Simulation of Collapse of Granular Columns Using

Simulation of Collapse of Granular Columns Using the Discrete Element Method

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Elnaz Kermani, S.M.ASCE1; Tong Qiu, M.ASCE2; and Tianbin Li3 Abstract: In this study, a three-dimensional (3D) numerical investigation of axisymmetric collapse of granular columns has been conducted using the discrete element method (DEM). The simulated granular columns have a constant initial radius of 5.68 mm and three aspect ratios: 0.55, 1.0, and 2.0. The columns consist of uniform spherical quartz particles with a diameter of 0.32 mm. In the DEM model, rotational velocities of particles are reduced by a factor at every time step to partially account for the additional rolling resistance due to the effect of particle shape and hysteretic contact behavior. The simple linear contact model is used; however, its performance is improved by using different stiffness values calculated by nonlinear Hertz–Mindlin contact model for each aspect ratio. The simulated final deposit heights, runout distances, and energy dissipation values are in good agreement with experimental observations reported in the literature. The effects of initial porosity and rotational resistance on the final deposit profile and energy dissipation at different aspect ratios are investigated through a parametric study. For different aspect ratios, a higher rotational resistance leads to higher final deposit height, shorter runout distance, and less energy dissipation. A lower value of initial porosity leads to higher final deposit height; however, the runout distance and evolution of normalized potential, kinetic, and dissipated energies versus time are insensitive to the initial porosity for the granular columns investigated. DOI: 10.1061/(ASCE)GM.1943-5622.0000467. © 2015 American Society of Civil Engineers. Author keywords: Collapse; Discrete element simulation; Energy dissipation; Granular flow; Porosity; Rotational resistance.

Introduction Flowlike landslides such as debris flows, debris avalanches, and rock avalanches (Cruden and Varnes 1996; Hungr and Evans 2004) typically involve rapid flow of solid grains and intergranular fluid (e.g., water and air) across irregular terrain. For the risk assessment of these landslides, the prediction of landslide runout is an essential element. To better understand the underlying physics, it is often necessary to initially study similar but simpler problems involving flow of dry granular materials where intergranular fluid and cohesion play negligible mechanical roles (Denlinger and Iverson 2004). For example, scaling laws on runout distance based on experiments of dry granular flows have been utilized to gain insights into the runout of large landslides on Mars (Lajeunesse et al. 2006). An interesting case of granular flow is the collapse of initially vertical three-dimensional (3D) axisymmetric columns of various granular materials (e.g., grains of salt, sand, sugar, rice, and glass beads), which has been investigated in several experimental studies (e.g., Lube et al. 2004, 2005, 2011; Lajeunesse et al. 2004). The main focus of these experiments is to link deposition pattern (e.g., final height, final profile, and runout distance) to the physical 1 Graduate Research Assistant, Dept. of Civil and Environmental Engineering, The Pennsylvania State Univ., University Park, PA 16802. E-mail: [email protected] 2 Assistant Professor, Dept. of Civil and Environmental Engineering, The Pennsylvania State Univ., University Park, PA 16802 (corresponding author). E-mail: [email protected] 3 Professor, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu Univ. of Technology, Chengdu 610059, P.R. China. E-mail: [email protected] Note. This manuscript was submitted on March 8, 2014; approved on November 10, 2014; published online on April 10, 2015. Discussion period open until September 10, 2015; separate discussions must be submitted for individual papers. This paper is part of the International Journal of Geomechanics, © ASCE, ISSN 1532-3641/04015004(12)/$25.00.

© ASCE

properties of granular materials and experiment setup (e.g., coefficient of friction, aspect ratio). It was observed that the flow behavior of these granular columns primarily depends on the initial aspect ratio a = hi / r i , where hi and r i are the initial height and radius of the granular column, respectively. Lube et al. (2004) observed three distinctive flow regimes. For the first regime, when 0 < a < 0:74, a circular undisturbed area at the top of the column remains at the initial height; when 0:74 < a < 1:7, the inner circular region is eroded gradually by outer moving particles, leaving a sharp cone at the center with its tip remaining at the initial height. For the second regime, corresponding to a > 1:7, the entire upper surface of the column starts to flow and the height of column decreases immediately; after some decrease in height, the horizontal upper surface erodes to form a dome. For the third regime, corresponding to a > 10, deformation of the upper surface starts when it reaches the flat flow front. Based on a regression analysis of their results, Lube et al. (2004) concluded that final runout distance, r f , can be expressed solely in terms of a and r i , suggesting that it is independent of many constitutive parameters, including friction angle, grain size, and underlying surface conditions; the particle friction has a negligible effect on the flow pattern except in the last instant when the flow comes to an abrupt stop. In a series of similar experiments based on dry glass beads and different underlying surface conditions in terms of roughness and erodibility, however, Lajeunesse et al. (2004) observed only two flow regimes and three deposit morphologies. For the first flow regime, when 0 < a < 0:74, the central region remains undisturbed at the original height, resulting in a truncated cone shape; when 0:74 < a < 3, the central region is eroded, leading to a perfect cone shape with a distinct tip at a reduced height. For the second regime, corresponding to a > 3, the whole top surface of the column begins to fall as a flow front forms at the bottom and propagates outward and the deposit shape forms a sombrero shape. According to the results, grain size and surface properties have no effect on deposit morphologies for small a values but the effect becomes significant as a increases. Comparing the observations

