Keywords

1 Introduction

Australian rail sector must continuously upgrade its track network and adopt innovative technology to reduce the cost as well as increased passenger comfort. Ballast layer plays a crucial role in transmitting the train loading to underlying track-formation layers at a reduced stress level [1]. The ballast grains experience significant degradation and breakage when subject to heavy hauls freight trains and moving. Due to ballast breakage, Australian rail industry spends a significant budget in terms of track maintenance, together with improving the ground before track construction where soft and saturated subgrade soils induce considerable difficulties in design and construction [2].

The degradation of ballast aggregates (breakage) significantly influences the shear strength of ballast. Sun et al. [3] conducted large-scale tests on ballast and found that confining pressure (i.e., lateral confinement) and frequency have a significant influence on the breakage of ballast under cyclic loading. Leng et al. [4] conducted cyclic tests on coarse-granular aggregates and the test results showed that there were two separate deformation stages of a rapid increase in deformation initially followed by a steady-state of plastic strain after a given loading cycle. These findings were in agreement with previous literature [5,6,7,8], among others. Previous studies found that particle breakage could happen even at small-stress level [9, 10]. In addition, previous studies indicated that increased lateral confinement for ballast resulted in decreased peak friction angles [11].

Research on the application of elastic inclusions (geosynthetics, geocell) for improved performance of tracks and subsequently decreased ballast breakage has been well documented [12,13,14,15,16,17,18], among others. Field trials carried out on fully instrumented tracks at Singleton, NSW Australia proved that rubber mat could reduce ballast deformation apart from reduced breakage [19, 20].

There is a lack of study on the use of recycled rubber mats for reducing the breakage of ballast grains from a micro-mechanical perspective. This paper reports the main findings of the study on the use of large-scale triaxial tests and discrete element modelling (DEM) on the breakage responses of railway ballast.

2 Large-Scale Triaxial Cyclic Tests

A large-scale triaxial testing apparatus designed and built at the University of Wollongong can accommodate a testing sample of 300 mm diameter and 600 mm high, as detailed in Fig. 1a. Materials tested in the laboratory including ballast, capping materials and recycled rubber mat as shown in Figs.1b–c. The ballast was cleaned and then passed through a set of standard sieves (53–9.5 mm) followed the Australian Standards (AS2758.7–2015) [21]. Upon drying, the ballast grains were coloured to help in quantifying the amount of breakage after each test. A mixture of sandy soil was adopted for a subgrade layer while a mixture of sand and crushed-rock was adopted for capping.

2.1 Sample Preparation and Testing Procedure

A tested sample was filled by a 200 mm thick subgrade at the bottom and compacted to a density of ρ = 18.5 kN/m3. A 100 mm-thick capping was then compacted to ρ = 20.5 kN/m3. A layer of rubber mats was then put on top of the capping. A ballast layer (300 mm thick) was placed and compacted to a ρ = 15.5 kN/m3. After preparing the specimen, it was filled with water and left overnight to saturate the specimen. Cyclic tests were started under qcyclic-max = 230kPa, qcyclic-min = 30kPa, frequencies: f = 10–40Hz and a constant confining pressure of σ3 = 20kPa. All tests were sheared up to N = 500,000 cycles. During these tests, the permanent vertical displacement, εa and volumetric strain, εv were measured. Ballast aggregates were recovered after each test, then sieved, and the changes in size distributions were recorded to investigate the amount of breakage.

Measured test data showed that the recycled rubber mats decreased deformation and breakage of ballast. This was because the mat could absorb cyclic and impact loads (i.e. energy) that led to less energy was transferred to ballast aggregates and thereby decreased breakage. The inclusion of rubber mat reduced the breakage up to 30.8% and 35.3% when subjected to frequencies of f = 10, 20 Hz, respectively. Detailed of test results were reported earlier by Indraratna et al. [22] and some of the test data were re-used in this study for the calibration of the DEM model.

Fig. 1.
figure 1

(a) a schematic section of triaxial apparatus; (b) mixed ballast in different colours following the gradation; (c) capping aggregates; and (d) recycled rubber mat (READS)

Full size image

3 Discrete Element Modeling

A discrete element method (DEM) has been widely used to study micro-mechanical responses of granular aggregates [23,24,25,26,27,28], among others. It is well documented that due to excessive computational time required to carry out cyclic tests in DEM, most of the previous DEM simulations were initiated to only a few thousand loading cycles. Given the excessively computational cost of simulating an actual track embankment in DEM having different layers of track substructure (i.e., ballast, capping, subgrade), thereby a coupled discrete-continuum modelling approach that utilises advantages of both numerical schemes with acceptable computational effort and accuracy is introduced in this study.

