Access the full text.
Sign up today, get DeepDyve free for 14 days.
Zijie Wang, G. Jing, Q. Yu, H. Yin (2015)
Analysis of ballast direct shear tests by discrete element method under different normal stressMeasurement, 63
Akbar Danesh, M. Palassi, A. Mirghasemi (2018)
Evaluating the influence of ballast degradation on its shear behaviourInternational Journal of Rail Transportation, 6
D. Nishiura, H. Sakai, A. Aikawa, Satori Tsuzuki, H. Sakaguchi (2017)
Novel discrete element modeling coupled with finite element method for investigating ballasted railway track dynamicsComputers and Geotechnics, 96
Hai Huang, S. Chrismer (2013)
Discrete element modeling of ballast settlement under trains moving at “Critical Speeds”Construction and Building Materials, 38
(2008)
Railway Ballast. China Railway Publishing House
N. Ngo, B. Indraratna, Cholachat Rujikiatkamjorn (2014)
DEM simulation of the behaviour of geogrid stabilised ballast fouled with coalComputers and Geotechnics, 55
S. Ji, Shanshan Sun, Ying Yan (2015)
Discrete Element Modeling of Rock Materials with Dilated Polyhedral ElementsProcedia Engineering, 102
M. Sol-Sánchez, N. Thom, F. Moreno-Navarro, M. Rubio-Gámez, G. Airey (2015)
A study into the use of crumb rubber in railway ballastConstruction and Building Materials, 75
E. Tutumluer, Y. Qian, Y. Hashash, J. Ghaboussi, D. Davis (2013)
Discrete element modelling of ballasted track deformation behaviourInternational Journal of Rail Transportation, 1
C. Ngamkhanong, S. Kaewunruen, C. Baniotopoulos (2017)
A review on modelling and monitoring of railway ballast
W. Zhai, Kaiyun Wang, Jianhui Lin (2004)
Modelling and experiment of railway ballast vibrationsJournal of Sound and Vibration, 270
Cheng Chen, G. McDowell, N. Thom (2012)
Discrete element modelling of cyclic loads of geogrid-reinforced ballast under confined and unconfined conditionsGeotextiles and Geomembranes, 35
Yunlong Guo, Chunfa Zhao, V. Markine, G. Jing, W. Zhai (2020)
Calibration for discrete element modelling of railway ballast: A reviewTransportation geotechnics, 23
Xu Zhang, Chunfa Zhao, W. Zhai (2016)
DEM Analysis of Ballast Breakage Under Train Loads and Its Effect on Mechanical Behaviour of Railway Track
Dinesh Gundavaram, S. Hussaini (2019)
Polyurethane-based stabilization of railroad ballast – a critical reviewInternational Journal of Rail Transportation, 7
Ivan Deiros, C. Voivret, G. Combe, F. Emeriault (2016)
Quantifying Degradation of Railway Ballast Using Numerical Simulations of Micro-deval Test and In-situ ConditionsProcedia Engineering, 143
Cheng Chen, B. Indraratna, G. McDowell, Cholachat Rujikiatkamjorn (2015)
Discrete element modelling of lateral displacement of a granular assembly under cyclic loadingComputers and Geotechnics, 69
W. Lim, G. McDowell (2005)
Discrete element modelling of railway ballastGranular Matter, 7
J. Eliáš (2014)
Simulation of railway ballast using crushable polyhedral particlesPowder Technology, 264
Junhua Xiao, De Zhang, Kai Wei, Zhe Luo (2017)
Shakedown behaviors of railway ballast under cyclic loadingConstruction and Building Materials, 155
X. Bian, Wei Li, Y. Qian, E. Tutumluer (2019)
Micromechanical Particle Interactions in Railway Ballast through DEM Simulations of Direct Shear TestsInternational Journal of Geomechanics
B. Indraratna, Qideng Sun, A. Heitor, J. Grant (2018)
Performance of Rubber Tire-Confined Capping Layer under Cyclic Loading for Railroad ConditionsJournal of Materials in Civil Engineering, 30
Shushu Liu, T. Qiu, Y. Qian, Hai Huang, E. Tutumluer, Shihui Shen (2019)
Simulations of large-scale triaxial shear tests on ballast aggregates using sensing mechanism and real-time (SMART) computingComputers and Geotechnics
(2002)
Railway geotechnics
(2017)
Numerical simulation and experiment study on the maco-meso mechanical behaviors of high-speed railway ballast
G. Jing, P. Aela, H. Fu (2019)
The contribution of ballast layer components to the lateral resistance of ladder sleeper trackConstruction and Building Materials
Xu Zhang, Chunfa Zhao, W. Zhai (2017)
Dynamic Behavior Analysis of High-Speed Railway Ballast under Moving Vehicle Loads Using Discrete Element MethodInternational Journal of Geomechanics, 17
M. Lu, G. McDowell (2006)
The importance of modelling ballast particle shape in the discrete element methodGranular Matter, 9
Association (1995) Manual for railway engineering 1995
Hai Huang, E. Tutumluer (2011)
Discrete Element Modeling for fouled railroad ballastConstruction and Building Materials, 25
N. Ngo, B. Indraratna, Cholachat Rujikiatkamjorn (2017)
Stabilization of track substructure with geo-inclusions—experimental evidence and DEM simulationInternational Journal of Rail Transportation, 5
J. Irazábal, F. Salazar, E. Oñate (2017)
Numerical modelling of granular materials with spherical discrete particles and the bounded rolling friction model. Application to railway ballastComputers and Geotechnics, 85
J. Harkness, A. Zervos, L. Pen, S. Aingaran, W. Powrie (2016)
Discrete element simulation of railway ballast: modelling cell pressure effects in triaxial testsGranular Matter, 18
Xu Zhang, Chunfa Zhao, W. Zhai (2019)
Importance of load frequency in applying cyclic loads to investigate ballast deformation under high-speed train loadsSoil Dynamics and Earthquake Engineering
B. Indraratna, T. Ngo (2018)
Ballast Railroad Design: SMART-UOW Approach
A. Bakhtiary, J. Zakeri, S. Mohammadzadeh (2020)
An opportunistic preventive maintenance policy for tamping scheduling of railway tracksInternational Journal of Rail Transportation, 9
Huiqi Li, G. McDowell (2018)
Discrete element modelling of under sleeper pads using a box testGranular Matter, 20
Ping Wang, Xiaochuan Ma, Jing-mang Xu, Jian Wang, Rong Chen (2019)
Numerical investigation on effect of the relative motion of stock/switch rails on the load transfer distribution along the switch panel in high-speed railway turnoutVehicle System Dynamics, 57
S. Lobo-guerrero, L. Vallejo (2006)
Discrete Element Method Analysis of Railtrack Ballast Degradation during Cyclic LoadingGranular Matter, 8
B. Indraratna, Wadud Salim (2003)
DEFORMATION AND DEGRADATION MECHANICS OF RECYCLED BALLAST STABILISED WITH GEOSYNTHETICSSoils and Foundations, 43
(2002)
Aggregates for railway ballast
Lei Xu, W. Zhai (2020)
Train–track coupled dynamics analysis: system spatial variation on geometry, physics and mechanicsRailway Engineering Science, 28
G. Jing, H. Fu, P. Aela (2018)
Lateral displacement of different types of steel sleepers on ballasted trackConstruction and Building Materials
Rail. Eng. Science (2020) 28(4):382–407 https://doi.org/10.1007/s40534-020-00216-9 Discrete element modelling of railway ballast performance considering particle shape and rolling resistance 1 2 1 2 3 • • • • • Yunlong Guo Chunfa Zhao Valeri Markine Can Shi Guoqing Jing Wanming Zhai Received: 11 May 2020 / Revised: 14 July 2020 / Accepted: 20 July 2020 / Published online: 27 August 2020 The Author(s) 2020 Abstract To simulate ballast performance accurately and assemblies in DSTs. In addition, the RRL contact model efficiently, the input in discrete element models should be can also provide accurate vertical and lateral ballast carefully selected, including the contact model and applied deformation under the cyclic loading in LPSTs. particle shape. To study the effects of the contact model and applied particle shape on the ballast performance Keywords Discrete element method Ballast performance (shear strength and deformation), the direct shear test Boundary condition Rolling resistance Direct shear test (DST) model and the large-scale process simulation test Lateral displacement (LPST) model were developed on the basis of two types of contact models, namely the rolling resistance linear (RRL) model and the linear contact (LC) model. Particle shapes are differentiated by clumps. A clump is a sphere assembly for one ballast particle. The results show that compared 1 Introduction with the typical LC model, the RRL method is more effi- cient and realistic to predict shear strength results of ballast Railways play a significant role in the transportation sys- tem worldwide and work in many sectors (urban rail, high- speed railway, heavy haul, intercity and metro) [1, 2]. & Can Shi Ballasted tracks, as the most widely used track type, consist shican@my.swjtu.edu.cn of rails, sleepers and the ballast layer [3, 4]. It possesses the Yunlong Guo advantages such as low construction cost, simple design yunlong.guo@tudelft.nl and construction, and easy maintenance [5]. Chunfa Zhao The ballast layer, a crucial component of ballasted track, cfzhao@swjtu.edu.cn provides resistances to sleepers, transmits and distributes Valeri Markine the loads