Fracture of disordered solids in compression as a critical phenomenon. III. Analysis of the localization transition

How cracks are hot and cool: a burning issue for paper

Material failure is accompanied by important heat exchange, with extremely high temperature - thousands of degrees - reached at crack tips. Such a temperature may subsequently alter the mechanical properties of stressed solids, and finally facilitate their rupture. Thermal runaway weakening processes could indeed explain stick-slip motions and even be responsible for deep earthquakes. Therefore, to better understand catastrophic rupture events, it appears crucial to establish an accurate energy budget of fracture propagation from a clear measure of various energy dissipation sources. In this work, combining analytical calculations and numerical simulations, we directly relate the temperature field around a moving crack tip to the part α of mechanical energy converted into heat. By monitoring the slow crack growth in paper sheets using an infrared camera, we measure a significant fraction α = 12% ± 4%. Besides, we show that (self-generated) heat accumulation could weaken our samples by microfiber combustion, and lead to a fast crack/dynamic failure/regime.

Nucleation phenomena in an annealed damage model: statistics of times to failure

In this paper we investigate the statistical behavior of an annealed continuous damage model. For different model variations we study distributions of times to failure and compare these results with the classical case of metastable nucleation in statistical physics. We show that our model has a tuning parameter, related to the degree of damage reversibility, that determines the model's behavior. Depending on the value of this parameter, our model exhibits statistical behavior either similar to classical reversible nucleation phenomena in statistical physics or to an absolutely different type of behavior intrinsic to systems with damage. This comparison allows us to investigate possible similarities and differences between damage phenomena and reversible nucleation.

Average crack-front velocity during subcritical fracture propagation in a heterogeneous medium

We study the average velocity of crack fronts during stable interfacial fracture experiments in a heterogeneous quasibrittle material under constant loading rates and during long relaxation tests. The transparency of the material (polymethylmethacrylate) allows continuous tracking of the front position and relation of its evolution to the energy release rate. Despite significant velocity fluctuations at local scales, we show that a model of independent thermally activated sites successfully reproduces the large-scale behavior of the crack front for several loading conditions.

Mean-field theory of localization in a fuse model

We propose a mean-field theory for the localization of damage in a quasistatic fuse model on a cylinder. Depending on the quenched disorder distribution of the fuse thresholds, we show analytically that the system can either stay in a percolation regime up to breakdown, or start at some imposed current, to localize starting from the smallest scale (lattice spacing), or instead go to a diffuse localization regime where damage starts to concentrate in bands of width scaling as the width of the system, but remains diffuse at smaller scales. Depending on the nature of the quenched disorder on the fuse thresholds, we derive analytically the phase diagram of the system separating these regimes and the current levels for the onset of these possible localizations. We compare these predictions to numerical results.

Interacting damage models mapped onto Ising and percolation models

We introduce a class of damage models on regular lattices with isotropic interactions between the broken cells of the lattice. Quasi-static fiber bundles are an example. The interactions are assumed to be weak, in the sense that the stress perturbation from a broken cell is much smaller than the mean stress in the system. The system starts intact with a surface-energy threshold required to break any cell sampled from an uncorrelated quenched-disorder distribution. The evolution of this heterogeneous system is ruled by Griffith's principle which states that a cell breaks when the release in potential (elastic) energy in the system exceeds the surface-energy barrier necessary to break the cell. By direct integration over all possible realizations of the quenched disorder, we obtain the probability distribution of each damage configuration at any level of the imposed external deformation. We demonstrate an isomorphism between the distributions so obtained and standard generalized Ising models, in which the coupling constants and effective temperature in the Ising model are functions of the nature of the quenched-disorder distribution and the extent of accumulated damage. In particular, we show that damage models with global load sharing are isomorphic to standard percolation theory and that damage models with a local load sharing rule are isomorphic to the standard Ising model, and draw consequences thereof for the universality class and behavior of the autocorrelation length of the breakdown transitions corresponding to these models. We also treat damage models having more general power-law interactions, and classify the breakdown process as a function of the power-law interaction exponent. Last, we also show that the probability distribution over configurations is a maximum of Shannon's entropy under some specific constraints related to the energetic balance of the fracture process, which firmly relates this type of quenched-disorder based damage model to standard statistical mechanics.

Correlation length exponent in the three-dimensional fuse network

We present numerical measurements of the critical correlation length exponent nu in the three-dimensional fuse model. Using sufficiently broad threshold distributions to ensure that the system is the strong-disorder regime, we determine nu to be nu=0.83+/-0.04 based on analyzing the fluctuations of the survival probability. This value is different from that of ordinary percolation, which is 0.88.

Origin of the universal roughness exponent of brittle fracture surfaces:stress-weighted percolation in the damage zone

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 nu approximately 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 zeta=2nu/(1+2nu)=8/11. This is in accordance with the numerical value zeta=0.75. Using the value for 3D percolation, nu=0.88, we predict the roughness exponent in the 3D fuse model to be zeta=0.64, in close agreement with the previously published value of 0.62+/-0.05. We furthermore predict zeta=4/5 for 3D brittle fractures, based on a recent calculation giving nu=2. This is in full accordance with the value zeta=0.80 found experimentally.