In underground rock engineering, the three-dimensional (3D) stress state governs the time-dependent behavior of surrounding rocks, thereby directly affecting the long-term safety and stability of engineering structures. This study presents a novel true triaxial time-dependent constitutive model that characterizes nonlinear creep behavior governed by coupling of viscoplastic deformation and time-dependent damage. The model advances the 3D Hoek-Brown yield function by incorporating the azimuthal orientation of cracks. The coupled model is implemented using a numerical procedure that features an explicit nonlinear integration algorithm for updating the time-dependent damage factor. The model's effectiveness is validated by its accurate prediction of true triaxial creep test results for Jinping marble. The comprehensive constitutive model advances continuum damage mechanics, providing a scientific basis for analyzing and evaluating the long-term safety and stability of deep hard rock engineering.
AmitranoDHelmstetterA (2006) Brittle creep, damage, and time to failure in rocks. Journal of Geophysical Research: Solid Earth111(B11).
2.
BaoHZhangKYanC, et al. (2020) Excavation damaged zone division and time-dependency deformation prediction: A case study of excavated rock mass at Xiaowan Hydropower Station. Engineering Geology272: 105668.
3.
BiswasSFernandez CastellanosDZaiserM (2020) Prediction of creep failure time using machine learning. Scientific Reports10(1): 16910.
4.
BorjaRIYinQZhaoY (2020) Cam-clay plasticity. Part IX: On the anisotropy, heterogeneity, and viscoplasticity of shale. Computer Methods in Applied Mechanics and Engineering360: 112695.
5.
CaoYJShenWQBurlionN, et al. (2018) Effects of inclusions and pores on plastic and viscoplastic deformation of rock-like materials. International Journal of Plasticity108: 107–124.
6.
ChenCXieHXuJ, et al. (2024) Creep behavior of sandstone under the coupling action of stress and pore water pressure using three-dimensional digital image correlation. International Journal of Damage Mechanics33(2): 147–173.
7.
ChenLWangCPLiuJF, et al. (2015) Damage and plastic deformation modeling of Beishan granite under compressive stress conditions. Rock Mechanics & Rock Engineering48(4): 1623–1633.
8.
ChenWKulatilakeP (2015) Creep behavior modeling of a marble under uniaxial compression. Geotechnical and Geological Engineering33: 1183–1191.
9.
ChenZHeCXuG, et al. (2019) A case study on the asymmetric deformation characteristics and mechanical behavior of deep-buried tunnel in phyllite. Rock Mechanics and Rock Engineering52(11): 4527–4545.
10.
ChoiHLimHJYunGJ (2023) An integrated unified elasto-viscoplastic fatigue and creep damage model with characterization method for structural analysis of nickel-based high-temperature structure. International Journal of Damage Mechanics32(1): 73–102.
11.
CorneliusRScottP (1993) A materials failure relation of accelerating creep as empirical description of damage accumulation. Rock Mechanics And Rock Engineering26(3): 233–252.
12.
FengX-TKongRYangC, et al. (2020) A three-dimensional failure criterion for hard rocks under true triaxial compression. Rock Mechanics and Rock Engineering53(1): 103–111.
13.
FengX-TWangZFZhouYY, et al. (2021) Modelling three-dimensional stress-dependent failure of hard rocks. Acta Geotechnica: 1–31.
14.
FirmePABrandaoNBRoehlD, et al. (2018) Enhanced double-mechanism creep laws for salt rocks. Acta Geotechnica13: 1329–1340.
15.
GuptaNMishraB (2020) Influence of stress-induced microcracks on viscoplastic creep deformation in Marcellus shale. Acta Geotechnica: 1–21.
16.
JiaCXuWWangS, et al. (2018) Experimental analysis and modeling of the mechanical behavior of breccia lava in the dam foundation of the Baihetan Hydropower Project. Bulletin of Engineering Geology & the Environment78(4): 2681–2695.
17.
JiangHJiangAZhangF, et al. (2023) Study on creep characteristics and fractional creep damage constitutive model of limestone under thermal-mechanical action. International Journal of Damage Mechanics32(2): 262–288.
