1. Introduction
Strengthening the impact resistance of structures to withstand rapid granular flows is of concern to design engineers aiming to prevent the harmful effects of landslide disasters [
1,
2]. Various types of mitigation structures have been designed to reduce the destructive power of such disasters, for example, check dams, flexible barriers, baffle arrays and slit dams [
3,
4,
5,
6,
7,
8,
9,
10]. The design of impact-resisting structures relies on good knowledge of the debris–barrier interaction; however, our understanding of this mechanism is still lacking [
11,
12,
13,
14,
15,
16,
17].
In nature, granular flows occur where loose overlying deposits are disturbed, and, in specific conditions, multiple small flow events may occur within a short time period [
7]. When a flow reaches a barrier and overflows, some of the flow material will be captured behind the barrier at the final stage. Often, before the deposited material is swept away, another flow surge may reach the barrier, creating a multiple-surge impact event. Although such events are common, current research focuses on single-surge impact [
4,
5,
10,
13,
17,
18,
19,
20]. Only several authors have addressed this issue. Albaba et al. [
21] simulated two surges of granular flow impacting on a flexible barrier and observed that the peak impact force decreased by 25% when compared with the force exerted by a single surge with the same total mass. Using a large-scale model test, Tan et al. [
7] investigated the dynamic response of a flexible barrier under the impact of three surges of debris flow, and concluded that the retaining capacity of the barrier deteriorated under multiple-surge impact. However, these studies have not elucidated the instinct mechanism during multiple-surge impact process. In addition, impact force estimation models also largely cover single-surge impact, such as hydraulic models [
22], with the multiple-impact model seldomly being solved. As a result, it is urgent to investigate the mechanism and explore a reliable force estimation model of multiple-surge impact.
The most challenging aspect of force estimation under the multiple-surge impact condition is the consideration of the dead zone, which is the static material deposited behind the barrier after impact. During debris–barrier interaction, the dead zone helps to dissipate the kinematic energy of the subsequent flow, which results in a lower impact force on the barrier [
13,
23]. Faug et al. [
19] studied the length of the dead zone, which is important for defense structure design. Song et al. [
6] also highlighted the importance of the dead zone in energy dissipation during interaction. However, these studies have not thoroughly provided quantitative analysis. Under multiple-surge impact, the interaction between the dead zone and subsequent surges is important, because the dead zone can serve as a cushioning layer. In engineering practice, granular layers are often used to protect structures from impact loading, providing added reinforcement, for example, in rock sheds [
18,
24] and rigid barriers [
25]. Therefore, it is also important to examine the cushioning effect of the dead zone during multiple-surge impact, which is not well covered in the literature, because it may offer some engineering implications.
In order to fill the gaps identified above, we calibrate a numerical model based on the discrete element method (DEM) using physical experimental results. This model is further used to investigate the multiple-surge impact process of granular flow on a rigid barrier. Specifically, the impact mechanism and the cushioning effect of the dead zone were quantitatively analyzed, and the evolution of the force impacting the barrier was investigated and discussed. Additionally, the possible solutions for multiple-surge impact force estimation are also discussed. Our results may provide useful information for better design of rigid barriers in the field.
2. Materials and Methods
For decades, researchers have developed various numerical methods to simulate flow-type landslides, such as rock avalanches and debris flow [
26]. Among these methods, the DEM has advantages over the continuum mechanics method [
3,
27] in modeling granular flows, as it can well address the discrete nature of such flows. The DEM is also widely used to investigate debris–barrier interaction [
9,
28] because it provides micro-scale information that cannot be obtained in physical modelling [
29]. Based on this information, we can determine the relation between the micro-mechanism and macro-behavior [
5,
17,
24].
In DEM simulation, the granular flow is represented by an assembly of spherical or non-spherical particles, and the single-particle motion (translation and rotation) is governed by Newton’s second law of motion. By calculating the contact force between the particles and solving Newton’s second law of motion, the particles’ position and velocity can be obtained step by step, providing a reliable model of the granular flow movement, impact and deposition.
