# The Effect of Functionally Graded Materials on Temperature during Frictional Heating: Under Uniform Sliding

^{*}

## Abstract

**:**

_{2}–Ti-6Al-4V and Al

_{3}O

_{2}–TiC composites. The influence of the gradient parameters of both materials on the evolution and spatial distributions of the temperature were investigated. The temperatures of the elements made of FGMs were compared with the temperatures found for the homogeneous ceramic materials.

## 1. Introduction

## 2. Statement of the Problem

- The bodies are related to the coordinate Cartesian system $Oxyz$, and their initial temperature distribution is homogeneous and equal to the ambient temperature ${T}_{a}$;
- At the initial time moment $t=0$, the bodies are pressed to each other with uniform pressure ${p}_{0}$ acting parallel to the $z$ axis and simultaneously start sliding with constant relative speed ${V}_{0}$ in the positive direction of the $x$ axis;
- Due to friction, on the contact surface $z=0$ heat is generated, which is absorbed by the elements of friction pair in the form of heat fluxes, causing an increase in their temperature $T(z,t)$ over the initial value ${T}_{a}$;
- During frictional heating, the sum ${q}_{1}+{q}_{2}$ of intensities of heat fluxes directed from the contact surface $z=0$ along the normal to the insides of the bodies, is equal to the specific power of friction ${q}_{0}=f{p}_{0}{V}_{0}$, where $f$ is the coefficient of friction. At the same time, the temperatures $T$ on the friction surfaces of both bodies are equal [32,33];
- Changes in the temperature gradients in the directions $x$ and $y$ are negligible and the gradient in the direction $z$ decreases, along with the distance from the contact surface;
- Thermal conductivity of materials ${K}_{l}$ are exponential functions of variable $z$, and their specific heat ${c}_{l}$ and density ${\rho}_{l}$, $l=1,2$ are constant [34]. Here and further, the lower index $l=1$ indicates the parameters and quantities relating to the first element, and $l=2$ to the second element.

## 3. Solution to the Problem

## 4. An Asymptotic Solution at the Initial Stage of Sliding

## 5. Numerical Analysis

_{2}and Al

_{3}O

_{2}and, along the thickness of the elements, they approach the core materials Ti-6Al-4V and TiC, respectively. The thermal properties of component materials are presented in Table 1.

_{2}–Ti-6Al-4V (l = 1) and ${K}_{2,0}=1.5\text{}{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$, ${K}_{2,1}=33.9\text{}{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ for the FGM Al

_{3}O

_{2}–TiC (l = 2). Substituting these coefficients into the formula (69) we obtain the dimensionless gradient parameters values ${\gamma}_{1}^{\ast}=1.28$, ${\gamma}_{2}^{\ast}=3.12$. Distribution of the thermal conductivity along the distance from the friction surface, for considered tribosystem is presented in the Figure 2. The positive roots of the nonlinear functional Equation (45) were found by means of the bisection method [43]. It was necessary to take at least 70 roots of Equation (45) in order to perform calculations according to Equations (50)–(55) with a relative accuracy of ${10}^{-3}$.

_{2}–Ti-6Al-4V ($l=1$) and Al

_{3}O

_{2}–TiC ($l=2$) during the sliding, are shown by the continuous curves in Figure 3, while the dashed lines in this figure illustrate the corresponding results obtained from the solutions (65)–(67) for the friction pair elements made of homogeneous materials ZrO

_{2}($l=1$) and Al

_{3}O

_{2}($l=2$). At a certain distance $\zeta $, the temperature monotonically increases over time (Fourier number $\tau $). The highest temperature is achieved on the contact surface $\zeta =0$. It can be seen that the elements of tribocouple made of homogeneous materials are heated more intensively during the sliding than the FGMs. Differences between the compared results increase over the time of heating. Taking into consideration notations (49), it can be established that, the maximum temperature rises are ${\Theta}_{\mathrm{max}}={604\text{}}^{\circ}\mathrm{C}$ and ${\Theta}_{\mathrm{max}}={765\text{}}^{\circ}\mathrm{C}$ achieved at the end of the process, for the friction pairs made of functionally graded and homogeneous materials, respectively.

_{3}O

_{2}) as compared with the temperature achieved in the FGM element Al

_{3}O

_{2}–TiC, which is caused by application of the core material (TiC) with high thermal conductivity and diffusivity.

_{2}–Ti-6Al-4V, while the parameter ${\gamma}_{2}^{*}$ value of the element Al

_{3}O

_{2}–TiC remains constant (Figure 6a).

_{2}, while in the second element ($l=2$), the temperature of the homogeneous material Al

_{3}O

_{2}remains higher than the temperature of the element made of FGM Al

_{3}O

_{2}–TiC, throughout the whole effective thickness. At a certain distance from the friction surface, increasing the material gradient parameters (enhancement of the volume fraction of the core material in the composite structure) causes a drop of the temperature in both FGMs used.

