# Influence of Thermal Sensitivity of Functionally Graded Materials on Temperature during Braking

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Statement to the Problem

- The braking process with constant deceleration is considered;
- At the initial time moment, the temperature of a brake is equal to the ambient temperature T
_{a}; - In the heat conduction equation, only the change in temperature gradient in the perpendicular direction to the disc-pad contact surfaces is taken into consideration;
- The thermal contact on the friction surfaces is perfect, i.e., the temperatures of its contact surfaces are equal, and the sum of frictional heat fluxes intensities, acting along the normal direction to the contact surface to the insides of the elements equal to the specific friction power;
- Due to the symmetry of the system with respect to the mid plane of the disc, when determining the brake temperature, the contact of one pad and a disc with half of its thickness is considered;
- The pads and the disc are made of two-component thermally sensitive functionally graded materials, in such a way that their friction surfaces are materials with low thermal conductivity (i.e., cermet), while the core materials are characterized by higher thermal conductivity (titanium alloys, aluminum, etc.);
- The thermal conductivity of the disc and pads materials increases exponentially with the distance from the contact surface;
- The whole initial kinetic energy of the vehicle is transformed into heat during braking, neglecting the small part of energy associated with wear on the contact surfaces of the disc and pads;

## 3. Solution with Temperature-Independent FGMs Properties

## 4. Volume Temperature

## 5. Numerical Analysis

_{2}O

_{3}(friction surface) and cooper Cu (core) [25]. The friction surface and core of the second element are manufactured of zirconium dioxide ZrO

_{2}and titanium alloy Ti-6Al-4V [14]. The temperature-dependent properties of these materials are as follows:

_{2}O

_{3}[26,27,28]

_{2}[27,30,31]

- (1)
- the values of the input parameters were given (Table 1), and then from Equations (8) and (9) the area of the nominal contact was calculated ${A}_{a}=0.0022\hspace{0.17em}{\mathrm{m}}^{2}$, specific friction power ${q}_{0}=3.87\hspace{0.17em}\mathrm{MW}\hspace{0.17em}{\mathrm{m}}^{-2}$, friction power ${Q}_{0}=8510\hspace{0.17em}\hspace{0.17em}\mathrm{W}$ and stop time ${t}_{s}=12.1\hspace{0.17em}\mathrm{s}$;
- (2)
- using the dependencies (40)–(51) the materials properties ${K}_{l,m}^{(0)}$, ${c}_{l,m}^{(0)}$ and ${\rho}_{l,m}^{(0)}$, $l,\hspace{0.17em}m=1,2$ at the initial temperature ${T}_{0}=20{\hspace{0.17em}}^{\circ}\mathrm{C}$ were established (Table 2);
- (3)
- the effective values of: the specific heat ${c}_{l}^{(0)}$, density ${\rho}_{l}^{(0)}$, thermal diffusivity ${k}_{l}^{(0)}$, the effective depths of heat penetration ${a}_{l}$ and the dimensionless gradient parameters of materials ${\gamma}_{l}^{*}$, $l=1,2$ were found from Equations (3) and (5)–(7). Then, the dimensionless parameters ${K}_{\epsilon}$ and ${\gamma}_{\epsilon}$ were determined from the Formulas (16) and (17), and also the weight ${G}_{l}$ and heat partition ratios ${\alpha}_{l}$, $l=1,2$ were calculated from the Equations (38) and (39) (Table 3);
- (4)
- the volume temperature values ${\vartheta}_{1}^{(0)}=471.97$ of the disc and ${\vartheta}_{2}^{(0)}=260.92$ the pad were obtained from the Equations (36) and (37);
- (5)
- the values of materials properties ${K}_{l,m}^{({\vartheta}_{l}^{(0)})}$, ${c}_{l,m}^{({\vartheta}_{l}^{(0)})}$, ${\rho}_{l,m}^{({\vartheta}_{l}^{(0)})}$, $l,m=1,2$, corresponding to the volume temperature ${\vartheta}_{l}^{(0)}$ were determined from the Formulas (40)–(51);
- (6)
- the steps (3)–(5) were repeated resulting in the corrected values for the volume temperature ${\vartheta}_{l}^{(1)}=624.93$, and ${\vartheta}_{2}^{(1)}=292.98$;
- (7)
- by means of the formula ${\vartheta}_{l}=0.5({\vartheta}_{l}^{(0)}+{\vartheta}_{l}^{(1)})$, $l=1,2$ final values of the volume temperature ${\vartheta}_{1}=548.45{\hspace{0.17em}}^{\circ}\mathrm{C},$ and ${\vartheta}_{2}=267.95{\hspace{0.17em}}^{\circ}\mathrm{C}$ were found;
- (8)
- based on the dependencies (40)–(51) the values of materials properties ${K}_{l,m}^{({\vartheta}_{l})}$, ${c}_{l,m}^{({\vartheta}_{l})}$,${\rho}_{l,m}^{({\vartheta}_{l})}$, $l,m=1,2$ corresponding to the volume temperature ${\vartheta}_{l}$ were established (Table 4) and other parameters necessary to perform the calculations (Table 5);
- (9)
- the temperature field ${\Theta}^{*}(\zeta ,\tau )$ (23)–(26), the temperature evolution ${\Theta}^{*}(\tau )$ (27), and temporal profiles of heat fluxes intensities ${q}_{l}^{*}(\tau )$, $l=1,2$ (30)–(32) were determined.