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from Lube et al. (2004) and Lajeunesse et al. (2004), it is noted that the former reported insignificant effect of the subsurface roughness and grain size on final deposit shape and runout distance for all aspect ratios, whereas the latter stated that the grain size and underlying surface properties play a significant role in the final deposit shape and height for high aspect ratios. More specifically, for 0:74 < a < 1:7, Lube et al. (2004) observed that the final height remains at the initial height; whereas Lajeunesse et al. (2004) observed that the height is reduced and is a function of the initial radius. Two-dimensional (2D) and 3D continuum-scale numerical simulations have been conducted to reproduce the experimental observations of Lube et al. (2004) and Lajeunesse et al. (2004) and to better understand the behavior of granular flows (e.g., Lagree et al. 2011; Chen and Qiu 2012; Mast et al. 2014; Zhang et al. 2014). Lagree et al. (2011) implemented rheological models into the Navier-Stokes equations to model the granular flow in two dimension, effectively treating the granular materials as a complex fluid. Chen and Qiu (2012) implemented an elasto plastic constitutive model into the smoothed particle hydrodynamics (SPH) formulations (Lucy 1977; Gingold and Monaghan 1977) to model the granular flow in 3D, treating the granular materials as elasto plastic solids. Chen and Qiu (2012) were able to reproduce the three flow regimes and the relationship between r f and a as observed by Lube et al. (2004). This suggests that a continuumscale method can be used to model many global behaviors of dense granular flow, which is consistent with the findings of many researchers (e.g., Drake 1991; Denlinger and Iverson 2004; Bui et al. 2008; Hungr 2008). However, some features of the deposit morphology such as sharp tips and curvature of the deformed profile could not be captured in continuum simulations. To capture these features, discrete and micromechanical behaviors at the particle level need to be adequately accounted for. Simulations based on the discrete element method (DEM; Cundall and Strack 1979) have been conducted (e.g., Zenit 2005; Staron and Hinch 2005, 2007; Cleary and Frank 2006; Lo et al. 2009; Lacaze and Kerswell 2009; Tapia-McClung and Zenit 2012; Girolami et al. 2012; Zhao et al. 2012; Huang et al. 2013; Kermani et al. 2014). Generally, these simulation results are in good qualitative agreement with 2D experiments (e.g., Lube et al. 2005; Balmforth and Kerswell 2005) and 3D experiments (e.g., Lube et al. 2004; Lajeunesse et al. 2004). To model rolling resistance in 3D simulations, Lo et al. (2009) utilized spherical particles attached to the underlying surface and Huang et al. (2013) utilized rolling resistance models by applying resistance torque at the contacts to capture the effects of nonspherical particles. Cleary and Frank (2006) conducted a 3D parametric study on the effect of different parameters, such as the static and dynamic friction coefficient, restitution coefficient, particle stiffness, rolling friction, and spin modification on the final height and runout distance. To explain the discrepancies between their numerical simulations and experimental observations, Cleary and Frank (2006) suggested that potential effects of particle shape and microliquid bridge cohesion should be considered. This study presents a 3D numerical study to simulate the collapse of granular columns using a commercial and research DEM code, PFC3D 4.0 (Particle Flow Code in three dimensions, Itasca 2008). In the DEM model, rotational velocities of particles are reduced by a factor at every time step to partially account for the additional rolling resistance due to the effect of particle shape and hysteretic contact behavior. The model utilizes a hybrid approach, where the linear contact model is used for parametric study but the contact stiffness values are calibrated based on the Hertz–Mindlin model for each aspect ratio. The goal of this study © ASCE

is to investigate the effects of particle rotational resistance and initial porosity on final deposit height, runout distance, and energy dissipation during collapse. These effects have not been adequately considered in previous numerical studies. Details of the DEM model will be presented, followed by comparisons among numerical results and experimental observations from Lube et al. (2004) and Lajeunesse et al. (2004) to demonstrate the importance of particle rotational resistance and initial porosity on final deposit profile and evolution of energy dissipation at different aspect ratios.

DEM Model The DEM introduced by Cundall and Strack (1979) has been widely used to simulate the behavior of granular media. A review of DEM and its applications has been published by Zhu et al. (2007, 2008) and O’Sullivan (2011). Typical applications of DEM include simulation of direct shear test (e.g., Cui and O’Sullivan 2006; Zhang and Thornton 2007), biaxial compression (e.g., Rock et al. 2008; Frost and Evans 2009), triaxial compression test (e.g., O’Sullivan et al. 2004; Cui et al. 2007), soil crust formation (e.g., Sjoblom 2014), railroad ballast (e.g., Huang and Tutumluer 2014), flow on inclined plane or pile (Urabe 2005; Brewster et al. 2005), flow in hoppers in 2D (Fraige and Langston 2004; Parisi et al. 2004) and in 3D (Ketterhagen et al. 2008; Datta et al. 2008), flow in mixers (Lemieux et al. 2008; Chaudhuri et al. 2006), and flow in drums and mills (e.g., Morrison et al. 2007; Portillo et al. 2007). Packings such as sand piles (Zhou et al. 1999, 2002) and collapse of granular columns in 2D (e.g., Staron and Hinch 2005; Zenit 2005) and in 3D (e.g., Lo et al. 2009; Huang et al. 2013) were also modeled using DEM. A DEM model is made up of a finite number of particles, often spherical balls, and a set of boundary elements known as walls. DEM in essence is a Lagrangian method in which particles are treated as separate entities and their motions are calculated by numerical integration of Newton's equations of motion. Interparticle forces are calculated using contact models based on contact properties (e.g., stiffness, friction, and damping) and overlap between particles. Detailed descriptions of DEM can be found in Itasca (2008) and O’Sullivan (2011) and hence are not presented herein. Numerical Sample Preparation DEM numerical samples utilized in this study were in the form of vertical columns with different aspect ratios and porosity values and were created by letting balls fall freely into a cylinder, which resembles pouring granular materials into cylinders as described by Lube et al. (2004) and Lajeunesse et al. (2004). Different initial porosities (i.e., initial density) were achieved by adjusting the inter particle friction coefficient to facilitate movement and rearrangement of particles during the sample preparation phase (Suiker and Fleck 2004; Huang et al. 2013) and subsequently by slightly compacting the column through a small downward movement of a temporary surface on top of the particles and allowing the formation of a new equilibrium state. After settlement and compaction, the granular columns were trimmed to desired heights, producing desired initial aspect ratios. In this study, numerical samples with combinations of a = 0:55, 1.0, and 2.0 and initial porosity n = 0:45, 0.42, and 0.40 were created. These porosity values correspond to relative density values of ∼15%, 33%, and 44%, respectively; based on the consistency descriptions of granular materials provided by Lambe and Whitman (1969), these samples can be described as very loose, loose, and medium.

