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Finite element modeling of thermomechanical problems under the vehicle braking process

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The braking phenomenon is an aspect of vehicle stopping performance where in kinetic energy due to speed of the vehicle is transformed into thermal energy produced by the brake disc and its pads. The heat must then be dissipated into the surrounding structure and into air flow around the brake system. The thermal friction field during the braking phase between the disc and the brake pads can lead to excessive temperatures. In our work, we presented numerical modeling using ANSYS software adapted in the finite element method, to follow the evolution of the global temperatures for the two types of brake discs, full and ventilated discs, during braking scenario. Also, numerical simulation of the transient thermal analysis and the static structural were performed here sequentially, with coupled thermo-structural method. Numerical procedure of calculation relies on important steps such that the Computational Fluid Dynamics (CFD) and thermal analysis have been well illustrated in 3D, showing the effects of heat distribution over the brake disc. This CFD analysis helped us in the calculation of the values of the thermal coefficients (h) that have been exploited in 3D transient evolution of the brake disc temperatures. Three different brake disc materials were tested and comparative analysis of the results was conducted in order, to derive the one with the best thermal behavior. Finally, the resolution of the coupled thermomechanical model allows us to visualize other important results of this research such as the deformations and the equivalent stresses of von Mises of the disc, as well as the contact pressure of the brake pads. Following our analysis and results we draw from it, we derive several conclusions. The choice allowed us to deliver the rotor design excellence to ensure and guarantee the good braking performance of the vehicles.

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a :

Vehicle deceleration (m/s2)

A c :

Surface area of the braking pad (m2)

A d :

Disc surface swept by a brake pad (m2)


Kinematic matrix

C p :

Specific heat (J/kg °C)


Material property matrix

d :

Displacement vector (m)

E :

Young’s modulus (MPa)

F :

Force vector (N)

F disc :

Rotor force (N)

g :

Gravitational acceleration (m/s2)

h :

Heat transfer coefficient (W/m2 °C)

k :

Thermal conductivity (W/m  °C)


Stiffness matrix

m :

Vehicle mass (kg)

p :

Pressure (Pa)

P :

Pressure for a single pad (MPa)

q 0 :

Heat flux entering the disk (W)

q x :

Conduction heat flux in x-direction (W/m2)

q y :

Conduction heat flux in y-direction (W/m2)

q z :

Conduction heat flux in z-direction (W/m2)

Q :

Internal heat generation rate per unit volume (W/m3)

R rotor :

Effective rotor radius (m)

R tire :

Tire radius (m)

S m :

Source term (kg)

t :

Time (s)

T :

Temperature (°C)

T 1 :

Specified surface temperature (°C)

T s :

Unknown surface temperature (°C)

t stop :

Time to stop (s)

T e :

Convective exchange temperature (°C)

v :

Velocity vector (m/s)

v 0 :

Initial speed of the vehicle (m/s)

v r :

Relative velocity vector (m/s)

x :

Space variable in x-direction (m)

y :

Space variable in x-direction (m)

z :

Space variable in x-direction (m)

z :

Braking effectiveness

α :

Thermal expansion coefficient (1/°C)


Strain vector

ε p :

Factor load distribution on the disk surface

µ :

Coefficient of friction disc/pad

ρ :

Mass density (kg/m3)


Stress vector (N/m2)

τ :

Stress tensor (Pa)

υ :

Poisson coefficient

ϕ :

Rate distribution of the braking forces between the front and rear axle

ω :

Angular velocity (rad/s)


Angular velocity vector (rad/s)


Computational fluid dynamic


Gray cast iron


Heat transfer coefficients

me :

Mechanical index

th :

Thermal index


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Correspondence to Ali Belhocine.

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Belhocine, A., Afzal, A. Finite element modeling of thermomechanical problems under the vehicle braking process. Multiscale and Multidiscip. Model. Exp. and Des. 3, 53–76 (2020). https://doi.org/10.1007/s41939-019-00059-w

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  • Brake disc
  • CFD
  • Heat flux
  • Heat transfer coefficient
  • Pad
  • Gray cast iron
  • von Mises stress
  • Contact pressure