## 4.1 Structural state model and percolation theory

The structural state transitional behavior shown in Fig. 1 can be explained by using the percolation theory (Malthe-Sorenssen 2015), where the critical clay fraction corresponds to the transition point of the structural state. Soils are mainly composed of non-clay minerals, clay minerals, and water. Among them, clay minerals (e.g., smectite, illite, kaolinite) can interact significantly with water. Thus, clay minerals and water can be treated as clay-water composites in the theoretical modeling of physical properties (Li and Wong 2016). According to Li and Wong (2016), the volume fraction of clay-water composites, \({f_{cw}}\) in the soil can be derived as a function of clay fraction, \({c_f}\) and porosity, :

$${f_{cw}}={c_f}(1 - n)+n$$

3

To visually characterize the structural state-changing behavior of soils using percolation theory, we randomly generate a L × L (50 × 50) lattice of points that are occupied with non-clay minerals and clay-water composites with different volume fractions (Fig. 4). Two sites are treated as connected if they are the nearest neighbors (4 neighbors on square lattice), and a set of connected sites generate a cluster. A cluster that is spanning is called the spanning cluster, and the system is percolating if there is a spanning cluster in the system. As shown in Fig. 4, different pieces of clay clusters are represented by different colors. As we increase the volume fraction of clay-water composites, \({f_{cw}}\), the system tends to be percolated by clay-water composites.

The plot of the probability for there to be a connected path from one side to another, as a function of \({f_{cw}}\) for various system sizes L is shown in Fig. 5. The critical volume fraction of clay-water composites \({f_{cw}}\) for the percolation behavior is determined as 0.59 accordingly. Given Eq. (3), the critical clay fractions \({c_f}\) for soils can be derived at values of 0.32 ( = 0.4) to 0.41 ( = 0.3) depending on their porosities. Such critical clay fraction values are comparable with the structural state parameter A used in Eq. (1) for governing the transition point of the curve.

## 4.2 Clay mineralogy

Non-clay fractions of soils are mainly composed of quartz, and the friction property of non-clay fraction is slightly affected by some external factors such as the applied normal stress and the salinity of the pore fluid. However, the clay matrix can have a major variation of the clay minerals, and the external factors will exert a considerable impact on the friction properties. According to the measured residual friction angles of the clay matrix, \({\phi _r}^{c}\) of a variety of soils with different clay mineralogy, the value of \({\phi _r}^{c}\) for clay matrix significantly decreases with the increase of plasticity index (PI) (Fig. 6). It should be noted that we only use results of high clay fraction samples (with a clay fraction of more than 0.7) to illustrate the relation, which is to characterize the strength of the clay matrix. When PI is more than 50%, there is little variation in \({\phi _r}^{c}\) with the increase in PI. The modeled result and measured data give a good correlation between the plasticity index and residual frictional angle as shown in Fig. 6, which is consistent with the relationship illustrated by Voight (1973) and Tsiambaos (1991). The value of the plasticity index is an indicator of smectite minerals illustrated in Fig. 7, the clay matrix will have a PI value higher than 50% when the smectite fraction in clay \({f_{sc}}\) is more than 0.1. Similar conclusions were made in the study of Tsiambaos (1991) based on the relations between residual strength and clay fraction or bentonite content of marls. The influence of the variation in clay mineral content on the residual strength was studied accordingly.

Taking water-saturated kaolinite and smectite minerals as an example, smectite particles will behave like the matrix filled in between kaolinite particles (Fig. 8). Even if at a small portion, those smectite particles still can prevent direct contact among kaolinite particles. Thus, the residual friction angle of clay matrix of different soils tends to have a constant value (5 degrees), because the smectite fraction in the clay of most landside soil is more than 0.1. Thus, the relation between residual friction angle and clay fraction for general landslide soils can be approximated using a single relation shown in Fig. 3.

