Evaluating the Influence of Particle Shape on Liquefaction Behavior Using Discrete Element Modeling

Transcription

1 Evaluating the Influence of Particle Shape on Liquefaction Behavior Using Discrete Element Modeling Alaa K. Ashmawy 1, Beena Sukumaran 2, and V. Vinh Hoang 1 1 Department of Civil and Environmental Engineering University of South Florida Tampa, Florida, USA 2 Department of Civil and Environmental Engineering Rowan University Glassboro, New Jersey, USA ABSTRACT New developments in liquefaction modeling of angular soils using the Discrete Element Method (DEM) are presented. A technique is introduced to simulate the two-dimensional projection of angular grains by clumping a series of overlapping discrete circular elements of equivalent properties. Representative particles from different types of sands are scanned using a digital image analyzer, and the corresponding particle outlines are fitted accordingly for the DEM simulation. Particle assemblies of varying degrees of angularity are then subjected to simulated undrained cyclic shear conditions to assess their liquefaction susceptibility. The influence of grain morphology on the cyclic response is evaluated. KEY WORDS: Liquefaction; grain morphology; particle shape; angularity; discrete element method; numerical modeling. INTRODUCTION Extensive damage caused by the flow of saturated deposits of loose sands has generated considerable interest, over the years, in understanding the behavior of these soils under undrained conditions. When subjected to undrained cyclic shear, loose sand deposits experience a gradual build up of pore water pressure, which leads to significant strength loss as a result of the decrease in effective stresses. Inherent factors affecting the flow behavior of sands include gain size, size distribution, shape, angularity, and surface roughness. Extrinsic factors include void ratio, fabric, initial effective stresses, and stress path (Sukumaran, 1996). Since the pioneering work of Seed and Lee (1966), the cyclic stress conditions causing liquefaction of sand have been thoroughly investigated using a variety of laboratory methods, such as cyclic triaxial, cyclic simple shear, and cyclic torsional shear. For loose sand specimens subjected to high cyclic shear stress amplitudes, the pore pressure increases over a relatively small number of cycles, resulting in a significant reduction in effective stress. The corresponding deformations increase in amplitude with loading cycles, and reach considerably large values (Seed and Lee, 1966; Vaid and Chern, 1985). In contrast, dense specimens are generally dilative, even at high levels of confining pressure. Build up of pore water pressure in this case results from alternate cycles of dilation upon shearing, and contraction upon stress reversal. The increase in pore water pressure is gradual, and liquefaction occurs at a relatively large number of cycles. Such a condition is defined as limited liquefaction or cyclic mobility since the corresponding strain amplitude remains limited, even near failure. The dependence of liquefaction susceptibility of sands and gravels on particle shape and angularity has been documented in the literature (Garga and McKay, 1984; Vaid et al, 1985; Hird and Hassona, 1990; Ashour and Norris, 1999). However, quantification of particle morphology and its direct influence on liquefaction susceptibility has been hindered by theoretical and practical limitations such as the lack of quantitative measures of particle shape, the experimental difficulties in defining grain morphology, and the inability to numerically model irregular particle geometries. Sukumaran and Ashmawy (2001) introduced a new image analysis-based method for characterizing particle shape and angularity. Particles sampled from nine natural and processed sands were scanned and digitized, and a new algorithm was presented to differentiate, quantitatively, between the shapes of various grains. It was envisioned that the digitized particle outlines as well as the new shape and angularity factors can be incorporated within a discrete element model framework to capture the influence of particle morphology on the mechanical response of granular assemblies. Discrete element modeling encompasses a vast array of methods to simulate the mechanical interaction between discontinuous bodies, both rigid and deformable. Different formulations have been proposed for a wide range of engineering applications (e.g., Cundall and Strack, 1979; Shi and Goodman, 1989; Mustoe, 1992), with variations in terms of numeric solution schemes, particle deformability and rotation, and contact detection. The Distinct Element Method was introduced by Cundall and Strack (1979) to model granular assemblies within the context of geotechnical engineering. The method relies on a timestepping explicit scheme, where equations of motion are solved over a finite time interval to obtain incremental displacements and rotations. Force-displacement laws are then invoked to calculate the resulting interparticle forces, which are fed back into the equations of motion over the next time increment (Itasca, 1999). The solution proceeds until a static or dynamic equilibrium threshold is reached. Paper No PCW-05 Ashmawy et al. 1