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Numerical simulations on these samples were conducted to investigate effects of particle rotational resistance and initial porosity on granular collapse for different aspect ratios. Fig. 1 shows the initial setup of a granular column for a = 1:0. The balls have a uniform diameter of 0.32 mm, which is the mean size of sand particles in Lube et al. (2004) and glass beads in Lajeunesse et al. (2004). The simulated columns have an initial radius r i = 5:68 mm, which is about 36 times of the particle radius, and consist of 10,200 to 40,100 particles. To initiate collapse, the cylindrical wall was lifted at a constant velocity of 0:02 m / s, which is consistent with the experiments by Lube et al. (2004) and Lajeunesse et al. (2004).

Material Properties Table 1 presents the material properties used in this study. Balls are assumed to be made of quartz with shear modulus G = 2:9E10 N/ m2 , Poisson’s ratio ν = 0:3, density ρs = 2650 kg / m3 , and friction coefficient μ = 0:445 (friction for quartz–quartz contact). According to Staron and Hinch (2007), the coefficient of restitution e, which can be defined as the ratio of the relative velocity between two contacting particles after and before the collision, significantly changes the system behavior when it gets close to 1, and this effect is more pronounced for large aspect ratios; however, for e less than 0.8, this effect becomes negligible. The amount of energy dissipation during collision increases as e decreases and e = 1 corresponds to elastic collision where no energy dissipation occurs. For the granular materials tested in Lube et al. (2004) and Lajeunesse et al. (2004), the e values are considered to be much less than 1. For example, the e value for glass beads is ∼0:75 (Zenit 2005), which is consistent with the results from several simple experiments conducted in this study involving measuring the rebound height of a glass bead falling onto various glass surfaces. Therefore, a single value of e = 0:75 is utilized in this study. The viscous damping coefficient, ξ, at contact is calculated as (Itasca 2008; O’Sullivan 2011) − ln(e) ξ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln2 (e) + π 2

(1)

z

Contact Stiffness Different contact models such as the linear model and Hertz– Mindlin model are readily available in PFC3D to model different interparticle contact behaviors. The linear model is based on constant normal and tangential stiffness; it is simple and computationally efficient, but it does not consider nonlinear contact behaviors. On the other hand, the Hertz–Mindlin model is a nonlinear model in which the contact stiffness depends on the elastic properties of particles, overlap, and normal contact force. In PFC3D, the Hertz–Mindlin model is based on an approximation of the theory of Mindlin and Deresiewicz (1953), which has been widely used in geotechnical engineering and geophysics in modeling small-strain nonlinear behaviors of granular assemblages (e.g., Dobry et al. 1982; Dvorkin and Nur 1996); however, it requires more computational time than does the linear model when applying to an assemblage with a large number of particles. In this study, a hybrid approach similar to the work of Teufelsbauer et al. (2009) is utilized, where the linear model is used for parametric study but the contact stiffness values are calibrated based on the Hertz–Mindlin model for each aspect ratio. This approach is discussed in detail in this section. Based on Mindlin and Deresiewicz (1953) and Itasca (2008), particle overlap u is a function of normal force F n (see Fig. 2) and normal secant contact stiffness kn can be expressed as pffiffiffiffiffiffi F n 2G × 2R 1 / 2 kn = = u (2) u 3 × (1 − ν) where R = ball radius. Fig. 3 presents the relationship between shear force F s and shear displacement δ according to the theory of Mindlin and Deresiewicz (1953). As δ increases, shear contact stiffness decreases until sliding occurs, as shown in Fig. 3. During granular collapse, shear displacement varies among contacts and rapidly changes within particle deposits. For example, within the central undisturbed region as observed by Lube et al. (2004), shear displacements at particle contacts are very small, whereas within the outer edge of the column, shear displacements are large and rapidly changing during collapse as particles slide over each other. It is believed that particle sliding is a more important mechanism than the pre sliding nonlinear behavior during granular collapse in capturing final runout distance. Therefore, shear secant contact stiffness k s corresponding to initiation of sliding (see

δ

Fn hi x

Fs

ri Fig. 1. Initial setup of a granular column (a = 1:0)

u

Table 1. Material Properties Used in DEM Simulation Parameters Particle diameter (D) Density (ρs ) Shear modulus (G) Poisson’s ratio (ν) Friction coefficient (μ) Viscous damping coefficient (ξ) © ASCE