18.
JinWArsonC (2018) Micromechanics based discrete damage model with multiple non-smooth yield surfaces: Theoretical formulation, numerical implementation and engineering applications. International Journal of Damage Mechanics27(5): 611–639.
19.
LeVTBuiHHNguyenGD, et al. (2023) Meso to macro connections to capture fatigue damage in cemented materials. International Journal of Fatigue176: 107890.
20.
LiY-DZhaoL-YLaiY-M (2023) A novel elastoplastic damage model for hard rocks under true triaxial compression: Analytical solutions and numerical implementation. International Journal of Geomechanics23(2): 04022268.
21.
MaXHaimsonBC (2017) Failure characteristics of two porous sandstones subjected to true triaxial stresses. Journal of Geophysical Research Solid Earth121(9): 6477–6498.
22.
MengLZhangSLiT, et al. (2025) Study on the creep constitutive model of layered rock considering anisotropic and damage factors after high temperature exposure. International Journal of Damage Mechanics34(3): 520–541.
23.
MiaoSPanP-ZZhaoS, et al. (2021) A new DIC-based method to identify the crack mechanism and applications in fracture analysis of red sandstone containing a single flaw. Rock Mechanics and Rock Engineering54(8): 3847–3871.
24.
PerzynaP (1966) Fundamental problems in viscoplasticity. In: PerzynaP (ed) Advances in Applied Mechanics. United States: Elsevier, Vol. 9, 243–377.
25.
PhanQTBuiHHNguyenGD, et al. (2024) Strain localization in the standard triaxial tests of granular materials: Insights into meso-and macro-scale behaviours. International Journal for Numerical and Analytical Methods in Geomechanics.
26.
PietruszczakSLydzbaDShaoJF (2004) Description of creep in inherently anisotropic frictional materials. Journal of Engineering Mechanics130(6): 681–690.
27.
RinaldiAKrajcinovicDMastilovicS (2007) Statistical damage mechanics and extreme value theory. International Journal of Damage Mechanics16(1): 57–76.
28.
SchaperyRA (1969) On the characterization of nonlinear viscoelastic materials. Polymer Engineering & Science9(4): 295–310.
29.
ShaoJFZhuQZSuK (2003) Modeling of creep in rock materials in terms of material degradation. Computers & Geotechnics30(7): 549–555.
30.
ShenW (2022) Approximate plastic yield criteria of geomaterials with pores and grains embedded in a porous matrix. International Journal of Plasticity153: 103275.
31.
ShenW (2024a) Effective yield strength of a saturated porous medium with a spheroidal meso-pore and spherical micro-pores. Rock Mechanics Bulletin3(1): 100097.
32.
ShenW (2024b) A micromechanics-based multiscale constitutive model for brittle to ductile behavior of porous geomaterial. Rock Mechanics and Rock Engineering57(9): 7205–7221.
33.
ShenW (2024c) A new yield criterion based constitutive model for porous materials. International Journal of Mechanical Sciences274: 109250.
34.
SongTFengX-TZhouY, et al. (2024) Anisotropic characteristics and creep model for thin-layered rock under true triaxial compression. Journal of Rock Mechanics and Geotechnical Engineering6(12): 4815–4834.
35.
SuGChenYJiangQ, et al. (2023) Spalling failure of deep hard rock caverns. Journal of Rock Mechanics and Geotechnical Engineering15(8): 2083–2104.
36.
SunMTangHWangM, et al. (2016) Creep behavior of slip zone soil of the Majiagou landslide in the three gorges area. Environmental Earth Sciences75: 1–12.
37.
SunXLiXZhengB, et al. (2020) Study on the progressive fracturing in soil and rock mixture under uniaxial compression conditions by CT scanning. Engineering Geology279: 105884.
38.
WangHCuiY-JVuMN, et al. (2024) Modelling the viscoplastic behaviour of Callovo-Oxfordian claystone with consideration of damage effect. Journal of Rock Mechanics and Geotechnical Engineering16(1): 303–316.
