Fracture of disordered solids and statistical physics of damage:
The localization of deformation in sheared compressed rocks is a problem widely studied experimentally by the mechanics and geophysics community, and a considerable amount of work has been performed by physicists to model various breakdown phenomena in heterogeneous materials. The genesis of macrofracture in disordered solids submitted to shear stresses is phenomenologically decomposed in three stages, corresponding to 1. the emergence of a large-scale spatial organization of microdefects starting from a homogeneous rock with random uncorrelated properties for the breaking thresholds of the microscopic brittle elements (grains, grain joints). 2. coalescence of the microdefects in the previously created clusters of microcracks, leading to macroscopic shear bands. 3. growth and dynamics of fracturate networks with spanning macrofractures. The initial creation of diffuse damage and the localization of it in clusters acquiring progressively macroscopic structure, is often approched by modelling lattices of elastic bonds (springs or beams), with a quenched disorder in the breaking thresholds, and following the history of the model as the imposed macroscopic strain is increased. The statistical characteristics of the damage formation process is then studied numerically. (see web pages on fracture, fiber bundles and fuse models).
An alternative approach was recently developped, based on a theory of statistical physics to describe the properties a network representing oriented microcracks developing in a sheared rock - the basic model allows in each cell of a regular network representing elementary grains, the absence of microcrack, or the presence of simple ellipsoidal microcrack with a few possible orientations.
The statistical properties of this system are studied via a formalism mapping it on a system of a "gas" of microcracks in interaction. This allows to obtain analytically approximate expressions for the probability distribution over the possible configurations of damage at a given level of macroscopic strain. A configuration of damage corresponds to the entire specification of the locations and orientations of all microfractures in a network, which by hypothesis started in an intact configuration. Its probability of occurence is the fraction of realizations of the initial quenched disorder that lead to this damage configuration at the specified strain, given that the system started intact at zero strain. The basic assumption to build such a formalism is that the probability distribution of the configurations of microcracks at mesoscopic scale maximizes Shannon's entropy under constraints coming from the energetics of the fracture processes. The interactions between oriented microcracks were analytically derived using elastostatics. This approach does not require the use of any particular constitutive relationship for the mateiral, which is a result of the collective process of microcrack formation - the micromechanical model for the interactions between these elementary objects is basic elastostatics. The expression of the probability distribution of the occurence of any configuration of a population of microcracks,allows to study analytically the statistical portperties of localization of damage without the use of numerical tools.
The major difference ot this theory with usual thermodynamics is that the disorder is entirely quenched ab initio, i.e. we do not rely on thermally activated fracturation processes. The validity of the basic assumption, maximization of Shannon's entropy under constraint by the probability distribution over defaults configurations, was demonstrated in the simplified case of a fiber bundle model with global load sharing rule , paradigm of fracture models. This allowed to show the existence of a so-far unnoticed phase transition in this system. Using the same quenched disorder - based statistical mechanics formalism is used in the more interesting case of an oriented damage network with elastostatic interactions [2,3,4,5], allowed to show the presence of a localization transition in this system, where en-echelon microcracks regroup in elongated clusters inclined on the main stress axis, corresponding to the future shear bands. The evolution of the geometric properties of these clusters are obtained via the study of the correlation function between microcracks, which exhibits a correlation length characteristic of the elongation of the clusters' large axis, which diverges as the critical localization macroscopic strain is approched. The figure below represents in a plane which contains the main macrostress axis, a line along which the correlation function has a constant value, close to the localization threshold.
contour of iso-value of the damage correlation function, and correlation length close to the localization transition
The derived value of the critical exponent describing the divergence of the autocorrelation length, was recently proved through arguments based on gradient percolation theory, to explain the observed roughness (or Hurst) exponent of 3D brittle fracture in disordered materials, H=0.8 .
The macroscopic mechanical properties of the material can also be derived in this framework, which predicts that this localization transition happens for low confing pressures, in the hardening regime (i.e. after a monotonic axial strain/ axial stress dependence), or conversely in the softening regime (after peak stress) at higher confining pressures. The continuous (critical) character of this transition guarantees for the universality of these generic results between different types of rocks, which can notably differ in microfracturing mechanism, i.e. present more inter- or intra-granular breaking. The predictions of this model are in accordance with triaxial tests compression measure, in the sense that thay reproduce qualitatively the global mechanical behavior observed for rocks around peak stress.
 Pride, S.R. and R. Toussaint,Thermodynamics of fiber bundles, Physica A, 312, p 159-171, 2002. click for Journal version or cond-mat/0209131
 Toussaint, R. and S.R. Pride, Fracture localization of disordered solids in compression as a critical phenomenon: I. Statistical physics formalism, Physical Review E, 66(3), 036135, 2002. click for Journal version or cond-mat/0209124.
 Toussaint, R. and S.R. Pride, Fracture localization of disordered solids in compression as a critical phenomenon: II.Model hamiltonian for a population of interacting cracks, Physical Review E, 66(3), 036136, 2002. click for Journal version or cond-mat/0209127
 Toussaint, R. and S.R. Pride, Fracture localization of disordered solids in compression as a critical phenomenon: III. Analysis of the localization transition, Physical Review E, 66(3), 036137, 2002. click for Journal version or cond-mat/0209129
 Toussaint, R., Fracturation of rocks in compression: The localization process as a critical phenomenon. PhD thesis, University of Rennes I, 2001. click for pdf version (4 Mo) or ps.zip version (1.2 Mo).
 Hansen, A. and J. Schmittbuhl: Origin of the Universal Roughness Exponent of Brittle Fracture Surfaces: Correlated Percolation in the Damage Zone, Physical Review E, 2003. click for Journal version or cond-mat/0207360.
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