European Journal of Mechanics / A Solids 85 (2021) 104085 Available online 3 August 2020 0997-7538/© 2020 Elsevier Masson SAS. All rights reserved. Non-linear anisotropic damage rheology model: Theory and experimental verifcation Ivan Panteleev a, * , Vladimir Lyakhovsky b , John Browning c, d , Philip G. Meredith e , David Healy f , Thomas M. Mitchell e a Institute of Continuous Media Mechanics UB RAS, Perm, Russia b Geological Survey of Israel, Jerusalem, Israel c Pontifcia Universidad Catolica de Chile, Santiago, Chile d Andean Geothermal Centre of Excellence, Universidad de Chile, Santiago, Chile e Department of Earth Sciences, University College London, London, UK f School of Geosciences, University of Aberdeen, Aberdeen, AB24 3UE, UK A R T I C L E INFO Keywords: True triaxial loading Non-linear elasticity Damage induced anisotropy Rheology Damage tensor ABSTRACT 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 modifed 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 compo- nents, 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 verifcation 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 con- ventional triaxial test data, and provided good fts 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 vari- ations in the tested Darley Dale sandstone. 1. Introduction Distributed rock damage, in the form of cracks, joints and other in- ternal faws, develops during rock-forming processes and affects phys- ical and mechanical rock properties. When rocks, ceramics and similar materials without pre-existing large, macroscopic damage zones are subjected, under relatively low pressure and temperature, to gradually increasing differential stress beyond their elastic limit, distributed brittle deformation develops in the bulk, associated with increasing crack density (damage). Gradual, distributed brittle failure is manifested macroscopically by changes in effective elastic properties and reduction of wave speeds. This process has been observed in numerous laboratory fracturing experiments (e.g., Jaeger and Cook, 1979; Lockner et al., 1992; Stanchits et al., 2006; Heap et al., 2009; Browning et al., 2017; Renard et al., 2019). The experimentally observed stress-strain curves are sometimes treated, all the way up to brittle instability, using linear elastic relations with effective moduli that depend on crack densities (e. g., O’Connell and Budiansky, 1974; Budiansky and O’Connell, 1976; Kachanov, 1992). The equivalent-linear approximation approach is the basis for many numerical codes (e.g., Bardet et al., 2000; Assimaki and * Corresponding author. E-mail address: pia@icmm.ru (I. Panteleev). Contents lists available at ScienceDirect European Journal of Mechanics / A Solids journal homepage: http://www.elsevier.com/locate/ejmsol https://doi.org/10.1016/j.euromechsol.2020.104085 Received 11 March 2020; Received in revised form 17 June 2020; Accepted 19 July 2020