Effects of Inter-Particle Frictional Coefficients on Evolution of Contact Networks in Landslide Process

During the process of landslide, its dynamic mechanism is important to understand and predict these kinds of natural hazard. In this paper, a new method, based on concepts of complex networks, has been proposed to investigate the evolution of contact networks in mesoscale during the sliding process of slope. A slope model was established using the discrete element method (DEM), and influences of inter-particle frictional coefficients with four different values on dynamic landslides were studied. Both macroscopic analysis on slope landslide and mesoanalysis on structure evolution of contact networks, including the average degree, clustering coefficient and N-cycle, were done during the process of landslide. The analysis results demonstrate that: 1) with increasing inter-particle frictional coefficients, the displacement of slope decreases and the stable angle of slope post-failure increases, which is smaller than the peak internal frictional angle; 2) the average degree decreases with the increase of inter-particle frictional coefficient. When the displacement at the toe of the slope is smaller, the average degree there changes more greatly with increasing inter-particle frictional coefficient; 3) during the initial stage of landslide, the clustering coefficient reduces sharply, which may leads to easily slide of slope. As the landslide going on, however, the clustering coefficient increases denoting increasing stability with increasing inter-particle frictional coefficients. When the inter-particle frictional coefficient is smaller than 0.3, its variation can affect the clustering coefficient and stable inclination of slope post-failure greatly; and 4) the number of 3-cycle increases, but 4-cycle and 5-cycle decrease with increasing inter-particle frictional coefficients.


Introduction
Slope failure is a common geological hazard, which is a changing progress with time and space, including initial stage, landslide stage and stable stage post failure [1]. Numerical methods have been widely used to predict the slope stability, study failure patterns and dynamic rules, including limit equilibrium method [2] [3], finite element method [4] and discrete element method [5]. The finite element method based on continuum mechanics is hardly employed to simulate the process of landslide, and although the discrete element method can be used to study the macroscopic rules of landslide, such as movement velocity and slide distance, it can't provide the bridging relationships between the contacting relationships between sliding blocks and their evolution rules in mesoscale and macroscopic movement rules during the process of landslide. Therefore, studies on mesostructures and evolution of contact networks are necessary to be done to provide a new method to analyze the macroscopic mechanism of landslide.
Discrete element method is a numerical method based on non-continuous medium analysis, which has been widely used in analysis of geotechnical problems [6] [7] [8] [9] [10]. Through defining the meso parameters of particles, it can be used to not only solve macroscopic problems, but also save much information about soil particles. Wang et al. [11] found that joint connectivity has great influences on stability of rockmass slope by defining different values of joint connectivity. Scholtès and Donze [12] studied the effects of joints with different angles of inclination on slope failure through acquiring particle velocity in the process of landslide. Chen and Liu [13] compared the numerical simulations with experimental results to study the properties of failure, and found that the angle at failure obtained by simulation is the same as that obtained by tests. Although DEM has been used to simulate the sliding process of landslide, macroscopic process of landslide should be analyzed physically in detail from the meso scale of soil grains and meso structure network, which will be explored primarily here in the viewpoint of complex network method.
Complex network is a method used to study the physical process or data, which is widely used in economics [14], sociology [15], medical science [16] and physics [17]. Recently, complex network has also been gradually employed to analyze the physical and mechanical properties of granular matter. Peters et al. [18] divided contact network into strong network and weak network according to the values of contact force among particles, and quantified the length and amounts of force chains. Walker and Tordesillas [19] investigated the deformational features of dense materials upon loading from the perspective of complex network, and studied the development of network structures and loss of connectivity in the process of compression. Tordesillas et al. [20]

Establishment of Slope Model
In the paper, PFC (particle flow code), one kind of DEM, is used to perform the sliding analysis of slope because it can simulate the landslide process and track the particles' state easily. For PFC in 2D, the particles are assumed to be rigid circles and the deformation can occur at the contacting points among these particles. The particles obey Newton' law of motion, and the computational principle can be found in references [5]. The contact-bonded model was used in the simulation, with particle sizes ranging from 3 mm to 6 mm, both normal stiffness and tangential stiffness are 50 MPa, contact bond strength at initial state is 50 MN/m, inter-particle frictional coefficient (f) at initial state is 0.9, and initial void ratio is 0.16. Similar to centrifuge model, an amplified gravity field is used in this simulation [22]. Under 200 g of gravitational field, a rectangular coordi-  According to Mohr-Coulomb criterion, the factors causing slope failure include reduction of internal frictional angles [23], loss of cohesive forces [24] and increase of pore water pressure [25]. In the following, the influences of inter-particle frictional coefficients on dynamic process of landslide of unbounded soil slope will be studied with variations of inter-particle frictional coefficients of 0.9, 0.5, 0.3

