Discrete Element Modeling for Granular Materials Guangcheng Yang Ph.D. candidate, School of Civil Engineering, Central South University, Changsha, 410075, China e-mail:
[email protected]
Xinghua Wang Professor, School of Civil Engineering, Central South University, Changsha, 410075, China
ABSTRACT Conventional numerical methods, such as finite element method (FEM) and finite difference method (FDM), are grid-based and macro-scale modeling approaches which are incapable of capturing microstructure phenomenon in soils. Discrete element method (DEM) models the system at the micromechanical level, which is a desirable technique to investigate microscale behavior in soils. The application of DEM to the simulation for mechanical behavior of granular materials is presented in this paper. The nonlinear Hertz contact model is implemented in the DEM code. A direct shear test is modeled and compared to experimental data, showing a good agreement between numerical simulation and lab observation. The validated model is used to simulate triaxial shear test and dynamic torsional shear test. The effects of shear strain along with loading frequency and particle friction on damping ratio are investigated..
KEYWORDS:
Discrete element method, granular material, torsional shear test.
INTRODUCTION Granular materials are the second most important material in nature (the first one being water). The study of how granular materials deform or flow from micromechanical perspective is of interest in applications as diverse as geotechnical engineering, pharmaceutical industry, energy production and other industrial processes. Computational modeling can provide insights into mechanisms of granular material behavior, playing an important role in understanding natural processes involved and predicting the outcomes of various events in specific conditions. Conventional computational methods, such as FEM and FDM (Zienkiewicz and Taylor. 2005; Clausen et al. 2006), are grid-based methods and based on macroscale continuum framework, which fail to capture the micromechanical behavior in granular materials (Cleary et al. 2004). Liquefaction, caused by the decrease of effective stress in saturated soil under dynamic loading, can result in significant loss of lives and properties. However, the process of the development of - 2463 -
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dynamic loading induced pore pressure is difficult to be predicted using macroscale approaches due to the limitation of constitutive models (Robertson and Wride, 1998). Another example of the importance that micro-mechanism needs to be investigated is the initiation and development of landslides (Cleary et al., 2004). Analytical methods and numerical techniques based on continuum framework have been widely applied to the evaluation of landslides (Ham et al. 2009). Nevertheless, the conventional methods play a limited role in helping understand under what circumstances the landslide may occur and how far the landslide will run from the initiation position, which are crucial considerations in risk assessment and protective-strategy formation. Another drawback associated with grid based methods is that they have difficulties in the simulation of problems involving extremely large deformation (Liu et al. 2007), which are commonly encountered with in geotechnical engineering. In FEM, the interpolation functions can produce highly accurate solutions if the meshes are of regular shape. The extremely large deformation may significantly degrade and distort the computational grids, leading to inaccurate solutions and in most cases the failure of computation. Various methods have been developed to overcome this drawback in FEM, among which, the adaptive remeshing method is a relatively mature one (Khoei and Lewis, 1999). However, this method requires the identification of updated deformed shape and mapping field variables to the deformed shape (Bui et al, 2008), which is obviously problematic when a complicated material model is used. As a purely Lagrangian approach, DEM tracks materials by using a set of discrete particles instead of grids, avoiding the numerical problems associated with mesh distortion. In addition, DEM models a system on a micro-mechanical level, capable of capturing micro-mechanisms in geotechnical material behavior. Originally developed by Cundell (1977) to solve rock mechanics problems in mining engineering, the application of DEM has been found in many other research areas and industries as diverse as pharmaceutical industry, agriculture, geotechnical engineering and energy production (Cleary 2004). DEM represents materials as discrete particles which match real material particle sizes and preferably their shapes (Hopkins et al. 1991). The computation algorithm involves the detection of particle collisions and calculation of contact forces. Once the net force on each particle is determined, the equations of motion are solved for the motion of particle system (Barker 1994). The developed 3D model in this paper is validated against a direct shear test. A good agreement between numerical simulation and experimental data is observed. The validated model is then used to simulate triaxial shear test and dynamic torsional shear test. The effects of shear strain along with loading frequency and particle friction on damping ratio are numerically investigated.
