# Metamodelling of wheel–rail normal contact in railway crossings with elasto-plastic material behaviour

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## Abstract

A metamodel considering material plasticity is presented for computationally efficient prediction of wheel–rail normal contact in railway switches and crossings (S&C). The metamodel is inspired by the contact theory of Hertz, and for a given material, it computes the size of the contact patch and the maximum contact pressure as a function of the normal force and the local curvatures of the bodies in contact. The model is calibrated based on finite element (FE) simulations with an elasto-plastic material model and is demonstrated for rail steel grade R350HT. The error of simplifying the contact geometry is discussed and quantified. For a moderate difference in contact curvature between wheel and rail, the metamodel is able to accurately predict the size of the contact patch and the maximum contact pressure. The accuracy is worse when there is a small difference in contact curvature, where the influence of variation in curvature within the contact patch becomes more significant. However, it is shown that such conditions lead to contact stresses that contribute less to accumulated plastic deformation. The metamodel allows for a vast reduction of computational effort compared to the original FE model as it is given in analytical form.

## Keywords

Metamodel wheel–rail contact mechanics Hertz switches & crossings FEM Plastic deformation## 1 Introduction

Railway turnouts (switches & crossings, S&C) are subjected to a severe load environment, in particular in the switch and crossing panels, see, e.g., [8, 18, 23]. High maintenance costs are generated because of the needs for repair and replacement of switch rails and crossings due to damage in the forms of wear, plastic deformation and breaking out of material caused by surface or subsurface initiated rolling contact fatigue cracking.

Field tests have been reported, where impact loads at crossings have been measured by instrumented wheelsets [9, 20] or by rail-mounted accelerometers [13]. Depending on vehicle speed and wheel–rail contact geometry, the impact load may be considerably higher than the nominal static wheel load [1]. Increasing vehicle speeds and axle loads, and wheels with severely worn profiles, induce contact conditions that generate higher magnitudes of wheel–rail contact forces and slip. Accelerated rail profile degradation and damage occur if the rail profiles are not corrected in time, since the deteriorated rail profiles induce contact conditions that further magnify the dynamic loads.

Optimisation of rail profiles [19, 31] and support stiffness [3] are two means to reduce the wheel–rail contact forces and damage in the crossing panel. Another approach is to use a higher strength material for the crossing. Common materials used in crossings include austenitic steels (such as the explosively hardened manganese steel Mn13), bainitic steels (B360) and high strength pearlitic rail steels (such as the fine-pearlitic rail grade R350HT). Based on field experience, each of the materials has different advantages in terms of resistance to the various types of damage. For example, the manganese steel has a higher rate of plastic deformation and hardening during the initial load cycles but at the same time a better adaptability of rail profile to meet the variation of worn wheel profiles in traffic. In addition, due to its work-hardening, the manganese steel has good toughness and wear resistance under severe impact loading conditions (see, e.g., [4]).

A cross-disciplinary simulation methodology, integrating several numerical tools, for predicting the damage in rail profiles of S&C by an iterative approach has been proposed by Johansson et al. [5]. In each iteration step, the methodology consists of four parts: (I) simulation of dynamic vehicle–track interaction to predict wheel–rail contact forces, creepages and contact positions; (II) analysis of wheel–rail contacts, using a three-dimensional finite element (FE) model with an elasto-plastic material model, to determine the size of the contact patches and stresses in the material due to the normal contact; (III) prediction of accumulated damage including plastic deformation, wear, and rolling contact fatigue; and (IV) updating of rail profiles, which are then used as input in the next iteration step. The methodology has been validated by comparing predicted rail profiles with those measured in the field at Härad in Sweden [5] and Haste in Germany [16]. A similar method based on an explicit FE model has been presented in [34].

The long-term objective of the described methodology is to enable efficient and accurate predictions of the influence of material selection on long-term S&C rail profile degradation to design better S&C. However, to represent the variation in traffic that a given crossing is subjected to, a distribution of load parameters such as wheel profile, train speed, axle load and wheel–rail friction coefficient needs to be accounted for. This in turn leads to a wide range of contact load scenarios with different magnitudes of wheel–rail contact load occurring at various positions along the crossing panel. Each of these scenarios needs to be accounted for by an FE simulation in part II of the methodology, which, due to the required combination of high mesh resolution in three dimensions and non-linear material model, is computationally expensive. Thus, there is a need to improve the methodology in terms of computational efficiency. Substituting all of the required FE simulations in each iteration step (part II above) with low-cost calculations using a calibrated analytical metamodel, with the distribution of output data calculated in step I as input, would reduce the computational cost significantly. A procedure to generate such a metamodel with limited loss in accuracy compared to the full elasto-plastic FE model is described in the present paper.

