Mikko J. Alava

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

Imbibition in disordered media

The physics of liquids in porous media gives rise to many interesting phenomena, including imbibition where a viscous fluid displaces a less viscous one. Here we discuss the theoretical and experimental progress made in recent years in this field. The emphasis is on an interfacial description, akin to the focus of a statistical physics approach. Coarse-grained equations of motion have been recently presented in the literature. These contain terms that take into account the pertinent features of imbibition: non-locality and the quenched noise that arises from the random environment, fluctuations of the fluid flow and capillary forces. The theoretical progress has highlighted the presence of intrinsic length-scales that invalidate scale invariance often assumed to be present in kinetic roughening processes such as that of a two-phase boundary in liquid penetration. Another important fact is that the macroscopic fluid flow, the kinetic roughening properties, and the effective noise in the problem are all coupled. Many possible deviations from simple scaling behaviour exist, and we outline the experimental evidence. Finally, prospects for further work, both theoretical and experimental, are discussed.

Structural transitions in scale-free networks.

Real growing networks such as the World Wide Web or personal connection based networks are characterized by a high degree of clustering, in addition to the small-world property and the absence of a characteristic scale. Appropriate modifications of the (Barabási-Albert) preferential attachment network growth capture all these aspects. We present a scaling theory to describe the behavior of the generalized models and the mean-field rate equation for clustering. This is solved for a specific case with the result C(k) approximately 1/k for the clustering of a node of degree k. This mean-field exponent agrees with simulations, and reproduces the clustering of many real networks.

Acoustic emission from paper fracture

We report tensile failure experiments on paper sheets. The acoustic emission energy and the waiting times between acoustic events follow power-law distributions. This remains true while the strain rate is varied by more than 2 orders of magnitude. The energy statistics has the exponent beta approximately 1.25+/-0.10 and the waiting times the exponent tau approximately 1.0+/-0.1, in particular, for the energy roughly independent of the strain rate. These results do not compare well with fracture models, for (brittle) disordered media, which as such exhibit criticality. One reason may be residual stresses, neglected in most theories.

Planar random networks with flexible fibers.

The transition in random fiber networks from two-dimensional to three-dimensional planar structure driven by increasing coverage (total fiber length per unit area) is studied with a deposition model. At low coverage the network geometry depends on the scale-free product of fiber length and coverage while at high coverage it depends on a scale-free combination of flexibility, width and thickness of the fibers. With increasing coverage the roughness of the free surface decouples from the substrate, faster when fibers are stiffer. In the high coverage region roughness decreases exponentially with increasing fiber flexibility.

Correlations in Bipartite Collaboration Networks

Collaboration networks are studied as an example of growing bipartite networks. These have been previously observed to exhibit structure such as positive correlations between nearest-neighbour degrees. However, a detailed understanding of the origin of such and the growth dynamics is lacking. Both of these issues are analysed empirically and simulated using various models. A new growth model is presented, incorporating empirically necessary ingredients such as bipartiteness and sublinear preferential attachment. This, and a recently proposed model of team assembly, both agree roughly with some empirical observations and fail in several others.

Patterns, Entropy, and Predictability of Human Mobility and Life

Cellular phones are now offering an ubiquitous means for scientists to observe life: how people act, move and respond to external influences. They can be utilized as measurement devices of individual persons and for groups of people of the social context and the related interactions. The picture of human life that emerges shows complexity, which is manifested in such data in properties of the spatiotemporal tracks of individuals. We extract from smartphone-based data for a set of persons important locations such as “home”, “work” and so forth over fixed length time-slots covering the days in the data-set (see also [1], [2]). This set of typical places is heavy-tailed, a power-law distribution with an exponent close to −1.7. To analyze the regularities and stochastic features present, the days are classified for each person into regular, personal patterns. To this are superimposed fluctuations for each day. This randomness is measured by “life” entropy, computed both before and after finding the clustering so as to subtract the contribution of a number of patterns. The main issue that we then address is how predictable individuals are in their mobility. The patterns and entropy are reflected in the predictability of the mobility of the life both individually and on average. We explore the simple approaches to guess the location from the typical behavior, and of exploiting the transition probabilities with time from location or activity A to B. The patterns allow an enhanced predictability, at least up to a few hours into the future from the current location. Such fixed habits are most clearly visible in the working-day length.

Kinetic roughening in slow combustion of paper.

Results of experiments on the dynamics and kinetic roughening of one-dimensional slow-combustion fronts in three grades of paper are reported. Extensive averaging of the data allows a detailed analysis of the spatial and temporal development of the interface fluctuations. The asymptotic scaling properties, on long length and time scales, are well described by the Kardar-Parisi-Zhang (KPZ) equation with short-range, uncorrelated noise. To obtain a more detailed picture of the strong-coupling fixed point, characteristic of the KPZ universality class, universal amplitude ratios, and the universal coupling constant are computed from the data and found to be in good agreement with theory. Below the spatial and temporal scales at which a crossover takes place to the standard KPZ behavior, the fronts display higher apparent exponents and apparent multiscaling. In this regime the interface velocities are spatially and temporally correlated, and the distribution of the magnitudes of the effective noise has a power-law tail. The relation of the observed short-range behavior and the noise as determined from the local velocity fluctuations is discussed.

Distributed graph coloring for self-organization in LTE networks

Primary Component Carrier Selection and Physical Cell ID Assignment are two important self-configuration problems pertinent to LTE-Advanced. In this work, we investigate the possibility to solve these problems in a distributive manner using a graph coloring approach. Algorithms based on real-valued interference pricing of conflicts converge rapidly to a local optimum, whereas algorithms with binary interference pricing have a chance to find a global optimum. We apply both local search algorithms and complete algorithms such as Asynchronous Weak-Commitment Search. For system level performance evaluation, a picocellular scenario is considered, with indoor base stations in office houses placed in a Manhattan grid. We investigate a growing network, where neighbor cell lists are generated using practical measurement and reporting models. Distributed selection of conflict-free primary component carriers is shown to converge with 5 or more component carriers, while distributed assignment of confusionfree physical cell IDs is shown to converge with less than 15 IDs. The results reveal that the use of binary pricing of interference with an attempt to find a global optimum outperforms real-valued pricing.

Shortest paths and load scaling in scale-free trees

The average node-to-node distance of scale-free graphs depends logarithmically on N, the number of nodes, while the probability distribution function of the distances may take various forms. Here we analyze these by considering mean-field arguments and by mapping the m=1 case of the Barabási-Albert model into a tree with a depth-dependent branching ratio. This shows the origins of the average distance scaling and allows one to demonstrate why the distribution approaches a Gaussian in the limit of N large. The load, the number of the shortest distance paths passing through any node, is discussed in the tree presentation.