Effects of Crushing Characteristics on Rheological Characteristics of Particle Systems

: A particle system’s large-deformation shear ﬂow exhibits obvious random characteristics, making accurate modeling of the particle system difﬁcult. Particle systems, which are frequently used in engineering, are prone to breakage, which introduces additional uncertainty into the system. The purpose of this study was to conduct ring-shear experiments on a variety of common engineering materials in order to quantify the effect of the dynamic crushing process of the particle system on the instability of shear ﬂow. Different shear fracture characteristics may result in a change in the volume trend of the system, from dilatancy to shrinkage. While the mean value of the crushable system’s stress ratio does not increase with shear rate, the stress ratio’s ﬂuctuation characteristic parameters are negatively correlated with shear rate. As particles become more easily sheared, the initial value of the stress ratio ﬂuctuation increases. The effect of shear rate on the ﬂuctuation in the system stress ratio is determined indirectly by the degree of system fragmentation. The study of the particle system’s ﬂuctuation characteristics will aid in developing a stochastic dynamic model for the landslide system in the future, allowing for improved prediction and prevention of landslide disasters.


Introduction
Debris flows are characterized by a high flow velocity, wide coverage, and the great harm that they cause. As examples, a landslide in Bijie, Guizhou, caused 27 deaths in August 2017 [1]. A landslide in Sichuan formed a barrier lake, and the region was severely affected in June 2015 [2], and a landslide at the Shenzhen landfill caused 77 deaths in 2015 [3]. A better understanding of landslides and the prevention of landslides requires landslides to be studied from the perspective of particle flow [4,5], while also requiring the establishment of an effective constitutive model.
Among rich theoretical systems related to particle flow, the model system most typically adopted is the µ(I) theoretical model system [6]. This model quantitatively describes the movement characteristics of a particle system in dense flow mode. In recent years, refined theoretical models based on µ(I) theory have been developed to better predict the behavioral characteristics of a particle system in the transition state [7][8][9]. It is worth noting, however, that existing theoretical models for particle flow analysis and modeling primarily focus on mean values of system mechanics, whereas the actual landslide impact force exhibits non-steady-state characteristics and can vary significantly [10,11]. There is still a dearth of research on these dynamic fluctuation characteristics [12,13]. Numerous studies have established the effect of volume fraction/shear velocity on fluctuations in ideal particle systems [14,15]. However, there is no unified theory to describe the macroscopic shear stress fluctuation in particle systems.
Meanwhile, in research on the constitutive structure of a large number of particles, particles have been simplified [16], and factors such as particle breakage have been less considered. An engineering project primarily utilizes quartz sand and calcareous sand [17,18]. During the shear flow process, significant fragmentation may occur [19]. The current state of knowledge regarding the factors affecting particle crushing is fairly comprehensive. The degree of particle breakage is affected by a variety of factors, including normal stress, shear rate, friction interface, and others [20,21]. A change in the system gradation will strongly affect the rheological properties [19,22]. There may be phenomena such as fluctuations in the shear residual strength [23], particle self-diffusion [24], and volume instability of the shear flow [19]. The effects of particle crushing on the rheological properties of a system need to be researched in more depth.
The shear flow properties of particles are most investigated by performing a ring-shear experiment. This experiment well simulates the entire process of the large deformation and flow of soil in actual landslides and is important to the study of particle flow [25]. To investigate the effect of particle crushing on the shear characteristic parameters of the particle system and to analyze the characteristics of the system shear fluctuation, the present study conducted a series of ring-shear experiments for three typical granular materials: glass beads, quartz sand, and calcareous sand. Glass beads are a non-broken particle system that is frequently employed as a control group in particle flow research. The experimental group consisted of the most frequently utilized quartz and calcareous sands in engineering. These two materials are the most often used coarse-grained materials in practical engineering and exhibit distinct crushing properties. The study of these two materials may aid in the comprehension of practical engineering cases. The sliding distance and velocity have the most intuitive effect on the shape of the landslide. It is a critical parameter in the calculation and forecast of landslides. As a result, we chose these two critical parameters as the research object for this study. The findings of this paper will contribute to the advancement of particle flow theory and increase public awareness of the peculiarities of landslide fluctuations.