or impacts from sleepers to the subgrade, as well v.l.markine@tudelft.nl as allows rapid drainage [6]. Generally, it is composed of Guoqing Jing blasted (quarried) rock aggregate, which is required to meet gqjing@bjtu.edu.cn certain characteristics such as narrow-graded (20–60 mm) Wanming Zhai and irregular particle shape, specific surface roughness, wmzhai@swjtu.edu.cn density, hardness, resistance to attrition and weathering [7]. Faculty of Civil Engineering and Geosciences, Delft Even though various railway ballast standards in terms of University of Technology, 2628CN Delft, The Netherlands particle size distribution or particle shape have already Train and Track Research Institute, State Key Laboratory of been formulated [7–9], their influences on ballast perfor- Traction Power, Southwest Jiaotong University, Chengdu mance (resilience, shearing strength, and settlement) have 610031, China not been sufficiently studied [10, 11]. School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China 123 Discrete element modelling of railway ballast performance considering particle shape and… 383 Laboratory or field tests are of limited use in studying be 4–6 times larger than the ballast particle (in laboratory the ballast performance, because the test conditions cannot tests), to ensure that the results are stable and unaffected be kept the same and many characteristics (e.g. ballast [19]. The boundary condition means that when a DEM density and sleeper type) cannot be controlled [12]. model represents only a part of the whole system (e.g. half- Additionally, due to the discrete nature of ballast, it is not sleeper track model for the whole ballasted track), the accurate or realistic to use the finite element method, which model boundary normally provides different reactions simulates the ballast layer as continuous layer [13]. The (displacements, forces). For example, when building the ballast performance keeps changing due to the ballast half-sleeper track model by DEM, the boundary of the degradation (abrasion and breakage) [12, 14–16]. In addi- ballast layer is mostly restricted (no displacements) [36]. tion, the sliding and rolling of individual ballast particles This will lead to false boundary-ballast reaction, since the also influence the performance of the ballast layer [17]. boundary imposes larger forces to the contact ballast par- The discrete element method (DEM) can overcome the ticles than in reality. When applying the dynamic loads, limitations of laboratory or field tests and the finite element such boundary condition will result false results due to models [18, 19]. As a powerful tool, it can (1) obtain all waves reflection effect. responses of the particles during simulations (e.g. velocity, To solve the issue of the boundary condition, the large- displacement, acceleration, and contact forces), (2) account scale process simulation test (LPST) model as described in for the properties of granular materials (density, size, and [18] was developed, in contrast to small-scale track model shape) [20], and (3) include the effects of breakage or (e.g. ballast box test model) [37]. It has five movable walls abrasion [17, 21–25]. at one side to provide consistent pressure stress during the Earlier studies have shown the feasibility of the DEM in cyclic loading, and in this way the boundary condition is evaluating the ballast performance [26–32]. However, included by moving the lateral walls and providing the there still exist some aspects for improvement. lateral deformation. On the one hand, the computational cost is the most Therefore, to develop an efficient and accurate method considerable limitation in developing DEM models that for DEM simulation, this work explores the effects of the may have millions of spheres (e.g. full-scale track model) rolling resistance linear (RRL) model on the ballast per- [21]. Larger number of particles means the increase in the formance of the direct shear test (DST) model and LPST total number of particle contacts, which results in great model. The shear strength and settlement of the RRL computational cost. This problem becomes more severe model is compared with those of the LC model. Specifi- when non-spherical particles are present in the DEM cally, the contact model of the spheres is the RRL, whereas models. The usage of the non-spherical particles can pro- the LC model is used for the non-spherical particles. vide more realistic load-deformation response [18, 33]. A non-spherical particle is generally made by a sphere assembly, named clump or cluster in the particle flow code 2 Methodology (PFC, commercial DEM software) [27, 34]. Using the non- spherical particles (sphere assembly) increases the spheres The DST model and LPST model are developed with the and the number of contacts (contact points between the commercial DEM software called Particle Flow Code in particles). The contacts are updated with every cycle 3D (PFC3D). The numerical results derived from these according to the force–displacement law, which finally models are compared with those from Ref. [18]. The increases the computation time considerably. adoption of two models can fully describe ballast perfor- In most cases, the contact method used in the earlier mance such as shear strength, resilience, settlement/per- models was elementary linear model (spring-damping manent deformation. Of these indicators, the shear strength model). By using the RRL, simple spheres can also be is most widely used and is measured generally by the DSTs possible to attain similar ballast performance, which can [12, 28, 38]. The settlement/permanent deformation is save a large amount of computational time. For example, it another key characteristic concerning the performance of was demonstrated that the linear rolling resistance contact ballast assemblies (especially in the field), and is measured model (using spheres as ballast particles) can obtain the by the cutting-edge LPSTs [18]. More importantly, this test same ballast lateral resistance results as those from field model is applicable to the lateral deformation of the ballast tests [35]. assemblies. On the other hand, even if the sleeper-ballast model uses the simple spherical or less-spherical particles, it has to be 2.1 DST developed in a large-scale manner (e.g. three-sleeper track model) due to the scale effect and the boundary condition. Figure 1 presents the setup of the DST and the corre- The scale effect means that the sample dimensions should sponding DEM models. The contact model parameters in Rail. Eng. Science (2020) 28(4):382–407 384 Y. Guo et al. Fig. 1 Schematic diagram of the applied methodology the DST model are calibrated using the DST results. The vertical and lateral hydraulic servo actuators can Afterwards, we compare the results obtained from two create the maximum loading of 30 t and 10 t, respectively different contact models, i.e. the RRL model and the LC (Fig. 1a). The vertical actuator can apply the normal force model. on the steel plate placed on the top of the upper box. This is utilised to provide a constant normal stress in ballast 2.1.1 Experimental samples. The lateral actuator is used to shift the lower box with a constant speed. In the DSTs, the ballast material is the commonly used The computer control system is utilised to measure aggregate of basalt rock produced in Quarry Pulandian, vertical and lateral displacements through the linear vari- Dalian, China. The ballast particles have a uniformed able differential transformers (LVDT). It also controls the shape, sufficient strength, and particle size distribution that application of the force or speed of the two hydraulic follow the British standard [7]. The ballast density is actuators and records the applied stress. 2530 kg/m . The ballast particles are placed in the shear box and The DST rig consists of three main parts: a steel square box, experience three steps. After placing ballast particles each two hydraulic servo actuators and a computer control system time, a vibratory compactor is used for compacting the (see Fig. 1a). The steel square consists of an upper steel square layer. After the third time of compaction, the steel plate box (inner size: 400 mm 9 400 mm 9 200 mm), a lower (weight of 25.64 kg) is placed on the top of the ballast steel square box (inner size: 400 mm 9 400 mm 9 200 mm) sample. Then, the direct shear tests are performed at a and a steel loading plate (size: 400 mm 9 400 mm 9 20 shearing speed of 2 mm/min under three different normal mm). There is a gap of 10 mm between the upper and lower stresses of 24, 54 and 104 kPa. The final horizontal boxes. Rail. Eng. Science (2020) 28(4):382–407 Discrete element modelling of railway ballast performance considering particle shape and… 385 displacement of the lower DST box is 80 mm (20% shear force is acting on it. To be more specific, the rolling fric- strain), which is adequate to obtain the peak shear stress. tion decides the maximum value that equals to the product of the rolling friction with the corresponding normal force. 