Time-dependent constitutive model for intact rocks with damage evolution law

The long-term stability of rocks is often compromised by time-dependent deformation and creep, which play a decisive role in the safety of tunnels, mines, dams and underground energy storage structures. Although numerous constitutive models have been developed, most existing approaches remain limited by three critical gaps: (i) the absence of a unified framework that simultaneously combines viscoelastic, viscoplastic and damage mechanisms; (ii) neglect of hardening laws in viscoplastic formulations which are essential to capture strength evolution during the creep and (iii) the inability to reliably predict tertiary creep and time-to-failure in intact rocks. To address these limitations, this study developed a novel constitutive model that integrated Schapery's nonlinear viscoelasticity, Perzyna-type viscoplasticity with isotropic hardening and an exponential damage evolution law, all formulated within the effective stress framework. A fully implicit algorithm with Newton-Raphson iterations ensured stable and accurate integration of viscoelastic, viscoplastic and damage responses. An important practical feature of the proposed framework is the staged calibration procedure. First, the viscoelastic parameters were identified from creep tests conducted below the long-term strength, hence, the viscoplastic parameters were calibrated from short-term creep tests above the long-term strength and finally, the damage parameters were determined through complete creep tests at two different stress magnitudes. The model then was calibrated and verified using experimental data obtained from Gosford sandstone under constant compressive creep loading. Independent verification at 60%, 80% and 90% of the uniaxial compressive strength (UCS) demonstrated that the proposed constitutive model can reproduce strain-time and strain rate-time behaviours quite well while it can accurately predict the onset of tertiary creep and time-to-failure without further re-calibration. The resulting data showed that the proposed model not only can predict the complete creep curve, it can also provide a predictive tool for time-to-failure assessment. Finally, it was demonstrated that the proposed model provides a thermodynamically consistent framework for simulating time-dependent behaviour of Gosford sandstone under constant compressive creep loading, with potential relevance to long-term stability assessment in rock engineering.