## 6. Conclusions

_{2}and aluminum oxide Al

_{3}O

_{2}. The volume fraction of ceramics in the materials decreases with the depth, in favor of the core materials.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

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

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

$f$ | Coefficient of friction (dimensionless) |

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

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

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

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

$p$ | Parameter of the Laplace transform (dimensionless) |

${p}_{0}$ | Contact pressure ($\mathrm{Pa}$) |

${q}_{l}$ | Intensity of the frictional heat flux ($\mathrm{W}{\mathrm{m}}^{-2}$) |

${q}_{0}$ | Specific power of friction ($\mathrm{W}{\mathrm{m}}^{-2}$) |

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

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

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

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

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

## Glossary

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

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

$\zeta $ | Thickness (dimensionless) |

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

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

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

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

$\tau $ | Time (dimensionless) |

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

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**Figure 2.**Distributions of the dimensionless thermal conductivities ${K}_{l}^{*}$, of FGM ZrO

_{2}–Ti-6Al-4V (l = 1) and Al

_{3}O

_{2}–TiC (l = 2) along the dimensionless distance $\zeta $ from the friction surface.

**Figure 3.**Evolution of the dimensionless temperature ${\Theta}^{\ast}(\zeta ,\tau )$ during sliding at different distances from the friction surface. Continuous curves represent FGMs: (

**a**) ZrO

_{2}–Ti-6Al-4V; (

**b**) Al

_{3}O

_{2}–TiC and the dashed curves represent homogeneous materials: (

**a**) ZrO

_{2}; (

**b**) Al

_{3}O

_{2}.

**Figure 4.**Distribution of the dimensionless maximum temperature rise ${\Theta}_{\mathrm{max}}^{*}$ reached at the end of friction process, along the distance $\zeta $ from the friction surface. Continuous curves represent FGMs ZrO

_{2}–Ti-6Al-4V (l = 1) and Al

_{3}O

_{2}–TiC (l = 2); the dashed curves represent homogeneous materials ZrO

_{2}(l = 1) and Al

_{3}O

_{2}(l = 2).

**Figure 5.**Evolutions of dimensionless temperature rise ${\Theta}_{}^{*}$ on the contact surface of the friction pair for various values of parameter: (

**a**) ${\gamma}_{1}^{\ast}$ for ${\gamma}_{2}^{\ast}=3.12$; (

**b**) ${\gamma}_{2}^{\ast}$ for ${\gamma}_{1}^{\ast}=1.28$. Continuous curves represent FGMs ZrO

_{2}–Ti-6Al-4V (l = 1) and Al

_{3}O

_{2}–TiC (l = 2), the dashed curves represent homogeneous materials ZrO

_{2}(l = 1) and Al

_{3}O

_{2}(l = 2).

**Figure 6.**Dependencies of the maximum dimensionless temperature rise ${\Theta}_{\mathrm{max}}^{*}$ on the dimensionless gradient of material: (

**a**) ${\gamma}_{1}^{\ast}$ for ${\gamma}_{2}^{\ast}=3.12$; (

**b**) ${\gamma}_{2}^{\ast}$ for ${\gamma}_{1}^{\ast}=1.28$. Continuous curves represent FGMs ZrO

_{2}–Ti-6Al-4V (l = 1) and Al

_{3}O

_{2}–TiC (l = 2), the dashed curves represent homogeneous materials ZrO

_{2}(l = 1) and Al

_{3}O

_{2}(l = 2).

**Figure 7.**Dependencies of dimensionless temperature rise ${\Theta}^{\ast}$ at the end of heating, on the dimensionless distance $\left|\zeta \right|$ from the contact surface for different values of parameters: (

**a**) ${\gamma}_{1}^{\ast}$ for ${\gamma}_{2}^{\ast}=3.12$; (

**b**) ${\gamma}_{2}^{\ast}$ for ${\gamma}_{1}^{\ast}=1.28$. Continuous curves represent FGMs ZrO

_{2}–Ti-6Al-4V (l = 1) and Al

_{3}O

_{2}–TiC (l = 2) and the dashed curves represent homogeneous materials ZrO

_{2}(l = 1) and Al

_{3}O

_{2}(l = 2).

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

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

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

$l=2$ | Al_{3}O_{2} | 1.5 | 4.98 |

TiC | 33.9 | 9.59 |

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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: Under Uniform Sliding. *Materials* **2021**, *14*, 4285.
https://doi.org/10.3390/ma14154285

**AMA Style**

Yevtushenko A, Topczewska K, Zamojski P.
The Effect of Functionally Graded Materials on Temperature during Frictional Heating: Under Uniform Sliding. *Materials*. 2021; 14(15):4285.
https://doi.org/10.3390/ma14154285

**Chicago/Turabian Style**

Yevtushenko, Aleksander, Katarzyna Topczewska, and Przemysław Zamojski.
2021. "The Effect of Functionally Graded Materials on Temperature during Frictional Heating: Under Uniform Sliding" *Materials* 14, no. 15: 4285.
https://doi.org/10.3390/ma14154285