# Numerical Analysis for Shear Behavior of Binary Interfaces under Different Bonded Conditions
## 2 Numerical Modeling of Binary Interfaces with Different Bonded Conditions
In PFC2D, the method of artificially setting the \"structural plane\" by controlling the distribution of particles will produce an \"internal lock\" during the movement of particles due to the characteristics of non-destructive particles, resulting in a completely unreasonable steep rise and steep fall of the shear stress-shear displacement curve of the unbonded flat joint. The smooth joint model is the solution to this problem. The smooth joint model is applied between particles so that the motion between particles allows overlap. The direction of mutual motion between particles will change from the tangential direction of the relative surface between particles to the tangential direction of the smooth joint model, and the \"internal lock\" phenomenon will no longer occur between particles. The generation of non-bonded binary interfaces is mostly formed by tension or sliding under the force. Considering that the model is sheared on the left, the tension region is located on the left of the interface, and therefore, the unbonded part is set on the left of the model. The bond ratio of the interface is achieved by placing spacer walls of different lengths between the upper and lower shear boxes, as shown in Figure 1. The bonded ratios were 0%, 25%, 50%, 75%, and 100. After the particles are generated in the shear box, a state of static equilibrium can be achieved by the applied forces and cyclic calculations. For the unbonded mode, the particles are fully bonded when given a planar nodal model since a hydrostatic equilibrium is reached. For the others, the particles on both sides of the wall are unable to make contact where the wall exists, thus forming a physical interface, and it is sufficient to give a smooth joint model at its corresponding location after deleting the spacing wall. Five levels of normal stress are set, respectively, for 1 MPa, 2 MPa, 3 MPa, 4 MPa, and 5 MPa. Shear loading of a binary interface is realized by applying a constant rate of horizontal displacement to the upper shear box wall. The direct shear model requires constant normal  
Figure 1: Particle modeling of binary interface in PFC. (**a**) Shear model of binary interface; (**b**) 0% bonded ratio; (**c**) 25% bonded ratio; (**d**) 50% bonded ratio; (**e**) 75% bonded ratio; (**f**) 100% bonded ratio. (# indicates the number of different shear box blocks. The red region means the degree of bonded ratio, other colors represent different locations of the cut box). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & \begin{tabular}{c} **Normal** \\ **Stiffness/GPa** \\ \end{tabular} & \
\begin{tabular}{c} **Tangential** \\ **Stiffness/GPa** \\ \end{tabular} &
\begin{tabular}{c} **Friction Coefficient** \\ \end{tabular} \\ \hline Smooth joint & 100 & 0.8 & 0.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Smooth joint model parameters. stress, and the model can be considered to have reached a state of constant stress loading when the difference between the target and actual stresses is sufficiently small, although the wall cannot be stressed directly in the PFC. Wall #6 in Figure 1 was set up as a servo wall to constantly update the force state through the Fish language so that the difference between the target stress and the actual stress is controlled to be small enough for the purpose of applying a relatively constant normal stress to the model. After that, the loading stop condition is set so that the upper wall can be loaded at a constant horizontal rate. While loading, horizontal displacement, stress, and crack of the model are monitored and recorded.