## 4.1 Comparison and stage division of shear stress in increasing shear

According to the experimental scheme designed above, the variation process of the first and second increasing shear stress of mud with different solid volume concentration with shear rate is shown in Fig. 3. a-e。 Fig. 3. f is the change process of viscosity with shear rate in the second increasing shear process.

For all solid volume concentration slurries, the dynamic change process of shear stress with shear rate in the two increasing shear processes is similar, and the shear stress crosses near the shear rate of 0.002s-1. When the shear rate is less than 0.002s-1, the first increasing shear stress is lower than the second stress, but when the shear rate exceeds 0.002s-1, the first increasing shear stress is higher than the second shear stress (Fig. 3. a-e).

The shear stress of mud with each solid volume concentration changes abruptly twice each time it is sheared. In the first and second increasing shear processes, the maximum shear stress appears near 0.225s-1 and 0.1s-1 respectively. However, the minimum value of shear stress appears near 1.83s-1 in both increasing shear. Take Nanyang expansive soil slurry with solid volume concentration of 37% as an example. In the first increasing shear process, the shear stress increases rapidly from 0 to 1647pa (shear stress increase stage) when the shear rate increases from 0 to 0.225s-1. The shear stress decreases from 1647pa to 908pa (shear stress reduction stage) when the shear rate increases from 0.225s-1 to 1.83s-1. The shear stress increases from 908pa to 1145pa (shear stress increase stage) when the shear rate increases from 1.83s-1 to 30s-1. In the second increasing shear process, The shear stress increases rapidly from 678pa to 990pa when the shear rate increases from 0 to 0.1s-1. The shear stress decreases from 990pa to 252pa (shear stress reduction stage) when the shear rate increases from 0.1s-1 to 1.83s-1. The shear stress increases from 252pa to 776pa (shear stress increase stage) when the shear rate increases from 1.83s-1 to 30s-1. Cheng C H (1986) described this change process as three stages: stress growth, stress attenuation and stable flow. Jeo (2017,2019) carried out rheological experiment on tailings waste by using mcr301 spherical lateral volume system. With the increase of shear rate, the shear stress also shows an increase stage and a decrease stage, and then reaches a stable flow state. Taking the lowest shear stress as the dividing point, the change process of shear stress is divided into failure stage and stable flow stage after failure. The mud shear zone is formed in the failure stage, and the shear stress first increases and then decreases with the shear rate. It is two completely different processes of the formation of the shear zone. The shear stress growth (strain hardening) process is a dynamic failure process of slurry microstructure. With the increase of failure degree, the required shear stress increases (Bonn 2009). The process of shear stress reduction (strain softening) is not a unique phenomenon in the process of mud shear. Moller et al. (2009) found this negative slope in the rheological test of colloidal gel materials (the shear stress decreases with the increase of shear rate). (G. Ovarlez, et al, 2015; H. Hafid, et al, 2015; O.H. Wallevik, et al, 2015; J. Spangenberg, et al, 2012; Qian Y & Kawashima S, 2018) it is confirmed that this process is the migration of coarse particles from high shear rate region to low shear rate region. Chen Xiaoqing (2006) found that there was a sudden increase of pore water pressure during the start-up of debris flow. Therefore, the migration of coarse particles and the increase of pore water pressure are the reasons for the decrease of shear stress (strain softening) with the increase of shear rate in the flow curve. Scotto di santolo et al. (2012) observed similar behavior in the study of debris flow mixture and assumed that it was caused by the crushing and liquefaction of debris flow suspension. The area where the shear rate is greater than 1.83s-1 (as shown in Fig. 3.a-e and the shaded part in Fig. 4) represents the third stage of the first increasing shear: stable flow stage. Although the change of shear stress with shear rate is not stable, and even the shear stress decreases continuously with the increase of shear rate (Fig. 3. b-e), it is likely to be the continuation of coarse particle migration and shear liquefaction process. In view of this, the variation process of mud shear stress of Nanyang expansive soil with shear rate is divided into: strain hardening stage, strain softening stage and stable flow stage (see Fig. 4).

