Scaling of interfaces in brittle fracture and perfect plasticity

Physica Scripta, 2003

The morphology of brittle fracture surfaces are self affine with roughness exponents that may be classified into a small number of universality classes. We discuss these in light of the recent proposal that the self affinity is a manifestation of the fracture process being a correlated percolation process. We also study numerically with high precision the roughness exponent in the two-dimensional fuse model with disorder both in breaking thresholds and conductances of the fuses. Our results are consistent with the predictions of the correlated percolation theory.

Physical Review Letters, 2006

The self-affine properties of post-mortem fracture surfaces in silica glass and aluminum alloy were investigated through the 2D height-height correlation function. They are observed to exhibit anisotropy. The roughness, dynamic and growth exponents are determined and shown to be the same for the two materials, irrespective of the crack velocity. These exponents are conjectured to be universal.

Physical Review B, 1991

We introduce and discuss the concept of scale-invariant disorder in connection with breakdown and fracture models of disordered brittle materials. We show that in the case of quenched-disorder models where the local breaking thresholds are randomly sampled, only two numbers determine the scaling properties of the models. These numbers characterize the behavior of the distribution of thresholds close to zero and to infinity. We review brieAy some results obtained in the literature and show how they fit into this framework. Finally, we address the case of an annealed disorder, and show via a mapping onto a quenched-disorder model, that our analysis is also valid there.

Physical Review Letters, 2008

The roughness exponent for fracture surfaces in the fuse model has been thought to be universal for narrow threshold distributions and has been important in the numerical studies of fracture roughness. We show that the fuse model gives a disorder dependent roughness exponent for narrow disorders when the lattice is influencing the fracture growth. When the influence of the lattice disappears, the local roughness exponent approaches local 0:65 0:03 for distribution with a tail toward small thresholds, but with large jumps in the profiles giving corrections to scaling on small scales. For very broad disorders the distribution of jumps becomes a Lévy distribution and the Lévy characteristics contribute to the local roughness exponent.

Roughness scaling laws for intergranular cracks deviate from self-affine (fractal-like) behavior at length scales related to the polycrystalline microstructure. We consider two versions of the same alloy material with many of the same microstructural length scales but differing in their processing history: one conventional and one grain boundary engineered. The engineered material, processed to contain a high fraction of ''special" grain boundaries, fails more slowly and more isotropically. We present evidence that the difference is determined by processes related to clusters of twin-related grains, shown through analysis of scales of the fracture roughness measured with confocal microscopy and the special grain boundary network determined by electron backscatter diffraction. Above the cluster scale, the fracture roughness exponents in the two materials are nearly indistinguishable (confirming theoretical predictions); below this scale conventional cracks exhibit correlations indicating consistently weak paths for crack propagation, suggesting percolation of ''random" boundaries.

Physical Review Letters, 2003

We suggest that the observed large-scale universal roughness of brittle fracture surfaces is due to the fracture propagation being a damage coalescence process described by a stress-weighted percolation phenomenon in a self-generated quadratic damage gradient. We use the quasistatic 2D fuse model as a paradigm of a mode I fracture model. We measure for this model, which exhibits a correlated percolation process, the correlation length exponent 1:35 and conjecture it to be equal to that of classical percolation, 4=3. We then show that the roughness exponent in the 2D fuse model is 2=1 2 8=11. This is in accordance with the numerical value 0:75. Using the value for 3D percolation, 0:88, we predict the roughness exponent in the 3D fuse model to be 0:64, in close agreement with the previously published value of 0:62 0:05. We furthermore predict 4=5 for 3D brittle fractures, based on a recent calculation giving 2. This is in full accordance with the value 0:80 found experimentally.

Physical Review Letters, 2013

We present a unified theory of fracture in disordered brittle media that reconciles apparently conflicting results reported in the literature. Our renormalization group based approach yields a phase diagram in which the percolation fixed point, expected for infinite disorder, is unstable for finite disorder and flows to a zero-disorder nucleation-type fixed point, thus showing that fracture has mixed first order and continuous character. In a region of intermediate disorder and finite system sizes, we predict a crossover with mean-field avalanche scaling. We discuss intriguing connections to other phenomena where critical scaling is only observed in finite size systems and disappears in the thermodynamic limit. PACS numbers: 62.20.mj,62.20.mm,62.20.mt,64.60.ae,05.70.Jk,45.70.Ht,64.60.F-Brittle fracture in disordered media intertwines two phenomena that seldom coexist, namely, nucleation and critical fluctuations. The usual dichotomy of thought between nucleated and continuous transitions makes the study of fracture interesting. Even more intriguing is the fact that crack nucleation happens at zero stress in the thermodynamic limit: smaller is stronger and larger is weaker. This makes the existence of critical fluctuation in the form of clusters and avalanches of all sizes even more mysterious. What kind of critical point governs a phase transition that happens at zero applied field (stress) in the thermodynamic limit, and what is the universality class of such a transition? How do self-similar clusters, extremely rough crack surfaces, and scale invariant avalanches ultimately give rise to sharp cracks and localized growth? These questions have been addressed previously via a host of different theories, such as those based on percolation and multifractals [1-4], spinodal modes and mean-field criticality , and classical nucleation . In this Letter, we present a theoretical framework based on the renormalization group and crossover scaling that unifies the seemingly disparate descriptions of fracture into one consistent framework.

Physical Review B, 2007

We present simulation results on fracture and random damage percolation in disordered two-dimensional ͑2D͒ lattices of different sizes. We systematically study the effect of disorder strength on the stress-strain behavior and on the evolving fracture pattern. In particular, the similarity of damage-cluster statistics between fracture and random percolation is investigated. For fracture in highly disordered systems, we confirm and extend our earlier results on the existence of a percolationlike damage regime, with accurate scaling laws for the cluster statistics, but we show that this regime vanishes at intermediate disorder strength. For low disorder, a qualitatively different and anisotropic damage pattern develops from the very beginning of loading. Both for low and high disorder strengths, macroscopic localization and strong damage anisotropy set in around the maximum-stress point, leading to the final crack formation. The surface roughness of the ultimate crack shows accurate size scaling, with a universal roughness exponent independent of the disorder strength but slightly dependent on the precise definition of the crack profile. The simulated roughness exponent is in good agreement with other numerical and experimental results on 2D systems and also close to the prediction of gradient percolation.

Physical Review E, 2006

We address the role of the nature of material disorder in determining the roughness of cracks which grow by damage nucleation and coalescence ahead of the crack tip. We highlight the role of quenched and annealed disorders in relation to the length scales d and ξc associated with the disorder and the damage nucleation respectively. In two related models, one with quenched disorder in which d ≃ ξc, the other with annealed disorder in which d ≪ ξc, we find qualitatively different roughening properties for the resulting cracks in 2-dimensions. The first model results in random cracks with an asymptotic roughening exponent ζ ≈ 0.5. The second model shows correlated roughening with ζ ≈ 0.66. The reasons for the qualitative difference are rationalized and explained.

Strength, fracture and complexity, 2005

We have implemented a numerical network of random fuses in three dimensions in order to study breakdown processes of brittle materials. With this model we have been able to measure critical exponents at breakdown, and in that sense gained information of the scaling laws involved in the morphology of fracture surfaces. We have studied both the scaling of roughness and behaviour of the correlation length ¡ for broad distributions in order to examine the linkage between them.