# The Effect of Functionally Graded Materials on Temperature during Frictional Heating at Single Braking

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## Abstract

**:**

## 1. Introduction

## 2. Statement to the Problem

- The materials of the pads and the disc are functionally graded with an exponential decrease in thermal conductivity along their thickness, with invariant specific heat and density;
- The initial temperature of all elements is the same and equal to the ambient temperature ${T}_{a}$;
- The whole work of friction goes to heating the bodies, while the wear of the friction surfaces is neglected;
- The free surfaces of the pads and the disc are adiabatic;
- The thermal and mechanical properties and coefficient of friction are independent of the temperature $T$;
- Only the change in the temperature gradient in the direction perpendicular to the friction surface is taken into account;
- The thermal contact of friction between the pads and the disc is perfect; the temperatures of their friction surfaces during braking are the same, and the sum of the intensity of the heat fluxes directed to both elements along the normal to the contact surface is equal to the specific friction power:$$q(t)={q}_{0}{q}^{*}(t),\text{}{q}_{0}=f{p}_{0}{V}_{0},\text{}{q}^{*}(t)={p}^{*}(t){V}^{*}(t),\text{}0\le t\le {t}_{s},$$
- Due to the symmetry with respect to the center plane of the disc, to establish the temperature of the braking system, it is sufficient to consider the contact scheme of one pad with a disc of half of its thickness.

## 3. Solution to the Problem

## 4. Dimensionless Form of Solution

## 5. Numerical Analysis

## 6. Conclusions

## 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 (${\mathrm{m}}^{2}$) |

${c}_{l}$ | Specific heat ($\mathrm{J}\text{}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$) |

$f$ | Coefficient of friction (dimensionless) |

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

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

${K}_{l}$ | Thermal conductivity ($\mathrm{W}\text{}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$) |

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

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

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

${q}_{0}$ | Nominal value of the specific power of friction ($\mathrm{W}\text{}{\mathrm{m}}^{-2}$) |

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

${t}_{i}$ | Time of the contact pressure increase ($\mathrm{s}$) |

${t}_{s}^{0}$ | Stop time at braking with constant deceleration ($\mathrm{s}$) |

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

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

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

$V$ | Velocity ($\mathrm{m}\text{}{\mathrm{s}}^{-1}$) |

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

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

$x,\text{}y,\text{}z$ | Spatial coordinates ($\mathrm{m}$) |

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

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

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

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

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

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

${\rho}_{l}$ | Density ($\mathrm{kg}\text{}{\mathrm{m}}^{-3}$) |

$\tau $ | Time (dimensionless) |

${\tau}_{i}$ | Time of contact pressure increase (dimensionless) |

${\tau}_{s}^{0}$ | Braking time at constant deceleration (dimensionless) |

${\tau}_{s}$ | Braking time (dimensionless) |

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

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**Figure 2.**Isotherms of the temperature rise $\mathsf{\Theta}(z,t)$ in the disc and the pad at ${t}_{i}=0.5\text{}\mathrm{s}$.

**Figure 3.**Evolutions of the temperature rise $\mathsf{\Theta}(z,t)$ during braking at ${t}_{i}=0.5\text{}\mathrm{s}$ for different distances from the friction surface: (

**a**) the disc; (

**b**) the pad.

**Figure 4.**Evolutions of the temperature rise $\mathsf{\Theta}(0,t)$ during braking for different values of the time ${t}_{i}$ of contact pressure increase.

**Figure 5.**Dependence of the maximum temperature rise ${\mathsf{\Theta}}_{\mathrm{max}}$ on the time ${t}_{i}$ of contact pressure increase.

**Figure 6.**Dependence of the maximum temperature rise ${\mathsf{\Theta}}_{\mathrm{max}}$ at ${t}_{i}=0.5\text{}\mathrm{s}$ on the dimensionless gradient of material: (

**a**) ${\gamma}_{1}^{*}$ for ${\gamma}_{2}^{*}=4.05$; (

**b**) ${\gamma}_{2}^{*}$ for ${\gamma}_{1}^{*}=1.28$.

Element Subscript | Material | $\mathbf{Thermal}\text{}\mathbf{Conductivity}\text{}\mathit{K}\text{}[{\mathbf{Wm}}^{-1}{\mathbf{K}}^{-1}]$ | $\mathbf{Thermal}\text{}\mathbf{Diffusivity}\text{}\mathit{k}\text{}\times \text{}{10}^{6}\text{}[{\mathbf{m}}^{2}{\mathbf{s}}^{-1}]$ |
---|---|---|---|

$l=1$ | ZrO_{2} | 2.09 | 0.86 |

Ti-6Al-4V | 7.5 | 3.16 | |

$l=2$ | ceramic | 3 | 1.15 |

aluminum alloy | 173 | 67.16 |

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**MDPI and ACS Style**

Yevtushenko, A.; Topczewska, K.; Zamojski, P.
The Effect of Functionally Graded Materials on Temperature during Frictional Heating at Single Braking. *Materials* **2021**, *14*, 6241.
https://doi.org/10.3390/ma14216241

**AMA Style**

Yevtushenko A, Topczewska K, Zamojski P.
The Effect of Functionally Graded Materials on Temperature during Frictional Heating at Single Braking. *Materials*. 2021; 14(21):6241.
https://doi.org/10.3390/ma14216241

**Chicago/Turabian Style**

Yevtushenko, Aleksander, Katarzyna Topczewska, and Przemysław Zamojski.
2021. "The Effect of Functionally Graded Materials on Temperature during Frictional Heating at Single Braking" *Materials* 14, no. 21: 6241.
https://doi.org/10.3390/ma14216241