2.1.2 DST model description The restriction is defined as rolling stiffness that is assumed as the clockwork spring (Fig. 2), and it increases with the The DST model (Fig. 1b) is utilised to measure the shear relative rotation. strength of the two contact models and four kinds of par- The four types of the particle shape used in the models ticle shapes. The porosity of the sample is 0.4, and the are a sphere, a 5-sphere clump, a 12-sphere clump and a particle size distribution (PSD) is based on the above 23-sphere clump. Note that one model corresponds to only experimental tests. Note that the PSD of all the models one type of particle shape. The clump particles are created remains the same. The model configuration is set as the with the identical template that was obtained by scanning experimental test configuration (Fig. 1a), including the box the real ballast particle [40]. size and the applied normal stresses. In addition, the normal stiffness and shear stiffness (the The basic contact mode of DEM is a kind of sphere– springs in Fig. 2) are another two parameters in the two sphere contact interactions. Even though in some models the contact models that considerably influence the calculation non-spherical particles (clumps) are used, the interaction in time. Figure 2 describes the LC model. the contact areas is still based on the sphere-sphere contact The calculation time is decided by the timestep calcu- model [39]. However, if the non-spherical particles are lated based on the two types of stiffness. Specifically, a present, the number of contact points increases and particle higher stiffness leads to a smaller timestep, causing more interlocking occurs, finally restraining the particle rotation. calculation time. The timestep is the smallest time period in On the other hand, if there are simple-shape particles simulation, in which the force–displacement law is applied (spheres) with certain rolling friction, it is also possible to to every updated contact. In other words, a particle moves result the same effect as the non-spherical particles [35]. at a speed in one timestep, and after the time is reached, the Therefore, the rolling friction [39]is used inthe DSTmodel. forces and displacements are updated. The specific intro- duction of the timestep can be found in [39]. For this, 2.1.3 Contact model and particle shape several values of these two parameters (shear and normal stiffnesses) are selected, and the results are compared for In order to determine whether the simple-shape (sphere) both efficient and accurate simulation. particles with the rolling friction can provide the same The properties of the DST model and contact model performance of the model as the complex-shape (clump) parameters are listed in Table 1, where four DST models respectively use four types of particle shapes, i.e. the particles, two types of contact model are utilised in the model, namely, the LC model and the RRL model. The sphere, 5-sphere clump, 12-sphere clump and 23-sphere models with the spheres use the RRL model, while the clump. For the DST model using spheres, the RRL model models with the clumps use the LC model. is utilised, and the particle-particle rolling friction coeffi- The RRL model has one more parameter (rolling fric- cient and the values of the two stiffness (normal and shear) tion) than the LC model. In other words, the only differ- are calibrated. For the DST models using non-spherical ence between the two contact models is the rolling friction. particles, the LC model is used and its results are compared The rolling friction will resist the particle rolling when a with that of the DST model using spheres. Fig. 2 Diagram of the normal stiffness and shear stiffness (modified after [39]) Rail. Eng. Science (2020) 28(4):382–407 386 Y. Guo et al. Table 1 Properties of DST model and contact model parameters Value With sphere Contact model type Rolling resistance linear contact model Particle type Sphere Density (kg/m ) 2530.0 Particle-particle friction coefficient 0.5 Particle-particle rolling friction coefficient 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6 5 6 7 8 Normal stiffness (N/m) 4 9 10 ,1 9 10 ,1 9 10 and 1 9 10 5 6 7 8 Shear stiffness (N/m) 4 9 10 ,1 9 10 ,1 9 10 and 1 9 10 Gravity (m/s ) 9.81 Damping 0.9 With clump Contact model type Linear contact model Particle type 5-sphere clump/12-sphere clump/23-sphere clump Density (kg/m ) 2530.0 Particle-particle friction coefficient 0.5 5 6 7 8 Normal stiffness (N/m) 4 9 10 ,1 9 10 ,1 9 10 and 1 9 10 5 6 7 8 Shear stiffness (N/m) 4 9 10 ,1 9 10 ,1 9 10 and 1 9 10 Gravity (m/s ) 9.81 Damping 0.9 Table 2 Model properties and parameters of large-scale process simulation test Value With sphere Contact model type Rolling resistance linear contact model Particle type Sphere Density (kg/m ) 2530.0 Particle-particle friction coefficient 0.5 Particle-particle rolling friction coefficient 0.3 5 5 5 6 Normal stiffness (N/m) 1 9 10 ,2 9 10 ,4 9 10 and 1 9 10 5 5 5 6 Shear stiffness (N/m) 1 9 10 ,2 9 10 ,4 9 10 and 1 9 10 Gravity (m/s ) 9.81 Damping 0.9 With clump Contact model type Rolling resistance linear contact model Particle type 5-sphere clump/12-sphere clump/23-sphere clump Density (kg/m ) 2530.0 Particle-particle friction coefficient 0.5 Particle-particle rolling friction coefficient 0.0 5 6 7 8 Normal stiffness (N/m) 4 9 10 ,1 9 10 ,1 9 10 and 1 9 10 5 6 7 8 Shear stiffness (N/m) 4 9 10 ,1 9 10 ,1 9 10 and 1 9 10 Gravity (m/s ) 9.81 Damping 0.9 The damping applied in the model is local damping (not comparison results. Moreover, it has been proved that damping at particle contacts), and the damping value is set different damping values have little influence on the shear according to Ref. [41]. Though there is no consensus in a strength results when the shear speed is very slow. High universal damping value, it has little influence on Rail. Eng. Science (2020) 28(4):382–407 Discrete element modelling of railway ballast performance considering particle shape and… 387 Fig. 3 Large-scale process simulation test and DEM model (Fig.3a reproduced from [18]) damping value tends to accelerate the formation of equi- by the overlapped spheres (clump), and the ballast particles librium state. are simulated with spheres or clumps (same as DST model). 2.2 Model description of large-scale process For the model with the spheres, the sample porosity is simulation test 0.354, which is larger than the one (0.338) in Ref. [18], whereas the models with three types of clumps have the The development of the LPST model refers to the LPST same porosity (0.338). Even though the porosity is differ- apparatus. As shown in Fig. 3a, the LPST apparatus was ent, the results have shown that their performances can still designed by Indraratna to develop physical simulation of be the same. ‘‘in situ’’ railway track. It can contain specimens that are The model properties and parameters are listed in 800 mm long, 600 mm wide, and 600 mm high [42]. Most Table 2, including density, friction, stiffness, rolling fric- importantly, one side of the apparatus is made by five tion, etc. The movable plates are simulated by walls that movable plates, which can provide consistent principal keep moving slightly to provide consistent principal stress stresses in the cyclic loading. More explanations on the (10 kPa). The maximum moving speed of the plates is set LPST apparatus can be found in Ref. [3]. as 10 mm/s. The LPST model shown in Fig. 3b includes sleeper, Four developed LPST models use four different types ballast layer and test box. The dimension of the specimen is of particle shapes, i.e. the sphere, 5-sphere clump, 800 mm 9 600 mm 9 475 mm, with the ballast thickness 12-sphere clump and 23-sphere clump. For the model (under the sleeper) of 325 mm. The sleeper is constituted with spheres, the RRL model is utilised and the values of Rail. Eng. Science (2020) 28(4):382–407 388 Y. Guo et al. 120 120 100 100 Experimental Experimental Normal and shear stiffness 1×10 Normal and shear stiffness 4×10 Normal and shear stiffness 1×10 Normal and shear stiffness 2×10 20 5 20 5 Normal and shear stiffness 5×10 Normal and shear stiffness 1×10 020 40 60 80 020 40 60 80 Displacement (mm) Displacement (mm) (a) Rolling friction coefficient 0.1 (b) Rolling friction