Impact of Ballast Fouling on the Mechanical Properties of Railway Ballast: Insights from Discrete Element Analysis

By Luyu Wang

Department of Civil Engineering and Applied Mechanics

McGill University Montréal, Québec, Canada July 2020

A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Master of Engineering

Abstract

Ballast fouling is a major factor that contributes to the reduction of the shear strength of railway ballast, which can further influence the stability of railway equipment. Depending on the railway function and location, the major sources of ballast fouling are the infiltration of foreign fine material and ballast material degradation. The discrete element method (DEM) is an efficient way to capture the mechanical behavior of particles under various loading conditions. It has been successfully applied to investigate the mechanical behavior of ballast assembly and fouled ballast in the previous practices.

In this study, a discrete element model is developed in PFC3D to simulate the stress-strain behavior of fresh ballast assembly tested in the direct shear test. A series of sensitivity analyses of the different parameters are conducted to better understand the corresponding effect on the stress- strain behavior of ballast assembly. An efficient approach to simulate the foreign material induced fouling of ballast and capture the mechanical behavior of the material at various levels of void contamination index (VCI) is proposed. The approach is based on the concept that the foreign fouling material will not only change the inter-particle friction angle but also the inter-particle contact stiffness. Therefore, both the effective modulus and the inter-particle friction coefficient are adjusted based on the validated fresh ballast contact model. Besides, empirical equations are developed to efficiently calculate the micro-mechanical parameters and then validated by comparing the simulation results with those measured in the experiments. Besides, ballast degradation as another important source of ballast fouling is simulated by changing the particle size distribution (PSD) to investigate the effect of ballast degradation on the shear strength property. The results indicate that the shear strength decreases and then increases with the increasing ballast degradation levels.

This study provides an efficient and accurate approach to simulate ballast fouling induced by foreign material in the discrete element model. At the same time, the effect of ballast degradation on the shear strength property is investigated with the micro-mechanical insight. The results will be beneficial for the planning of future full-scale test to simulate ballast fouling and help researchers better understand the impacts of ballast degradation. Conclusions and limitations are presented in Chapter 5. More sets of micro-mechanical parameters are recommended to be examined in future research to prove that the proposed approach can not only capture the mechanical behavior of fouled ballast assembly but also the volumetric behavior. The particle surface texture modification induced by ballast degradation is also needed to be considered in future research to carry out a comprehensive investigation of the ballast degradation impacts.

Résumé

L'encrassement du ballast est un facteur majeur qui contribue à la réduction de la résistance au cisaillement du ballast ferroviaire, ce qui peut influencer davantage la stabilité de l'équipement ferroviaire. Selon la fonction et l'emplacement du chemin de fer, les principales sources d'encrassement du ballast sont l'infiltration de matières étrangères fines et la dégradation des matériaux de ballast. La méthode des éléments discrets (DEM) est un moyen efficace de saisir le comportement mécanique des particules dans diverses conditions de chargement. Elle a été appliquée avec succès pour étudier le comportement mécanique de l'assemblage du ballast et du ballast encrassé dans les pratiques précédentes.

Dans cette étude, un modèle d'élément discret est développé en utilisant PFC3D pour simuler le comportement contrainte-déformation d'un assemblage de ballast frais testé dans le test de cisaillement direct. Une série d'analyses de sensibilité des différents paramètres est réalisée pour mieux comprendre l'effet correspondant sur le comportement contrainte-déformation de l'assemblage de ballast. Une approche efficace pour simuler l'encrassement induit par les matières

étrangères du ballast et saisir le comportement mécanique du matériau à différents niveaux d'indice de contamination par les vides (VCI) est proposée. L'approche est basée sur le concept que le matériau d'encrassement étranger modifiera non seulement l'angle de frottement inter-particules mais également la rigidité de contact inter-particules. Par conséquent, le module effectif et le coefficient de frottement inter-particules sont ajustés en fonction du modèle de contact de ballast frais validé. De plus, des équations empiriques sont développées pour calculer efficacement les paramètres micromécaniques puis validées en comparant les résultats de simulation avec ceux mesurés dans les expériences. En outre, la dégradation du ballast comme autre source importante d'encrassement du ballast est simulée en modifiant la distribution de la taille des particules (PSD)

III pour étudier l'effet de la dégradation du ballast sur la propriété de résistance au cisaillement. Les résultats indiquent que la résistance au cisaillement diminue puis augmente avec l'augmentation des niveaux de dégradation du ballast.

Cette étude fournit une approche efficace et précise pour simuler l'encrassement du ballast induit par des matières étrangères dans le modèle à éléments discrets. En même temps, l'effet de la dégradation du ballast sur la résistance au cisaillement est étudié avec la compréhension micromécanique. Les résultats seront utiles pour la planification de futurs essais à grande échelle pour simuler l'encrassement du ballast et aider les chercheurs à mieux comprendre les impacts de la dégradation du ballast. Les conclusions et les limites sont présentées au chapitre 5. Il est recommandé d'examiner plus d'ensembles de paramètres micromécaniques dans de futures études de recherches afin de démontrer que l'approche proposée peut non seulement saisir le comportement mécanique de l'ensemble de ballast encrassé, mais également le comportement volumétrique. La modification de la texture de la surface des particules induite par la dégradation du ballast doit également être prise en compte dans les recherches futures pour mener une étude approfondie des impacts de la dégradation du ballast.

Acknowledgments

This study would not have been completed without the help from many people. First and foremost,

I would like to express my indebted appreciation to my supervisors, Professor Mohamed A.

Meguid and Professor Hani S. Mitri, for their expert supervision, encouragement, patience, and advice throughout this research project.

I also wish to express my sincerest gratitude to my colleagues in Geo-Group and special thanks to

Dr. Gao Ge for his technical support and valuable suggestions.

In addition, I would like to acknowledge the financial support of the scholarships from the Henan

Polytechnical University, China. and McGill University. It would not have been possible to finish this work without their generous support. I am really grateful for their support.

Last but not least, I would like to express my appreciation to my father Lidong Wang, mother

Luqing Fang, and girlfriend Yijie Zhao for their constant support, belief, and encouragement.

Table of Contents

Abstract ...... I

Acknowledgments...... V

Table of Contents ...... VI

List of Figures ...... IX

List of Tables ...... XII

List of Symbols ...... XIII

1 Introduction ...... 1

1.1 Background ...... 1

1.2 Research Motivation ...... 4

1.3 Research Objectives ...... 5

1.4 Thesis Outline ...... 6

2 Literature Review ...... 8

2.1 Introduction ...... 8

2.2 Ballast Fouling ...... 8

2.2.1 Sources of Ballast Fouling ...... 8

2.2.2 Ballast Fouling Measurement ...... 9

2.2.3 Effects of Ballast Fouling ...... 12

2.3 Discrete Element Method and PFC 3D ...... 15

2.3.1 Introduction to DEM and PFC3D ...... 15

2.3.2 Calculation Cycles and Time Step ...... 16

2.3.3 Linear Contact Model in PFC3D ...... 17

2.3.4 Particle Modeling in DEM ...... 19

2.4 Approaches of Ballast Fouling in DEM ...... 21

2.5 Summary ...... 24

3 Methodology ...... 26

3.1 Introduction ...... 26

3.2 Numerical Model Set-up in PFC3D ...... 27

3.2.1 Direct Shear Box Set-up ...... 27

3.2.2 Particle Modeling ...... 31

3.3 Sensitivity Analyses of Different Factors in DEM Model ...... 35

3.3.1 Shear Velocity ...... 35

3.3.2 Number of Spheres in Clump ...... 37

3.3.3 Porosity ...... 42

3.3.4 Effective Modulus ...... 44

3.3.5 Interparticle Friction Coefficient ...... 46

3.4 Fresh Ballast DEM Model...... 48

3.5 Summary ...... 54

4 DEM Simulation for Ballast Fouling ...... 55

4.1 Introduction ...... 55

4.2 Model Simulation Subjected to Different VCI Ratios ...... 55

4.3 Relationship between VCI Ratio and Micro-mechanical Parameter ...... 60

4.4 Relationship between VCI Ratio and Shear Strength ...... 64

4.5 Effect of Ballast Degradation on Stress-strain Behavior ...... 68

4.6 Summary ...... 73

5 Conclusion and Limitation ...... 75

VII

5.1 Conclusion ...... 75

5.2 Limitations ...... 76

Reference ...... 77

VIII

List of Figures

Figure 1-1 Typical track structures...... 2

Figure 1-2 Typical section view of ballasted track structure (Indraratna, et al., 2011b)...... 2

Figure 1-3 Contribution to settlement in rail track (Selig and Waters, 1994)...... 3

Figure 2-1 Various sources of ballast fouling (Selig and Waters, 1994)...... 9

Figure 2-2 Gradation variation from fresh to highly fouled ballast condition with increased FI

(modified from Indraratna et al. (2011c) and Selig and Waters (1994))...... 10

Figure 2-3 Shear strength results under different degrees of coal dust induced ballast fouling

(Huang et al., 2009)...... 13

Figure 2-4 Shear strength results under different level of ballast degradation (Danesh et al., 2018b).

...... 14

Figure 2-5 PFC3D model calculation cycle (Itasca, 2019)...... 17

Figure 2-6 Typical LCM and geometry...... 19

Figure 2-7 DEM simulation result against experimental result (Huang and Tutumluer, 2011). .. 22

Figure 2-8 Particle size distributions of clean and degraded ballast aggregates (Y. Qian et al., 2015).

...... 24

Figure 3-1 Direct shear box and ballast assembly in PFC3D...... 27

Figure 3-2 Particle size distribution curve in the experimental test...... 28

Figure 3-3 Schematic diagram of forces acting in the direct shear box...... 30

Figure 3-4 Ballast scan process...... 33

Figure 3-5 Different clumps created by bubble-pack algorithm under different ρ and φ...... 34

Figure 3-6 Irregularly shaped clumps used in this research...... 35

Figure 3-7 Shear stress-strain curves for different shear rate applied...... 37

Figure 3-8 Clump group library...... 40

Figure 3-9 Shear stress-strain behavior of each group...... 40

Figure 3-10 Coordination number of each group...... 41

Figure 3-11 Shear stress-strain behaviors at different porosity properties...... 43

Figure 3-12 Shear strength variation with porosity properties...... 43

Figure 3-13 Coordination number variation with porosity properties...... 44

Figure 3-14 Shear stress-strain behavior at different E*...... 45

Figure 3-15 Shear stress-strain behaviors at different μ...... 47

Figure 3-16 Shear stress-strain curves fitting for fresh ballast...... 49

Figure 3-17 Contact force distribution at εs = 0%, No. of contact = 26013, Max. contact force =

248.98N...... 51

Figure 3-18 Contact force distribution at εs = 3%, No. of contact = 20617, Max. contact force =

345.90N...... 51

Figure 3-19 Contact force distribution at εs = 7%, No. of contact = 21065, Max. contact force =

538.73N...... 52

Figure 3-20 Contact force distribution at εs = 13.3%, No. of contact = 19382, Max. contact force

= 426.86N...... 52

Figure 3-21 Particle displacement at εs = 3%...... 53

Figure 3-22 Particle displacement at εs = 13.3%...... 53

Figure 4-1 Concept explanation on the method to simulate fouled ballast...... 57

Figure 4-2 Comparisons between DEM simulation results and experimental results...... 59

Figure 4-3 Inter-particle friction coefficient variation with VCI ratio...... 61

Figure 4-4 Effective modulus variation with VCI ratio...... 61

Figure 4-5 Comparisons between DEM simulation result and experimental result for VCI 70%.