In our study, we adopt a commercial software named EDEM to conduct simulations, in which the micro-contact force could be calculated by the Hertz–Mindlin (no-slip) contact model for its computational efficiency. The model calculates the normal force (
) by Hertz’s theory and the tangential force (
) by Mindlin’s no-slip model:
here, the subscript
n is the normal direction, and
t is the tangential direction;
K denotes the elastic stiffness constant, and
D is the damping coefficient;
u represents the overlapping or relative displacement between two particles in contact;
is the relative velocity; and
is the coefficient of the Coulomb friction. Equation (2) shows that the tangential force is limited by Coulomb’s law of friction and accounts for the gross sliding movement between two particles in contact. A rolling torque is adopted to address the rolling friction, expressed as:
where
is the rolling friction,
is the distance between the contact point and the center of mass, and
is the unit angular velocity of the particle at the contact point.
The numerical model (
Figure 1) is based on the physical test conducted by Jiang and Towhata [
14]. The flume is inclined at 40°, and a 0.4 m-high rigid barrier is set at the end of the flume, perpendicular to the flume base. The flume sidewall is 0.35 m high and 0.3 m wide, and the total flume length is 2.19 m. Initially, we used spherical particles with a particle diameter of 10–20 mm to model the granular soil, forming a rectangular deposition body, 0.15 m high, 0.44 m long and 0.3 m wide. The total mass of the particles is 27 kg, and the bulk density of the initial disposition body is 13.6 kN/m
3, similar to that in the experiment of Jiang and Towhata [
14]. The flume and barrier were represented in the model by a wall element. The DEM input parameters are listed in
Table 1, which are obtained by experimental measurement and calibration or from other studies.
The proposed numerical model was validated by comparing the flow kinematics and impact dynamics of granular flow with that observed in laboratory experiments of Jiang and Towhata [
14]. The comparison results are presented in
Figure 2, which verifies that our DEM model could generally capture the four stages of flow evolution process reported by Jiang and Towhata [
14]. (1) The flow starts from a dam break failure of a rectangular debris deposition with obvious particle motion at flow surface (
Figure 2a,a1,b,b1). (2) Then, a mature flow is developed: the flow length is increased, while the flow depth becomes smaller, especially at the flow front where the particles collision is significant (
Figure 2c,c1). (3) When the flow front reaches the barrier, some discrete particles become agitated because of impact and rebound. Meanwhile, a portion of flow deposits behind barrier, and the dead zone develops with ongoing debris–barrier interaction (
Figure 2d,e,d1,e1). (4) Finally, all particles settle behind the barrier that formed a trapezoidal deposition morphology (
Figure 2f,f1). We also compared the Froude number (
, where
v and
H denote the average velocity and depth of the flow front, respectively) of the numerical and experiment results. Our result,
= 7.8, was close to the experimental results (6.7). In addition, the comparison of time-dependent impact force is shown in
Figure 3. It is noticed the peak value, residual value and evolution trend of impact force registered in physical tests are reasonably reproduced. These comparisons indicate that the proposed numerical model can be used reliably to investigate the multiple-surge impact behavior of granular flow. More details of calibration process of the DEM model can be accessed in
Appendix A.
Based on the calibrated model, we first consider two scenarios: a single-surge event and double-surge impact. The latter scenario follows the methodology of Albaba et al. [
21]. Initially, surge1, which has half the volume of the single-surge scenario, was formed. This flow reaches the rigid barrier and forms the dead zone, then another surge (surge2) with a volume equal to that of surge1 is released and interacts with the dead zone and the barrier. The total volume of surge and the slope angle varied in the modeling to investigate the effect of the morphology of the dead zone on the multiple-surge impact mechanism. The simulation program is presented in
Table 2.