This paper is organized in the following way: Sect. 2 describes tools available for the modelling of wheel–rail contact; the FE model used in this study is presented in Sect. 3; two alternative metamodels that are used to replace the FE simulations are described and compared in Sect. 4; Sect. 5 investigates the influence of constant curvature assumed for the contact geometry on the accuracy of the metamodels; Sect. 6 presents a demonstration of the suggested approach, and finally, concluding remarks are stated in Sect. 7.

The following notation convention has been adopted in this paper: small Latin characters in bold (e.g., \(\varvec{a}\)) denote vectors, while capital characters (e.g., \(\varvec{A}\)) or small Greek characters (e.g., \(\varvec{\sigma }\)) in bold denote tensors of the second order, unless stated differently by the context.

## 2 Wheel–rail contact in the crossing panel

One key aspect in the simulation of rail profile degradation is to determine the correct contact pressure distribution in the wheel–rail contact. An overview of available contact models can be found in the literature (see, e.g., [26]). Several tools exist for solving the normal contact problem, for example, solutions based on (1) the theory of Hertz [12]; (2) Kalker’s variational method [7]; and (3) FE simulation. The first two approaches are applicable for elastic material response only. In addition, they rely on the assumption that the bodies in contact are large compared with the dimensions of the contact patch, such that they can be considered as infinite half-spaces. The same holds for most fast methods for vehicle dynamics simulations (see, e.g., [25]). Furthermore, the theory of Hertz assumes that the geometry of each contact surface can be approximated by an elliptic paraboloid. Kalker’s method imposes no restriction on the wheel–rail contact geometry, while the FE simulation approach allows for both arbitrary geometry and inelastic material response.

A comparison between calculated contact conditions for elastic material when applying an in-house implementation of Kalker’s variational method [21] or an FE model, using either the nominal geometry (accounting for the variation in curvature within the contact patch) or the simplified geometry (constant radius of curvature), will be presented in Sect. 5. It will be shown that there is excellent agreement between the two methods when the nominal geometry is used. However, the aim of the procedure presented in this paper is to account for plasticity due to wheel–rail contact in a parameterised and numerically efficient way. If plasticity is overlooked and the contact pressure is computed based on the elastic material response, the stress in the rail will, at some contact locations, be so high that it will (in part III of the simulation methodology) lead to a drastic overestimation of the accumulated plastic strain or to an absence of the equilibrium solution when the estimated stress exceeds the ultimate stress. Therefore, a plasticity model together with a simplified geometry of the wheel–rail contact will be proposed in this paper.

## 3 FE model of wheel–rail contact

### 3.1 FE mesh

The modelled quarter of the cut-out piece of the rail is shown in Fig. 6. The upper side of the block, which is pressed against the wheel, has constant curvature \(1 / R_{x}^{\text {r}}\) in the \(yz\) plane and zero curvature in \(xz\) plane (the rail has a constant inclination in the running direction). Its bottom side is fixed, while the two front sides in the \(xz\) and \(yz\) planes are the planes of symmetry, i.e. the out-of-plane deformation is set to zero. The back sides in the \(xz\) and \(yz\) planes are set to be free boundaries, because the identification of the stiffness of linear springs that would correspond to physical boundary conditions was considered superfluous, since Fig. 7 shows negligible effect stemming from the free boundaries. The edge length \(L\) was chosen to assure that the area \(A\) of the contact patch and the maximum contact pressure \(p_{0}\) for the elastic wheel and rail are in close agreement with those provided by the contact theory of Hertz. In the following, the FE model with constant curvature illustrated in Fig. 6 will be referred to as “simplified FE model”.

### 3.2 Material modelling of R350HT

Identified material parameter values (Poisson’s ratio \(\nu\) is not calibrated) for the Ohno–Wang model for rail grade R350HT

| \(\nu\) | \(\sigma _\mathrm{y}\) | \(C_1\) | \(\gamma _{1}\) | \(m_1\) | \(C_2\) | \(\gamma _{2}\) | \(m_2\) |
---|---|---|---|---|---|---|---|---|

\(\left[ \text {GPa} \right]\) | [-] | [MPa] | [MPa] | [-] | [-] | [MPa] | [-] | [-] |

183 | 0.3 | 484 | \(2.23\cdot 10^5\) | 1320 | 2.2 | \(4.25\cdot 10^4\) | 2.62 | 2 |

