thermomechanical simulation of wear and hot bands in a...

EB2013-MS-007

THERMOMECHANICAL SIMULATION OF WEAR AND HOT BANDSIN A DISC BRAKE BY ADOPTING AN EULERIAN APPROACHRashid, Asim*, Strömberg, NiclasDepartment of Mechanical Engineering, Jönköping University, Sweden.

Keywords: Eulerian framework, frictional heat, hot band, wear history, pad wear, repeated braking

Abstract: In this paper frictional heating of a disc brake is simulated while taking wear into account. By performingthermomechanical finite element analysis, it is studied how the wear history will influence the development of hotbands. The frictional heat analysis is based on an Eulerian formulation of the disc, which requires significantlylower computational time as compared to a standard Lagrangian approach. A real disc-pad system to a heavytruck is considered, where complete three-dimensional geometries of the ventilated disc and pad are used in thesimulations. A sequential approach is adopted, where the contact forces are computed at each time step taking thewear and thermal deformations of the mating parts into account. After each brake cycle, the wear profile of the padis updated and used in subsequent analysis. The results show that when wear is considered, different distributionsof the temperature on disc are obtained for each new brake cycle. After a few braking cycles two hot bands appearon the disc surface instead of only one. These results are in agreement with experimental observations.

1. INTRODUCTION

Disc brakes are used to adjust the speed of a vehicle by pressing a set of pads against a rotating disc. It converts thekinetic energy of the moving vehicle mainly into heat. This heat causes the disc and the pad surface temperatureto rise in a short period of time. Due to relative sliding, both the pad and disc wear, which affects the behavior ofdisc-pad system over time. Since the pad material is softer as compared to the brake disc, the wear of the pad isdominant [1]. Higher temperature of the pad surface during braking also affects life of the pad negatively due toincreased wear rate.

Tribological contact in disc brakes has been studied both by physical experiments and numerical simulations. Bothtechniques have their own roles and importance to understand the disc-pad system fully. Lee and Barber [2] per-formed an experimental investigation of thermoelastic instability in disc brakes. They observed that temperatureinitially rises faster at the inner and outer radii of the pad after many repeated test runs. They attributed this behaviorto the concentrated wear at the center of the pad during previous runs. Eriksson et al. [3] provided a comprehensiveoverview of different processes, both at micro and macro scale, causing the contact surface variations. Panier etal. [4] performed an experimental investigation of railway disc brakes and proposed a classification of hot spotsobserved on a brake disc surface based on thermographs. They also studied the influence of pad stiffness and padcontact length on hot spots development. Österle et al. [5] performed a pin-on-disc test and showed that a thirdbody with a different structure and composition from the first bodies was trapped in a gap between the pin and thedisc. The pin was cut from a commercial brake pad and the disc material was cast iron. Hong et al. [6] comparedthe friction and wear characteristics of three friction materials with different binder resins. In their study, the wearrate below a critical temperature showed a slow increase, but above it the wear rate increased rapidly. Furthermorebelow the critical temperature binder played a minor role in the wear resistance of the friction material, but above itthe wear rate was strongly influenced by the thermal decomposition of the resin.

Contact pressure distribution is an important parameter for disc-pad systems. For the experimental study of the dis-tribution of contact pressure, pressure sensitive films [1] have been used. These methods can only be used for staticanalysis. Due to wear and other thermomechanical changes, contact pressure distribution does not stay constant dur-ing braking operation so numerical simulations become an obvious choice to determine the evolution of the contactpressure. Many researchers have used numerical simulations to enhance the understanding of the disc-pad system.Dufrénoy and Weichert [7] implemented a two-dimensional (2D) fully coupled thermomechanical algorithm takingwear into account. Kao et al. [8] developed a three-dimensional (3D) FE model capable of performing fully coupledthermomechanical analysis. They took the effect of wear on contact pressure distribution into consideration. Theyused this model to study hot judder in a disc brake. Koetniyom et al. [9] performed sequentially coupled thermo-mechanical finite element analysis of disc brakes under repeated braking conditions. They considered only a smallsegment of the disc taking the cyclic symmetry into account and assumed a uniform heat flux. In [10], Dufrénoyand Weichert developed an uncoupled 3D FE model. They simulated only one-twelfth of the disc by considering theaxial and rotational symmetries of the disc and used temperature dependent material data. Gao et al. [11] developeda fully coupled 3D thermomechanical FE model to investigate the fatigue fracture in disc brakes. They assumedthat thermal properties of the materials for disc and pad are invariant with temperature. Abubakar and Ouyang [1]performed wear simulation of a brake pad by using a commercial FEA software and compared their results with