...... 63

Figure 4-6 Comparison between and model predicted shear strength and experimental shear strength...... 64

Figure 4-7 Strength envelopes of DEM model for different VCI ratios...... 66

Figure 4-8 Linear relation between hyperbolic constant a and b and shear strength reduction. ... 67

Figure 4-9 Definition of breakage index and PSD variations...... 68

Figure 4-10 Ballast PSD curves applied to degradation simulation...... 69

Figure 4-11 Shear strength variation with breakage index...... 71

Figure 4-12 Coordination number variation with breakage index...... 72

Figure 4-13 Porosity variation with breakage index...... 73

List of Tables

Table 2-1 Categories of ballast fouling defined by FI and PVC...... 10

Table 3-1 Micro-mechanical contact constants adopted for fresh ballast result simulation...... 48

Table 3-2 Micro-mechanical parameters of ballast assembly in previous DEM simulations...... 49

Table 4-1 Micromechanical parameters adopted for different VCI ratios simulation...... 58

Table 4-2 Collection of micro-mechanical parameters and VCI ratios...... 61

Table 4-4 Summary of shear strength value in simulation and experiment test...... 65

Table 4-5 Particle size distribution characteristics of the 12 gradations and corresponding breakage index, porosity and coordination number...... 70

XII

List of Symbols

HSR High-Speed Rail

DEM Discrete Element Method

PSD Particle Size Distribution

PFC3D Particle Flow Code in Three Dimensions

FI Fouling Index

PVC Percentage Void Contamination

VCI Void Contamination Index

LCM Linear Contact Model

PPC Particle-Particle Contact

PWC Particle-Wall Contact

E* effective modulus

μ inter-particle friction coefficient n porosity

Bi breakage index

φ angle parameter in bubble-pack algorithm

ρ ratio parameter in bubble-pack algorithm k* normal-to-shear stiffness ratio kn normal contact stiffness ks shear contact stiffness

R, L contact geometric parameters

푠 퐹푚푎푥 shear resistance

XIII

푠 퐹𝑖 shear contact force

푛 퐹𝑖 normal contact force

K contact spring stiffness m mass of the smallest particle

Δt critical timestep

FN normal force on shear plane

FS shear force on shear plane

Fn1 normal force act on left wall of upper section

Fn2 normal force act on right wall of upper section

Fs1 shear force act on top plate

Fs2 shear force act on front wall of upper section

Fs3 shear force act on back wall of upper section

σn normal stress on shear plane

σs shear stress on shear plane

D length of shear box

B width of shear box v shear velocity

* E P effective modulus for particle to particle contact

* E W effective modulus for particle to wall contact

Φ inter-particle friction angle before ballast fouled

Φ’ inter-particle friction angle after ballast fouled

K1 inter-particle contact stiffness before ballast fouled

K2 inter-particle contact stiffness after ballast fouled

XIV

푃4 percentages of ballast particle by weight passing the No.4 sieve

푃200 percentages of ballast particle by weight passing the No.200 sieve

V1 void volume between re-compacted ballast particles

V2 total volume of re-compacted fouling material ef void ratio of fouling material eb void ratio of fresh ballast

Gs.b specific gravity of ballast

Gs.f specific gravity of fouling material

Mf dry mass of fouling material

Mb dry mass of fresh ballast

I inertial number da average particle size

P confining pressure

ρd particle density

τf shear strength a, b hyperbolic constants

F(d) particle size cumulative distribution by mass

α fractal dimension d particle size dm minimum particle size dM maximum particle size

1 Introduction

1.1 Background

The railway track system is a critical part of the trade and transportation corridors of a country.

For the countries with a developed railway network, the railway has become the most popular transportation compared with others to deliver a large number of heavy goods and passengers from one transportation hub to another. In Japan, 24.6 billion passengers as almost 72.5% of the total passengers chose railway for their domestic travel in 2016 (Statistics Bureau Ministry of Internal

Affairs and Communications Japan, 2018). In Canada, more than 331.7 million tonnes of the industrial raw materials were transported through railway in 2018 which consists of coal, grain, chemicals, petroleum products, forest products, etc. (Transport Canada, 2019). Besides, the railway plays a key role in reducing emissions and mitigating global warming. According to

Transport Canada (2019), transportation by rail contributes to the reduction of greenhouse gas emissions because a 100-car freight train carrying 10,000 tonnes of goods can replace 300 trucks.

At the same time, in 2018, 25,700 tons of standard coal and 6,468 tons of the sulfur dioxide emission have been saved in China from the previous year due to the electrified railways had been widely constructed and upgraded (National Railway Administration of People's Republic of China,

2019).

The railway track structure usually has two formats, one is the ballasted track structure (Figure

1-1a) and the other is the slab track structure (Figure 1-1b). The slab track structure has been widely constructed in recent decades on high-speed rail (HSR) around the world. However, the ballasted track structure has 3 major advantages comparing with the slab track structure which are lower capital cost; easy to construction, and enable the track to be adjusted (Gillet, 2010).

Therefore, the ballasted track structure as a conventional track structure still occupies the major

1 proportion of railway mileage in most regions. Besides, the ballasted track structure also has been used in some special segments of the HSR such as long-span bridges, poor geological regions, and the railway hub station (X. Zhang et al., 2017).

(a) ballasted track structure. (b) slab track structure.

Figure 1-1 Typical track structures.

The superstructure and the substructure are the two basic units of a typical ballasted track structure.

The superstructure includes the rails, fasteners, and sleepers that rest on and are embedded in the substructure which consists of ballast, sub-ballast, and subgrade layers as shown in Figure 1-2.

Figure 1-2 Typical section view of ballasted track structure (Indraratna, et al., 2011b).

Ballast layer with a typical thickness from 250 to 450mm is placed at the top of the substructure where the sleepers are embedded. The layer is formed by the uniformly graded coarse granular aggregate and a great number of voids. The major functions of ballast layer are concluded by Selig and Waters (1994) as follows.

1. Provide stable support for the track and resist vertical (e.g. train loading), lateral (e.g.

curving and rail buckling), and longitudinal (e.g. train acceleration and braking) forces

applied to the sleepers.

2. Provide enough settlement to absorb the energy transmitted from the superstructure to the

underlying layers and protect the layers from the excessive permanent settlement as shown

in Figure 1-3.

3. Provide immediate water drainage for track structure through the voids.

Figure 1-3 Contribution to settlement in rail track (Selig and Waters, 1994).

As ballast ages, the process consists of the ballast degradation and foreign fine materials infiltration fill the voids within the ballast layer due to the repeated loading, local environment pollution, and material spilling from the wagon. These processes are regarded as ballast fouling (Selig and Waters,

1994). Ballast fouling is one of the primary reasons to prevent the ballast layer from fulfilling its functions as described above which can further impact the overall performance of the railway system (Selig and Waters, 1994). As a solution, periodic maintenance of ballast layer is required to mitigate the impacts of ballast fouling and to restore the design properties. However, due to ballast layer are especially vulnerable to ballast fouling, a large amount of money is spent on the maintenance of the railway track and nearly 40% of the overall funds are spent on solving ballast

3 fouling induced problems each year (Kerr, 2014). Therefore, ballast fouling should be investigated to reduce maintenance costs and optimize ballast usability.

1.2 Research Motivation

The discrete element method (DEM) (Cundall and Strack, 1979) as a particle-based numerical modeling method has been previously used to investigate mechanical behaviors of fresh ballast assembly (Gao and Meguid, 2018a; Hossain et al., 2007; McDowell et al., 2006; O'Sullivan et al.,

2008; Stahl and Konietzky, 2011). It was also used to investigate fouled ballast (Huang and

Tutumluer, 2011; Indraratna et al., 2014). In previous studies, ballast fouling induced by the infiltration of foreign fine material was simulated using two approaches to obtain corresponding mechanical behavior. In the first approach, small particles are injected into the ballast assembly

(Indraratna et al., 2014; Xu et al., 2016; Yanli, 2010). This approach accurately captures the mechanical behavior of fouled ballast assembly. However, it is computationally intensive due to the decrease in particle size and increase of number of particles (Cundall and Strack, 1979; Hossain et al., 2007). Therefore, this approach may be deemed unsuitable. The second approach relies on reducing the ballast particle surface friction angle without injection of new fine particles (Huang and Tutumluer, 2011; Huang et al., 2009). In this case, the computational time is significantly reduced and the influence of fine particles is reflected as lubricant between the ballast particles, thus affecting the inter-particle friction angle (Huang and Tutumluer, 2011; Rujikiatkamjorn et al.,

2012). The limitations of this approach are the calibration process is complex (Xu et al., 2016) and the simulation result is not as accurate as expected especially the elastic-plastic behavior of the ballast cannot be accurately reproduced. Given the above, it is necessary to develop an approach that can efficiently and accurately simulate the mechanical behavior of foreign material induced ballast fouling to benefit the future full-scale test in their planning to simulate the ballast fouling.

Mechanical ballast degradation, as another major source of ballast fouling, produces approximately 76% of the fouling material under normal conditions (Al-Qadi et al., 2008). In previous studies, ballast degradation was simulated by using breakable particles which are subjected to dynamic loading conditions in two-dimensional DEM model (Hossain et al., 2007;

Indraratna et al., 2020; Indraratna et al., 2010b; Lobo-Guerrero and Vallejo, 2006; J. Qian et al.,

2016). The degraded ballast mechanical behavior under static load conditions in three-dimensional was rarely investigated in DEM model. Furthermore, in the laboratory test, opposite conclusions about the effect of ballast degradation on the shear strength behavior have been reported. Danesh et al. (2018b) indicated that the shear strength is constantly decreased with the increasing level of ballast degradation. However, Y. Qian et al. (2015) concluded that the shear strength is increased by ballast degradation. Therefore, it is necessary to apply the DEM simulation to investigate the effect of ballast degradation on the shear strength property and attempt to explain the corresponding mechanism at the microscopic level.

1.3 Research Objectives

The main objectives of the research are:

1. Review the recent studies on ballast fouling analyses from experimental studies and

numerical studies.

2. Conduct sensitivity analyses on DEM model variables (i.e. shear velocity, number of

spheres in a clump, porosity, effective modulus, and interparticle friction coefficient) to

investigate their effects on the stress-strain behavior of ballast assembly.

3. Reproduce the stress-strain behavior of fresh ballast assembly and foreign material fouled

ballast assembly at different levels by using a new approach to keep the balance between

computational time and result accuracy.

4. Develop empirical equations between the degrees of foreign material induced ballast

fouling and corresponding micro-mechanical parameters based on the extensive results to

efficiently determine the required parameters at a given level of ballast fouling.

5. Investigate the effect of foreign material on the shear strength behavior of fouled ballast

assembly and driven the normalized shear strength reduction equations.

6. Investigate the effect of ballast degradation on the shear strength behavior of ballast

assembly from microscopic point of view.

1.4 Thesis Outline

This thesis has 5 chapters and beside this introduction chapter, the structure of the remaining chapters is as follows. Chapter 2 presents a literature review on the sources, quantification, and effect of ballast fouling. The basic knowledge of DEM and PFC3D consists of the calculation cycle, contact constitutive model, and particle shape evolution are introduced. The previous approaches about ballast fouling simulations in DEM model are reviewed.

In Chapter 3, the DEM model set-up process and irregularly shaped particle modeling process are presented. The sensitive analyses on shear velocity, the number of spheres in a clump, porosity (n), effective modulus (E*), and interparticle friction coefficient (μ) are conducted. The DEM model and values of E* and μ for the fresh ballast contact model are validated by comparing the DEM simulation results with the experimental results.

In Chapter 4, the stress-strain behavior of ballast fouling due to coal dust is reproduced by using the validated DEM through a new approach. The simulation results are validated by comparing with the experimental results under the same boundary conditions. Empirical equations are developed to efficiently determine the E* and μ values at a given ballast fouling level. The effect of foreign material on the shear strength reduction is investigated and the normalized shear strength reduction equations at different levels of ballast fouling are determined. Besides, ballast degradation is simulated in DEM model by changing the particle size distribution curves. The effect of ballast degradation on the shear strength property is evaluated at the microscopic point of view.

Chapter 5 presents the conclusions and limitations of this research work and the recommendations for future research.