coefficient 0.1 25 25 Experimental Normal and shear stiffness 4×10 20 20 Normal and shear stiffness 2×10 Normal and shear stiffness 1×10 15 15 10 10 Experimental Normal and shear stiffness 1×10 6 0 Normal and shear stiffness: 1×10 Normal and shear stiffness: 5×10 -5 -5 020 40 60 80 020 40 60 80 Displacement (mm) Displacement (mm) (c) Rolling friction coefficient 0.1 (d) Rolling friction coefficient 0.1 120 120 100 100 80 80 60 60 Experimental Experimental Rolling friction 0.1 Rolling friction 0.1 40 40 Rolling friction 0.2 Rolling friction 0.2 Rolling friction 0.3 Rolling friction 0.3 Rolling friction 0.4 Rolling friction 0.4 20 20 Rolling friction 0.5 Rolling friction 0.5 Rolling friction 0.6 Rolling friction 0.6 020 40 60 80 0 20406080 Displacement (mm) Displacement (mm) 5 5 (e) Normal and shear stiffness 1 ×10 (f) Normal and shear stiffness 2 ×10 Fig. 4 Shear stress and deformation results of the DST simulation with sphere under the normal stress 24 kPa Rail. Eng. Science (2020) 28(4):382–407 Dilation (mm) Shear stress (kPa) Shear stress(kPa) Dilation (mm) Shear stress (kPa) Shear stress (kPa) Discrete element modelling of railway ballast performance considering particle shape and… 389 Experimental Experimental Rolling friction 0.1 Rolling friction 0.1 Rolling friction 0.2 Rolling friction 0.2 Rolling friction 0.3 Rolling friction 0.3 Rolling friction 0.4 Rolling friction 0.4 Rolling friction 0.5 Rolling friction 0.5 Rolling friction 0.6 Rolling friction 0.6 0 2040 6080 020 40 60 80 Displacement (mm) Displacement (mm) 5 6 (h) (g) Normal and shear stiffness 4 ×10 Normal and shear stiffness 1 ×10 25 25 Experimental Experimental Rolling friction 0.1 Rolling friction 0.1 Rolling friction 0.2 Rolling friction 0.2 20 20 Rolling friction 0.3 Rolling friction 0.3 Rolling friction 0.4 Rolling friction 0.4 Rolling friction 0.5 Rolling friction 0.5 Rolling friction 0.6 15 Rolling friction 0.6 -5 -5 020 40 60 80 020 40 60 80 Displacement (mm) Displacement (mm) 5 5 (i) Normal and shear stiffness 1 ×10 (j) Normal and shear stiffness 2 ×10 30 40 Experimental Experimental Rolling friction 0.1 Rolling friction 0.1 Rolling friction 0.2 Rolling friction 0.2 Rolling friction 0.3 Rolling friction 0.3 Rolling friction 0.4 Rolling friction 0.4 Rolling friction 0.5 20 Rolling friction 0.5 Rolling friction 0.6 Rolling friction 0.6 -5 -5 020 40 60 80 020 40 60 80 Displacement (mm) Displacement (mm) 5 6 (k) (l) Normal and shear stiffness 4 ×10 Normal and shear stiffness 1 ×10 Fig. 4 continued Rail. Eng. Science (2020) 28(4):382–407 Shear stress (kPa) Dilation (mm) Dilation (mm) Dilation (mm) Shear stress (kPa) Dilation (mm) 390 Y. Guo et al. Experimental Experimental Rolling friction 0.1 Rolling friction 0.1 40 Rolling friction 0.2 Rolling friction 0.2 Rolling friction 0.3 Rolling friction 0.3 Rolling friction 0.4 Rolling friction 0.4 Rolling friction 0.5 20 Rolling friction 0.5 Rolling friction 0.6 Rolling friction 0.6 020 40 60 80 020 40 60 80 Displacement (mm) Displacement (mm) (a) Normal stress 24 kPa (b) Normal stress 54 kPa 24 kPa, experimental 54 kPa, experimental 104 kPa, experimental 24 kPa, numerical 54 kPa, numerical 104 kPa, numerical Experimental Rolling friction 0.1 Rolling friction 0.2 Rolling friction 0.3 Rolling friction 0.4 Rolling friction 0.5 Rolling friction 0.6 0 20406080 020 40 60 80 Displacement (mm) Displacement (mm) Normal stress 104 kPa (c) (d) Rolling friction 0.3 Experimental Experimental Rolling friction 0.1 Rolling friction 0.1 Rolling friction 0.2 Rolling friction 0.2 Rolling friction 0.3 Rolling friction 0.3 Rolling friction 0.4 Rolling friction 0.4 20 Rolling friction 0.5 Rolling friction 0.5 Rolling friction 0.6 Rolling friction 0.6 -5 -5 0 20406080 020 40 60 80 Displacement (mm) Displacement (mm) (e) Normal stress 24 kPa (f) Normal stress 54 kPa Fig. 5 Shear stress and dilation results of the DST simulation under the normal stress of 24, 54 or 104 kPa the two stiffness (normal and shear) are calibrated. For the DST model with spheres. The applied cyclic loading the LPST model with non-spherical particles, the LC frequency is 20 Hz and it is a sinusoidal loading from 50 model is used and the results are compared with those of to 460 kPa. Rail. Eng. Science (2020) 28(4):382–407 Shear stress (kPa) Dilation (mm) Shear stress (kPa) Dilation (mm) Shear stress (kPa) Shear stress (kPa) Discrete element modelling of railway ballast performance considering particle shape and… 391 24 kPa, experimental Experimental 54 kPa, experimental Rolling friction 0.1 104 kPa, experimental Rolling friction 0.2 24 kPa, numerical Rolling friction 0.3 54 kPa, numerical 20 104 kPa, numerical Rolling friction 0.4 Rolling friction 0.5 Rolling friction 0.6 6 15 -2 -4 -5 020 40 60 80 020 40 60 80 Displacement (mm) Displacement (mm) (g) Normal stress 104 kPa (h) Normal stressand rolling friction 0.3 Fig. 5 continued In Fig. 4e–h and i–l, the shear stress and deformation 3 Results and discussion results with increasing rolling friction are presented. From the shear stress results, it can be observed that the peak shear 3.1 DST simulation results stress considerably increase with the rolling friction, and 3.1.1 Contact model ballast assemblies with higher rolling friction needs larger shear displacement to reach the peak shear stress. Another Figure 4 presents the shear stress and deformation results fact is that with the higher stiffness, the peak shear stress increases at a faster rate than the rolling friction. From the of the DST model with sphere. The rolling friction coef- ficient is set as 0.1, and the normal stress is 24 kPa. deformation results, it can be seen that the deformation increases with the rolling friction, and high stiffness can cause From Fig. 4a and b, it can be seen that the peak shear large deformation change under the increasing rolling fric- stress increases with the stiffness; however, it increases tion. Through comparing the experimental results with sim- slightly and stays at around 60–70 kPa after the stiffness is ulation ones in Fig. 4, we find that the stiffness of 4 9 10 can over 4 9 10 . Moreover, the peak shear stress is reached with shorter horizontal box displacement, when the contact be chosen as the most suitable value for the DST model. 5 7 Based on the above results, both normal and shear stiffness is increased from 5 9 10 to 1 9 10 . Figure 4c and d presents the deformation results at different stiffness stiffness take the value of 4 9 10 . Nevertheless, we design the following simulation conditions to validate this values. They illustrate that lower stiffness will cause sig- nificant shear contraction, and higher stiffness can lead to value. The DST simulations under different normal stresses are performed and the results of shear stress and defor- deformation results more similar to the experimental test, 5 5 6 7 i.e. 4 9 10 ,5 9 10 ,1 9 10 and 1 9 10 . Based on the mation are shown in Fig. 5. Figure 5a–c present the shear stress results under the above results, it can be seen that the shear peak stress normal stress of 24, 54 and 104 kPa, respectively, and increases with the stiffness, but for the model with sphere Fig. 5d presents the shear stress with the rolling friction of and low rolling friction (0.1) it does not agree with the 0.3. From the figure, it can be seen that with the stiffness of experimental shear peak stress. In addition, it is reasonable that the peak shear stress appears when the shear dis- 4 9 10 , the shear stress results under three normal stresses are consistent with the experimental ones. More impor- placement is 30 mm. Particularly, lower stiffness leads to less computation tantly, it is shown that the rolling friction value of 0.3 can be selected for the following simulations that change par- time. For example, using the spheres with the stiffness at 5 7 1 9 10 and 1 9 10 take the computation time of 433 and ticles with different shapes (clumps). 