# Modeling Shearing and Bending Behavior of Kiwifruit Branches Using the Discrete Element Method
## 2 Materials and Methods
### Discrete Element Modeling of Kiwifruit Branch  
The kiwifruit branch model was constructed using the DEM software PFC3D 6.0 (Itasca Consulting Group, Inc., Minneapolis, MN, USA). This software employs two primary elements--balls and walls--to simulate the mechanical and dynamic behavior of materials. Agricultural materials are represented by individual balls or ball clusters, while agricultural machinery or its components are modeled by wall assemblies. Previous studies have utilized the DEM to create branch model with small particles. The parallel bond model is composed of linear model, dashpot, and parallel bond (Figure 3). It allows adjacent particles to bond together when their distance falls within a certain range and was employed to simulate the cohesion within the kiwifruit branch tissues. Parallel bonds are capable of transmitting both force and moment between the balls. The force-displacement law for the parallel bond model updates the contact force and moment as detailed in Equation (5):  
\[\left\{ \begin{array}{l}F_{c}=F_{\text{l}}+F_{d}+F\\ M_{c}=M\end{array} \right. \tag{5}\]  
where \(F_{\text{c}}\) is contact force, \(M_{\text{c}}\) is contact moment, \(F_{\text{l}}\) is linear force, \(F_{\text{d}}\) is dashpot force, \(F\) is the parallel bond force, and \(M\) is the parallel bond moment. Figure 3: Behavior and components of parallel bond model: (**a**) bonded (**b**) unbonded. The parallel bond force is resolved into normal and shear forces, and the parallel bond moment is resolved into twisting and bending moments according to Equation (6):  
\[\left\{ \begin{array}{l}F_{b}=-F_{n}\cdot n+F_{s}\\ M=M_{t}\cdot n+M_{b}\end{array} \right. \tag{6}\]  
where \(F_{n}\) is the normal force, \(F_{t}\) is the shear force, \(M_{t}\) is the twisting moment, and \(M_{b}\) is the bending moment. Parallel bond provides mechanical behavior of a finite-sized piece of cement-like material deposited between the two contacting balls. Relative motion between the balls could cause force and moment to develop within the bond material. This force and moment then act on the two contacting balls. Bonds break and particles separate when the stress surpasses the predefined thresholds of tensile strength or cohesion and the bond material is removed from the model, thus simulating the branch's breaking behavior. The unbonded parallel bond model is equivalent to the linear model [17]. Considering the complexity of kiwifruit branches, modeling their actual structure is challenging. In this study, to facilitate the establishment of the DEM model, the rind, which has a relatively minor impact on mechanical properties, was disregarded, and the branch was simplified to an annular structure composed of a cluster of particles. Initially, a box with dimensions of 100 mm in length, 14 mm in width, and 14 mm in height was created and filled with balls. The diameter of these balls was uniformly distributed, ranging from 0.6 to 0.8 mm. Subsequently, a cylindrical assembly of balls with a diameter of 10.5 mm was formed by selectively removing particles located outside the cylindrical region. The medullary space was then simulated by removing a central cylindrical volume of particles with a diameter of 3.0 mm. The final branch model is depicted in Figure 4a, with the bonding keys of the parallel bond model between particles illustrated in Figure 4b. The fundamental parameters of the DEM model are listed in Table 1. The friction coefficient between the ball and wall and the ball local damping coefficient friction angle and bond gap were derived from prior studies [12; 25]. The other 6 parameters, which are likely to affect the mechanical properties of the kiwifruit branch, are currently undergoing calibration and will be discussed in the subsequent section.