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physical tests. They considered the real surface topography of the pad while building the finite element model bymeasuring height distributions with a gauge. Söderberg and Andersson [12] performed a simulation of wear andcontact pressure distributions of the brake pad using a general purpose finite element analysis software. Vernerssonand Lundén [13] studied the behavior of brakes numerically for repeated brake cycles. They used a 2D fully coupledFE model while considering the coefficient of friction as being constant and a temperature dependency of the wearrate. They found out that wear of the pad strongly depends on the stiffness of the friction material and its mounting.

Today, the prevalent way to simulate frictional heating of disc brakes in commercial softwares is to use the La-grangian approach in which the finite element mesh of a disc rotates relative to a brake pad. Although this approachworks well, it is not feasible due to extremely long computational times. Particularly, for simulating repeated brak-ing this approach is of little importance for practical use. Sometimes two-dimensional FE models are used to reducethe computational time but this approach is not sufficient to model complex behavior. The rotational symmetryof the disc makes it possible to model it using an Eulerian approach, in which the finite element mesh of the discdoes not rotate relative to the brake pad but the material flows through the mesh. This requires significantly lowercomputational time as compared to the Lagrangian approach. Nguyen et al. [14] developed an Eulerian algorithmfor sequentially coupled thermal mechanical analysis of a solid disc brake. First they performed a 3D contact cal-culation to determine the distribution of the pressure. Then a sequentially coupled analysis is implemented by firstperforming a transient heat transfer Eulerian analysis followed by a steady-state mechanical analysis. Recently,Strömberg [15] developed a finite element approach using an Eulerian framework for simulation of frictional heat-ing in sliding contacts. In his approach, the fully coupled problem is decoupled in one mechanical contact problemand a frictional heat problem. For each time step the thermoelastic contact problem is first solved for the temperaturefield from the previous time step. Then, the heat transfer problem is solved for the corresponding frictional power.In another paper [16] this approach was implemented for simulating frictional heating in disc-pad systems.

In this work, frictional heating of a disc brake, while taking the wear into account, is simulated by implementing anEulerian approach. A toolbox developed by Strömberg, which is based and described in his earlier work [16] butnow extended to include wear of the pad, is used to perform the frictional heat analysis. In this Eulerian approachthe contact pressure is not constant, but varies at each time step taking into account the wear and thermomechanicaldeformations of the disc and the pad. This updated contact pressure information is used to compute wear, and heatgeneration and its flow to the contacting bodies at each time step. In such manner, the wear and nodal temperaturesare updated accurately and their history is recorded at each time step. Then a Python script is used to write the wearand temperature history to an output file for subsequent use. The disc-pad system is simulated for several brakecycles. After each brake cycle pad geometry reflects the material removed by accumulated wear and this updatedgeometry of the pad is used in subsequent brake cycles. Because the finite element mesh of the disc does not rotaterelative to the pad, the contact region is always well defined and a node-to-node based approach can be adopted.This allows the mesh to be refined only in the region where the brake pad is in contact with the disc, which results inlower computational time. The output file with temperature history can be used e.g. in a sequentially coupled stressanalysis.

The results show the appearance of two hot bands on the disc surface after several brake cycles which cannot bepredicted when wear is ignored. The Eulerian approach has proved tremendously cheap in terms of computationaltime when compared to a fully coupled Lagrangian approach. This is demonstrated by presenting numerical results.

2. FRICTIONAL HEAT ANALYSIS

The workflow of the approach used for frictional heat analysis is shown in Fig. 1. An input file, which contains themeshed geometry with appropriate boundary conditions and loads is required for the frictional heat analysis. Duringthis analysis linear thermo-elasticity is adopted and the problem is decoupled in two parts. In the first part, for a

Input file

In-house software

ODB file

Figure 1: Workflow of sequential approach.

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Specify initial temperatures of

disc and pad.

Thermoelastic contact problem

is solved while taking the wear

into account and contact pressure

distribution is determined.

Wear gaps are updated.

Heat transfer problem is solved

and new nodal temperatures are

determined.

Figure 2: Sequential approach used during frictional heat analysis to determine temperature history.

Table 1: Material properties for frictional heat analysis.

