# Coupled Thermal Stress Analysis of Volvo Truck Disc Brake

Authors: Welteji Bena; Goftila Sirata
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

This paper deals with the coupled thermal structural analysis developed in Volvo heavy truck disc brake as a function of disc thickness and braking time. A ventilated type of disc brake was employed by having vents (holes) on its surface, which needs a cooling system. When the brake pedal is applied, the friction between the disc and pad tremendously increases due to the heat generated in this system. This heat was partitioned into two contacting bodies (disc rotor and pad). Both analytical and numerical approaches were used to carry out the analysis. Analytically, the transient temperature obtained by the partial solution approach and the stress field were evaluated by the elastic von Mises criterion. In the numerical approach, temperature and stress response were analyzed by the explicit standard ABAQUS. From both results, high thermal gradient and von Mises stress distribution were observed at the sharp edges of the disc, which causes a decrement in braking efficiency and an overall drop in braking performance.


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## Body

## 1. Introduction

The disc brake is one of the components of the vehicle that slows or stops a wheel's rotation while it is in motion. To stop the wheel, friction material in the form of brake pads is forced on both sides of the disc. The braking system is performed by a combination of different components of the braking assembly. These are the disc rotor that rotates with the wheel; the caliper assembly attached to the steering knuckle; and the friction materials (disc pads) that are mounted to the caliper assembly [1, 2]. The ventilated type of disc brakes is widely used compared to the solid type. It has vents (holes) on its surface which are used for high cooling performance and provide wide-ranging brake torque (Figure 1 [3]).Figure 1
Disc pads assembly with forces applied to the disc.During braking, the vehicle's kinetic energy is converted into heat energy by friction between the rotor and the brake pad, which is then dissipated into the surroundings via heat transfer [3, 4]. Heat dissipated at the rubbing surface of the rotor and brake pad creates a temperature gradient through the disc’s thickness as well as compressive stresses on both sides of the disc. After releasing the brake, the outer surface is cooled and generates tensile stresses, resulting in surface cracks on the disc, thermal deformation, and degradation of the material of the pads [4, 5].Several scholars have been researching on disc brake thermomechanical coupling analyses. Deshpande et al. [5] studied on the analysis of heat conduction in the disc brake system. Green’s equation based on time and space was used to calculate heat generation at the contact interface. This heat is ideally dissipated into the environment to avoid temperature rise and friction coefficient between the disc brake and pad. Dissipated heat depends on the rate of heat input, thermal conductivity of the disc material, and disc brake geometry. Higher thermal conductivity materials transmit heat more quickly [6]. The ventilated type of disc geometry has high cooling performance compared to the solid type because it has a number of vents (holes) on its surface. The larger the number of ventilation holes, the lower the maximum temperature variation [4, 6, 7].Riyadi S. and Tauviqirrahman [8] discussed factors affecting the interface temperature, including sliding speed, braking load, and friction material type, which were all taken into consideration. The influence of real contact area on local interface temperatures was investigated using FEA, and it was observed that the maximum temperature at the friction interface does not increase linearly with decreasing contact area ratio. Popescu et al. [9] investigated the thermal and mechanical behavior of the brake system in the case of the emergency braking of a mine hoist model using FEA (COMSOL Multiphysics). From the results, they concluded that the active components (the shoes and pads) had higher temperatures than the passive components (the drum and disc). The pad’s temperature was higher than the temperature of the shoe’s active components. Jian and Shui [10] presented a numerical and experimental analysis of a ventilated disc brake's transient temperature field during hard braking. There is a thermal gradient in the radial, circumferential, and axial directions, and the temperature field distribution is not symmetrical. The temperature gradient of the contact areas is larger than that of the noncontact areas, and the simulation results are in agreement with the experimental results. Kang and Chao [11] reported the thermal stress of Volvo heavy trucks in steady state and transient conditions using FEM (ABAQUS) and an analytical solution (Limpert equation). They observed thermal stress due to heat flux generation has a major influence on the fatigue stress and thermal cracking is highly localized. It has a distinct gradient in both the axial and radial directions.The influence of the material properties on the thermoelastic behavior of disc brake has been studied by many scholars. The influence of boron carbide on the mechanical properties and bonding strength of B4C/Nickel 63 coatings of brake disc was investigated by Ramesh et al. [12] based on numerical (ANSYS) and experimental analysis. The result shows that good bonding strength and mechanical characteristics were achieved when B4C content was 20% to 30%. Thermophysical characteristics of three types of gray cast iron disc brake (FG15, FG20, and FG25AL) were introduced by Belhocine and Abdullah [13]. Materials with lower thermal conductivity induce significant thermal gradients, which raise the surface temperature of the brake disc. They concluded that gray cast iron FG15 has better thermal performance. Eltoukhy et al. [14] modelled transient analysis of thermoelastic contact problems for disc brakes of various types using the finite-element analysis (FEA) approach. They focused on the effect of the discs' material properties on thermoelastic behavior. It was found that their proposed numerical model showed a much-closed fit with experimental results found in the literature. Singh and Shergill [15] presented a work on disc brake thermal analysis utilizing Finite element analysis (Comsol) to estimate temperature distribution. Cast iron, aluminium, and ceramics disc brake materials and conduction, convection, and radiation heat transfer mode were considered. Cast iron is used in disc brakes because it provides moderate cooling at low temperatures when compared to other materials.The objective of this study is to conduct the coupled thermal structural analysis of ventilated type Volvo truck disc brake through the investigation of thermoelastic couplings such as heat flux generated, von Mises stress, contact pressure, and deformations as the function of disc thickness and braking time. Finally, the analytical results were compared with the numerical results carried out by the commercial tool ABAQUS version 2020, and the obtained results are discussed.


