# Bonded-Particle Model with Nonlinear Elastic Tensile Stiffness for Rock-Like Materials
## 2 Method Description
### Contact Model  
The contact particles are bonded together by the bonds, which can resist normal force, shear force and bending moment. As shown in Figure 1a, particle \(p\) and particle \(q\) are in contact with each other at point A at the original states. \(X_{p}^{0}\) and \(X_{q}^{0}\) are their coordinates respectively. \(dir_{pq}^{0}\) and \(dir_{qp}^{0}\) are the vectors pointing from their centers to the contact point A respectively. At time \(t\), the two particles move to new positions \(X_{p}^{t}\) and \(X_{q}^{t}\), as shown in Figure 1b. The contact point A on particle \(p\) move to point B and the point A on particle \(q\) move to point C. The vector \(dir_{pq}^{0}\) and the vector \(dir_{qp}^{0}\) rotate to \(dir_{pq}^{t}\) and \(dir_{qp}^{t}\), respectively. In addition, \(\theta_{p}\) and \(\theta_{q}\) are rotation angles of particles \(p\) and \(q\) in time \(t\) respectively, defining anticlockwise as positive direction. The contact forces and moment applied to particle \(p\) from particle \(q\) via the bond are normal force \(F_{pq}^{nr}\), shear force \(F_{pq}^{tn}\) and moment \(M_{pq}\), as shown in Figure 1c. The shear force and moment are determined by the following equations:  
\[F_{pq}^{tn}=-k_{s}\varepsilon_{pq}^{s} \tag{1}\]  
\[M_{pq}=-k_{m}(\theta_{p}-\theta_{q}) \tag{2}\]  
where \(k_{s}\) and \(k_{m}\) are shear stiffness and bending stiffness respectively, \(dis_{pq}^{0}\) is the original distance between \(p\) and \(q\), \(\varepsilon_{pq}^{s}\) is shear strain of the bond between \(p\) and \(q\) which is the ratio of deformation in tangential direction to \(dis_{pq}^{0}\). A nonlinear elastic tensile stiffness model is proposed to describe the relationship between normal force and normal strain undergoing tension. Then the force in normal direction is expressed as:  
\[F_{pq}^{nr}= \begin{cases}k_{n}be_{0}^{n}\Big{(}1-\exp(-\varepsilon_{pq}^{n}/ be_{0}^{n})\Big{)}&\text{if }\varepsilon_{pq}^{n}>0\\ k_{n}\varepsilon_{pq}^{n}&\text{if }\varepsilon_{pq}^{n}\leq 0\end{cases}  \tag{3}\]  
where \(k_{n}\) is normal stiffness, \(\varepsilon_{pq}^{n}\) is normal strain of the bond between \(p\) and \(q\), the \(b\) is shape coefficient of the nonlinear elastic model and \(\varepsilon_{0}^{n}\) is the ultimate normal tensile strain. Figure 2 presents the relationship between normal force and normal strain with different value of \(b\). The value of \(b\) has great effect on the normal force versus normal strain curve and the ultimate normal tensile force, \(F_{m}^{nr}\). \(\varepsilon_{pq}^{n}\) and \(\Delta\varepsilon_{pq}^{s}\) which is the increment of \(\varepsilon_{pq}^{s}\) in each time step are given by:  
\[\varepsilon_{pq}^{n}=\frac{dis_{pq}^{t}-dis_{pq}^{0}}{dis_{pq}^{0}} \tag{4}\]  
\[\Delta\varepsilon_{pq}^{s}=\frac{r_{p}\dot{\theta}_{p}+r_{q}\dot{\theta}_{q}+( \dot{\mathbf{u}}_{p}-\dot{\mathbf{u}}_{q})\bullet\mathbf{U}_{pq}}{dis_{pq}^{0} }\Delta t \tag{5}\]  
where \(dis_{pq}^{t}\) is the distance between the centers of \(p\) and \(q\) at time \(t\), \(r_{p}\) and \(r_{q}\) are the radii of particles \(p\) and \(q\) respectively, \(\mathbf{u}_{p}\) and \(\mathbf{u}_{q}\) are the centroid displacements of particles \(p\) and \(q\) respectively, \(\mathbf{U}_{pq}\) is an  
Figure 1: Bond between particles and its forces and moment: (**a**) original positons of \(p\) and \(q\); (**b**) positions of \(p\) and \(q\) at time \(t\); (**c**) contact forces and moment on \(p\). unit vector obtained by rotating the unit vector which points to the center of \(q\) from the center of \(p\) 90 degrees in the anticlockwise direction, \(\Delta t\) is time step. Another modification to the BPM is that the bonds are assumed as many tiny elastic beams, which distribute between two contact particles in a region wide \(2 wr_{m}\) and are parallel to the center line of the two particles, as presented in Figure 3, where the \(w\) is bond width coefficient, \(r_{m}\) is the smaller one of \(r_{p}\) and \(r_{q}\).