Effect of the Intermediate Principal Stress on Pre-peak Damage Propagation in Hard Rock Under True Triaxial Compression

It is of foremost importance to understand the mechanisms of damage propagation in rock under true triaxial stress. True triaxial compression tests reported in the literature do reflect the effect of the intermediate principal stress (σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _2$$\end{document}), but predictive models are still lacking. In this paper, an enhanced version of the Discrete Equivalent Wing Crack Damage (DEWCD) model initially proposed in Jin and Arson (in Int J Solids Struct 110:279–293, 2017) is calibrated and tested to bridge this gap. The original DEWCD model can predict most mechanical non-linearities induced by damage but it cannot capture dilatancy effects accurately. To overcome this limitation, a dependence of the energy release rate on the first and third stress invariants is introduced in the damage potential. The enhanced DEWCD model depends on eight constitutive parameters. An automated calibration procedure is adopted to match pre-peak stress-strain curves obtained experimentally in Feng et al. (in Rock Mech Rock Eng 52(7):2109–2122, 2019) during true triaxial compression. The model successfully captures the differences in deformation and damage in the three principal directions of loading and accurately predicts that an increase of compression σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _2$$\end{document} yields a decrease of the intermediate (tensile) deformation, a triggering of damage at a lower value of σ1-σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _1-\sigma _2$$\end{document}, as well as a decrease of cumulated damage in the direction of σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _2$$\end{document} and an increase of cumulated damage in the direction of σ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _3$$\end{document} at the stress peak (pre-softening). During the true triaxial compression stage, a higher intermediate principal stress hinders dilatancy such that the volumetric strain at the peak of σ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _1$$\end{document} changes from dilation to shrinkage. The enhanced DEWCD model shows good performance in axis-symmetric compression and true triaxial compression, both for monotonic and cyclic loading. A comparison of three true triaxial stress paths at constant/variable mean stress/Lode angle suggests that: (i) the mean stress controls damage hardening and the sign of the volumetric strain rate at damage initiation, (ii) the second stress invariant is the primary control factor of the direction of the irreversible deviatoric strain rate during triaxial loading and of the sign of the total volumetric strain rate at failure; (iii) the Lode angle controls the direction of the total deviatoric strain rate.