_{2}O

_{3}-Cu element. This result is also confirmed by the parameter values ${a}_{l}$, $l=1,2$ presented in Table 3 and Table 5.

_{2}O

_{3}-Cu. The linear change in ${q}_{l}^{*}(\tau )$ is the result of the specific friction power ${q}^{*}(\tau )$ (8), which decreases linearly during braking with a constant deceleration, and the requirement to meet the boundary condition ${q}_{1}^{*}(\tau )+{q}_{2}^{*}(\tau )={q}^{*}(\tau )$, $0\le \tau \le {\tau}_{s}$. The influence of thermal sensitivity on the intensity of heat fluxes is much smaller than on the temperature. For thermally sensitive materials, the maximum values of the intensity of heat fluxes are ${q}_{1,\mathrm{max}}^{*}=0.864$ and ${q}_{2,\mathrm{max}}^{*}=0.136$, and for constant properties of the materials, we have ${q}_{1,\mathrm{max}}^{*}=0.895$ and ${q}_{2,\mathrm{max}}^{*}=0.105$.

## 6. Conclusions

- the influence of thermal sensitivity on the temperature of FGMs may be more significant than in the case of homogeneous materials;
- for the selected friction pair, taking into account the thermal sensitivity caused an almost threefold reduction in the maximum temperature in comparison to the appropriate temperature values, found with the same properties of the materials;
- the influence of thermal sensitivity on the intensity of heat fluxes directed from the friction surface to the interior of the friction pair elements is insignificant. This means that to estimate the amount of heat absorbed by the individual elements of the friction pair, appropriate solutions to linear problems can be used.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${a}_{l}$ | Effective depth of heat penetration ($\mathrm{m}$) |

${A}_{a}$ | Area of the nominal contact region (${\mathrm{m}}^{2}$) |

${c}_{l,m}$ | Specific heat capacity ($\mathrm{J}\hspace{0.17em}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$) |

$f$ | Coefficient of friction (dimensionless) |

${G}_{l}$ | Weight of the friction elements (kg) |

${J}_{k}(\cdot )$ | The Bessel functions of the first kind of the kth order |

${k}_{l,m}$ | Thermal diffusivity (${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$) |

${K}_{l,m}$ | Thermal conductivity ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$) |

$p$ | Contact pressure ($\mathrm{Pa}$) |

${p}_{0}$ | Nominal value of the contact pressure ($\mathrm{Pa}$) |

${R}_{e}$ | External radius of the pads (m) |

${R}_{i}$ | Internal radius of the pads (m) |

$q$ | Specific power of friction ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-2}$) |

q_{0} | Nominal value of the specific power of friction ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-2}$) |

${Q}_{0}$ | Nominal friction power ($\mathrm{W}$) |

$t$ | Time ($\mathrm{s}$) |

${t}_{s}$ | Stop time ($\mathrm{s}$) |

$T$ | Temperature (${}^{\circ}\mathrm{C}$) |

${T}_{0}$ | Initial temperature (${}^{\circ}\mathrm{C}$) |

$V$ | Velocity ($\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$) |

${V}_{c},\hspace{0.17em}{V}_{\rho}$ | Volume fractions of the material phases |

${V}_{0}$ | Initial velocity ($\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$) |