## 2. Material and Methods

The disc brake rotor is made from gray cast iron material, which provides good wear resistance and excellent thermal conductivity besides the low cost of the production, which eases the dissipation of heat generated by the friction of the pads during a stop and the capacity of damping vibrations [16]. The stress-strain properties of gray cast iron are also influenced by its composition, i.e., graphite flakes and metal matrix [9]. In addition, the content of graphite provides important properties such as dimension stability under different heating conditions and high vibrating damping [10]. Table 1 shows the mechanical and thermophysical properties of the disc-pad materials.Table 1
Mechanical and thermophysical properties of disc brake and pad materials
S.NoMaterial propertiesSymbolUnitPadDisc1Thermal conductivitykw/m ̊ C5572Densityρkg/m3140072503Specific heatcJ/kg ̊ C10004604Poisson’s ratiov0.250.285Thermal expansionα10−6 / ̊ C1010.386Elastic modulusEGPa11387Coefficient of frictionμ0.430.43


### 2.1. Braking Conditions and Assumptions

For thermomechanical analysis, the following braking conditions were assumed:(1)
The average stopping distance of a truck is 73.45 m, with a deceleration rate of4.89m/s2 in 5.48 seconds(2)
The ambient temperature ofT∞=27℃(3)
Travelling speed of96Km/hr under fully air disc brake under the best road condition(4)
Angular speed of the disc123rad/secSeveral assumptions are made to simplify the complexity of the analysis and to obtain a reasonable output from the result of the analysis.(I)
The disc material is considered as homogeneous and isotropic because material properties were assumed to be constant(II)
All kinetic energy at the disc brake rotor surface is converted into frictional heat or heat flux since the rate of internal heat generation and stored energy contributes to a lesser amount(III)
Due to short brake time, radiation heat transfer is neglected, hence relatively low temperature(IV)
The temperature only varies along with its thickness, and constant heat flux is applied radially(V)
Inertia and body force effects are negligible during the analysis(VI)
The contact pressure is uniformly distributed over all friction surfaces; hence the heat generation of the mid plane is considered as symmetric


## 2.1. Braking Conditions and Assumptions

For thermomechanical analysis, the following braking conditions were assumed:(1)
The average stopping distance of a truck is 73.45 m, with a deceleration rate of4.89m/s2 in 5.48 seconds(2)
The ambient temperature ofT∞=27℃(3)
Travelling speed of96Km/hr under fully air disc brake under the best road condition(4)
Angular speed of the disc123rad/secSeveral assumptions are made to simplify the complexity of the analysis and to obtain a reasonable output from the result of the analysis.(I)
The disc material is considered as homogeneous and isotropic because material properties were assumed to be constant(II)
All kinetic energy at the disc brake rotor surface is converted into frictional heat or heat flux since the rate of internal heat generation and stored energy contributes to a lesser amount(III)
Due to short brake time, radiation heat transfer is neglected, hence relatively low temperature(IV)
The temperature only varies along with its thickness, and constant heat flux is applied radially(V)
Inertia and body force effects are negligible during the analysis(VI)
The contact pressure is uniformly distributed over all friction surfaces; hence the heat generation of the mid plane is considered as symmetric