2 Literature Review

2.1 Introduction

In this chapter, a comprehensive literature review on ballast fouling, DEM, and previous approaches of simulating ballast fouling in DEM are presented. Ballast fouling is reviewed in section 2.2 including the factors induced ballast fouling, ballast fouling quantification, and the effect of ballast fouling on the ballast assembly. Section 2.3 reviews the basic concepts of the DEM and PFC3D including the calculation mechanism, contact constitutive model, and particle shape evolution. The overview of the previous simulation approaches about ballast fouling in DEM and the limitations of each approach are introduced in section 2.4.

2.2 Ballast Fouling

2.2.1 Sources of Ballast Fouling

Ballast fouling is defined as the voids between the ballast particles within the ballast layer that are filled with materials in a relatively small size (Huang and Tutumluer, 2011). It is one of the primary causes of track deterioration. The sources of ballast fouling were summarized by Selig and Waters

(1994) based on the numerous field studies in North America and are listed as follows.

1. Ballast degradation.

2. Infiltration from ballast surface.

3. Sleeper (Tie) wear breakage.

4. Infiltration from underlying granular layers.

5. Subgrade infiltration.

The corresponding proportion of each source contributes to the ballast fouling under normal conditions is shown in Figure 2-1. It can be concluded that ballast degradation is the major source of the ballast fouling compared to foreign materials infiltration (Selig and Waters, 1994). However, the composition of ballast fouling components varies with the railway local environment and railway function. Danesh et al. (2018a) introduced that 60% of ballast fouling is generated by sandy fine infiltration in the desert regions. Feldman and Nissen (2002) indicated that 70-95% of ballast fouling is from fine coal particle infiltration due to wagon spillage at the transportation line for coal.

Figure 2-1 Various sources of ballast fouling (Selig and Waters, 1994).

2.2.2 Ballast Fouling Measurement

The degrees of fouling of ballast can be quantitatively represented by three different indices, namely Fouling Index (FI) (Selig and Waters, 1994), Percentage Void Contamination (PVC)

(Feldman and Nissen, 2002) and Void Contaminant Index (VCI) (Indraratna et al., 2010a). Selig and Waters (1994) proposed FI based on the gradation of numerous ballast samples obtained from various North American track sites (Figure 2-2). The definition of FI is as follows.

FI = 푃4 + 푃200 Equation 2-1

9 where 푃4 and 푃200 are percentages of ballast particle by weight passing the No.4 sieve (4.75 mm) and the No.200 sieve (0.075 mm), respectively. Besides, Selig and Waters (1994) related the categories of ballast fouling with FI as listed in Table 2-1.

Table 2-1 Categories of ballast fouling defined by FI and PVC.

Category FI (%) PVC (%) Clean <1 0 to < 20 Moderately clean 1 to < 10 - Moderately fouled 10 to < 20 20 to < 30 Fouled 20 to < 40 - Highly fouled ≥ 40 ≥ 30

Figure 2-2 Gradation variation from fresh to highly fouled ballast condition with increased FI (modified from Indraratna et al. (2011c) and Selig and Waters (1994)).

FI can efficiently represent the weight of small size particle percentage within the ballast assembly.

However, fouling material induced volumetric changes in the ballast voids cannot be efficiently reflected by FI, because the same weight of fouling materials have different specific gravities and occupy different volume in the voids (Indraratna et al., 2011c).

Feldman and Nissen (2002) proposed percentage void contamination (PVC) to measure the void volume change within the ballast assembly due to fouling material. The definition of PVC is as follows.

푉2 PVC = × 100% Equation 2-2 푉1 where 푉1 is the void volume between re-compacted ballast particles and 푉2 is the total volume of re-compacted fouling material (particle size smaller than 9.5 mm). Besides, Feldman and Nissen

(2002) introduced the PVC defined categories of ballast fouling condition as shown in Table 2-1

PVC can capture the void volume changes within the ballast assembly due to the fouling material.

However, by using the total volume of fouling material may overestimate the ballast fouling condition because the voids can exist between the fouling material for some relatively larger size fouling particles (Indraratna et al., 2011c).

Indraratna et al. (2010a) proposed the void contaminant index (VCI) to represent the degree of ballast fouling by considering the advantages and disadvantages of FI and PVC, respectively. VCI captures the influence of the void ratio, the specific gravity, and the mass of the foreign fouling materials on the ballast fouling result. The definition of VCI is as follows.

1 + 푒푓 퐺푠∙푏 푀푓 VCI = × × × 100 Equation 2-3 푒푏 퐺푠∙푓 푀푏 where 푒푓 is the void ratio of fouling material; 푒푏 is the void ratio of fresh ballast; 퐺푠∙푏 is the specific gravity of ballast; 퐺푠∙푓 is the specific gravity of fouling material; 푀푓 is the dry mass of fouling material; and 푀푏 is the dry mass of fresh ballast. VCI ratio accurately represents the percentage of the void volume within the fresh ballast assembly that firmly occupied by the fouling material. Besides, the VCI ratio is sensitive to the types of fouling material because it included the

11 specific gravity of the fouling material (Tennakoon et al., 2012). To this end, VCI is selected in this research to represent the degree of foreign material induced ballast fouling.

2.2.3 Effects of Ballast Fouling

Ballast fouling generates the unfavorable effects on ballast layer workability and further influence the railway system performance (Huang and Tutumluer, 2011). The previous studies on the effect of ballast fouling on the ballast layer are sorted into two categories based on the sources of ballast fouling which are foreign material infiltration and ballast degradation, the corresponding effects are introduced as follows.

The effect of foreign material on the mechanical behavior of ballast layer depends on the types of fouling material, degrees of fouling, and water content. Huang et al. (2009) conducted a series of large-scale direct shear tests to investigate the effect of different fouling materials on the shear strength behavior of ballast assembly. They indicated that the shear strength of the ballast assembly steadily decreases with the increasing degree of ballast fouling as Figure 2-3 shows. Wet fouling was found to accelerate this trend. Besides, the coal dust is the worst fouling material compared with clay soil and mineral finer in terms of shear strength reduction. They concluded that the fouling material acts as the lubricant that decreases the ballast inter-particle friction angle which can further decrease the load-bearing capacity of ballast layer. This observation is further analyzed by Indraratna et al. (2011a) that the lubricating effect of the fouling material also increases the dilation behavior of the ballast assembly as it can facilitate sliding and rolling of ballast particle.

Figure 2-3 Shear strength results under different degrees of coal dust induced ballast fouling (Huang et al., 2009).

Indraratna et al. (2013) conducted a triaxial test on fresh ballast and ballast fouled by clay. They concluded a same relationship between the degree of ballast fouling and shear strength as Huang et al. (2009) introduced. In addition, the external material generates the cushioning effect between the ballast particle which can reduce the amount of particle breakage and inter-particle contact stresses. As a result, the stiffness of the ballast assembly is decreased by foreign material. This observation is consistent with the result reported by Budiono et al. (2004) that the fine particle adversely affects the stiffness of track structure because the interaction between ballast particle is changed as the intruded fouling material.

Ballast degradation is the major source of ballast fouling and its impacts on the mechanical behavior of ballast assembly were investigated in the previous laboratory studies. Danesh et al.

(2018b) conducted a direct shear test on the degraded (crushed) ballast assembly which generated by different compression loads and the levels of ballast degradation was measured by using the

Fouling Index (FI). They indicated that the shear strength is constantly decreased with the increased ballast degradation level. In addition, the shear strength is rapidly decreased at the initial

13 stage of ballast degradation but the reduction rate becomes slow at the relatively high levels of ballast degradation as shown in Figure 2-4. Whereas, a contrary result was reported by Y. Qian et al. (2015) that ballast degradation increased the shear strength of ballast assembly compared with fresh ballast. They attributed this phenomenon to that the better gradation was obtained after the ballast degraded and the shear strength is correspondingly increased.

Figure 2-4 Shear strength results under different level of ballast degradation (Danesh et al., 2018b).

Besides, the effect of ballast degradation on the deformation of ballast layer was also investigated.

Y. Qian et al. (2014) conducted a triaxial test on the degraded ballast assembly. They prepared the degraded ballast particles by using the Los Angeles abrasion test which considered both size and surface texture changes induced by ballast degradation. They indicated that a higher permanent settlement is generated by ballast degradation due to finer and smoother particles that have been generated and the interparticle locking was gradually loosed. A similar result was reported by

Lobo-Guerrero and Vallejo (2006) that ballast degradation induced the increase of permanent deformation.

In conclusion, foreign fouling material generates lubricating and cushioning effect between the ballast particles to decrease the inter-particle friction angle and the inter-particle contact stress. As a result, the load-bearing capacity and material stiffness of the ballast layer are decreased. In addition, different materials generate different degrees of shear strength reduction under the same level of ballast fouling and shear strength reduction in wet conditions is greater than in dry conditions. In the end, ballast degradation increases permanent settlement, however, the influence on shear strength property of ballast layer is still unclear.

2.3 Discrete Element Method and PFC 3D

2.3.1 Introduction to DEM and PFC3D

Discrete Element Method (DEM) is a particle-based numerical modeling method and was firstly developed by Cundall (1971) for investigating rock mechanics problems and later applied to the granular materials mechanical study by Cundall and Strack (1979). In DEM, finite displacements and rotations of discrete bodies including complete detachment are allowed and new contacts are recognized automatically as the calculation progresses (Cundall and Hart, 1992). DEM provides a way to investigate the complex mechanical behavior of granular material assembly under various loading conditions at both microscopic and macroscopic levels. At the same time, ballast assembly as a typical granular material that moves independently and that interaction only occur at contact points is well simulated by using DEM.

In DEM, the interaction between particles is determined in a dynamic process with the state of equilibrium developing whenever the internal forces are balanced. The contact forces and displacements of particles are calculated by tracking the movements of each particle. These movements are induced by the disturbance caused by the specified wall and/or particle displacement, and body forces which are propagated through contacted particles. The dynamic

15 behavior of the particle is represented in the timestep algorithm. During one timestep, the velocity and acceleration of the particle are assumed as constants and the disturbance only propagates from the particle to its immediate neighbors. As a result, the force acting on any particle is determined exclusively by its interaction with the particles.

The commercial software Particle Flow Code in three dimensions (PFC3D) can be used to generate the stressed assemblies through the displacement of wall and interaction of particles by using DEM.

In PFC3D, the rigid spheres are used to represent single, breakable and unbreakable particles (Lim et al., 2005; J. Qian et al., 2016; Thornton, 2000), and the rigid wall is used to apply the velocity boundary condition to the assemblies for the purpose of compaction and confinement.

2.3.2 Calculation Cycles and Time Step

In a PFC3D model, the interaction force between particles and the displacement of each particle is determined through the force-displacement law and Newton’s second law within each timestep.

The principle of a calculation cycle is shown in Figure 2-5. At the beginning of the timestep, the contacts are detected and updated from the particle and boundary positions. Then, the contact force within the specified contact constitutive model is updated through the force-displacement law based on the relative displacement between the two contacted bodies. Newton’s second law is then applied to each particle to update its velocity and position based on the resultant unbalanced force and moment originate from the contact forces and the other forces acting on the particle. At the same time, the boundary positions are updated based on the specified boundary velocities.

Figure 2-5 PFC3D model calculation cycle (Itasca, 2019).

The particle kinematic behavior in the PFC3D model is determined in the timestep unit. The timestep should not exceed the critical timestep to guarantee that the disturbance only propagates from the particle to its immediate neighbor (Djordjevic, 2005). The critical timestep ∆푡 in a PFC3D model is related with the smallest particle size and its contact stiffness as follows.

푚 ∆푡 = √ Equation 2-4 퐾 where m is the mass of the smallest particle of the sample, and K is the contact spring stiffness

(O'Sullivan and Bray, 2004). The critical timestep length is increased by increasing the particle size or decreasing the contact stiffness, which in turn the computational time of the overall simulation is saved (Hossain et al., 2007; B. Mishra and Murty, 2001).

