# Numerical Analysis for Shear Behavior of Binary Interfaces under Different Bonded Conditions
## 2 Numerical Modeling of Binary Interfaces with Different Bonded Conditions  
Before proceeding with the modeling, the basic assumptions of the PFC\({}^{\text{2D}}\) need to be introduced. In PFC\({}^{\text{2D}}\), models can be built to simulate the motion and interaction between various physical components and display the state and result of the motion and interaction. Basic assumptions include:  
(1) The particle is an indestructible circular entity;  
(2) The entities interact with each other in the form of force and moment;  
(3) Contact occurs at overlapping interfaces between rigid entities;  
(4) The interaction between entities meets [PERSON]'s law of motion, and the motion state of the wall can be determined by the user. The calculation of particle flow completes the iteration update of the force and displacement state of the particle in each finite time step. If the time step is small enough, the disturbance will not travel farther from any one particle than its neighbors in a single step. The simulation process for the direct shear test is divided into three main stages: model building, parameter assignment and direct shear loading. (1) Modeling  
Walls were generated in PFC2D to constitute the shear box for the direct shear test, with dimensions of 100 mm \(\times\) 100 mm, consisting of eight walls. Particles with a reasonable size ratio are generated in the shear box, and loads are applied to these randomly distributed and partially overlapping particles until a static equilibrium state. Also, specific shapes of walls are generated between the upper and lower shear boxes as needed, and the particles inside the shear boxes are divided into two parts, which can be grouped together to simulate different rocks in the subsequent parameter assignments. (2) Parameter assignment  
There are three places in the binary interface direct shear model that need to be assigned values separately: upper rock particles, lower rock particles and interface. The physical properties used for the calibration of the rock parameters in this paper are provided by the literature [8], including mudstone and limestone, with an interface friction coefficient of 0.55. The calibration parameters are shown in Table 1, and a comparison of the calibration results is shown in Table 2, from which it can be seen that the simulation results and the tests have good consistency. After assigning the calibrated parameters in the rock in the direct shear model, a smooth joint model is generated at the interface of rocks. The parameters required for the smooth joint model include normal stiffness, tangential stiffness and friction coefficient, which are determined by comparing the agreement of the simulation curves to the test curve. The parameter calibration results of the smooth joint model in this paper are shown in Table 3. It is worth mentioning that the smooth joint model simulates the shear behavior \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rocks** & **Results** & **Poisson Ratio** & \
\begin{tabular}{c} **Tangent** \\ **Modulus/GPa** \\ \end{tabular} &
\begin{tabular}{c} **Uniaxial** \\ **Compressive** \\ **Strength/MPa** \\ \end{tabular} \\ \hline \multirow{2}{*}{Mudstone} & Test & 0.320 & 4.7 & 11.50 \\  & Simulation & 0.321 & 4.7 & 11.39 \\ \hline \multirow{2}{*}{Limestone} & Test & 0.114 & 42.6 & 51.80 \\  & Simulation & 0.114 & 42.8 & 51.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Rock physical parameters and parameter calibration results [8]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rocks** & **Rigidity** & **Bonded** & **Strength** & **Max Particle** & **Particle Size** \\  & **Ratio** & **Modulus/GPa** & **Parameters/MPa** & **Size/mm** & **Ratio** \\ \hline Mudstone & 8 & 0.455 & 4.9 & 0.9 & 1.5 \\ Limestone & 1.5 & 3.30 & 17.8 & 0.9 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of rock calibration parameters. of a flat surface considering dilatation without considering the local particle contact direction.