## 3. Analytical Analysis

### 3.1. Mechanism of Heat Transfer in Vehicle Disc Brake

The majority of heat is transferred during braking by conduction through the brake disc and pads.Gradually, the temperature in the disc increases, heat is conducted to the surroundings, and cooling increases by convection air flowing through the vanes and by radiation (Figure2 [12]). The upper and lower surfaces of the disc are subjected to convection, while the remaining surfaces, such as the circumferential, inner, and external radius areas of the disc, are considered insulated [17]. The temperature or heat flow at the boundary region is defined by boundary conditions. To develop the boundary condition, consider the energy balance at the surface.Figure 2
Heat transfer mechanism in disc brake system.Heat supply = heat loss(1)qcond+qsup=qconv+qrad.Writing the equations for heat conduction, convection, and radiation, it became(2)−k∂T∂n+qsup=hT−T∞+εσT4−T∞4,where q is heat flux, T is surface temperature, T∞ is ambient temperature, ε is the emissivity of the surface, and σ is the Stefan–Boltzmann constant, 5.6697×10−8w/m2k4.In assumption, radiation heat transfer is neglected due to the relatively low temperature attained in frictional heating [13] and the external heat supply (qsup) term is no longer relevant. So, (2) is reduced to(3)k∂T∂n+hT−T∞=0.In this paper, different types of linear boundary conditions are considered for the solution of heat conduction problems. The initial condition specifies the temperature distribution in the medium at the origin of the time coordinate (that is,t = 0), and the boundary conditions specify the temperature or the heat flow at the boundaries of the region. Three different types of linear boundary conditions are considered:(1)
The situation when the temperature distribution is prescribed at the boundary surface, that is,T = f (z, t) on the surface, is referred to as a boundary condition at a specified surface temperature.(2)
Boundary condition at specified heat flux, where heat flux is prescribed at the surface is defined ask∂q/∂n=fz,t.(3)
Boundary condition at convective heat transfer is a condition which is readily obtained from equation (3) by setting the radiation term and the heat supply equal to zero. k∂T/∂n+hT=hT∞r,t on the surface of the disc brake. A cylindrical coordinate system has been assumed in order to show the transient temperature field T (r, z, t) of a rotating disc. The boundary condition of transient temperature under a heat conduction problem was given [14, 18, 19]:(4)∂2T∂r2+1r∂T∂r+∂2T∂z2=1α∂T∂t,rd≤r≤rp,0≤z≤δd,t>0.


### 3.2. Pressure Distribution for Thermal Analysis

The nature of the contact and pressure distribution at the interface between the friction material pad and the disc rotor affects the performance of a disc brake. For the calculation of heat flux, uniform pressure distribution and uniform wear are considered. When the pad is new and short enough, the pressure is distributed uniformly in the contact area, that is,P = Pmax. After several braking actions when the pad is run down, uniform wear is taken into account and Pmax=Prp/r [20], where rp, P, and Pmaxare the inner radius of the pad, pressure at the radial position, and the maximum pressure on the pad-disc contact surface, respectively. Figure 3 shows the boundary condition of the disk for both types of pressure distribution, and since the thermal problem in the disk is symmetric in the z direction, only half of the disk (Z=δ/2) is considered in the analytical analysis.Figure 3
Boundary conditions of the disk. (a) Uniform wear and (b) uniform pressure.
(a)and (b)Figure4 shows pressure distribution condition versus braking time. Initially, the pressure increases slightly, but at the end of the braking period, high pressure is applied to the pad to stop the motion of the vehicle. The displacement of the pads induces contact pressure on the disc and the angular speed decreases as the time of braking increases according to the relationship: ωt=ω01−t/th,0≤t≤tb. Initially, the angular speed of the disc brake is 123rad/sec at t = 0 from the vehicle operating condition given in Table 1.Figure 4
Pressure distribution at different braking times.


### 3.3. Heat Partition Factor and Energy Input

The heat partition coefficientγ refers to the amount of heat generated which is “absorbed” by the disc and the amount “transferred” or “dissipated” to the pads and surroundings. It is a function of the thermal properties of the bodies, the contact geometry, and the sliding speed [21‐23].(5)γ=ξdSdξpSp+ξdSd,where ξp and ξd are the thermal effectivities of the pad and the disk, and Sp and Sd are frictional contact surfaces of the pad and the disk, respectively. Thermal diffusivity of the pad (ξp and the disc (ξdare calculated by making use of values given in Table 1.(6)ξp=kpρpcp=5×1400×1000=2645.75jm2℃1Sec,(7)ξd=kdρdcd=57×7250×460=13787.50j/m2℃1Sec.The frictional contact surfaces of the pad,Sp, and the disc, Sd were calculated as follows [24, 25]:(8)sp=ϕ∫rpRprdr=ϕ2Rp2−rp2=6520.2132−0.1242=0.017004m2,(9)sd=2π∫rdRdRdr=πRd2−rd2=π0.2182−0.1042=0.115263m2.Inserting the result of equations (6)–(9) into equation (5) gives partitioning coefficient of disc rotor and brake pad of γ=0.97


### 3.4. Heat Generation due to Friction at the Contact Interfaces

During the braking system, mechanical energy is converted to calorific energy, which is characterized by total disc and pad heating during the braking period. Figure5 shows the contact surface element of the disc and the pad. Heat is generated at the interface between the disc and the pad due to friction, and this heat is partitioned between the two bodies.Figure 5
Contact surface element of two components. (a) The disk and (b) the pad.
(a)(b)The heat generated at the brake friction interface can be transferred to the brake rotor as well as the brake pads [3, 5, 26].(10)dE˙=dP=VdFf=ωμpϕ0r2dr,dE˙=dE˙d+dE˙d=1−γdP+γdP,where dE˙ is the rate of heat generated due to friction between two sliding components, V is the relative sliding velocity, and dFf is the frictional force. The terms dE˙p and dE˙d are the amount of absorbed heat by the pad and the disk, respectively.


