# 3-D Numerical Study on Progressive Failure Characteristics of Marbles under Unloading Conditions
## 1 Introduction
For instance, [PERSON] and [PERSON] [39] applied discrete element method to investigate the crack initiation, propagation, and coalescence in rock-like specimens with two flaws under uniaxial compression. [PERSON] and [PERSON] [52] developed the enriched numerical manifold method to model cracking behaviors of rock masses. [PERSON] et al. [54; 55] proposed the grain-based discrete element method to explore dynamic behaviors of fracturing and damage evolution of rock materials at the grain scale. [PERSON] et al. [41] adopted three-dimensional discrete element method to conduct investigation of the fracture behavior of thermally degraded rocks. [PERSON] et al. [45; 46] adopted a typical discrete element method code to simulate the progressive fatigue failure of concrete subjected to the multi-level cyclic loading, the micro-scale cracks can be characterized and the results show consistency with lab testing. Discrete element method, which discretizes the computational domain into a collection of particles with different shapes, are commonly adopted to simulate the failure of geomaterials. Compared with other numerical methods, the random collections of particles in discrete element methods are much closer to rocks and soils. Although there exist many literatures on discrete element simulations of rock failure behaviors [39; 40; 41; 42; 43; 44; 45; 46; 47; 56; 57; 58; 59], the existing investigations on failure characteristics of rock and rock masses are mainly focused on the single compressive loading conditions. Failure characteristics and the mechanisms of rock and rock masses under unloading conditions are rarely explored using discrete element simulations, which are necessary to improve understanding the disaster mechanism of rock masses during tunnel excavations and mineral mining, as shown in Figure 1. [PERSON] et al. [60] used the PFC2D to carry out a series of studies on the unloading process of pre-cracked marble samples based on the physics experiment. The results reveal the effects of unloading rate and crack inclination on the strength and cracking performance of pre-cracked marble samples. However, the rock mass unloading process in engineering is more complicated, and requires further study. Therefore, numerical explorations on failure characteristics of rock masses under loading and unloading conditions are necessary, which is not only meaningful to revealing the unloading failure mechanism for scientist, but also beneficial to geotechnical engineers. Most current researches focus on mechanical and failure behaviors of intact rock specimens under unloading conditions, but there are few systematic studies on the progressive failure processes of pre-fissured rock specimens under unloading conditions [61]. Although the failure characteristics with different unloading rates are summarized, the influence of initial confining pressures on mechanical and failure behaviors is not carried out. The existing literature provides that failure mechanism of rock specimens under unloading conditions is very different from that under loading conditions, which indicates the necessary to research the unloading mechanical and failure characteristics of pre-fissured rock specimens. In this work, particle-based discrete element method (PB-DEM) was applied to investigate the mechanical failure characteristics of pre-fissured rock specimens under the triaxial compression unloading conditions. Microscopic parameters in PB-DEM for marble were calibrated using trial-and-error approach based on the previous experiments. Then, different fissured marble specimens with various fissure length \(a\), fissure inclination angle \(\alpha\), ligament angle \(\beta\), and ligament length \(b\) under the conventional triaxial loading and unloading conditions were simulated using PB-DEM. Effects of fissure distribution, initial confining pressures, and unloading rates on the mechanical and failure characteristics of fissured marble specimens were numerically investigated.