1242 s, respectively. In the same test condition, using the sphere, 5-sphere clump, 12-sphere clump and 23-sphere 3.1.2 Particle shape clump take the computation time of 51, 80, 306 and 3.1.2.1 Shear stress and deformation In Fig. 6, the shear 400 min, respectively. This means using the spheres is 8 stress and deformation of the model with the spheres are times efficient at most. Rail. Eng. Science (2020) 28(4):382–407 Dilation(mm) Dilation (mm) 392 Y. Guo et al. Experimental Experimental Sphere 40 Sphere 5-Sphere clump 5-Sphere clump 12-Sphere clump 12-Sphere clump 24-Sphere clump 20 24-Sphere clump 0 0 020 40 60 80 020 40 60 80 Displacement (mm) Displacement (mm) (a) Normal stress 24 kPa (b) Normal stress 54 kPa Experimental Sphere 300 25 5-Sphere clump 12-Sphere clump 24-Sphere clump 250 20 150 10 Experimental Sphere 5-Sphere clump 100 5 12-Sphere clump 24-Sphere clump -5 0 20406080 020 40 60 80 Displacement (mm) Displacement (mm) (c) Normal stress 104 kPa (d) Normal stress 24 kPa Experimental Experimental Sphere Sphere 5-Sphere clump 5-Sphere clump 12-Sphere clump 12-Sphere clump 24-Sphere clump 24-Sphere clump -2 -5 020 40 60 80 020 40 60 80 Displacement (mm) Displacement (mm) (e) Normal stress 54 kPa (f) Normal stress 104 kPa Fig. 6 Shear stress and dilation results of the DST simulation with different particles Rail. Eng. Science (2020) 28(4):382–407 Shear stress (kPa) Shear stress (kPa) Dilation (mm) Dilation (mm) Shear stress (kPa) Dilation (mm) Discrete element modelling of railway ballast performance considering particle shape and… 393 Fig. 7 Force chain results of the DST simulation under the normal stress 104 kPa and shearing displacement 20 mm compared with those of the ones with the clumps (5-, 12- or 23-sphere clump, as some particles link each other to 23-sphere) and the experimental one. For the spheres, the become one big particle. The rolling friction has the same stiffness is 4910 and the rolling friction is 0.3 (the RRL effects as strengthening the contacts and acting as the model). For the clumps, the stiffness is 4 9 10 and no interlocks. rolling friction is applied (LC model). From Fig. 6a–c, it can For further testing the vertical settlement and lateral be observed that for the RRL model (i.e. applying rolling deformation of the sphere with rolling friction, the LPST friction), the simple sphere and complex shapes (clumps) model is developed. The simulation results are compared have similar shear stress results. In addition, it can be with the clumps and the results from Ref. [18]. observed that the shear stress of the 12-sphere clump is almost the same as that of the 23-sphere clump, but under the 3.1.2.2 Contact force analysis The contact force analysis normal stresses of 54 and 104, their peak shear stress values is crucial for observing the differences of different particle are lower than that of the 5-sphere clump. shapes and contact models at the mesoscopic level. Most of From Fig. 6d–f, it can be observed that the deformation the earlier studies utilised the criterions at the macroscopic results for using the sphere can better accord with the level, such as the shear strength in the DST (or triaxial test) experimental results than those for using the clump. The [38], and the friction angle in the hopper discharge [37]. 5-sphere clump deformation is higher than the deformation They compared the shear stress and strain or the repose of other two types of clump. The 23-sphere clump has the angle to present that the parameters in the model can be most realistic shape, but it provides the lowest deformation, confirmed. However, different parameters can similarly which is much lower than the experimental ones. match the same test results. In other words, large difference It is indicated that spheres with rolling friction can in the parameters of the contact model may still lead to replace complex-shaped particles (clumps). Interestingly, it similar response. For this reason, the analysis at the is found that the 23-sphere clump sample provides lower mesoscopic level is necessary to perform with respect to shear stress than the 5-sphere clump sample. This means the contact force chain, contact force distribution, and the interlocks of the 5-sphere clump is stronger than the coordination number. Rail. Eng. Science (2020) 28(4):382–407 394 Y. Guo et al. 90 Sphere, 92.6° 90 Sphere, 34.1° 45 5-Sphere clump, 87.6° 5-Sphere clump, 29.9° 120 60 120 60 12-Sphere clump, 100.8° 100 12-Sphere clump, 35.1° 23-Sphere clump, 88.7° 23-Sphere clump, 40.1° 150 30 25 150 30 60 0 0 180 0 0 0 180 0 210 330 210 330 240 300 240 300 270 270 l bl (a) Displacement 0 mm (b) Displacement 20 mm p p 90 Sphere, 29.7° 90 Sphere, 34.8° 140 5-Sphere clump, 35.9° 5-Sphere clump, 39.6° 120 60 120 60 12-Sphere clump, 30.6° 100 12-Sphere clump, 34.4° 23-Sphere clump, 37.5° 23-Sphere clump, 36.3° 80 150 30 150 30 60 0 0 180 0 0 0 180 0 80 210 330 210 330 240 300 240 300 270 270 (c) Displacement 40 mm (d) Displacement 60 mm 90 5-Sphere clump, 43.5° 90 Sphere, 34.1° 12-Sphere clump, 43.4° 5-Sphere clump, 43.5° 50 120 60 120 60 120 23-Sphere clump, 44.4° 12-Sphere clump, 43.4° 23-Sphere clump, 44.4° 100 40 30 150 30 150 30 0 0 180 0 0 0 180 0 40 15 210 330 30 210 330 240 300 50 240 300 (e) Displacement 80 mm (f) Enlarged figure θ = 92.6° 120 60 150 30 0 0 180 0 210 330 240 300 (g) Example of the diagram Fig. 8 Distributions of the particle normal contact forces under the normal stress 24 kPa Rail. Eng. Science (2020) 28(4):382–407 Enlarge Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Discrete element modelling of railway ballast performance considering particle shape and… 395 24 kPa, sphere 54 kPa, sphere 104 kPa, sphere 24 kPa, 5-sphere clump 54 kPa, 5-sphere clump 104 kPa, 5-sphere clump 24 kPa, 12-sphere clump 54 kPa, 12-sphere clump 104 kPa, 12-sphere clump 24 kPa, 23-sphere clump 6 54 kPa, 23-sphere clump 104 kPa, 23-sphere clump 20 40 60 80 Shearing displacement (mm) (a) Positions for coordination number calculation (b) Coordination number comparison Fig. 9 Coordination number comparison of the sphere and clump 3.1.2.3 Contact force chain The contact force chain is 29.9/30.6/36.3); afterwards, the value slightly increases. used for observing the force transmit and the shear band. Note that the lowest primary orientation with the spheres is The force chain results of the DST simulation are shown in approximately the same as that with the clumps except for Fig. 7, where the shear band with the spheres are wider the 12-sphere clump. than that with the clumps. The largest contact force values What is more, the contact force with the spheres is 2–2.5 are close, but the model with the spheres has large average times larger than that with the clumps. However, the contact forces and clear force chain. This can also be average contact forces are approximately the same for the observed in the other conditions with the shearing dis- models with the clumps (Fig. 8e, f). For example, in placements of 80 mm and normal stresses of 54/24 kPa Fig. 8b, the largest average contact force with the sphere is (Fig. 14,‘‘Appendix’’). This is because using the spheres around 100 N, and the smallest is around 40 N. Corre- reduce the numbers of particles and contacts, and then each spondingly, the models with clumps produce the maximum contact contributes larger forces. and minimum values of 40–50 N and 15–25 N, respec- For easy comparison, contact force anisotropy and their tively. It can also be seen that the average contact forces distribution are shown in Fig. 8g by the rose diagrams. In increase with the shearing displacement. Fig. 8, the contact force under the normal stress of 24 kPa It is significant to find that the average contact force of with 5 different shearing displacements (0 and 80 mm) is the spheres is 2–2.5 times larger than that of the clumps. presented, and all the rose diagrams are given in Fig. 15. This is because the contacts of the spheres are approxi- In Fig. 8g, the average contact force is calculated from mately half of the clumps, and thus every contact bear the projected forces. The contact forces are projected to the more shearing stress. Alternatively, every contact of the YZ plane, and the Y-axis directs the shearing direction, as spheres are strengthened. The contact number of each shown in Fig. 1b. The YZ plane is chosen as the shearing particle can be presented by the coordination number, direction has the most apparent contact force chain change which will be discussed in the following section. during shearing. The average contact force is calculated by averaging the 3.1.2.4 Coordination number The comparison of coor- forces within a certain angle range (every five degrees). dination number change is shown in Fig. 9b. The coordi- Specifically, the forces have a direction vector that has an nation number is the average number of active contacts for angle to the Y-axis. While 360 are divided every 5 into 72 each particle. The coordination number is calculated by the ranges, the forces with the direction vectors in one range use of the particles that lie at the shearing zone within the are averaged. The points in every ranges are connected to four measurement spheres, as shown in Fig. 9a. As the form one closed curve like the black curve in Fig. 8g. The shearing zone is the most important position to produce the red curve in Fig. 8g is obtained by smoothing the closed shearing stress, the particles at the