# Discrete element simulation of railway ballast: modelling cell pressure effects in triaxial tests

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## Abstract

The paper investigates reproducing the effects of confining pressure on the behaviour of scaled railway ballast in triaxial tests in discrete element models (DEM). Previous DEM work, using a standard Hertzian elastic contact law with an elastic–perfectly plastic tangential slip model, has been unable to replicate the behaviour observed in laboratory tests across a range of confining pressures without altering both the material stiffness and the inter-particle friction. A new contact law modelling damage at the contacts between particles is introduced. Particle contact is via spherically-capped conical asperities, which reduce in height if over-stressed. This introduces plasticity to the behaviour normal to the contact surface. In addition, the inter-particle friction angle is varied as a function of normalized contact normal force. At relatively low normal forces the friction angle must be increased for peak mobilized friction angles to match the laboratory data, an effect that is attributed to interlocking at the scale of surface roughness. Simulation results show close agreement with laboratory data.

## Keywords

DEM Confining pressure Triaxial test Railway ballast Contact damage## 1 Introduction

Triaxial test results on railway ballast and other granular materials (e.g., [3, 9, 13, 17, 20, 25]) have shown that, when other factors are held constant, the peak mobilized strength and volumetric strain are strongly influenced by the confining cell pressure applied. In particular, at low confining pressure the peak mobilised shear strength is significantly greater. This has particular relevance in the context of railway ballast. Railway ballast is placed as a near surface layer and operates entirely within a low horizontal confining stress regime, and is required to perform acceptably over a large range of vertical to horizontal stress ratios.

Summary of laboratory and simulated results for triaxial tests

Lab | Model | Lab | Model | |
---|---|---|---|---|

15 kPa | 15 kPa | 200 kPa | 200 kPa | |

\(\phi _{peak}\) (\(\circ \)) | 47 | 47 | 45 | 51 |

\(\epsilon _a\) at onset of dilation (%) | 0.3 | 0.3 | 3.2 | 0.3 |

Peak \(\epsilon _{vol}\) (%) | 0 | 0 | 1 | 0.1 |

- 1.
The model is too stiff or is otherwise incorrectly calibrated. Contacts between real ballast particles are not Hertzian, as the particles have rough, non-spherical surfaces, so there is some scope for uncertainty in the choice of stiffness magnitude. However, the 10 GPa shear modulus used for the Hertzian contact stiffness in the simulation is already considerably less than that of granite, which is in the region of 17–29 GPa [12] (based on a Poisson’s ratio of 0.2).

- 2.
The ballast in the laboratory is experiencing some sort of damage that increases with cell pressure. Visual and sieve analysis before and after triaxial testing revealed no discernible particle breakage and no measurable change in particle size distribution. This was also observed in tests on full scale ballast [1]. Therefore, if damage is occurring to the material used in these tests, it must take the form of very small-scale crushing or abrasion of contacting asperities, perhaps reducing the effective surface roughness.

- 1.
Particle shape and size distribution (PSD).

- 2.
Initial void ratio.

- 3.
Particle contact stiffness.

- 4.
Inter-particle friction angle.

Particle size distribution used in simulation

% Passing by weight (network rail specification) | Sieve size (mm) | One-third scale sieve size (mm) |
---|---|---|

100 | 62.5 | 20.83 |

85 | 50 | 16.67 |

17.5 | 40 | 13.33 |

12.5 | 31.5 | 10.5 |

1.5 | 22.4 | 7.47 |

The effects of varying the inter-particle friction angle, for a constant shear modulus of \(10\,\hbox {GPa}\) and a constant confining pressure of 200 kPa, are shown in Fig. 5. These results were obtained using the discrete element model and the specimen of potential particles [11] presented in [2]. Reducing the interparticle friction angle has little effect on the initial response, which remains stiffer than the laboratory test. Also, the critical state strength is not significantly altered, with all three simulations being slightly weaker than the laboratory test. However, the peak strength increases with increasing inter-particle friction angle and is accompanied by an increase in the rate of dilation. Thus it is clear that a change of interparticle friction angle alone cannot correct the fit to the laboratory data at the higher confining pressure of 200 kPa (it would also spoil the fit at lower confining pressures).

