© J. Frangieh et al., Published by EDP Sciences, 2023

1 Introduction

In most modern railway trains, different types of braking systems are used to either reduce the moving velocity of the train or even to stop it. Among the different types, frictional brakes including disc brakes and tread brakes are commonly used. During the braking event, a static brake component (shoe and lining) is pushed into contact with a rotating component (wheel and disc), transforming the kinetic energy into mechanical thermal energy [1]. This phenomenon results in wear, including abrasion of brake lining and disc releasing particulate matter (PM) of various sizes directly to the atmosphere increasing the concentration of toxic particles in airborne PM particularly less than 10 m PM10 [2]. Some of the debris and particles are being kept in the contact interface forming the third-body layer, while others are ejected out of the tribological circuit [3–5], notably in the form of airborne particles. Thus, wear depends on the ability of the tribological system to maintain its friction performance without sacrificing from its volume [6,7]. The brake wear particles and debris rate depend not only on the driving behavior and severity of brake [8], but also on the material used in the braking system [9]. During the braking phenomenon, the brake disc may reach a temperature as high as 800 C on the area of friction, thus the material used should have good thermal stability and the ability to bear thermal fatigue [10].

Due to its complexity, the developments of brakes are still carried out in a very empirical way. Therefore, it requires a lot of experiments and tests. In general, these tests are carried out at the real scale of the braking system which results in an expensive and time consuming testing. Using smaller scale experiments seems relevant. It is more convenient to perform tribological studies on a reduced-scale braking system rather than testing the original scale system. A smaller scale testing will be representative of the real situation if it could expose the same problem happening at the original scale. However, the performances of both scales are not always equivalent.

The wear mechanisms are complex, multi-physical and multiscale giving a high importance to the thermomechanical phenomena affecting thus the ability to maintain the third body in the interface as well as its physico-chemical evolution [11]. In their article, Cristol-Bulthé, Desplanques and Degallaix showed the effect of thermomechanical behavior on the tribological circuit at the interface. Therefore, reaching equality in surface temperature is not always sufficient to reproduce the brake at a different scale. In fact, from a thermomechanical point of view, the elevation of temperature in the brake components due to the diffusion of the heat generated in the friction contact will lead to a thermal expansion limiting thus the surface of contact between the pad and the disc. Localising contact pressure on certain zones of the disc will result in an increased temperature in these zones and formes the so-called hot-bands and hot-spots. Higher temperature and contact pressure at the localization will induce higher wear in the area.

2 Thermal approach to disc brake scale reduction

In order to build a representative thermal model, the strategy of scale change with thermal similarity is tested using analytical and numerical point of view. The thermal problem and the heat equation in disc brakes was solved by Newcomb and was used in the approach of thermal scale change [12,13].

2.1 Scale reduction

Thermal similarity will occur when the reduced-scale and the original scale encounters the same thermal surface loading during braking, the surface temperature of the brake components in contact with each other is the same. As the heat loss by convection and radiation in such situations is minimal in front of the heat flux generated by friction at the interface, it could be neglected and assumed that all the mechanical energy is transformed into heat flux in the contact ***$\left(\varphi (t\right)=P{S}_s{V}_0\left(1-\frac{t}{t_f}\right))$. The thermal scale change approach was developped by Roussette[7,13]. In his approach, Roussette has introduced the ‘Energy scale factor’ denoted as ‘k’ that is characterized as the proportion of energy dissipated when braking at the reduced scale compared to the energy dissipated when braking at the full scale.

$$\text{Energy\hspace{0.17em}scale\hspace{0.17em}factor\hspace{0.17em}}k=\frac{Q_1}{Q_r}=\frac{{\left(P{S}_d{V}_0{t}_F\right)}_1}{{\left(P{S}_d{V}_0{t}_F\right)}_r}$$(1)