2.3.3 Linear Contact Model in PFC3D

The general behavior of the particle assembly is calculated in a PFC3D model through each contact and corresponding contact constitutive model. The linear contact model (LCM) is one of the contact constitutive model in PFC3D that has been widely used to simulate the ballast particle interaction in previous studies (Gao and Meguid, 2018a; Hossain et al., 2007; Indraratna et al.,

2014; Stahl and Konietzky, 2011; X. Zhang et al., 2017). In those studies, LCM was used to represent the particle to particle contact (PPC) as well as particle to wall contact (PWC). In the

LCM, the divider, spring, slider, and dashpot components are built between the contacted bodies

(Figure 2-6) to control the interaction between the contact bodies. The divider is to detect the occurrence of the contact, and the dashpot is to control the viscous behavior of the particle.

The spring controls the linear elastic behavior. The corresponding contact force and relative displacement are determined from the force-displacement law. The spring stiffnesses in the normal

(perpendicular to the contact surface) and tangential (parallel to the direction of the contact surface) directions are determined through the deformability method in PFC3D. In this method, the effective modulus E* and the normal-to-shear spring stiffness ratio k* are specified and related to the spring stiffness at normal and tangential directions as follows.

휋푅2퐸∗ 푘 = Equation 2-5 푛 퐿

푘 푘 = 푛 Equation 2-6 푠 푘∗

min⁡(푅 , 푅 )⁡⁡Particle⁡to⁡Particle⁡Contact 푅 = { 1 2 Equation 2-7 푅1, Particle⁡to⁡Wall⁡Contact

푅 + 푅 , Particle⁡to⁡Particle⁡Contact 퐿 = { 1 2 Equation 2-8 푅1, Particle⁡to⁡Wall⁡Contact where kn is the spring stiffness in the normal direction, ks is the spring stiffness in the tangential direction, R and L are the geometric parameters of the contact as shown in Figure 2-6 (a) and (c).

The contact stiffnesses are automatically scaled based on the individual particle size and the specified E* and k* values. Hence, the mechanical behavior of the assemblies with different particle sizes are associated with the emergent properties elastic modulus and the Poisson’s ratio in a macroscopic manner (Gao and Meguid, 2018a).

(a) Geometric parameters in (b) Rheological components (c) Geometric parameters in

PPC. in a linear contact model. PWC.

Figure 2-6 Typical LCM and geometry.

The slip behavior between the particles in contact is controlled by the slider in LCM. The interparticle friction coefficient μ is applied to define the occurrence of slip behavior as follows:

푠 푛 퐹푚푎푥 = 휇|퐹𝑖 | Equation 2-9

푠 푠 푡푟푢푒, |퐹𝑖 | > 퐹푚푎푥 푠푙𝑖푝 = { 푠 푠 Equation 2-10 푓푎푙푠푒, |퐹𝑖 | ≤ 퐹푚푎푥

푠 푛 푠 where the 퐹푚푎푥 is the shear resistance, 퐹𝑖 is the normal contact force at specific timestep i, 퐹𝑖 is the shear contact force at specific timestep i. The slip occurs when the shear contact force is greater than the shear resistance. The higher interparticle friction coefficient μ is, the slip behavior is more difficult to occur between the contacted bodies. Therefore, the shear strength and stiffness of granular material are increased (Wiącek et al., 2012).

2.3.4 Particle Modeling in DEM

Particle shape in a DEM model can cause a significant impact on the mechanical behavior of the assemblies (Lu and McDowell., 2007). Cundall and Strack (1979) represented the particle by using the circular particle in a two-dimension analysis at the very beginning. With the computational power increased, the spherical particle was then widely applied in the three-dimension analysis

(Cui and O'Sullivan, 2006; Lu and McDowell, 2008; Suiker and Fleck, 2004; Thornton, 2000).

The spherical particle provides more valuable information and is easier to relate with the real granular than the circular particle. However, both the circular and spherical particles produce the excessively rolling and insufficient interlocking during the test. Therefore, the mechanical behavior is far from the real situation (Iwashita and Oda, 1999). Lin and Ng (1997) conducted a

3D triaxial compression test by using elliptical and spherical shaped particle assembly, respectively. They found that the higher shear strength, greater initial stiffness, less particle rotation, more dilation, and coordination number (average number of contacts per particle) from the elliptical particle assembly results, which represents more realistic interaction is simulated by elliptical assembly.

In PFC3D, the single rigid sphere is the basic element and used to represent the particle. In addition, complex irregular-shape of the particle with breakable or unbreakable characteristics can also be created by using cluster or clump command in PFC3D through overlapping a different number of spheres. Lim and McDowell (2005) conducted a box test to simulate the traffic loading on the spherical particle assembly and cubic particle assembly that the particle is formed by clumping 8 spheres. They indicated that the cubic particle assembly is stiffer than the spherical one and less deformation is produced due to the interlocking mechanism is more realistically simulated. Lu and

McDowell (2008) contacted a triaxial test on the spherical, cubic, and triangular particle assembly, respectively. They found the triangular particle assembly shows the highest shear strength and initial stiffness, which indicated that the shear strength and initial stiffness of particle assembly are proportional with the particle angularity. Similar finding was reported by D. Mishra and Mahmud

(2017). Later, with digital technology and computational power developed, the particle with irregularly shaped was attempted to simulate in the PFC3D model. Stahl and Konietzky (2011) and

Huang and Tutumluer (2011) used the image aid technology to measure the major geometries of

20 the real ballast particle to create the irregularly shaped particle in PFC3D. They successfully reproduced the laboratory test measured the mechanical behavior of ballast assembly by using the

PFC3D model and the created particle model. Nowadays, ballast particle with angular and rough surface texture is accurately captured by a 3D scanner. The irregularly shaped particle is generated within the scanned contour by using the automatic clump generation and bubble-pack algorithm in PFC3D (Taghavi, 2011). As a result, more realistic ballast particle shape and more realistic interaction between particles are generated by using those advanced technologies (H. Li and

McDowell, 2018; Ling et al., 2020). Taking these advantages, the irregular shaped particle generated based on the scanned surface of real ballast particle is used in this research to represent the ballast particle model in PFC3D.

2.4 Approaches of Ballast Fouling in DEM

Ballast fouling that consists of ballast degradation and external material infiltration was attempted to be simulated in the DEM model in the previous studies by using limited approaches. Huang and

Tutumluer (2011) simulated the ballast assembly fouled by coal dust by only reducing the fresh ballast interparticle friction angle based on the concept of foreign material that generates the lubricating effect between fresh ballast. The shear strengths were well predicted in the model by comparing the simulation results with those measured in the laboratory experiment. However, the cushioning effect of the fouling material was neglected in this approach. As a result, the calibration process is complex (Xu et al., 2016) and the simulation result is not as accurate as expected especially the elastic-plastic behavior of the fouled ballast cannot be accurately reproduced as

Figure 2-7 shows. Z. Wang et al. (2015) simulated a direct shear test on ballast assembly to investigate the effect of inter-particle friction angle on the stress-strain behavior. They indicated that the inter-particle friction angle can efficiently modify the shear strength property of the ballast

21 assembly. However, rare changes in elastic-plastic behavior were observed after inter-particle friction angle changed. Similar results were also reported by Powrie et al. (2005), Lu and

McDowell (2008), and Thornton and L. Zhang (2010).

Figure 2-7 DEM simulation result against experimental result (Huang and Tutumluer, 2011).

Indraratna et al. (2014) proposed another one approach to simulate the coal dust-induced ballast fouling by injecting the small spheres into the ballast assembly. In their approach, the micro- mechanical parameters (e.g. contact stiffness and inter-particle friction coefficient) between the ballast particles were kept as constant as the fresh ballast condition but the reduced parameters were specified to the injected small spheres. This approach accurately predicts the ballast mechanical and volumetric behaviors by comparing the simulation results with the experimental results reported by Indraratna et al. (2011a). However, this approach is computationally intensive due to the decrease in particle size and increase of number of particles. Indraratna et al. (2014) reported that one fouled ballast simulation took approximately 500 hours. Hence, this approach is

22 difficult to be applied for larger-scale applications, especially for the full-scale track simulations in the future. Starfield and Cundall (1988) suggested that it is not necessary to develop a very large and complicated model considering all the uncertainties and input parameters making it too difficult to simulate.

Ballast degradation includes abrasion of sharp edges and split of particle into two or more pieces.

It was mainly treated as a dynamic process by using breakable particle and subjected to dynamic loading condition in DEM model in previous studies (Hossain et al., 2007; Indraratna et al., 2020;

Indraratna et al., 2010b; Lobo-Guerrero and Vallejo, 2006; J. Qian et al., 2016). Ballast degradation under static load conditions was alternatively simulated by changing the particle size distribution (PSD) curve because it shifts the PSD curve from uniformly graded to broader graded

(Buddhima Indraratna et al., 2005; Y. Qian et al., 2014). This approach was applied in a simulated triaxial test prepared by Y. Qian et al. (2015). Fresh ballast and degraded ballast without fines were represented by two different PSD curves as shown in Figure 2-8 and then tested by using DEM.

However, it should be mentioned that degraded ballast with fines conditions is difficult to simulate in the DEM model because it would dramatically increase the computational expense.

Figure 2-8 Particle size distributions of clean and degraded ballast aggregates (Y. Qian et al., 2015).

2.5 Summary

Ballast fouling is one of the primary reasons for track deterioration. Ballast degradation contributes to the largest proportion of ballast fouling and followed by foreign material infiltration. However, the proportion of sources of ballast fouling is varied with railway function and location. Ballast fouling levels are measured by different indices, namely FI, PVC, and VCI. The foreign material infiltration induced ballast fouling is accurately measured by using VCI. The foreign material within the ballast layer can generate the lubricating and cushioning effect to reduce the load- bearing capacity and material stiffness of the ballast layer. The degree of the influence of foreign material on mechanical behavior of ballast assembly depends on the types of material, the degree of fouling level, and water content. In addition, ballast degradation increases the ballast layer permanent settlement, however, the influence on shear strength property of ballast layer is still unclear.

DEM is an effective numerical modeling method to investigate the ballast mechanical behavior under different load conditions from the microscopic and macroscopic point of view. The foreign material induced ballast fouling has been simulated in DEM model through two approaches but either computational time-costly or inaccurately. Besides, ballast degradation can alternatively simulate in DEM model by decreasing the particle size of fresh ballast.

Based on the above literature review, an approach that can efficiently and accurately simulate the stress-strain behavior of the foreign material induced ballast fouling in DEM model is desired.

Moreover, it is imperative to investigate the effect of ballast degradation on the shear strength property of the ballast assembly by using the DEM model and attempt to explain the corresponding mechanism at the microscopic level.

3 Methodology

3.1 Introduction

Direct shear test is one of the most widely used laboratory tests to obtain the stress-strain behavior of granular materials. However, it is often not feasible to conduct this test on ballast particles by using standard geotechnical engineering laboratories because of the relatively large ballast particle size (nominal maximum particle size is 63.5mm). Indraratna et al. (2011a) conducted an experimental direct shear test on ballast assembly through a specially made shear box measuring

300 mm × 300 mm × 195 mm. In their test, the stress-strain behavior of the ballast assembly at different levels of coal dust-induced ballast fouling and under different normal stresses was measured. In this chapter, the direct shear box is simulated in PFC3D to reproduce the shear box used in experiment and the ballast assembly is simulated based on real ballast samples and similar physical parameters as the experiment. A series of sensitivity analyses is conducted to investigate the effect of different factors (shear velocity, number of spheres in the clump, porosity, effective modulus, and inter-particle friction coefficient) on the stress-strain behavior of ballast assembly.

The micro-mechanical parameters consist of effective modulus and the inter-particle friction coefficient for the fresh ballast contact model are calibrated by comparing the simulated shear stress-strain behavior with the experimental result. Besides, the microscopic behaviors including contact force distribution and displacement of ballast particles are observed to better understand the macroscopic behavior of ballast assembly. This simulated direct shear test is applied to obtain the stress-strain behavior of ballast assembly at different levels of foreign material induced ballast fouling and ballast degradation presents in Chapter 4.