3.1 Coupled DEM-FEM

Ballast with varied geometries and sizes were digitalised in DEM by bonding many cylindrical together via parallel bond (Fig. 2a). The breaking (disconnection) of those bonds was approximately considered to represent particle breakage. Micro-mechanical parameters adopted for the DEM analysis were selected by calibration with laboratory test data as presented in Table 1. Figure 2b shows a schematic diagram of a coupled DEM-FEM model for the large-scale triaxial test, where the subgrade, capping and rubber mat was simulated by the continuum method. The schematic diagram of exchanging of forces (Fn, Fs) and displacement \(\left( {\dot{X}_{i}^{{\left[ E \right]}} } \right)\) between a discrete grain and continuum element at its interface is presented in Figs. 2c-d.

Contact forces at their interface can be calculated as:

$$ F_{i}^{{\left[ c \right]}} = F_{n}^{{\left[ c \right]}} + F_{s}^{{\left[ c \right]}} $$
(1)

where, normal force \(\left( {F_{n}^{{\left[ c \right]}} } \right)\) and shear force increment \(\left( {\Delta F_{s}^{{\left[ C \right]}} } \right)\) can be computed by:

$$ F_{n}^{{\left[ C \right]}} = K^{n} U^{n} n_{i} ;\;\;\;\;\;{\text{and}}\;\;\;\Delta F_{s}^{{\left[ C \right]}} = - K^{s} \left( {\Delta X_{i}^{{\left[ C \right]}} - \Delta X_{i}^{{\left[ C \right]}} n_{i} } \right) $$
(2)

Forces and moments on discrete particles can be determined by:

$$ F_{i}^{{\left[ P \right]}} \leftarrow F_{i}^{{\left[ P \right]}} - F_{i}^{{\left[ C \right]}} ;\;\;\;{\text{and}}\;\;M_{i}^{{\left[ P \right]}} \leftarrow M_{i}^{{\left[ P \right]}} - e_{{ijk}} \left( {X_{j}^{{\left[ C \right]}} - X_{j}^{{\left[ P \right]}} } \right)F_{k}^{{\left[ C \right]}} $$
(3)

Relative velocity (displacement) at the interface \(\left( {V_{i} } \right)\) is given by:

$$ V_{i} = \dot{X}_{{i,E}}^{{\left[ C \right]}} - \dot{X}_{{i,E}}^{{\left[ P \right]}} = \dot{X}_{{i,E}}^{C} - \left[ {\dot{X}_{i}^{{\left[ P \right]}} + e_{{ijk}} \omega _{j}^{{\left[ P \right]}} \left( {X_{k}^{{\left[ C \right]}} - X_{k}^{{\left[ P \right]}} } \right)} \right] $$
(4)

where, \(\dot{X}_{{i,E}}^{{\left[ C \right]}}\)and \(\dot{X}_{{i,E}}^{{\left[ P \right]}}\) are the velocities of elements and discrete grains. \(\dot{X}_{i}^{{\left[ P \right]}}\)and \(\omega _{j}^{{\left[ P \right]}}\)are the translation and rotation of a grain, and \(e_{{ijk}}\)is the permutation symbol.

The velocity of continuum elements at the interface is then calculated as:

$$ \dot{X}_{{i,E}}^{{\left[ C \right]}} = \sum N_{j} \dot{X}_{{i,E}}^{j} $$
(5)

where, \(N_{j}\)is a shape function, determined by: \(N_{j} = \left( {1 + \xi _{o} } \right)\left( {1 + \eta _{o} } \right)/4,j = 1,2,3,4;\,\,and\;\xi _{o} = \xi _{i} \xi ,\eta _{o} = \eta _{i} \eta ;\xi _{i}\) and \(\eta _{i}\) are coordinates of nodes. Shear and normal forces are distributed to the nodal force, \(F_{i}^{{\left[ {E,j} \right]}}\) via shape function \(N_{j}\):

$$ F_{i}^{{\left[ {E,j} \right]}} = F_{i}^{{\left[ E \right]}} + F_{i}^{{\left[ C \right]}} N_{j} $$
(6)

During load application, contact forces DEM domain (Zone 1) is transferred to continuum domain (Zone 2); and the displacements of continuum elements, are transferred back to the DEM – Zone 1 as boundary conditions. More detail on the model calibration, application of loading and boundary conditions, and laboratory test results can be found in [22].

