Experimental Investigation and Micromechanics-Based Analytical Modeling of Crep and Relaxation Behaviors of Beishan Granite  
[PERSON]  
1 Institute of Rock Mechanics, Ningbo University, Ningbo 315211, China 1*  
researchers [23; 24] have investigated the stress-relaxation behaviors of discontinuities under cyclic loading via shear-relaxation tests. Others have proposed advanced techniques, such as digital image correlation, for identifying damage and quantifying full-field strain at various scales. For example, [PERSON] et al. [25] studied the effects of initial damage state on the time-dependent behaviors of sandstone by conducting a creep-test scheme with controlled initial damage.  
At the same time, constitutive models of the short- and long-term mechanical properties of multiple types of rock have also been widely studied, both theoretically and numerically [26; 27; 28; 29]. [PERSON] et al. [30] and [PERSON] et al. [31] presented a mathematical framework for modeling creep in frictional materials, and assumed creep deformation to be associated with progressive evolution of such materials' micro-structures. [PERSON] et al. [32] subsequently proposed a unified elastic-plastic and viscoplastic damage model capable of describing instantaneous and time-dependent plastic deformations at various time scales. More importantly, ongoing development of [PERSON] et al. [32]'s micro-mechanical plastic-damage constitutive model has not only provided more accurate descriptions of the time-dependent behaviors of rock, but has also laid an important foundation for future studies and investigations.  
In the present work, driven by the need of a more accurate and comprehensive description of the long term behaviors of the host rock to ensure the integrity and safety of the deep-ground structures, the time-dependent behaviors of Beishan granite are first studied experimentally, through conventional triaxial compression tests and single-stage creep and relaxation tests at various levels of confining pressures and initial deviatoric stress. Then, a unified micromechanics-based plastic damage (UMBPD) analytical model is formulated for simulating instantaneous and time-dependent creep and relaxation behaviors. Specifically, based on Mori-Tanaka homogenization, the representative elementary volume of Beishan granite can be treated as a matrix-inclusion system composed of a solid matrix and randomly oriented penny-shaped microcracks. Second, we will apply a unified plastic-yield criterion to describe time-independent and time-dependent deformation under different levels of high-deviatoric stress. Then, by studying long-term mechanical behaviors, the researchers will ascertain an appropriate time-dependent criterion of Beishan granite for damage due to subcritical cracking, which will be used to simultaneously simulate creep deformation and stress relaxation; and lastly, the simulation results obtained from the constitutive analytical model will be compared against our experimental results.  
Throughout the paper, the following notion on tensorial product of any second order tensors \(A\) and \(B\) will be taken: \((A\otimes B)_{ijkl}=A_{ij}B_{kl}\). By using the second order identity tensor \(\delta\), the fourth order identity tensor \(\mathbb{I}\) and the fourth order hydrostatic projector \(\mathbb{J}\) are expressed in the components form as \(I_{ijkl}=\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)/2\) and \(J_{ijkl}=\delta_{ij}\delta_{kl}/3\), respectively. The fourth order deviatoric projection tensor \(\mathbb{K}\) is then introduced.