Numerical investigation of force transmission in granular media using discrete element method

Vietnam Journal of Mechanics, VAST, Vol 42, No (2020), pp 153 – 171 DOI: https://doi.org/10.15625/0866-7136/14787 NUMERICAL INVESTIGATION OF FORCE TRANSMISSION IN GRANULAR MEDIA USING DISCRETE ELEMENT METHOD Thong Chung Nguyen1 , Lu Minh Le1 , Hai-Bang Ly2 , Tien-Thinh Le3,∗ Vietnam National University of Agriculture, Hanoi, Vietnam University of Transport Technology, Hanoi, Vietnam Duy Tan University, Da Nang, Vietnam ∗ E-mail: letienthinh@duytan.edu.vn Received: 19 January 2020 / Published online: 10 May 2020 Abstract In this paper, a numerical Discrete Element Method (DEM) model was calibrated to investigate the transmission of force in granular media To this aim, DEM simulation was performed for reproducing the behavior of a given granular material under uniform compression The DEM model was validated by comparing the obtained shear stress/normal stress ratio with results published in the available literature The network of contact forces was then computed, showing the arrangement of the material microstructure under applied loading The number and distribution of the contacts force were also examined statistically, showing that the macroscopic behavior of the granular medium highly depended on the force chain network The DEM model could be useful in exploring the mechanical response of granular materials under different loadings and boundary conditions Keywords: granular mechanics, discrete element method, force chain, compression test INTRODUCTION A granular medium is composed of separate particles that move without dependence and interact with other particles via contact points [1] Typical granular materials could be found in civil engineering, such as geotechnical engineering, mining or energy production, chemical, pharmaceutical, and agricultural industries [2–4] Research and development of machinery/device for processing granular materials have been considerably increased over the past ten years, requiring above all a good knowledge of interactions between particulate systems itself and with machine parts [5] For instance, the coefficient of friction has been introduced, measured to characterize the dissipation of energy when the particles collide [6] These particulate interactions have been investigated for many years using analytical, semi-analytical, or experimental approaches [3,7,8] Despite all the efforts, it is not always possible to carry out a large number of configurations c 2020 Vietnam Academy of Science and Technology 154 Thong Chung Nguyen, Lu Minh Le, Hai-Bang Ly, Tien-Thinh Le taking into account all the possible parameters [6] Moreover, experimental works might not have the required ability to investigate the local interactions, particularly in terms of transmission of stress, collapse of force chain under deformation and so on [9] It clearly showed that a more robust manner is thus required for better understanding and characterizing the mechanical properties of granular materials [10] From a numerical simulation point of view, the mechanics of granular media can be modeled by either continuum [11–13] or discrete [14–16] approaches More precisely, in a discrete approach, the Discrete Element Method (DEM) has been primarily employed to simulate granular materials [10, 17] As an example, Than et al [18] have developed a DEM model for investigating the plastic response of wet granular material under compression Also, based on DEM technique, Xie et al [19] have pointed out the influence of interlayer on the strength and deformation of layered rock specimens in uniaxial tests In another study, Tran et al [2] have employed DEM algorithm to simulate the behavior of concrete under triaxial loading Xu et al [20] have proposed a comparison between DEM simulation and experiments while investigating the mechanical behavior of sea ice Lommen et al [17] have studied the relationship between particle stiffness and bulk material behavior in a numerical simulation context Furthermore, the combination of DEM and other numerical techniques has been performed by Dratt and Katterfeld [21] The authors have combined DEM with Finite Element Method (FEM) for investigating the dynamic deformation of machine parts in contact with particle flow Besides, Zhou et al [22] have combined DEM with Computational Fluid Dynamics (CFD) for modeling granular flow in hydraulic conveyor So far, studies involving DEM technique could strongly improve the investigation of mechanical properties of particulate systems by enabling an access to the local behavior in a granular media Such numerical simulation technique could also save time and cost compared with complex experiments in the design and development of machinery involving particulate systems In this study, DEM model was developed for investigating the transmission of stress in granular media under the compression force To this aim, the following steps were adopted as a methodology First, a set of DEM parameters for the granular media was collected in the available literature, involving dimensional, gravimetric, mechanical, and interaction properties Precisely, the DEM parameters were the size distribution, shape, mass density, Young’s modulus, Poisson’s ratio, shear modulus, coefficient of static friction, coefficient of rolling friction and coefficient of restitution In a second step, a compression test was designed and performed using DEM simulations Simultaneously, local mechanical information of particles was recorded, including the stress, force chain transmission and so on The obtained results allowed exploring the ability of DEM technique in a mechanical context Moreover, the features of DEM method were exposed to monitoring and analyzing the displacements and forces of all particles in the considered granular media Numerical investigation of force transmission in granular media using discrete element method 155 MATERIALS AND METHODS 2.1 Brief introduction to DEM DEM was developed based on the simulation of the motion of separate particles in a granular medium [23] Such motion is determined by solving Newton’s translational and rotational equations of motion for individual particles The translational equation of motion is given as below [24] mi dvi = dt ∑ Fij + mi g , (1) j where mi is the mass of particle i, vi is the velocity, t is the time, Fij is the force of contact acting on the particle i from the particle j, and g is the gravity The rotational equation of the motion is expressed as follow [23] Ii dωi = dt ∑ Tij , (2) j where Ii is the moment of inertia, ω i is the angular velocity, and Tij is the torque acting on the particle i from the particle j In a DEM model, the contact force is commonly modeled by spring, dashpot, and frictional slider [25, 26] One of the most used contact models is the Hertz–Mindlin model [27], involving various parameters such as Young’s modulus, Poisson’s ratio, shear modulus, coefficient of static friction, coefficient of rolling friction and coefficient of restitution [28] These coefficients, relating the relationships between particle/particle and particle/wall, were introduced to characterize the loss of energy when the particles interact Based on this principle, DEM simulation could reflect the interactions occurring inside the granular media [18] Underlying assumptions of DEM model include isotropy and elasticity of the considered particles On the other hand, the spherical element is the fundamental element in a DEM model The description of DEM model is well documented in Lommen et al [17] and Xie et al [19] One of the first applications of DEM was carried out by Cundall and Strack for investigating the mechanics of rock and soil [1] Recently, the fast growth of computational capacity makes it more and more practical to employ numerical methods for solving engineering problems [16] To date, many works using DEM technique for investigating the mechanical properties of granular materials have been published [2,20,29–31] 2.2 Description of compression test The compression test used in this study is schematized in Fig Granular material with characteristics introduced in Tab was filled into a box container of 400 × 100 × 300 mm The initial height of the granular medium was 280 mm, exhibiting more than 47.000 