# 3-D Numerical Study on Progressive Failure Characteristics of Marbles under Unloading Conditions
## 1 Introduction  
In nature, geologic tectonic activities result in the formation of different scale of discontinuous structures, such as pores, fissures, joints, and faults, in rock masses, which have significant influence on the mechanical properties, stability, and safety of rock masses in geotechnical engineering, such as mineral engineering, foundation pit engineering, tunnel engineering, and deep energy engineering [1; 2; 3]. On the other hand, rock masses with discontinuities are located in a three-dimensional state in situ before being excavation. Since there exist different high in-situ geo-stresses, i.e., \(\sigma_{1}>\sigma_{2}>\sigma_{3}\
eq 0\), excavations in rock masses result in stress redistribution around the cavities, which can be considered as an unloading process. The unloading conditions significantly affect the mechanical behaviors of rock masses, which may lead to slabbing/spalling and rock-bursts [1; 2]. Therefore, it is important and necessary to investigate the mechanical and failure behaviors of rock masses under unloading conditions, which is not only beneficial to understanding the unloading failure mechanism of rock masses, but also contributes to geotechnical engineering projects. In the past decades, numerous laboratory investigations have been conducted to explore the deformation and failure characteristics of rocks, especially granite and marble, under unloadingconditions in uniaxial and conventional triaxial compression tests. There is a great randomness in the discontinuous structure of natural rock. For the reproducibility of the experiment, pre-fissured rock materials are often adopted in experiments. [PERSON] [4] defined wing and secondary cracks based on the fracture initiation and propagation characteristics in laboratory experimental observations. Some scholars also investigated crack propagation and coalescence behaviors and mechanism of rock specimens with two or multiple pre-existing fissures under uniaxial and conventional triaxial compressive loading conditions [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. However, laboratory experiments for rock and pre-fissured rock specimens under compressive loading conditions are expensive, time consuming, and not environmentally-friendly. Furthermore, the internal failure characteristics and deep physically-meaningful phenomena are difficult to observe due to lack of monitoring techniques [19; 20]. In the past decades, as computer and storage technologies develop, different kinds of numerical methods, which can be classified into continuum and discontinuous approaches, were proposed to investigate the fracture problems in solids. Compared with the laboratory tests, numerical methodologies are cheap, repeatable, and easier to reveal the failure mechanism. For the continuum numerical methods, finite element with remeshing techniques [21; 22], smeared elements [23; 24; 25], extended finite element method [26; 27], peridynamics [28; 29; 30; 31; 32; 33; 34; 35], and phase field method [36; 37] have been successfully developed to investigate the failure evolutions of rock and rock masses under compressive loading. For example, [PERSON] et al. [30] developed stress-based failure criteria in the non-ordinary state-based peridynamics to investigate the crack initiation, propagation, and coalescence in the pre-fissured rock specimens under uniaxial and biaxial compressive loading. [PERSON] et al. [38] proposed a 3D numerical mesoscale model to describe the time-independent deformation and fracturing of brittle rocks that accounts for material heterogeneity and local material degradation. [PERSON] et al. [36; 37] implemented the phase field model into COMSOL to study the fracture behavior of rock specimens with pre-existing fissures subjected to the uniaxial compressive loads. For discontinuous numerical methods, discrete element method [39; 40; 41; 42; 43; 44; 45; 46; 47], displacement discontinuity method [48; 49; 50], numerical manifold method [51; 52; 53], etc., were developed to simulate the failure processes of brittle solids.