### 3.5. Heat Flux Entering into the Disc

At the contacting surfaces of the brake system, heat flux is obtained by dividing the rate of thermal energy by the respective surface contact area for each component. Based on the uniform wear assumption, an imperfect thermal contact type was modelled [5]. The friction heat flux generated in the interface of the disc brake and pad can be expressed as(11)qpr,t=dE˙pdSp=1−γμrPtωt=1−γμPmaxrpω01−ttb,qdr,t=dE˙ddSd=γωμpϕ0r2dr2πrdr=ϕ02πγμptrωt=ϕ02πγμPmaxrdω01−ttb.


### 3.6. Thermal Stresses in the Disc Brake

Thermal stresses are caused by thermomechanical mismatches between two materials, nonuniform temperature distribution in a material, and anisotropy in thermal expansion in a material [27]. During hard braking, the material of the disc and pad will expand, but when the disc cools down again it is compressed, which results in thermal stress induced [24].


#### 3.6.1. Stress Distributed along Circumferential Direction

The rise in the temperature in the circumferential direction causes the material to try to expand freely [28]. Transient stress, σθ, response as a function of disc thickness, Z, becomes(12)σθ=−E1−vαΔT=−αE1−vqoδk−16+αtδ2+z22δ2−2∑n=0−1nλn2cosλnzδe−λn2α/∂2t.


#### 3.6.2. Axial Stress in the Disc Brake System

Equal pressure applied to both sides of the disc suppresses the free thermal expansion of the rotor in the axial direction. The axial stress (𝜎𝑧𝑧) is obtained from the temperature difference at the outer surface (T1) and inner surface (T2) of the disc.(13)σz=−αE1−vT1t−T2t.


#### 3.6.3. Radial Stress Distribution

The inner radius of the rotor is fixed in order to prevent the radial movement of the rotor, and the external radius of the disc is free from stress and displacement. The thermal stress is obtained by superposing the stress in equation (12) with the stress due to the application of an equal and opposite distribution of force on the edges. When ΔT is the temperature gradient, then the stress is induced in the disc by the tensile force of intensity α (ΔT) E at the ends:(14)FR=∫0δαEΔTdz,whereas a uniformly distributed radial tensile stress formed in the disc can be expressed by(15)σrr=−αEΔT1−v+1δ1−v∫0δαEΔTdz.Then, by inserting equation (12) into compressive radial stress, the radial thermal stress is obtained as(16)σrr=−αE1−vqoδk−16+αtδ2+z22δ2−2∑n=0−1nλn2cosλnzδe−λn2α/∂2t+1δ1−vαEqoδk∫0δ−16+αtδ2+z22δ2−2∑n=0−1nλn2cosλnzδe−λn2α/∂2tdz.


### 3.7. von Mises Stress Analytical Analysis

To estimate the thermal stresses created by temperature distributions, the elastic von Mises stress (σe) is defined in equation (17). This equation relates the three principal stresses and determines yield (plastic deformation onset) in metals.(17)σe=12σr−σθ2+σθ−σz2+σz−σr2,where σr, σθ, and σz are principal stresses in a cylindrical axis system.


## 3.1. Mechanism of Heat Transfer in Vehicle Disc Brake

The majority of heat is transferred during braking by conduction through the brake disc and pads.Gradually, the temperature in the disc increases, heat is conducted to the surroundings, and cooling increases by convection air flowing through the vanes and by radiation (Figure2 [12]). The upper and lower surfaces of the disc are subjected to convection, while the remaining surfaces, such as the circumferential, inner, and external radius areas of the disc, are considered insulated [17]. The temperature or heat flow at the boundary region is defined by boundary conditions. To develop the boundary condition, consider the energy balance at the surface.Figure 2
Heat transfer mechanism in disc brake system.Heat supply = heat loss(1)qcond+qsup=qconv+qrad.Writing the equations for heat conduction, convection, and radiation, it became(2)−k∂T∂n+qsup=hT−T∞+εσT4−T∞4,where q is heat flux, T is surface temperature, T∞ is ambient temperature, ε is the emissivity of the surface, and σ is the Stefan–Boltzmann constant, 5.6697×10−8w/m2k4.In assumption, radiation heat transfer is neglected due to the relatively low temperature attained in frictional heating [13] and the external heat supply (qsup) term is no longer relevant. So, (2) is reduced to(3)k∂T∂n+hT−T∞=0.In this paper, different types of linear boundary conditions are considered for the solution of heat conduction problems. The initial condition specifies the temperature distribution in the medium at the origin of the time coordinate (that is,t = 0), and the boundary conditions specify the temperature or the heat flow at the boundaries of the region. Three different types of linear boundary conditions are considered:(1)
The situation when the temperature distribution is prescribed at the boundary surface, that is,T = f (z, t) on the surface, is referred to as a boundary condition at a specified surface temperature.(2)
Boundary condition at specified heat flux, where heat flux is prescribed at the surface is defined ask∂q/∂n=fz,t.(3)
Boundary condition at convective heat transfer is a condition which is readily obtained from equation (3) by setting the radiation term and the heat supply equal to zero. k∂T/∂n+hT=hT∞r,t on the surface of the disc brake. A cylindrical coordinate system has been assumed in order to show the transient temperature field T (r, z, t) of a rotating disc. The boundary condition of transient temperature under a heat conduction problem was given [14, 18, 19]:(4)∂2T∂r2+1r∂T∂r+∂2T∂z2=1α∂T∂t,rd≤r≤rp,0≤z≤δd,t>0.