Disc Pad PlateThermal conductivity [W/mK] 47 0.5 46Young’s modulus [GPa] 92.9 2.2 210Poisson’s ratio [-] 0.26 0.25 0.3Thermal expansion coefficient [10−5/K] 1.55 1 1.15Density [kg/m3] 7200 1550 7800Heat capacity [J/kgK] 507 1200 460

given temperature distribution the contact problem is solved while taking the wear of the pad into account to obtainthe nodal displacements and contact pressure distribution. The new contact pressure distribution is used to updatethe wear gaps. In the second part, for the obtained contact pressure distribution the energy balance is solved and newnodal temperatures are determined. These equation systems are then solved sequentially and, wear and temperaturehistories are developed. The nodal temperatures determined at a time step are taken into account in the next timestep to update the deformed geometry of the disc and pad. This is shown schematically in Fig. 2. The wear andnodal temperature history is then written in an output file (called ODB file) by using a Python script. Details aboutthe governing equations can be found in [16].

Three parts are considered for the frictional heat analysis. Materials assumed for the disc and the back plate are castiron and steel, respectively. Friction material used as brake pad is a composite. Temperature independent materialproperties used for these parts are listed in Table 1.

3. NUMERICAL RESULTS

The assembly of the disc-pad system considered in this paper is shown in Fig. 3. This is an assembly of a disc-pad system of a heavy Volvo truck. The outer diameter and thickness of the disc are 434 [mm] and 45 [mm],respectively. The ventilated disc is geometrically symmetric about a plane normal to the z-axis. It is assumedthat thermomechanical loads applied to the system are symmetric so only half of this assembly is considered forthe simulation and symmetry constraints are applied on the nodes lying on the symmetry plane. Some detailedgeometry at the inner radius has been removed to simplify the model as that is not important for this analysis. Thedisplacements along x and y directions of the nodes located at the inner radius of the disc are set to zero. All thesurfaces of the disc, except the one lying on the the symmetry plane are considered to lose heat by convection.

The brake pad is supported by a steel plate at the back side as shown in Fig. 4. Some detailed geometry of the backplate which is not necessary for the simulation has been removed. Two cylindrical pins apply a normal force on theback surface of the back plate which transmits it to the pad. Displacements at the back surface of the back plate,other than along the force direction, are fixed. Furthermore temperature is set to zero on the back surface.

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The disc is meshed such that it has smaller elements where it contacts the pad as shown in Fig. 5 (only a smallportion of the disc is shown). This is an advantage of the Eulerian approach because the finite element mesh of thedisc does not rotate relative to the brake pad but the material flows through the mesh. The heat flux generated at theinterface of the stationary pad and the disc is considered with convective heat transfer in the disc. In a Lagrangianapproach a fine mesh should be applied on the complete surface of the disc because the finite element mesh of thedisc rotates relative to the brake pad or some adaptive strategy should have to be applied. All the parts consideredfor the simulation are meshed with 4-node linear tetrahedron elements in HyperMesh (HyperWorks 10.0). Thesemeshed parts are then used to prepare input file with boundary conditions and loads in Abaqus/CAE. The discassembly is meshed with 269438 elements that has 64957 nodes and 185697 degrees of freedom.

Now the results of frictional heat simulations will be described for two different cases. In the first case, a brakeapplication is simulated for one cycle and wear is not considered. Figure 6 shows the surface temperature as afunction of time and disc radius for this case. The nodes of the disc chosen for this plot are located at 180◦ awayfrom the middle of the pad. A brake force of 24.5 [kN] is applied for 45 [s] on the back surface of back plate. Theangular velocity of the disc is 45 [rad/s] and held constant throughout the simulation. This loadcase corresponds to atruck moving downhill with a constant speed. The force is ramped up by using a log-sigmoid function during 20 timeincrements and then held constant for next 70 increments with time step = 0.5 [s]. The friction coefficient is µ = 0.3,contact conductance coefficient is ϕ = 0.1 [W/NK] and convection coefficient is set to 50 [W/m2K]. The brakeforce generates an average brake moment of 1240 [Nm] after the ramping up. The total CPU time is 4272 [s] on aworkstation with Intel Xeon X5672 3.20 GHz processor. In the graph it can be seen that temperature is not uniformlydistributed over the disc instead a narrow band with relatively higher temperature appears in approximately middleof the disc surface.