Fs

Value 0.32 mm 2,650 kg / m3 2:9E10 N/ m2 0.3 0.445 0.0912

Fn Fig. 2. Overlap and shear displacement of two particles in contact 04015004-3

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Fig. 3) is taken as the equivalent shear contact stiffness for the linear model and is calculated as (Mindlin and Deresiewicz 1953; Itasca 2008) 1 / 3 4 3G2 (1−ν)R ks = Fn1 / 3 (3) 3(2 − ν) where F n can be calculated as


pffiffiffiffiffiffi 2G × 2R 3 / 2 Fn = u 3 × (1 − ν)

(4)

Fig. 3 indicates that use of ks results in an exact match between the Hertz–Mindlin model and the linear model in terms of shear displacement needed to initiate sliding; however, the proposed approach underestimates the energy dissipated during initiation of sliding as compared with the Hertz–Mindlin model, which is shown as the shaded area in Fig. 3. For the case of aspect ratio of 1, the underestimated energy dissipation is ∼16% of the energy dissipation needed to initiate sliding. It is expected that energy dissipation due to particle–particle sliding and collision is more significant than pre sliding energy dissipation; therefore, this simple treatment is unlikely to alter the overall energy dissipation behavior. Based on Eqs. (2)–(4), particle–particle contact normal and shear stiffness in the linear model can be determined from the Hertz–Mindlin model for a given u. During collapse, u varies among contacts and rapidly changes within deposits, resulting in variable stiffness values. For computational efficiency, however, one set of stiffness values needs to be assigned to the balls and wall during the entire collapse of each granular column. To represent an entire collapse process, the average overlap for ball–ball and ball–wall contacts during the initial, intermediate, and final stages of collapse are obtained by running PFC3D simulations using the Hertz–Mindlin model. The contact stiffness values

Fs

corresponding to the average overlap are calculated and utilized for the subsequent parametric study. In PFC3D, contact stiffness is calculated assuming that two contacting entities (e.g., ball–ball, ball–wall) act in series; therefore, the stiffness of each ball is twice the ball–ball contact stiffness (Itasca 2008). Table 2 presents the obtained average values of u, kn , and k s for granular columns with a = 0:55, 1.0, and 2.0. The stiffness of the bottom wall can be similarly determined based on the average ball–wall overlap, ball–wall contact stiffness, and ball stiffness and are also presented in Table 2. Table 2 indicates that u, k n , and ks increase as a increases. It is important to note that kn and ks values are not the same for different aspect ratios owing to the underlying physics; this treatment is different than previous parametric studies (e. g., Lo et al. 2009; Girolami et al. 2012) in which the same stiffness values are used for different aspect ratios. Rotational Resistance Rolling/rotational resistance is widely utilized in DEM simulations to account for energy loss due to effects of hysteretic contact behaviors (e.g., micro sliding, visco elasticity, and plasticity) and asperities at real contacts; it has also been used to partially account for the effect of nonspherical particle shape (Ai et al. 2011; Wensrich and Katterfeld 2012). Rotational resistance has been modeled by implementing resisting moments at particle contacts (e.g., Iwashita and Oda 1998, 2000; Zhou et al. 1999, 2002; Jiang et al. 2005; Li et al. 2005a, b; Liu et al. 2012; Girolami et al. 2012; Zhang et al. 2013; Huang et al. 2013) or directly reducing angular velocities of particles (e.g., Poisel and Preh 2008; Teufelsbauer et al. 2009). Owing to its efficiency and simplicity, the approach proposed by Teufelsbauer et al. (2009) is simplified and utilized in this study. In Teufelsbauer et al. (2009), angular velocity ω of a particle is reduced to account for rotational resistance as follows: ωit + 1 = Ap ωit

Initiation of Sliding

A=

Fs = μ × Fn

1 1 + cs

(5a) (5b)

where A = a retarding coefficient; p = a parameter related to the time scale needed to reduce ω; c = contact number; i = direction (i.e., i = x, y, z); s = a parameter that governs sensitivity of the retarding coefficient to contact number; and superscripts t and t + 1 = current and next time steps, respectively. Teufelsbauer et al. (2009) compared their DEM simulations and experiments of flow–obstacle interaction in rapid granular avalanches of sand. Based on their study, s is in the range of [0.5, 1] and p is ∼0:003. Rolling resistance is dependent on contact number, and this physics is preserved in Eqs. (5a and 5b); however, it is computationally inefficient to implement it in PFC3D. Granular column collapse is dominated by dense granular flow where contact numbers of the majority of the particles are within a small range. For PFC3D simulations with uniform spherical particles, Ap values are found to be within a small range with an average value of

ks 1

δ Fig. 3. F s versus δ based on Mindlin and Deresiewicz (1953)

Table 2. Equivalent kn and k s for Different a Balls a 0.55 1 2 © ASCE

Walls

u (m)

kn (N/m)

ks (N/m)

u (m)

kn (N/m)

ks (N/m)