## 3.2. Pressure Distribution for Thermal Analysis

The nature of the contact and pressure distribution at the interface between the friction material pad and the disc rotor affects the performance of a disc brake. For the calculation of heat flux, uniform pressure distribution and uniform wear are considered. When the pad is new and short enough, the pressure is distributed uniformly in the contact area, that is,P = Pmax. After several braking actions when the pad is run down, uniform wear is taken into account and Pmax=Prp/r [20], where rp, P, and Pmaxare the inner radius of the pad, pressure at the radial position, and the maximum pressure on the pad-disc contact surface, respectively. Figure 3 shows the boundary condition of the disk for both types of pressure distribution, and since the thermal problem in the disk is symmetric in the z direction, only half of the disk (Z=δ/2) is considered in the analytical analysis.Figure 3
Boundary conditions of the disk. (a) Uniform wear and (b) uniform pressure.
(a)and (b)Figure4 shows pressure distribution condition versus braking time. Initially, the pressure increases slightly, but at the end of the braking period, high pressure is applied to the pad to stop the motion of the vehicle. The displacement of the pads induces contact pressure on the disc and the angular speed decreases as the time of braking increases according to the relationship: ωt=ω01−t/th,0≤t≤tb. Initially, the angular speed of the disc brake is 123rad/sec at t = 0 from the vehicle operating condition given in Table 1.Figure 4
Pressure distribution at different braking times.


## 3.3. Heat Partition Factor and Energy Input

The heat partition coefficientγ refers to the amount of heat generated which is “absorbed” by the disc and the amount “transferred” or “dissipated” to the pads and surroundings. It is a function of the thermal properties of the bodies, the contact geometry, and the sliding speed [21‐23].(5)γ=ξdSdξpSp+ξdSd,where ξp and ξd are the thermal effectivities of the pad and the disk, and Sp and Sd are frictional contact surfaces of the pad and the disk, respectively. Thermal diffusivity of the pad (ξp and the disc (ξdare calculated by making use of values given in Table 1.(6)ξp=kpρpcp=5×1400×1000=2645.75jm2℃1Sec,(7)ξd=kdρdcd=57×7250×460=13787.50j/m2℃1Sec.The frictional contact surfaces of the pad,Sp, and the disc, Sd were calculated as follows [24, 25]:(8)sp=ϕ∫rpRprdr=ϕ2Rp2−rp2=6520.2132−0.1242=0.017004m2,(9)sd=2π∫rdRdRdr=πRd2−rd2=π0.2182−0.1042=0.115263m2.Inserting the result of equations (6)–(9) into equation (5) gives partitioning coefficient of disc rotor and brake pad of γ=0.97


## 3.4. Heat Generation due to Friction at the Contact Interfaces

During the braking system, mechanical energy is converted to calorific energy, which is characterized by total disc and pad heating during the braking period. Figure5 shows the contact surface element of the disc and the pad. Heat is generated at the interface between the disc and the pad due to friction, and this heat is partitioned between the two bodies.Figure 5
Contact surface element of two components. (a) The disk and (b) the pad.
(a)(b)The heat generated at the brake friction interface can be transferred to the brake rotor as well as the brake pads [3, 5, 26].(10)dE˙=dP=VdFf=ωμpϕ0r2dr,dE˙=dE˙d+dE˙d=1−γdP+γdP,where dE˙ is the rate of heat generated due to friction between two sliding components, V is the relative sliding velocity, and dFf is the frictional force. The terms dE˙p and dE˙d are the amount of absorbed heat by the pad and the disk, respectively.


## 3.5. Heat Flux Entering into the Disc

At the contacting surfaces of the brake system, heat flux is obtained by dividing the rate of thermal energy by the respective surface contact area for each component. Based on the uniform wear assumption, an imperfect thermal contact type was modelled [5]. The friction heat flux generated in the interface of the disc brake and pad can be expressed as(11)qpr,t=dE˙pdSp=1−γμrPtωt=1−γμPmaxrpω01−ttb,qdr,t=dE˙ddSd=γωμpϕ0r2dr2πrdr=ϕ02πγμptrωt=ϕ02πγμPmaxrdω01−ttb.


## 3.6. Thermal Stresses in the Disc Brake

Thermal stresses are caused by thermomechanical mismatches between two materials, nonuniform temperature distribution in a material, and anisotropy in thermal expansion in a material [27]. During hard braking, the material of the disc and pad will expand, but when the disc cools down again it is compressed, which results in thermal stress induced [24].


