Non-linear anisotropic damage rheology model: Theory and experimental verification

Ivan Panteleev*, Vladimir Lyakhovsky, John Browning, Philip G. Meredith, David Healy, Thomas M. Mitchell

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


We extend the isotropic non-linear damage rheology model with a scalar damage parameter to a more complex formulation that accounts for anisotropic damage growth under true triaxial loading. The model takes account of both the anisotropy of elastic properties (associated with textural rock structure) and the stress- and damage-induced anisotropy (associated with loading). The scalar, isotropic model is modified by assuming orthotropic symmetry and introducing a second-order damage tensor, the principal values of which describe damage in three orthogonal directions associated with the orientations of the principal loading axes. Different damage components, accumulated under true triaxial loading conditions, allows us to reproduce both stress-strain curves and damage- and stress-induced seismic wave velocity anisotropy. The suggested model generalization includes a non-classical energy term similar to the isotropic non-linear scalar damage model, which allows accounting for the abrupt change in the effective elastic moduli upon stress reversal.

For calibration and verification of the model parameters, we use experimental stress-strain curves from deformation of dry sandstone under both conventional and true triaxial stress conditions. Cubic samples were deformed in three orthogonal directions with independently controlled stress paths. To characterize crack damage, changes in ultrasonic P-wave velocities in the three principal directions were measured, together with the bulk acoustic emission output. The parameters of the developed model were constrained using the conventional triaxial test data, and provided good fits to the stress-strain curves and P-wave velocity variations in the three orthogonal directions. Numerical simulation of the true triaxial test data demonstrates that the anisotropic damage rheology model adequately describes both non-linear stress-strain behavior and P-wave velocity variations in the tested Darley Dale sandstone.

Original languageEnglish
Article number104085
Number of pages14
JournalEuropean Journal of Mechanics A/Solids
Early online date3 Aug 2020
Publication statusPublished - 1 Jan 2021

Bibliographical note

The contributions by Browning, Meredith, Healy and Mitchell were supported by UKRI NERC awards NE/N003063/1 and NE/N002938/1, funding for which is gratefully acknowledged. The contribution by Panteleev was supported by Russian Science Foundation (project № 19-77-30008).


  • True triaxial loading
  • Non-linear elasticity
  • Damage induced anisotropy
  • Rheology
  • Damage tensor
  • ROCK


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