particles At the top of the container, a compression plate is placed The latter can move freely along the vertical direction (z-axis) A confinement force is exerted to the compression plate, which compresses the granular medium uniformly under a constant loading Such compression force is a constant normal one applying to the particles, Nguyen Chung Thong, Le Minh Lu, Ly Hai Bang and Le Tien Thinh On the other hand, the spherical element is the fundamental element in a DEM model The description of DEM model is well documented in Lommen et al [17] and Xie et al [19] One of the first applications of DEM was carried out by Cundall and Strack for investigating the mechanics of rock and soil [1] Recently, the fast growth of computational capacity makes it more and more practical to employ numerical methods for solving engineering problems [16] To date, many works using DEM technique for investigating the mechanical properties of granular materials have been published [2,20,29–31] 2.2 Description of compression test 156 Thong test Chung Nguyen, MinhisLe,schematized Hai-Bang Ly, Le The compression used in thisLustudy inTien-Thinh Fig Granular material with characteristics introduced in Table was filled into a box container of 400x100x300 mm The initial height of the granular medium was 280 mm, exhibiting more than 47.000 particles At the top of the whereas the forcea compression acting onplate theis upper part x-direction perpendicular to the norcontainer, placed The latterin canthe move freely along theisvertical direction (z-axis) confinement is exerted to the compression plate,awhich compresses granular medium mal force,Awhich wasforce previously mentioned Such design of thethe test allows characterizuniformly under a constant loading Such compression force is a constant normal one applying to the ing the transmission oftheforce theongranular medium locally, under compression particles, whereas force in acting the upper part in the x-direction is perpendicular to the normal using a force, which was previously mentioned Such a design of the test allows characterizing the transmission numerical DEM approach of force in the granular medium locally, under compression using a numerical DEM approach Fig Design of compression instudy this study Fig Design of compression testtest in this 2.3 DEM input parameters In this study, the mechanical behavior of agricultural granular materials was investigated, such as 2.3 DEM input parameters dry soybean grains (Glycine max variety, moisture content lower than 10%) to develop and design the In thisseeding study, the The mechanical behavior of agricultural granular materials was invesmachine microscopic parameters of soybean particles are commonly represented based on four categories, in the following tigated, such as dry assoybean grains (Glycine max variety, moisture content lower than The first includes such as the true density The second category of soy10%) to develop andcategory design the gravimetric seeding properties machine The microscopic parameters includes dimensional properties, especially size (i.e., equivalent diameter) and shape The third category bean particles are commonly represented basedYoung’s on four categories, as in the includes mechanical properties, such as shear modulus, modulus, and Poisson’s ratio Thefollowing last category includes the interaction properties, such asproperties friction (coefficient of static friction particle/particle The first category includes gravimetric such as the true density The secand particle/wall, coefficient of rolling friction particle/particle and particle/wall), restitutiondiameter) ond category includes dimensional properties, especially size (i.e., equivalent (coefficient of restitution particle/particle, and particle/wall) It should be noticed that the calibration of and shape The third category includes mechanical properties, such as shear modulus, Young’s modulus, and Poisson’s ratio The last category includes the interaction properties, such as friction (coefficient of static friction particle/particle and particle/wall, coefficient of rolling friction particle/particle and particle/wall), restitution (coefficient of restitution particle/particle, and particle/wall) It should be noticed that the calibration of all microscopic parameters for soybean grains is not an easy task 32] Thus, in this study, the microscopic parameters (i.e., DEM input parameters) of particles were taken from the available literature of Ghodki et al [32], as it was reported for the same variety of soybean Moreover, Ghodki et al [32] have admitted a single sphere modeling for the shape of particles, which allowed reducing the computational time considerably compared to multi-spheres or superquadric approaches [33] It should be noticed that such single sphere modeling was selected based on the shape characterization of the considered particles [32] In this study, the LIGGGHTS R code (stand for Open Source Discrete Element Method Particle Simulation) was used for the DEM simulations [34] A no-cohesion nonlinear Hertz–Mindlin model was used for simulating the contact between particle-particle and Numerical investigation of force transmission in granular media using discrete element method 157 particle-wall, as recommended by various works, such as Raji et al [25], or Horabik et al [35] Tab indicates the details of DEM simulation performed in this study, including the DEM input parameters collected from the available literature [32] The simulations Table Parameters of DEM simulations in this study Parameter Description and value Unit Sliding friction: Hertz-Mindlin Contact model Rolling friction: constant directional torque Cohesion: none m/s2 Gravity 9.81 Particle shape model Spherical Time step 1e-5 s Particle size 6.24 mm True density of particles 1220 kg/m3 Young’s modulus of particles 50 MPa Poisson’s ratio of particles 0.26 Shear modulus of particles 19.84 MPa Young’s modulus of wall 3000 MPa Poisson’s ratio of wall 0.37 Shear modulus of wall 1095 Coefficient of static friction particle/particle 0.26 Coefficient of static friction particle/wall 0.30 Coefficient of restitution particle/particle 0.17 Coefficient of restitution particle/wall 0.35 Coefficient of rolling friction particle/particle 0.08 Coefficient of rolling friction particle/wall 0.08 Length of container 400 mm Width of container 100 mm Number of particles 47362 Initial fill height 280 mm Final height 200 mm Mesh of wall Triangular type (STL) Number of elements (container and plate) 15604 Element area Average: 7.06e-5 m2 Minimum angle Average: 54.15 ˚ Aspect ratio Average: 1.05 Velocity of compression plate 10−1 MPa m/s 158 Chung Nguyen, Lu Minh Le, Hai-Bang Ly, Tien-Thinh Le NguyenThong Chung Thong, Le Minh Lu, Ly Hai Bang and Le Tien Thinh were performed using a(container Lenovo Intel of Chung Thong, LeThinkPad Minh Lu, Lu, Ly L420 Hai Bang andCore Le Tien Thinh Chung Thong, Le Minh Ly Hai Bang and Lei5-2520M Tien Thinh 2.50 GHz, 58 Gb NumberNguyen of Nguyen elements and plate) 15604 RAM, whereas the post-treatments were performed by using Matlab R2018a 2[36] and Element area Average: 7.06e-5 m Paraview 5.4.1 [37] Number of elements (container and and plate) Number ofangle elements (container plate) 15604 15604 Minimum Average: 54.15 ° In order to ensure the relevance of the selected set of DEM input 2parameters, in2 Element area Average: 7.06e-5 m Element area Average: 7.06e-5 m ratio1, size characterization and siloAverage: 1.05tests were performed More dicatedAspect in Tab discharge Minimum Average: 54.15 Minimum angle ° ° Velocity ofangle compression plate allowedAverage: 10-154.15 m/s of parprecisely, the size characterization obtaining a particle size distribution Aspect Average: Aspect ratioratio Average: 1.051.05 ticles (for generating particle diameter in DEM simulations), whereas the silo discharge Velocity of compression Velocity of compression plateplate 10-110-1 m/sm/s test allowed checking the efficiency of friction coefficients (i.e., static and rolling frictions In order to ensure the relevance of the selected set of DEM input parameters, indicated in Table particle/particle) Brief details of these two investigations are following 1, size characterization and silo discharge tests were performed More precisely, the size characterization The