Fig. 2.
figure 2

(a) Simulated ballast grains; (b) coupled DEM-FEM model; and (c) interaction of a grain and an element; and (d) force and moment exchanges (modified after [24])

Full size image
Table 1. Micro-mechanical parameters adopted for DEM analysis
Full size table

3.2 Predicted Stress-Strain Behaviour of Ballast

Figure 3 presents predicted cyclic stress-strain from the coupled DEM-FEM model at varied load cycles N subjected to a frequency, f = 20 Hz. The predicted strains remarkably increase up to about 5.5% at the first N = 1000 load cycles, followed by slightly increased strains up to N = 5,000 cycles. After this cycle, the ballast remains relatively unchanged to the end of the simulation (N = 10,000 cycles). This indicated that ballast aggregates exhibited remarkable compression and re-arrangement during the initial stage of loading, but after achieving a threshold compression, any subsequently applied loads would resist further deformation and instead, accelerate ballast breakage.

Fig. 3.
figure 3

Applied cyclic stress versus axial strain εa at varying load cycle, N

Full size image

3.3 Simulated Ballast Breakage

The accumulation of bond breaking with increased N subjected to 4 frequencies of f = 10–40 Hz is presented in Fig. 4. It is noted that the breaking of bonds (Br) within a simulated particle was considered to represent particle breakage. It is worth mentioning that Br is different from the actual ballast breakage measure in the laboratory, the Br could supply an indication of the extent of breakage occurs. From DEM-FEM simulation, the Br increases with an increase in load frequency, f and this trend are similar to the ballast breakage measured in the laboratory data. Subjected to a given f, the number of broken bonds increases significantly within the first N = 5,000 cycles; and then keeps almost unchanged at subsequent loading cycles. The inclusion of a recycled rubber mat underneath the ballast layer leads to a decreased Br; and this could be attributed to the energy-absorbing characteristics of the rubber mat.

Fig. 4.
figure 4

(adopted from Indraratna et al. 2020 – with permission from ASCE).

Predicted contact bond breaking at different frequencies, f = 10–40 Hz with and without recycled rubber mat

Full size image

3.4 Contact Force Distributions

Figure 5 presents the distributions of contact forces in the ballast layer and vertical stresses predicted for capping and subgrade layers. Vertical stress contours in the capping and subgrade layers at different load cycles N are also presented where the contour was captured under f = 20 Hz. It is seen that the applied cyclic loads are distributed to particles through contact force chains where each contact force is represented at the contact point having a thickness proportional to the force magnitude. Upon cyclic loading, large contact forces appeared and concentrated beneath a top-loading plate. It is clearly seen that the inclusion of rubber mat could increase the contact areas among ballast particles and underneath layers, widen the distribution of vertical stresses; and this in turn decrease magnitude of contact forces the interfaces. Predicted vertical stress at the capping is large at the interfaces with the ballast grains, whereby it is predicted to significantly decrease with depth. An increased applied load cycle N, leads to an increased number of contact force and increased contact force magnitude. This could be associated with the densification and re-arrangement of particles.

Fig. 5.
figure 5

(adopted from Indraratna et al. 2020).

Distributions of contact forces in ballast layer and vertical stresses predicted for capping and subgrade: (a) without rubber mat; and (b) with a rubber mat

Full size image

4 Conclusions

The following findings can be established:

  • Measured laboratory test data proved that the axial deformation, εa increased with an increased N and the higher frequency applied the higher εa was measured. Ballast grain showed a large axial strain, εa within the first N = 1000 cycles, followed by a slight increase in the εa at a decreased rate.

  • The inclusion of a recycled rubber mat resulted in decreased axial strain and breakage of ballast. This was because the mat absorbed cyclic and impact loads and thereby reduced particle breakage of up to 30.8% and 35.3% when subjected to f = 10, 20Hz.

  • A coupled DEM-FEM model was introduced to investigate the applied cyclic load-deformation responses of ballast, where discrete grains were modelled by the DEM and the continuum media (REAM, capping and subgrade) were modelled by a continuum media. From the coupled discrete-continuum model (coupled DEM-FEM), the predicted amount of broken bonds (i.e., mimicking particle breakage) increased with increased frequency (f = 10–40 Hz) and it decreased with the inclusion of a rubber mat.

  • The force distributions were studied, and they showed that the forces distributed non-uniformly across the ballast specimens and the inclusion of a rubber mat decreased the magnitude of contact forces which also implied for a reduced particle breakage.