2.22E − 10 3.07E − 10 5.00E − 10

14,500 17,000 22,000

12,000 14,000 18,000

2.00E − 10 2.70E − 10 4.10E − 10

13,000 15,000 18,000

11,000 12,500 15,000

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Results and Discussion The DEM model was utilized to simulate collapse of granular columns. The goal was to investigate the effect of particle rotational resistance and initial porosity on final deposit morphology (e.g., height and runout distance) and energy dissipation. During these simulations, deposit height was continuously monitored; the final profile was obtained when the deposit height became relatively constant. Fig. 4 shows deposit height versus time of a typical simulation, which demonstrates that final deposit height hf is reached after a sufficient amount of time tf . The value of t f is recorded for each simulation and is used for time normalization. Final runout distance r f is defined by position of the farthest particles from the center of the cylinder with a minimum of one contact with the main deposit, which is consistent with the selection of r f in Lube et al. (2004) and Lajeunesse et al. (2004). Fig. 5 shows the plan view of the deposit before and after removing outlier particles that have no contact with the main deposit. Fig. 6 shows hf and rf of a typical final deposit profile. Fig. 7 presents the initial column and final deposit profile for a = 0:55, 1.0, and 2.0 (n = 0:4, Ap = 0:95). Table 3 presents a quantitative comparison among DEM simulation results and experimental results by Lube et al. (2004) and Lajeunesse et al. (2004). For a = 0:55, the central region of the column remains undisturbed during collapse, and a truncated cone shape with a flat surface at the top is observed from the DEM simulation as shown

in Fig. 7(a), which is consistent with observations from Lube et al. (2004) and Lajeunesse et al. (2004). According to the experiments, the final deposit height should remain at the initial height (i.e., hf = hi ); however, DEM simulations yielded hf = 0:97hi . The slight decrease in final height may be due to a combination of several factors. First, the DEM model utilized equivalent contact stiffness values based on the average overlap in the entire column during various stages of collapse, which may have underpredicted the overall stiffness of the center region of the deposit that remained at the initial height, causing the final height to be slightly smaller in DEM simulations. Second, some micromechanical behaviors such as particle interlocking and force chain development in the experiments may have not been fully reproduced in the DEM simulation owing to differences in the number of particles, particle size distribution, and particle shape. Third, the initial porosity value (i.e., n = 0:4) in the DEM simulation is different than the experiments. The granular columns in Lajeunesse et al. (2004) have porosity values in the range of 0.35–0.38. Lube et al. (2004) observed that hf and r f are insensitive to how the cylinder was filled (i.e., porosity); however, hf is found to be sensitive to n in the DEM simulations, which will be discussed later. Regarding rf , Table 3 shows that the simulated r f is smaller than that observed by Lube et al. (2004) and Lajeunesse et al. (2004) by 5% and 2%, respectively. For a = 1:0, the central region is eroded toward the end of collapse, leading to a cone-shaped deposit with a distinct tip as shown in Fig. 7(b), which is consistent with the observations of Lube et al. (2004) and Lajeunesse et al. (2004) in a qualitative sense. Regarding hf , Lube et al. (2004) observed hf = hi , whereas Lajeunesse et al. (2004) observed hf < hi . Table 3 shows that the simulated results are generally consistent with the latter with a 7% difference in hf and an exact match in r f . For a = 2:0, a similar trend as a = 1:0 was observed and the final deposit has a cone-shaped profile, which is consistent with observations of the two experiments in a qualitative sense. Table 3 shows that the simulated hf lies between the observations of both experiments, and the simulated rf is smaller than that from Lube et al. (2004) and Lajeunesse et al. (2004) by 15% and 2%,

hi

Deposit Height


∼0:99 based on the values of p and s from Teufelsbauer et al. (2009). To improve computational efficiency, a constant value of Ap is utilized in this study. In order to evaluate the effect of rotational resistance on final deposit morphology and energy dissipation, three values of Ap , namely 0.90, 0.95, and 0.99, are considered in this parametric study. These values are intended to include varying degrees of rotational resistance arising from effects of particle shape, particle size distribution, and different test conditions among the experiments of Lube et al. (2004), Lajeunesse et al. (2004), and Teufelsbauer et al. (2009).

(a)

hf

(b)

rf

Fig. 5. Determination of r f : (a) plan view of untreated deposit; and (b) plan view of deposit after removing outlier particles

z hf tf

x rf

Time Fig. 4. Deposit height versus time of a typical simulation © ASCE

Fig. 6. Final deposit profile with hf and r f 04015004-5

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(a)

(c)

(b)

Fig. 7. Initial column and final deposit profile for various a: (a) a = 0:55; (b) a = 1:0; and (c) a = 2:0

Table 3. Comparison of Experimental and Numerical Results for a = 0:55, 1.0, and 2.0 (n = 0:4, Ap = 0:95) a DEM experimental results

Dimensions of final profile (mm)

Initial values DEM simulations

hi hf rf hf rf hf DEM / hf EXP r f DEM / rf EXP hf rf hf DEM / hf EXP r f DEM / rf EXP

Lube et al. (2004)

Lajeunesse et al. (2004)

0.55

1.0

2.0

3.10 5.71 11.25 3.00 3.90 4.68 9.02 11.47 15.70 3.10 5.71 5.60 9.52 12.76 18.47 0.97 0.68 0.84 0.95 0.90 0.85 3.10 4.20 4.20 9.22 11.47 16.10 0.97 0.93 1.11 0.98 1.00 0.98

respectively. Fig. 7 and Table 3 indicate that the DEM model can capture the main features of the granular collapses qualitatively and quantitatively. Table 4 presents a comparison of numerical results from the current study and relevant studies from the literature. Table 4 indicates that 2D simulations from Staron and Hinch (2005) using DEM and from Lagree et al. (2011) using a continuum method provided generally comparable results in final height as 3D simulations; however, they drastically overestimated final runout distances for all aspect ratios because of the lack of consideration of particle–particle interaction in the third dimension. Among 3D DEM simulations performed on the tests of Lube et al. (2004), Lo et al. (2009) provided slightly better estimate in final height and runout distance than the current study for a = 0:55 and 1.0; however, the current study provided better results for a = 2:0, particularly in final height. For the tests of Lajeunesse et al. (2004), the current study provided better results than Lo et al. (2009) for a = 1:0 and 2.0. It should be noted that Lo et al. (2009) utilized spherical particles attached to the underlying surface to model rolling resistance along the non smooth bottom boundary, whereas no special treatment of the bottom boundary was needed in the current study. Effect of Rotational Resistance on Deposit Morphology Tables 5 and 6 show hf and r f for a column with a = 1:0 and n = 0:4 for different Ap values and a comparison between © ASCE