2 Until recently, limitations in computer speed and power represented a major obstacle to the widespread implementation of DEM in practice. In recent years, with the advent of high-speed desktop computers, DEM has gained considerable momentum as a powerful tool in studying microstructural phenomena in granular assemblies. The first comprehensive DEM software package, PFC 2D, was introduced in 1995 by Cundall and his co-workers, to model two-dimensional circular elements (disks or spheres) with linear or Hertz-Mindlin contact models (Itasca, 1999). A three-dimensional version of the software was later released. Despite its limitations, 2D modeling of discrete systems remains substantially more practical than 3D in terms of image capture, particle and fabric reconstruction, and computational effort. In this paper, a new method for incorporating non-circular particle geometries in DEM analysis is introduced. Digitized particle outlines of various natural and processed sands (Sukumaran and Ashmawy, 2001) are converted into software-compatible input to generate random assemblies of angular particles. The new method is used to study the influence of particle shape on liquefaction susceptibility of sands. DEM MODELING OF ANGULAR PARTICLES A broad range of approaches have been proposed in earlier research studies to address the issue of non-circular particle modeling in 2D, with varying degrees of success. One class of methods utilizes mathematical functions to describe non-circular outlines, such as ellipses (e.g., Ting et al, 1993; Ng, 1994), super-quadratics (e.g., Williams and Pentland, 1992; Cleary, 2000), and continuous circular segments (e.g., Potapov and Campbell, 1998). While computationally efficient compared to other methods, these functions do not allow the simulation of randomly shaped particles. Another approach relies on approximating the particle shape using polygons (Barbosa and Ghaboussi, 1992; Matuttis et al, 2000), which provides more realistic representations of natural particle shapes, but is computationally intensive if highly irregular shapes are simulated. A third approach is to combine several circular outlines into a cluster to form more complex shapes. To accomplish this, Jensen et al (1999) imposed unlimited tensile and shear strength conditions and high contact stiffness between a set of bonded circles. Thomas and Bray (1999) presented a different scheme where kinematic restrictions are imposed on disks within each cluster to prevent relative rotation and translation. In both methods, the number of disks within a cluster was limited to three or four to decrease computation time. Only non-overlapping elements were used within each cluster to avoid high repulsion forces, or to simplify the kinematic computations. Consequently, the simulated geometries did not resemble those of actual sand particles. In order to model angular particles of arbitrary shapes in the current study, a set of subroutines were introduced into PFC 2D using a software-specific programming language, Fish. Like most other DEM codes, the default discrete elements within PFC 2D are limited in terms of shape to circular disks or balls. However, the algorithm developed in this study makes use of the clump logic, a built-in function that allows temporary or permanent rigid bonding between several disk elements, without detecting contacts or calculating contact forces between elements belonging to the same clump. Therefore, this approach is very efficient from a computational standpoint. The new method relies on clumping a number of overlapping disk elements to best represent the non-uniform shape of natural particles. First, two-dimensional outlines of a series of particles are obtained using a digital microscope or scanner. For each particle, overlapping circles are then inscribed within the outline to capture the shape, as illustrated in Fig. 1. The number of overlapping circles, which represent disk elements, depends on the degree of non-uniformity in Figure 1. Disk elements inscribed within a particle outline to capture the shape. particle shape and angularity, the desired level of geometric accuracy, and the required computation time limit. Typically, ten to fifteen disks will adequately capture the shape of a particle. Because each particle contains a number of overlapping disk elements, the density of each disk must be scaled to ensure that the mass of the particle remains proportional to the area, regardless of the number of disks or level of overlap. In the present study, an approximate method was adopted whereby the density of the overlapping disk elements forming the particle is scaled as follows: Ap ρ d = ρ p (1) ΣAd where ρ d is the density of the disk elements, A p is the area of the particle, ΣA d is the sum of the areas of the disk elements, and ρ p is the density of the soil particle. While this method ensures that the mass of the particle is proportional to the area, it neither guarantees equivalency in terms of rotational inertia, nor in terms of center of mass. In situations where dynamic moments and rotational accelerations are significant, modifications need to be made by either adjusting the relative mass of the disk elements, or superimposing inertia balancing elements at specific points. To simulate an assembly of angular grains, particles are first generated as discrete circular elements within the desired range of grain sizes. This operation is performed using the default element generator in the software. Then, by invoking the shape conversion algorithm described above, each circular particle is transformed into an angular equivalent through replacement with a corresponding set of disk elements, selected