*G*, for a constant inter-particle friction angle of \(40^{\circ }\). With a value of \(G=1\,\hbox {GPa}\), the initial stiffness of the response is much improved. However, the peak strength, rather than reducing as desired, has increased along with the dilation rate. At \(G=100\,\hbox {MPa}\) the peak mobilized friction angle is reduced, but the initial response is too soft. Once again, it seems that the critical state response is not significantly affected.

Summary of laboratory and simulated results for cyclic loading tests

Test | Residual modulus\(^\mathrm{a}\) \(E_r\) (MPa) | Axial strain\(^\mathrm{b}\) \(\epsilon _a \) |
---|---|---|

Laboratory | 125.3 | 0.0134 |

Simulation \(\hbox {G}=0.5\,\hbox {GPa}\) | 16.0 | 0.0132 |

Simulation \(\hbox {G}=10\,\hbox {GPa}\) | 91.9 | 0.00847 |

Although this choice of parameters yields quite a close match for the monotonic tests, universally changing the interparticle friction angle to match the data at different confining pressures is not a workable solution for more general loading cases in which the confining pressure is not held constant. Furthermore, although there is some room for manoeuvre in the choice of stiffness (ballast particles are not spherical and particle contacts are therefore not perfectly Hertzian), the very low value used (about one fiftieth of the value for granite) is difficult to justify. This is borne out in a comparison of cyclic loading tests shown in Fig. 8, for the laboratory and simulation data. These results are summarized in Table 3. This shows that a much stiffer model (G of at least 10 GPa) is required to match the per-cycle deflection observed in the laboratory scaled triaxial test.

One explanation for the observed behaviour is that plastic damage occurs at contacts as they are loaded. Cavarretta et al. [5] showed that for coarse grained sand the inter-particle contact stiffness included an irrecoverable proportion of deflection beyond which the deflection vs. stress plot showed Hertzian contact behaviour. This would make the contacts appear initially soft in monotonic loading (as the asperities are crushed), but stiffer as loading progressed and stiffen cyclic loading behaviour if the model was subsequently unloaded and reloaded.

## 2 Conical damage model

*R*. This asperity contacts an elastic half plane via a Hertzian contact with the rounded tip. A minimum radius of curvature \(R_{min}\) is specified, as shown in Fig. 9a. The maximum stress, \(\sigma _0\), for a Hertzian contact (which occurs at the centre of the contact) can be expressed as a function of the radius of contact,

*R*, the normal force,

*P*, and the overlap of the spherical cap with the elastic half plane, \(\delta \), as:

*R*is given by:

*R*, and offset, \(\delta _c\), are calculated. The new normal force,

*P*, is calculated on the basis of the new overlap between the two particles, \(\delta \), (which will have reduced as the surface of the asperity has receded) and the new contact stiffness, \(k_n\), which has changed due to the change of contact radius:

### 2.1 Calibration of the conical damage model

- 1.
Select the elastic shear modulus,

*G*. (Poisson’s ratio, \(\nu \), was set to 0.2 for all simulations). - 2.Select \(R_{min}\): it was assumed that very little crushing occurs at a confining pressure of 15 kPa, thus the behaviour is dominated by the influence of \(R_{min}\) on the stiffness of the contacts.
- (a)
Using the contact force data at peak strength for the best fit simulation at a cell pressure of 15 kPa (Fig. 2), from Eq. 4 calculate the value of the contact radius required to support the load, based on the elastic modulus chosen in (1). The proportion of non-crushing contacts as a function of contact radius, for a shear modulus of 5 GPa, is shown in Fig. 10.

- (b)
Choose a value of \(R_{min}\) so that most of the contacts will not crush.

- (a)
- 3.Select \(\alpha \): with all other parameters held constant, the cone angle, \(\alpha \), determines the rate of plastic deformation normal to the contact plane in response to a normal force increment beyond the elastic limit. An initial estimate for alpha can be obtained by studying the simulation data for the best fit at a cell pressure of 200 kPa with purely elastic behaviour normal to the contact plane (Fig. 7). In this case, the fit to the laboratory data was obtained by using a low inter-particle stiffness, resulting in relatively large particle overlaps. By reapportioning these particle overlaps between plastic and elastic deformations, a more realistic material stiffness can be used, and a value of \(\alpha \) may be determined as follows:A value of alpha can be calculated for each contact. Next, select a single value that satisfies the largest number of contacts. For example, in the case of \(G=5\,\hbox {GPa}\) and \(R_{min}=4\,\hbox {mm}\), a histogram of values of \(\alpha \) for all particle contacts is shown in Fig. 11, which shows a peak at \(\alpha \approx 78^{\circ }\).
- (a)
At the peak of mobilized friction, examine the contact forces,

