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

The properties of the Hamiltonian developed in Paper II are studied showing that at a particular strain level a ``localization'' phase transition occurs characterized by the emergence of conjugate bands of coherently oriented cracks. The functional integration that yields the partition function is then performed analytically using an approximation that employs only a subset of states in the functional neighborhood surrounding the most probable states. Such integration establishes the free energy of the system, and upon taking the derivatives of the free energy, the localization transition is shown to be continuous and to be distinct from peak stress. When the bulk modulus of the grain material is large, localization always occurs in the softening regime following peak stress, while for sufficiently small bulk moduli and at sufficiently low confining pressure, the localization occurs in the hardening regime prior to peak stress. In the approach to localization, the stress-strain relation for the whole rock remains analytic, as is observed both in experimental data and in simpler models. The correlation function of the crack fields is also obtained. It has a correlation length characterizing the aspect ratio of the crack clusters that diverges as (\xi \sim (\ep_{c}-\ep)^{-2}) at localization.Comment: 11 pages, 3 figure

Mechanical instability of saturated soils sampled in the Oran coast, Algeria.

La liquéfaction est un risque naturel important associé aux séismes. Certains de ses effets dévastateurs incluent le basculement et le naufrage des bâtiments et des ponts, ainsi que la destruction des pipelines. L'ingénierie géotechnique conventionnelle suppose que la liquéfaction se produit via une pression interstitielle élevée. Nous montrons, en utilisant des simulations et des expériences, un autre mécanisme de liquéfaction dans des sols sableux saturés, sans pression de fluide interstitielle élevée. L’objectif de ce travail, est de suivre et de caractériser le déplacement vertical d'un intrus sur une masse de sol saturée lors d'expériences en laboratoire. A partir des résultats, Il a été démontré que la liquéfaction a un impact direct sur le déplacement vertical de l’intrus. On peut en conclure que le mouvement de l'intrus dépend essentiellement de l'accélération imposée et de la densité relative du sol

Interacting damage models mapped onto Ising and percolation models

We introduce a class of damage models on regular lattices with isotropic interactions, as e.g. quasistatic fiber bundles. 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 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, that damage models with 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.Comment: 16 pages, 3 figure

Pattern formation during air injection into granular materials confined in a circular Hele-Shaw cell

We investigate the dynamics of granular materials confined in a radial Hele-Shaw cell, during central air injection. The behavior of this granular system, driven by its interstitial fluid, is studied both experimentally and numerically. This allows us to explore the associated pattern formation process, characterize its features and dynamics. We classify different hydrodynamic regimes as function of the injection pressure. The numerical model takes into account the interactions between the granular material and the interstitial fluid, as well as the solid-solid interactions between the grains and the confining plates. Numerical and experimental results are comparable, both to reproduce the hydrodynamical regimes experimentally observed, as well as the dynamical features associated to fingering and compacting. © 2006 The American Physical Society

Thermally activated crack fronts propagating in pinning disorder: simultaneous brittle/creep behavior depending on scale

We study theoretically the propagation of a crack front in mode I along an interface in a disordered elastic medium, with a numerical model considering a thermally activated rheology, toughness disorder, and long range elastic interactions. This model reproduces the large scale dynamics of the crack front position in fast or creep loading regimes, but also the small scale self-affine behavior of the front. Two different scaling laws are predicted for the front morphology, with a Hurst exponent of 0.5 at small scales, and a logarithmic scaling law at large scales, consistently with experiments. The prefactor of these scaling laws is expressed as function of the temperature, and of the quenched disorder characteristics. The cross-over between these regimes is expressed as function of the quenched disorder amplitude, is proportional to the average energy release rate, and to the inverse of temperature. This model captures as well the experimentally observed local velocity fluctuations probability distribution, with a high velocity tail $P(v) \sim v^{-2.6}$. This feature is shown to arise when the quenched disorder is sufficiently large, whereas smaller toughness fluctuations lead to a lognormal-like velocity distribution. Overall, the system is shown to obey a scaling determined by two distinct mechanisms as function of scale: namely, the large scales display fluctuations similar to an elastic line in an annealed noise excited as the average front travels through the pinning landscape, while small scales display a balance between thresholds in possible elastic forces and quenched disorder

Fracture of disordered solids in compression as a critical phenomenon: II. Model Hamiltonian for a population of interacting cracks

To obtain the probability distribution of 2D crack patterns in mesoscopic regions of a disordered solid, the formalism of Paper I requires that a functional form associating the crack patterns (or states) to their formation energy be developed. The crack states are here defined by an order parameter field representing both the presence and orientation of cracks at each site on a discrete square network. The associated Hamiltonian represents the total work required to lead an uncracked mesovolume into that state as averaged over the initial quenched disorder. The effect of cracks is to create mesovolumes having internal heterogeneity in their elastic moduli. To model the Hamiltonian, the effective elastic moduli corresponding to a given crack distribution are determined that includes crack-to-crack interactions. The interaction terms are entirely responsible for the localization transition analyzed in Paper III. The crack-opening energies are related to these effective moduli via Griffith's criterion as established in Paper I.Comment: 9 pages, 1 figur

Coupled air/granular flow in a linear Hele-Shaw cell

We investigate experimentally the pattern formation process during injection of air in a noncohesive granular material confined in a linear Hele-Shaw cell. We characterize the features and dynamics of this pattern formation on the basis of fast image analysis and sensitive pressure measurements. Behaviors are classified using two parameters-injection pressure and plate opening-and four hydrodynamic regimes are defined. For some regions of the parameter space, flows of air and grains are shown to be strongly coupled and instable, and lead to channelization within the granular material with obvious large-scale permeability variations. © 2008 The American Physical Society

Mixing of a granular layer falling through a fluid

We analyze the granular Rayleigh-Taylor instability of densely packed grains immersed in a compressible or an incompressible fluid using numerical simulations and two types of experiments. The simulations are based on a two-dimensional (2D) molecular dynamics model and the experiments have been carried out in systems of grains immersed in water/glycerol (incompressible fluid) and in air (compressible fluid). The variation of the interstitial fluid is shown to generate different dynamical patterns and mixing properties of the granular systems. The results have been quantified using 2D autocorrelation functions, the power spectrum of the velocity field and velocity field histograms. Excellent agreement is found between the numerical simulations and the experiments. © 2010 The American Physical Society