Where Q₁ and Q_r are the dissipated energy at scale 1 and reduced scale respectively, μ is the friction coefficient, P is the contact pressure, V₀ is the initial sliding velocity and tp as the braking duration. To replicate the thermal loading experienced in the disc pad contact at full scale, an effective approach involves maintaining the same similitude factor k, for both the rubbing surfaces of the disc (S_d1 and S_dr) and the friction surfaces of the pad (S_p1 and S_pr), both at reduced and full scale [14].

$$k=\frac{Q_1}{Q_r}=\frac{S_{d1}}{S_{dr}}=\frac{S_{p1}}{S_{pr}}$$(2)

The ‘Characteristic ratio of rubbed surfaces’ denoted as ‘n’ is defined as the ratio of the contact surfaces at scale 1 over the ratio of contact surfaces at reduced scale.

$$\text{Characteristic\hspace{0.17em}ratio\hspace{0.17em}of\hspace{0.17em}rubbed surfaces }n=\frac{S_{d1}/S_{p1}}{S_{dr}/S_{pr}}.$$(3)

During the design of the tribometer braking components, it is aimed to target a characteristic ratio of rubbed surfaces equal to 1. The energy scale change strategy makes it possible to obtain identical surface temperature in both scales Q₁ = kQ_r [7,13,14]. However, achieving simultaneous compliance with these three rules is highly challenging, making it difficult to accurately replicate the thermal conditions experienced at full scale [14]. To overcome this drawback, a scale conversion rule has been developed based on Newcomb’s model. This rule provides the average temperature variation of the friction surface ∆θ(t) during a stop braking event leading to the following equation:

$${\left(P\cdot V_0\cdot \sqrt{t_f}\right)}_r=\gamma {\left(P\cdot V_0\cdot \sqrt{t_f}\right)}_1$$(4)

where ***$\gamma =\frac{1+A}{n+A}$ the characteristic constant of the scale change and ***$A=\frac{S_{pr}{\xi }_p}{S_{dr}{\xi }_d}$ and ξ the effusivity of the material.

Equation (4) ensures that for any given triplet P, V₀, and t_f at the original scale, the same surface temperature variations at a reduced scale could be obtained [7,13].

2.2 Comparison between both scales

In this part, an analytical and a numerical comparison between the initial scale and a reduced scale were carried out. Reduced scale braking parameters were generated from the original scale brake parameters using the scale change approach.

Fig. 1 Schematic representation of both disc.

2.2.1 Discs geometries

An unventilated solid disc is used for the original scale having an external diameter of 640 mm and a thickness of 45 mm. The mean radius of friction between the disc and the pads is 245 mm. For scale 1 disc brake, both faces of the disc are used to perform the braking application. Whereas, the reduced scale disc used on the tribometer is single sided having an external diameter of 217 mm, a thickness of 22.5 mm and a mean friction radius of 100 mm.

2.2.2 Materials properties

Both discs are produced from steel (28CrMoV5). The mechanical and thermal material properties used are extracted from the thesis of Mann [15]. The model is represented as an elasto-plastic material of type “kinematic hardening”.

2.2.3 Braking parameters and initial conditions

A normal high speed train brake is used for all calculations and cases in this paper. A stop brake is simulated where the train initial speed is 200 km/h. The weight carried by each disc brake is 2 tons. A clamping force of 50 000 N is used to stop the train in a duration of 10 s. Tables 1 and 2 show the different parameters of the studied brake at both scales.

Using equation (4), the parameters were transposed to the reduced scale system. In this case, the characteristic constant of scale change is equal to 1. The transposition was made by fixing the braking duration and sliding speed at the mean contact radius, thus, the pressure was chosen accordingly.

Fig. 2 Analytically calculated mean surface temperature.

2.2.4 Analytical results

The average surface temperature of both discs were computed analytically using the solution of Newcomb for disc brake applications [12] and were represented in Figure 2. It can be observed in Figure 2 that the brake realised on the tribometer has reached the same level of temperature as the real train disc brake which validate the scale change using the thermal approach from an analytical point of view.

Fig. 3 Numerical models and boundary conditions.

