Static and dynamic impact forces on a rigid barrier due to dry debris flow simulated by a DEM-based granular column collapse

. Geophysical flows like debris flows require protective barriers so as to scale down their disastrous effects. The impact force exerted on these barriers depends upon several factors like the velocity and height of flowing debris, the density and other associated characteristics of flowing mass, and the height of the sediment deposited behind the barrier. These factors are found to vary with time during the debris flow process, thereby causing significant temporal variations in the magnitude of impact force. Assessing the impact force on a barrier is a prerequisite for its design. In this work, the impact of dry debris flow, emanating from a granular column collapse, on a rigid barrier has been investigated using 3-D Discrete Element Method (DEM). It has been found that the impact force is predominantly influenced by dynamic thrust at the initial stages, which, with elapse time, subsequently transits to the sole contribution of static pressure. The applicability of the existing hydraulic models comprising hydrostatic and hydrodynamic formulae has also been analysed in this exercise. Furthermore, the empirical coefficients associated with these formulae have been evaluated for the dry debris flow impacting the rigid barrier.


Introduction
Debris flows constitute a class of devastating geophysical flows that mostly comprises inhomogeneous material mixture. The flowing debris engulfs everything that comes in its trajectory, thus leading to volume augmentation during the flow [1]. Due to such increased bulk volume, the flowing debris characteristically exhibits high velocity and pressure, thereby posing a compounded threat to lives, infrastructure and environment. Debris flow control structures in form of protective barriers are often used to create obstruction along the pathway of the flowing debris, thereby aiding in reducing the potential damage [1]. These protective structures should be designed for adequate strength or capacity so as to restrain the dislodged mass moving with high kinetic energy. In turn, as the flowing mass is hindered by the barrier, the obstructed debris transfer a temporally varying impact force to such structures [2]. Thus, estimation of the impact force is a prerequisite for the design of such barriers [1] and requires a comprehensive understanding of the mechanism and dynamics of the debris flow. Impact force exerted by the flowing debris depends on various factors like velocity and height of flowing mass along with the density, particle size and other associated characteristics of debris [3]. Recent studies have also revealed that initial inclination of flow front, intergranular friction and porosity of flowing mass, and mass length significantly influence the impact force, wherein the inclination of the front plays a dominant role * Corresponding author: mousumi.ju06@gmail.com with the maximum impact force experienced for an initial inclination angle of 90º [4]. Continuum-based numerical analyses have been extensively conducted in the past to assess the impact of flowing debris on the barriers [2]. The studies indicate that the kinetic energy carried by the flow, which reflects the flow velocity, has a direct implication on the magnitude of the thrust imparted on the obstructing barrier. Furthermore, the heights of flow and sediment (settled mass) also affects the impact force exerted on the barrier. These analyses confirm that the impact force essentially comprises two components, a static part and a dynamic part. The deposited mass behind the barrier structure primarily contributes to the static part; whereas, the flowing mass striking against the structure is responsible for the dynamic component. However, such continuum-based impact force estimation often assumes a single phase idealisation for the flowing debris mass, wherein the variability in flow is taken into account through empirical constants, which are difficult to calibrate.
Dry debris flows have predominant granular behaviour with only frictional inter-particle contacts controlling the mobility [1]. Such a flow can be easily replicated by causing an instantaneous collapse of granular column and can be investigated numerically using Discrete Element Method (DEM) [5]. DEM is a numerical modelling technique based on Langrangian approach and deals with the particle level responses [5]. It helps in comprehending large deformation problems such as debris flows with detailed and intricate insights. This approach has been recently used for understanding the micro and macro mechanical behaviour in reference to the dynamic interaction between dry granular flow and rigid barrier [4,6]. The present study aims to assess the characteristics of the impact of dry debris flow on a rigid barrier using 3D -DEM with due consideration to estimation of its dynamic and static components. In this regard, applicability of the existing continuum based hydraulic models has also been analysed along with the estimation of associated empirical parameters.

Methodology
In this study, a DEM platform has been used to investigate a dry debris flow simulated through the collapse of a granular column. The simulations have been carried out using LIGGGHTS [7]. The efficacy of the granular column collapse simulations has been validated against the experimental results from Shi et al. [8,9].
In the present study, a granular column of aspect ratio (height/width) as 4 has been allowed to collapse freely over a planar horizontal frictional bed. The dry debris flow emanating from the collapsing mass has been made to impact a rigid barrier placed at a certain distance away from the granular column. Fig. 1 represents the initial geometry of the granular column and the final settled state of the particles behind the rigid barrier. Table 1 depicts the geometrical details of the column and barrier. It also includes the micromechanical properties of the column particles, which have been calibrated iteratively against the granular column collapse experiment reported in Shi et al. [8]. Table 1. Micro-mechanical properties and the geometry details used in this study.

