# Bonded-Particle Model with Nonlinear Elastic Tensile Stiffness for Rock-Like Materials
## 1 Introduction  
Discrete element method (DEM) is proved to be a very effective method in modelling rock-like materials [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. The discontinuous nature of DEM makes DEM very suitable for dealing with the initiation and propagation of micro cracks in rock-like materials. Within DEMs, the bonded-particle model (BPM) is widely used to simulate the deformation and fracture of rock-like materials [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. Since the BPM was first proposed [1], it has been rapidly developed. A clumped particle model was proposed in [2] based on the BPM, which improves the predictive capability of the DEM by successfully predicting the whole failure envelope and increasing the ratio of uniaxial compressive strength to tensile strength (UCS/TS) making it closer to that of rocks. A damage-rate law was first included by [PERSON] [3] to extend the BPM and to make it more suitable for simulating the static-fatigue behavior of granite. [PERSON] [19] introduced the Mohr-Coulomb failure criterion into the BPM without considering the bending stiffness, and successfully reproduced the mechanical behavior of frictional cohesive materials. A progressive failure model was applied later in [21], which makes the BPM moreproper in simulating the failure behavior of rock with different brittleness. Shear tests were modeled in [11] by using the BPM and the development of micro-cracks during shear test under normal stress was investigated. In order to increase UCS/TS to a practical level, many efforts have been made in the past decade. Through investigating the effect of porosity and crack density on macro parameters of materials, [PERSON] [6] found that UCS/TS can be increased by decreasing porosity and increasing crack density. Increasing the interaction range of particles [12] and using specimens containing larger macro-pores [9] are also proved to be effective in increasing UCS/TS. [PERSON] [14] modified the BPM by partially considering the contribution of moments to the stress and got satisfactory results in capturing high UCS/TS. Besides the BPM, there are many other models proposed for rock-like materials. A model assuming that the adjacent particles interact over a width rather than a point was proposed by [PERSON] [22; 23] and was used to study the effect of rolling resistance. Recently the twisting resistance was included in a three dimensional contact model by [PERSON] [24]. A model with hardening damage response was introduced to study the mechanical behaviors of concrete under high confining pressures in [25]. [PERSON] [26] proposed an energy-based failure criterion which proved to be more suitable for numerical simulation of particulate materials. [PERSON] [27] investigated the performance of specimens with weak bonds or heterogeneous in modelling rocks. Most of the work related to DEM for rock focuses on reproducing the failure process of rock materials and capturing proper macro parameters of given rock by using proper micro parameters in the model. The relationships of micro and macro parameters in the DEM were investigated in [4; 28; 29]. The effect of particle size was studied in [30; 31; 32]. The effect of the particle size distribution and different methods to calculate the stiffness were discussed in [33]. Although a lot of investigations of rock deformation and failure based on DEM have been made and some attempts to model rock cutting process using DEM were made [10; 34; 35], to the best of our knowledge, there still is not any published work discussing the nonlinear deformation of rock-like materials under tensile loading using DEM. Many experimental results show that the strain-stress curve of rock under uniaxial tensile loading is nonlinear [8; 36; 37; 38; 39; 40]. And the Young's modulus and Poisson's ratio in tension are found to be different from that in compression [38; 41; 42; 43].