of particles was using aparticle home-made imaging Insize order to ensure the relevance of the selected setconducted of DEM input parameters, indicated in Table In order to characterization ensure relevance the selected of DEM input parameters, indicated in Table allowed obtaining a the particle size of distribution ofset particles (for generating diameter in DEM silo 1, size characterization and discharge tests were performed More precisely, the size characterization 1, size characterization and silo discharge tests were performed More precisely, the size characterization platform (4.42 MP/cm pixel density Fujifilm X-E2S camera with a Fujinon XF18-55mm simulations), whereas the silo discharge test allowed checking the efficiency of friction coefficients (i.e., allowed obtaining a particle size distribution of an particles (for generating particle diameter in mm DEM(recallowed obtaining a frictions particle size distribution of Brief particles (forof generating diameter inper DEM F2.8-4 R OIS lens), allowed obtaining image resolution of 16 pixels static andLM rolling particle/particle) details these twoparticle investigations are following simulations), whereas the silo discharge allowed checking the efficiency of friction coefficients simulations), whereas the silo discharge test test allowed checking the efficiency of friction coefficients (i.e.,(i.e., ommended forcharacterization characterizing particles greater than two mm in size are [38]) Soybean grains static and rolling frictions particle/particle) Brief details of these are following sizefrictions of particles was conducted using ainvestigations home-made imaging platform (4.42 static andThe rolling particle/particle) Brief details of these two investigations following were randomly selected for capturing images (about 900 grains were tested) Fig 2(a) MP/cm² pixel density Fujifilm camera with using a Fujinon XF18-55mm F2.8-4 R LM OIS lens), The size characterization ofX-E2S particles was conducted using a home-made imaging platform (4.42 Thethe size characterization of particles was conducted a home-made imaging platform (4.42 shows raw image, whereas Fig 2(b) presents the processed binary image indicating allowed obtaining an image resolution of 16 pixels mm (recommended for particles MP/cm² pixel density Fujifilm X-E2S camera with aper Fujinon XF18-55mm F2.8-4 R LM lens), MP/cm² pixel density Fujifilm X-E2S camera with a Fujinon XF18-55mm F2.8-4 R characterizing LM OISOIS lens), the equivalent diameter of each particle The equivalent diameter was computed based greater than mm in size [39]) Soybean grains were randomly selected for capturing images (about allowed obtaining an image resolution of pixels 16 pixels (recommended characterizing particles allowed obtaining an image resolution of 16 per per mmmm (recommended for for characterizing particles on the obtained area of the particle Using 900 equivalent diameters, the particle size dis900 grains were tested) Fig 2a shows the raw image, whereas Fig 2b presents the processed greater than mm in size [39]) Soybean grains were randomly selected for capturing images (about greater than mm in size [39]) Soybean grains were randomly selected for capturing images (about binary image indicating thein equivalent each particle The equivalent diameter was computed tribution is shown Fig exhibiting an average ofFig 6.33 and a standard deviation 900 grains were tested) Fig 2a diameter shows image, whereas 2bmm presents processed binarybased 900 grains were tested) Fig 2a 2(c), shows the the rawofraw image, whereas Fig 2b presents the the processed binary the obtained area of the particle 900 equivalent diameters, thewas particle size based distribution image indicating the equivalent diameter of each particle The equivalent diameter was computed based inis image indicating theis equivalent diameter ofUsing each particle Thediameter equivalent diameter computed ofon 0.46 mm It seen that the average particle obtained by image analysis on the obtained area of the particle Using 900 equivalent diameters, the particle size distribution issoyshown in Fig 2c, exhibiting an average of 6.33 mm and a standard deviation of 0.46 mm It is that on the obtained area of the particle Using 900 equivalent diameters, the particle size distribution is seen this study was very close to the result obtained by Ghodki et al [32] for the same shown in Fig 2c, exhibiting an average of by 6.33 mm a standard deviation ofwas 0.46 mm Itseen is seen that result shown in Fig 2c,particle exhibiting an average of 6.33 mm andand aanalysis standard deviation of 0.46 mm It isclose thatthe the average diameter obtained image in this study very to bean variety (i.e., 6.24 mm).obtained Finally,bythe particle size distribution was used for generating the average particle diameter image analysis in this study was very close tothe the result size theobtained average particle diameter by image analysis in this study was very close to the result by Ghodki et al obtained [33] for the same soybean variety (i.e., 6.24 mm) Finally, particle particle diameter in DEM simulations obtained by Ghodki al [33] for the same soybean variety mm) Finally, particle obtained by Ghodki et al.et [33] for the same soybean variety (i.e.,(i.e., 6.246.24 mm) Finally, the the particle sizesize distribution was used for generating particle diameter in DEM simulations distribution for generating particle diameter in DEM simulations distribution waswas usedused for generating particle diameter in DEM simulations (a) (b) (c) Fig Size characterization ofofparticles inin this study: (a)(a) raw image, (b) processed image with an equivalent Fig.Fig Size characterization of particles in this study: (a) raw image, (b) processed image with an equivalent 2.2.Size characterization particles this study: raw image, (b) processed image with an equivalent diameter of particle, (c)(c) particle distribution image analysis diameter of each particle, and and (c)and particle size size distribution fromfrom image analysis diameter ofeach each particle, particle size distribution from image analysis Fig Size characterization of particles in this study: (a) raw image, (b) processed image with an Regarding the a aflat-bottomed rectangular silo of 160 and 100 of length and and equivalent diameter of each and (c) particle size from analysis Regarding the discharge test,test, aparticle, flat-bottomed rectangular silo ofdistribution 160 and 100and mmmm ofimage length and Regarding the discharge discharge test, flat-bottomed rectangular silo of 160 100 mm of length width, respectively, together with a circular orifice ofmm 50 mm of diameter, was prepared Aof kgsoybean of soybean width, respectively, together with a circular orifice of 50 of diameter, was prepared A kg width, respectively, together with a circular orifice of 50 mm of diameter, was prepared A kg of soybean particles was randomly selected filled the silo, exhibiting fill height of 100 In the DEM particles waswas randomly selected and and filled intointo the silo, exhibiting a filla height of 100 mm.mm In the DEM particles randomly selected and filled into the silo, exhibiting a fillcritical height 100 mm In the DEM Regarding the discharge test, a flat-bottomed rectangular silo ofof 160 100 mm of simulation, same procedure applied friction plays inand the rheology simulation, the the same procedure waswas applied As As friction plays the the mostmost critical rolerole in the rheology simulation, the same procedure was applied As friction plays the most critical role in the rheology length and width, respectively, together with a circular orifice of 50 mm of diameter, was behavior of granular materials [32], the efficiency of the selected coefficients of static and rolling behavior of granular materials [32], the efficiency of the selected coefficients of static and rolling behavior ofAgranular materials [32], the efficiency of this thethis selected of static rolling frictions particle/particle (see Table 1) were checked based on test Tocoefficients this in DEM simulation, prepared kg of soybean particles was randomly selected and filled into theand silo, exfrictions particle/particle (see Table 1) were checked based on test To this aim,aim, in DEM simulation, frictions particle/particle (see Table 1) were checked based on