METHODOLOGY Calculation algorithm DEM simulates the mechanical behavior of a system comprised of a collection of arbitrarily shaped and arranged particles. The particles displace independent of one another, and only interact at the contacts and interfaces between particles. The computation involves the application of Newton’s second law to particles and a force-displacement law at contacts. Newton’s second law is used to determine the motion of each particle arising from the contact and body forces acting on it, while the force-displacement law is used to update the contact forces arising from the relative motion at each contact. The calculation cycle requires the repeated application of the law of motion to each particle and a force-displacement law to each contact. The calculation cycle is
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illustrrated in Figure 1. At the start of each h time step, tthe set of conntacts is deteected from thhe know wn particle possitions. The fo orce-displacem ment law is thhen applied too each contacct to update thhe contacct forces baseed on the relattive motion or o overlappingg distance bettween the twoo entities at thhe contacct and the contact constitu utive model. Next, N the law w of motion iss applied to eeach particle tto updatte its velocity and position n based on thee resultant forrce and momeent arising froom the contact forcess and body fo orces acting on n the particle..
Fig gure 1: Calcculation cyclle for DEM
Force-dis F splaceme ent law The T force-disp placement law w relates thee relative dissplacement bbetween two particles at a contacct to the conttact force actiing on the parrticles. As shhown in Figurre 2, the forcee-displacemennt law applies at a co ontact and can n be describeed in terms o f a contact pooint, , lyinng on a contact plane that is defineed by a unit normal n vectorr, . The conntact point iss within the innterpenetratioon volum me of the two o particles. The T normal veector on a coontact betweeen two particcles is directeed along the line betw ween ball cen nters. The contact force iss decomposedd into a norm mal componennt acting g in the direcction of the normal vector,, and a shearr component acting in the contact planne. The force-displacement law relates r thesee two compoonents of foorce to the correspondinng components of thee relative disp placement viaa the normal and shear stifffnesses at thhe contact. Thhe momeent arising fro om the normaal and shear fo orce can be caalculated by: =
×
(1)
wheree F is the force vector, R is the contaact position vvector with rrespect to thee center of thhe particcle, and indiccates the indeex of contact points. p
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Figu ure 2: Contaact law betweeen two partticles (after IItasca, 2003)) After A the calcu ulation of forrce and moment on each ccontact pointt on a particlee, the resultannt force and rotation moment m on th he particle aree determined..
Law of motio on The T motion off a single rigid d particle is determined d byy the resultannt force and m moment vectorrs acting g on it, descrribed in term ms of the tran nslational mootion of a pooint in the paarticle and thhe rotatio onal motion of o the particlee. The translaational motionn of the centeer of mass is described witth respecct to its positiion, , velociity, , and accceleration, . The rotatioonal motion off the particle is descriibed with resp pect to its an ngular velocity y, , and anngular accelerration, . Thhe equations oof motio on can be expressed as two vector equations: oone relates tthe resultantt force to thhe translational motio on; and the other relates th he resultant m moment to thhe rotational motion, whicch can bee expressed as: a =
(22) =
and wheree the mass, m and
are a the resultaant force and d rotational m moment on paarticle , respectively; is the moment of o inertia with h respect to thhe mass centeer of the particcle.
(33) is
The T leap-frog time integraation method d is employeed to updatee the particlee velocity annd position.
MODEL L VALID DATION The T developed d numerical model m is valid dated by a dirrect shear testt conducted bby Yan (20100). The particle p assem mblage is prepared by rando omly generatiing particles iin a containerr with 10 cm iin length h, 10 cm in width w and 3.71 1 cm in heigh ht, as shown inn Figure 3. A vertical forcce is applied oon the to op of shear bo ox, which rem mains unchang ged during exxperiment. Thhe soil specim men undergoees static shearing with h bottom shear box movin ng horizontallly in a very sm mall velocityy of 3 mm/minn. The particle p frictio on is assumed d to be 0.7 in this t simulatioon. Figure 4 shhows the com mparison of thhe experriment and sim mulation resu ults for shear stresses undeer various noormal forces. The numerical resultts indicate thaat the shear strength s of so oil increases as the increaase of normaal force, whicch
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follow ws the trend d observed in n experimen nts. The com mputational sshear stress vs. horizonttal displaacement curvees show a goo od agreementt with that obttained from eexperiments.
Figuree 3: Numerical model off the direct sshear test (affter Yan, 20110)
Figuree 4: Shear stress and horrizontal displlacement (affter Yan, 20110)
APPLICA A ATION OF O NUM MERICAL L MODE EL TO CO OMPLEX X SIMU ULATION NS Triaxia al shear ttest In n DEM, con ntact models play an im mportant role in the accuuracy of sim mulations. Thhe param meters in these contact models, m such as normal aand shear stiiffness, frictiion coefficiennt betweeen particles, are based on n micro-mech hanical perspeective, whichh are difficultt to be directlly obtain ned from geo otechnical exp periments. Th herefore, the parameter caalibration is nnecessary prioor to sim mulation of larrger problemss containing the t same mateerials as the ssample. Triaxial shear test is a com mmonly used method m to dettermine strength parameteers in a macroo-fashion. A D DEM modelinng of triaaxial shear tesst is carried out o to match th he micro-scalle parameterss with the maccro-scale onees.