${W}_{0}$ | Initial kinetic energy of the system ($\mathrm{J}$) |

$z$ | Spatial coordinate in axial direction ($\mathrm{m}$) |

lower $l$ | Number of the main ($l=1$) and frictional ($l=2$) elements of the friction pair |

lower $m$ | Number of the component material $m=1,\hspace{0.17em}2$ of selected friction element |

${\alpha}_{l}^{}$ | Heat partition ratio (dimensionless) |

$\beta $ | Cover angle of the pads (rad) |

${\gamma}_{l}^{}$ | Parameter of material gradient (${\mathrm{m}}^{-1}$) |

${\gamma}_{l}^{*}$ | Parameter of material gradient (dimensionless) |

${\Theta}_{l}$ | Temperature rise (${}^{\circ}\mathrm{C}$) |

${\Theta}_{l}^{*}$ | Temperature rise (dimensionless) |

${\Theta}_{0}$ | Temperature scaling factor (${}^{\circ}\mathrm{C}$) |

${\rho}_{l,m}$ | Density ($\mathrm{kg}\hspace{0.17em}{\mathrm{m}}^{-3}$) |

$\tau $ | Time (dimensionless) |

${\tau}_{s}$ | Time of braking (dimensionless) |

$\zeta $ | Spatial coordinate in axial direction (dimensionless) |

${\vartheta}_{l}$ | Volume temperature (${}^{\circ}\mathrm{C}$) |

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**Figure 1.**Dependencies of the dimensionless thermal conductivities ${K}_{l,m}^{*}$ on temperature $T$.

**Figure 2.**Dependencies of the dimensionless specific heat capacities ${c}_{l,m}^{*}$ on temperature $T$.

**Figure 4.**Evolutions of the dimensionless temperature ${\Theta}^{*}(\zeta ,\tau )$ during braking at different distances $\zeta $ from the surface of friction with (solid lines) and without (dashed lines) taking into account the thermal sensitivity of the materials: (

**a**) Al

_{2}O

_{3}—Cu; (

**b**) ZrO

_{2}—Ti-6Al-4V.

**Figure 5.**Distribution of the dimensionless temperature ${\Theta}_{\mathrm{max}}^{*}(\zeta )={\Theta}^{*}(\zeta ,{\tau}_{\mathrm{max}})$ reached at the time moment $\tau ={\tau}_{\mathrm{max}}$ along the distance $\zeta $ from the surface of friction with (solid lines) and without (dashed lines) taking into account the thermal sensitivity of the materials.

**Figure 6.**Isotherms of the dimensionless temperature ${\Theta}^{*}(\zeta ,\tau )$ for: (

**a**) thermally sensitivity materials; (

**b**) materials with properties at the initial temperature.

**Figure 7.**Temporal profiles of the dimensionless heat fluxes ${q}_{l}^{*}(\tau )$, $l=1,2$ during braking with (continuous lines) and without (dashed lines) taking into account the thermal sensitivity of the materials. Dotted lines represent the dimensionless specific power of friction ${q}^{*}$.

Friction Coefficient $\mathit{f}$ | Nominal Pressure ${\mathit{p}}_{0},\hspace{0.17em}\mathbf{MPa}$ | Initial Sliding Speed ${\mathit{V}}_{0}^{},\hspace{0.17em}{\mathbf{ms}}^{-1}$ | Initial Kinetic Energy ${\mathit{W}}_{0}^{},\hspace{0.17em}\mathbf{kJ}$ | Outer Radius ${\mathit{R}}_{\mathit{e}}^{},\hspace{0.17em}\mathbf{mm}$ | Inner Radius ${\mathit{R}}_{\mathit{i}}^{},\hspace{0.17em}\mathbf{mm}$ | Initial Temperature ${\mathit{T}}_{0}^{},\hspace{0.17em}{}^{\xb0}\mathbf{C}$ |
---|---|---|---|---|---|---|

0.27 | 0.602 | 23.8 | 103.54 | 37.5 | 26.5 | 20 |

Element Index | Material Index | Material | Thermal Conductivity ${\mathit{K}}_{\mathit{l},\mathit{m}}^{(0)},\hspace{0.17em}\hspace{0.17em}\mathit{W}{\mathbf{m}}^{-1}{\mathbf{K}}^{-1}$ | Specific Heat Capacity ${\mathit{c}}_{\mathit{l},\mathit{m}}^{(0)},\hspace{0.17em}\mathbf{J}\hspace{0.17em}\mathbf{k}{\mathbf{g}}^{-1}{\mathbf{K}}^{-1}$ | Density ${\mathit{\rho}}_{\mathit{l},\mathit{m}}^{(0)},\hspace{0.17em}\hspace{0.17em}\mathbf{kg}{\mathbf{m}}^{-3}$ |
---|---|---|---|---|---|