## 4.3 Applied normal stress

Experimental results by Eid et al. (2016) indicate that the residual friction angle of clay matrix decreases with the increase in the applied normal stress (Fig. 9). Such a decreasing trend is applicable for soils with different clay mineralogy. We use the PI value to represent the difference in clay mineralogy. It can be found that the normal effective stress has a major impact on the residual strength of the clay matrix when the stress is less than 200 kPa. The value of \({\phi _r}^{c}\) tends to be constant when the applied normal stress is more than 200 kPa. It is because at lower stress level (i.e. <200 kPa), clay particles have edge-to-face type contact which plays a vital role during external loading. However, more face-to-face types of contact are generated at higher stress levels, which leads to a lower value of fraction angle. That is why we only include the results with large normal stress (more than 200 kPa) in Fig. 2c. It is to minimize the impact of the applied normal effective stress. From an engineering point of view, it is on the safe side to use the lower bound value of \({\phi _r}^{c}\).

## 4.4 Effect of pore fluid salinity

According to Maio and Fenellif (1994), the residual strength of clays is affected by their mineral composition, and by the nature of their constituent pore fluid illustrated in Fig. 10, the residual strength of pure clay (clay matrix) mainly increases with an increase of the pore fluid salinity, which yields similar conclusions as compared to the findings in the study by Moore (1991). The shear strength of kaolin is not affected by the solutions used, whereas the residual strength of bentonite (rich in smectite) varies greatly because of the abundance of diffusion double layers. At a salinity of 6 moles, the water is fully saturated with sodium chloride solution. Except for pure bentonite, most of the clay matrices display a stable residual strength when the salinity is more than 1 mole.

We performed a statistical analysis on the pore fluid salinity of 39 soil samples from various depths of worldwide sites. Among them, 15 samples are from eastern Canada sensitive clay slopes (Locat and St-Gelais 2014), 9 samples are from Yama-ashi, Higashi-shiroishi and Ariake-kantaku in Japan (Egashira and Ohtsubo 1982), and 12 samples are from various locations of Norway (Rosenqvist 1953; Bjerrum 1954). As shown in Fig. 11, the pore fluid salinity displays a lognormal distribution. Most soils have a pore fluid salinity less than 0.1 mole, which indicates that the salinity is not a significant factor to be considered in analyzing the residual strength of soils.

## 4.5 The inclusion of coarse-grained particles

The proposed approach of estimating residual friction angle of land-sliding soils was only applied to soils not containing coarse-grained particles (e.g., gravel-sized particles and sand), and the recommended structural state parameters A and B values are suggested as 0.41 and 0.26, respectively. However, some land-sliding zones also contain a substantial amount of coarse-grained particles (Wen et al. 2007). Based on the measured results from Wen et al. (2007), we use the proposed structural state approach to derive parameters A and B for soils containing coarse-grained particles (Fig. 12). In the experimental results by Wen et al. (2007), sandy soil samples, which are gravels free (i.e. gravel particle size larger than 2 mm removed) were used in reversal direct shear tests. Whereas in-situ direct shear tests were carried out on soils containing gravel size particles. Shown in Fig. 12, it is illustrated that the proposed structural state approach yields a good correlation between the modeled result and the experimental data of reversal direct shear tests, where gravel-free soils were used. It is also should be noted that the derived structural state parameters (A and B) are different from those from soils only containing fine-grained particles as displayed in Fig. 2. Thus, the inclusion of coarse-grained particles will significantly affect the values of structural state parameters, but the proposed approach is still effective if gravel-sized particles are not involved. For those in-situ direct tests result on soils containing gravel-sized particles, our modeled result does not match well with the measured data (Fig. 12). The variation in results of in-situ direct tests can be related to a large uncertainty in in-situ direct shear tests. More in-situ test data are needed for further investigation. Thereby, we do not suggest using the proposed structural state approach to estimate the residual strength of land-sliding soils with significant gravel-sized particles.