3.2 Numerical Model Set-up in PFC3D

3.2.1 Direct Shear Box Set-up

In this research, a shear box measuring 300mm × 300mm × 195mm consisting of upper and lower sections is generated in PFC3D (Figure 3-1). Each section consists of five rigid walls, which are the front, back, left, right, and top/bottom walls. Two horizontal walls are attached as the wing on each section at mid-height on the left and right sides of the box to prevent any particle from escaping the box during shear.

Figure 3-1 Direct shear box and ballast assembly in PFC3D.

The expansion method which can create isotropic and homogenous ballast assembly is applied

(Jiang et al., 2003) to generate a total number of 8424 particles in the shear box with typical ballast density 2700 kg/m3 and diameters from 7.5 mm to 40 mm according to the particle size distribution

(Figure 3-2) reported by Indraratna et al. (2011a). The formed particles are based on 9 irregular shapes and are randomly positioned and orientated within the shear box without overlap. The particles are then rearranged based on the cycled calculation of unbalanced forces to satisfy the

27 required porosity (n) and the equilibrium state inside the shear box. The porosity of the ballast assembly after initial packing is taken as 0.45, which is similar to the reported porosity of the laboratory test.

100%

80%

60%

40% % Passing %

20%

0% 1 10 100 Particle Size (mm) Figure 3-2 Particle size distribution curve in the experimental test.

Once the particle generation process is completed and the equilibrium state is satisfied, normal stress is applied at the top and bottom of the sample (see Figure 3-3a) and kept constant by adjusting the position and velocity of the top and bottom walls through the servo-control feature in PFC3D (Itasca, 2019). It should be noted that the pre-defined shear plane is kept at mid-height of the ballast assembly through moving the top and bottom walls to prevent the progressive failure due to the non-uniformity of stresses and strains that develops in the ballast assembly during the shear process (Shibuya et al., 1997).

The lower section of the box is then moved horizontally under a constant rate to apply the shear force on the ballast assembly. A shear velocity of 4 mm/s is selected by comparing the shear stress- strain results to the shear velocity in different orders of magnitudes from 0.04 mm/s to 40 mm/s

(the detail comparison process is shown in section 3.3.1) to save the computational time and to

28 avoid the unduly disturbance on the stress-strain behavior of ballast assembly (Indraratna et al.,

2014; S. Liu, 2006; X. Wang and J. Li, 2014). Finally, like the laboratory test, the ballast assembly is sheared to a shear strain εs = 13.3% (approximately 40 mm of horizontal displacement).

The forces acting on the shear box are shown in Figure 3-3. The normal force (Fn) acting on the shear plane is equal to the normal load applied to the sample due to the gravity-free environment within the box. The shear force (Fs) acting on the shear plane is equal to the sum of the horizontal forces in the upper section and calculated in Equation 3-1 as follows.

퐹푠 = 퐹푛1 + 퐹푛2 + 퐹푠1 + 퐹푠2 + 퐹푠3 Equation 3-1 where Fn1 and Fn2 are the normal forces acting on left and right walls of the upper section, Fs1 is the surface shear force on the top plate, Fs2 and Fs3 are the surface shear force on the front and back walls of upper section as shown in Figure 3-3b.

(a) 2D view.

(b) 3D view.

Figure 3-3 Schematic diagram of forces acting in the direct shear box.

The normal and shear stress 휎푛 and 휎푠 acting on the shear plane are calculated based on Equation

3-2 and Equation 3-3 as follows.

퐹푁 휎 = Equation 3-2 푛 퐷(퐵 − 푣푡)

퐹푆 휎 = Equation 3-3 푠 퐷(퐵 − 푣푡) where 퐷 and 퐵 are the length and width of the shear box as shown in Figure 3-3b. The lower section moves at a constant velocity 푣. Thus, after a time 푡 has elapsed, the shear plane area is

퐷(퐵 − 푣푡).

In conclusion, the DEM model set-up process is summarized as follows.

1. Set up the test apparatus according to the defined dimensions.

2. Generate the ballast assembly according to the specified parameters.

3. Apply the constant normal pressure on the ballast assembly.

4. Move the lower section of the shear box to apply the shear force and measure the

corresponding stress-strain behavior.

3.2.2 Particle Modeling

The irregularly shaped clump (particle) is applied to represent the ballast particle in this DEM model. A clump is defined as a group of spheres with different sizes clustered into one unbreakable particle that acts like a single particle in the DEM model (Gao and Meguid, 2018b). Irregularly shaped clumps in a DEM model can realistically reflect the mechanical interaction between ballasts because they provide sufficient interlocking and avoid excessive rolling (Guo et al., 2020).

The particle shape characteristics (i.e. angularity and flakiness) can also affect the shear stress-

31 strain behavior of ballast assembly (Bian et al., 2019; Danesh et al., 2020; Kozicki et al., 2012;

Lekarp et al., 2000; Selig and Waters, 1994). In this thesis, 9 ballast particles which are randomly collected from an operating freight line in Quebec, Canada are used to generate irregularly shaped clumps to mitigate the influence of the particle shape and the shape characteristics on the mechanical behavior of the ballast assembly. The bubble-pack algorithm in PFC3D (Itasca, 2019) is used to generate the irregularly shaped clump model based on scanned real ballast contour

(Ferellec and McDowell, 2010; Taghavi, 2011).

The irregularly shaped clump in DEM model is generated in the following steps.

1. The real ballast particle (Figure 3-4a) is scanned by a 3D scan machine and a

stereolithography (STL.) file which records the contour information of the ballast (Figure

3-4b) is obtained and loaded into PFC3D by using the geometry command (Itasca, 2019).

2. The clump generation and bubble-pack algorithm in PFC3D (Itasca, 2019) are used to

overlap a different number of spheres and to fill the space enclosed by the contour of the

real ballast particle.

3. The two parameters which consist of the angle (φ, range from 0° to 180°) and the ratio (ρ,

range from 0.0 to 1.0) in the bubble-pack algorithm are specified by the user to modify the

coordinate and size of the spheres within the clump as shown in Figure 3-5.

(a) Ballast particle

(b) Corresponding contour file

Figure 3-4 Ballast scan process.

Figure 3-5 Different clumps created by bubble-pack algorithm under different ρ and φ.

In the bubble-pack algorithm, the angle φ is defined as the maximum intersection angle between two contact spheres and has a positive relationship with the smoothness of the particle surface.

The parameter ρ represents the ratio of the smallest sphere to the largest sphere; smaller ρ implies finer particle shear feature is reflected (Taghavi, 2011). Normally, a greater number of spheres are required to generate a more realistic clump created by the bubble-pack algorithm as shown in

Figure 3-5. However, the computational time is inevitably increased by the larger number of spheres. Considering the computational power limitation, the 9 irregularly shaped clumps with an average of 13.7 spheres are produced based on the specified φ and ρ values as Figure 3-6 shows.

The effect of the number of spheres in clump on the stress-strain behavior is evaluated in section

3.3.2.

Figure 3-6 Irregularly shaped clumps used in this research.

3.3 Sensitivity Analyses of Different Factors in DEM Model

3.3.1 Shear Velocity

Shear velocity controls the lower section moving rate and can directly influence the computational time of one simulation. However, an overly-fast shear velocity can generate unduly disturbance on the ballast assembly and therefore impact the stress-strain behavior (Indraratna et al., 2014).

Ideally, the ballast assembly should be tested under a quasi-static state to prevent any disturbance on the sample during the shear process (Indraratna et al., 2014; S. Liu, 2006; X. Wang and J. Li,

2014). The dimensionless parameter called inertial number (I) was introduced to define the quasi- static condition and determined by Equation 3-4 (Jop et al., 2006).

푣푑 I = 푎 Equation 3-4 √푃 ⁄휌푑 where v is the shear velocity, da is the average particle diameter, P is the confining pressure and ρd is the particle density. The inertial number should be smaller than 10-3 to satisfy the quasi-static state (MiDi, 2004). As a result, the shear velocity is calculated as small as 10-4 mm/s in our DEM model. However, this velocity is unrealistic to be applied in the model because it may cost hundreds of days to complete one simulation. Therefore, a series of sensitivity analyses with different magnitudes of shear rate are conducted to find a reasonable shear velocity which can keep the balance between computational time and minimized influence on stress-strain behavior.

Similar approaches can be found in X. Wang and J. Li (2014) and Gao and Meguid (2018a).

In this sensitivity analysis, 4 different shear velocities which are 40 mm/s, 4 mm/s, 0.4 mm/s, and

0.04 mm/s are applied to evaluate the effects of shear velocity on the stress-strain behavior. The

15 kPa normal stress is applied in these simulations to guarantee the inertial number cannot be larger under the greater normal stress. The stress-strain behavior at each shear velocity is shown in Figure 3-7.

Figure 3-7 Shear stress-strain curves for different shear rate applied.

It can be seen that if the shear rate is as small as 4 mm/s, the stress-strain behavior of the ballast assembly does not differ from the smaller shear rate. It is worth mentioning that 0.04 mm/s is close to the shear rate applied in the experimental test (Indraratna et al., 2011a). Therefore, the shear velocity at 4 mm/s is chosen in all direct shear test simulations in this thesis with the normal stress

15 kPa, 27 kPa, 51 kPa, and 75 kPa, and particle density 2700 kg/m3 and the average particle diameter about 24.00 mm. As a result, the shear velocity in other simulations cannot generate any further disturb on the ballast assembly due to the inertial number cannot be greater than the value in this section.

3.3.2 Number of Spheres in Clump

The irregularly shaped clumps are applied in this research which are generated using automatic clump generation and bubble-pack algorithm in PFC3D. As mentioned in section 3.2.2, the clumps that generated based on the same contour file are different by using various combinations of the angle (φ) and the ratio (ρ) values in the bubble-pack algorithm. The clump with a more realistic

37 shape feature is created by a greater number of spheres through the greater φ and lower ρ values, correspondingly the more realistic contact is generated. However, the simulations are extremely time-consuming if the clump is as realistic as the scanned ballast particle. A reasonable number of spheres in the clump are desired to keep an acceptable balance between computational time and realistic simulation of ballast particles.

In this section, a series of sensitivity analyses on the effect of number of spheres in the clump on the stress-strain behavior and coordination numbers 퐶푛 (average number of contacts per particle) is conducted. Figure 3-8 shows 3 created clump groups which are labeled 1, 2, and 3 with the number of the average spheres being 10.2, 13.7 and 21.0, respectively, and created using varied Φ and ρ values. The ballast assembly that formed based on each clump group is tested under 75 kPa normal stress. Figure 3-9 and Figure 3-10 show corresponding stress-strain behavior and coordination number variation (measured after shear progressed) of each clump group, respectively.

(a) Group 1 with the average number of 10.2 spheres.

(b) Group 2 with the average number of 13.7 spheres.

Figure 3-8 Clump group library.

Figure 3-9 Shear stress-strain behavior of each group.

Figure 3-10 Coordination number of each group.

As can be seen in Figure 3-9, the slope of each curve is proportional with the average number of spheres in the clump, which represents the stiffness of the ballast assembly is constantly increased.

It can be attributed to that more contacts are formed within the ballast assembly and the distribution of the contact force becomes more uniform, as Figure 3-10 proves that the coordination number is constantly increased by the increase of the number of spheres. This observation is consistent with the result reported by Gu and Yang (2013). However, an interesting phenomenon is observed that clump group 2 with an average number of 13.7 spheres shows greater shear strength compared with clump group 3. It may be attributed to that angle parameter φ which controls the surface smoothness of the clump is constantly increased in order to increase the number of spheres in each clump, which in turn the smoother surface has been generated. As a result, the interlocking mechanism of the smoother surface is reduced and the shear strength of the ballast assembly is correspondingly decreased. Based on the above observation, clump group 2 is selected in the

41 following simulations, as they showed the highest shear strength and a relatively shorter computational time than clump group 3.