randomly from a library of available particle shapes. The algorithm is formulated such that the resulting angular particle is equal in area to the original particle it is replacing, thus preserving the original particle size distribution. A random initial rotation between 0 and 360º is applied to each transformed particle to ensure uniformity of the particle orientations within the assembly. Alternatively, anisotropic assemblies can be produced by placing the particles at preferred orientations. The new algorithm was used to generate Fig. 2, which shows a random assembly of circular particles, together with the transformed irregularly shaped equivalents. In order for the system to remain stable throughout the solution, it is necessary that the computational time step, t, remains below a critical value of 2/ω max, where ω max is the highest eigen frequency of the system. Because ω max changes every time the particle positions change, the eigen frequencies must be recalculated every time step by solving the global stiffness matrix, which is highly inefficient. Instead, an approximate critical time step, t c, is calculated from: c { m k } t = min / (2) i i Paper No PCW-05 Ashmawy et al. 2

3 identical spheres is 0.35, which corresponds to a Hexagonal Close Packed (HCP) or Face-Centered Cubic (FCC) structure. The minimum void ratio for the equivalent 2D packing is Nevertheless, 2D models offer useful insight into micromechanical phenomena that would otherwise be difficult to model in 3D because of computational and practical obstacles. MATERIALS Figure 2. Random assembly of eight circular particles (left) and the transformed equivalent particles (right). where m i and k i are the mass and equivalent stiffness of disk element i, respectively (Itasca, 1999). To ensure stability, the actual time step during computation is taken to be 0.8 t c. Because each angular particle consists of a clump of smaller disk elements with scaled (smaller) densities, the critical time step calculated from Eq. 2 is highly over-conservative, and the corresponding computation time is unnecessarily long. To rectify this problem, the critical time step is adjusted to correspond to the mass and stiffness of the clump, not the individual disk elements. The efficiency of the solution is thus preserved without compromising the numerical stability of the system. While it is possible to use the Hertz-Mindlin contact model, its applicability is strictly limited to spherical particles. Therefore, the linear-stiffness contact model was used in this analysis, but the development of more advanced non-linear contact models is warranted. The particle digitization and generation process has been automated through a set of software macro functions. First, the digitized particle outline is fitted with a number of inscribed circles using CAD software. This is the only aspect in the process that requires real-time user input. The central coordinates and diameter of each circle are then extracted and exported into a spreadsheet, where they are resized to a unit area. The output is saved in a format compatible with the DEM software, and the data is archived into the particle library. It is important to recognize the limitations of two-dimensional DEM models in terms of soil fabric and particle arrangement. Because particles in 2D are representative of tabulated cylinders or disks (Fig. 3), rotation can occur only around the out-of-plane axis. Contacts between the particles are established along out-of-plane lines over the full thickness of the disk, as opposed to single contact points in 3D. In addition, because of the geometry of the particles, the range of possible void ratios for a 2D assembly is significantly smaller than that of the equivalent 3D assembly. For instance, the theoretical maximum void ratio for uniform-size spheres is 0.91, which corresponds to a Simple Cubic (SC) lattice structure, compared to 0.27 for the corresponding 2D assembly. Similarly, the minimum void ratio for an assembly of Figure 3. Typical shapes of tabulated cylinders (thin disks) for 2D modeling of granular particles. For the purpose of this study, five different types of natural and processed sands were considered, in addition to perfectly rounded glass beads. Digitized particle outlines for the materials were obtained from Sukumaran and Ashmawy (2001). Table 1 summarizes the relevant properties of the various materials. It is evident from the mean grain size, D 50, and the uniformity coefficient, C u, that all materials are highly uniform (poorly graded) and fall within the typical particle size range of liquefiable sands. The parameters SF and AF represent, respectively, the average shape and angularity factors for each material. These parameters are basically indicative of the level of non-uniformity in particle shape and sharpness of the corners (Sukumaran and Ashmawy, 2001). Table 1. Relevant properties of granular materials. Material D 50 (mm) C u SF% AF% Rounded Glass Beads Michigan Dune Sand Syncrude Tailings Sand Daytona Beach Sand Ottawa Angular Sand Ottawa Rounded Sand Since the current study is aimed at comparing the liquefaction response of otherwise similar materials of different particle shapes, the grain size distribution and particle density were kept constant in all simulations, with a mean grain size, D 50, of 0.4 mm and a uniformity coefficient, C u, of 1.5. The only difference among the various simulations was in particle shapes, which were