## 4.2 Effect of solid volume concentration on rheological parameters of shear stable flow

Proper rheological model is the key to analyze debris flow movement. From the above experimental results, it is difficult to use a unified model to describe the increase and decrease of shear stress in different shear rate ranges in the process of increasing shear rate. The failure stage is the formation process of shear zone and the focus of debris flow initiation process. The rheological characteristics of mud in stable flow stage are more suitable for the analysis of debris flow movement process. The commonly used rheological models in this stage mainly include Bingham model( \({\tau }={\tau }_{c}+k\dot{\gamma }\)), Herschel – Bulkley model(\({\tau }={\tau }_{c}+k{\dot{\gamma }}^{n}\)) and Power law model(\({\tau }=k{\dot{\gamma }}^{n}\)). According to the analysis in Section 4.1, the second increasing shear stable flow stage (shear rate is more than 1.83s-1) is selected as the representative to analyze the variation process of shear stress with shear rate (see Fig. 5).

In order to analyze the applicable rheological model and the influence of solid volume concentration on rheological parameters, the variation process of shear stress and shear rate is plotted in the double logarithmic coordinate system (Fig. 5). Figure 5 shows that the slope decreases with the decrease of solid volume concentration, and there is a good power-law relationship between shear stress and shear rate: \({\tau }=k{\dot{\gamma }}^{n}\)), and has a good correlation. In order to distinguish from the analysis of deceleration shear process in the next section, the rheological parameters of this stage (stable flow) are marked as: \({\text{k}}_{1}\),\({n}_{1}.\) The rheological parameters\({\text{k}}_{1}\) and \({n}_{1}\) of slurries with different solid volume concentrations are summarized in Table 2. In the stable flow stage, the rheological parameters \({\text{k}}_{1}\) and \({\text{n}}_{1}\) have an exponential relationship with the solid volume concentration (\({C}_{s}\)), where\({\text{k}}_{1}=0.1582{e}^{{19.473C}_{s}}\), \({\text{n}}_{1}=0.0017{e}^{{14.376C}_{s}}\)) (see Fig. 6. a and b). That is, the relationship between shear stress and shear rate related to solid volume concentration in the accelerated shear stable flow stage can be expressed as:

$${\tau }=0.1582{e}^{{19.473C}_{s}}\bullet {\dot{\gamma }}^{0.0017{e}^{{14.376C}_{s}}}$$

Table 2

Rheological parameters \({\text{k}}_{1}\),\({\text{n}}_{1}\)

Solid volume concentration(\({C}_{s}\)×100%) | \({\text{k}}_{1}\) | \({\text{n}}_{1}\) |

0.37 | 310.86 | 0.2919 |

0.34 | 105.59 | 0.2533 |

0.31 | 51.162 | 0.1625 |

0.29 | 36.701 | 0.1142 |

0.27 | 29.835 | 0.069 |

Jeong (2009) conducted a coaxial cylinder rheological experiment on Jonquiere clay (Eastern Canada) using rotovisco rv-12 rheometer. When the shear rate is in the range of 1 to 500s-1, the rheological curves of slurries with different liquid index (liquid index) in the double logarithmic coordinate system are a group of basically parallel lines, and the viscosity decreases with the increase of liquid index. The liquid index increases with the increase of water content, and the solid volume concentration decreases with the increase of liquid index. Figure 6A shows that the rheological parameter \({\text{k}}_{1}\) of Nanyang expansive soil slurry increases with the increase of solid volume concentration, which is consistent with Jeong (2009). However, the mud flow index \({n}_{1}\) of Nanyang expansive soil decreases with the decrease of solid volume concentration (Fig. 6. b), which is inconsistent with the result of Jeong (2009) that the flow index is 0.09.

