Phani Kumar V. V. Nukala

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

Crack roughness and avalanche precursors in the random fuse model.

We analyze the scaling of the crack roughness and of avalanche precursors in the two-dimensional random fuse model by numerical simulations, employing large system sizes and extensive sample averaging. We find that the crack roughness exhibits anomalous scaling, as recently observed in experiments. The roughness exponents (zeta, zeta(loc) ) and the global width distributions are found to be universal with respect to the lattice geometry. Failure is preceded by avalanche precursors whose distribution follows a power law up to a cutoff size. While the characteristic avalanche size scales as s(0) approximately L(D) , with a universal fractal dimension D , the distribution exponent tau differs slightly for triangular and diamond lattices and, in both cases, it is larger than the mean-field (fiber bundle) value tau=5/2 .

Statistical properties of fracture in a random spring model

Using large-scale numerical simulations, we analyze the statistical properties of fracture in the two-dimensional random spring model and compare it with its scalar counterpart: the random fuse model. We first consider the process of crack localization measuring the evolution of damage as the external load is raised. We find that, as in the fuse model, damage is initially uniform and localizes at peak load. Scaling laws for the damage density, fracture strength, and avalanche distributions follow with slight variations the behavior observed in the random fuse model. We thus conclude that scalar models provide a faithful representation of the fracture properties of disordered systems.

Size effects in statistical fracture

We review statistical theories and numerical methods employed to consider the sample size dependence of the failure strength distribution of disordered materials. We first overview the analytical predictions of extreme value statistics and fibre bundle models and discuss their limitations. Next, we review energetic and geometric approaches to fracture size effects for specimens with a flaw. Finally, we overview the numerical simulations of lattice models and compare with theoretical models.

A fast and efficient algorithm for Slater determinant updates in quantum Monte Carlo simulations

We present an efficient low-rank updating algorithm for updating the trial wave functions used in quantum Monte Carlo (QMC) simulations. The algorithm is based on low-rank updating of the Slater determinants. In particular, the computational complexity of the algorithm is O(kN) during the kth step compared to traditional algorithms that require O(N(2)) computations, where N is the system size. For single determinant trial wave functions the new algorithm is faster than the traditional O(N(2)) Sherman-Morrison algorithm for up to O(N) updates. For multideterminant configuration-interaction-type trial wave functions of M+1 determinants, the new algorithm is significantly more efficient, saving both O(MN(2)) work and O(MN(2)) storage. The algorithm enables more accurate and significantly more efficient QMC calculations using configuration-interaction-type wave functions.

Submatrix updates for the continuous-time auxiliary-field algorithm

(Dated: October 19, 2010)We present a sub-matrix update algorithm for the continuous-time auxiliary field method thatallows the simulation of large lattice and impurity problems. The algorithm takes optimal advantageof modern CPU architectures by consistently using matrix instead of vector operations, resulting in aspeedup of a factor of ≈ 8 and thereby allowing access to larger systems and lower temperature. Weillustrate the power of our algorithm at the example of a cluster dynamical mean field simulation ofthe N´eel transition in the three-dimensional Hubbard model, where we show momentum dependentself-energies for clusters with up to 100 sites.

Time-parallel multiscale/multiphysics framework

We introduce the time-parallel compound wavelet matrix method (tpCWM) for modeling the temporal evolution of multiscale and multiphysics systems. The method couples time parallel (TP) and CWM methods operating at different spatial and temporal scales. We demonstrate the efficiency of our approach on two examples: a chemical reaction kinetic system and a non-linear predator-prey system. Our results indicate that the tpCWM technique is capable of accelerating time-to-solution by 2-3-orders of magnitude and is amenable to efficient parallel implementation.

Role of disorder in the size scaling of material strength.

We study the sample-size dependence of the strength of disordered materials with a flaw, by numerical simulations of lattice models for fracture. We find a crossover between a regime controlled by the disorder and another controlled by stress concentrations, ruled by continuum fracture mechanics. The results are formulated in terms of a scaling law involving a statistical fracture process zone. Its existence and scaling properties are revealed only by sampling over many configurations of the disorder. The scaling law is in good agreement with experimental results obtained from notched paper samples.

Percolation and localization in the random fuse model

We analyse damage nucleation and localization in the random fuse model with strong disorder using numerical simulations. In the initial stages of the fracture process, damage evolves in an uncorrelated manner, resembling percolation. Subsequently, as the damage starts to accumulate, current enhancement at the tips of the microcracks leads eventually to catastrophic failure. We study this behaviour, quantifying the deviations from percolation and discussing alternative scaling laws for damage. The analysis of damage profiles confirms that localization occurs abruptly, starting from a uniform damage landscape. Finally, we show that the cumulative damage distribution follows the normal distribution, suggesting that damage is uncorrelated on large length scales.

An efficient algorithm for simulating fracture using large fuse networks

The high computational cost involved in modelling of the progressive fracture simulations using large discrete lattice networks stems from the requirement to solve a new large set of linear equations every time a new lattice bond is broken. To address this problem, we propose an algorithm that combines the multiple-rank sparse Cholesky downdating algorithm with the rank-p inverse updating algorithm based on the Sherman–Morrison–Woodbury formula for the simulation of progressive fracture in disordered quasi-brittle materials using discrete lattice networks. Using the present algorithm, the computational complexity of solving the new set of linear equations after breaking a bond reduces to the same order as that of a simple backsolve (forward elimination and backward substitution) using the already LU factored matrix. That is, the computational cost is O(nnz(L)), where nnz(L) denotes the number of non-zeros of the Cholesky factorization L of the stiffness matrix A. This algorithm using the direct sparse solver is faster than the Fourier accelerated preconditioned conjugate gradient (PCG) iterative solvers, and eliminates the critical slowing down associated with the iterative solvers that is especially severe close to the critical points. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.