shearing zone have the curve for observing the primary orientation more easily. most obvious movements. The radius of the measurement The primary orientation is the purple line drawn by eval- spheres is 0.1 m, and the coordinates of the measurement uating the direction of the red curve. Specifically, the spheres are (0.1, 0.18, 0.2), (0.3, 0.18, 0.2), (0.1, 0.3, 0.2) purple line separates the area into two equal ones. and (0.3, 0.3, 0.2). From this figure, it can be seen that with the increase of From Fig. 9, it can be seen that the coordination number the shearing displacement, the primary orientation increases as the particle shape is more complex (from decreases from around 90 (0 mm) to the lowest (29.7/ spheres to clumps). Moreover, with the increase of the Rail. Eng. Science (2020) 28(4):382–407 Coordination number 396 Y. Guo et al. Fig. 10 Particle rotation of using the spheres or clumps Rail. Eng. Science (2020) 28(4):382–407 Discrete element modelling of railway ballast performance considering particle shape and… 397 First 10 cycles 50th cycles 100th cycles 1000th cycles 0 0.5 1.0 1.5 2.0 8.5 9.0 Vertical settlement (mm) (a) Figure reproduced from [18] 5 6 7 8 4×10 1×10 1×10 1×10 300 300 200 200 100 100 0 0 0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Vertical settlement (mm) Vertical settlement (mm) (c) Sphere with rolling friction (b) Sphere with rolling friction 5 7 4×10 1×10 1×10 1×10 300 300 02468 10 12 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Vertical settlement (mm) Vertical settlement (mm) (d) 5 -Sphere clump without rolling friction (e) 5 -Sphere clump without rolling friction Fig. 11 Applied stress vs vertical displacement in the first 15 cycles normal stress (24, 54, and 104 kPa), the coordination 3.1.2.5 Particle rotation In order to confirm the effects number also increases, as the assemblies are more com- of the rolling resistance on the particle rotation, the sphere pacted. The 23-sphere clumps can produce approximately model with the RRL model is compared with the clump twice coordination number than the spheres. The coordi- model with the LC model, as shown in Fig. 10. This fig- ure illustrates the projection of all the particles’ rotation on nation number results demonstrate that the contacts of the spheres are less than those of the clumps. the Y–Z Plane, and particularly, the Y-axis is the DST box- shearing direction. The circles in the DST box represent the magnitude and position of the particle rotation, and the Rail. Eng. Science (2020) 28(4):382–407 Applied stress (kPa) Applied stress (kPa) Applied stress (kPa) Applied stress (kPa) Applied stress(kPa) 398 Y. Guo et al. 5 6 1×10 1×10 4×10 1×10 200 200 0 2 4 6 8 1012141618 02468 10 12 Vertical settlement (mm) Vertical settlement (mm) (f) 12 -Sphere clump without rolling friction (g) 12 -Sphere clump without rolling friction 7 8 5 6 1×10 1×10 4×10 1×10 0 5 10 15 20 Vertical settlement (mm) Vertical settlement (mm) (h) 23 -Sphere clump without rolling friction (i) 23-Sphere clump without rolling friction Fig. 11 continued 3.2 Large-scale process simulation test model circle colour helps to distinguish the rotation magnitude (Fig. 10a). The figure shows part of the results that are 3.2.1 Stiffness and particle shape obtained under the normal stress 104 kPa, and all the particle rotation results are given in Fig. 16 (‘‘Appendix’’). The particle rotation is calculated by Eq. (1)[39]. In the Figure 11 presents the applied stress vs vertical displacement equation, the Euler angles are utilised to calculate the particle with four kinds of particles and different normal and shear rotation (i.e. /, h and w), which present the precession rota- stiffnesses. From the figure, it can be observed that the elastic tion, nutation rotation and intrinsic rotation, respectively. deformation and plastic deformation reduce as the stiffness qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi increase. In addition, the results at the stiffness of 4 9 10 2 2 2 P ¼ ðÞ h þðÞ w þðÞ u : ð1Þ cannot accord with the results in Ref [18], where the elastic deformation and plastic deformation are within 0.5 mm From the figures, it can be observed that the particle (Fig. 11a). After the comparison, it is found that the results rotation of the sphere model is almost the same as that of the that correspond to the spheres or 5-sphere clumps with the clump model. Specifically, the largest rotation (over 180) 7 8 stiffness of 1 9 10 or 1 9 10 N/m (Fig. 11c, e) can appears at the similar positions, which are the left side of the approximately accord with those in [18]. This proves that upper shear box and the right side of the lower shear box, and using one set of contact model parameters may not be fit to all both of them are near the shearing interface. In addition, the tests, despite testing on the same material. Even though most of the large circles (green, purple, and red) appear the DST is a well-known method for confirming the along the diagonal line of the shear box and the line is approximately perpendicular to the contact chain direction. Rail. Eng. Science (2020) 28(4):382–407 Appliedstress (kPa) Applied stress(kPa) Applied stress (kPa) Applied stress (kPa) Discrete element modelling of railway ballast performance considering particle shape and… 399 Plate 1 Plate 1 Plate 1 2.0 Plate 2 Plate 2 Plate 2 Plate 3 Plate 3 Plate 3 5 1.5 Plate 4 Plate 4 Plate 4 Plate 5 Plate 5 Plate 5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.0 0.2 0.6 0.8 1.0 Time (s) Time (s) Time (s) (c) Sphere with rolling friction; stiffness 1×10 5 (b) Sphere with rolling friction; stiffness 1×10 (a) Sphere with rolling friction; stiffness 4×10 1.0 Plate 1 Plate 1 Plate 1 Plate 2 Plate 2 Plate 2 0.8 Plate 3 Plate 3 Plate 3 Plate 4 Plate 4 Plate 4 5 Plate 5 Plate 5 Plate 5 0.6 3 0.4 3 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Time (s) Time (s) 8 8 6 (d) Sphere with rolling friction and stiffness 1×10 (e) 5-Sphere clump; stiffness 4×10 (f) 5-Sphere clump; stiffness 1×10 1.0 Plate 1 6 Plate 1 Plate 1 1.6 Plate 2 Plate 2 Plate 2 0.8 Plate 3 Plate 3 5 Plate 3 Plate 4 Plate 4 Plate 4 1.2 Plate 5 Plate 5 4 Plate 5 0.6 0.8 3 0.4 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Time (s) Time (s) (g) 5-Sphere clump; stiffness 1×10 8 5 (h) 5 -Sphere clump; stiffness 1×10 (i) 12 -Sphere clump; stiffness 4×10 Plate 1 Plate 1 6 3.0 Plate 1 Plate 2 Plate 2 Plate 2 Plate 3 Plate 3 5 Plate 3 2.5 Plate 4 Plate 4 Plate 4 Plate 5 4 Plate 5 2.0 Plate 5 3 1.5 2 1.0 0.5 0.0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Time (s) Time (s) 6 8 (j) 12-Sphere clump; stiffness 1×10 (l) 12 -Sphere clump; stiffness 1×10 (k) 12-Sphere clump; stiffness 1×10 Plate 1 8 Plate 1 Plate 1 Plate 2 Plate 2 Plate 2 Plate 3 4 5 Plate 3 Plate 3 Plate 4 6 Plate 4 Plate 4 Plate 5 4 Plate 5 Plate 5 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Time (s) Time (s) 5 (n)23-Sphere clump; stiffness 1×10 7 (m) 23 -Sphere clump; stiffness 4×10 (o) 23 -Sphere clump; stiffness 1×10 4.0 Plate 1 Plate 2 Plate 3 3.0 DEM (N=1) Plate 4 DEM (N=100) Plate 5 DEM (N=1000) 2.0 Lab (N=1) Lab (N=100) Lab (N=1000) 1.0 Lab (N=250000) 0.0 010 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 Lateral displacement (mm) Time (s) 8 (q) Figure reproduced from [18] (p) 23-Sphere clump; stiffness 1×10 Fig. 12 Lateral displacement vs time of the five movable plates and the reference form literature [18] Rail. Eng. Science (2020) 28(4):382–407 Lateral displacement (mm) Lateral displacement (mm) Lateral displacement (mm) Lateral displacement (mm) Lateral displacement (mm) Lateral displacement (mm) Lateral displacement (mm) Lateral displacement (mm) Lateral displacement (mm) Distance above Lateral displacement (mm) Lateraldisplacement (mm) subballast (mm) Lateral displacement (mm) Lateral displacement (mm) Lateraldisplacement (mm) Lateral displacement (mm) Lateral displacement (mm) 400 Y. Guo et al. Plate 1 Plate 1 Plate 2 Plate 1 50 Plate 2 Plate 2 25 Plate 3 Plate 3 Plate 3 Plate 4 Plate 5 Plate 4 Plate 4 40 40 Plate 5 Plate 5 20 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Time (s) Time (s) 5 6 7 (c) Sphere with rolling friction; stiffness 1×10 (a) Sphere with rolling friction; stiffness 4×10 (b) Sphere with rolling friction; stiffness 1×10 35 70 50 Plate 1 Plate 1 Plate 2 30 Plate 2 Plate 3 Plate 1 Plate 3 Plate 4 Plate 2 Plate 4 Plate 5 Plate 3 25 Plate 5 Plate 4 Plate 5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Time (s) Time (s) 8 5 6 (d) Sphere with rolling friction; stiffness 1×10 (e) 5 -Sphere clump; stiffness 4×10 (f) 5-Sphere clump; stiffness 1×10 30 80 40 Plate 1 Plate 1 Plate 2 Plate 2 60 Plate 3 Plate 3 Plate 1 30 Plate 4 Plate 