randomly selected from the corresponding shape library for each material. The particle-to-particle friction coefficient, µ, was taken to be equal to 0.25, corresponding to a true friction angle, δ µ, of 14º. It is noted that δ µ is fundamentally different from the angle of internal friction of the granular assembly, φ, which is a global shearing resistance parameter for the assembly. While φ is affected by a number of factors including void ratio of the assembly, particle shape and particle arrangement (fabric), the value of δ µ is only a function of soil mineral and surface moisture conditions. Values of δ µ for quartz, compiled by Mitchell (1993) from various sources, range from 6º to 31º. The value of 14º, adopted in the present study, is approximately equal to the average over the full range of δ µ values reported by Horn and Deere (1962). LIQUEFACTION MODELING Typical loading conditions for small scale liquefaction experiments are illustrated in Fig. 4. Only the stress increments, which alternate between positive and negative values, are shown in the Figure. The cyclic triaxial test is by far the most common laboratory method used in practice to evaluate the liquefaction response of soils. After the sample is initially subjected to an all-around consolidation pressure (not shown in the Figure), cyclic loading is imposed by alternating the vertical stress increment, σ v, in a periodic fashion within a positive/negative range of values. The test is simple to perform, but fails to replicate typical loading conditions stemming from earthquake loading in the Paper No PCW-05 Ashmawy et al. 3

4 σ v T field. In particular, variations occur in the mean total stress within each cycle as a result of the successive increase and decrease in vertical stresses. Consequently, the gradual buildup of pore water pressure is attributed not only to changes in shear stresses, but also to variations in mean total stress. Similarly, results from cyclic direct shear tests (Fig. 4-b) are considerably biased due to boundary conditions and a stress path that are non-representative of field conditions. Cyclic simple shear tests (Fig. 4-c) are more difficult to perform, but have the advantage of best replicating earthquake loading conditions. In addition, the application of the cyclic shear stresses, τ, causes no changes in mean total stress. The gradual buildup of pore water pressure is, therefore, attributed only to the shear component of loading. An identical stress path can be achieved in a cyclic triaxial configuration by simultaneously varying the vertical and horizontal cyclic stresses in equal magnitudes and opposite directions (Fig. 4-d). The difference between simple shear and pure shear loading is only in the direction of principal stresses and the stress rotations during loading. In the present study, liquefaction simulations were carried out by applying cyclic pure shear loading conditions, similar to those shown in Fig. 4-d, over a representative volume of grains. The size of the sample was selected so as to minimize boundary effects while avoiding excessive computation time. Typical geotechnical testing standards (e.g., ASTM) stipulate that, in order to minimize boundary effects, the width of the sample must be equal to, at least, six to ten times the diameter of the largest particle. Since the maximum particle diameter in all simulations was 0.5 mm or less, the size of the 2D representative volume (sample size) was taken to be 5 5 mm. Even though it would be inconceivable from a practical standpoint to prepare and test such a small sample, the numerical results are still valid to the extent of the validity of the specified ASTM criteria. Two different methods of sample preparation were considered: dry pluviation and static compaction. A detailed description of the simulation steps is presented next. τ σ h σ v (a) (b) (c) (d) Figure 4. Typical loading conditions for small-scale liquefaction experiments: (a) cyclic triaxial; (b) cyclic direct shear; (c) cyclic simple shear; and (d) cyclic pure shear. comparisons could be made regarding the influence of grain shape on the results, as all other parameters were kept constant. The dry pluviation simulation sequence is shown in Fig. 5. Particles were first suspended in space with large enough separation gaps to prevent contact forces from developing, and to allow subsequent free fall under gravity. Next, the particles were subjected to gravitational acceleration and allowed to settle under their own weights. The uneven top surface of the sample was leveled by cropping excess particles above the desired height of the sample. While the depositional process is identical, differences in particle shapes resulted in a different soil fabric and void ratio for each material. However, since no additional forces were imposed on the sample beyond the self-weight of the grains, the relative density is close to zero in all cases. Therefore, dry pluviation represents a soil assembly formed by a particular sedimentation process, regardless of soil fabric. The samples were then subjected to an isotropic confining pressure of 40 kpa, with no restrictions on volume change. The load was applied gradually through servo-controlled movement of perfectly-smooth boundary plates. Because stresses are inherently undefined in discrete element models, constant stress boundaries are difficult to implement. Therefore, stresses were defined in this study as the total forces acting on the boundary