4.2 Difference in erosion characteristics as determined from traditional and new model
The interpretation of erosion characteristics from traditional model (Equation 1) was based on the relationship of erosion rate with hydraulic stress, as depicted in Fig. 7 (a). Erosion coefficient and critical shear stress can be inferred from the slope and horizontal intercept of curve, respectively. In the current study, the\(\dot{\varepsilon}-\tau\) curves were also presented, as shown in Fig. 7 (b). However, the temporal variation of the two curves is reversed so that the physical definition of the horizontal intercept is changed. Critical shear stress is the hydraulic shear stress when the erosion happens initially (Wall and Fell 2004b). However, the horizontal intercept in the current study is the hydraulic shear stress at the end of experiment. Therefore, this quantity is newly defined as equilibrium shear stress. Critical shear stress was proposed to formulate the criteria of hydraulic earth construction to prevent erosion (Arulanandan and Perry 1983). Similarly, equilibrium shear stress would be suggested to the consideration of post-construction risk assessment. As indicated in Table 3, the determination of equilibrium shear stress also showed the stability of the measurement approach with a nearly constant value of 0.08 Pa.
Even though erosion characteristics could be obtained from the\(\dot{\varepsilon}-\tau\) curves, some errors of the interpretation from the curves happened due to the theoretical discretization of model application. As described in Fig. 6, erosion coefficient could be deduced from the continuous \(R^{4}-t\) curves. This approach is recommended by the current study. Only after \(R(t)\) is known from experiments, other quantifies can be calculated. The model of \(R(t)\)has no issue of discretization since the instantaneous data of \(R(t)\)is collected and analyzed. In contrast, the measurement of erosion rate considering the average soil loss at a time interval results into the deviation induced by discretization. Besides, equation [22] and Fig. 4 suggest that variation of erosion rate with time should follow the concave function. Therefore, it implies that the instantaneous erosion rate should be theoretically and slightly higher than the measured erosion rate.
The comparison between the traditional and new model for obtaining erosion characteristics was analyzed, as shown in Fig. 8. Traditionally, erosion characteristics can be developed from the\(\dot{\varepsilon}-\tau\) curves based on equation [1]. The\(\dot{\varepsilon}-\tau\) curves plotted with measured data should be fitted by linear relationships. However, some subtle fluctuations appeared in curves due to the problem of discretization. With the interpretation from \(R^{4}-t\) curves, the theoretical\(\dot{\varepsilon}-\tau\) curves were predicted. Measured\(\dot{\varepsilon}-\tau\) curves generally coincided with the prediction model. Table 4 summarized the outcomes of the two models. The values of erosion coefficient provided from \(R^{4}-t\) model were slightly higher than those developed from the\(\dot{\varepsilon}-\tau\) curves. However, the performance of fitting of \(R^{4}-t\) model is better than the traditional model, as shown in the comparison of the determination coefficient (R2). Besides, the results from traditional model showed higher variance among the HETs.
Comparison between predicted and measured erosion behavior (from Fig. 8) can be generalized as shown in Fig. 9. Erosion process during HET can be categorized into three phases, namely initial phase , normal phase , and final phase . At the initial phase, the measurement is lower than the idealization of the model because it needs some time to initiate erosion with the existence of critical shear stress. The second phase is expected as the erosion variation is almost close to the theoretical model. At the final phase, the erosion coefficient is strengthened, which results into the rapid failure of the soil body and achieves the equilibrium of hydraulic shear stress and erosion resistance of the soil. Fell et al. (2003) also categorized the erosion process of infrastructure failure into four phases, includinginitiation , continuation , progression , andformation . These two distinction methods are similar and complementary to each other. The initial phase and final phase share the same definition with the first and fourth phases in the study of Fell et al. (2003), respectively. The normal phase is parallel to the combined phases of continuation and progression. The concept of three phases is suitable for small-scale experiments, while the four-phase analysis is used in large-scale tests. Both two models are necessary to understand the erosion process from laboratory tests to field surveillance. Results from both current experiments and previous studies suggested that the erosion coefficient would slightly increase with elapsing time. This weakness of erosion resistance is attributed by the reduction of Young’s modulus and possible enhancement of permeability during the HETs (Parron Vera et al. 2014; Jiang and Soga 2019). Progressive internal erosion reduces the sliding resistance at interparticle contact, triggering macroscopic deformations and even failure of the soil specimen (Hicher 2013). This explains the nonlinear relationships between applied shear stress and the erosion rate in the study of (Khanal et al. 2016). Therefore, it is significant to understand the stage of the erosion variation to react the erosive responses.