3.3.3 Porosity

The porosity of the ballast assembly before the shear process is modified through 2 approaches in

DEM model. One approach is using porosity command in the PFC3D when assembling the ballast particle. Porosity as a user-specified parameter should be satisfied after the sample is packed. The other one is specifying the interparticle friction coefficient (μ) during the normal stress application process. The porosity of ballast assembly is modified based on the user-specified μ values due to the particle sample is consolidated when applying normal stress. Katagiri et al. (2010) indicated that the densest sample was generated when μ = 0, and the sample became loose when the μ was increased. However, it is unfeasible to precisely control the porosity of ballast assembly due to μ is a parameter that indirectly influences the final porosity. As a result, after simulating different μ values and comparing the simulated stress-strain behavior with the experiment result, μ is determined as 0.46 during applying the normal stress, besides, μ is subsequently changed to different values during the shear process to simulate different inter-particle actions. As a result, the porosity of the ballast assembly in this research is only changed through the specified porosity value.

In this sensitivity analysis, five different porosity values are specified during the ballast assembly packing process, which are 0.3, 0.35, 0.4, 0.45, and 0.5. These porosities are selected based on the investigation prepared by Shenton (1978) that the typical ballast porosity varies from 0.38 to 0.52 due to different packing conditions. The stress-strain behavior, shear strength, and the coordination number variations under 51kPa normal stress of each ballast assembly in different porosity values are shown in Figure 3-11, Figure 3-12, and Figure 3-13, respectively.

Figure 3-11 Shear stress-strain behaviors at different porosity properties.

Figure 3-12 Shear strength variation with porosity properties.

Figure 3-13 Coordination number variation with porosity properties.

As can be seen in Figure 3-11, the initial slope of each curve (before the shear strain of 4%) is generally decreased with the increase of porosity. Figure 3-12 shows the shear strength is steadily decreased by the increase of the porosity value. This observation is agreed with the result reported by Selig and Waters (1994) and Lekarp et al. (2000) that the greater density results in both stiffer and stronger behavior of granular material. This can attribute to that an intensive contact skeleton within the ballast assembly is created to resist the applied shear force. As can be seen in Figure

3-13, the coordination number is inversely proportional with the porosity value. In the end, the porosity of 0.45 is used in the following simulations which is consistent with the laboratory experiment (Indraratna et al., 2011a).

3.3.4 Effective Modulus

* Effective modulus (E ) controls the contact stiffness (kn and ks) between the particles as well as between particles and walls as introduced in section 2.3.3. In the DEM model, the contact stiffness

44 and interparticle friction coefficient (μ) are regarded as the predominant parameters influencing the stress-strain behavior of the simulated material (Iwashita and Oda, 1999; O’Sullivan, 2011).

* The effective modulus normally consists of 2 components, one is E P that controls the particle to

* particle contact stiffness, the other one is E W that controls the particle to wall contact stiffness. In

* * * this DEM model, E W is always twice larger than E P to simulate a steel wall condition, and E p is represented by using E* in the following sections.

In this section, the sensitivity analysis for the effect of effective modulus (E*) on the stress-strain

* 6 behavior of ballast assembly is conducted. Five different E values are selected, which are 1.0e

Pa, 5.0e6 Pa, 1.0e7 Pa, 1.8e7 Pa and 5.0e7 Pa. The responses under the normal stress 51 kPa are shown in Figure 3-14.

Figure 3-14 Shear stress-strain behavior at different E*.

As expected, the effective modulus (E*) has a significant influence on the stress-strain behavior of the ballast assembly. The initial slope and ultimate shear stress of each curve are proportional with

45 effective modulus and the strain of the ultimate shear stress shows a negative relationship with E*.

These observations are consistent with J. Liu et al. (2015). However, it should be mentioned that the material stiffness (initial slope) is the mechanical behavior that E* directly influences, whereas, the increase in ultimate shear stress should be regarded as a side effect of E* modification as E* only controls the contact stiffness. In the end, this sensitivity analysis is beneficial to the fresh ballast micro-mechanical parameters calibration work in section 3.4.

3.3.5 Interparticle Friction Coefficient

The inter-particle friction coefficient μ is the other predominant parameter influencing the stress- strain behavior of the ballast assembly. As introduced in section 2.3.3, the slip behavior between the contact bodies (particle to particle or particle to wall) is controlled by μ. It should be noted that the inter-particle friction coefficient of the contacts between particle and wall is specified as 0 to simulate the frictionless condition on the internal surface of the shear box. Similar approach can be found in J. Liu et al. (2015) and Marketos and Bolton (2005).

Sensitivity analysis of the effect of inter-particle friction coefficient (μ) on stress-strain behavior is conducted in this section. Five different μ values are selected which are 0.1, 1.1, 2.1, 3.1, and

4.1, respectively. The corresponding stress-strain behaviors under the normal stress 51 kPa are shown in Figure 3-15.

Figure 3-15 Shear stress-strain behaviors at different μ.

As can be seen in Figure 3-15 that the distinct difference exists between the curves when the inter- particle friction coefficient (μ) is lower than 2.1. However, the difference becomes smaller when

μ is at a relatively high level. This trend is consistent with the corresponding inter-particle friction angles which are 5.7○, 47.7○, 64.5○, 72.1○, and 76.3○, respectively. This observation is similar to the finding reported by Lu and McDowell (2008). In addition, the initial slope and ultimate shear stress of each curve show the positive relationship with μ and the strain of ultimate shear stress shows a negative relationship with μ. A similar observation has been found in the sensitivity analysis of the effective modulus (E*). It can be concluded that as the predominant parameters, both E* and μ positively control the initial slope and ultimate shear stress behavior. However, μ directly modifies the ultimate shear stress as μ controls the slip behavior, and the increase in initial slope should be regarded as a side effect during this process. In the end, this sensitivity analysis is beneficial to the fresh ballast micro-mechanical parameters calibration work in section 3.4.

3.4 Fresh Ballast DEM Model

In this section, The DEM model is validated with the results of experimental direct shear tests on ballast assembly subjected to three different normal stresses, namely 27 kPa, 51 kPa, and 75 kPa, respectively to simulate the shear stress-strain behavior of fresh ballast assembly. The micro- mechanical parameters (Table 3-1) for the fresh ballast contact model are determined by trial and error until the simulation result is identical to the experiment result.

Table 3-2 shows the fresh ballast micro-mechanical parameters of some previous studies. It should be noted that a relatively small effective modulus E* compared with the previous experience is selected in this DEM simulation to extend the critical timestep length. As a result, the computational time for one simulation is reduced to approximately 13 hours which is significantly shorter than the similar simulation prepared by Indraratna et al. (2014). However, it should be acknowledged that the relatively fast shear rate is another major reason that results in such short computational time. The decreased effective modulus E* leads to reduced stiffness of ballast assembly. Therefore, the inter-particle friction coefficient μ as another predominant parameter is increased to compensate for the reduction of stiffness.

Table 3-1 Micro-mechanical contact constants adopted for fresh ballast result simulation.

Value Parameters PPC PWC

* 7 7 Effective modulus, E (Pa) 1.8e 3.6e

Stiffness ratio, k* 1.0 1.0

Interparticle friction coefficient, μ 4.1 0

PPC – Particle to particle contact; PWC – Particle to wall contact

Table 3-2 Micro-mechanical parameters of ballast assembly in previous DEM simulations.

No. Reference E* (Pa) μ k* 1 B. Wang et al. (2017) 4.5e9 0.5 2.5 Gao and Meguid 2 3.18e8 0.32 1.5 (2018a) 3 Yan et al. (2015) 7.0e11 0.5 2.0 4 Nie et al. (2019) 1.0e8 0.5 1.3 E*- Effective modulus; μ – Inter-particle friction coefficient; k*- Normal to shear stiffness ratio

Figure 3-16 Shear stress-strain curves fitting for fresh ballast.

Figure 3-16 shows a comparison between DEM simulation and experimental results for fresh ballast. As can be seen, the computed shear stress-strain behavior is generally in agreement with the laboratory test results reported by Indraratna et al. (2011a). It can also be seen that the DEM simulation accurately reproduces the strain-softening effect when the applied normal stress is 27 kPa and 51 kPa normal stress. This is not the case however when the applied normal stress 75 kPa, where the DEM model continues to exhibit strain hardening behavior thus overestimating the shear

49 strength. This may be attributed to ballast particle breakage in the laboratory under such high stress

(75 kPa), which could not be accurately captured by the DEM simulation (Indraratna et al., 2005;

Shi et al., 2020). Thus, it can be concluded that the DEM model and the fresh ballast contact model are validated with the experimental results of Indraratna et al. (2011a) as it captures the complete shear stress-strain behavior of fresh ballast assembly reasonably accurately when the applied normal stress is up to 51 kPa. It should be mentioned that a typical heavy haul train of 20-30-t axle loads indicates the confining pressure to the ballast layer is in the range of 10-40 kPa and rarely exceeds 60 kPa (Ngo et al., 2017). The validated micro-mechanical properties will be used subsequently as a basis to derive the reduced properties for different levels fouled ballast DEM models which are presented in Chapter 4.

Beside the macroscopic behavior, DEM also provides a way to observe the granular material microscopic behavior. Figure 3-17 to Figure 3-20 illustrates the contact force network among the ballast particles at shear strains of 0%, 3%, 7%, and 13.3%, respectively under normal stress of 51 kPa. The contact force between the particles as well as between particles and walls are represented by lines whose thickness is proportional to the magnitude of the force. It can be observed that at the initial shear stage (εs = 0%), the contact forces are distributed uniformly within the ballast assembly and transmitting along the vertical direction from the top to the bottom of the shear box when normal stress is applied. However, as the shear process continues, the orientation of the contact forces are modified from vertically to diagonally (from bottom left to top right) and formed a distinct shear bond within the shear box. It indicates that the shear resistance of the ballast assembly is mainly supplied by this shear bond. Furthermore, the macroscopic mechanical behavior at each shear stage is accordingly related to the contact force distribution. Initially, the peak shear stress value is observed at the shear strain of 7%, as a result, the maximum contact

50 force and the second-most total number of contacts are observed at this shear stage. Secondly, the shear stress at the shear strain of 13.3% is higher than the stress at the shear strain of 3%, accordingly, the maximum contact force at 13.3% shear strain is greater than the one at 3% shear strain and there is only a small difference in the overall number of contacts. In the end, the number of contacts within the shear box is constantly decreased from the initial stage to the end which proves that the ballast particle contact skeleton is continually broken with the shear progressed.

Figure 3-17 Contact force distribution at εs = 0%, No. of contact = 26013, Max. contact force = 248.98N.

Figure 3-18 Contact force distribution at εs = 3%, No. of contact = 20617, Max. contact force = 345.90N.

Figure 3-19 Contact force distribution at εs = 7%, No. of contact = 21065, Max. contact force = 538.73N.

Figure 3-20 Contact force distribution at εs = 13.3%, No. of contact = 19382, Max. contact force = 426.86N.

In Figure 3-21 and Figure 3-22, the displacement of particles at the shear strains of 3% and 13.3% under 51 kPa normal stress is represented by vectors. It can be observed that the particles in the lower section of the box are moved horizontally due to the specified movement of the lower box.

However, the displacement directions of most particles in the upper section of the box are modified from horizontally to vertically as shearing progressed. It reveals that the dilation behavior of the ballast assembly during the shear process is induced by those particle movements.

Figure 3-21 Particle displacement at εs = 3%.

Figure 3-22 Particle displacement at εs = 13.3%.

In conclusion, by using the above DEM model, and the micro-mechanical parameters setting, the stress-strain behaviors of the fresh ballast assembly under different normal stresses are accurately reproduced. By investigating the micro-mechanical behavior of the ballast assembly, the orientations of the contact force in the shear box are varied from vertical to diagonal with the shear progressed. In addition, the shear bond provides the major shear resistance to the ballast assembly.

In the end, the displacement variation of the ballast particle in the upper section induces the dilation behavior of the ballast assembly at the end of the shear process.

3.5 Summary

In this chapter, a simulated direct shear test has been conducted in PFC3D in order to obtain the stress-strain behavior of the ballast assembly by using DEM. An isotropic and homogeneous ballast packing is generated within the box by using the expansion method. The irregularly shaped clump which created based on the real ballast particle shape is applied to represent the ballast particle in this simulation. The physical properties of the ballast assembly are similar to the experimental sample prepared by Indraratna et al. (2011a). The linear contact model in PFC3D is applied to reflect the interaction between particles as well as between particles and walls.