### 3.6.1. Stress Distributed along Circumferential Direction

The rise in the temperature in the circumferential direction causes the material to try to expand freely [28]. Transient stress, σθ, response as a function of disc thickness, Z, becomes(12)σθ=−E1−vαΔT=−αE1−vqoδk−16+αtδ2+z22δ2−2∑n=0−1nλn2cosλnzδe−λn2α/∂2t.


### 3.6.2. Axial Stress in the Disc Brake System

Equal pressure applied to both sides of the disc suppresses the free thermal expansion of the rotor in the axial direction. The axial stress (𝜎𝑧𝑧) is obtained from the temperature difference at the outer surface (T1) and inner surface (T2) of the disc.(13)σz=−αE1−vT1t−T2t.


### 3.6.3. Radial Stress Distribution

The inner radius of the rotor is fixed in order to prevent the radial movement of the rotor, and the external radius of the disc is free from stress and displacement. The thermal stress is obtained by superposing the stress in equation (12) with the stress due to the application of an equal and opposite distribution of force on the edges. When ΔT is the temperature gradient, then the stress is induced in the disc by the tensile force of intensity α (ΔT) E at the ends:(14)FR=∫0δαEΔTdz,whereas a uniformly distributed radial tensile stress formed in the disc can be expressed by(15)σrr=−αEΔT1−v+1δ1−v∫0δαEΔTdz.Then, by inserting equation (12) into compressive radial stress, the radial thermal stress is obtained as(16)σrr=−αE1−vqoδk−16+αtδ2+z22δ2−2∑n=0−1nλn2cosλnzδe−λn2α/∂2t+1δ1−vαEqoδk∫0δ−16+αtδ2+z22δ2−2∑n=0−1nλn2cosλnzδe−λn2α/∂2tdz.


## 3.6.1. Stress Distributed along Circumferential Direction

The rise in the temperature in the circumferential direction causes the material to try to expand freely [28]. Transient stress, σθ, response as a function of disc thickness, Z, becomes(12)σθ=−E1−vαΔT=−αE1−vqoδk−16+αtδ2+z22δ2−2∑n=0−1nλn2cosλnzδe−λn2α/∂2t.


## 3.6.2. Axial Stress in the Disc Brake System

Equal pressure applied to both sides of the disc suppresses the free thermal expansion of the rotor in the axial direction. The axial stress (𝜎𝑧𝑧) is obtained from the temperature difference at the outer surface (T1) and inner surface (T2) of the disc.(13)σz=−αE1−vT1t−T2t.


## 3.6.3. Radial Stress Distribution

The inner radius of the rotor is fixed in order to prevent the radial movement of the rotor, and the external radius of the disc is free from stress and displacement. The thermal stress is obtained by superposing the stress in equation (12) with the stress due to the application of an equal and opposite distribution of force on the edges. When ΔT is the temperature gradient, then the stress is induced in the disc by the tensile force of intensity α (ΔT) E at the ends:(14)FR=∫0δαEΔTdz,whereas a uniformly distributed radial tensile stress formed in the disc can be expressed by(15)σrr=−αEΔT1−v+1δ1−v∫0δαEΔTdz.Then, by inserting equation (12) into compressive radial stress, the radial thermal stress is obtained as(16)σrr=−αE1−vqoδk−16+αtδ2+z22δ2−2∑n=0−1nλn2cosλnzδe−λn2α/∂2t+1δ1−vαEqoδk∫0δ−16+αtδ2+z22δ2−2∑n=0−1nλn2cosλnzδe−λn2α/∂2tdz.


## 3.7. von Mises Stress Analytical Analysis

To estimate the thermal stresses created by temperature distributions, the elastic von Mises stress (σe) is defined in equation (17). This equation relates the three principal stresses and determines yield (plastic deformation onset) in metals.(17)σe=12σr−σθ2+σθ−σz2+σz−σr2,where σr, σθ, and σz are principal stresses in a cylindrical axis system.


## 4. Finite Element Method (FEM)

### 4.1. Transient Temperature Distribution

The ABAQUS computational tool is used to solve transient thermal FEM analysis by inputting the disc material in Table1 and the braking condition. From the results, the maximum temperature produced on the rubbing surface of the disc was 358.9°C, and on the opposite surface, the minimum temperature of 84.97 C was observed. The temperature varies with the braking period at the contact interface and disc thickness, and a high thermal gradient occurs at the final braking (Figure 6).Figure 6
Transient temperature distribution in the disc brake by FEM approach.


### 4.2. Circumferential Stresses

The intensity of heat (heat flux) is high at the contact zone and it slightly decreases as it moves away from the rubbing surface (Figure7). At fixed time moments, the highest value of circumferential stress, σθθ (positive) obtained is 11 MPa at r = rd = 104 mm for t = 5.48 sec. However, the value of σθθdecreases as the radial distance r increases, and the lowest circumferential stresses (compressive stresses) of −1358 MPa were obtained at r = Rd = 218 mm.Figure 7
Circumferential stress across disc thickness.