plate, divided by the width of the plate. The stiffness of the plates was taken to be one order of magnitude smaller than the particle-to-particle contact stiffness to better simulate a constant stress boundary. The process of sample consolidation under 40 kpa isotropic confinement replicates in-lab saturation and initial consolidation in preparation for cyclic shearing and liquefaction. As a result of isotropic loading, the samples contracted, with the vertical and horizontal strains being almost equal. The volumetric strain at the end of consolidation was calculated for each sample, and the void ratio was computed accordingly. Sample Preparation Static Compaction The second sample preparation method is intended to reproduce samples with the same size distribution and void ratio and a comparable grain arrangement, irrespective of deposition process. To this end, a random assembly of circular particles was generated within the desired grain size distribution range, with an initial void ratio of 0.3. The model boundary plates were fixed to prevent any changes in volume, and the system was brought to equilibrium, eventually giving the soil Sample Preparation Dry Pluviation Dry pluviation, also known as dry screening, is one of the methods specified by ASTM for preparing cyclic triaxial samples. In the present study, a random assembly of particles was generated within the grain size distribution range specified earlier in the paper (see Materials section). To avoid any variations in results due to minor differences in grain size distribution, a single random assembly was initially generated for the Rounded Glass Beads and used in all subsequent analyses. For each of the sand materials (Table 1), this particle assembly was transformed into angular equivalents through the shape conversion algorithm described earlier in the paper. Therefore, valid (a) (b) (c) Figure 5. Simulation of dry pluviation process: (a) angular particles are generated; (b) particles settle under gravity; (c) particles outside desired boundaries are removed. Paper No PCW-05 Ashmawy et al. 4

5 (a) (b) (c) Figure 6. Identical grain size distributions prepared at the same void ratio and arranged to similar initial fabrics: (a) Glass Beads; (b) Ottawa Rounded; and (c) Ottawa Angular. fabric shown in Fig. 6-a. The shape conversion algorithm was then invoked for each of the sand materials listed in Table 1, and a corresponding angular assembly was created accordingly. Because of the irregular shape of the natural grains, and since a random rotation angle was assigned to each particle during conversion, some overlap occurred between neighboring particles. This was more significant in the case of highly angular particles, such as Daytona Beach Sand, Syncrude Tailings Sand, and Ottawa Angular Sand. Consequently, large unbalanced forces were produced in the particle assembly immediately after conversion. By subsequently allowing the system to equilibrate, the particles rotated and shifted from their original positions. Therefore, changes occurred in the soil fabric, depending on particle shape and angularity. Relatively rounded materials, such as Ottawa Rounded Sand (Fig. 6-b) exhibited a fabric which closely resembles that of Rounded Glass Beads (Fig. 6-a), while materials with a higher level of angularity, such as Ottawa Angular Sand (Fig. 6-c) produced a somewhat different fabric. Each of the soils was then statically compacted to an all-around confining stress of 40 kpa, by gradually moving the boundary plates inward, as in the case of dry pluviation. In this study, the target void ratio at the end of static compaction was set to a constant value of for all materials, although any other value within the range of limiting void ratios could be used. While a single static loading cycle was needed to reach the desired void ratio in the case of Syncrude Tailings Sand, several isotropic unloading/reloading cycles were required for the other materials in order to reach the desired void ratio at a stress of 40 kpa. On the other hand, Ottawa Rounded and Michigan Dune sands decreased too much in volume during the first loading cycle, yielding void ratios smaller than Undrained Cyclic Loading It is not possible to directly model pore water pressures and drainage conditions in DEM models. To replicate fully undrained conditions, restrictions were imposed on the model to prevent any volume change from occurring inside the specified boundaries. Cyclic pure shear conditions (Fig. 4-d) were simulated through successive cycles of shear stress-controlled loading. Simultaneous control of shear stress amplitude and sample volume was accomplished using a cyclic loading algorithm, which was incorporated as a subroutine in the software. The algorithm sequence consists of first assigning a negative (downward) velocity, V y, to the top plate, and an equal positive velocity to the bottom plate. Concurrently, the right and left plates are set in outward motion (i.e., to the right and left, respectively), at a velocity V x. Since the total volume of the sample must remain constant during undrained loading, a relationship exists between V x and V y as follows: b Vy = V x (3) a where a and b are the instantaneous width and height of the sample, respectively. The velocities are adjusted by means of a gain factor to prevent dynamic instability of the system at all times. The algorithm was written such that the instantaneous velocity is inversely proportional to the