviour characterized by the Gutenberg–Richter law. The applicability of self-organized criticality to microfracturing has been questioned,, however, as power laws alone are not unequivocal evidence for it. Here we present a scalar model of microfracturing which generates power-law behaviour in properties related to acoustic emission, and a scale-free hierarchy of avalanches characteristic of self-organized criticality. The geometric structure of the fracture surfaces agrees with that seen experimentally. We find that the critical steady state exhibits plastic macroscopic behaviour, which is commonly observed in real materials.

First-Order Transition in the Breakdown of Disordered Media

We study the approach to global breakdown in disordered media driven by increasing external forces. We first analyze the problem by mean-field theory, showing that the failure process can be described as a first-order phase transition, similarly to the case of thermally activated fracture in homogeneous media. Then we quantitatively confirm the predictions of the mean-field theory using numerical simulations of discrete models. Widely distributed avalanches and the corresponding meanfield scaling are explained by the long-range nature of elastic interactions. We discuss the analogy of our results to driven disordered first-order transitions and spinodal nucleation in magnetic systems. [S0031-9007(97)02501-5]

Scaling exponents for barkhausen avalanches in polycrystalline and amorphous ferromagnets

We investigate the scaling properties of the Barkhausen effect by recording the noise in several soft ferromagnetic materials: polycrystals with different grain sizes and amorphous alloys. We measure the Barkhausen avalanche distributions and determine the scaling exponents. In the limit of vanishing external field rate, we can group the samples in two distinct classes, characterized by exponents tau = 1.50+/-0.05 or tau = 1.27+/-0.03, for the avalanche size distributions. We interpret these results in terms of the depinning transition of domain walls and obtain an expression relating the cutoff of the distributions to the demagnetizing factor which is in quantitative agreement with experiments.