this test To this aim, in DEM simulation, the coefficient of static friction was a 0.18-0.34 range with a step of 0.04, whereas the the coefficient static friction was varied in DEM ain0.18-0.34 range with asame step of 0.04, whereas hibiting a fillofheight of 100 mm Invaried the simulation, the procedure wasthe applied the coefficient offriction staticwas friction was varied in a -0.18-0.34 aA step of 0.04, whereas the of rolling varied between 0.05 0.14 arange step ofwith 0.03 macroscopic property, coefficient of rolling varied between 0.05 - 0.14 withwith a step of 0.03 Aofmacroscopic property, Ascoefficient friction plays thefriction mostwas critical role in the rheology behavior granular materials [39], coefficient of rolling friction was varied between 0.05 0.14 with a step of 0.03 A macroscopic the final mass retained in the silo after discharged, was chosen to make comparisons between experiment the final mass retained in theselected silo after discharged, wasof chosen to make comparisons betweenparticle/particle experimentproperty, the efficiency of the coefficients static and rolling frictions final mass retained in the silo after discharged, was chosen to make comparisons between experiment and DEM simulations andthe DEM simulations and DEM simulations Numerical investigation of force transmission in granular media using discrete element method 159 (see Tab 1) were checked based on this test To this aim, in DEM simulation, the coefficient of static friction was varied in a 0.18–0.34 range with a step of 0.04, whereas the coefficient of rolling friction was varied between 0.05–0.14 with a step of 0.03 A macroscopic property, the final mass retained in the silo after discharged, was chosen to make comparisons between experiment and DEM simulations RESULTS Nguyen Chung Thong,AND Le MinhDISCUSSIONS Lu, Ly Hai Bang and Le Tien Thinh Nguyen Chung Thong, Le Minh Lu, Ly Hai Bang and Le Tien Thinh 3.1 Validation of numerical model RESULTS AND DISCUSSIONS In this3.1 section, the numerical DEM model is compared with experimental work in the model RESULTS AND DISCUSSIONS Validation of numerical literature3.1 to Validation evaluateofthe effectiveness of the model 3(a) presents the initial assembly numerical modelDEM model is comparedFig In this section, the numerical with experimental work in the literature of particles in the box container, described in Section 2.2, whereas Fig 3(b) shows the to evaluate effectiveness of theDEM model Fig is 3acompared presents the assembly of particles in the box In thisthe section, the numerical model withinitial experimental work in the literature in Section 2.2,model whereas Fig.3a3b shows initial force chainofnetwork thethe medium, initial force chaindescribed ofofthe medium, as wellthe a visualization ofof in the compression tocontainer, evaluate thenetwork effectiveness the Fig presents theas initial assembly particles box as asdescribed a visualization of the compression and itsthe triangular mesh inmesh Section 2.2, whereas Fig.plate 3b shows initial force chain network of the medium, plate andcontainer, itswell triangular as well as a visualization of the compression plate and its triangular mesh (a) Fig (b) Fig Visualization of: (a) particle assembly at initial configuration and (b) initial force chain network and Fig Visualization of: (a) particlecompression assembly at initial configuration (b) initial force chain network and plate with triangularand mesh Visualization of: (a) particle assembly and (b) initial force compression plate at withinitial triangularconfiguration mesh chain As recommended byand various works in theplate literature [10,40], the coefficient static friction network compression with triangular mesh ofstatic As recommended by various in of thetheliterature [10,40], coefficient particle/particle is characterized by works the ratio shear stress to thethe normal stress,of while thefriction granular particle/particle is characterized shear stressthe to the normal stress, while the granular material is subjected to loading.byInthe thisratio caseofofthe compression, stresses in the x-axis (tangential to the material subjected to loading this case of compression, the stresses in the40], x-axis (tangential toby thetheof static directionis of compression) and In z-axis (normal to the of compression) were calculated As recommended by various works in thedirection literature [10, the coefficient direction of compression) andforces z-axisMore (normal to the such direction of compression) werebased calculated the corresponding wall reaction precisely, reactions were calculated on theby reaction friction particle/particle ismesh characterized bysuch the ratio[40] ofFigs the4ashear stress to the normal corresponding reaction forces Moreinprecisely, reactions were calculated on the forces in eachwall triangular element contact with particles andbased 4b show thereaction evolution forces in each triangular mesh element contact withThe particles [40] Figs 4aIn andthe 4bshear show the evolution stress, while the granular material is subjected to loading this case oftocompression, of normal stress and shear stress overin elapsed time comparison between stress normal ofstress normal stress and shear stress over elapsed time The comparison between the shear stress toin normal ratio and the work of Ghodki etto al the [33] direction for the considered granular material isand shown Fig.(normal 4c the stresses in the x-axis (tangential of compression) z-axis to stress ratio and the[33], workthe of Ghodki et al [33] for the considered is shown in Fig 4c.the In Ghodki et al inter-particle friction coefficient of granular 0.26 wasmaterial calibrated by combining the direction of compression) were calculated by the corresponding wall reaction forces Inexperimental Ghodki et al [33], the inter-particle friction coefficient of 0.26 was calibrated by combining the angle of repose test and DEM simulation (calibration result was indicated in Section 3.2 experimental of repose test simulation (calibration was indicated in increase Section More precisely, reactions were calculated based onresult thethe reaction in3.2 each trianin Ghodkisuch etangle al [33]) As can beand seenDEM in Fig 4c, the normal stress on wall starts forces to when inthe Ghodki et al [33]) As can be with seen the in Fig 4c, the normal A stress onovershoot the wall starts to observed, increase when compression plate contacts particle assembly small is also due to gular mesh element in with particles [40].AFigs 4(a) and 4(b)observed, showdue theto evolution the platecontact contactsparticles with theand particle assembly overshoot is plate also thecompression first interactions between compression plate.small The compression is vertically moved first interactions between particles and compression plate The The compression plateelements, is vertically of normalthe shear stress over elapsed time comparison between the shear instress order to and compress the granular material under constant velocity As for discrete themoved particles in order to compress the granular material under of constant discrete the particles granular stress to normal stress and the work Ghodki etAs al.for[32] forelements, the considered arranged in order toratio respond to the loading Finally, the velocity granular medium reaches a convergence in both arranged in order to respond to the loading Finally, the granular medium reaches a convergence in both the normal and shear stress Such convergence exhibits the the equilibrium of the granular mediumcoefficient under material is inshear Fig.stress 4(c).Such In convergence Ghodki etexhibits al [32], inter-particle of theshown normal and the equilibrium of the granularfriction medium under constant loading As shown in Fig 4c, the ratio of shear stress to normal stress at equilibrium state under constant loading As shown in Fig 4c, the ratio of shear stress to normal stress at equilibrium state under 0.26 was calibrated byiscombining the experimental angle of repose and DEM simulaconstant loading highly correlated compared with the work of Ghodki et al [33]test for the considered constant loading is highly correlated compared with the work of Ghodki et al [33] for the considered granular material, showing a high effectiveness of the proposed numerical DEM model tion (calibration result was indicated in Section 3.2 in Ghodki et al [32]) As can be seen granular material, showing a high effectiveness of the proposed numerical DEM model 160 Thong Chung Nguyen, Lu Minh Le, Hai-Bang Ly, Tien-Thinh Le in Fig 4(c), the normal stress on the wall starts to increase when the compression plate NguyenisChung Le Minh Lu, Ly Hai Bangfirst and Le Tien Thinh contacts with the particle assembly A small overshoot alsoThong, observed, due to the interactions between particles and compression plate The compression plate is vertically moved in order to compress the granular material under constant velocity As for discrete elements, the particles arranged in order to respond to the loading Finally, the granular medium reaches a convergence in both the normal and shear stress Such convergence exhibits the equilibrium of the granular medium under constant loading As shown in Fig 4(c), the ratio of shear stress to normal stress at equilibrium state under constant loading is highly correlated compared with the work of Ghodki et al [32] for the considered granular material, showing a high effectiveness of the proposed numerical DEM model.Nguyen 7 Chung Thong, Le Minh Lu, Lu, Ly Hai Bang andand Le Tien Thinh Nguyen Chung Thong, Le Minh Ly Hai Bang Le Tien Thinh Fig Evaluation of: (a) normal stress, (b) shear stress, and (c) shear stress / normal stress ratio over time (a) Normal Fig 8 (b) Tangential (c) Ratio In addition, the results of the silo discharge test are presented in Fig Visualization of dischar flow at different colored layers in a slice view mode is presented in Fig 5a, showing the retention zo in the silo and Fig 5b the stress/normal difference Δm between mass retained in the silo from DE Evaluation of: (a) normal stress, (b)flat-bottomed shear stress, (c)shows shear simulations and experiment, in function of the friction coefficients particle/particle It is shown that t stress difference ratio over time Δm could vary between and 30 g The mass retained in the experiment was 176.7 g It seen that the couple of (0.26, 0.08) allowed Nguyen Le Lu, Minh Haiand Bang Tien Thinh obtaining the smallest value of Δm (2.1 g) Thus, t Nguyen ChungChung Thong,Thong, Le Minh Ly Lu, HaiLy Bang Le and TienLe Thinh efficiency of the selected friction coefficients was confirmed, allowed having more confident results Evaluation of: normal (a) normal stress, (b) shear stress, (c) shear stress / normal stress time Fig.Fig Evaluation of: (a) stress, (b) shear stress, and and (c) shear stress / normal stress ratioratio overover time In addition, results of the discharge presented in Fig Visualization of discharge In addition, the the results of the silosilo discharge testtest are are presented in Fig Visualization of discharge at different colored layers a slice view mode is presented in Fig showing retention zone flowflow at different colored layers in ainslice view mode is presented in Fig 5a, 5a, showing the the retention zone in the flat-bottomed 5b shows difference between mass retained in the from DEM in the flat-bottomed silo.silo Fig.Fig 5b shows the the difference ΔmΔm between mass retained in the silosilo from DEM simulations experiment, in function of the friction coefficients particle/particle is shown simulations andand experiment, in function of the friction coefficients particle/particle It isItshown thatthat the the difference could between and 30The g The mass retained in the experiment 176.7 difference ΔmΔm could varyvary between and 30 g mass retained in the experiment waswas 176.7 g Itg.isIt is couple of (0.26, 0.08) allowed obtaining smallest value of Δm Thus, seenseen thatthat the the couple of (0.26, 0.08) allowed obtaining the the smallest value of Δm (2.1(2.1 g) g) Thus, the the efficiency of the selected friction coefficients confirmed, allowed having more confident results efficiency of the selected friction coefficients waswas confirmed, allowed having more confident results (a) Fig (a) (a) (b) (b) (b) Fig Results of silo discharge test: (a) visualization of particle flow at different colored layers and retention Fig.(b) evolution Results ofofsilo discharge (a) coefficient visualization of particle flow at different colored layers and retention in functiontest: of the of static friction particle/particle ofcolored 5.zone, Results silo Δm discharge test: (a)thevisualization of particle flow and at coefficient different zone, (b)of evolution of Δm in function coefficient of static friction particle/particle and coefficient of layers rollingof friction particle/particle and retention zone, (b) evolution of ∆m in particle/particle function of the coefficient of static friction rolling friction 3.2 Investigation of transmissionand of force particle/particle coefficient of rolling friction particle/particle 3.2 Investigation of transmission of force In this section, the numerical DEM model was used to investigate the transmission of force in the In this section, the numericalFig DEM modelthe wasevolution used to investigate transmission of force granular medium under compression shows of particle the velocity (z-velocity, x- in the granular under compression Fig.at6different shows the evolution of compression particle velocity xvelocity, and medium velocity magnitude, respectively) positions of the plate.(z-velocity, Fig velocity, and velocity magnitude, respectively) at different theforce compression plate Fig presents the corresponding configurations of the granular medium,positions includingofthe chain network presents the corresponding configurations of the granular forceparticles chain network (z-direction, x-direction, and force magnitude, respectively) It ismedium, seen thatincluding the most the moving x-direction, and force magnitude, It is seen that the moving particles are (z-direction, those in contact with the compression plate As therespectively) compression plate was moved in most the z-direction, the are velocity wasthe dominant compared to As other thoseininz-direction contact with compression plate thedirections compression plate was moved in the z-direction, the velocity in z-direction was dominant compared to other directions Numerical investigation of force transmission in granular media using discrete element method 161 In addition, the results of the silo discharge test are presented in Fig Visualization of discharge flow at different colored layers in a slice view mode is presented in Fig 5(a), showing the retention zone in the flat-bottomed silo Fig 5(b) shows the difference ∆m between mass retained in the silo from DEM simulations and experiment, in function of the friction coefficients particle/particle It is shown that the difference ∆m could vary between and 30 g The mass retained in the experiment was 176.7 g It is seen that the couple of (0.26, 0.08) allowed obtaining the smallest value of ∆m (2.1 g) Thus, the efficiency of the selected friction coefficients was confirmed, allowed having more confident results 3.2 Investigation of transmission of force In this section, the numerical DEM model was used to investigate the transmission of force in the granular medium under compression Fig shows the evolution of particle velocity (z-velocity, x-velocity, and velocity magnitude, respectively) at different positions of the compression plate Fig presents the corresponding configurations of the granular medium, including the force chain network (z-direction, x-direction, and force magnitude, respectively) It is seen that the most moving particles are those in contact with the compression plate As the compression plate was moved in the z-direction, the Nguyenwas Chung Thong, Le Minh Lu, Ly Haito Bang and directions Le Tien Thinh velocity in z-direction dominant compared other Fig Visualization of the velocity field of particles in the granular medium at different positions of the compression plate.of The colorbar was adapted for particles each velocityinfield order to explore the mostat appropriate Fig Visualization the velocity field of theingranular medium different positions vision effect of the compression plate The colorbar was adapted for each velocity field in order to explore the Regarding the force chain most network (Fig 7), at vision initial configuration (without loading from appropriate effect compression plate), the force chains with low