difference between current and peak shear stress amplitudes. Therefore, the cyclic shear stress function is periodic and almost sinusoidal in shape. Although Eq. 3 represents a closed form solution of the relationship between V x and V y, the total volume was still checked periodically to ensure that the volumetric strain remains equal to zero. If a small difference is detected due to numerical approximations, the position of the boundary plates is adjusted to correct the discrepancy. As described earlier, the boundary stress is defined here as the total force acting on a boundary plate, divided by the width of the plate. Therefore, the average shear stress acting on the sample, τ xy, is calculated from: sx sy τ xy = (4) 2 where s x and s y are the normal stresses acting on the vertical and horizontal boundary plates, respectively. Since no volume change occurs during loading, the magnitudes of s x and s y change as a function of loading cycles due to fabric rearrangement. In essence, s x and s y represent the effective stresses acting on the sample boundaries. It follows that the mean effective stress, p', is obtained from: s x + s y p = (5) 2 The difference between initial and instantaneous values of p' is equal to the pore water pressure, u. Dilative soils exhibit an increase in mean effective stress while contractive soils show a decrease. A stress path can be plotted by tracking the changes in τ xy and p' during cyclic loading. Particle assemblies prepared by dry pluviation and static compaction. were both subjected to simulated undrained cyclic loading. By controlling the cyclic shear stress amplitude in each run, plots of the cyclic stress ratio, CSR, versus number of cycles to liquefaction, N, were obtained. Similar plots are conventionally generated in the laboratory to evaluate the liquefaction susceptibility of a soil. The cyclic stress ratio is defined as the ratio between the amplitude of cyclic shear stress, τ cyc, and the initial effective mean stress, p o '. Although liquefaction is broadly defined as the total loss in effective stresses (i.e., p' = 0), such a condition is impossible to attain from a theoretical standpoint. Instead, in the present study, the onset of liquefaction was assumed to have taken place when the pore pressure ratio reached 90% or the strain exceeded 15%, whichever occurred first. The pore pressure ratio, R u, is equal to the ratio between the current pore water pressure, u, and the initial effective mean stress, p o '. RESULTS AND DISCUSSION In order to evaluate the numerical model, a number of parameters were monitored and their histories saved during the runs. These include soil fabric and particle arrangement, interparticle contact forces, principal (boundary) stresses, pore water pressure, and axial strain. Initial isotropic consolidation was found to produce a meta-stable fabric that collapses rapidly during the first stages on loading, causing a rapid increase in pore water pressure. Once a more stable structure is Paper No PCW-05 Ashmawy et al. 5

6 reached (typically within a few loading cycles), the cyclic behavior of the assembly becomes more predictable. Earlier and recent experimental observations (Mulilis et al, 1977; Ladd, 1977; Vaid and Sivathayalan, 2000) indicate that the initial soil fabric has a strong influence on the number of cycles to liquefaction, which is in agreement with the current findings. Additional comparisons between numerical results from this study and experimental data in the literature confirm that the model was highly successful in replicating all mechanisms and conditions leading to liquefaction. Soil Fabric and Interparticle Forces Particle shape was found to have a strong influence on the initial soil fabric, namely in terms of void ratio, particle arrangement, and interparticle contacts. The pluviated void ratios of materials with rounded particles (Glass Beads, Michigan Dune, and Ottawa Rounded) were in the range of 0.24 to Higher void ratios, ranging between 0.28 and 0.29, were observed for the angular sands (Syncrude Tailings, Daytona Beach, and Ottawa Angular). In effect, the range of possible void ratios (maximum to minimum) is much broader for angular materials due to their irregular geometric shapes. Angular particles can form both a highly collapsible fabric due to the internal arching phenomenon known as the honeycomb effect, and a stable tight fabric due to particle interlocking. To illustrate this phenomenon, Fig. 7 shows close up views of two zones within the simulated Ottawa Angular Sand assembly, at the end of the dry pluviation phase. Examination of the interparticle contact forces revealed that the coordination number (number of contacts per particle), as well as the magnitude and distribution of the forces, is highly dependent on particle shape. Figure 8 shows the vectors of the interparticle forces for three materials of varying degrees of angularity. For reference purposes, these are the same materials shown in Fig. 6, but the sample preparation method is different. In Fig. 8, the thickness of each vector is proportional to the force it represents. Because of the irregular geometry of angular particles, the coordination number is large. In fact, it is not uncommon for two adjacent particles to touch at more than one point, which is an impossible condition for Rounded Glass Beads. As a result, the contact forces are larger for rounded particles, and clearly defined force chains are formed. In contrast, the