39.
WangHXuZHuaiH, et al. (2025) Experimental study on the mechanical properties of red sandstone with fractures under different loading rates. International Journal of Damage Mechanics34(3): 438–467.
40.
WangQHeMLiS, et al. (2021) Comparative study of model tests on automatically formed roadway and gob-side entry driving in deep coal mines. International Journal of Mining Science and Technology31(4): 591–601.
41.
WangSXuWChenH, et al. (2022) The time-dependent behaviour and failure mechanism of dacite under unloading condition. European Journal of Environmental and Civil Engineering26(15): 7756–7770.
42.
WangSXuWWangW, et al. (2018) Experimental and numerical investigations on the mechanical behavior of fine-grained sandstone. International Journal of Geomechanics18(2): 04017150.
43.
WangSYangSZhangQ (2023) A novel time-dependent damage constitutive model for hard rock materials. Geomechanics and Geophysics for Geo-Energy and Geo-Resources9(1): 118.
44.
WangSSXuWY (2020) A coupled elastoplastic anisotropic damage model for rock materials. International Journal of Damage Mechanics29(8): 1222–1245.
45.
WangSSZhaoLYZhangWL (2022) An enhanced constitutive model for quasi-brittle rocks with localized damage. Acta Geotechnica17(11): 5223–5238.
46.
WangZPanPZuoJ, et al. (2023) A generalized nonlinear three-dimensional failure criterion based on fracture mechanics. Journal of Rock Mechanics and Geotechnical Engineering15(3): 630–640.
47.
WuSZhangSZhangG (2018) Three-dimensional strength estimation of intact rocks using a modified Hoek-Brown criterion based on a new deviatoric function. International Journal of Rock Mechanics and Mining Sciences107: 181–190.
48.
WuYLiXZhangL, et al. (2023) Heterogeneity induced strain localization in block-in-matrix-soils subjected to uniaxial loading using real-time CT scanning. Journal of Rock Mechanics and Geotechnical Engineering15(8): 1951–1959.
49.
YangS-QChengL (2011) Non-stationary and nonlinear visco-elastic shear creep model for shale. International Journal of Rock Mechanics and Mining Sciences48(6): 1011–1020.
50.
YangS-QTangJ-ZWangS-S, et al. (2022) An experimental and modeling investigation on creep mechanical behavior of granite under triaxial cyclic loading and unloading. Rock Mechanics and Rock Engineering55(9): 5577–5597.
51.
ZhangSGuangZ (2020) Strength and deformability of a low-porosity sandstone under true triaxial compression conditions. International Journal of Rock Mechanics and Mining Sciences127.
52.
ZhaoJFengX-TZhangX, et al. (2018b) Time-dependent behaviour and modeling of Jinping marble under true triaxial compression. International Journal of Rock Mechanics and Mining Sciences110: 218–230.
53.
ZhaoJFengX-TZhangX-W, et al. (2018a) Brittle-ductile transition and failure mechanism of Jinping marble under true triaxial compression. Engineering Geology232: 160–170.
54.
ZhaoJFengXTZhangXW, et al. (2019) Brittle and ductile creep behavior of Jinping marble under true triaxial stress. Engineering Geology258: 105157.
55.
ZhaoLYZhuQZXuWY, et al. (2016) A unified micromechanics-based damage model for instantaneous and time-dependent behaviors of brittle rocks. International Journal of Rock Mechanics and Mining Sciences84: 187–196.
56.
ZhuZYangSRanjithPG, et al. (2023) A comprehensive review on mechanical responses of granite in enhanced geothermal systems (EGSs). Journal of Cleaner Production383: 135378.
57.
ZuoJLiuHLiH (2015) A theoretical derivation of the Hoek–Brown failure criterion for rock materials. Journal of Rock Mechanics and Geotechnical Engineering7(4): 361–366.
58.
ZuoJ-pLiH-tXieH-p, et al. (2008) A nonlinear strength criterion for rock-like materials based on fracture mechanics. International Journal of Rock Mechanics and Mining Sciences45(4): 594–599.