DEM simulation results and experimental observations from Lube et al. (2004) and Lajeunesse et al. (2004). An Ap value of 1.0 corresponds to the case of no rotational resistance; as Ap values decrease, rotational resistance increases (see Eq. 5(a)). Table 5 indicates that an increase in rotational resistance results in an increase in hf and a decrease in r f . Table 6 shows that for Ap of 0.9 and 0.95, the simulated hf and r f are in good agreement with Lajeunesse et al. (2004) with a perfect match in r f for Ap = 0:95 and an excellent match in hf for Ap = 0:90. Fig. 8 shows deposit height versus normalized time, t / tf , for a column with a = 1:0 and n = 0:4 considering different rotational resistance values. Fig. 8 indicates that for the case of no rotational resistance (i.e., Ap = 1), the decrease in height occurs in two stages. Most of this decrease occurs abruptly over a short period at the beginning of collapse, followed by a prolonged smooth decrease in height. As the rotational resistance increases, these two stages become less distinctive, and decrease in height with time becomes smoother. Similar conclusions can be drawn from corresponding plots and tables for a = 0:55 and a = 2:0 and hence the results are not presented herein. Effect of Initial Porosity on Deposit Morphology Table 7 presents the effect of n on final deposit profile for columns with a = 1:0 and different Ap values. Fig. 9 presents the effect of n on deposit height versus time for a = 1:0 and Ap = 0:95. Table 8 presents the effect of n on final deposit profile for columns with different a values and Ap = 0:95. Tables 7 and 8 and Fig. 9 indicate that final deposit height is sensitive to porosity and a denser column results in a higher final deposit height for all aspect ratios. In a denser column, the center region is more stable owing to more interlocking between particles, which results in higher hf . However, Tables 7 and 8 suggest that final runout distance is insensitive to porosity for all aspect ratios. This trend can be explained by borrowing the concept of critical void ratio (Casagrande 1936) from soil mechanics in the continuum framework. Under shear deformation, dense granular materials are dilative (i.e., n increases) and loose granular materials are contractive (i.e., n decreases). Under the same confining stress, the same granular materials will reach the same critical void ratio (i.e., porosity) under large shear deformations regardless of the initial porosity. For a granular column, collapse is accompanied by the materials undergoing flow-type failure with large deformation. Once the critical void ratio in the material is mobilized after some displacement, which is small compared with the final runout distance, the flow

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Table 4. Comparison of Current Study and Numerical Results from Literature a Experimental tests

Numerical results

Lube et al. (2004)

Current study Staron and Hinch (2005) 2D Lagree et al. (2011) 2D

Lajeunesse et al. (2004)

Current study Staron and Hinch (2005) 2D Lagree et al. (2011) 2D Lo et al. (2009) 3D

0.55

1.0

2.0

hf / hf EXP rf / r f EXP hf / hf EXP rf / r f EXP hf / hf EXP rf / r f EXP hf / hf EXP rf / r f EXP hf / hf EXP rf / r f EXP hf / hf EXP rf / r f EXP hf / hf EXP rf / r f EXP hf / hf EXP rf / r f EXP

0.97 0.95 1.0 1.41 0.96 1.31 1.0 0.97 0.97 0.98 1.0 1.46 0.96 1.36 1.0 1.0

0.68 0.90 0.65 1.56 0.67 1.43 0.63 0.96 0.93 1.00 0.88 1.73 0.91 1.59 0.85 1.07

0.84 0.85 0.84 1.84 0.90 1.65 0.68 0.83 1.11 0.98 1.12 2.11 1.19 1.90 0.9 0.95

7

Table 5. Effect of Ap on Final Deposit Profile (a = 1:0, n = 0:4)

p

A 0.90 0.95 0.99 1.00

Ap in DEM simulations

6

Dimensions Lube et al. Lajeunesse of final profile (2004) et al. (2004) hf (mm) rf (mm)

5.71 12.76

4.20 11.47

1.0

0.99

0.95

0.90

Deposit Height (mm)


Lo et al. (2009) 3D


1.82 3.33 3.90 4.11 17.77 12.30 11.47 11.04

Table 6. Effect of Ap on DEM/Experimental Results for Final Deposit Profile (a = 1:0, n = 0:4) Ap in DEM simulations Experimental tests Lube et al. (2004) Lajeunesse et al. (2004)

DEM/experimental results

1.0

hf DEM / hf EXP r f DEM / r f EXP hf DEM / hf EXP

0.32 0.58 0.68 0.72 1.39 0.96 0.90 0.87 0.43 0.79 0.93 0.98

r f DEM / r f EXP

1.55 1.07 1.00 0.96

Energy Dissipation Potential and kinetic energies of DEM particles were monitored to investigate energy dissipation during collapse. Potential energy of each column, Ept , was calculated as N

(6)

i=1

where m = mass of particle; z = height of particle’s centroid; g = acceleration of gravity; N = total number of particles; subscript i = particle index; and superscript t = time. Kinetic energy of each column, Ekt , was calculated as N 1 N 1 Ekt = ∑ mi (vit )2 + ∑ I i (ωit )2 i=12 i=12

© ASCE

4 3

0.99 0.95 0.90

becomes insensitive to the initial porosity, and this leads to similar runout distance.