contact forces in the case of angular materials are smaller and distributed uniformly across the assembly. Stress Paths All simulated samples in this study were subjected to pure shear loading conditions. Since no changes occur in mean total stress throughout the cyclic loading phase, the corresponding stress path is a vertical line of length (2 τ cyc ), centered around a horizontal coordinate of (p o '). The effective stress path, however, depends on the magnitude of pore water pressure that gradually builds up in the sample as a result Figure 8. Interparticle force vectors for dry pluviated materials: (a) Glass Beads; (b) Ottawa Rounded; and (c) Ottawa Angular. The thickness of vector is proportional to the force. of cycling shearing. Figure 9 shows the simulated effective stress paths for Ottawa Rounded Sand and Daytona Beach Sand, at cyclic stress ratios, CSR, of 5% and 16%, respectively. It is evident from the results, when validated against a vast body of experimental data in the literature, that the model was indeed successful in capturing the various important behavior phases and phenomena leading to liquefaction. The results also illustrate the strong influence of particle shape on the liquefaction susceptibility of sands. Ottawa Rounded Sand (Fig. 8-a) exhibited a highly contractive response, characterized by a consistently positive increase in pore water pressure increase during shearing and unloading. With the exception of a few cycles where conditions seemed to stabilize, the sample underwent a steady and rapid increase in pore pressure ratio. Over the last two cycles, the pore water pressure increased dramatically, resulting in full liquefaction of the granular assembly. In contrast, the behavior of the more angular Daytona Beach Sand (Fig. 8-b) was characterized by successive cycles of dilation upon shearing, and contraction upon stress reversal. Sudden changes in soil fabric and local collapse occurred only around p' = 37 to 30 kpa. Otherwise, the response indicates an overall stable fabric, as evident from the relatively slow increase in pore pressure ratio. Over the last few cycles, the stress path followed the phase transformation pattern identified in experimental studies (e.g., Vaid and Chern, 1985). q (kpa) q (kpa) (a) (b) (c) (a) (b) (a) (b) p' (kpa) Figure 7. Close up views of (a) collapsible fabric, and (b) stable fabric, from dry pluviation simulation of Ottawa Angular Sand. Figure 9. Simulated effective stress paths for sands compacted to the same initial void ratio: (a) Ottawa Rounded Sand, at CSR=5%, and (b) Daytona Beach Sand at CSR=16%. Paper No PCW-05 Ashmawy et al. 6

7 Liquefaction Susceptibility Although the terms liquefaction susceptibility and liquefaction potential are often used interchangeably in the literature, a difference exists in terms of their basic definitions. According to Youd and Perkins (1978), liquefaction susceptibility represents the level of inability of a soil to resist cyclic shear. It is only a function of inherent particle properties, soil fabric, void ratio, and initial conditions. In contrast, liquefaction potential denotes the likelihood that a particular soil will liquefy under a given seismic shaking condition. Liquefaction potential, therefore, depends on external cyclic loading parameters as well as the susceptibility of the soil to liquefaction. As such, it is appropriate in the present study to address the influence of particle morphology on liquefaction susceptibility. Plots of cyclic stress ratio, CSR, versus the number of cycles to liquefaction, N, are shown in Fig. 10-a and 10-b for simulated samples prepared by dry pluviation and static compaction, respectively. Both are plotted to the same vertical scale for direct comparison. It is recalled that the void ratio of dry-pluviated samples is not constant, but is rather close to the maximum index value due to the sample preparation procedure. At a large number of cycles, the CSR causing liquefaction ranges between 0.06 and 0.12 for all sands. However, no discernible trend can be identified with respect to the dependence of liquefaction susceptibility on particle shape. It is concluded that, at the maximum index density, the susceptibility of soils to liquefaction is not a function of grain morphology. CSR CSR (a) (b) Glass Beads Michigan Dune Ottawa rounded Daytona Beach Syncrude Tailings Ottawa angular N (cycles) Figure 10. Cyclic stress ratio (CSR) versus number of cycles to liquefaction (N) for simulated samples prepared by: (a) dry pluviation, and (b) static compaction. In contrast, the results shown in Fig 10-b indicate that a strong relationship exists between particle shape and liquefaction susceptibility for soils prepared at a void ratio of Absent from the plot is Michigan Dune Sand, which consolidated to a void ratio of during the first cycle of static compaction. The void ratio of Ottawa rounded sand was equal to 0.228, which is also smaller than the target value of 0.245, albeit by a small amount. By comparing the results in Fig. 10-b in light of the shape and angularity factors in Table 1, it is observed that Ottawa Angular Sand, which possesses the highest level of irregularity in terms of angularity and shape, exhibits the highest resistance to liquefaction. In contrast, the highest liquefaction susceptibility corresponds to Glass Beads and Ottawa Rounded Sand, which are highly rounded in shape. The asymptotic CSR value varies significantly, with values as high as 