# Discrete element modelling of scaled railway ballast under triaxial conditions

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## Abstract

The aim of this study is to demonstrate the use of tetrahedral clumps to model scaled railway ballast using the discrete element method (DEM). In experimental triaxial tests, the peak friction angles for scaled ballast are less sensitive to the confining pressure when compared to full-sized ballast. This is presumed to be due to the size effect on particle strength, whereby smaller particles are statistically stronger and exhibit less abrasion. To investigate this in DEM, the ballast is modelled using clumps with breakable asperities to produce the correct volumetric deformation. The effects of the quantity and properties of these asperities are investigated, and it is shown that the strength affects the macroscopic shear strength at both high and low confining pressures, while the effects of the number of asperities diminishes with increasing confining pressure due to asperity breakage. It is also shown that changing the number of asperities only affects the peak friction angle but not the ultimate friction angle by comparing the angles of repose of samples with different numbers of asperities.

## Keywords

Discrete element modelling Triaxial tests Railway ballast## 1 Introduction

The majority of railway tracks in the world are still using ballast in their design because of their relatively low cost compared to concrete slab tracks [1]. In order to design ballasted track and maintain it properly, it is important to fully understand the mechanical behaviour of the ballast material. With regards to most geomaterials, the triaxial test is one of the most useful laboratory tests for investigating the deformation and strength of railway ballast [2, 3, 4, 5, 6, 7, 8, 9]. Indraratna et al. [2] performed large scale triaxial tests on latite basalt under various confining pressures and found the angle of internal friction was a function of the confining pressure and particle breakage was more pronounced at higher confining pressure. Aursudkij et al. [6] also performed large scale triaxial tests on limestone particles under confining stresses of 10, 30 and 60 kPa and the sample with lowest confining stress was found to give the largest amount of dilation but the smallest amount of breakage. Additionally, as a powerful numerical tool to understand the micromechanical behaviour of granular material, the discrete element method (DEM) [10] has shown some success in simulating triaxial tests on granular materials [11, 12, 13, 14, 15, 16, 17, 18, 19]. However, it is a challenge to consider both irregular particle shape and ballast degradation in DEM modelling of railway ballast; e.g. Lobo-Guerrero and Vallejo [20, 21] published work on DEM of ballast degradation but only circular particles were used, and the studies where the complex ballast particle shapes were modelled, particle breakage was ignored [22, 23, 24, 25]. Considering that most ballast degradation is not attributable to particle splitting but instead primarily particle abrasion [4, 5], Lu and McDowell [14, 15] introduced a tetrahedral shaped clump with small breakable asperities (Fig. 1) to represent a ballast particle, which as far as the authors are aware, was the DEM study of abrasion with an irregular shaped particle. Although this asperity breakage model cannot represent bulk fracture, it conserves mass and presents the fragment movements as a result of the crushing mechanism. Ballast particles are usually angular,uniformly graded [26], tend to be blocky and equi-dimensional—so the tetrahedral shape was chosen to fit these criteria. By using this model they successfully explained firstly the difference in shear strength at low and high confining pressure, and the associated reduction in dilation at high confining pressure observed on full-sized ballast particles.

## 2 DEM model

The discrete element method (DEM) considers a granular material like ballast as an assembly of objects interacting through a contact law. The contact interaction is generally modelled using a linear stiffness governing the overlap between the contacting objects. This normal and shear contact forces are characterised by normal and shear stiffness coefficients \(k_{n}\) and \(k_{s}\) respectively. A friction coefficient *f* based on the Mohr–Coulomb criterion limits the ratio between the shear and normal components of the contact force. Although more complex contact models have been developed, this basic linear elastic model is embedded in the code PFC3D [29] and used in the present study. The acceleration of the object is first deduced from the contact forces acting on it using Newton’s second law and then its velocity and displacement are deduced by integration over a time increment. The displacements of the objects lead to new overlaps between them from which new contact forces arise forming a full DEM cycle calculation. The motion of the objects is determined using a succession of time increments. In order to avoid non-physical oscillation within the assembly of objects, non-viscous damping reduces their accelerations by 70 % (by default) to allow further dissipation of energy in addition to friction at contacts [29].

## 3 Triaxial test conditions

Sample parameters

Particle number | 750 |
---|---|

Density | 2600 \({\mathrm{kg/m}}^{3}\) |

Clump/asperity normal and shear stiffness | \(10^{9}\,{\mathrm{N/m}}\) |

Clump/asperity friction | 0.5 |

Vertical wall normal stiffness | \(10^{9}\,{\mathrm{N/m}}\) |

Horizontal wall normal stiffness | \(10^{8}\,{\mathrm{N/m}}\) |

Wall shear stiffness | 0 |

Wall friction | 0 |

## 4 Simulations without crushing

The initial voids ratio, asperity number and parallel bond parameters

Initial voids ratio | Number of asperities | Normal and shear stiffness coefficients \(k_{pb n}\) and \(k_{pb s}\) | Normal and shear strengths \(\sigma _{n}\) and \(\sigma _{s}\) | |
---|---|---|---|---|