Ept = ∑ mi gzit

5

(7)

2 1

0

0.2

0.4

0.6

0.8

1

Normalized Time, t t f Fig. 8. Effect of Ap on deposit height versus normalized time (a = 1:0, n = 0:4)

where v = translational velocity of particle; and I = polar moment of inertia of particle. Strain energy stored in particles is neglected because it is small compared with the potential and kinetic energy (Zenit 2005). Initial energy of each column, Eo , is the potential energy of the vertical column at rest (i.e., t = 0). Once collapse is initiated, potential energy converts into kinetic energy while a portion of it dissipates owing to friction (i.e., slip work), damping, and rotational resistance. Knowing the potential and kinetic energy at any given time, energy dissipation, Edt , can be calculated as Edt = E o − Ept − Ekt

(8)

Values of potential, kinetic, and dissipated energy were traced and normalized by E o of the column for each simulation.

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Table 7. Effect of n and Ap on Final Deposit Profile (a = 1:0)

Table 8. Effect of n on Final Deposit Profile for Different a (Ap = 0:95)

Ap n 0.40 0.42


1.0

0.99

0.95

0.90

n

hf rf hf rf hf rf

1.82 17.77 1.70 18.00 1.67 17.90

3.33 12.30 3.18 12.32 3.00 12.20

3.90 11.47 3.58 11.40 3.45 11.44

4.11 11.04 3.90 11.10 3.61 11.00

0.40


0.55

1.0

2.0

hf rf hf rf hf rf

3.00 9.02 2.95 9.03 2.85 9.01

3.90 11.47 3.58 11.40 3.45 11.44

4.68 15.70 4.60 15.70 4.40 15.56

0.42 0.45

t

Table 9. Effect of Ap on Edf / Eo for Different n (a = 1)

6

Ap

n

t Edf

0.40

/ Eo

DEM simulations

0.42 Deposit Height (mm)


0.45

a

0.45

5

n

0.99

0.95

0.9

0.40 0.42 0.45

0.67 0.67 0.68

0.61 0.62 0.62

0.58 0.57 0.59

Eq. (9) from Lajeunesse et al. (2004)

0.63

t

Table 10. Effect of n and a on Edf / Eo (Ap = 0:95) a t Edf

4

/ Eo

DEM simulations

Eq. (9) from Lajeunesse et al. (2004)

3

0

0.2

0.4 0.6 0.8 Normalized Time, t t f

1

Fig. 9. Effect of n on deposit height versus normalized time (a = 1:0, Ap = 0:95)

Lajeunesse et al. (2004) investigated final normalized t dissipated energy, Edf , versus a and provided the following semi-empirical equations pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a t Edf / Eo = pffiffiffi 12 tan 2 θr − a2 , a < 0:74 (9a) 2 6 3 tan θr t

Edf / E o = 1 −

0:74 , a > 0:74 2a

(9b)

where θr = angle of repose and is considered to be 21° (Lajeunesse et al. 2004). Table 9 presents the effect of Ap and n on t t Edf / E o for a = 1:0. Table 9 shows that Edf / E o is dependent on Ap value but is relatively independent of n value. An increase in rotational resistance results in a decrease in energy dissipation and t consequently a decrease in Edf / E o . Table 9 indicates that, for tf p A = 0:95, the simulated Ed / Eo value has an excellent match with the value calculated using Eqs. (9a and 9b). Fig. 10 shows the effect of a and n on evolution of normalized potential, kinetic, and dissipated energies versus time for Ap = 0:95 within the first 0.2 s of collapse. Fig. 10 suggests that this evolution is insensitive to n, which is consistent with Table 9. Fig. 10 indicates that, for all cases, energy dissipation occurs in two stages. The first stage corresponds to a short period immediately following the collapse, where potential energy decreases and dissipated energy increases rapidly. This stage is responsible for © ASCE

n

0.55

1.0

2.0

0.40 0.42 0.45

0.39 0.40 0.42 0.43

0.61 0.62 0.62 0.63

0.76 0.77 0.77 0.81

most of the energy dissipation. The second stage corresponds to a prolonged slow decrease in potential energy and increase in dissipated energy until constant values are reached. Fig. 10 shows t that, as a increases, final dissipated energy (i.e., Edf / E o ) increases and more energy is dissipated in stage 1. Table 10 presents t a summary of Edf / E o values for Ap = 0:95 and all combinations of a and n along with those calculated using Eq. (9). Table 10 indicates that the DEM model can capture the final dissipated energy for the three aspect ratios considered. Fig. 10 also indicates that, for all cases, normalized kinetic energy (i.e., Ekt / E o ) has a rapid increase at the beginning of collapse followed by a decrease after reaching its maximum value, all within a short duration. The duration and the maximum value of Ekt / Eo increase as a increases, which suggests that more potential energy converts into kinetic energy at the beginning of collapse as the height of column increases. Zenit (2005) and Tapia-McClung and Zenit (2012) demonstrated the evolution of potential, kinetic, and dissipated energies for various aspect ratios of 2D columns prepared under self-weight induced compaction. Their results are consistent with Fig. 10 in both a qualitative and quantitative sense. In particular, Tapia-McClung and Zenit (2012) observed that particle shape (i.e., elongation ratio) has negligible effect on the evolution of energies. Although the effect of porosity was not explicitly investigated, 2D columns prepared under self-weight induced compaction are likely to have different initial porosities for particles of different shapes owing to different degrees of particle interlocking. Therefore, Tapia-McClung and Zenit (2012)’s results also imply that the evolution of energies is independent of initial porosity, which is consistent with Fig. 10. Fig. 11 shows the evolution of the ratio of frictional energy loss (Eft ) to total dissipated energy, Eft / Edt , within the first 0.2 s of collapse for a = 0:55, 1.0 and 2 with n = 0:4. Similar to

