Collective Damage Growth Controls Fault Orientation in Quasibrittle Compressive Failure

Véronique Dansereau, Vincent Démery, Estelle Berthier, Jérôme Weiss, and Laurent Ponson

Phys. Rev. Lett. 122, 085501 – Published 27 February 2019

Abstract

The Mohr-Coulomb criterion is widely used in geosciences and solid mechanics to relate the state of stress at failure to the observed orientation of the resulting faults. This relation is based on the assumption that macroscopic failure takes place along the plane that maximizes the Coulomb stress. Here, this hypothesis is assessed by simulating compressive tests on an elastodamageable material that follows the Mohr-Coulomb criterion at the mesoscopic scale. We find that the macroscopic fault orientation is not given by the Mohr-Coulomb criterion. Instead, for a weakly disordered material, it corresponds to the most unstable mode of damage growth, which we determine through a linear stability analysis of its homogeneously damaged state. Our study reveals that compressive failure emerges from the coalescence of damaged clusters within the material and that this collective process is suitably described at the continuum scale by introducing an elastic kernel that describes the interactions between these clusters.

Received 8 March 2018
Revised 12 December 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.085501

Physics Subject Headings (PhySH)

Continuum mechanics Elastic deformation Elastic modulus Elasticity Fracture Material failure Rocks

Random & disordered media

Linear response theory

Condensed Matter, Materials & Applied PhysicsGeneral PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Véronique Dansereau¹, Vincent Démery^2,3, Estelle Berthier^4,5, Jérôme Weiss⁶, and Laurent Ponson⁴

¹Nansen Environmental and Remote Sensing Center, N-5006 Bergen, Norway
²Gulliver, CNRS, ESPCI Paris, PSL Research University, 10 rue Vauquelin, 75005 Paris, France
³Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France
⁴Institut Jean Le Rond d’Alembert (UMR 7190), CNRS, Sorbonne Universités, 75005 Paris, France
⁵Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA
⁶Univ. Grenoble Alpes, CNRS, ISTerre, 38000 Grenoble, France

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Issue

Vol. 122, Iss. 8 — 1 March 2019

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Images

Figure 1
Compressive test simulations. (a) The prescribed boundary conditions are superimposed to a snapshot of the field of the level of damage $d$ simulated after peak load [timing indicated by the red vertical line in (b)]. The material properties in this simulation are $ϕ = 30 °$ and $ν = 0.3$ and the disorder parameters, $η = 0.05$ and $a = 1$ . No lateral confinement is applied. The orientation of the fault $θ_{loc}$ is determined by a projection histogram method [16]. (b) The corresponding stress-strain (black) and damage rate (gray) curves are given by the solid lines. The dotted lines show the same quantities for a simulation using identical loading and material properties and a stronger disorder ( $η = 0.5$ and $a = 1$ ). [(b), inset] Macroscopic maximum and minimum principal stresses, $Σ_{1}$ , $Σ_{2}$ , (colored dots) estimated at the onset of damage localization (i.e., at peak load) in a set of five simulations using the same material properties as in (a) and (b) and different confining ratios (biaxial compression for $R > 0$ and biaxial compression-tension for $R < 0$ ). The black solid lines represent the MC criterion for a homogeneous material with cohesion $\bar{τ_{c}}$ . Open circles are used for the disorder parameters $η = 0.05$ and $a = 1$ and filled circles for the parameters $η = 0.5$ and $a = 1$ .
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Figure 2
(a) Mean localization angle $θ_{loc}$ as a function of the internal friction angle $ϕ$ for an ensemble of 25 simulations with minimal disorder using identical boundary and loading conditions. No confinement is applied and $ν = 0.3$ . The black dashed line shows the MC prediction $θ_{MC}$ , the dotted line, the angle of the most unstable mode $θ_{LS}$ , and the dashed-dotted line the angle of maximal stress redistribution $θ_{\max}$ . The error bars represent $\pm 1$ standard deviation from the mean. Mean localization angle for (b) different values of Poisson’s ratio without confinement and (c) different values of confinement ratio for $ν = 0.3$ .
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Figure 3
Localization angle measured from the compression simulations as a function of the internal friction angle for (a) weak disorder ( $η = 0.05$ ) and (b) strong disorder ( $η = 0.5$ ) and different values of $a$ . No confinement is applied and $ν = 0.3$ . Mean $θ_{loc}$ for $a = 1$ and $η = 0.5$ (strong disorder) and (c) different values of $ν$ without confinement and (d) different confinement ratios for $ν = 0.3$ . The maximum confinement ratio, $R_{\max}$ [16], is 58% for $ϕ = 15 °$ , 33% for $ϕ = 30 °$ , 17% for $ϕ = 45 °$ , and 7% for $ϕ = 60 °$ . The black dashed line shows $θ_{MC}$ , the dotted lines $θ_{LS}$ , and the dashed-dotted line $θ_{\max}$ .
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