## 1 Introduction
FEM is difficult to describe the initiation and propagation of cracks, making it nearly impossible to model an actual rock breaking process [21; 22]. DEM can simulate the macro-constitutive characteristics and micro-mechanical properties of rock materials [23; 24; 25; 26; 27]. Due to its ability to simulate real cracks, it has gained more attention in rock cutting issues [14; 28; 29; 30; 31; 32; 33; 34]. [PERSON] et al. [35] used PFC2D to analyze the rock failure process under wedge indenters, and revealed that the rock fragmentation process was synchronized with the loading and unloading process of cutter forces. [PERSON] et al. [36] investigated the failure mechanism of indenters with different wedge angles during rock indentation using the discrete element method, and found that as the wedge angle increased, the fracture zone size and the indentation force increased, promoting the formation of subvertical crack. [PERSON] et al. [37] compared the rock breaking performances caused  
Figure 1: Button cutter and its rock breaking effect. by arc and wedge cutters with two blades. The results showed that arc blades can produce larger rock breaking areas and longer lateral rocks than wedge blades. Although two-dimensional discrete element models can yield relatively high-resolution crack patterns, they are based on plane stress or plane strain assumptions, which still have significant limitations in characterizing real three-dimensional geometry and predicting indentation forces of buttons. Only a few researchers have used the three-dimensional discrete element method to investigate the rock breaking mechanism of cutters and validated the simulation results with laboratory tests [38; 39; 40; 41; 42]. [PERSON] et al. [43] constructed a large 3D model of cutter rock fragmentation to simulate the actual rock breaking process of cutters using the MatDEM software. [PERSON] et al. [44] simulated the interaction between soil and different structure parameters of sweep tillage tools using DEM, and found that structure parameters had major effects on tillage forces and soil movement for shallow working. When using DEM to simulate 3D rock cutting, the increase in model dimensions and degrees of freedom leads to greater computational demand for numerical simulation [45]. Thus, researchers have to sacrifice model element resolution to achieve reasonable simulation time; that is, the particle radius in the existing 3D simulations mainly ranged from 1.3 and 6.0 mm. Due to the small contact area between the carbide button and the rock, the difference in the indenter profiles cannot be accurately characterized. In addition, the 3D crack propagation patterns in the rock breaking process cannot be observed explicitly. The objective of this research is to numerically investigate the 3D rock breaking mechanism and failure process caused by five different carbide buttons (spherical, saddle, wedge, conical, and parabolic buttons) widely used in button cutters, and further to determine an optimal button for the design of small diameter button cutter used to cut Beishan granite. To overcome the computational bottleneck, GPU-based discrete element code was used. In the following sections, the principle of numerical methods and model setup details are first introduced. The rationality of numerical results is then validated by comparing it with an indentation experiment. Next, the rock breaking processes of five different buttons are investigated from both micro and macro perspectives. Then, the cutting efficiency of the five buttons are compared in terms of crack pattern and energy consumption. Finally, the conclusion of this study are drawn.