Fig. 4 Numercial models and boundary conditions.

3 Thermomechanical approach to disc brake scale reduction

In the previous section, it was observed that the geometry can affect the thermal behaviour of the material by affecting the way it diffuses heat into its body and to the surroundings. In this section, the mechanical behaviour due to the increase in body temperature will be discussed.

3.1 Thermomechanical problem in disc brakes

Due to thermal expansion of the disc, the contact will be lost in some zones while localization of contact pressure will occur in some other zones. In the zone where the contact is localised, the surface will encounter an increase in temperature leading to an increase in surface wear at that spot. At first, the localization will happen at the outer radius of the friction surface as the sliding velocity is slightly greater. Then the material will expand leading to a possible migration of the localizations to other areas.

As mentioned earlier, contact dynamics and the thermomechanical phenomena have high influence on wear. In order to reproduce the braking phenomena at the tribometer scale, the same kinetics of localization should be reproduced. This approach requires a complete understanding of the thermomechanical processes occurring during braking, with respect to the contact loading, the thermal localizations, the kinematics of contact opening and closing, and the couplings with the tribological circuit and the wear processes.

The aim of this work is to reproduce the thermomechanical behaviour of the train disc on the reduced scale disc. Finding additional correlation in the mechanical behaviour will lead to a feasible approach of scale change from the thermomechanical perspective.

A new disc geometry for the tribometer was introduced. While designing this disc, slopes of the friction surface were one of the most important criteria that were took into consideration. A mechanical numerical analysis was realised on the brakes discussed is Section 2.2. The deformations of the surfaces due to thermal expansion were analysed and compared.

Fig. 5 Numerical models and boundary conditions.

Fig. 6 Mean surface temperature from FE models.

3.1.1 Original scale brake

As shown in Figure 3, the train disc is considered to be clamped on its hub and normal forces are applied to the friction surface representing the clamping force exerted by the calliper. Figure 3b illustrates the temperature profile in the high speed train disc after 5.5s (maximum temperature is reached).

From Figure 3b, it can be observed that the heat is diffused in the body of the disc in a symmetrical way with respect to its mean radius of friction. This symmetrical diffusion of heat will lead to a uniform surface distortion due to expansion. The hub design and the flexibility in the pad system will compensate for the axial rotation of the surface meaning that localization would occur in the middle of the contact area. This is not the case on the reduced scale due to the high rigidity of the system. Thus, the reduced scale disc is supposed to have similar thermomechanical behavior (favored radial deformation)

3.1.2 Design 1

Regarding the reduced scale disc, it is considered to be bolted (bonded area ) with the spindle. The contact between the spindle and the disc are considered friction-less. For this model also, maximum temperature was reached after 11.5s and figures below illustrate the temperature profile and the numerical model’s boundary conditions.

Figure 4b reveals the temperature gradient in the reduced scale disc. In contrary to the original scale disc, the heat is diffused to the inner radius of the disc resulting in a nonuniform, umbrella shaped surface distortion. Axial expansion was favored on the radial one, thus, leading to the rotation of the surface to the exterior. At the beginning of the brake, the hot-band will be formed at the outer radius of the friction surface due to the higher sliding velocity then will migrate to the inner radius of the friction surface. This thermomechanical phenomena is not representative of the original scale and will lead to different tribological mechanisms.

3.1.3 Design 2

In the aim of having a better representation of the scale one behavior on a smaller scale and having a better control of the expansion, a new disc design was developed. During the design of the new disc, it was intended to favor the radial deformation on the axial deformation. In this way, when the disc reaches high temperatures, it will tend to expand radially conserving the contact at the middle of the friction surface. By comparing surfaces slopes of different designs going through the same thermal solicitation, the following design gave the best results overall. The disc has a mean friction radius of 87 mm. Figure 7b shows the numerical model generated from the design 2 of the disc.