Materials and Methods of Ring-Shear Test
A ring-shear instrument (Figure 1a) from Geotechnical Consulting & Testing Systems Company was used in the experiment. This instrument provides a uniform shear rate and an unlimited rotation angle [26]. The sample had an outer diameter of 150 mm, inner diameter of 100 mm, and height of 20 mm (which met the requirement of height > 10 × particle diameter). The experiment was conducted for three typical granular materials-namely, glass beads, quartz sand, and calcareous sand. Samples with an initial particle size of 2-2.5 mm were obtained by pre-screening the material. The normal stress in the ring-shear experiment was set at 100 kPa, and the sampling rate was 10-100 Hz depending on the duration of the experiment. In the early stage, the filling mass of each material was determined from the results of preliminary experiments to ensure that each sample had the same initial height under a positive pressure of 100 kPa. It was finally determined that each group of glass beads and quartz sand had a mass of 330 g and that each group of calcareous sand had a mass of 270 g. The initial material shape is shown in Figure 1b. The partition filling method was adopted to ensure uniform filling in the experiment, as shown in Figure 1c. The experimental process was designed following a previous study using the same instrument [27]. The normal stress was first applied once the filling was complete. Shear loading began once the system completed pre-consolidation, and the volume no longer changed. The velocity and shear distance gradients were established, and a total of 40 groups of tests were conducted, as illustrated in Table 1.

Quantitative Description of Fracture Characteristics of the Different Particle Systems
The degree of particle fragmentation was not high under the experimental conditions of the present study. If the traditional analysis method [28] is used to quantitatively analyze the degree of system fragmentation with Br, there will be a small difference in the degree of fragmentation between each group of samples, and it will be difficult to distinguish better. Therefore, this research used the classification discussion method to

Quantitative Description of Fracture Characteristics of the Different Particle Systems
The degree of particle fragmentation was not high under the experimental conditions of the present study. If the traditional analysis method [28] is used to quantitatively analyze the degree of system fragmentation with Br, there will be a small difference in the degree of fragmentation between each group of samples, and it will be difficult to distinguish better. Therefore, this research used the classification discussion method to summarize and discuss the screening data. The particle system after shearing was divided into three groups-namely, a group of unbroken particles (2-2.5 mm in diameter), a group of partially broken particles (1-2 mm in diameter), and a group of completely broken particles (<1 mm in diameter). The percentage mass content for each group is presented in Figure 2. In the initial stage of shearing, the particles were broken apart violently, the content of unbroken particles decreased rapidly, and the content of partially broken particles rose rapidly. Beyond a shearing displacement of approximately 360 • , the disintegration of the particles tended to become gentle. A higher shear rate resulted in a higher degree of system fragmentation. In subsequent sections, the "percentage of partially broken particles" is used as a quantitative indicator to describe the degree of particle breakage. It should be highlighted that appropriate analysis investigation of particle breakage will require a significant number of additional tests. summarize and discuss the screening data. The particle system after shearing was divided into three groups-namely, a group of unbroken particles (2-2.5 mm in diameter), a group of partially broken particles (1-2 mm in diameter), and a group of completely broken particles (<1 mm in diameter). The percentage mass content for each group is presented in Figure 2. In the initial stage of shearing, the particles were broken apart violently, the content of unbroken particles decreased rapidly, and the content of partially broken particles rose rapidly. Beyond a shearing displacement of approximately 360°, the disintegration of the particles tended to become gentle. A higher shear rate resulted in a higher degree of system fragmentation. In subsequent sections, the "percentage of partially broken particles" is used as a quantitative indicator to describe the degree of particle breakage. It should be highlighted that appropriate analysis investigation of particle breakage will require a significant number of additional tests.  Figure 3 shows representative axial displacement curves for the three particle types. The glass bead system showed a trend of dilatancy, with the change in the sample volume being approximately 2%. In the initial stage of shearing, the particles rearranged appreciably, the close occlusion between particles was broken, and the volume of the system Water 2022, 14, 532 5 of 10 increased rapidly. When the shear displacement reached approximately 120 • , the system tended to be stable, and the height of the sample hardly changed. In the fragile system, there was severe dilatation in the early stage of shearing that gradually changed to severe shrinkage. The volume peaked at a shear displacement of approximately 20 • . This conclusion is basically consistent with previous studies [19]. The dilatancy peak of the calcareous sand system was higher than that of the quartz sand system, which may be related to the more complex shape and larger slenderness ratio of the calcareous sand particles. The two crushable systems fractured strongly as the shearing displacement increased. The dominant factor of the change in the system volume changed from particle rearrangement to particle fragmentation. On the one hand, with the development of the crushing of the system, the sphericity of the particles gradually increased, making the particles more compact and reducing the volume. On the other hand, the small particles/silt sand produced in the crushing fell into the pores formed by the large particles, which increased the compactness of the system. The degree of particle fragility strongly affected the volume change. The volume change in the quartz sand particles with relatively high hardness under shear displacement shown in the figure was approximately 4.6%, whereas the volume change in the more easily broken calcareous sand reached 6.8% under the same test conditions. Figure 3 shows representative axial displacement curves for the three particle types. The glass bead system showed a trend of dilatancy, with the change in the sample volume being approximately 2%. In the initial stage of shearing, the particles rearranged appreciably, the close occlusion between particles was broken, and the volume of the system increased rapidly. When the shear displacement reached approximately 120°, the system tended to be stable, and the height of the sample hardly changed. In the fragile system, there was severe dilatation in the early stage of shearing that gradually changed to severe shrinkage. The volume peaked at a shear displacement of approximately 20°. This conclusion is basically consistent with previous studies [19]. The dilatancy peak of the calcareous sand system was higher than that of the quartz sand system, which may be related to the more complex shape and larger slenderness ratio of the calcareous sand particles. The two crushable systems fractured strongly as the shearing displacement increased. The dominant factor of the change in the system volume changed from particle rearrangement to particle fragmentation. On the one hand, with the development of the crushing of the system, the sphericity of the particles gradually increased, making the particles more compact and reducing the volume. On the other hand, the small particles/silt sand produced in the crushing fell into the pores formed by the large particles, which increased the compactness of the system. The degree of particle fragility strongly affected the volume change. The volume change in the quartz sand particles with relatively high hardness under shear displacement shown in the figure was approximately 4.6%, whereas the volume change in the more easily broken calcareous sand reached 6.8% under the same test conditions.    Figure 4a,b shows axial displacement curves for the calcareous sand and quartz sand system under different degrees of shearing and crushing. A nonlinear negative relation between the volume compression of the system and the degree of fragmentation is suggested by this figure. The Spearman correlation coefficient was used to quantify this relationship. The relation coefficient between fragmentation degree and volume compression of the system was −0.8071/−0.9562 (calcareous sand/quartz sand), which proves a strong negative relationship. The correlation coefficient between velocity and volume compression of the system was 0.2400/−0.0981 (calcareous sand/quartz sand), which indicates that there is no significant relation between shear velocity and volume compression.

