## 4 Effects of Particle Shape, Initial Void Ratio, and Inter-Particle Friction Angle
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Aspect Ratio of Particles**} & **Friction Coefficient in Specimen Preparation** & **Initial Void Ratio** \\ \hline \multirow{4}{*}{\(1\)} & 0.001 & 0.588 \\ \cline{2-3}  & 0.01 & 0.598 \\ \cline{2-3}  & 0.15 & 0.670 \\ \cline{2-3}  & 0.4 & 0.701 \\ \cline{2-3}  & 0.5 & 0.744 \\ \hline \multirow{4}{*}{\(1.3\)} & 0.001 & 0.493 \\ \cline{2-3}  & 0.01 & 0.530 \\ \cline{2-3}  & 0.15 & 0.591 \\ \cline{2-3}  & 0.4 & 0.646 \\ \cline{2-3}  & 0.5 & 0.727 \\ \hline \multirow{4}{*}{\(1.7\)} & 0.001 & 0.499 \\ \cline{2-3}  & 0.01 & 0.560 \\ \cline{2-3}  & 0.15 & 0.600 \\ \cline{2-3}  & 0.4 & 0.709 \\ \cline{2-3}  & 0.5 & 0.775 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the test program. vertical stress. However, the difference of \(K_{0}\)-value between spherical particle and clump particle with \(AR=1.3\) is much larger than that between particle with \(AR=1.3\) and particle with \(AR=1.7\). Thus, it can be inferred that the change of \(K_{0}\)-value caused by the aspect ratio is more drastic when the aspect ratio is relatively small. For granular materials, particle shape, inter-particle friction coefficient, and initial void ratio control their strength parameters (peak strength and critical state strength). By the combination of the variations of these three factors, enough wide range of strength parameters can be obtained to study the correlation between strength parameters and \(K_{0}\)-value. Figure 15 shows the variation of \(K_{0}\)-values with the critical state friction angle and peak friction angle for all specimens in this section. The prediction line of the [PERSON]'s [44] simplified formula is also drawn in the figure. From the figure, it is more general to adopt the peak friction strength parameter in this simplified formula. The critical friction strength is constant as the initial void ratio changes, contradicting the phenomenon that \(K_{0}\)-value varies with initial void ratio. However, there is still a certain gap between the  
Figure 14: \(K_{0}\) versus initial void ratios for specimens at different aspect ratios. Figure 13: \(K_{0}\) versus initial void ratio for specimens with different \(\mu\) values. measured values and the prediction results using peak strength parameters. [PERSON] and [PERSON] [11], [PERSON] [4], and [PERSON] and [PERSON] [7] have also indicated that [PERSON]'s formula does not provide a concise estimation of \(K_{0}\)-value. Therefore, we do not think \(K_{0}\) can be predicted precisely enough by just one strength parameter. Based on the results presented in this section, we try to give the following formula for estimating the \(K_{0}\)-value of granular materials as a function of critical state friction angle (\(\varphi_{c}\)), particle shape, and the relative density of initial state (\(Dr\)). In the calculation of \(Dr\), \(e_{max}\) and \(e_{min}\) are obtained by the method proposed by [PERSON] and [PERSON] [45]:  
\[K_{0}=S(15-Dr-15\times sin\varphi_{c})-1 \tag{14}\]  
where \(S\) is the particle shape factor. For particles with \(AR\) = 1.0, 1.3, and 1.7, the shape factor is 0.154, 0.179, and 0.183, respectively. In the formula above, we adopt critical state friction angle instead of peak friction angle, because the later one is affected by too many factors such as initial void ratio, confining pressure, mineral composition, etc., to establish a more accurate formula of multiple factors. The critical state friction angle is generally accepted to be controlled only by mineral composition [46]. Additionally, relative density was used to describe the initial state characteristics of the material [47], and particle shape parameter \(S\) was used to characterize the effect of particle shape on \(K_{0}\) value. Figure 16 illustrates the fitting effect of the formula. As we can see from the figure, the proposed formula is accurate enough to predict the DEM simulation results in this section.