A series of sensitivity analyses is conducted to investigate the effect of the different factors on the stress-strain behavior. It can be concluded that the shear velocity of 4mm/s is reasonable to save the computational time and avoid unduly disturbance on the ballast assembly. The increased number of spheres in the clump results in a more realistic interaction between particles but the shear strength is not constantly increased due to the smoother surface is generated. The shear strength of ballast assembly is inversely proportional with the porosity of the ballast assembly. The effective modulus and interparticle friction coefficient both generate a positive influence on the material stiffness and the shear strength.

In the end, the DEM model predicted stress-strain behaviors of fresh ballast under different normal stresses are compared with the experimental results reported by Indraratna et al. (2011a). A good agreement between the simulation results and experimental results is observed which proves that the mechanical behavior of the fresh ballast assembly is accurately captured by the DEM model and the model is qualified for the analysis in the next chapter.

4 DEM Simulation for Ballast Fouling

4.1 Introduction

In this chapter, an efficient approach to simulate the stress-strain behavior of coal dust-induced fouling ballast assembly for different VCI ratios is presented. In this approach, the micro- mechanical parameters (effective modulus E* and inter-particle friction coefficient μ) of the fresh ballast contact model determined in Chapter 3 are reduced and the simulated stress-strain results are validated by comparison with experimental results under the different normal stresses.

Empirical equations are determined and validated by comparing the simulation results with experimental results. Those equations can efficiently determine the required micro-mechanical parameters for a given VCI ratio within this DEM model. Besides, the effect of foreign fouling material on the shear strength reduction is investigated and the normalized shear strength reduction for different VCI ratios and under different normal stresses is determined. In the end, ballast degradation is simulated by changing the particle size distribution (PSD) curve and measured by using the breakage index (Bi). The degraded ballast sample is tested through the direct shear test to investigate the effect of degraded ballast on the shear strength property. At the same time, variations of microscopic properties of the degraded ballast assembly are introduced to better understand the mechanism that influencing the shear strength property.

4.2 Model Simulation Subjected to Different VCI Ratios

In previous studies, ballast fouling has been simulated in the DEM model by two approaches. As introduced in section 2.4, those approaches have advantages and disadvantages. In order to simulate ballast fouling more efficiently and with acceptable accuracy, a new approach with shorter computational time and well simulation result is proposed. It captures the shear stress- strain behaviors of foreign fouling material induced fouled ballast assembly under different VCI

55 ratios by reducing the effective modulus E* and the interparticle friction coefficient μ of the validated fresh ballast model instead of injecting numbers of fine particles into the ballast assembly.

This approach is developed based on the concept that the fouling material accumulated in the voids generates lubricating (Huang and Tutumluer, 2011) and cushioning (Indraratna et al., 2013) effects between ballast particles which will reduce the interparticle friction angle from Φ to Φ’ and interparticle contact stiffness from K1 to K2 as Figure 4-1 shown.

Fresh ballast, inter-particle friction angle Φ Fouled ballast, inter-particle friction angle Φ’ (a) Inter-particle friction coefficient reduced to simulate lubricating effect.

Fresh ballast, inter-particle contact stiffness Fouled ballast, inter-particle contact stiffness

K1 K2 (b) Inter-particle contact stiffness reduced to simulate cushioning effect.

Figure 4-1 Concept explanation on the method to simulate fouled ballast.

In this study, the approach is applied to simulate the stress-strain behavior of fouled ballast assembly at VCI 20%, 40%, and 95%, respectively. The reduced effective modulus E* and inter- particle friction coefficient μ at the given VCI ratio are listed in Table 4-1. These are determined by using trial and error until the simulation results are identical to the laboratory results. Figure

4-2 shows the comparison between the DEM simulations and experimental results which reported

57 by Indraratna et al. (2011a) for three different normal stresses namely 27 kPa, 51 kPa, and 75 kPa at VCI of 20%, 40%, and 95%.

Table 4-1 Micromechanical parameters adopted for different VCI ratios simulation.

Parameters VCI20% VCI40% VCI95% PPC 5.5e6 5.25e6 5.0e6 * Effective modulus E (Pa) PWC 1.1e7 1.05e7 1.0e7 Interparticle friction coefficient PPC 3.85 3.55 2.05 μ PWC 0 0 0 PPC - Particle to particle contact; PWC – Particle to wall contact

(a) VCI 20%,

(b) VCI 40%

As can be seen from Figure 4-2, the simulation results using this approach are in good agreement with the experimental results. The strain-softening behavior in the post-peak range is reasonably well captured by the DEM simulations at all levels of VCI when the applied normal stress is 27 kPa and 51 kPa. For higher normal stress (75 kPa), the shear stress-strain behavior is well-predicted to a shear strain of approximately 8% for VCI of 40% and 95% (Figure 4-2b and c). The predicted behavior is slightly higher than the experimental result for the VCI of 20% (Figure 4-2a). A similar situation is observed for VCI of 95% under 27 kPa (Figure 4-2c) before the shear strain of 2% where the experimental shear stress is almost zero. It should be noted that the shear stress is overestimated in the post-peak range by the DEM simulation for all levels of VCI when the applied stress is 75 kPa. This may be attributed to particle breakage which may take place under the highest bearing pressure of 75 kPa after the peak shear strength is reached (Indraratna et al., 2005). Further, it is difficult to use the rigid particles in the DEM simulation to accurately capture this behavior

(Ngo et al., 2014; Shi et al., 2020). In conclusion, it can be said that the shear stress-strain behavior of the fouled ballast assembly under different levels of VCI is reasonably well simulated by the method of reducing the effective modulus E* and the inter-particle friction coefficient μ of a validated fresh ballast contact model. The computational time is further reduced with this method as the critical timestep is larger compare with the simulation of fresh ballast assembly.

4.3 Relationship between VCI Ratio and Micro-mechanical Parameter

In this section, the effect of fouling material on the effective modulus E* and inter-particle friction coefficient μ variation is investigated. The parameters E* and μ which are used to simulate the ballast fouling at different VCI ratios are listed in Table 4-2. In addition, the corresponding data points and the best fit line are plotted in Figure 4-3 and Figure 4-4.

Table 4-2 Collection of micro-mechanical parameters and VCI ratios.

E*p (Pa) μ VCI 1.8e7 4.1 0% 5.5e6 3.85 20% 5.25e6 3.55 40% 5.0e6 2.05 95%

4.5

3.5

2.5

Interparticle Coefficient Interparticle Friction 1.5 0% 20% 40% 60% 80% 100% VCI

Figure 4-3 Inter-particle friction coefficient variation with VCI ratio.

20 ×105 18 16 14 12 10 8 6

4 Effective (Pa) Modulus Effective 2 0 0% 20% 40% 60% 80% 100% VCI

Figure 4-4 Effective modulus variation with VCI ratio.

As can be seen in Figure 4-3, the inter-particle friction coefficient μ is steadily decreased with the increase of the VCI ratio. The effect of fouling material on the inter-particle friction coefficient μ is represented by a linear equation as shown in Equation 4-1. Figure 4-4 shows the relationship between effective modulus E* and VCI ratios. One-term power regression method is applied to represent the best fit line as shown in Equation 4-2. The curve illustrates that the effective modulus rapidly decreases at the initial stage of ballast fouling. However, once the VCI ratio is above 20%, the decrease rate is slower and becomes relatively insignificant.

휇 = −2.218VCI + 4.247;⁡푅2 = 0.973 Equation 4-1

퐸∗ = ⁡4.894푒6푉퐶퐼−0.1411;⁡푅2 = 0.996 Equation 4-2

Based on Figure 4-3 and Figure 4-4, it can be concluded that the fouling material accumulated in the voids between ballast particle gradually modifies the interparticle friction angles, however, the contact stiffness is significantly decreased once the fouling material intrudes into the contacted particles and the modification of contact stiffness becomes insignificant with the continue increased level of ballast fouling. This observation is well explained by Figure 4-1b that the contact stiffness is instantly modified once the fouling material exists between contacted ballast particles.

Empirical equations between VCI and μ as well as VCI and E* can be used to conveniently calculate the required E*, μ for a DEM simulation at a given VCI ratio. To validate these two equations, DEM simulation results based on the calculated E*, μ values are compared with the experimental results at VCI 70% subjected to normal stresses of 15 kPa, 27 kPa, 51 kPa, and 75 kPa. The calculated E*, μ are 5.15MPa and 2.7, respectively at VCI of 70%. The comparison between simulation results and experimental results reported by Indraratna et al. (2011a) is shown in Figure 4-5.

Figure 4-5 Comparisons between DEM simulation result and experimental result for VCI 70%.

It can be observed that there is a good agreement between the simulation and experimental results.

In addition, by plotting the simulated shear strength against the experimental shear strength as shown in Figure 4-6, the slope of the best fit line indicates that these two shear strengths are in good agreement. However, it should be noted that while the required E* and μ at a given VCI ratio are conveniently determined from Equation 4-1 and Equation 4-2, such empirical equations are only valid for this DEM model.

Figure 4-6 Comparison between and model predicted shear strength and experimental shear strength.

4.4 Relationship between VCI Ratio and Shear Strength

The strength envelope is an important result obtained from the direct shear test. It can visualize the difference between simulation results and experiment results at the shear strength value. The shear strength results obtained through DEM simulation and experiment are summarized in Table

4-3. Figure 4-7 plots of normal stress (σn) versus shear strength (τf) for both tests under different

VCI ratios.

Table 4-3 Summary of shear strength value in simulation and experiment test.

Normal Shear Strength (kPa) Test types Stress (kPa) 0% VCI 20% VCI 40% VCI 70%VCI 90%VCI DEM 27 54.53 43.95 39.64 38.28 35.95 51 93.35 76.73 68.45 69.41 65.88 75 121.50 104.77 95.33 93.36 87.60 Experimental 27 52.10 45.58 41.66 36.29 35.28 51 92.89 82.92 74.00 67.19 65.70 75 112.32 99.03 91.30 82.52 79.81

Shear strength envelopes are generated using power fitting curves of the DEM simulation data points and the shear strength should be zero in the absence of normal stress considering the ballast is unbound granular material (J. Liu et al., 2015; Tian et al., 2005). The power equation of each shear strength envelop under different ballast fouling levels is as follows:

0.7692 2 휏푓 = 4.427휎푛 , 푅 = 0.99, 퐹푟푒푠ℎ⁡푏푎푙푙푎푠푡 Equation 4-3

0.8429 2 휏푓 = 2.762휎푛 , 푅 = 0.99,⁡ 푉퐶퐼20% Equation 4-4

0.8592 2 휏푓 = 2.335휎푛 , 푅 = 0.99,⁡ 푉퐶퐼40% Equation 4-5

0.8540 2 휏푓 = 2.304휎푛 , 푅 = 0.99,⁡ 푉퐶퐼70% Equation 4-6

0.8471 2 휏푓 = 2.283휎푛 , 푅 = 0.99,⁡ 푉퐶퐼95% Equation 4-7

The strength envelopes accurately represent the simulation shear strength values as proved by

Equation 4-3 to Equation 4-7 and illustrate a good agreement with the experimental shear strength as shown in Figure 4-7.

Figure 4-7 Strength envelopes of DEM model for different VCI ratios.

The strength envelopes in Figure 4-7 suggest that the ballast shear strength is proportional with the applied normal stress and inversely proportional with the VCI ratio. Meanwhile, the effect of fouling material on shear strength reduction is weaker as the VCI ratio increases. This can be attributed to the loss of contacts between the ballast particles due to the fouling material infiltration when the VCI approaches 100%. This observation is in line with the result found by Indraratna et al. (2011a) which also represents that the proposed approach is effective in simulating foreign material induced ballast fouling.

The normalized shear strength reduction of the fouled ballast assembly under different VCI ratios and normal stresses conditions is determined through a hyperbolic relationship that was developed by Indraratna et al. (2011a) as follows.