4 Plate 2 Plate 5 Plate 5 Plate 3 Plate 4 Plate 5 0 5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Time (s) Time (s) 7 8 5 (g) 5-Sphere clump; stiffness 1×10 (h) 5-Sphere clump; stiffness 1×10 (i) 12-Sphere clump; stiffness 4×10 35 120 Plate 1 Plate 2 Plate 3 30 100 Plate 4 Plate 1 Plate 5 Plate 1 Plate 2 Plate 2 25 80 Plate 3 Plate 3 Plate 4 Plate 4 Plate 5 20 60 Plate 5 15 40 10 20 5 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Time (s) Time (s) 6 8 (j) 12-Sphere clump; stiffness 1×10 (l) 12 -Sphere clump; stiffness 1×10 (k) 12-Sphere clump; stiffness 1×10 100 100 Plate 1 Plate 1 Plate 1 Plate 2 Plate 2 Plate 2 80 80 Plate 3 Plate 3 Plate 3 Plate 4 Plate 4 Plate 4 Plate 5 60 Plate 5 60 Plate 5 40 40 20 20 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Time (s) Time (s) 6 7 5 (n) 23 -Sphere clump; stiffness 1×10 (o) 23-Sphere clump; stiffness 1×10 (m) 23-Sphere clump; stiffness 4×10 Plate 1 Plate 2 Plate 3 Plate 4 Plate 5 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) (p) 23-Sphere clump; stiffness 1×10 Fig. 13 Lateral stress vs time of the five movable plates Rail. Eng. Science (2020) 28(4):382–407 Lateral stress(kPa) Lateral stress (kPa) Lateral stress (kPa) Lateral stress (kPa) Lateral stress(kPa) Lateral stress (kPa) Lateral stress (kPa) Lateral stress (kPa) Lateral stress(kPa) Lateral stress (kPa) Lateral stress (kPa) Lateral stress (kPa) Lateral stress (kPa) Lateral stress (kPa) Lateral stress (kPa) Lateral stress(kPa) Discrete element modelling of railway ballast performance considering particle shape and… 401 parameters in the numerical models, an extra test should be 4 Conclusions applied for confirming if the parameters are suitable for all tests. To increase the DEM simulation efficiency, the DEM Additionally, it also demonstrates that the sphere model models for the DST and the LPST are developed and with rolling friction can have the same or even better per- applied to analyse the ballast performance in terms of shear formance compared with the clump model. Particularly, the strength and deformation. The efficiency of different con- hypothesis of the interlocks (Sect. 3.1.2) can be proved again tact model types and particle shapes are studied. The by comparing the results in Fig. 11c, e, g and i. Specifically, numerical results are compared with the experimental the spheres with rolling friction has less vertical deformation results and results from the literature. From the results and than the clumps. Moreover, the 5-sphere clump has less discussion, the following conclusions can be summarised: deformation than the 12-sphere clump and 23-sphere clump. 1. Using spheres and linear rolling resistance model with 7 8 The stiffness cannot be 1 9 10 or 1 9 10 due to two properly chosen parameters, it is possible to simulate facts: (1) the initial stage (before applying loadings) of ballast performance accurately. The parameters can be ballast particle in the numerical simulation plays an confirmed by comparing the modelling results with the important role on the first a few cycles. This can be illus- experimental tests. trated from Fig. 11, showing that the first cycle has the 2. The RRL model can limit the particle movements by largest deformation than the other following cycles. In enhancing the forces at the contacts between particles, addition, there is large deformation in the first 5 cycles; complex shape particles with the LC model can afterwards, the deformation becomes small and stable, achieve the same performance in this way. which means the contacts between the sleeper and the ballast 3. The macroscopic ballast performance (e.g. shear particles become more and the ballast particles near the strength) is dependent on the particle contact at the sleeper are rapidly compacted. (2) The elastic deformation mesoscopic level (i.e. coordination number). The and plastic deformation values have a large range due to the performance differences of the different particle discrete nature of railway ballast [43, 44]. In response to this, shapes are mainly decided by the coordination number. the lateral displacement and stress results of five movable 4. After calibrating the contact model parameters of a test plates are presented and compared with the results in [18]. model, the numerical results can be quite approximate to the experimental ones; nevertheless, the calibrated 3.2.2 Lateral displacement and stress parameters may not be available for other test models. 5. The DEM models with spheres and the RRL model can The lateral displacement results of the five movable plates present similar macro performance with those with are shown in Fig. 12, and their stress results are shown in clumps if model parameters have suitable values. Fig. 13. From Fig. 12a–d, it can be seen that the lateral Nevertheless, these models have quite different parti- displacements reduce with the stiffness increase. Accord- cle scale performance; e.g. there are still large ing to Fig. 12q, after 100 cycles, the lateral displacements discrepancies among the particle performances (move- of all five movable plates are within 5 mm. To match the ments) in these models. test lateral displacements, the stiffness values of 4 9 10 and 1 9 10 are not suitable for the LPST model. The LPST model was tested in 15 loading cycles, and it Particularly, the stress from the five movable plates is necessary to observe the long-term deformation perfor- become a constant (10 kPa) for the stiffness of 1 9 10 ,as mance after thousands of cycles. Additionally, the spheres shown in Fig. 13d, h, l, and p. This demonstrates that the with the rolling friction can lead to the same results as sphere with rolling friction can provide adequately reliable those from the tests; however, the detailed reasons and results. To be more specific, in Fig. 13d, the stresses mesoscopic mechanics at the particle contacts can be become stable from 32 kPa to 10 kPa after one cycle, while analysed deeper in a way of the DEM simulations. It should the other particle shapes stabilize from much higher values, be emphasised that the contact model parameters need 75 kPa (5-sphere clump), 105 kPa (12-sphere clump) and further investigations to settle the most suitable ones. 185 kPa (23-sphere clump). In addition, the 12-sphere Finally, the particle degradation has not been considered in clump and 23-sphere clump need more cycles to become this work, and further studies will be performed in this stable, 11 and 9 cycles, respectively. respect. Acknowledgements The research was supported by the China Scholarship Council and the Natural Science Foundation of China (Grant No. 51578469). We also would like to acknowledge the sup- port of the Chinese Program of Introducing Talents of Discipline to Rail. Eng. Science (2020) 28(4):382–407 402 Y. Guo et al. Universities (111 Project, Grant No. B16041). We want to thank the indicated otherwise in a credit line to the material. If material is not support during my work at the International Joint Laboratory on included in the article’s Creative Commons licence and your intended Railway Engineering System Dynamics in Southwest Jiaotong use is not permitted by statutory regulation or exceeds the permitted University. We would like to thank Xu Zhang from Guangdong use, you will need to obtain permission directly from the copyright University of Technology, China and Shunying Ji from Dalian holder. To view a copy of this licence, visit http://creativecommons. University of Technology, China, for their contribution to this work. org/licenses/by/4.0/. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as Appendix long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this See Figs. 14, 15, 16, and Table 3. article are included in the article’s Creative Commons licence, unless Fig. 14 Force chain results of the DST simulation Rail. Eng. Science (2020) 28(4):382–407 Discrete element modelling of railway ballast performance considering particle shape and… 403 Fig. 14 continued Rail. Eng. Science (2020) 28(4):382–407 404 Y. Guo et al. 90 90 90 120 60 120 60 120 60 50 θ = 43.5° θ = 43.4° 120 50 θ = 34.1° 150 30 150 30 40 150 30 80 30 180 0 180 0 180 0 210 330 210 330 40 210 330 160 240 300 240 300 240 300 270 270 270 Sphere, 24 kPa, 80 mm 5-Sphere clump, 24 kPa, 80 mm 12-Sphere clump, 24 kPa, 80 mm 90 90 300 120 120 60 120 60 120 60 250 θ = 48.3° 100 θ = 44.4° θ = 41.8° 200 80 150 30 150 30 150 30 150 60 100 40 50 20 180 0 0 0 180 0 180 0 10 50 20 100 40 150 60 210 330 210 330 210 330 200 80 250 100 240 300 240 300 240 300 300 120 270 270 23-Sphere clump, 24 kPa, 80 mm Sphere, 54 kPa, 80 mm 5-Sphere clump, 54 kPa , 80 mm 90 90 90 120 450 120 60 120 60 120 60 θ = 43.5° 350 θ = 45.2° θ = 41.4° 80 300 150 30 150 30 250 150 30 50 200 40 150 180 0 180 0 180 0 40 40 150 210 330 210 330 210 330 80 300 90 350 240 300 110 240 300 240 300 120 450 270 270 270 12-Sphere clump, 54 kPa , 80 mm 23-Sphere clump, 54 kPa, 80 mm Sphere, 104 kPa, 80 mm 90 90 90 200 200 120 60 120 60 120 60 180 180 160 160 θ = 44.5° θ = 43.3° 160 θ = 44.3° 140 140 120 120 150 30 150 30 150 30 100 100 80 80 60 60 40 40 40 180 0 180 0 180 0 40 40 60 60 60 80 80 100 100 210 330 210 330 120 210 330 120 120 140 140 160 160 180 180 240 300 240 300 200 240 300 200 200 270 270 270 5-Sphere clump, 104 kPa, 80 mm 12-Sphere clump, 104 kPa, 80 mm 23-Sphere clump, 104 kPa, 80 mm Fig. 15 Distributions of the particle contact forces Rail. Eng. Science (2020) 28(4):382–407 Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Average contact force (N) Discrete element modelling of railway ballast performance considering particle shape and… 405 Fig. 16 Particle rotation illustration of the DST models Rail. Eng. Science (2020) 28(4):382–407 406 Y. Guo et al. References 1. Ngo NT, Indraratna B, Rujikiatkamjorn C (2017) Stabilization of track substructure with geo-inclusions—experimental evidence and DEM simulation. Int J Rail Transp 5(2):63–86 2. Xu L, Zhai W (2020) Train–track coupled dynamics analysis: system spatial variation on geometry, physics and mechanics. Railw Eng Sci 28(1):36–53 3. Indraratna B, Ngo T (2018) Ballast railroad design: SMART- UOW approach. CRC Press, Boca Raton 4. Li D, Hyslip J, Sussmann T et al (2002) Railway geotechnics. CRC Press, Boca Raton 5. Gundavaram D, Hussaini SKK (2019) Polyurethane-based sta- bilization of railroad ballast: a critical review. Int J Rail Transp 7(3):219–240 6. Bakhtiary A, Zakeri JA, Mohammadzadeh S (2020) An oppor- tunistic preventive maintenance policy for tamping scheduling of railway tracks. Int J Rail Transp. https://doi.org/10.1080/ 23248378.2020.1737256 7. BS EN 13450:2002 (2013) Aggregates for railway ballast. British Standards Institution, London 8. TB/T2140-2008 (2008) Railway Ballast. China Railway Pub- lishing House, Beijing 9. A.R.E. Association (1995) Manual for railway engineering 1995. American Railway Engineering Association, Mitchellville 10. Zhai W, Wang K, Lin J (2004) Modelling and experiment of railway ballast vibrations. J Sound Vib 270(4):673–683 11. Ji S, Sun S, Yan Y (2015) Discrete element modeling of rock materials with dilated polyhedral elements. Procedia Eng 102:1793–1802 12. Danesh A, Palassi M, Mirghasemi AA (2018) Evaluating the influence of ballast degradation on its shear behaviour. Int J Rail Transp 6(3):145–162 13. Wang P, Ma X, Xu J et al (2018) Numerical investigation on effect of the relative motion of stock/switch rails on the load transfer distribution along the switch panel in high-speed railway turnout. Veh Syst Dyn 57(2):226–246 14. Li H, McDowell GR (2018) Discrete element modelling of under sleeper pads using a box test. Granul Matter 20(2):26 15. Lobo-Guerrero S, Vallejo LE (2006) Discrete element method analysis of rail track ballast degradation during cyclic loading. Granul Matter 8(3–4):195–204 16. Elia´sˇ J (2014) Simulation of railway ballast using crushable polyhedral particles. Powder Technol 264:458–465 17. Liu S, Qiu T, Qian Y et al (2019) Simulations of large-scale triaxial shear tests on ballast aggregates using sensing mechanism and real-time (SMART) computing. Comput Geotech 110:184–198 18. Chen C, Indraratna B, McDowell G et al (2015) Discrete element modelling of lateral displacement of a granular assembly under cyclic loading. Comput Geotech 69:474–484 19. Ngo NT, Indraratna B, Rujikiatkamjorn C (2014) DEM simula- tion of the behaviour of geogrid stabilised ballast fouled with coal. Comput Geotech 55:224–231 20. Tutumluer E, Qian Y, Hashash YMA et al (2013) Discrete ele- ment modelling of ballasted track deformation behaviour. Int J Rail Transp 1(1–2):57–73 21. Jing G, Aela P, Fu H (2019) The contribution of ballast layer components to the lateral resistance of ladder sleeper track. Constr Build Mater 202:796–805 22. Xiao J, Zhang D, Wei K et al (2017) Shakedown behaviours of railway ballast under cyclic loading. Constr Build Mater 155:1206–1214 Rail. Eng. Science (2020) 28(4):382–407 Table 3 Coordination number of the DST models Shearing 24 kPa, 54 kPa, 104 kPa, 24 kPa, 54 kPa, 104 kPa, 24 kPa, 54 kPa, 104 kPa, 24 kPa, 54 kPa, 104 kPa, displacement sphere sphere sphere 5-sphere 5-sphere 5-sphere 12-sphere 12-sphere 12-sphere 24-sphere 24-sphere 24-sphere (mm) clump clump clump clump clump clump clump 0 4.9 5.0 5.1 7.4 7.9 8.6 7.8 8.5 9.1 8.5 9.3 9.9 20 4.4 4.7 5.2 7.5 8.2 9.2 7.7 8.6 9.7 8.5 9.6 10.9 40 4.4 4.6 5.0 7.1 7.8 8.9 7.7 8.3 9.5 8.5 9.5 11.2 60 4.2 4.6 4.9 7.2 7.8 8.5 7.7 8.6 9.7 8.4 9.4 10.4 80 4.3 4.4 4.6 7.0 8.0 8.7 7.3 8.4 9.6 8.0 9.8 10.7 Discrete element modelling of railway ballast performance considering particle shape and… 407 23. Zhang X, Zhao C, Zhai W (2016) Dynamic behavior analysis of of railway track. In: Proceedings of the 7th international con- high-speed railway ballast under moving vehicle loads using ference on discrete element methods, vol 188, pp 1323–1333 ´ ˜ discrete element method. Int J Geomech 17(7):04016157 35. Irazabal J, Salazar F, Onate E (2017) Numerical modelling of 24. Lim WL, McDowell GR (2005) Discrete element modelling of granular materials with spherical discrete particles and the railway ballast. Granul Matter 7(1):19–29 bounded rolling friction model: application to railway ballast. 25. Deiros I, Voivret C, Combe G et al (2016) Quantifying degra- Comput Geotech 85:220–229 dation of railway ballast using numerical simulations of micro- 36. Huang H, Chrismer S (2013) Discrete element modeling of bal- deval test and in-situ conditions. Procedia Eng 143:1016–1023 last settlement under trains moving at ‘‘critical speeds’’. Constr 26. Zhang X, Zhao C, Zhai W (2019) Importance of load frequency Build Mater 38:994–1000 in applying cyclic loads to investigate ballast deformation under 37. Chen C, McDowell GR, Thom NH (2012) Discrete element high-speed train loads. Soil Dyn Earthq Eng 120:28–38 modelling of cyclic loads of geogrid-reinforced ballast under 27. Jing G, Fu H, Aela P (2018) Lateral displacement of different confined and unconfined conditions. Geotext Geomembr types of steel sleepers on ballasted track. Constr Build Mater 35:76–86 186:1268–1275 38. Wang Z, Jing G, Yu Q et al (2015) Analysis of ballast direct shear 28. Bian X, Li W, Qian Y et al (2019) Micromechanical particle tests by discrete element method under different normal stress. interactions in railway ballast through DEM simulations of direct Measurement 63:17–24 shear tests. Int J Geomech 19(5):04019031 39. Itasca C, PFC (particle flow code in 2 and 3 dimensions), version 29. Lu M, McDowell GR (2006) The importance of modelling ballast 5.0 [user’s manual], Minneapolis, 2014 particle shape in the discrete element method. Granul Matter 40. Guo Y, Zhao C, Markine V et al (2020) Calibration for discrete 9(1–2):69–80 element modelling of railway ballast: a review. Transp Geotech 30. Nishiura D, Sakai H, Aikawa A et al (2018) Novel discrete ele- 23:100341 ment modeling coupled with finite element method for investi- 41. Zhang X (2017) Numerical simulation and experiment study on gating ballasted railway track dynamics. Comput Geotech the maco-meso mechanical behaviors of high-speed railway 96:40–54 ballast, Ph.D. thesis, Southwest Jiaotong University, Chengdu (in 31. Harkness J, Zervos A, Le Pen L et al (2016) Discrete element Chinese) simulation of railway ballast: modelling cell pressure effects in 42. Indraratna B, Salim W (2003) Deformation and degradation triaxial tests. Granul Matter 18(3):1–13 mechanics of recycled ballast stabilised with geosynthetics. Soils 32. Ngamkhanong C, Kaewunruen S, Baniotopoulos C (2017) A Found 43(4):35–46 review on modelling and monitoring of railway ballast. Struct 43. Indraratna B, Sun Q, Heitor A et al (2018) Performance of rubber Monit Maint 4(3):195–220 tire-confined capping layer under cyclic loading for railroad 33. Huang H, Tutumluer E (2011) Discrete element modelling for conditions. J Mater Civ Eng 30(3):06017021 fouled railroad ballast. Constr Build Mater 25(8):3306–3312 44. Sol-Sa´nchez M, Thom NH, Moreno-Navarro F et al (2015) A 34. Zhang X, Zhao C, Zhai W (2017) DEM analysis of ballast study into the use of crumb rubber in railway ballast. Constr breakage under train loads and its effect on mechanical behaviour Build Mater 75:19–24 Rail. Eng. Science (2020) 28(4):382–407
Railway Engineering Science – Springer Journals
Published: Aug 27, 2020
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.