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n = 0 .40 0 .42 0 .45

n = 0 .40 0 .42 0 .45 E tp Ekt Edt

1

E tp Ekt Edt

Eo Eo Eo

1

Eo Eo Eo a = 1 .0

a = 0 .55 0.8 Normalized Energy

Normalized Energy

0.6

0.4

0.2

0 (a)

0.6

0.4

0.2

0

0.05

0.1 Time (s)

0.15

0

0.2

0

0.05

0.1 Time (s)

(b)

0.15

0.2

n = 0 .40 0 .42 0 .45 E tp Ekt Edt

1

Eo Eo Eo

a = 2.0 0.8 Normalized Energy


0.8

0.6

0.4

0.2

0

0

0.05

0.1 Time (s)

(c)

0.15

0.2

Fig. 10. Normalized potential, kinetic, and dissipated energy versus time within the first 0.2 s of collapse: (a) a = 0.55; (b) a = 1.0; and (c) a = 2.0

Fig. 10, the evolution of this ratio occurs in two stages. The first stage corresponds to a short period immediately following the collapse and, for all cases, Eft / Edt has a rapid increase at the beginning followed by a decrease after reaching its maximum value, all within a short duration. The second stage corresponds to a prolonged slow increase in Eft / Edt until constant values are reached. In this stage, Eft / Edt increases as a increases owing to © ASCE

more frictional work caused by longer runout distance. Fig. 11 indicates that frictional energy loss accounts for more than 50% of total energy loss during the first stage for all aspect ratios, and 40–50% of total energy loss during the second stage. The rest of energy dissipation is caused by viscous damping at particle contact, particle collision, and reduction in angular velocity due to rotational resistance.

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•

0.6 0.5 0.4 E tf


Edt

0.3 a

0.2 n = 0 .40

0.1

0 .55

Acknowledgments

1 .0

Support of this study is provided by the U.S. National Science Foundation under Grant No. CMMI-1131383, the Open Fund of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Chengdu University of Technology) under Grant No. SKLGP2012K001, and the Mid-Atlantic Universities Transportation Center (MAUTC). These supports are gratefully acknowledged. The authors are also very appreciative of Dr. Matthew Evans at Oregon State University for his help in speeding up the DEM simulations.

2 .0 0

Energy dissipation during collapse occurs in two stages for all aspect ratios. The first stage corresponds to a short period immediately following the collapse, where energy dissipation occurs rapidly. This stage is responsible for most of the energy dissipation. The second stage corresponds to a prolonged slow accumulation in energy dissipation until reaching a maximum value that increases with aspect ratio. Kinetic energy has a rapid increase at the beginning of collapse, followed by a decrease after reaching its maximum value, all within a short duration. The duration and maximum value of normalized kinetic energy increase as aspect ratio increases.

0

0.05

0.1 Time (s)

0.15

0.2

Fig. 11. Ratio of frictional energy loss to dissipated energy versus time within the first 0.2 s of collapse for various a (n = 0:4)

Conclusions A 3D numerical study to simulate the axisymmetric collapse of granular columns has been conducted using a commercial and research DEM code. The simulated granular columns have combinations of aspect ratios: 0.55, 1.0 and 2.0, and initial porosity values: 0.40, 0.42, and 0.45. The DEM model utilizes a rotational resistance factor to reduce angular velocity of uniform spherical particles to partially account for rotational resistance arising from the effect of particle shape, particle size distribution, inter locking, and hysteretic contact behaviors. The goal of this study is to investigate the effect of rolling resistance and initial porosity on final deposit morphologies (e.g., height and runout distance) and energy dissipation. As a result of this investigation, the following conclusions are reached: • Simulated final deposit heights and runout distances for different aspect ratios from the DEM model with a rotational resistance factor of 0.95 are in good agreement with the experimental observations from Lube et al. (2004) and Lajeunesse et al. (2004). Simulated energy dissipation for a rotational resistance factor of 0.95 also shows excellent agreement with semi-empirical equations from Lajeunesse et al. (2004). Therefore, the DEM model can capture the main features of granular collapses qualitatively and quantitatively compared with available experimental observations. This finding suggests that a simple treatment of rolling resistance by reducing the rotational velocities of particles by a factor at each time step is adequate in modeling the final deposit morphology after the collapse of granular columns. • Parametric study shows that higher rotational resistance leads to higher final deposit height, shorter runout distance, and less energy dissipation. Parametric study also indicates that final deposit height is sensitive to initial porosity; however, final runout distance and evolution of normalized potential, kinetic, and dissipated energies versus time are insensitive to initial porosity. Lower initial porosity leads to higher final deposit height for granular columns with different aspect ratios, which can be attributed to the increase in stability at the central region of denser columns. © ASCE

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