Calibration of Parameters of Biaxial Tests
Here studies are focused on variations of inter-particles frictional coefficients affecting dynamic process of landslide when bond strength between soil particles quickly reduces to be zero. Therefore, it is the first step to measure the mechanical behaviors of unbounded soils with different inter-particle frictional coefficients for biaxial compression tests. The generated sample is 0.8 m in height and 0.4 m in length including 4000 particles, and the contact bond strength between soil particles is zero, the initial void ratio is 0.16. The inter-particle frictional coefficients are 0.9, 0.5, 0.3, and 0.2, respectively, and lateral confining pressures employed are 50 kPa, 100 kPa, 200 kPa, and 400 kPa, respectively, and when the axial compressional strain reaches 20% with loading rate of 1%/min, it stops. Figure 3 present the deviatoric stress-axial strain curves and volumetric strain-axial strain curves of different inter-particle frictional coefficients at the same lateral confining pressures respectively, and Figure 4 presents the relationship between internal frictional angles in peak values and inter-particle frictional coefficients. From Figure 2 and Figure 3, we can find that with increasing of inter-particle frictional coefficients, the peak deviator stress also increase with more heavily strain softening, and the samples dilate more heavily;

Figure 2 and
and dilatancy becomes greater with increasing inter-particle frictional coefficient under the same lateral confining pressures. The reason for this phenomena is that: the samples are relative dense at the beginning of applying deviatoric stress, and only particles displace, rearrange and dilate leading to bear the external load; the particles with higher inter-particle frictional coefficient can bear more external loading between their contacting points, followed by stress decrease accompanied by dilatancy; with increasing axial strain, the samples contract within small axial stress and dilate until failure, which result loose zone along a band in the sample and slip along this band, thus the inter-particle frictional coefficient hardly affects the residual strength and samples with higher inter-particle frictional coefficients have bigger stress reduction. From Figure 4, we know that the internal frictional angles at peak deviator stress increase and the shear strength    of the samples also increase with increasing inter-particle frictional coefficient.

Displacement Analysis
As shown in Figure 1, Line 1 is located in the top of slope and Line 2 is located in the middle of slope. Take Point O as origin of coordinate as shown in Figure   1, we can trace the displacement paths on the particles located on the grid nodes, as shown in Figure 5. From Figure 5, we know that the particles of grid nodes close to slope surface have bigger displacements, and after failure, the particles displace to new positions lower than the initial ones. When the inter-particle fric-

Analysis on Geometrical Shape of Slope Post-Failure at a New Stable State
After the slopes fail, the particles slide and adjust their position timely and arrive at their new positions, and thus the geometrical shape of the slope at the new stable state can be formed as shown in Figure 6. From Figure 6, we know that the slope surface of slope at stable state after failure does not go through the toe of initial slope, but about the middle of surface of initial slope. When f is 0.2, the stable angle of slope post-failure is 18.4˚, which is slightly smaller than its internal frictional angle 20.5˚ at peak deviatoric stress. These phenomena appear for other values of f 0.3, 0.5 and 0.9. When f is different, slope slides more greatly Engineering  and the volume of landslide is huger with decreasing f, which leads to smaller angle of slope formed at stable state after failure.

Displacement at Slope Toe
The deformation of slope during landslide is very important for dynamic analysis, which will be analyzed here at slope toe here, as shown in Figure 7. From  its speed arrive at a new stable value. Therefore, when f is 0.9, the movement at slope toe can be divided into three stages including slow movement, quickly increasing rate of movement and reducing movement rate to a stable value; but, when f is 0.5, 0.3 and 0.2, respectively, the movement at slope toe can be divided into two stages including quick movement and the decreasing speed tending to a new stable value. The reason for this is maybe that: at f = 0.9, at the initial stage, particles near slope crest will first slide and drive the soil particles below them to slide gradually, and this process needs some time for the particles adjusting their stress state to balance external loads transferred from adjacent particles, which leads to small amounts of particles slide towards slope toe due to their bigger internal frictional angles at peak deviatoric stress and higher shear strength; with increasing computational steps, more particles slide, which cause more particles quickly slide to slope toe and thus form a new stable state.

Evolution Analysis of Contact Network in the Process of Landslide
In the contact network, a vertex denotes a single particle, and an edge linking the center of two particles denotes there is a contact between them. When landslide occurs, the particles will rearrange and adjust their positions, which can reflect the evolution of contact networks of these particles, such as loss of old contacts and formation of new ones. Recurring to some concepts from complex network, including the average degree of contact network, clustering coefficient and N-cycle, we can investigate the mesostructures varying with the process of landslide, which can help in understanding the dynamic mechanism of landslide.