l = 1 | m = 1 | Al_{2}O_{3} | 37.24 | 727.29 | 3990.92 |

m = 2 | Cu | 402.65 | 147.35 | 8947.92 | |

l = 2 | m = 1 | ZrO_{2} | 1.94 | 452.83 | 6102.16 |

m = 2 | Ti-6Al-4V | 6.87 | 538.08 | 4431.79 |

Element Index | l = 1 | l = 2 |
---|---|---|

${c}_{l}^{(0)},\hspace{0.17em}\mathrm{J}\hspace{0.17em}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ | 437.3 | 495.5 |

${\rho}_{l}^{(0)},\hspace{0.17em}{\mathrm{kgm}}^{-3}$ | 6469.4 | 5267 |

${k}_{l}^{(0)}\times \hspace{0.17em}{10}^{6},\hspace{0.17em}{\mathrm{m}}^{2}\hspace{0.17em}{\mathrm{s}}^{-1}$ | 13.2 | 0.743 |

${\gamma}_{l}^{*}$ | 2.381 | 1.266 |

${a}_{l}^{},\hspace{0.17em}\mathrm{mm}$ | 21.854 | 5.193 |

${G}_{l}^{},\hspace{0.17em}\mathrm{kg}$ | 0.3127 | 0.0605 |

${\alpha}_{l}^{}$ | 0.896 | 0.104 |

Element Index | Material Index | Material | Thermal Conductivity ${K}_{l,m}^{({\vartheta}_{l})},\hspace{0.17em}\hspace{0.17em}{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ | Specific Heat Capacity ${c}_{l,m}^{({\vartheta}_{l})},\hspace{0.17em}\mathrm{J}\hspace{0.17em}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ | Density ${\rho}_{l,m}^{({\vartheta}_{l})},\hspace{0.17em}\hspace{0.17em}{\mathrm{kgm}}^{-3}$ |
---|---|---|---|---|---|

l = 1 | m = 1 | Al_{2}O_{3} | 10.19 | 1097.93 | 3945.59 |

m = 2 | Cu | 367.15 | 401.89 | 8690.20 | |

l = 2 | m = 1 | ZrO_{2} | 1.99 | 552.67 | 6069.84 |

m = 2 | Ti-6Al-4V | 9.57 | 615.44 | 4399.06 |

Element Index | l = 1 | l = 2 |
---|---|---|

${c}_{l}^{({\vartheta}_{l})},\hspace{0.17em}\mathrm{J}\hspace{0.17em}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ | 749.7 | 584.9 |

${\rho}_{l}^{({\vartheta}_{l})},\hspace{0.17em}\hspace{0.17em}{\mathrm{kgm}}^{-3}$ | 6317.2 | 5233.8 |

${k}_{l}^{({\vartheta}_{l})}\times \hspace{0.17em}{10}^{6},\hspace{0.17em}{\mathrm{m}}^{2}\hspace{0.17em}{\mathrm{s}}^{-1}$ | 2.15 | 0.65 |

${\gamma}_{l}^{*}$ | 3.585 | 1.583 |

${a}_{l}^{},\hspace{0.17em}\mathrm{mm}$ | 8.834 | 4.854 |

${G}_{l}^{},\hspace{0.17em}\mathrm{kg}$ | 0.1234 | 0.0562 |

${\alpha}_{l}^{}$ | 0.863 | 0.137 |

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**MDPI and ACS Style**

Yevtushenko, A.; Topczewska, K.; Zamojski, P.
Influence of Thermal Sensitivity of Functionally Graded Materials on Temperature during Braking. *Materials* **2022**, *15*, 963.
https://doi.org/10.3390/ma15030963

**AMA Style**

Yevtushenko A, Topczewska K, Zamojski P.
Influence of Thermal Sensitivity of Functionally Graded Materials on Temperature during Braking. *Materials*. 2022; 15(3):963.
https://doi.org/10.3390/ma15030963

**Chicago/Turabian Style**

Yevtushenko, Aleksander, Katarzyna Topczewska, and Przemysław Zamojski.
2022. "Influence of Thermal Sensitivity of Functionally Graded Materials on Temperature during Braking" *Materials* 15, no. 3: 963.
https://doi.org/10.3390/ma15030963