*P*, and particle overlaps, \(\delta \). - (b)
Calculate the radius of curvature,

*R*, required to support the force at each of the (non-zero force) contacts, using Eq. 4. - (c)
From Eq. 6, calculate the elastic deformation, \(\delta _e\), that would result from the application of this force, using the radius,

*R*, and the shear modulus,*G*. - (d)
Calculate the plastic displacement, \(\delta _p\), that would give rise to the same overall contact displacement, \(\delta \), as \(\delta _p = \delta - \delta _e\). This minimum value of \(\delta _p\) is limited to zero for the purpose of the next calculation.

- (e)Use geometry to calculate the angle \(\alpha \) for each contact such that the new contact radius,
*R*, is obtained for a plastic reduction in height of the asperity, \(\delta _p\), as:$$\begin{aligned} \alpha = \arcsin \left( {\frac{R-R_{min}}{R-R_{min}+\delta _p}}\right) \end{aligned}$$(9)

- (a)
- 4.Run a cyclic loading test and examine the resulting resilient modulus. Observation of the results for different parameter values shows that:Adjust the shear modulus, by iteration, to obtain the correct residual modulus and then repeat steps (1)–(4) to update \(R_{min}\), \(\alpha \) and
- (a)
The relationship between shear modulus and resilient modulus is rather non linear.

- (b)
The resilient modulus is not very sensitive to \(\alpha \) or the interparticle friction angle, \(\phi \).

- (c)
The plastic axial strain is affected by \(\alpha \) and \(\phi \).

*G*. - (a)

## 3 Variable friction model

### 3.1 Variable friction model: a first attempt

*a*and

*b*are constants.

- 1.
The residual friction angle, \(\phi _{residual}\) was set to \(28^{\circ }\). The coefficients \(\alpha \) and \(R_{min}\) were set to the values obtained in Sect. 2.1.

- 2.
The magnitude of the normal force above which the friction coefficient is constant was set to 80 N (see Fig. 16). This results in most of the contacts at a confining pressure of 15 kPa having an increased friction coefficient, and many of the higher-load carrying contacts at 200 kPa the residual value (see Fig. 13).

- 3.
A trial value for

*b*was chosen. The parameter*a*was calculated from Eq. 11 with \(P=80\,\hbox {N}\) (from step 2) and \(\mu =tan(\phi _{residual})\). Simulations at confining pressures of 15 and 200 kPa were then carried out. This step was repeated, adjusting*b*, to obtain the best fit to the experimental data.

Parameters for power law friction simulation

Parameter | Value |
---|---|

\(\alpha \) | \(78^{\circ }\) |

\(R_{min}\) | \(4\,\hbox {mm}\) |

\(\phi _{residual}\) | \(28^{\circ }\) |

| 1.1037 |

| 1.2 |

G | \(5\,\hbox {GPa}\) |

### 3.2 A damage-dependent friction model

- 1.
The normal force should be expressed in a non-dimensional form so that the model can be applied at different scales.

- 2.
There is no link between the state of damage and the inter-particle friction angle.

*P*, normalized by the critical force, \(P_c\), i.e. the value of normal force at which plastic deformation will occur for a given sphere radius and set of elastic properties:

*R*is the radius of the sphere,

*K*is a hardness coefficient related to the Poisson’s ratio of the material by \(K = 0.454 + 0.41\nu \) and \(H=2.8Y\), according to [27], where

*Y*is the yield strength of the material.

Kogut and Etsion were studying the behaviour of metals and their approach may not be directly applicable to rock contacts. However, several authors have reported increased friction with rough surfaces (see for example [7, 8, 23]). In the case of railway ballast, it is possible to imagine that surface roughness, on a smaller scale than the asperities considered in this paper, could provide a resistance to lateral movements through interlocking. At low normal forces, the interlocking effect would dominate, becoming less significant with increasing normal force. However, the lateral strength of the interlocked region depends on the contact area, which in the case of nominally Hertzian contacts is a function of the effective radius of contact. For constant surface roughness and normal load, an increase in the nominal radius of contact will increase the area of contact and could generate a greater degree of interlocking. This provides a possible link between the conical damage model (see Sect. 2), which alters the effective contact radius, and the frictional behaviour.