The Evolution of Average Degree
The degree of a vertex [26] is the basic characteristic of network, which means  (1) in which N is the number of particles, a ij is adjacent matrix, which means that a ij = 1 when particles i and j contact, otherwise a ij = 0.
The evolution of average degree with time steps at different values of f is shown in Figure 8.

The Evolution of Clustering Coefficient
The clustering coefficient, also called transitivity [27] is a typical property of network, which describes the probability of adjacent vertices for the points in network. Three vertices adjacent each other form a triangle, which can resist rotation among them, so the clustering coefficient can be used to predict the stability of complex structure. The clustering coefficient of a vertex (or a particle in granular matter) is defined as the ratio of the number of triangles embodying it to that of a connected triple centering it, in which a triangle is a set of three vertices with edges between each pair of vertices and a connected triple is a set of

N-Cycle
N-cycle is the shortest closed path with N in length in an undirected network [28]. In network formed in granular matter, the particles contact each other and form many structures with closed cycles, which sustain together the stability of network. For the three types of structures cycles shown in Figure 11, 3-cycle has the most stability, 4-cycle has the most unstability, and 5-cycle force in between, and the reason for this is that compared with the force cycles having odd number N, the force cycles with even number N rotate more easily [20]. The failure process of slope is the rearrange process of particles, which will result the loss of old contacts and formation of new contacts, and both structure and number of cycles will change timely. At different values of f, the evolution of 3-cycle, 4-cycle and 5-cycle with computational time steps are shown in Figure 12. From Figure 12, we can find that the 3-cycle has the greatest number and 5-cycle has the least one. At initial stage of landslide, the number of 3-cycle decreases, but the numbers of 5-cycle and 4-cycle (except for f = 0.9) also increase. With computational time steps increase, however, the numbers of 3-cycle, 4-cycle and 5-cycle tend to be stable. The reason for this is that: before initiating landslide, the network structure of soil particles of slope is shown as Figure 13(a), where the number of 3-cycle is most and the slope has the most stability; during the process of landslide, the network Especially, however, when f = 0.9, the number of 4-cycle decreases with increasing time steps, the reason for which is that: there are less amounts of particles sliding in the process of landslide for the slope with f of 0.9, and its network structure of particles is shown in Figure 13

Analysis on Relationships Bridging Meso and Macro Parameters
During the process of landslide, the relationships between meso parameters and macro parameters can be obtained based on the above analysis on the evolution of degree, clustering coefficient and N-cycle. The evolution relationship between the average degree and displacement of slope toe is shown in Figure 14, from which we can find that during the process of landslide, with increasing f the average degree decreases fast within small displacement of slope toe, and then when the displacement of slope toe becomes bigger, the average degree decreases. Therefore, monitoring the variation of the average degree of soil particles can predict the slide displacement of slope.
When the slope reaches a new stable state post-failure, the relationship between clustering coefficients and inclination of angle at stable state of slope can be obtained and shown in Figure 15. From Figure 15, we can find that with increasing f, both inclination angle and clustering coefficient increase at a new stable state. When f is smaller than 0.3, the variation of f affects greatly on clustering coefficient and inclination angle; however, when f is more than 0.3, the variation of f affects slightly on clustering coefficient and inclination angle.

Conclusions
Based on discrete element method, the landslide processes of slope with four sets of inter-particle frictional coefficients are simulated and the evolution of meso structures is analyzed in the perspective of complex network. The conclusions can be summarized as follows.
1) The inter-particle coefficient has a great influence on the displacement of slope and evolutions of meso structure parameters, including the average degree, the clustering coefficient, and N-cycle (N = 3, 4, 5), during the sliding process.
2) With increasing inter-particle frictional coefficients, the displacement decreases and the inclination angle formed at stable state of slope increases, the average degree becomes smaller at initial stage of landslide and decreases when reaching a new stable state, the clustering coefficients decrease all the time.
3) For the same computational time step, with decreasing f, the number of 3-cycle decreases, and those of 4-cycle and 5-cycle increase, leading to the decreasing stability of soil slope.
In this study we explore the evolution of meso parameter during landslide Engineering process in a first step to establish the relationships bridging meso and macro parameters, such as evolution of average degree, clustering coefficient and evolution of N-cycle with displacement of slope toe. These meso parameters should be connected with the physical mechanism of landslide in further study to evaluate the evolution and movement of landslide.