## 4 Metamodelling

### 4.1 Sampling

The simulation data that need to be sampled are the results of the FE simulations in terms of maximum contact pressure \(p_{0i}^\mathrm{s}\) and semi-axes \(a_i^\mathrm{s},\, b_i^\mathrm{s}\) (superscript s for sampling) of the wheel–rail contact described in Sect. 3. These data will be used to calibrate the metamodel. The number of samples per output is assumed to be *M*, i.e. \(i=1,\ldots ,M\). For a given pair of wheel and rail profiles, and for a given relative lateral displacement \(\varDelta y\) between wheel and rail (recall Fig. 3), the contact positions on wheel and rail are solved in the KPF module of GENSYS. The range of relative lateral wheel–rail displacements \(\varDelta y\) in the crossing panel is constrained in one direction by the contact between the wheel flange and the gauge side of the crossing rail, and in the opposite direction by the contact between the back of the wheel flange (on the other wheel in the wheelset) and the check rail. The range of relative wheel–rail displacements considered in this study is set to \(\varDelta y \in \left[ -5,5 \right]\) mm. Note that a positive value of \(\varDelta y\) means that the wheel flange has moved from the nominal wheel–rail contact position towards the rail. For each predicted contact position on wheel and rail, the curvature is calculated by fitting two circles (see Sect. 3) and thereby obtaining \(R_{x}^{\text {w}}, R_{x}^{\text {r}}\). The other wheel radius \(R_{y}^{\text {w}}\) is calculated by knowing the location of the contact point and the wheel radius at the running circle.

### 4.2 Metamodels

#### 4.2.1 Polynomial metamodel

#### 4.2.2 Hertzian-based metamodel

### 4.3 Calibration of metamodels

*h*and performing a minimisation of this function. The objective function is chosen in a staggered fashion with a relative error for each of the responses:

*h*for the polynomial metamodel is performed using the “Powell” algorithm, see [22], and for the Hertzian-based metamodel using the “Nelder-Mead” algorithm, see [15].

The metamodels were implemented in Python with an extensive use of algorithms and routines for scientific computing provided by the SciPy library [6].

## 5 Assumption of constant local curvature

Influence of contact geometry on maximum contact pressure \(p_{0}\) and maximum von Mises stress \(\sigma ^{\text {Mises}}\) in case of elasto-plastic model for rail grade R350HT

\(\varDelta y\) | \(p_{0,\text {plastic}}^{\text {FEM}}\) | \(p_{0\text {plastic}}^{\text {FEMs}}\) | \(\frac{\text {FEM} - \text {FEMs}}{\text {FEM}}\) | \(\max \sigma ^{\text {Mises}}\) | \(\max \sigma ^{\text {Mises}}_{\text {s}}\) | \(\frac{\text {FEM} - \text {FEMs}}{\text {FEM}}\) |
---|---|---|---|---|---|---|

\(( \text {mm} )\) | (MPa) | (MPa) | (%) | (MPa) | (MPa) | (%) |

\(-5\) | 1569 | 1548 | 1.3 | 771 | 804 | \(-4.3\) |

\(-4\) | 1593 | 1532 | 3.8 | 761 | 800 | \(-5.1\) |

\(-3\) | 1533 | 1447 | 5.6 | 755 | 767 | \(-1.6\) |

\(-2\) | 1556 | 1252 | 19.5 | 744 | 705 | \(-6.9\) |

\(-1\) | 1488 | 1054 | 29.2 | 728 | 636 | 12.6 |

0 | 1488 | 997 | 33.0 | 705 | 610 | 13.5 |

1 | 1404 | 950 | 32.3 | 678 | 583 | 14.0 |

2 | 1179 | 875 | 25.8 | 616 | 554 | 10.1 |

3 | 956 | 784 | 18.0 | 516 | 504 | 2.3 |

4 | 1133 | forbidden | – | 585 | – | – |

5 | 1590 | 1204 | 24.3 | 649 | 689 | \(-6.2\) |

Computed rail and wheel curvatures at the points of contact for different wheel–rail lateral displacements

\(\varDelta y\) | \(1 / R_{x}^{\text {r}}\) | \(1 / R_{x}^{\text {w}}\) | \(\left| 1/R_{x}^{\text {r}} - 1/R_{x}^{\text {w}} \right|\) |
---|---|---|---|

\(( \text {mm} )\) | (m\(^{-1}\)) | (m\(^{-1}\)) | (m\(^{-1}\)) |

\(-5\) | \(-10.4\) | \(-1.5\) | 8.9 |

\(-4\) | \(-10.2\) | \(-1.8\) | 8.4 |

\(-3\) | \(-9.0\) | \(-2.1\) | 6.9 |

\(-2\) | \(-6.8\) | \(-2.3\) | 4.5 |

\(-1\) | \(-5.1\) | \(-2.6\) | 2.5 |

0 | \(-5.0\) | \(-3.0\) | 2 |

1 | \(-5.0\) | \(-3.2\) | 1.8 |

2 | \(-5.0\) | \(-3.6\) | 1.4 |

3 | \(-5.1\) | \(-4.2\) | 0.9 |

5 | \(-11.2\) | \(-7.3\) | 3.9 |

## 6 Results

An evaluation of the adopted approach and results of application of the calibrated metamodels are presented in this section. The metamodels were calibrated using the simplified (constant curvature) FE model with the elasto-plastic material model and with the sampling points (integer values of \(\varDelta y\)) on cross sections A and C. Furthermore, the sampling was performed for four different normal contact loads \(F_{n} \in \left\{ 0.5 N, N, 1.5N, 2 N \right\}\), where \(N = 111.5\) kN is the nominal static wheel load. The values of the calibrated model parameters are given in Appendix B. The metamodel will be evaluated for section B.

