## 1 Introduction  
The yield stress is a special property associated with non-Newtonian fluids, such as clay suspensions, concrete cements, crude oil, foams, food, paints, pastes, and polymers [1; 2; 3; 4; 5; 6]. In general, the yield stress is defined as the shear stress at the point where the applied stress exceeds the critical shear rate; in this case, the material starts to flow. Materials may deform elastically due to structural changes below the yield stress, but they may deform greatly and flow like liquids above the yield stress. There are numerous methods that can be employed to determine the yield stress of non-Newtonian fluids: model fitting, a slump test, an inclined plane test, a stress ramp test, a stress growth test, oscillation amplitude sweep, and creep tests. However, the yield stress is strongly dependent on the test conditions and techniques employed [7; 8]. In addition, many non-Newtonian fluids are thixotropic [9; 10]. Structural changes with time are unavoidable during shear. Time-dependent rheological characteristics make it difficult to find a robust method for determining the true yield stress [11]. This subject is still under debate in cement and concrete research (such as building materials) and disaster-prevention measures (such as debris flow analysis for estimating future catastrophic hazards). The dynamic and static yield stresses can be determined by equilibrium flow curves and stress growth methods. The dynamic yield stress is the minimum shear stress in the flow curve with a steady-state flow condition, while the static yield stress is the peak shear stress measured with a constant shearing time [12]. It is also well-known that the dynamic yield stress is associated with flow stopping, but the static yield stress is associated with flow starting [13]. According to previous studies [6], the dynamic and static yield stresses are related to the state before and after microstructural changes in materials. In addition, the conception of the time depending on the value of the yield stress has been pointed out by numerous studies [14; 15; 16]. As a result, it is expected that the value of dynamic yield stress is much lower than that of static yield stress with respect to the structural states. It is well-known that the yield stress is related to the structural change, that is, thixotropy. Rheological properties are essential in the fluid industry and natural disaster-related sciences. The rheological properties are examined for various materials, such as clay suspensions, muds, sand-rich materials, and reconstituted pyroclastic debris flows [13; 17; 18; 19; 20]. Water is typically a viscous liquid, while particle-bearing liquids typically behave as non-Newtonian fluids, indicating more complex rheological characteristics. Few studies have focused on the geotechnical and rheological properties of post-failure materials in mass movements, which can result in rapid complex fluid-like behavior as a kind of soil liquefaction. In the field of debris flows, a multiphase process in large particle-bearing debris flows is of paramount importance [21] because they are very dense and exhibit strong interactions between particles in the interstitial fluid. Rheological tests are often considered large particle-laden flows, e.g., the mean grain size of materials tested in rheometers has often been a typical sand with particle sizes from 0.075 to 2 mm in diameter. However, the change in rheological properties is significant when the solid fraction is slightly changed [13]. The disposal of mine tailings and waste rock materials involving sand- and gravel-size particles in open pits is a problem that needs to be managed. Severe erosion and mass movements in abandoned mine deposits can result in physicochemical contamination at and near the mine deposits [22]. In this case, the determination of flow properties is crucial for debris flow rheology [19] because the runout distances and speeds of debris flows are mainly estimated by yield stress and viscosity [23].