# Modeling Shearing and Bending Behavior of Kiwifruit Branches Using the Discrete Element Method
## 3 Results and Discussion
### Determination of Optimal Values for the DEM Parameters  
In the response surface test, the range identified during the steepest ascent experiment was employed. To estimate experimental error, five tests were conducted at the central point, totaling 21 tests. The scheme and outcomes of the test are presented in Table 7. The \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{4}{c}{**Test Factors**} & \multirow{2}{*}{**Shear Strength (MPa)**} & \multirow{2}{*}{**Elastic Modulus (GPa)**} & \multirow{2}{*}{**Relative Error (\%)**} \\ \cline{2-2} \cline{5-6}  & **B** & & & & & \\ \hline
1 & \(2\times 10^{9}\) & \(1\times 10^{8}\) & \(1\times 10^{8}\) & 1 & 13.49 & 1.17 & 30.1 \\
2 & \(2.2\times 10^{9}\) & \(1.8\times 10^{8}\) & \(1.8\times 10^{8}\) & 1.1 & 32.37 & 1.35 & 7.2 \\
3 & \(2.4\times 10^{9}\) & \(2.6\times 10^{8}\) & \(2.6\times 10^{8}\) & 1.2 & 50.71 & 1.41 & 38.9 \\
4 & \(2.6\times 10^{9}\) & \(3.4\times 10^{8}\) & \(3.4\times 10^{8}\) & 1.3 & 75.48 & 1.57 & 85.1 \\
5 & \(2.8\times 10^{9}\) & \(4.2\times 10^{8}\) & \(4.2\times 10^{8}\) & 1.4 & 107.81 & 1.72 & 97.4 \\
6 & \(3\times 10^{9}\) & \(5\times 10^{8}\) & \(5\times 10^{8}\) & 1.5 & 29.57 & 1.76 & 144.6 \\ \hline \hline \end{tabular}  
* B: Effective modulus of bond; D: Cohesion of bond; E: Effective modulus between ball and wall; F: Normal-to-shear stiffness ratio.  
\end{table} 
Table 6: Scheme and results of steepest ascent test. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Source** & **Sum of Squares** & **F-Value** & _p-Value_ \\ \hline Response & Elastic modulus (GPa) & & \\ Model & 0.872 & \(11.153\) & 0.009 \\ A-emod & 0.001 & 0.085 & 0.783 \\ B-phemod & 0.482 & 36.962 & 0.002 \\ C-ten & 0.016 & 1.197 & 0.324 \\ D-coh & 0.000 & 0.001 & 0.979 \\ E-bfemod & 0.154 & 11.796 & 0.019 \\ F-kratio & 0.129 & 9.883 & 0.026 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._shear strength values ranged from 13.45 to 72.02 MPa, while the elastic modulus values varied from 1.03 to 1.53 GPa. Analysis of variance (ANOVA) of the models is shown in Table 8. Both models were highly significant with \(p\)-value smaller than 0.01. All four factors under examination significantly influenced the elastic modulus. The cohesion of the bond and the effective modulus between the ball and the wall significantly affected shear strength. Regarding interaction effects, only factors D\({}^{2}\) and E\({}^{2}\) were found to significantly impact shear strength. Based on these results, the models for shear strength and elastic modulus were formulated, as presented in Equations (8) and (9), respectively. In these equations, only significant factors were retained; non-significant factors were omitted. \[ \begin{split}\text{y}_{1}=4.06118-1.24953\times\text{B}+0.328864 \times\text{D}+0.301449\times\text{E}-4.27666\times\text{F}-0.0942861\times \text{D}^{2}-0.0587426\times\text{E}^{2}\\ \text{R}^{2}=0.98\end{split}  \tag{8}\]  
\[ \begin{split}\text{y}_{2}=-47.57-7.26\times\text{D}+4.45\times \text{E}\\ \text{R}^{2}=0.97\end{split}  \tag{9}\]  
The examined models for elastic modulus y\({}_{1}\) and shear strength y\({}_{2}\) exhibited very high coefficients of determination (R\({}^{2}\)), with values of 0.98 and 0.97, respectively. The R\({}^{2}\) value indicated how much variability in the values of the mechanical response can be explained by the experimental variables. The analysis of the F-value of each factor presented in Table 8 revealed the order of importance of the four factors on shear strength as E \(>\) B \(>\) F \(>\) D and for elastic modulus as E \(>\) D \(>\) B \(>\) F. According to the developed optimization regression model of each index, and considering the significance and importance order of the factors, we have selected the two most influential factors for each index to investigate their impact on elastic modulus and shear strength (Figure 6). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.