In the second case, brake application is simulated for several cycles and material removed due to wear in each cycleis considered in subsequent braking operations. During each brake cycle, the wear coefficient is set to 10−10 [m2/N]and rest of the parameters are same as for the first case. The total CPU time for a single cycle is 4289 [s] on aworkstation with Intel Xeon X5672 3.20 GHz processor. Each brake cycle requires almost the same CPU time foreach simulation. In Fig. 7, temperature of the disc surface is shown at the end of brake operation for first cycle.A ring of high temperatures, called a hot band, is evident in the middle of the disc. Figure 8a shows the surfacetemperature as a function of time and disc radius for the first cycle of brake application. The nodes of the discchosen for this plot are located at 180◦ away from the middle of the pad. In the graph it can be seen that during thecycle there is only one hot band on the disc surface.

By intuition it can be thought that the high temperature ring should form near the outer radius of the disc. Butthe ring appeared approximately in the middle of the disc surface. It might be understood by studying the contactpressure plots at different time steps as shown in Fig. 10. In Fig. 10a the contact pressure plot for the first timeincrement or at the moment when the pad comes into contact with the disc is shown. It can be seen that the contactpressure is not the highest at the outer radius of the pad. The region where contact pressure is higher generates moreheat and causes further expansion of the disc and the pad material near this area which in turn causes higher contactpressure. In the meantime convex bending caused by thermal deformation of the pad and the back plate, as shownin Fig. 9, also plays a major role in concentration of contact pressure towards the middle of the pad surface. This

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Figure 3: An assembly of the disc-pad system, also showing the cylindrical pins used to push the back plate.

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Back Plate Brake Pad

Figure 4: Brake pad with back plate.

convex bending can be explained by the expansion of the pad surface material due to the increase in temperatures.The frictional heat causes the pad surface temperature to rise in a short period of time as compared to the innerregion of the pad and the back plate as shown in Fig. 9. Consequently, the surface expands more than the innerregion of the pad and the back plate which results in the convex bending. These phenomena combined with theramping up of brake force in later increments, causes the higher contact pressure in an area which is away from theouter radius of disc as shown in Fig. 10d.

In Fig. 11 contact pressure plots are shown for further time steps when the brake force is held constant for the firstcycle and Fig. 12 shows the wear on the pad for corresponding time steps. It can be seen that the contact pressurekeeps on concentrating towards the middle of the pad with increasing time increments. It can also be observed thatwear is higher in the areas where contact pressure is higher.

Figure 8b shows the surface temperature as a function of time and disc radius for the 41st cycle of brake application.It can be seen that in the beginning there are two hot bands which converge to one as the temperature increases withtime. In Figure 13 which shows temperature of the disc surface at 13th time increment for the 41st cycle of brakeapplication, two hot bands can be seen. In Fig. 14, temperature of the disc surface is shown at the end of brakeoperation for the 41st cycle of brake application. By comparing with Fig. 7, it can be concluded that after 41 brakecycles the maximum temperature has decreased and the hot band becomes wider at the end of brake operation. Theappearance of two bands can be explained by the shifting of high contact pressure areas. Due to the concentratedwear in the middle of the pad during repeated brakings, a depression appears when the pad cools down and returnsto its undeformed state at the end of a brake operation. So during next brake cycle, the high contact pressure firstbuilds on the outer regions of the pad surface. In Fig. 15 accumulated wear of the pad is shown at the end of the40th brake cycle. Fig. 16 shows the distribution of contact pressure during the 41st cycle. It can be seen that contactpressure first builds on the outer regions which are less worn out and then due to thermomechanical deformations ofthe pad, as discussed before, moves to the middle of the pad surface with increasing time increments. By comparingthe results of the first case with those obtained for the first brake cycle of the second case, it can be concluded thatfor a pad without wear history there is no noticeable influence during braking due to wear. But accumulated weardoes have a significant influence on the distribution of temperature after some brake cycles.

4. DISCUSSION

The temperatures predicted by the in-house software have been compared with the temperatures recorded by athermal imaging camera during a physical test and found to be relatively higher. Moreover, two hot bands predictedafter repeated brake cycles are not as distinct as observed in the thermographs. These differences could be dueto temperature independent material data, friction coefficient, and wear coefficient used during the frictional heatanalysis. For more realistic results, temperature dependent material data should be used. Furthermore, the friction

Step: Step−1Increment 44: Step Time = 22.00

ODB: Job−9.odb Abaqus/Standard 6.9−EF1 Thu Apr 28 09:38:30 W. Europe Daylight Time 2011

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Figure 5: Mesh of the disc.