## 4.3 Rheological properties of decreasing shear

The desilting process of debris flow is the process of debris flow fluid from stable flow state to static state. The change process of shear stress of debris flow slurry with the decrease of shear rate is the fundamental problem that must be concerned about the desilting condition and accumulation range of debris flow. Therefore, after the first speed increasing shear, the shear rate immediately decreases from 30s-1 to 0 in a logarithmic manner. The dynamic change process of shear stress and viscosity with the decrease of shear rate is shown in Fig. 7a, b。

In the process of decelerating shear, the shear stress decreases with the decrease of shear rate. The relationship between mud shear stress and shear rate is power-law. In the double logarithmic coordinate system, the shear stress, viscosity and shear rate of mud with each solid volume concentration are a group of nearly parallel parallel lines (Fig. 7. a, b). The relationship between shear stress and shear rate can be expressed as:

$${\tau }={\text{k}}_{2}\bullet {\dot{\gamma }}^{{n}_{2}}$$

Where \({\text{k}}_{2}\) and \({n}_{2}\) are the rheological parameters in the deceleration shear stage. The rheological parameters \({\text{k}}_{2}\) and \({n}_{2}\) of slurries with different solid volume concentrations are listed in Table 3. Parameter \({\text{k}}_{2}\) is closely related to the slurry solid volume concentration, which increases in a power-law relationship with the increase of solid volume concentration (see Fig. 8). The relationship between parameter \({\text{k}}_{2}\) and slurry solid volume concentration can be expressed as:\({\text{k}}_{2}={0.0144e}^{30.031{C}_{s}}\)。In the range of 27% − 37% of solid volume concentration, the value of rheological parameter \({n}_{2}\) is between 0.0271–0.0333 (see Table 3), which is less affected by solid volume concentration. Since the relationship between shear stress and shear rate is a group of approximately parallel lines (Fig. 7. a), the rheological parameter \({n}_{2}\) can be taken as its average value (0.0289) in the analysis of desilting process. According to the above analysis, in the process of deceleration shear, the shear stress can be expressed by solid volume concentration and shear rate. The formula is as follows:

$${\tau }={0.0144e}^{30.031{C}_{s}}\bullet {\dot{\gamma }}^{0.0289}$$

Table 3

Rheological parameters \({\text{k}}_{2}\),\({n}_{2}\)

Solid volume concentration(\({C}_{s}\)×100%) | \({\text{k}}_{2}\) | \({n}_{2}\) |

0.37 | 1072.7 | 0.0333 |

0.34 | 368.68 | 0.0284 |

0.31 | 137.11 | 0.0278 |

0.29 | 82.065 | 0.0277 |

0.27 | 55.469 | 0.0271 |

average value | —— | 0.0289 |

The above analysis shows that there is a great difference between the increasing shear stress and decreasing shear stress of Nanyang expansive soil slurry with the change of shear rate. There are two sudden change points of shear stress in the increasing shear process, although it is difficult to use a unified model to express the change process of shear stress with the shear rate. However, there is a good power-law relationship between shear stress and shear rate in the steady flow stage of increasing shear rate and decelerating shear process. A unified power-law model can be used to describe the variation process of shear stress with shear rate.

## 4.4 Characteristic value of mud shear stress

The stress conditions such as starting, maintaining stable flow and deposition of debris flow are the basic problems often concerned in the whole process analysis of debris flow. According to the above analysis of the variation process of shear stress and viscosity with shear rate (two increasing shear and one decreasing shear), Nanyang expansive soil slurry is a thixotropic non-Newtonian fluid with yield stress (H.A. Barnes, 2007). Bonn (2017) described in detail the physical behavior of thixotropic non-Newtonian fluid materials with yield stress, and summarized the experimental techniques that can be used for yield stress testing. For thixotropic mud, the determination of yield stress depends on the accurate experimental scheme (Joshi, 2018). At present, there is a consensus on distinguishing the yield stress measured by the solid to liquid (or yield) transition from the yield stress measured during the liquid to solid transition. Mud yield stress can be described as static or dynamic yield stress (Bonnecaze and Brady 1992; Pham et al. 2008). The static yield stress can be regarded as the static yield stress according to the shear stress under the actual shear rate through the continuous shear test (Bonn, 2017). Although the determination of static yield stress depends on the shear rate applied and the "sample age" (Benzi et al. 2021a, 2021b). However, in the analysis of debris flow start-up process, the stress value represents the shear stress that must be overcome by debris flow start-up. In this experiment, two increasing shear were carried out, and the stress development process was similar (Fig. 3). Before the formation of the shear band (stable flow), both experienced the process of strain hardening and strain softening (Fig. 4). There was a peak shear stress, which can be used to represent the maximum shear stress that the mud must overcome from the formation of the shear band to the stage of stable flow, that is, the static yield stress. Although there is a peak stress in each increasing shear, and the peak stress of the first increasing shear is much greater than that of the second. However, for the analysis of debris flow start-up process, it can be considered that the mud is the primary shear, so it is more reasonable to use the first stress peak to analyze the debris flow start-up process. Therefore, the maximum shear stress of the first increasing shear is taken as the static yield stress of Nanyang expansive soil slurry( \({\tau }_{1}\))。Static yield stress of Nanyang expansive soil slurry with each solid volume concentration(\({\tau }_{1}\)) See Table 4.