The prestress occurring between the disc and its support allows a better control of the deformation. This prestress could be controlled by controlling the imposed displacement of the bolt connection (Fig. 7). The same braking situation was used in the simulation, a thermal analysis was performed and the mean surface temperature was compared with the train scale disc and design 1. Figure 5 shows the symmetry of the temperature gradient with respect to the mean radius of friction.

The same level of temperature was approximately reached using this design, however, the time needed to reach maximum temperature is different as the geometry plays a big role in the dissipation ability of the heat in its body. Under the surface of friction, the thickness of the disc is 22mm helping in the fast diffusion of the heat. This thickness could be changed to adapt to different braking conditions.

Figure 7 is a representation of the deformation due to thermal expansion. Dashed lines represent the deformed shape of the disc and the location of the hot band at the end of the braking phenomena. It can be observed that the disc at scale 1 and the design 2 have similar thermal localisation. However, in order to highlight the difference in the thermomechanical behavior between all designs, the slopes induced by the deformation of the contact surface were evaluated. First, the displacement in both radial and axial (out of plane) were extracted from the FE model. Then the slopes were calculated by setting three points on the surface (A: inner radius of contact, B: mean radius of contact, C: outer radius of contact). In addition, the maximum difference in the out of plane displacement was evaluated.

As mentioned before, the rotation of the disc at scale 1:1 could be neglected due to the flexibility in the assembly of the pads and their ability to be follow the orientation of the contact surfaces of the disc. By comparing results obtained in Table 3, design 2 shows comparatively smaller slope angles than design 1. Rotation is not an option that the pad on the tribometer could perform, thus, the slopes induced by thermal expansion has a large influence on the contact between the pad and the disc. Design 2 shows slope angles more similar to the one calculated on the original scale. In addition, by comparing the maximum difference in axial position between both reduced scale designs, it can be seen that the ∆_y of design 2 is much smaller than the design 1 thus resulting in a better performing disc. The tribological mechanisms would be more realistic and consequently wear and particle emissions.

Fig. 7 Thermomechanical localisations.

Fig. 8 Slopes calculation schema.

4 Conclusion

In conclusion, the thermal approach to scale reduction in disc brakes has been introduced. Analytical and numerical thermal simulations validate this approach by demonstrating that the equality in surface temperature level for both scales was achieved.

Reaching thermal similarity based on the thermal scale change strategy is not always representative. Heat diffusion in the disc will affect its thermomechanical deformation. Having similar thermomechanical mechanisms will lead to a better representation of wear at reduced scale tribometer.

A new design for the disc of the tribometer was introduced with an intention of favoring the radial deformation. It showed thermomechanical behavior closer to the full scale configuration. It means that the tribological mechanisms (particles trapping, compaction, etc.) would be more realistic and consequently wear and particle emissions.

For a better understanding of the real situation, it is necessary to understand the thermomechanical processes happening during the braking event. Future works will be focused on an in-depth look at the wear mechanisms experimentally under different braking conditions. Another work is to develop a coupled thermomechanical simulation (representation of both disc and pad) revealing contact localization and temperature levels at the hot zones. This will insure a better understanding of the thermomechanical phenomena at the interface.

Conflict of interest

The authors declare that they have no known competing interests and have full control of all primary data and agree to allow the journal to review their data if requested.

Funding

This project is financed by the French government as part of the Future Investments Program, now integrated into France 2030, and operated by ADEME.

Acknowledgments

The authors gratefully acknowledge the Nord-Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, The ministry of Higher Education and Research, and the National Center for Scientific Research for their continuous support.

References

All Tables

All Figures

	Fig. 1 Schematic representation of both disc.
In the text

	Fig. 2 Analytically calculated mean surface temperature.
In the text

	Fig. 3 Numerical models and boundary conditions.
In the text

	Fig. 4 Numercial models and boundary conditions.
In the text

	Fig. 5 Numerical models and boundary conditions.
In the text

	Fig. 6 Mean surface temperature from FE models.
In the text

	Fig. 7 Thermomechanical localisations.
In the text

	Fig. 8 Slopes calculation schema.
In the text