*R*calculated in Eq. 4. It is important to note that the normal force,

*P*can never exceed the critical force \(P_c\), as this would result in damage to the cone, an increase in the contact radius and, consequently, in the critical force, \(P_c\). As a result, the friction coefficient, \(\mu \), has a lower limit given by \(\mu =\beta \).

#### 3.2.1 Model calibration

The parameters \(\beta \) and \(\gamma \) were derived from the values of residual friction, \(\phi _{residual}\), and the exponent, \(1/b-1\) (see Eq. 11) determined in Sect. 3.1. Thus, \(\beta =\tan (28^\circ )\) and \(\gamma =1/1.2-1=-0.1\dot{6}\). The cone angle, \(\alpha \), and shear modulus, *G* were initially set to the values obtained in Sect. 2.1.

*R*, from Eq. 4 into 5 to give a relationship between the normal force and the plastic settlement, \(\delta _c\), as:

*G*, was then adjusted to obtain a good fit to the resilient modulus measured from the laboratory cyclic loading data. At each stiffness, \(R_{min}\) was recalculated using Eq. 16 and \(\alpha \) was adjusted to maintain a constant value of

*A*in Eq. 18. Finally, small trial-and-error adjustments were made to \(R_{min}\) and \(\alpha \) to obtain the correct monotonic loading response across the range of confining pressures.

Parameters for damage-dependent friction simulation

Parameter | Value |
---|---|

\(\alpha \) | \(81^{\circ }\) |

\(R_{min}\) | 12 mm |

\(\beta \) | 0.53171 |

\(\gamma \) | −0.16667 |

G | 8 GPa |

## 4 Summary and conclusions

A discrete element contact law has been proposed to model the behaviour observed in laboratory triaxial tests of scaled railway ballast at a range of confining pressures from 15 to 200 kPa. It was shown not to be possible to match the laboratory behaviour using the standard Hertzian elastic contact law with an elastic-perfectly plastic tangential slip model, presented in [2]. This suggests that some damage must be occurring at particle contacts. A new contact model was proposed, based on the supposition that particles contact at asperities. The asperity contact was modelled by a spherically-capped cone. In an elastic regime, the contact is Hertzian with a radius defined by the cap radius. If the yield stress is exceeded the cap is crushed and forms a lower cap with a larger radius, able to support the load. This model was shown to provide good agreement with the initial loading behaviour in monotonic laboratory tests for the different confining pressures tested and also satisfies the requirement for a stiffer unloading response. However, the damage model did not significantly alter the peak strength, which reduces with confining pressure in the laboratory tests. Therefore a supplementary model was proposed that varies the inter-particle friction coefficient as a function of a normalized load, which is the contact normal force divided by the critical force, \(P_c\) (the load at which plastic failure will start). For contact normal loads less than \(P_c\), the friction is increased; an effect attributed to interlocking at the surface roughness scale. For virgin monotonic loading of a contact, the friction will decrease until the load is equal to \(P_c\) and the friction coefficient reaches a constant minimum value. A continued increase in loading will result in plastic damage and a consequent increase in \(P_c\) to the current load value. Subsequent unloading results in an increase in friction coefficient.

Simulation results show excellent agreement with the laboratory data for all monotonic triaxial tests. Good agreement was obtained with cyclic loading data over the first few cycles, in terms of both the resilient response and the axial strain. In later cycles, the simulation displays larger plastic axial strain than the laboratory results and this remains a subject for further investigation.

## Notes

### Acknowledgments

The authors are grateful for the financial support of the Engineering and Physical Sciences Research Council (EPSRC) through the Programme Grants: Track 21 (EP/H044949/1) and Track to the Future (EP/M025276/1). We would also like to acknowledge the work of Sharif Ahmed, who created the particle shapes that were used in this work and in the original study [2]. Data supporting the results presented in this paper are openly available from the University of Southampton repository at doi: 10.5258/SOTON/393834. This work was carried out alongside the University of Nottingham as part of the ‘Track to the Future’ project. Each research group worked independently to model the behaviour of a scaled ballast with reference to the same laboratory data [3]. The work at Nottingham, in a paper [21] submitted for review simultaneously with this one, models particle abrasion via the breakage of small asperities.

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