Experimental Results on the Compression Characteristics of the Particle System
Water 2022, 14, x FOR PEER REVIEW 6 of 11 compression of the system was 0.2400/−0.0981 (calcareous sand/quartz sand), which indicates that there is no significant relation between shear velocity and volume compression.

Stress Ratios of the Different Particle Systems
We hope that the normal stress remains stable when we analyze the residual shear strength of a system. However, owing to the feedback-adjustment mechanism of the ringshear system, it is theoretically impossible to keep the normal stress absolutely constant. Figure 5 shows a representative stress curve of the entire shearing process for the quartz sand. It is seen that the normal stress fluctuates greatly, which will introduce large systematic error into the analysis of the shear stress fluctuation characteristics. However, analysis of part of the curve (presented in the insets) shows that the fluctuations in the normal stress and shear stress due to the system error are largely synchronized. Therefore, this paper does not directly study the shear stress fluctuation curve but analyzes the stress ratio to avoid systematic errors. It is noted that taking the stress ratio as the object of discussion makes the variance in the subsequently used data very small. However, this in no way means that the system fluctuates very little. In the representative experiment, the maximum shear stress was 36.02% higher than the mean value, but the stress ratio variance corresponding to this system was only 0.000414. Figure 6 shows the relationship between the degree of system fracture and the stress ratio for quartz sand and calcareous sand. It is seen that the system stress ratio did not change with an increase in the fracture degree. The correlation coefficient between stress ratio and fracture degree was −0.1723/−0.3036 (quartz sand/calcareous sand), which demonstrates a weak relation between these two variables. Previous studies have revealed that as the degree of fracture increases, the system stress ratio may also increase [29]. This phenomenon was not observed in the present experiment, possibly because under the experimental conditions of the present study, the degree of particle breakage was not high, and the small fine particle part was insufficient to change the overall shear residual strength of the system. The shear rate had no notable effect on the mean value of the system stress ratio. Figure 7 shows the relationship between the degree of crushing and the fluctuation in the stress ratio for quartz sand and calcareous sand. It is seen that for the same material system, a higher degree of fragmentation resulted in a smaller fluctuation in the system stress ratio. The correlation coefficient between the variance of stress ratio and fracture degree was −1.0/−0.7 (quartz sand/calcareous sand), which demonstrates a strong relation between these two variables. The fluctuation in the initial stress ratio increased as the particles were more easily broken.