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As sh hown in Figu ure 5, the num merical particcle assemblagge containingg 2886 DEM particles witth diameeter ranging from fr 1.5 to 2 mm m confined in a 40 mm bby 20 mm cyllinder containner.
Figuree 5: Particle assemblage for triaxial ttest The T confining g pressure iss set to 1 Mp pa and remaains unchangged during nuumerical tesst. As a Lagrangian approach, a it iss difficult to apply a presssure boundaryy on discrete particles. Thhis o boundary is i accounted for by applying a movingg wall on the vertical sidee of the modeel. type of The velocity v of thee moving walll is determineed by the conntact force bettween particlees and the waall in ord der to produce desirable confining prressure. A hoorizontal walll boundary, moving alonng verticcal direction, is applied on n the top to produce p the ddeviatoric strress in the triiaxial test. Thhe volum metric strain and axial strrain under ellastic load iss shown in F Figure 6, shoowing a lineaar relatio onship. As th he increase off axial deviato oric stress, thhe soil samplee undergoes a shear failurre, indicaating a nonelaastic deformaation. Figure 7 shows that the axial devviatoric stresss changes nonnlinearrly with the axial a strain un nder the elasstic-plastic loaad. The horizzontal red linne is confininng pressu ure, which iss constant du uring simulattion. The tri--axial test siimulation inddicates a goood repressentation of granular g materrial behavior under shear.
Figure 6: Volumetric V strain vs. axiaal strain for elastic load (After Itascaa, 2003)
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Figuree 7: Axial deeviatoric streess vs. axial strain (Afteer Itasca, 20003)
Dyn namic torsional sshear tes st Soil materials exhibit intern nal friction beetween particcles as the maaterials underrgo non-elasttic deform mation. The energy dissip pated in non-elastic deform mation can bbe determinedd by hysteressis loop under cyclic loads. The cyclic c torsion nal shear testt is a most ccommonly ussed method tto mine damping g in geotechn nical materialss. Figure 8 shhows a typicall hysteresis looop. and determ are th he area of hy ysteresis loop p and triangle, respectivelly. The damp mping ratio is calculated bby equatiion (4) shown n below.
Figure 8: Typical T hysteeresis loop (aafter Kramerr, 1996)
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=
(44)
4
A DEM model is developed to simulatee dynamic torrsional shear test. As show wn in Figure 9, the particle p assem mblage, contaained in a cylinder c conttainer, consissts of 8015 particles witth diameeter ranging from f 0.75 mm m to 4.0 mm m. The dynam mic torsional shear test is carried out bby applying a harmon nic rotation att the top of asssemblage, ass shown in Fiigure 10. Thee shear strain is calcullated using th he following equation: e =
∙ /
(55)
a the rotatio onal angle, cy ylinder radiuss and specimeen height, resspectively. Thhe wheree , and are confin ning pressuree is set at 7ee5 Pa, remain ning unchangged during thhe test. Tablee 1 shows thhe materrial propertiess and loading parameters in n the simulatiions.
Fig gure 9: Particle assemblaage for dynaamic torsionaal shear test
=
=
∙
( ∙ )
∙ /
Fixed at the bottom
Figure 10: Soil co olumn subjeccted to a torssional excitaation at the ttop
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Table 1: Model parameters Porosity Sample size Density Harmonic Excitation function
0.4 =5 × 17.5 2700 / ( ) = ∙ =2 ∙ = ∙ 2 ∙ ∙ cos( ) is the amplitude in torsion
×
where
The effects of shear strain along with loading frequency and particle friction coefficient on damping ratio are numerically investigated.
Effect of shear strain As the soil is plastically deformed, the energy dissipation is caused by internal particle friction as well as particle rearrangement. Therefore, the damping ratio increases as shear strain increases (Kramer, 1996). This behavior is simulated by applying harmonic torsion with different amplitudes at the top of the soil column, leaving all other parameters constant. As shown in Figure 11, higher torsional shear strain results in larger hysteresis loop area indicating higher damping ratio. The computed damping ratio for torsional strain of 0.06% is 0.063, whereas the damping is 0.29 for torsional strain of 0.35%.
Effect of particle friction coefficient The simulations are conducted for various particle friction coefficients. Figure 12 presents the variation of damping ratio with particle coefficient. It is shown that the damping ratio decreases as the particle friction increases, which is opposite to what our intuition may indicate. The energy dissipation in coarse sand is caused by particle rearrangement, whereas higher particle friction limits the movement of particles. As a consequence, the damping ratio for sand with higher friction is lower.
