In this thesis thermal stresses in a brake disc during a braking operation are simulated. The simulations are performed by using a sequential approach where the temperature history generated during a frictional heat analysis is used as an input for the stress analysis. The frictional heat analysis is based on the Eulerian method, which requires signicantly lower computational time as compared to the Lagrangian approach. The stress analysis is performed using a temperature dependent material model both with isotropic and kinematic hardening behaviors. The results predict the presence of residual tensile stresses in circumferential direction for both hardening behaviors. These residual stresses may cause initiation of radial cracks on the disc surface after a few braking cycles. For repeated braking an approximately stable stress-strain loop is obtained already after the 1st cycle for the linear kinematic hardening model. So, if the fatigue life data for the disc material is known, its fatigue life can be assessed. These results are in agreement with experimental observations available in the literature. The simulation results predict one hot band in the middle of the disc for a pad with no wear history. It is also shown that convex bending of the pad is the major cause of the contact pressure concentration in middle of the pad which results in the appearance of a hot band on the disc surface. The results also show that due to wear of the pad, different distributions of temperature on the disc surface are obtained for each new brake cycle and after a few braking cycles, two hot bands appear on the disc surface. This sequential approach has proved tremendously cheap in terms of computational time so it gives the freedom to perform multi-objective optimization studies. Preliminary results of such a study are also presented where the mass of the back plate, the brake energy and the maximum temperature generated on the disc surface during hard braking are optimized. The results indicate that a brake pad with lowest possible stiffness will result in an optimized solution with regards to all three objectives. Another interesting result is the trend of decrease in maximum temperature with an increase in back plate thickness. Finally an overview of disc brakes and related phenomena is presented as a literature review.

1. Simulation of Thermal Stresses in a Brake Disc
Asim Rashid
Department of Mechanical Engineering
Jönköping University, Sweden
LICENTIATE SEMINAR
Asim.Rashid@ju.se
August 29, 2013

2. Overview
 Background
 Workflow
 Boundary Conditions
 Results
 Comparison

3. Background
Disc Brake
Used to decelerate or stop a vehicle
Converts kinetic energy to heat
Hot bands appear on disc surface
(Patented design; Hulten and Dagh, 2006),

4. Background
Buckling
Coning
Fade
Vibrations and noise
Cracks on disc surface

5. Numerical Approaches
1. Axisymmetric
2D geometry
2. Rotational symmetric
A sector of disc
3. Fully coupled in Lagrangian framework
Complete 3D geometry
4. Sequentially coupled in Eulerian framework
Complete 3D geometry

6. Background

7. Sequential Approach
In-house Software Abaqus
Mesh of Disc and Pad
Temperature History
1. Frictional heat analysis
• In-house software developed by Niclas Strömberg [Strömberg, 2011]
• Eulerian framework
• Linear thermo-elasticity
2. Stress Analysis
• Abaqus
• von Mises plasticity model

8. Workflow-Frictional Heat Analysis
1. Input file with boundary conditions and loads (Abaqus)
2. Analysis in In-house software
3. ODB file with temperature history
In-house Software
ODB File
Input File

9. Workflow-Stress Analysis
1. ODB file with temperature history
2. Stress analysis in Abaqus
Frictional heat analysis
Stress analysis
In-house Software
ODB File
Input File
Abaqus
ODB file

10. In-house Software
Specify initial temperatures of
disc and pad.
Thermoelastic contact problem is solved and
contact pressure distribution is determined.
Heat transfer problem is solved and new
nodal temperatures are determined.
t=t+∆t
t=0

11. Brake force
Log-sigmoid function to ramp up the force on pad

12. Preparing models
Geometry of the disc is symmetric
Loads on disc assumed symmetric
Three simplified components considered during simulation

13. Meshing : Eulerian approach
1. Fine mesh at the contact
2. Node to node contact between disc and pad
3. 4-node linear tetrahedron elements