amplitude were created at the bottom of the granular medium, showing the influence of the weight of particles at the top level However, at initial configuration, the force chains generally had no specific orientations, i.e., the contact forces were uniformly distributed in the medium When the compression plate contacts with the medium at heights of 270, 260, and 230 mm, the force chains were progressively created, also in increasing amplitude The contact forces in the z-direction were significant compared to those in the x-direction This is also proved when regarding the velocity field (Fig 6) This exciting result showed how the compression forces were transmitted through the particulate system The orientations of force chains are mainly parallel to the vertical axis, which is the direction of the compression loading The transmission network also provides information on the structural arrangement, related to the change of the microstructure to respond to the 162 Thong Chung Nguyen, Lu Minh Le, Hai-Bang Ly, Tien-Thinh Le Regarding the force chain network (Fig 7), at initial configuration (without loading from compression plate), the force chains with low amplitude were created at the bottom of the granular medium, showing the influence of the weight of particles at the top level However, at initial configuration, the force chains generally had no specific orientations, i.e., the contact forces were uniformly distributed in the medium When the compression plate contacts with the medium at heights of 270, 260, and 230 mm, the force chains were progressively created, also in increasing amplitude The contact forces in the z-direction were significant compared to those in the x-direction This is also proved when regarding the velocity field (Fig 6) This exciting result showed how the compression forces were transmitted through the particulate system The orientations of force chains are mainly parallel to the vertical axis, which is the direction of the compression loading The transmission network also provides information on the structural arrangement, related to the 10 Nguyen Chung Thong, Le Minh Lu, Ly Hai Bang and Le Tien Thinh change of the microstructure to respond to the loading Fig Visualization of force chain network in the granular medium at different positions of compression plate The colorbar was adapted each case in in order explore themedium most appropriate vision effect Fig Visualization of force chainfor network the togranular at different positions of compression plate The colorbar was adapted for each case in order to explore the most appropriate vision effect Fig 8(a) presents the increase of number of contact forces in function of fill height, normalized to the number of contact force at initial configuration (i.e., 100%), whereas Fig 8(b) shows the evolution of the number of contact forces in function of elapsed time Fig Evaluation of the number of contact forces in function of (a) fill height and (b) elapsed time Fig 8a presents the increase of number of contact forces in function of fill height, normalized to the number of contact force at initial configuration (i.e., 100%), whereas Fig 8b shows the evolution of the number of contact forces in function of elapsed time It is seen that the number of contact forces Numerical investigation of force transmission in granular media using discrete element method 163 It is seen that the number of contact forces linearly increased, as expected, because of the uniform movement of the compression plate At the equilibrium state, the number of contact forces was increased by about 170% and remained a horizontal asymptotic, as seen in Fig 8(b) The average and standard deviation values of the probability density distribution of the force chain network are also presented in Fig 9(a), and 9(b), respectively, in function of fill height As the compression is in the z-direction, the mean value of the contact force in the z-direction was the highest However, contact forces exhibit approximately the similar standard deviation values in all the directions Finally, Fig 9(c) presents the statistical distribution of the magnitude of contact forces, including their average and standard deviation Statistically, the contact forces increase in both amplitude Fig Visualization of force chain network in the granular medium at different positions of compression plate and7.Fig standard deviation Visualization of force chain network in the granular medium at different positions of compression plate The colorbar was adapted for each case in order to explore the most appropriate vision effect The colorbar was adapted for each case in order to explore the most appropriate vision effect Fig Evaluation ofnumber the number of contact forces in function heightand and(b) (b)elapsed elapsedtime time Fig Evaluation of (a) the of contact forces in function of of (a)(a) fillfill height (b) Fig.presents 8a presents the increase of number of contact forces functionofoffill fillheight, height,normalized normalized to to Fig 8a the increase of number of contact forces in in function the of contact at initial configuration (i.e., 100%), Fig.8b8bshows shows the evolution of the number of contact at initial (i.e., 100%), whereas the evolution of Fig number Evaluation offorce theforce number ofconfiguration contact forces in function ofwhereas (a) fillFig height and (b) elapsed time the number of contact forces in function of elapsed time It is seenthat thatthe thenumber numberofofcontact contact forces forces the number of contact forces in function of elapsed time It is seen The force vectors presented in Fig were employed to calculate the total force exerted on each particle in the system At the maximum compression point of 200 mm of height, Fig 10(a) presents the histogram of the number of particles in contact with a given particle, whereas the histogram of total force exerted on all the particles is shown in Fig 10(b) The total force exerted on a given particle was calculated by the sum of all the contact forces of its surrounding particles It can be noticed that at the maximum compression point of 200 mm, each particle was exposed to an average of surrounding particles, whereas the average of total force exerted was 33 N, with a standard deviation of about 13 N In Appendix, the measurement of the critical breakage force that causes the soybean grains to crack is presented Results showed that the critical compressive force was in the range of 50-70 N (approximately 1.5 mm of particle deformation) Based on the results obtained (Fig 10(b)) and the measurement of breakage force, it can be seen that the 200 mm compression point was a critical limit for maintaining the bond between particles If the compression increased further, the total force exerted on particles would also increase, leading to the destruction of particle bonds Nguyen Chung Thong, Le Minh Lu, Ly Hai Bang and Le Tien Thinh 11 linearly increased, as expected, because of the uniform movement of the compression plate At the equilibrium state, the number of contact forces was increased by about 170% and remained a horizontal asymptotic, as seen in Fig 8b The average and standard deviation values of the probability density distribution of the force chain network are also presented in Fig 9a, and 9b, respectively, in function of fill height As the compression is in the z-direction, the mean value of the contact force in the z-direction was the highest However, contact forces exhibit approximately the similar standard deviation values in all the directions Finally, distribution of the magnitude contact forces, 164 Fig 9c presents the statistical Thong Chung Nguyen, Lu MinhofLe, Hai-Bang Ly, Tien-Thinh Le including their average and standard deviation Statistically, the contact forces increase in both amplitude and standard deviation 12 12 Nguyen Chung Thong, LeLe Minh Lu,Lu, LyLy Hai Bang and LeLe Tien Thinh Nguyen Chung Thong, Minh Hai Bang and Tien Thinh Fig Evaluation of of contact force at different fillfill heights: (a)(a) average value, (b)(b) standard deviation value, and Fig Evaluation contact force at different heights: average value, standard deviation value, and (a) (b) (c)(c) contact force magnitude contact force magnitude The force vectors presented in in Fig were employed to to calculate thethe total