The dynamic yield stress represents the critical stress value of the material gradually entering the deposited solid state from the flowing liquid state. Theoretically, it is the shear stress value when the shear rate approaches zero []. However, in the actual test, it is impossible to obtain the shear stress of mud at zero shear rate. Usually, the shear stress when the shear rate is close to 0 or the intercept of the stress axis is taken as the dynamic yield stress value.

In this experiment, when the shear rate increases from 0 to 30s-1 (the first increasing shear), the shear rate decreases from 30s-1 to 0 (decelerating shear). According to the above method, the intercept of the shear stress axis or the shear stress value with the shear rate close to 0 can be taken as the dynamic yield stress of the mud. From the variation process of increasing shear stress with shear rate (Fig. 3), it can be seen that the shear rate is in the range of 0-1.83s-1, which is the formation process of mud shear zone. During the formation stage of shear zone, the mud is in an unstable flow state (Jeo 2017, Leo 2020). Therefore, the dynamic yield stress determined by the shear rate of 0 or the intercept from the shear stress axis cannot be used for the analysis of the stable movement process of debris flow (Pell, 2018). According to the analysis of the development process of twice increasing shear stress (Section 4.1), when the shear rate exceeds the critical shear rate (1.83s-1), the shear zone is formed and the mud enters a stable flow state. The shear stress value corresponding to the critical shear rate represents the minimum stress value to maintain the stable movement of mud, which is more practical (Jeo, 2017). In order to distinguish the concept of dynamic yield stress, the shear stress corresponding to the critical shear rate is called stable flow yield stress(\({\tau }_{2}\)). From the start-up to the stable flow stage of debris flow, the mud must undergo a complex shear process, and the original microstructure of the mud is damaged by multiple shear. Therefore, in this paper, the shear stress corresponding to the critical shear rate in the secondary increasing shear process is taken as the stable flow yield stress. Stable flow yield stress of Nanyang expansive soil slurry with each solid volume concentration(\({\tau }_{2}\)) See Table 4. In order to compare the difference between static yield stress and stable flow yield stress, for Nanyang expansive soil slurry with various solid volume concentrations, the difference(\(\varDelta {\tau }\)) between static yield stress(\({\tau }_{1}\)) and stable flow yield stress ( \({\tau }_{2}\))is shown in Table 4.

Table 4

Characteristic value of shear stress

solid volume concentration /% | 37 | 34 | 31 | 29 | 27 |

Static yield stress(\({\tau }_{1})\)/Pa | 1606 | 550 | 207 | 125 | 76 |

Steady flow yield stress(\({\tau }_{2})\)/Pa | 352 | 125 | 53 | 39 | 30 |

\(\varDelta {\tau }\) /Pa | 1254 | 425 | 154 | 86 | 46 |

The effect of solid volume concentration on static yield stress (\({\tau }_{1}\)) and steady flow yield stress (\({\tau }_{2}\)) is shown in Fig. 9.

Table 4 and Fig. 9 confirm that the static yield stress and stable flow yield stress of mud increase exponentially with solid volume concentration (Fig. 9). The static yield stress is much greater than the steady flow yield stress, and this difference decreases with the decrease of slurry solid volume concentration (Table 4). (J.J. Assaad et al 2006)。The static yield stress and stable flow yield stress of mud have the following relationship with the volume concentration of mud solid: static yield stress:, \({R^2}=0.9937\);Steady flow yield stres, :\({\tau _2}=0.0301{e^{24.84{C_s}}}\)༌ \({R^2}=0.9663\)。