### 6.1 Quadratic metamodel

For different positions of contact point on cross section C, a comparison of the responses \(a, b\) and \(p_{0}\) predicted by the quadratic metamodel and the simplified FE model is shown in Fig. 16. An analogous comparison for cross section B, which the metamodel was not calibrated for, is shown in Fig. 17. The figures reveal a significant spread in contact patch size and maximum contact pressure depending on the lateral position of the point of contact. In addition, it can be seen how non-uniformly the contact points on the rail are located when shifting the wheel by a constant value, since the data points are not uniformly spaced. Furthermore, it is evident that the polynomial metamodel may lose precision for certain contact point locations. This phenomenon can be explained by the fact that the response surface at these locations cannot be accurately approximated by a quadratic function. Another important observation is that the metamodel is able to predict responses for a cross section it was not calibrated for (cross section B), see Fig. 17.

Errors in quadratic metamodel

Error | \(a\), (%) | \(b\), (%) | \(p_{0}\), (%) |
---|---|---|---|

Minimum | 0.4 | 0.0 | 0.2 |

Average | 4.6 | 7.0 | 1.7 |

Maximum | 11.5 | 24.3 | 7.0 |

### 6.2 Hertzian-based metamodel

Errors in Hertzian-based metamodel

Error | \(a\) (%) | \(b\) (%) | \(p_{0}\) (%) |
---|---|---|---|

Minimum | 0.1 | 0.3 | 0.0 |

Average | 2.0 | 2.0 | 2.1 |

Maximum | 5.2 | 5.9 | 5.1 |

### 6.3 Coefficient of determination

\(R^{2}\) values for different models (closer to one is better)

Model | \(R^{2}_{a}\), (−) | \(R^{2}_{b}\), (−) | \(R^{2}_{p_{0}}\), (−) |
---|---|---|---|

Quadratic | 0.94 | 0.82 | 0.99 |

Hertzian-based | 0.99 | 0.99 | 0.99 |

The major benefit of using the metamodel can be seen when replacing the FE contact simulations in the simulation methodology briefly described in Sect. 1. Each FE simulation requires approximately 5 min on a computer with a 6-core “haswell” CPU and 128 GB of RAM. This leads to a significant computational effort when considering that thousands of wheel–rail contact scenarios need to be simulated in each iteration step.

## 7 Conclusions

Metamodels for predicting the size of the contact patch and the maximum contact pressure in simulation of wheel–rail contact in railway crossings have been presented. The first type of model is inspired by response surface methodology and is, therefore, represented by a polynomial. The second type is based on the Hertzian contact theory for two bodies in contact. The performance of both types of metamodels has been quantified and it was found that the Hertzian-based metamodel is more accurate.

The metamodel can take into account plastic deformations and, therefore, is an enhancement of the Hertzian solution. The error stemming from the assumption made in the theory of Hertz regarding the geometry of objects in contact was quantified by comparing with the results from an FE model with the true contact geometry. For the cases considered, the discrepancy ranged from 1 to 33% in maximum contact pressure and from 2 to 13% in maximum von Mises stress. The accuracy was found to be poor for the cases with a smaller difference in contact curvature. However, more importantly from the material degradation point of view is that the cases with a larger difference in contact curvature are captured with a higher precision, i.e. smaller errors were observed for cases with higher values of von Mises stress. This enables the use of the simplified geometry in the simulation methodology [5]. In future work, the presented metamodel will be incorporated in the methodology aiming at robust and efficient predictions of long-term material deterioration in S&C components. The metamodel will facilitate attaining the goal of the methodology since a considerable amount of computational effort is saved by replacing the FE contact simulations by a closed-form solution. In this context, a closed-form solution is a solution expressed in terms of input parameters and simple mathematical operations. Furthermore, the simplification of the geometry allows for the straightforward parameterisation needed for creating a metamodel. There is an obvious trade-off between the accuracy of the single contact simulation and the robustness and efficiency of the overall approach. A further investigation is needed to draw conclusions about the impact of the made assumption.

## Notes

### Acknowledgements

The performed work is part of the activities of the Centre of Excellence CHARMEC (CHAlmers Railway MEChanics). It is partly financed within the European Horizon 2020450 Joint Technology Initiative Shift2Rail through contract no. 730841. The authors would like to thank voestalpine VAE GmbH and the Swedish Transport Administration (Trafikverket) for their support. The simulations were performed on resources at Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC).

## Supplementary material

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