### 4.3. Radial Stresses in Disc Brake

The rubbing surface is subjected to compressive stresses while the portion far from the contact face is free to expand and it is subjected to tensile (Figure8). The maximum compressive stress obtained σrr=−933 MPa at the distance of r = 135 mm, and the highest positive value radial stress of σrr=161 MPa occurred at the radial distance of r = 147 mm in the braking time of t = 5.48 sec.Figure 8
Contour plot of radial stress across disc thickness.


### 4.4. von Mises Stress Distributions

Under a hydraulic pressure of 1.47 MPa, the von Mises stress distribution in the disc-pad normal contact surface is obtained as a function of disc thickness. The maximum value obtained from FEM is 1411 MPa and the minimum value that occurs is 17 MPa. From the contour plot of Figure9, it is clearly seen that the maximum and minimum von Mises stress distributions are observed at the sharp edges, which is the main cause of the subsurface fatigue crack initiation.Figure 9
Contour plot of von Misses stress distribution without temperature effect.


## 4.1. Transient Temperature Distribution

The ABAQUS computational tool is used to solve transient thermal FEM analysis by inputting the disc material in Table1 and the braking condition. From the results, the maximum temperature produced on the rubbing surface of the disc was 358.9°C, and on the opposite surface, the minimum temperature of 84.97 C was observed. The temperature varies with the braking period at the contact interface and disc thickness, and a high thermal gradient occurs at the final braking (Figure 6).Figure 6
Transient temperature distribution in the disc brake by FEM approach.


## 4.2. Circumferential Stresses

The intensity of heat (heat flux) is high at the contact zone and it slightly decreases as it moves away from the rubbing surface (Figure7). At fixed time moments, the highest value of circumferential stress, σθθ (positive) obtained is 11 MPa at r = rd = 104 mm for t = 5.48 sec. However, the value of σθθdecreases as the radial distance r increases, and the lowest circumferential stresses (compressive stresses) of −1358 MPa were obtained at r = Rd = 218 mm.Figure 7
Circumferential stress across disc thickness.


## 4.3. Radial Stresses in Disc Brake

The rubbing surface is subjected to compressive stresses while the portion far from the contact face is free to expand and it is subjected to tensile (Figure8). The maximum compressive stress obtained σrr=−933 MPa at the distance of r = 135 mm, and the highest positive value radial stress of σrr=161 MPa occurred at the radial distance of r = 147 mm in the braking time of t = 5.48 sec.Figure 8
Contour plot of radial stress across disc thickness.


## 4.4. von Mises Stress Distributions

Under a hydraulic pressure of 1.47 MPa, the von Mises stress distribution in the disc-pad normal contact surface is obtained as a function of disc thickness. The maximum value obtained from FEM is 1411 MPa and the minimum value that occurs is 17 MPa. From the contour plot of Figure9, it is clearly seen that the maximum and minimum von Mises stress distributions are observed at the sharp edges, which is the main cause of the subsurface fatigue crack initiation.Figure 9
Contour plot of von Misses stress distribution without temperature effect.


## 5. Discussion

### 5.1. Transient Temperature Distribution

Thermal gradient carried out at the surface contact of disc and pad at different period of time,0≤t≤5.48Sec. The maximum of temperature 359.8°C and 358.9°C is observed analytically and by FEM, respectively. As illustrated in Figure 10, the minimum value of temperature occurred at the first braking and the maximum result was formed at the final braking time.Figure 10
Transient temperature distribution at different braking times.


### 5.2. Circumferential and Axial Stresses

The transient temperature distribution was predicted as input data for the analysis of coupled stresses (thermal and mechanical stresses) under each time step. Figure11(a) shows circumferential stress results across disc thickness. The minimum circumferential stress was −1467 MPa and −1358 MPa analytically and by FEM, respectively, with 8% of error. The maximum circumferential stress obtained by FEM is 11 MPa and by analytical is 10.47 MPa.Figure 11
. (a) Circumferential stresses and (b) axial stresses along disc thickness.
(a)(b)In the case of axial stresses, the temperature distribution does not interrupt the axial stress because the direction of the temperature distribution is parallel to the axial stress. Therefore, variation in axial stress was not observed. The axial stress developed in the disc was totally compressive since the compressive force was applied on both sides of the rubbing surface. As shown in Figure11(b), axial stress with a negligible variation of −1467 MPa and −1317 MPa is developed in disc brake by analytical and FEM, respectively.


### 5.3. Comparison of Radial Stresses

The surface of the disc is exposed to both tensile stress (0 to 7.8 mm) and compressive stress (7.8 mm to 18 mm thickness). As shown in Figure12, the maximum tensile stress of 161 MPa by FEM and 215 MPa was obtained by analytical at the lower surface. The minimum stresses obtained by FEM and analytical results are -933 MPa and -1243 MPa, respectively. As the thickness approaches the mid of the disc, both results match each other, but at the surface the variation is large which might be resulted from big step time and coarse mesh size.Figure 12
Radial stresses result by FEM and analytical approach.


