## Abstract

We used the differential stress-induced damage (DSID) model to predict anisotropic crack propagation under tensile and shear stress. The damage variable is similar to a crack density tensor. The damage function and the damage potential are expressed as functions of the energy release rate, defined as the thermodynamic force that is work-conjugate to damage. Contrary to the previous damage models, flow rules are obtained by deriving dissipation functions by the energy release rate, and thermodynamic consistency is ensured. The damage criterion is adapted from the Drucker–Prager yield function. Simulations of biaxial stress tests showed that: (1) three-dimensional states of damage can be obtained for three-dimensional states of stress; (2) no damage propagates under isotropic compression; (3) crack planes propagate in the direction parallel to major compression stress; (4) damage propagation hardens the material; (5) stiffness and deformation anisotropy result from the anisotropy of damage. There is no one-to-one relationship between stress and damage. We demonstrated the effect of the loading sequence in a two-step simulation (a shear loading phase and a compression loading phase): the current state of stress and damage can be used to track the effect of stress history on damage rotation. We finally conducted a sensitivity analysis with the finite element method, to explore the stress conditions in which damage is expected to rotate around a circular cavity subject to pressurization or depressurization. Simulation results showed that: (1) before damage initiation, the DSID model matches the analytical solution of stress distribution obtained with the theory of elasticity; (2) the DSID model can predict the extent of the tensile damage zone at the crown, and that of the compressive damage zone at the sidewalls; (3) damage generated during a vertical far-field compression followed by a depressurization of the cavity is more intense than that generated during a depressurization of the cavity followed by a vertical far-field compression.

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## Acknowledgments

This study was conducted at the Georgia Institute of Technology, as part of a research program on Finite Element Modeling of Hydraulic Fracturing. Funding was provided by ConocoPhillips, Houston, Texas.

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## Notation: List of Parameters

### Notation: List of Parameters

Symbol | Name | Dimensions | SI units |
---|---|---|---|

\(\varvec{\Omega }\) | Damage tensor | \({\mathrm{M}} ^{0}{\mathrm{L} }^{0}{\mathrm T}^{0}\) | – |

\(N\) | Number of cracks | \({\mathrm M} ^0{\mathrm L} ^0{\mathrm T} ^0\) | – |

\(d_{k}\) | Volumetric fraction of the cracks | \({\mathrm M} ^0{\mathrm L} ^0{\mathrm T} ^0\) | – |

\(\mathbf {n}_{k}\) | Normal direction of the kth crack | \({\mathrm M} ^{0}{\mathrm L}^0{\mathrm T}^{0}\) | – |

\(r_{i}\) | Radius of the ith crack plane | \({\mathrm M}^{0}{\mathrm L}^{1}{\mathrm T}^{0}\) | mm |

\(e_{i}\) | Thickness of the ith crack plane | \({\mathrm M} ^0{\mathrm L} ^1{\mathrm T} ^0\) | mm |

\(\varvec{\varepsilon }\) | Total strain | \({\mathrm M} ^0{\mathrm L} ^0{\mathrm T} ^0\) | – |

\(\varvec{\varepsilon }^{el}\) | Pure elastic strain | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(\varvec{\varepsilon }^{ed}\) | Elasto-damage strain | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(\varvec{\varepsilon }^{id}\) | Irreversible strain | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(\varvec{\varepsilon }^{E}\) | Total elastic strain | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(\varvec{\sigma }\) | Stress | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | MPa |

\(\mathbf {Y}\) | Damage conjugated force | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | MPa |

\(\dot{\varvec{\Omega }}\) | Damage rate | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(\varvec{\varepsilon }^{el}\) | Pure elastic strain | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(G_{\mathrm{s}}\) | Gibbs free energy | \(\mathrm M ^1\mathrm L ^2\mathrm T ^{-2}\) | J |

\(\mathbb {S}_{0}\) | Initial compliance tensor | \(\mathrm M ^{-1}\mathrm L ^1\mathrm T ^2\) | GPa\({^{-1}}\) |

\(a_{i}\) | Material parameters accounting for stiffness due to damage | \(\mathrm M ^{-1}\mathrm L ^1\mathrm T ^2\) | GPa\({^{-1}}\) |

\(\nu _{0}\) | Initial Poisson’s ratio | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(E_{0}\) | Initial Young’s modulus | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | GPa |

\(\varvec{\delta }\) | second-order identity tensor | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(f_{\mathrm{d}}\) | Damage function | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | MPa |

\(J^*\) | Second invariant of the deviatoric part of the physical damage force | \(\mathrm M ^2\mathrm L ^{-2}\mathrm T ^{-4}\) | MPa\({}^{2}\) |

\(I^*\) | first invariant of the physical damage force | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | MPa |

\(\alpha\) | Material constant to control the shape of the cone | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(C_{0}\) | Initial damage threshold | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | MPa |

\(C_{1}\) | Damage hardening variable | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | MPa |

\(\mathbb {P}_{1}\) | Projection tensor to make the damage driving force parallel to stress | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(\mathbb {P}_{2}\) | Projection tensor to account for the damage rate direction | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(\sigma ^{(p)}\) | pth eigenstress | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | MPa |

\(\mathbf {n}^{(p)}\) | pth principal direction | \(\mathrm M ^0\mathrm L ^0\mathrm T ^0\) | – |

\(g_{\mathrm{d}}\) | Damage potential | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | MPa |

\(C_{2}\) | Hardening variable in damage potential | \(\mathrm M ^1\mathrm L ^{-1}\mathrm T ^{-2}\) | MPa |

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Xu, H., Arson, C. Mechanistic Analysis of Rock Damage Anisotropy and Rotation Around Circular Cavities.
*Rock Mech Rock Eng* **48, **2283–2299 (2015). https://doi.org/10.1007/s00603-014-0707-5

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### Keywords

- Rock mechanics
- Continuum damage mechanics
- Thermodynamics
- Finite element method
- Damage rotation
- Anisotropy
- Circular cavity