14. Boundary Conditions: Disc
•Translations of nodes lying on Inner surface are fixed in X and Y directions
•Translations of nodes lying on Symmetry surface are fixed in Z direction
•All surfaces except the symmetry surface are considered to lose heat by
convection

15. Boundary Conditions: Pad Assembly
•Normal force is applied on back surface of support plate
•Displacements of the support plate other than along the force direction are fixed
•Temperature is set to zero on back surface

16. Frictional Heat Analysis In-house Software
Brake Time= 20s
Brake Force= 24.5 KN
Velocity= 45 rad/s (constant)
Run Time= 2,08 hours
Number of elements = 270,194
Intel Xeon X5570 @ 2.93 GHz

17. Material Properties
Figure: True stress as a function of true plastic strain for different temperatures
20oC 200oC 400oC
600oC 800oC 1120oC

18. Frictional Heating
Nodal temperatures on the disc surface

19. Stress Analysis
Circumferential stresses on the disc surface

20. Stress Analysis
Circumferential plastic strains on the disc surface

21. Stress Analysis
Radial stress on the disc surface

22. Stresses cause
∆T1
∆T2
∆T3
∆T4
x
y
𝜕𝑇
𝜕𝑦

23. Stress Analysis-Repeated braking
Temperature history: Repeated braking

24. Material Model
Isotropic hardening Kinematic hardening

25. Material Model: Linear Hardening

26. Material Model
Implemented
• Linear kinematic hardening
• Same behavior in tension and compression
• von Mises yield criterion
Experimental observations
• Kinematic hardening gives better agreement [Josefson B.L. 1995]
• Different behavior in tension and compression [Koetniyom S.2000]
• von Mises yield criterion in tension and compression[Koetniyom S.2002]

27. Stress Analysis
Circumferential stresses vs. strain with linear kinematic hardening

28. Stress Analysis
Circumferential stresses after braking during first cycle

29. Stress Analysis Abaqus
Effective Plastic Strain at the end of third bake cycle with linear kinematic hardening model
•von Mises Plasticity
•Temperature dependent material data
•Run Time= 1.8 hours

30. Comparison
Run Time (hours) Brake Time(s)
Sequential Approach 3.8 45
Fully Coupled Approach 480 5

31. Frictional Heat Analysis-Wear
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.
t=t+∆t
t=0

32. Frictional Heat Analysis
Brake Time= 45s
Brake Force= 24.5 KN
Brake Moment= 1240 Nm
Velocity= 45 rad/s (constant)
Run Time= 1.18 hours
Number of elements = 269,438
Intel Xeon X5672 @ 3.2 GHz
t = 45 s

33. Frictional Heat Analysis
Two bands of high temperatures during brake application at the 41st cycle
t = 6.5 s

34. Temperature as a function of time and disc radius

35. Temperature as a function of time and disc radius
Thermography camera Simulation: Cycle 41
Temperature independent
 Wear coefficient
 Coefficient of friction
 Thermal expansion coefficient

36. Convex bending

37. Contact pressure
First brake cycle-Ramping of force

38. Contact pressure
First brake cycle-Constant force

39. Wear on pad surface
First brake cycle

40. Wear on pad surface
Accumulated wear on the pad surface at the end of the 40th cycle

41. Contact pressure
41st brake cycle

42. Expansion and Wear: simplified model of pad
Before brake
During brake: middle column expands
more due to higher temperature
After cooling: middle column wears
more due to more elongation

43. Multi-objective optimization

44. Multi-objective optimization

45. References
Hulten, J. and Dagh, I. (2006) Brake Disc for a Vehicle Disc Brake, 29 August, US Patent
7,097,010.
Strömberg, N. (2011) An Eulerian approach for simulating frictional heating in discpad
systems. European Journal of Mechanics - A/Solids, 30(5):673-683, 2011.

46. Thanks for your attention!
Questions?