Scaling laws during localization of brittle/ductile systems
Résumé
Complex multiscale fracture patterns are found to develop during fracturing process, as a consequence of strength and heterogeneities enhanced by fracture-to-fracture interactions. This spatial complexity was found to be ubiquous in geological environments. Statistical analysis on these systems demonstrates that it can be quantified, at the 1st-order, by both basic scaling laws: a fractal correlation between fault positions, and a power-law distribution of fault lengths. The scaling exponents are varying according to the nature of the fracturing process (jointing or faulting), and/or to the stress context (body forces vs. tectonic forces; compression vs. extension vs. strike-slip). The rationale of these basic observations still remains an issue. Our view on these basic processes is mainly based on some sandbox experiments where the deformation is applied to a thin brittle-ductile plate whose strength is of the order of gravity forces. The work is also motivated by having a better understanding of lithosphere deformation in continental collision. At the macroscopic level, deformation results in a large-scale localization whose deformation regime (compressional, wrenching or extensional) depends on the balance between gravity forces and layer strengths. At finer level, large-scale shear bands are actually made up of a series of faults - actually shear bands that develop in the sand layer with geometrical properties similar to tectonic faults -, whose geometrical complexity is primarily dependent on the brittle-to ductile strength ratio . is a function of the compression velocity, silicone viscosity, layer thicknesses and densities was found to well quantify the main transition between a pure ductile regime where no large fault develops, to a pure-brittle regime where localization takes place in a few very large faults. In between (0.5< <10), faults organize in a complex multi-scale pattern that progressively leads to large-scale localization in two main conjugate shear bands densely fractured. A fine analysis of the deformation field highlights the very nature of the localization process. Thanks to high-resolution measures of displacement, a scaling analysis of the average deformation intensity was performed to highlight the coalescence of locally deformed zone into a large-scale structure. This analysis reveals two additional modes of localization in the intermediate brittle regime (0.5< <10), which differs in the scaling of the mean deformation intensity at small scales: a "ductile-but-localizing" regime where the deformation intensity becomes scale-independent at small scales, and a "brittle" regime where the deformation intensity increases continuously when decreasing scale. The latter regime directly addresses the issue of the homogenization of local heterogeneities since the basic notion of mean deformation is not univocally defined. In the former regime, the viscosity of the underlying ductile layer introduces a correlation length scale below which strain heterogeneities are smoothed. Localization occurs from structures whose size is large enough to become insensitive to viscosity effects. The large-scale localization rate is a function of . Finally we discuss the relationship between the characteristics of the fault patterns (length, density) as a function of , and of the strain field.