Stress Ratios of the Different Particle Systems
We hope that the normal stress remains stable when we analyze the residual shear strength of a system. However, owing to the feedback-adjustment mechanism of the ringshear system, it is theoretically impossible to keep the normal stress absolutely constant. Figure 5 shows a representative stress curve of the entire shearing process for the quartz sand. It is seen that the normal stress fluctuates greatly, which will introduce large systematic error into the analysis of the shear stress fluctuation characteristics. However, analysis of part of the curve (presented in the insets) shows that the fluctuations in the normal stress and shear stress due to the system error are largely synchronized. Therefore, this paper does not directly study the shear stress fluctuation curve but analyzes the stress ratio to avoid systematic errors. It is noted that taking the stress ratio as the object of discussion makes the variance in the subsequently used data very small. However, this in no way means that the system fluctuates very little. In the representative experiment, the maximum shear stress was 36.02% higher than the mean value, but the stress ratio variance corresponding to this system was only 0.000414.  Figure 6 shows the relationship between the degree of system fracture and the stress ratio for quartz sand and calcareous sand. It is seen that the system stress ratio did not change with an increase in the fracture degree. The correlation coefficient between stress ratio and fracture degree was −0.1723/−0.3036 (quartz sand/calcareous sand), which demonstrates a weak relation between these two variables. Previous studies have revealed that as the degree of fracture increases, the system stress ratio may also increase [29]. This phenomenon was not observed in the present experiment, possibly because under the experimental conditions of the present study, the degree of particle breakage was not high, and the small fine particle part was insufficient to change the overall shear residual strength of the system. The shear rate had no notable effect on the mean value of the system stress ratio.   Figure 7 shows the relationship between the degree of crushing and the fluctuation in the stress ratio for quartz sand and calcareous sand. It is seen that for the same material system, a higher degree of fragmentation resulted in a smaller fluctuation in the system stress ratio. The correlation coefficient between the variance of stress ratio and fracture degree was −1.0/−0.7 (quartz sand/calcareous sand), which demonstrates a strong relation between these two variables. The fluctuation in the initial stress ratio increased as the particles were more easily broken.

Discussion of Shear Velocities
Many previous studies have used the shear rate as an independent variable with which to analyze the system stress ratio. The degree of system fragmentation is also re-(b) (a) Figure 7. Relation of the variance of the stress ratio and the degree of particle breakage for different particle systems: (a) quartz sand and (b) calcareous sand.

Discussion of Shear Velocities
Many previous studies have used the shear rate as an independent variable with which to analyze the system stress ratio. The degree of system fragmentation is also related to the shear rate, and thus when analyzing the system stress ratio, it is necessary to decouple the particle crushing and shear rate in clarifying the dominant factors affecting the system. In this section, the unbroken particle system is taken as the control group in discussing the effect of the shear rate on the stress ratio for systems with different crushing characteristics, and a reasonable explanation is given from the perspective of particle crushing. Figure 8 shows the relationship between shear rate and stress ratio in different particle systems. It can be seen in Figure 8a that in the non-broken particle system, the stress ratio and the shear rate have a certain negative relation (relation coefficient equals to -0.7211), and this negative relation characteristic may be related to the weakening of inter-particle occlusion. In Figure 8b, there is no significant relationship between the stress ratio of the crushable particle system and the shear rate. The relation coefficient between velocity and stress ratio is -0.3107/-0.2028 (calcareous sand/quartz sand). Therefore, it can be considered that the crushing factor will increase the system stress ratio to a certain extent. Under the experimental conditions of this study, the increase in the system stress ratio caused by crushing and the decrease in the system stress ratio caused by the weakened occlusion basically cancel each other out. Figure 7. Relation of the variance of the stress ratio and the degree of particle breakage for different particle systems: (a) quartz sand and (b) calcareous sand.