force exerted The force vectors presented Fig were employed calculate total force exertedononeach each particle in in thethe system AtAt thethe maximum compression point ofof 200 mm ofof height, Fig 10a particle system maximum compression point 200 mm height, Fig 10apresents presentsthethe histogram of of thethe number of of particles in in contact with a given histogram number particles contact with a givenparticle, particle,whereas whereasthethehistogram histogramofoftotal total force exerted onon allall thethe particles is is shown in in Fig 10b The total force exerted onona given force exerted particles shown Fig 10b The total force exerted a givenparticle particlewas was calculated byby thethe sum of of allall thethe contact forces of of itsits surrounding particles It It can bebe noticed calculated sum contact forces surrounding particles can noticedthat thatat atthethe maximum compression point of of 200 mm, each particle was maximum compression point 200 mm, each particle wasexposed exposedto toananaverage averageofof8 8surrounding surrounding particles, whereas thethe average of of total force exerted was 3333 N,N, with a standard deviation ofof about 1313 N.N particles, whereas average total force exerted was with a standard deviation about In In thethe Appendix, thethe measurement of of thethe critical breakage force that causes thethe soybean grains toto Appendix, measurement critical breakage force that causes soybean grains crack presented Results showed that criticalcompressive compressiveforce forcewas wasininthetherange rangeofof50-70 50-70NN crack is is presented Results showed that thethecritical (approximately mm particledeformation) deformation).Based Basedononthetheresults resultsobtained obtained(Fig (Fig.10b) 10b)and andthethe (approximately 1.51.5 mm of ofparticle (c) measurement breakage force, it can seen that 200 mm compression pointwas wasa critical a criticallimit limit measurement of of breakage force, it can bebe seen that thethe 200 mm compression point maintaining bond between particles compression increased further, total force exerted forfor maintaining thethe bond between particles If If thethe compression increased further, thethe total force exerted Fig Evaluation of contact force at different fill heights: (a) average value, (b) standard deviation particles would also increase, leading destruction particle bonds onon particles would also increase, leading to to thethe destruction ofof particle bonds value, and (c) contact force magnitude Fig Evaluation contact force at Height = 200 mm: number particles in contact, distribution Fig 10.10 Evaluation of of contact at Height = 200 mm: (a)(a) number of of particles in(b) contact, (b)(b) distribution ofof (a) force total force exerted particles total force exerted onon particles Fig 10 Evaluation of contact force at Height = 200 mm: (a) number of particles in contact, (b) distribution of total force exerted on particles 3.3 Discussions 3.3 Discussions The main findings this work could summarized followings: The main findings of of this work could bebe summarized asas thethe followings: calibration procedure a numerical DEM model a granularassembly assemblywas was • •AA calibration procedure of of a numerical DEM model forfor a granular presented and validated with experimental data; presented and validated with experimental data; The DEM model was developed, allowed investigating transmissionofofforce force • • The DEM model was developed, allowed investigating thethetransmission granular medium under compression loading; in in thethe granular medium under compression loading; The force chain network was statistically quantified, showing responseofofthethe • • The force chain network was statistically quantified, showing thethe response granular medium under loading granular medium under loading Numerical investigation of force transmission in granular media using discrete element method 165 3.3 Discussions The main findings of this work could be summarized as the followings: - A calibration procedure of a numerical DEM model for a granular assembly was presented and validated with experimental data; - The DEM model was developed, allowed investigating the transmission of force in the granular medium under compression loading; - The force chain network was statistically quantified, showing the response of the granular medium under loading Indeed, for a given granular medium under loading, the force chain network could be considered as a load-bearing system [41] The force chain network allows the granular material to adapt itself in order to support different loadings and boundary conditions From a material point of view, the force chain network characterizes the microstructure of the granular material It has been pointed out that the granular material can change its microstructure in order to bear the given loadings [42] Moreover, the weak network of particles surrounds the force chain network has been reported as the principal energy dissipation source by various works in the literature [43] Therefore, the results of this study confirm the role of the force chain network in granular mechanics as such network governs the mechanical response of the materials In order to explore the influence of microscopic particle parameters on the force chain network, especially from friction point of view, DEM simulation for compression test was repeated in changing the value of the coefficient of static friction particle/particle and coefficient of rolling friction particle/particle, respectively The results of such a sensitivity analysis are presented in Fig 11 Figs 11(a) and 11(b) show the deviation of the number and magnitude of contact forces in function of the deviation of coefficient of static friction particle/particle, respectively (coefficient of static friction particle/particle varied in the range of [0.20, 0.26 and 0.32]) On the other hand, Figs 11(c) and 11(d) show the deviation of the number and magnitude of contact forces in function of the deviation of the coefficient of rolling friction particle/particle, respectively (coefficient of rolling friction particle/particle varied in the range of [0.05, 0.08 and 0.11]) It could be observed that changing the value of microscopic parameters of particles modified the force chain network in terms of both number and magnitude of contact forces, with different amplitudes at different heights A higher friction coefficient allowed obtaining a more significant number and magnitude of contact forces, which was also observed in different studies in the literature [44, 45] Besides, it can be seen that the coefficient of static friction particle/particle exhibited a more critical role than the rolling friction coefficient in the compression problem, especially in terms of the magnitude of contact forces The reason is that the rolling resistance is mainly considered in dynamic impact problems, as pointed out by Zhang et al [46] Certainly, further investigations should be carried out in order to explore the failure of the granular materials, i.e., buckling of force chains Such an investigation could indicate the stability of the force chain network and which parameters that the stability depends on On the other hand, the relationship between the force chain network and the energy of the particulate system should also be examined to clarify the energy particle/particle varied in range of [0.20, 0.26 and 0.32]) On other hand, Figs 11c and 11d the and magnitude ofand forces inother of thethe deviation of thethe thedeviation deviationofvaried ofthe thenumber and magnitude ofcontact contact forces infunction function of deviation ofshow particle/particle innumber thethe range of [0.20, 0.26 0.32]) On thethe hand, Figs 11c and 11d show the deviation of the number and magnitude of contact forces in function of the deviation of coefficient of rolling friction particle/particle, respectively (coefficient of rolling friction coefficient of rolling friction particle/particle, respectively (coefficient of rolling friction the deviation of the number and magnitude of contact forces in function of the deviation of thethe coefficient rolling friction respectively (coefficient rolling friction particle/particle inin the ofparticle/particle, 0.11]) bebeobserved changing thethe