# Bonded-Particle Model with Nonlinear Elastic Tensile Stiffness for Rock-Like Materials
## 5 Conclusions  
In this paper, a new model is proposed for rock-like materials, which is named as NET-BPM model. The NET-BPM model, which adopts nonlinear tensile stiffness and a new form of bonds in its contact model, has the following advantages. (1) It is very difficult to gain a large ratio of UCS/TS more than 20 in conventional BPM model [1; 2; 20]. However, compared to our model, a large range of UCS/TS ratio (from 5 to 45 in the presented simulations) can be obtained by using different \(b\) and \(\varepsilon_{0}^{s}/\varepsilon_{0}^{n}\); (2) The nonlinear behavior of some rocks under direct tension, which is observed in many experiments [8; 36; 37; 38; 39; 40], can be reproduced well in our model with a proper \(b\); (3) The differences between Young's modulus, Poisson's ratio of specimens, when they are under compression and tension, which are observed in experiments [38; 41; 49; 50], are captured well in the NET-BPM model. e.g., the value of \(E_{c}/E_{t}\) ranging from 1.0 to 1.6 can be obtained; (4) It is possible to change the width of the bond without changing the stiffness of the bond in the NET-BPM model, which helps to obtain different UCS without affecting TS and \(E_{c}\). The relationships between micro and macro parameters in our model were investigated through simulating uniaxial compression tests and direct tension tests. And the tests were performed upon specimens sharing identical particle arrangement. The results show that there is no significant difference for the effect of \(k_{s}/k_{n}\) on rock's strength, [PERSON]'s modulus and Poisson's ratio between our model and typical DEM model [28]. Larger \(k_{s}/k_{n}\) will results in larger \(E_{c}\), UCS, TS and UCS/TS, and smaller \(v_{c}\). Besides \(k_{s}/k_{n}\), \(k_{m}/(k_{n}R)\) also has great effects on UCS, TS and UCS/TS, and the effects will become larger with the increasing of \(k_{s}/k_{n}\). It should be noted that the larger \(k_{m}/(k_{n}R)\) will results in greater UCS, TS and UCS/TS, while it still has ignorable effects on [PERSON]'s modulus and Poisson's ratio. In present investigation, the parameter \(b\) has great effects on strength, Young's modulus and Poisson's ratio of specimens in tensile tests but small effects in uniaxial compressive tests. In conclusion to this, parameter \(b\) has great influences on TS, UCS/TS, \(E_{c}/E_{t}\) and \(v_{c}/v_{t}\). However, as the increasing of \(b\), those effects will become smaller. In addition, the effect of the bond width coefficient \(w\) on UCS is obvious in the NET-BPM. When \(w\) increases, UCS will decrease, but TS, \(E_{c}\) and \(v_{c}\) will remain unchanged. Moreover, the ultimate shear strain has great effects on UCS, but very small effect on TS, \(E_{c}\), \(E_{t}\), \(v_{c}\) and \(v_{t}\). Increasing ultimate shear strain will increase UCS and results in larger UCS/TS. The LDB granite and a quartzite under uniaxial compression tests and direct tension tests were modelled by the NET-BPM model. All the properties, including \(E_{c}\), \(E_{t}\), \(v_{c}\), UCS, TS and UCS/TS from simulations are in good agreement with those from experiments. In addition, the stress-strain curves of LDB granite and the quartzite are also reproduced correctly. All those results indicate that NET-BPM is a very suitable method for modelling the deformation and fracture of rock-like materials. Some work in the subsequent studies are listed as follows. (1) The effects of particle size and particle size ratio on the simulation results, which are not concerned in this work, need to be investigated. (2) More simulations like biaxial and Brazilian tests need to be performed to validate our model. (3) The 3-D NET-BPM model needs to be established to broaden the application scope of the NET-BPM model. This work was supported by the National Natural Science Foundation of China (Grant No. 51179104). Thanks [PERSON] for language help. [PERSON] developed the original concept, proposed the new model and wrote the paper.