# Use of Functionally Graded Material to Decrease Maximum Temperature of a Coating–Substrate System

^{*}

## Abstract

**:**

_{2}—Ti-6Al-4V, applied on a cast iron substrate. In order to explain the effect of FGM on temperature, corresponding analysis was carried out for the coating made of a homogeneous (ZrO

_{2}) material.

## 1. Introduction

_{2}O), aluminum oxide (Al

_{2}O

_{3}), silicon carbide (SiC), tungsten carbide (WC), and yttria-stabilized zirconia (YSZ) are employed for coatings of the friction elements in braking systems [3,4,5,6,7,8]. Ceramic intrinsically possess low fracture toughness, so TBCs are susceptible to mechanical or thermal stress-induced failures such as delamination or spallation of the top coating under harsh thermal conditions [9,10,11]. Especially, cracks of the brittle ceramic coating may be initiated in the friction elements of braking systems, when the concentration of subsurface tensile stresses appear due to the frictional heating [12]. Many properties of ceramic TBC are very different from that of metallic substrate, so their direct application may lead to cracking due to the thermal expansion mismatch at the interface of the core and the outer layer [13]. The interface of the coating-substrate system has been proved to be the most critical location for failure [1].

## 2. Statement to the Problem

## 3. Solution to the Problem

## 4. Verification of the Solution

## 5. Heating the Coating Surface by a Heat Flux with Linearly Decreasing Intensity in Time

## 6. Numerical Analysis

_{2}that smoothly transforms into a titanium alloy Ti-6Al-4V in the structure of the coating material. The element substrate is made of cast iron ChNMKh. Essential for calculations, properties of these materials at the ambient temperature ${T}_{0}=20{}^{\circ}\mathrm{C}$ are included in Table 1.

_{2}, fulfills well the assumed a priori role of dissipating heat from the heated surface of the FGC and thus lowering its temperature. It should be noted that the described effect of lowering the temperature as a result of using FGM is maintained inside the layer ($0\le \zeta <1$). However, starting from the interface and further into the substrate ($\zeta \ge 1$), we observe the opposite behavior of the temperature evolution—the temperature of the substrate with the FGC is higher than when using a homogeneous coating (Figure 3b). This is due to the significantly higher thermal conductivity of cast iron compared to the titanium alloy (Table 1). The temperature of the substrate is much (more than an order of magnitude) lower than the temperature of the coating.

## 7. Conclusions

- Deposition of functionally graded coating on the homogeneous substrate allow to effectively lower the temperature on the heated surface;
- FGC is the main adsorbent of frictional heat generated. As a result, values of temperature achieved in the substrate are much lower than that obtained in the coating temperature level;
- The temporal profile of the heat flux intensity has a noticeable impact on the spatial-temporal distribution of isotherms only in the coating;
- Gradient parameter of the FGC has a crucial influence on the maximum temperature for the selected coating–substrate system;
- Obtained asymptotic solutions are useful for the express estimation of the temperature of the FGC-substrate system at small and large values of the Fourier number;
- The proposed mathematical model can be utilized as an effective tool for simulating the temperature mode of homogeneous bodies with functionally graded coating.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

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

$d$ | Thickness of FGC ($\mathrm{m}$) |

${\mathrm{I}}_{\mathrm{n}}(\cdot )$ | Modified Bessel functions of the nth order of the first kind |

${\mathrm{J}}_{\mathrm{n}}(\cdot )$ | Bessel functions of the nth order of the first kind |

${\mathrm{K}}_{\mathrm{n}}(\cdot )$ | Modified Bessel functions of the nth order of the second kind |

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

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

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

${q}_{0}$ | Nominal value of the heat flux intensity ($\mathrm{W}{\mathrm{m}}^{-2}$) |

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

${t}_{s}$ | Final moment of the heating process ($\mathrm{s}$) |

$T$ | Temperature (°C) |

${T}_{0}$ | Initial temperature (°C) |

$\mathrm{v}$ | Volume fraction of the material phases (dimensionless) |

${\mathrm{Y}}_{\mathrm{n}}(\cdot )$ | Bessel functions of the nth order of the second kind |