0.2 for angular sands and as low as 0.03 for rounded materials. This large difference reinforces the strong influence of grain morphology on the undrained cyclic response of sands. CONCLUSIONS A new method for simulating particles of arbitrary shapes in discrete element modeling (DEM) was presented. The method relies on representing each particle as a series of overlapping, rigidly connected, distinct elements of equivalent properties. Modifications were applied to the computation time step in order to increase the speed of computation without compromising the numerical stability or solution accuracy. A large number of individual particles from six different natural, processed, and manufactured granular materials were digitized, converted into software-compatible input, and stored in a software library for DEM modeling. Simulations of cyclic shear tests were carried out on representative volumes of the six materials with the same size distribution to evaluate the dependence of liquefaction susceptibility on particle morphology. The findings indicate that, at the maximum void ratio, the susceptibility to liquefaction is independent of particle shape. However, the influence of particle morphology on liquefaction susceptibility was significant in the case of sands prepared at the same void ratio. ACKNOWLEDGEMENTS This research study was supported, in part, by the National Science Foundation, Grant No. INT The assistance of Dr. Thomas G. Davis in automating particle digitization is greatly appreciated. REFERENCES Ashour, M., and Norris, G. (1999). Liquefaction and undrained response evaluation of sands from drained formulation, J Geotech & Geoenv Eng, ASCE, Vol 125, No 81, pp Barbosa, R., and Ghaboussi, J. (1992). Discrete finite element method, Eng Comput, Vol 9, No 2, pp Cleary, P.W. (2000). DEM simulation of industrial particle flows: case studies of dragline excavators, mixing in tumblers and centrifugal mills, Powder Tech, Vol 109, No 1, pp Cundall, P.A., and Strack, O.D.L. (1979). Discrete numerical model for granular assemblies, Géotechnique, Vol 29, No 1, pp Garga, V.K. and McKay, L D. (1984). Cyclic triaxial strength of mine tailings, J Geotech Eng, Vol 110, No 8, pp Hird, C.C., and Hassona, F.A.K. (1990). Some factors affecting the liquefaction and flow of saturated sands in laboratory tests, Eng Geol, Vol 28, No 1-2, pp Horn, H.M., and Deere, D.U. (1962). Frictional characteristics of minerals, Géotechnique, Vol 12, No 4, pp Paper No PCW-05 Ashmawy et al. 7

8 Itasca Consulting Group (1999). Particle Flow Code in Two Dimensions. Software Version 2.0. Jensen, R.P., Bosscher, P.J., Plesha, M.E., and Edil, T.B. (1999). DEM simulation of granular media-structure interface: effects of surface roughness and particle shape, Int J Numer & Analytical Methods in Geomech, Vol 23, No 6, pp Ladd, R.S. (1977). Specimen preparation and cyclic stability of sands, J Geotech Eng Div, Vol 103, No 6, pp Matuttis, H.G., Luding, S., and Herrmann, H.J. (2000). Discrete element simulations of dense packings and heaps made of spherical and nonspherical particles, Powder Tech, Vol 109, No 1, pp Mitchell, J.K. (1993). Fundamentals of Soil Behavior. Second Edition. 437 pp. Mulilis, J.P., Seed, H.B., Chan, C.K., Mitchell, J.K., and Arulanandan, K. (1977). Effects of sample preparation on sand liquefaction, J Geotech Eng Div, Vol 103, No 2, pp Mustoe G.G.W. (1992). A generalized formulation of the discrete element method, Eng Comput, Vol 9, No 2, pp Ng, T.T. (1994). Numerical simulations of granular soil using elliptical particles, Comput & Geotech, Vol 16, No 2, pp Potapov, A.V., and Campbell, S.C. (1998). A fast model for the simulation of non-round particles, Granular Matter, Vol 1, No 1, pp Seed, H. B. and Lee, K. L. (1966). Liquefaction of saturated sands during cyclic loading, J Soil Mech & Foundations Div, ASCE, Vol 92, No SM6, pp Shi, G.H., and Goodman, R.E. (1989). Generalization of twodimensional discontinuous deformation analysis for forward modeling, Int J Numer & Analytical Methods in Geomech, Vol 13, No 4, pp Sukumaran, B. (1996). Study of the effect of particle characteristics on the flow behavior and strength properties of particulate materials, Ph.D. thesis, Purdue University, 197 p. Sukumaran, B., and Ashmawy, A.K. (2001). Quantitative characterization of the geometry of discrete particles, Géotechnique, Vol 51, No 7, pp Thomas, P.A., and Bray, J.D. (1999). Capturing nonspherical shape of granular media with disk clusters, J Geotech & Geoenv Engrg, Vol 125, No 3, pp Ting, J.M., Khwaja, M., Meachum, L.R., and Rowell, J.D. (1993). Ellipse-based discrete element model for granular materials, Int J Num & Analytical Methods in Geomech, Vol 17, No 9, pp Vaid, Y.P., and Chern, J.C. (1985). Cyclic and monotonic undrained response of saturated sands, Advances in the Art of Testing Soils Under Cyclic Conditions, V. Khosla, Editor, ASCE, pp Vaid, Y.P., Chern, J.C., and Tumi, H. (1985). Confining pressure, grain angularity, and liquefaction, J Geotech Eng, Vol 111, No 10, pp Vaid,Y.P., and Sivathayalan, S. (2000). Fundamental factors affecting liquefaction susceptibility of sands, Can Geotech J, Vol37, No. 3, pp Williams, J.R., and Pentland, A.P. (1992). Superquadrics and modal dynamics for discrete elements in interactive design, Eng Comput, Vol 9, No 2, pp Youd, T.L., and Perkins, D.M. (1978) Mapping liquefaction-induced ground failure potential, J Geotech Eng Div, Vol 104, No 4, pp Paper No PCW-05 Ashmawy et al. 8