Effect of torsional frequency The influence of loading frequency on damping ratio is shown in Table 2. It is indicated that the damping ratio decreases as the frequency increases at low loading frequency, which is consistent with the experimental observation from Shibuya et al (1995). The impacts of loading frequency on damping ratio at medium and high frequency are not investigated in our simulation since higher frequency requires extremely small time step which is computationally demanding.
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(a) Hysterresis loop fo or torsional sshear strain 00.35%
(b) Hysterresis loop fo or torsional sshear strain 00.06% Figure 11: 1 Damping g under diffeerent shear sstrain
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0.08
Damping ratio
0.06 0.04 0.02 0.00 0.2
0.3
0.4
0.5
0.6
0.7
0.8
Particle friction coefficient
Figure 12: Influence of particle friction coefficient on damping ratio Table 2: Effect of torsional frequency on damping ratio Frequency ( ) Strain ( ) Damping ratio 200 50
0.35% 0.35%
0.29 0.31
CONCLUSIONS The application of DEM to the simulation for mechanical behavior of granular materials is presented in this paper. A direct shear test is modeled and compared to experimental data, showing a good agreement between numerical simulation and lab observation. The validated model is used to simulate triaxial shear test and dynamic torsional shear test. The effects of shear strain along with loading frequency and particle friction on damping ratio are investigated. The numerical results show that larger shear strain results in higher damping ratio, which is consistent with previous experimental observation. Furthermore, both loading frequency and particle friction have impacts on damping ratio of granular materials. However, the effects of loading frequency are not as significant as our intuition may indicate. The most unexpected results are the damping ratio decreases as the particle friction increases for coarse sand. As a result, the damping ratio is not an intrinsic material property. On the contrary, it depends on strain magnitude as well as loading frequency, particle shape and confining pressure. The numerical results suggest that DEM could be a powerful method to investigate microscale behavior in granular materials.
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REFERENCES 1. Barker, G. C. (1994) “Computer simulations of granular materials.” Granular matter: an interdisciplinary approach (ed. A. Mehta). Springer. 2. Bui, H.H., Fukgawa, R., Sako, K. and Ohno, S. (2008) “Lagrangian mesh-free particle method (SPH) for large deformation and post-failure flows of geomaterial using elasticplastic soil constitutive model.” International Journal for Numerical and Analytical Methods in Geomechanics, 32 (12), 1537–1570. 3. Clausen, J., Damkidle, L. and Andersen, L. (2006) “Efficient return algorithms for associated plasticity with multiple yield planes.” International Journal for Numerical Methods in Engineering, 66, 1036–1059. 4. Cleary, P.W. and Parkash, M. (2004) “Discrete-Element Modelling: Methods and Applications in the Environmental Sciences.” Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 362(1822), 2003-2030. 5. Cleary, P. W. (2004) “Large scale industrial DEM modeling.” Engineering Computation. 21, 169–204. 6. Cundall, P. and Strack, O. (1979) “A discrete numerical model for granular assemblies.” Geotechnique, 29, 47–65. 7. Ham, G., Rohn, J., Meier, T. and Czurda, K. (2009) “Finite element simulation of a slow moving natural slope in the upper-austrain alps using a visco-hypoplastic constitutive model.” Geomorphology, 103, 136–142. 8. Hopkins, M., Hibler, W. and Flato, G. (1991), “On the simulation of the sea ice ridging process.” Journal of Geophysics Research, 96, 4809–4820. 9. Itasca (2003). Particle Flow Code, PFC3D, release 3.0. Itasca Consulting Group, Inc., Minneapolis, Minnesota. 10. Khoei, A.R. and Lewis, R.W. (1999) “Adaptive finite element remeshing in a large deformation analysis of metal powder forming.” International Journal for Numerical Methods in Engineering, 45, 801-820. 11. Kramer, S.L. (1996), Geotechnical Earthquake Engineering, Prentice Hall. 12. Liu, J., Chen, H., Ewing, R. and Qin, G. (2007) “An efficient algorithm for characteristic tracking on two dimensional triangular meshes.” Computing, 80, 121–136. 13. Robertson, P.K. and Wride, C.E. (1998) “Evaluating Cyclic Liquefaction Potential using the cone penetration test.” Canadian Geotechnical Journal, Ottawa, 35(5), 442-459. 14. Yan, Y. and Ji, S. (2010) “Discrete element modeling of direct shear tests for a granular material”, International Journal for Numerical and Analytical Methods in Geomechanics, 34, 978-990. 15. Zienkiewicz and Taylor (2005) “The Finite Element Method for Solids and Structural Mechanics (5 ed.)” Elsevier
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