### 5.4. von Mises Stress along with Disc Thickness

The disc brake is extremely exposed to compressive stress rather than tensile stress. Radial and circumferential stresses vary with thickness, with a maximum value at the surface of the disc (Figure13(a)). The von Mises stress across disc thickness under the applied brake pressure of 1.47 MPa as shown in Figure 13(b). The maximum stress of 1411 MPa by FEM and 1912 MPa by analytical is read from the results. The maximum von Mises stress is high on the terminal edge of the inside radius of the disc in contact with the pad (2 mm and 4 mm from the contact surface) because the stress concentration is high at the sharp edges.Figure 13
. (a) Stress components and (b) von Misses stress distribution within disc brake system.
(a)(b)


## 5.1. Transient Temperature Distribution

Thermal gradient carried out at the surface contact of disc and pad at different period of time,0≤t≤5.48Sec. The maximum of temperature 359.8°C and 358.9°C is observed analytically and by FEM, respectively. As illustrated in Figure 10, the minimum value of temperature occurred at the first braking and the maximum result was formed at the final braking time.Figure 10
Transient temperature distribution at different braking times.


## 5.2. Circumferential and Axial Stresses

The transient temperature distribution was predicted as input data for the analysis of coupled stresses (thermal and mechanical stresses) under each time step. Figure11(a) shows circumferential stress results across disc thickness. The minimum circumferential stress was −1467 MPa and −1358 MPa analytically and by FEM, respectively, with 8% of error. The maximum circumferential stress obtained by FEM is 11 MPa and by analytical is 10.47 MPa.Figure 11
. (a) Circumferential stresses and (b) axial stresses along disc thickness.
(a)(b)In the case of axial stresses, the temperature distribution does not interrupt the axial stress because the direction of the temperature distribution is parallel to the axial stress. Therefore, variation in axial stress was not observed. The axial stress developed in the disc was totally compressive since the compressive force was applied on both sides of the rubbing surface. As shown in Figure11(b), axial stress with a negligible variation of −1467 MPa and −1317 MPa is developed in disc brake by analytical and FEM, respectively.


## 5.3. Comparison of Radial Stresses

The surface of the disc is exposed to both tensile stress (0 to 7.8 mm) and compressive stress (7.8 mm to 18 mm thickness). As shown in Figure12, the maximum tensile stress of 161 MPa by FEM and 215 MPa was obtained by analytical at the lower surface. The minimum stresses obtained by FEM and analytical results are -933 MPa and -1243 MPa, respectively. As the thickness approaches the mid of the disc, both results match each other, but at the surface the variation is large which might be resulted from big step time and coarse mesh size.Figure 12
Radial stresses result by FEM and analytical approach.


## 5.4. von Mises Stress along with Disc Thickness

The disc brake is extremely exposed to compressive stress rather than tensile stress. Radial and circumferential stresses vary with thickness, with a maximum value at the surface of the disc (Figure13(a)). The von Mises stress across disc thickness under the applied brake pressure of 1.47 MPa as shown in Figure 13(b). The maximum stress of 1411 MPa by FEM and 1912 MPa by analytical is read from the results. The maximum von Mises stress is high on the terminal edge of the inside radius of the disc in contact with the pad (2 mm and 4 mm from the contact surface) because the stress concentration is high at the sharp edges.Figure 13
. (a) Stress components and (b) von Misses stress distribution within disc brake system.
(a)(b)


## 6. Conclusion

In this work, the coupled thermal structural analysis of heavy truck ventilated type of disc brake was presented. Both analytical and FEM approaches have been employed to study temperature gradient, stresses, and deformation with the function of disc thick thickness and braking time.In the thermal gradient, the maximum transient temperature of 359.8°C was obtained at the end of braking and the minimum temperature was found at the first braking. At fixed time moments,t = 5.48Sec, the highest circumferential tensile stress σθθ of 11 MPa at inner radius and the lowest compressive stress of −1358 MPa was obtained at r = Rd = 218 mm. Hence, the value of circumferential stress decreases with the increment of radial distance. In radial stress analysis, the portion far from the contact surface is free to expand and it is subjected to tensile stress, but the rubbing surface is exposed to compressive stress. The highest tensile stress of 161 MPa by FEM and 215 MPa analytically, and the lowest compressive stress of −933 MPa (FEM) and −1243 MPa (analytical) were obtained from the results of the analysis. In contrast, the axial stress was constant throughout the disc thickness and, in the results, −1467 MPa and −1317 MPa were developed in the disc brake by analytical and FEM, respectively.In von Mises analysis, under the applied brake pressure of 1.47 MPa, the maximum value of 1411 MPa by FEM and 1912 MPa by analytical was found on the terminal edge of the inside radius of the disc in contact with the pad (2 mm and 4 mm from the contact surface) because the stress concentration is high at the sharp edges, which is the main cause of the subsurface fatigue crack initiation. The obtained solutions have been numerically verified with finite-element analysis (FEA), where simulations have been performed for disc brake materials to study the thermal effects. The results of thermal gradient, deformation, and stresses induced obtained by analytical approaches have good agreement with FEA results.



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*Source: 7945264-2022-07-13.xml*