# Steady-State Thermal Analysis of Functionally Graded Rotating Disks Using Finite Element and Analytical Methods

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## Abstract

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## 1. Introduction

## 2. Material Gradation Relation

## 3. Methodology and Problem Formulation

- In-house finite element program;
- APDL;
- Analytical solution.

#### 3.1. Method #1: In-House FE Program

#### 3.2. Method #2: APDL

#### 3.3. Method #3: Analytical Solution

## 4. Thermal Stress and Strain Relations

## 5. Numerical Results and Discussion

#### 5.1. Material Property Variations in an FG Disk

#### 5.2. Numerical Results of the Three Methods

#### 5.3. Material Gradation Parametric Study

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison between a coated material and FGM used to resist thermal loads [5].

**Figure 4.**Variations in material properties in FG rotating disks along the radius: (

**a**) thermal conductivity; (

**b**) thermal expansion; (

**c**) Young’s modulus.

**Figure 5.**Temperature distribution using three methods for ${\mathrm{m}}_{\mathrm{YM}}=0.3539$, ${\mathrm{m}}_{\mathrm{TE}}=-0.23$, and ${\mathrm{m}}_{\mathrm{TC}}=-1.3575$.

**Figure 6.**Thermal strain of the FG disk versus the non-dimensional radius for ${\mathrm{m}}_{\mathrm{YM}}=0.3539$, ${\mathrm{m}}_{\mathrm{TE}}=-0.23$, and ${\mathrm{m}}_{\mathrm{TC}}=-1.3575$.

**Figure 7.**Thermal stress of the FG disk versus the non-dimensional radius for ${\mathrm{m}}_{\mathrm{YM}}=0.3539$, ${\mathrm{m}}_{\mathrm{TE}}=-0.23$, and ${\mathrm{m}}_{\mathrm{TC}}=-1.3575$.

**Figure 8.**Effect of the ${\mathrm{m}}_{\mathrm{TC}}$ on the temperature distribution in FGMs for ${\mathrm{m}}_{\mathrm{YM}}=0.3539$ and ${\mathrm{m}}_{\mathrm{TE}}=-0.23$.

**Figure 9.**Effect of the ${\mathrm{m}}_{\mathrm{TE}}$ on the thermal strain distribution for ${\mathrm{m}}_{\mathrm{YM}}=0.3539$ and ${\mathrm{m}}_{\mathrm{TC}}=-1.3575$.

**Figure 10.**Effect of the ${\mathrm{m}}_{\mathrm{YM}}$ on the thermal stress distribution for ${\mathrm{m}}_{\mathrm{TE}}=-0.23$ and ${\mathrm{m}}_{\mathrm{TC}}=-1.3575$.

**Table 1.**Material properties of the inner and outer surfaces [8].

Material Properties | $\mathbf{E}\left(\mathbf{G}\mathbf{P}\mathbf{a}\right)$ | $\mathbf{\upsilon}$ | $\mathbf{\alpha}\left(\frac{1}{\mathbf{K}}\right)$ | $\mathbf{K}\left(\frac{\mathbf{W}}{\mathbf{m}\mathbf{K}}\right)$ |
---|---|---|---|---|

Ceramic (outer surface) | $117.0$ | $0.3$ | $7.11\times {10}^{-6}$ | $2.036$ |

Metal (inner surface) | $66.2$ | $0.3$ | $10.3\times {10}^{-6}$ | $18.1$ |

${\mathbf{K}}_{\u2022}$ | ${\mathbf{m}}_{\mathbf{T}\mathbf{C}}$ | ${\mathbf{\alpha}}_{\u2022}$ | ${\mathbf{m}}_{\mathbf{T}\mathbf{E}}$ | ${\mathbf{E}}_{\u2022}$ | ${\mathbf{m}}_{\mathbf{Y}\mathbf{M}}$ |
---|---|---|---|---|---|

$2.036$ | $-1.3575$ | $7.11\times {10}^{-6}$ | $-0.23$ | $117$ | $0.3539$ |

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**MDPI and ACS Style**

Shahzamanian, M.M.; Shahrjerdi, A.; Sahari, B.B.; Wu, P.D.
Steady-State Thermal Analysis of Functionally Graded Rotating Disks Using Finite Element and Analytical Methods. *Materials* **2022**, *15*, 5548.
https://doi.org/10.3390/ma15165548

**AMA Style**

Shahzamanian MM, Shahrjerdi A, Sahari BB, Wu PD.
Steady-State Thermal Analysis of Functionally Graded Rotating Disks Using Finite Element and Analytical Methods. *Materials*. 2022; 15(16):5548.
https://doi.org/10.3390/ma15165548

**Chicago/Turabian Style**

Shahzamanian, M. M., A. Shahrjerdi, B. B. Sahari, and P. D. Wu.
2022. "Steady-State Thermal Analysis of Functionally Graded Rotating Disks Using Finite Element and Analytical Methods" *Materials* 15, no. 16: 5548.
https://doi.org/10.3390/ma15165548