Details of the rigid barrier
Barrier height (m) 0.096 Distance of the barrier from the column (m) 0.12

Results and Inferences
The variation of total impact force (F) is plotted with respect to time in Fig. 2. In order to calculate the impact force, the barrier has been discretised with multiple mesh elements in LIGGGHTS [7]. At any time instant, the total impact force (F) is a cumulative sum of the product of impact pressure over each mesh element with the area of that element, thereby, causing an undulating yet less noisy plot for F. The beginning of the plot represents the time at which the confining gate of the column has been instantaneously removed, i.e., when the collapse of the granular column has begun and the material have started to flow towards the barrier. The touchdown impact of the flowing particles on the rigid barrier occurs at a later time (~ 0.19 s) that is demarcated by a sudden rise in the impact force. In the succeeding time instants, the undulating impact peaks are witnessed as the continually flowing particles transfers the dominant dynamic impact to the barrier. Further, the dynamic impacts gradually decrease with the complementing attainment of a static impact force from the particles settled behind the barrier, as can be noted approximately beyond a time instant of 0.4 s.   It can be inferred from Fig. 3(a) that initially the impact pressure on the barrier is primarily governed by the dynamic thrust of the flowing particles, which is reflected by higher pressure magnitudes on the lower level of the barrier. As the particles start settling behind the barrier, the dynamic component experiences a transition to the static counterpart, and thereby with time, the impact pressure exerted over the lower levels of barrier reduces due to the predominant contribution from the static lateral pressures (Fig. 3b). However, there is yet a remnant effect of dynamic impact in the lower levels as the succeeding flow thrusts behind the settled particles and indirectly transmits a reduced dynamic impact over the barrier. Additionally, these flowing particles, having high kinetic energy, jump over the settled particles and exert an extended dynamic impression over the barrier at the immediate upper level. This is apparent from the kink illustrated in Fig. 3b, which has been discussed explicitly in [2]. As time further elapses, more and more particles start settling behind the barrier, thereby, augmenting the contribution of static pressure over the extended height of barrier. In such scenario, the direct impact of dynamic pressures from the flowing particles contracts meagrely in higher elevations of the barrier (as apparent from Fig.  3d), and eventually converges to the imparted timeindependent static pressure.
In order to assess the impact of debris flows, certain hydraulic models are available that include the relations for estimating hydrostatic and hydrodynamic components of the impact force. The expression for hydrostatic part (Eqn. 1) is based upon the height of deposition behind the barrier; whereas, the relation for hydrodynamic component estimation (Eqn. 2) is pivoted around the flowing velocity and height [3].
where Fs and Fd are the hydrostatic and hydrodynamic components of impact forces, respectively, k and α are the empirical coefficients for hydrostatic and hydrodynamic formulae respectively, ρ is the density of the flowing mass, hs is the height of flow with static behaviour, i.e., the height of settled mass behind the barrier, hf is the height of flow, v is the flowing velocity and b is the width of the barrier. Fig. 4 shows the variation of average flow velocity (v) with time. For determining the average flow velocity, a threshold velocity has been first identified. By averaging out the particle velocities with magnitude greater than the threshold velocity, average flow velocity has been obtained. The threshold velocity has been fixed at 1% of maximum of average particle velocity.  particles with velocity greater than the threshold velocity. Similarly, Fig. 5 also illustrates the variation in the height of flow with static behaviour, i.e., average height of settled debris mass behind the barrier (hs) with time. The same has been ascertained by considering the height of all those particles that are near to the barrier and have velocity lesser than the threshold velocity.   Fig. 6. While estimating the values of empirical coefficients, certain truncation criteria have been considered for v, hf and hs, in order to eliminate any unrealistic or spurious values mostly for extreme time instants. It can be observed from Fig. 6 that for initial time instants, where the impact is predominantly dynamic (hf > hs), the value of α is lesser than or nearly equal to 1, whereas the value of k is significantly large. Since the initial impact force is mostly governed by the flow velocity, the hydrodynamic formula (Eqn. 2) suitably represents the impact force that is indicated by a lesser α value. However, as the time progresses, the contribution of static pressure enhances. As hs builds up and hf gradually diminishes over the time, the k value decreases and α value increases. Finally, for the time instants of 0.4 s and beyond, the value of k coefficient becomes near to 1 while that of α coefficient is extremely high. During these later times, the force experienced by the barrier is derived solely from the static impression of the deposited mass behind the barrier (as seen from Fig. 5). Hence, the lateral stresses on the barrier could be better represented by hydrostatic formula (Eqn. 1), as indicated by lesser k value (nearly equal to 1). Thus, the range of k and α coefficients, as well as the applicability of the type of hydraulic models, is dependent on the predominant forcing behaviour (dynamic or static) that changes and evolves with time; similar inference has been reported by Leonardi et al. [10]. Based on the present study, the range of k values is estimated as 0.93 -67.52 and that of α values is estimated as 0.27 -13.62. Since the design of such barriers is governed by maximum impact force (indicated in Fig. 2) [1], the estimated values of k and α coefficients at the time instant of maximum impact force as illustrated in Table 2. Within the limits of this particular study, the obtained values of the empirical coefficients are in reasonable agreement with the available ranges as mentioned by Lee and Jeong [2]. It is to be noted that the impact force exerted by a dry debris mass has been investigated here. However, a true debris flow may also contain fluid part leading to a more complex impact force variation over the rigid barrier, that is beyond the scope of this study.