particle/particle varied therange range of[0.05, [0.05,0.08 0.08and and 0.11]).It Itcould could observed that changing coefficient ofofvaried rolling friction particle/particle, respectively (coefficient ofofthat rolling friction particle/particle varied in the range of [0.05, 0.08 and 0.11]) It could be observed that changing value of microscopic parameters of particles modified the force chain network in terms of both number value of microscopic parameters of particles modified the force chain network in terms of both number particle/particle varied in the range of [0.05, 0.08 and 0.11]) It could be observed that changing thethe value ofmicroscopic microscopic parameters particles modified force chain network terms of both number and ofofcontact forces, with amplitudes at different heights AA higher friction andmagnitude magnitude contact forces, withdifferent different amplitudes at different heights higher friction value of parameters ofof particles modified thethe force chain network in in terms of both number and magnitude of contact forces, with different amplitudes at different heights A higher friction coefficient allowed obtaining a more significant number and magnitude of contact forces, which was coefficient allowed obtaining a more significant number and magnitude of contact forces, which was and magnitude of contact forces, with different amplitudes at different heights A higher friction coefficient allowed obtaining a more significant number and magnitude of contact forces, which was also observed in different studies in the literature [44,45] Besides, it can be seen that the coefficient of also observed in different studies in the literature [44,45] Besides, it can be seen that the coefficient coefficient allowed obtaining a more significant number and magnitude of contact forces, which was of also observed differentstudies studies [44,45] Besides, itrolling can seen that the coefficient of static friction exhibited aliterature more role than friction coefficient ininthe static frictionparticle/particle particle/particle exhibited a morecritical critical role thanthe rolling friction coefficient also observed inin different inin thethe literature [44,45] Besides, itthe can bebe seen that the coefficient ofthe static friction particle/particle exhibited aofmore critical rolethan the rolling friction coefficient in compression problem, especially ininterms thethe magnitude ofthan contact forces The reason is isthat thethe compression problem, especially terms of magnitude of contact forces The reason that static friction particle/particle exhibited a more critical role the rolling friction coefficient in the 166 Thong Chung Nguyen, Lu Minh Le, Hai-Bang Ly, Tien-Thinh Le compression problem, especially terms magnitude contact forces The reason that the rolling resistance is is mainly considered in dynamic impact problems, asas pointed out byby Zhang etis al.al [46] rolling resistance mainly considered in dynamic impact problems, pointed out Zhang et [46] compression problem, especially ininterms ofofthethemagnitude ofofcontact forces The reason is that the rollingresistance resistanceis is mainly considered dynamic impact problems, pointed Zhang [46] rolling mainly considered inin dynamic impact problems, asas pointed outout byby Zhang et et al.al [46] (a) (b) Fig 11.11 Evaluation of of thethe number of of contact forces in in function of:of: (a)(a) coefficient of of static friction Fig Evaluation number contact forces function coefficient static friction particle/particle, (c) coefficient rolling friction particle/particle; evaluation ofcoefficient thethe magnitude offriction contact forces particle/particle, (c) coefficient of rolling friction particle/particle; evaluation of magnitude of contact forces Fig 11.11 Evaluation ofof theof number ofof contact forces in in function of:of: (a)(a) coefficient of of static Fig Evaluation the number contact forces function static friction (c) (d) particle/particle, (c)(c) coefficient ofof rolling friction particle/particle; evaluation of of thethe magnitude of of contact forces particle/particle, coefficient rolling friction particle/particle; evaluation magnitude contact forces Fig 11 Evaluation of the number of contact forces in function of: (a) coefficient of static friction particle/particle, (c) coefficient of rolling friction particle/particle; evaluation of the magnitude of contact forces in function of: (b) the coefficient of static friction particle/particle, (d) coefficient of rolling friction particle/particle dissipation Finally, more complex mechanical tests should be conducted, which could establish the relationship between the microscale parameters and macroscopic responses of the granular media CONCLUSIONS In this work, numerical simulation of the compression test for a granular medium has been presented to investigate the force transmission The numerical DEM model was calibrated, taking into account the microscale parameters of the granular medium, especially the particle size distribution, mechanical properties, coefficient of static friction, coefficient of rolling friction and coefficient of restitution The DEM model was Numerical investigation of force transmission in granular media using discrete element method 167 compared with experimental data, showing a good capability of simulation The force chain network of the granular medium was calculated, showing the crucial role of such a network on the macroscopic mechanical response of the medium Contact forces were analyzed statistically, presenting its evaluation in function of the applied load Results showed that the more the compression applied, the higher number of contacts forces was created, along with an increase of amplitude and standard deviation In addition, further works should be conducted and applied for different granular materials and also under different loading conditions Moving walls with constant lateral confinement should be investigated in further studies as they can affect the settlement of the sample in the zdirection and the static/dynamic states of the sample at the time step Finally, the failure of granular materials should be investigated regarding the collapse of the force chain network at the micro-level ACKNOWLEDGMENT The authors gratefully acknowledge the Vietnam National University of Agriculture for supporting this research REFERENCES [1] P A Cundall and O D L Strack A discrete numerical model for granular 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Agriculture In each test, three particles were positioned in the machine to form an equilateral triangle, as showing in Fig A1(a) Three tests were finally conducted, as shown in Fig A1(b) for displacement - compression force curves The breakage of particles was observed at 50-70 N for most of the particles (i.e., a shortening higher than 1.5 mm compared to the average particle diameter of 6.24 mm) It should be noticed that the critical value should also be selected based on the germination Nguyenafter Chung Thong, Le Minh Lu,higher Ly Haithan Bang85-90%) and Le Tien Thinh rate of particles being deformed (i.e., Consequently, 50 N was 15 finally chosenNguyen as a critical of soybean particles 15 Chungbreakage Thong, Le force Minh Lu, Ly Hai Bang and Le Tien Thinh (a) (b) Fig A1 Measurement of critical breakage force of particles: (a) compression test, (b) displacement - Fig.A1 A1.Measurement Measurement critical breakage force particles: compression displacement Fig of of critical 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Chareyre, L Sibille, Localized fluidization in granular materials: Theoretical geometric shapes in thermal environment, Vietnam Journal of Mechanics 40 (2018) 199–215  ... monitoring and analyzing the displacements and forces of all particles in the considered granular media Numerical investigation of force transmission in granular media using discrete element method. .. medium under loading granular medium under loading Numerical investigation of force transmission in granular media using discrete element method 165 3.3 Discussions The main findings of this work... velocity in z-direction was dominant compared to other directions Numerical investigation of force transmission in granular media using discrete element method 161 In addition, the results of the