## 3 Results
### Shear Rate-Dependent Flow Behavior: Dynamic and Static Yield Stresses
#### 3.2.2 Stress Growth Test: Time Dependency  
Flow-like sediments are well-known as time-dependent materials (i.e., thixotropic materials). The time dependency of the rheological behavior of waste materials at a constant shear rate was \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Bingham**} & \multicolumn{2}{c}{**Bilinear**} & \multicolumn{2}{c}{**Modified Bingham**} & \multicolumn{2}{c}{**Herschel–Bulkley**} & \multicolumn{2}{c}{**Power Law**} \\ \hline \(\dot{\gamma}\) & \(\tau_{c}\) & \(\eta_{h}\) & \(\dot{\gamma}_{\sigma}\) & \(c\) & \(\eta_{pN}\) & \(m\) & \(\tau_{c\text{-}HB}\) & \(K\)-\(\eta_{\text{HB}}\) & \(n\)-\(\eta_{\text{HB}}\) & \(K\) & \(n\) \\ \hline
0.1 & 22.39 & 1.34 & 0.47 & 10.43 & 49.4 & 10.43 & 23.0 & 0.83 & 1.14 & 11.04 & 0.52 \\
0.01 & 19.75 & 1.46 & 0.08 & 1.52 & 257.05 & 1.52 & 18.0 & 0.69 & 1.13 & 13.01 & 0.42 \\
0.001 & 15.31 & 1.10 & 0.04 & 0.61 & 384.21 & 0.61 & 15.0 & 1.96 & 0.84 & 10.88 & 0.45 \\
0.0001 & 17.02 & 1.09 & 0.03 & 0.43 & 674.49 & 0.43 & 12.0 & 1.78 & 0.96 & 11.50 & 0.45 \\
0.00001 & 15.33 & 1.13 & 0.01 & 0.14 & 1703.47 & 0.14 & 12.0 & 1.90 & 0.91 & 8.98 & 0.53 \\ \hline \hline \end{tabular} Note: \(\dot{\gamma}_{\sigma}\) and c are bilinear constants; \(\eta_{\text{sh}}=\) pseudo-Newtonian viscosity obtained from the Bilinear model; \(m=\) model constant, \(\tau_{c\text{-}HB}=\) Herschel–Bulkley yield stress; \(n\)-\(\eta_{\text{HB}}=\) flow behavior index in the Herschel–Bulkley model; \(K\) and \(n=\) flow behavior indices in the Power law model.  
\end{table} 
Table 1: Rheological properties obtained from different rheological models. Figure 3: Shear rate dependency of rheological behavior at relatively low shear rates: (**a**) Shear stress-shear rate relationship and (**b**) viscosity-shear rate relationshipexamined. The applied shear rates ranged from 0.001 to 1 s\({}^{-1}\). The shear rate was continuously applied, and the torques were measured for 350 s, except at 0.5 s\({}^{-1}\). Most rheological tests for coarse-grained sediments are performed for relatively short periods of time, such as less than 1 min. If the shear rate is less than 10\({}^{-3}\) s\({}^{-1}\), the test is difficult to carry out with a time constraint. Figure 4 shows the test results of the stress growth method: shear stress with time and viscosity with time. There is significant fluctuation during shearing. This large scatter (fluctuation) is observed for each flow curve. This feature may be due to interactions between large particles. The scatter may be minimized when the fine content is increased. In Figure 4a, the flow behavior is quite different when different continuous shear rates are applied at constant values during shearing. Typically, two types of shear behavior can be found for a given range of shearing times: strain hardening and strain softening behavior. The first type has a peak value (i.e., the yield stress) in shear stress with time, and the second type has a peak and a steady-state value. The latter is very similar to the residual shear stress in soil mechanics. It can be defined as the minimum shear resistance of the materials along the previously existing slip surface. Strain hardening is dominant for relatively low initial shear rates (i.e., shear rates of 10\({}^{-3}\) and 10\({}^{-2}\) s\({}^{-1}\)), but strain softening is dominant for relatively high initial shear rates (i.e., shear rates of 10\({}^{-1}\) and 1 s\({}^{-1}\)). The transitional behavior occurs between 10\({}^{-2}\) and 10\({}^{-1}\) s\({}^{-1}\). The shear stress fluctuation (vertical variation in shear stress for a given time) is large. This result may occur because of particle-particle interaction via the nonslip boundary system, resulting in collisions and fragmentation in coarse-grained sediments. The viscosity is also very sensitive to the shearing time (Figure 4b).