Simulation 1 | 0.72 | 0 | No bonds | No bonds |

Simulation 2 | 0.88 | 0 | No bonds | No bonds |

Simulation 3 | 1.02 | 0 | No bonds | No bonds |

Simulation 4 | 0.72 | 8 | \(1.8\times 10^{13}\) Pa/m | \(5\times 10^{7}\) Pa |

Simulation 5 | 0.72 | 8 | \(1.8\times 10^{13}\) Pa/m | \(5\times 10^{8}\) Pa |

Simulation 6 | 0.72 | 8 | \(1.8\times 10^{13}\) Pa/m | \(5\times 10^{6}\) Pa |

Simulation 7 | 0.72 | 8 | \(1.8\times 10^{13}\) Pa/m | \(5\times 10^{5}\) Pa |

Simulation 8 | 0.72 | 1 | \(1.8\times 10^{13}\) Pa/m | \(5\times 10^{7}\) Pa |

Simulation 9 | 0.72 | 3 | \(1.8\times 10^{13}\) Pa/m | \(5\times 10^{7}\) Pa |

Simulation 10 | 0.72 | 6 | \(1.8\times 10^{13}\) Pa/m | \(5\times 10^{7}\) Pa |

## 5 Simulations with breakage

Further simulations were conducted with 8 breakable asperities (simulations 4-7). The bond parameters are given in Table 2. The parallel bond stiffnesses \(k_{pb n}\) and \(k_{pb s}\) were chosen as the same values as in Lu and McDowell [15] and the bond strength was given a range of values to investigate its effect. Figure 6 shows the mobilised angles of friction and volumetric behaviour for simulation 4 compared to simulation 1 with no asperities. Introducing breakable asperities reduces the dilation and leads to a better modelling of the volumetric deformation at both stress levels because of the introduction of particle abrasion. In addition, the volumetric deformation at high confining pressure is affected more by the presence of the crushable asperities than that at low confining pressure, due to more broken asperities at the higher confining pressure. However, there is roughly 10\(^{\circ }\) difference in the peak mobilised friction angles at low and high confining pressure according to Fig. 6, which is only observed in the experimental results on full size ballast [2] but not on scaled size ballast [27]; this must because at the stress levels being considered, the smaller scaled ballast particles are statistically stronger [35, 36] and influenced less by the range of applied confining pressures. In order to model better the mobilised strength of the scaled ballast under the applied confining pressures, the effect of changing the bond strengths for the 8-asperity clump (simulations 5–7) and the number of asperities (simulations 8–10, with the same bond strength as simulation 4) will now be investigated.

Figure 7 shows the effect of changing the bond strength for a clump with 8 asperities. It can be seen that the peak mobilised friction angle increases and the sample becomes more dilative with increasing asperity bond strength at both stress levels. Figure 7c, f shows the percentage of broken asperity bonds during monotonic loading. As expected, the percentage of broken asperities increases with decreasing bond strength. It is clearly observed that the number of broken asperities affects both the mobilised friction and the volumetric deformation of the sample, which is consistent with Lu and McDowell [15]: the higher the number of broken asperities during shearing, the lower is the mobilised friction angle and the less dilative the sample. However, it appears that no single bond strength for the 8-asperity clump is capable of capturing the mobilized strength and dilation at both 15 and 200 kPa. Thus the effect of the number of the crushable asperities on the behaviour was investigated in simulations 8–10.

If the peak strength and dilation of the sample is governed by the behaviour of the asperities, then it would be anticipated that the addition of more small surface asperities would increase the peak strength and dilation at low stress, but that effect would reduce at high stress levels due to the breakage of these asperities. This is clearly shown in Fig. 8. It is evident that the use of three asperities (simulation 9) is able to capture the essential behaviour of the scaled ballast over the range of applied confining pressures. One problem with the simulations, however, is establishing the effect of the asperities on the ultimate or critical state angle of friction because in each of the simulations the sample has not reached a critical state; indeed they are still dilating and in each case at an approximately constant rate with axial strain, giving a wide peak strength as a function of axial strain. Thus, in order to simulate conditions at large strains to establish the critical state angles, all that is required is the angle of repose in each case- in this test particles are indeed rolling down the soil slope at large shear strains. If the size of the asperities is sufficiently small, this should have a negligible effect on the ultimate friction angle \(\upphi _{crit}\), but still be sufficient to provide additional interlock and higher peak strength at low stress levels and low shear strains. Figure 9 shows a simple heap test for each clump shape in order to determine the effect of the number of asperities on the angle of repose. Figure 9 shows an approximately equal angle of repose in each case. This is to say, by changing the number of surface asperities, the tetrahedral clump samples exhibit different peak strengths at low confining pressure, but still provide an approximately constant ultimate friction angle \(\upphi _{crit}\).Therefore it is possible to achieve a quantitative modelling of the peak mobilised friction angle and volumetric strain at each stress level for scaled ballast by considering the number of surface asperities, without adversely affecting the modelling of ultimate conditions.

## 6 Conclusion

This paper demonstrates that it is possible to reproduce the behaviour of ballast particles at different scales in DEM using tetrahedral clumps with breakable asperities. It has been shown that introducing asperities increases the peak shear strength and dilation of sample but the effect reduces with increasing confining pressure due to particle abrasion (breaking of asperities). By examining the effect of asperity strength and the number of asperities on the mobilised peak strength and dilation, it has been possible to model the behaviour of scaled ballast over a range of confining pressures under triaxial conditions. However, under such conditions the axial strains are insufficient for the samples to reach ultimate conditions and so the effect of the number of asperities on the angle of repose was determined to ensure that not only could the peak strength and volumetric behaviour be correctly captured, but that the critical state angle of friction was not significantly affected. The results have shown the importance of the modelling of asperity breakage if the correct peak strength and associated dilation together with ultimate conditions are to be correctly captured under different confining pressures. Future work will examine the effect of a flexible membrane on the triaxial test simulation results, although this is computationally much more intensive.

## Notes

### Acknowledgments

The authors would like to thank EPSRC for funding this work and to John de Bono for his helpful comments on this paper.

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