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

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

${\gamma}^{\ast}$ | Dimensionless parameter of FGM gradient |

$\mathsf{\Lambda}$ | Temperature rise scaling factor (°C) |

$\epsilon $ | Dimensionless coefficient of thermal activity of friction couple |

$\mathsf{\Theta}$ | Temperature rise (°C) |

${\mathsf{\Theta}}^{\ast}$ | Dimensionless temperature rise |

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

$\tau $ | Dimensionless time |

${\tau}_{s}$ | Dimensionless final time of heating |

$\zeta $ | Dimensionless spatial coordinate in axial direction |

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**Figure 3.**Evolutions of dimensionless temperature rise ${\mathsf{\Theta}}^{\ast}(\zeta ,\tau )$ for selected values of dimensionless spatial variable $\zeta $ in: (

**a**) coating, (

**b**) substrate. Solid lines—FGC, dashed lines—homogeneous ZrO

_{2}coating.

**Figure 4.**Comparison of time profiles of dimensionless temperature rise ${\mathsf{\Theta}}^{\ast}(\zeta ,\tau )$ in the FGC, obtained using the exact (68), (69) (solid lines) and asymptotic (dashed lines) solutions: (

**a**) for small (76), (77), (

**b**) for large (80), (81) Fourier number $\tau $, for selected values of dimensionless spatial variable $\zeta $.

**Figure 5.**Effect of FGC gradient ${\gamma}^{\ast}$ on dimensionless temperature rise: (

**a**) evolution ${\mathsf{\Theta}}^{\ast}(0,\tau )$ (68), (69) for selected values of ${\gamma}^{\ast}$, (

**b**) dependency ${\mathsf{\Theta}}_{\mathrm{max}}^{\ast}\equiv {\mathsf{\Theta}}^{\ast}(0,0.5)$ on ${\gamma}^{\ast}$.

**Figure 6.**Evolution of dimensionless temperature rise ${\widehat{\mathsf{\Theta}}}^{\ast}(\zeta ,\tau )$ (113), (114) for selected dimensionless values of spatial variable $\zeta $ in: (

**a**) coating, (

**b**) substrate. Solid lines—FGC, dashed lines—homogeneous coating made of ZrO

_{2}.

**Figure 7.**Dimensionless isolines of the temperature rises: (

**a**) ${\mathsf{\Theta}}^{\ast}(\zeta ,\tau )$ (68), (69), (

**b**) ${\widehat{\mathsf{\Theta}}}^{\ast}(\zeta ,\tau )$(113), (114). Solid lines—FGC, dashed lines—homogeneous coating made of ZrO

_{2}.

Material | Thermal Conductivity ${\mathbf{Wm}}^{-1}{\mathbf{K}}^{-1}$ | Specific Heat Capacity $\mathbf{J}{\mathbf{kg}}^{-1}{\mathbf{K}}^{-1}$ | Density $\mathbf{kg}{\mathbf{m}}^{-3}$ |
---|---|---|---|

ZrO_{2} | ${K}_{1,1}=1.94$ | ${c}_{1,1}=452.83$ | ${\rho}_{1,1}=6102.16$ |

Ti-6Al-4V | ${K}_{1,2}=6.87$ | ${c}_{1,2}=538.08$ | ${\rho}_{1,2}=4431.79$ |

ChNMKh | ${K}_{2}=52.17$ | ${c}_{2}=444.6$ | ${\rho}_{2}=7100$ |

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**MDPI and ACS Style**

Yevtushenko, A.; Topczewska, K.; Zamojski, P.
Use of Functionally Graded Material to Decrease Maximum Temperature of a Coating–Substrate System. *Materials* **2023**, *16*, 2265.
https://doi.org/10.3390/ma16062265

**AMA Style**

Yevtushenko A, Topczewska K, Zamojski P.
Use of Functionally Graded Material to Decrease Maximum Temperature of a Coating–Substrate System. *Materials*. 2023; 16(6):2265.
https://doi.org/10.3390/ma16062265

**Chicago/Turabian Style**

Yevtushenko, Aleksander, Katarzyna Topczewska, and Przemysław Zamojski.
2023. "Use of Functionally Graded Material to Decrease Maximum Temperature of a Coating–Substrate System" *Materials* 16, no. 6: 2265.
https://doi.org/10.3390/ma16062265