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Figure 7: After the brake application at the first cycle, a ring of high temperature develops on the disc surface.

coefficient of a brake pad is generally dependent on temperature, velocity and contact pressure [17] but in this workit is assumed to be constant at µ = 0.3 to represent an average behavior. Similarly, the wear coefficient is generallydependent on temperature and velocity [6, 18] but in this work it is assumed to be constant at 10−10 [m2/N]. Ina very near future, we will extend this work such that a temperature dependent behavior of the friction and wearcoefficients is included in the proposed method. At present the in-house software assumes constant angular velocityof the disc that corresponds to a vehicle moving downhill with a constant speed but in the future it could also beextended to non-constant angular velocities.

5. CONCLUDING REMARKS

In this work frictional heat analysis of a disc brake has been performed taking into account wear of a pad. Thisanalysis is performed in an in-house software based on the Eulerian approach. It has been shown that brakinghistory affects the evolution of temperature distribution during a brake cycle. The analysis predicts concentratedwear in the middle of the pad which results in the appearance of two hot bands after repeated brake cycles.

It has been shown that other than the local factors e.g. thermal expansion, convex bending of the pad and the backplate also plays a major role in the contact surface evolution. Phenomenon of convex bending has been describedin other works [2, 3], to the best of our knowledge, but no experimental observation or numerical simulation resultshave been presented to support it. In this paper it has been shown with numerical simulations that convex bendingplays a major role in the concentration of contact pressure to the middle of pad.

This method has proved tremendously cheap in terms of computational time when compared to the Lagrangianapproach. In the future this approach can be used to study the influence of different geometries of the pad and thedisc on the maximum temperature with a reasonable simulation time. It can be very useful when studying newdesigns for real disc brake systems.

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Figure 8: Temperature as a function of time and disc radius with the consideration of wear.

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0 56112167223279335390446502558614669

Figure 9: Thermally induced deformations of the pad and back plate during brake operation shown in differentprojections. The deformation is exaggerated for visual clarity.

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0.0000.0050.0100.0140.0190.0240.0290.0330.0380.0430.0480.0520.057


ODB: Brake3P_PN.odb Abaqus/Standard 6.9−EF1 Wed Jan 25 15:03:56 W. Europe Standard Time 2012

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0.000 8.316 16.633 24.949 33.265 41.582 49.898 58.215 66.531 74.847 83.164 91.48099.797



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Figure 10: Nodal contact forces represented as pressure plots on the pad surface shown at different time steps forthe first cycle during ramping up of the brake force. The legend is given in [N].

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Figure 11: Nodal contact forces represented as pressure plots on the pad surface shown at different time steps forthe first cycle while the force is held constant. The legend is given in [N].

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Figure 12: Wear on the pad surface, shown in [m], at different time steps for the first cycle.

Viewport: 1 ODB: E:/DiscBrakes/Results/No_..._Simulation_1/Brake3P.odb

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Figure 13: Two bands of high temperatures on the disc surface at t = 6.5 [s] during the 41st cycle of brake application.

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Viewport: 1 ODB: E:/DiscBrakes/Results/No_..._Simulation_1/Brake3P.odb

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Figure 14: After the brake application at the 41st cycle, a ring of high temperatures develops on the disc surface.

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ODB: Brake3P_WEAR.odb Abaqus/Standard 6.9−EF1 Wed Feb 01 14:19:08 W. Europe Standard Time 2012

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Figure 15: Accumulated wear on the pad surface, shown in [m], at the end of the 40th cycle.

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0.000 1.813 3.627 5.440 7.253 9.06610.88012.69314.50616.32018.13319.94621.759


ODB: Brake3P_PN.odb Abaqus/Standard 6.9−EF1 Wed Feb 01 14:15:36 W. Europe Standard Time 2012

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0.000 12.129 24.258 36.386 48.515 60.644 72.773 84.902 97.030109.159121.288133.417145.546



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Figure 16: Nodal contact forces represented as pressure plots on the pad surface shown at different time steps forthe 41st cycle. The legend is given in [N].

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6. ACKNOWLEDGEMENT

This project was financed by Vinnova (FFI-Strategic Vehicle Research and Innovation) and Volvo 3P.

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[4] Panier, S., Dufrénoy, P., and Weichert, D. “An experimental investigation of hot spots in railway disc brakes”.Wear, 256(7-8):764–773, 2004.

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