(휏푝)퐹표푢푙푒푑−푏푎푙푙푎푠푡 (휏푝)퐹푟푒푠ℎ⁡푏푎푙푙푎푠푡 푉퐶퐼 = − Equation 4-8 휎푛 휎푛 푎 × 푉퐶퐼 + 푏

66 where (휏푝)퐹표푢푙푒푑−푏푎푙푙푎푠푡 is the shear strength of fouled ballast sample, (휏푝)퐹푟푒푠ℎ⁡푏푎푙푙푎푠푡 is the shear strength of fresh ballast sample, 휎푛 is the normal stress, a and b are hyperbolic constants that depend on the normal stress. The linear relationship between the reduction of shear strength and

VCI ratio is obtained by rearranging Equation 4-8 as follow:

푉퐶퐼 ∙ 휎푛 = 푎 × 푉퐶퐼 + 푏 Equation 4-9 (휏푝)퐹푟푒푠ℎ⁡푏푎푙푙푎푠푡 − (휏푝)퐹표푢푙푒푑−푏푎푙푙푎푠푡

The best fit lines under different normal stresses through the linear regression method are illustrated in Figure 4-8, which are based on the data points calculated from Equation 4-9. The hyperbolic constants a and b are the slope and y-intercept of each regression line which are varied with normal stress. The normalized shear strength reduction for a given VCI ratio and under the specified normal stress is determined through the completed Equation 4-9.

VCI

푝

푛

휏

휎

∆

× VCI

Figure 4-8 Linear relation between hyperbolic constant a and b and shear strength reduction.

4.5 Effect of Ballast Degradation on Stress-strain Behavior

The impact of ballast degradation on the shear strength behavior of ballast assembly under the static loading condition is examined in this section. The ballast degradation causes the particle size distribution (PSD) curve of fresh ballast gradually swing to the degraded ballast PSD curve when assuming the maximum particle size is unchanged as shown in Figure 4-9 (Indraratna et al., 2005;

Y. Qian et al., 2014). The degree of ballast gradation is measured by using breakage index (Bi)

(Einav, 2007) which is defined as the area between fresh ballast PSD and degraded ballast PSD to the area of fractal PSD as shown in Figure 4-9. Equation 4-10 is applied to determine the fractal

PSD.

3−훼 3−훼 푑 − 푑푚 F(d) = 3−훼 3−훼 Equation 4-10 푑푀 − 푑푚 where F(d) is particle size cumulative distribution by mass, α is the fractal dimension and usually taken as 2.6, d is the particle size, dm represents the minimum particle size which is taken as

0.074mm, dM is the maximum particle size.

Figure 4-9 Definition of breakage index and PSD variations.

100% G1 G2 G3

80% G4 G5 G6

G7 G8 G9 60% G10 G11 G12 40%

Fractal Passing by Mass(%) Passing by 20%

0% 0.01 0.10 1.00 10.00 100.00 Particle Size(mm) Figure 4-10 Ballast PSD curves applied to degradation simulation.

In order to simulate ballast degradation efficiently, the PSD curve in the validated DEM model in

Chapter 3 is changed to more uniformly-graded with the greater minimum particle size to represent the initial fresh ballast condition and then the particle size is gradually reduced to simulate the ballast degradation process. As can be seen in Figure 4-10, G1 represents the ballast assembly at fresh condition, and the rest 11 PSD curves represent the ballast particle size at different levels of ballast degradation. The corresponding characteristics of each PSD curve are shown in Table 4-4.

The maximum ballast particle size of each gradation is taken as 37.5 mm to satisfy the sample size ratio which defines as the minimum length of the sample divided by the maximum particle size and should be greater than 5 to minimize the boundary effect (Indraratna et al., 1993). It should be mentioned that the effective modulus E* and inter-particle friction coefficient μ are kept constant as the validated fresh ballast contact model in Chapter 3 because the material stiffness and inter- particle friction coefficient are assumed not to be modified.

Table 4-4 Particle size distribution characteristics of the 12 gradations and corresponding breakage index, porosity and coordination number.

Coordination Gradation Bi (%) Cu DM D60 D50 D30 D10 Dm Porosity No. 1 0.00 1.16 37.50 34.85 34.19 32.83 30.00 27.00 0.3872 3.3292 2 2.82 1.20 37.50 33.68 32.72 30.83 28.00 20.00 0.3780 3.5312 3 5.85 1.37 37.50 32.79 31.62 28.92 24.00 18.00 0.3624 3.2272 4 8.12 1.45 37.50 31.91 30.51 27.50 22.00 16.00 0.3585 3.0328 5 10.01 1.57 37.50 31.32 29.78 26.42 20.00 14.50 0.3600 2.9357 6 11.79 1.66 37.50 30.74 29.04 25.42 18.50 13.00 0.3472 2.7009 7 14.27 1.93 37.50 30.42 28.65 23.67 15.75 12.50 0.3326 2.8360 8 16.55 2.10 37.50 30.08 28.23 22.27 14.31 10.00 0.3498 2.1972 9 18.97 2.37 37.50 29.64 27.67 20.97 12.50 9.00 0.3220 2.1343 10 22.46 2.63 37.50 28.98 26.85 17.79 11.00 8.50 0.3131 2.0202 11 26.37 2.91 37.50 28.30 26.00 16.94 9.71 4.50 0.2897 1.7487 12 28.72 3.12 37.50 28.11 25.36 15.58 9.00 4.00 0.3050 1.5823

Note: Bi: breakage index (%); Cu: coefficient of uniformity, determined by: Cu= d60/d10. dM:

maximum ballast size used in this study; dm: minimum ballast size used in this study; d10, d30,

d50, d60: diameters in millimeters at which 10%, 30%, 50% and 60% by weight of ballast passes through the sieve.

160 140 y = 1196.1x2 - 535.2x + 142.63 120 R² = 0.9028 100 80 60

40 Shear Strength (kPa) Shear Strength 20 0 0% 5% 10% 15% 20% 25% 30% Breakage Index

Figure 4-11 Shear strength variation with breakage index.

The ballast assemblies generated based on these 12 PSD curves are tested under 51 kPa normal stress and the corresponding shear strength properties are shown in Figure 4-11. The best fit line is generated by using the second-order polynomial regression method to reflect the shear strength changing tendency. As can be seen, the shear strength of the ballast assembly first decreases then increases as the result of the increased ballast degradation. This conclusion has combined the findings reported by Danesh et al. (2018b), that the shear strength is constantly decreased but in different reduction rates as the result of an increase of the breakage index; and the finding reported by Y. Qian et al. (2015) that the shear strength of the degraded ballast assembly is greater than the fresh ballast assembly.

The conclusion is better explained at the microscopical level. In Figure 4-12, the coordination number of the particles that measured after shear progressed is constantly decreased as the result of increasing the breakage index, it represents those small size particles are work as interceptors to disrupt the large particle contact skeleton (Boler et al., 2014) which can reduce the interlocking mechanism between large particles. However, it cannot be denied that the denser ballast assembly

71 is also packed before the shear process with the increasing breakage index as Figure 4-13 shows.

The effect of reduction in interlocking is to decrease the shear strength, while the effect of denser ballast assembly is to increase the shear strength as shown in sensitivity analysis in section 3.3.3.

In the case of ballast degradation at the initial stage, the effect of interlocking overweighs the effect of denser ballast assembly. As a result, the net effect is a decrease in the shear strength at the initial ballast degradation stage. However, with the ballast degradation continues, the effect of denser ballast assembly gradually dominates the shear strength behavior and the increasing tendency in the shear strength is observed at the corresponding ballast degradation level.

4.0

3.5 y = -6.9303x + 3.5633 R² = 0.9526 3.0

2.5

2.0

Coordination Number Coordination 1.5

1.0 0% 5% 10% 15% 20% 25% 30% Breakage Index

Figure 4-12 Coordination number variation with breakage index.

0.41 0.39 y = -0.3163x + 0.3858 0.37 R² = 0.9316 0.35 0.33

Porosity 0.31 0.29 0.27 0.25 0% 5% 10% 15% 20% 25% 30% Breakage Index

Figure 4-13 Porosity variation with breakage index.

4.6 Summary

In conclusion, an efficient approach is proposed to accurately simulate the stress-strain behavior of the fouled ballast assembly subjected to different VCI ratios by changing the effective modulus

E* and inter-particle friction coefficient μ. In this approach, the computational time is further saved compared with the validated DEM model in Chapter 3. Based on the observation of the variation of E* and μ with the VCI ratio, the fouling material accumulated in the voids between ballast particles can steadily decrease the inter-particle friction angle but rapidly decrease the inter- particle contact stiffness. In addition, empirical equations are used to present the relationship between the VCI ratio and micro-mechanical parameters. Those equations are validated by comparing the simulation results with the experimental results at VCI of 70%.

The effect of external material on the shear strength reduction is investigated based on the simulation results which reveals that the effect is becoming weaker with the increasing of fouling material amount (VCI ratio). This observation is in line with the findings reported by Indraratna et

73 al. (2011a). Moreover, the normalized shear strength reduction at a given VCI ratio and normal stress of this DEM model is calculated based on the corresponding equations.

The effect of ballast degradation on the shear strength property is investigated at the microscopic level. The result indicates that the shear strength of the ballast assembly first decreases then increases as the result of the increased ballast degradation. This behavior can attribute to the effects of both interlocking mechanism reduction and ballast assembly density increased.

5 Conclusions and Limitations

5.1 Conclusions

In this thesis, a simulated direct shear test has been developed in PFC3D in order to obtain the stress-strain behavior of the ballast assembly by using DEM. A series of sensitive analyses are conducted to investigate the effect of the different factors on the stress-strain behavior. The reduction of effective modulus E* and inter-particle friction coefficient μ are adopted to simulated foreign material induced ballast fouling that saves significant computational time. The stress-strain behaviors of fouled ballast under different normal stresses are compared with those results measured in the laboratory experiment to validate the proposed ballast fouling simulation approach.

Empirical relations between the micro-mechanical parameters and the VCI ratio are determined to help calculate the required parameters for the DEM simulation. These are validated by comparing numerical and experimental results. Shear strength envelopes for different levels of VCI ratio are generated to investigate the effect of foreign fouling material on the shear strength reduction.

Further, hyperbolic equations are derived to calculate normalized shear strength reduction due to ballast fouling under specified normal stress. In the end, particle size is decreased to simulate ballast degradation in the DEM model. The effect of the ballast degradation on the shear strength is examined. Several conclusions can be drawn from this study.

1. By reducing the effective modulus E* and inter-particle friction coefficient μ instead of

injecting fine particles into a fresh ballast DEM model to simulate the foreign material

induced ballast fouling, the shear stress-strain behavior is predicted with reasonable

accuracy and the computational time of one simulation is reduced.

2. Foreign fouling material accumulated in the voids between ballast particles can steadily

decrease the inter-particle friction angle but rapidly decrease the inter-particle contact

stiffness.

3. The empirical equations between the micro-mechanical parameters and VCI are

determined and validated to help calculate the required E* and μ values for a given DEM

simulation.

4. The shear strength of ballast assembly is inversely proportional with VCI ratios. The effect

of ballast fouling material on shear strength reduction is weaker for relatively high VCI

ratios.

5. Ballast degradation affects the shear strength; it causes the shear strength to initially

decrease then increase with higher degradation levels.

5.2 Limitations

It must be stated that the simulation results of fresh and fouling ballast assembly are based on a test program that depends solely on a single repetition, and the simulation results will vary to some extent for example by using different particle shapes. Besides, the volumetric behavior of ballast assembly and the smoother surface of particles caused by the ballast fouling are not considered in this study. In the future, more sets of micro-mechanical parameters should be examined to prove that the proposed approach can not only capture the mechanical behavior but also the volumetric behavior of fouled ballast assembly. In addition, the effect of ballast degradation on particle surface texture should be considered by using particles with smoother surfaces to more comprehensively investigate the effect of ballast degradation on the mechanical behavior of ballast assembly.

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