Discussion of Shear Velocities
Many previous studies have used the shear rate as an independent variable with which to analyze the system stress ratio. The degree of system fragmentation is also related to the shear rate, and thus when analyzing the system stress ratio, it is necessary to decouple the particle crushing and shear rate in clarifying the dominant factors affecting the system. In this section, the unbroken particle system is taken as the control group in discussing the effect of the shear rate on the stress ratio for systems with different crushing characteristics, and a reasonable explanation is given from the perspective of particle crushing. Figure 8 shows the relationship between shear rate and stress ratio in different particle systems. It can be seen in Figure 8a that in the non-broken particle system, the stress ratio and the shear rate have a certain negative relation (relation coefficient equals to -0.7211), and this negative relation characteristic may be related to the weakening of interparticle occlusion. In Figure 8b, there is no significant relationship between the stress ratio of the crushable particle system and the shear rate. The relation coefficient between velocity and stress ratio is -0.3107/-0.2028 (calcareous sand/quartz sand). Therefore, it can be considered that the crushing factor will increase the system stress ratio to a certain extent. Under the experimental conditions of this study, the increase in the system stress ratio caused by crushing and the decrease in the system stress ratio caused by the weakened occlusion basically cancel each other out. (b) (a) Figure 8. Relationship between the shear rate and stress ratio for the different particle systems: (a) glass beads, (b) quartz sand and calcareous sand. Figure 9 shows the relation between the shear rate and the fluctuation in the stress ratio in different particle systems. The relation coefficient between velocity and stress ratio variance was −0.5952/−0.9871/−0.8305. With the increase in the shear rate, there was no obvious change in the fluctuation in the stress ratio of the non-broken particle system. In the crushable particle system, the fluctuation in the stress ratio was negatively related to the shear rate. This conclusion is consistent with the relevant conclusions in Section 3. It can be considered that in a crushable system, the main influencing factor of the fluctuation in the stress ratio is the particle crushing effect rather than velocity effect. ratio in different particle systems. The relation coefficient between velocity and stress ratio variance was −0.5952/−0.9871/−0.8305. With the increase in the shear rate, there was no obvious change in the fluctuation in the stress ratio of the non-broken particle system. In the crushable particle system, the fluctuation in the stress ratio was negatively related to the shear rate. This conclusion is consistent with the relevant conclusions in Section III. It can be considered that in a crushable system, the main influencing factor of the fluctuation in the stress ratio is the particle crushing effect rather than velocity effect. Figure 9. Relation of the variance of the stress ratio and the shear rate for different particle systems.

Conclusions
There are uncertainties in the shear flow deformation of a particle system. The quantitative examination of the uncertainty in this phase is critical for comprehending the overall landslide process. By quantitatively defining the unstable properties of the particle system, it is anticipated that a probabilistic model of the entire landslide process will be constructed in the future, allowing for more precise prediction of crucial factors such as the landslide's impact velocity. The findings of ring-shear experiments with various particle systems were analyzed quantitatively in this work to determine the effect of particle system fragmentation features on shear flow instability. The main conclusions drawn from the results of the study are as follows.
(1) The shear rate has a greater effect on the degree of system fragmentation, with a higher shear rate resulting in a higher degree of system fragmentation. (2) Differences in granular materials result in large differences in the trend of the volume change during the shearing process. The unbroken system is slightly dilatant, whereas the breakable system has dilatancy in the initial stage of shearing and then large shear shrinkage. Particles that are more fragile have greater shear shrinkage. There is a nonlinear negative relation between the volume compression of the system and the degree of fragmentation. (3) The particle crushing effect increases the system stress ratio to a certain extent. In the unbroken system, the mean value of the stress ratio is negatively correlated with the shear speed, whereas in the breakable system, the breaking effect and the shear speed effect cancel each other out, and the mean stress ratio thus remains relatively stable. (4) The particle crushing effect reduces the fluctuation in the system stress ratio. However, the fluctuation in the initial stress ratio increases as the particles are more easily broken.  Figure 9. Relation of the variance of the stress ratio and the shear rate for different particle systems.

Conclusions
There are uncertainties in the shear flow deformation of a particle system. The quantitative examination of the uncertainty in this phase is critical for comprehending the overall landslide process. By quantitatively defining the unstable properties of the particle system, it is anticipated that a probabilistic model of the entire landslide process will be constructed in the future, allowing for more precise prediction of crucial factors such as the landslide's impact velocity. The findings of ring-shear experiments with various particle systems were analyzed quantitatively in this work to determine the effect of particle system fragmentation features on shear flow instability. The main conclusions drawn from the results of the study are as follows.
(1) The shear rate has a greater effect on the degree of system fragmentation, with a higher shear rate resulting in a higher degree of system fragmentation. (2) Differences in granular materials result in large differences in the trend of the volume change during the shearing process. The unbroken system is slightly dilatant, whereas the breakable system has dilatancy in the initial stage of shearing and then large shear shrinkage. Particles that are more fragile have greater shear shrinkage. There is a nonlinear negative relation between the volume compression of the system and the degree of fragmentation. (3) The particle crushing effect increases the system stress ratio to a certain extent. In the unbroken system, the mean value of the stress ratio is negatively correlated with the shear speed, whereas in the breakable system, the breaking effect and the shear speed effect cancel each other out, and the mean stress ratio thus remains relatively stable. (4) The particle crushing effect reduces the fluctuation in the system stress ratio. However, the fluctuation in the initial stress ratio increases as the particles are more easily broken.