# Numerical Modeling of the Effect of Randomly Distributed Inclusions on Fretting Fatigue-Induced Stress in Metals

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}, MgO, Al

_{2}MgO

_{4}, CaSO

_{4,}TiB

_{2}, Al

_{3}Ti and refractory brick (Al, Si, O). Typical geometric form of inclusions are particles, films or group of films and rods [24]. As a result, stress concentration will appear at the interface between different materials or voids, which may give rise to shorter fatigue life. Thus, in addition to contact area, the subsurface area can also be affected by the heterogeneity, and in some cases it is strongly affected [25]. There are many ways to analyze the behavior of heterogeneous materials. Previous researchers have proposed several models to predict the effect of inclusions and defects on metal fatigue strength. Murakami and Endo [25] proposed an engineering guide to predict the fatigue strength of components with heterogeneity. Some experimental results about high-strength steels showed that fatigue failures were mostly caused by the inclusions inside the matrix [26,27,28,29,30,31].

## 2. Line Contact under Partial Slip

## 3. Finite Element Model and Validation

_{2}CuMg and Al

_{2}O

_{3}that are very commonly embedded in aluminum alloy 2024-T3 [55]. Their SEM observations showed that inclusions in material are highly discrete and randomly distributed. So we model the heterogeneity of materials by representative volume element method using DIGIMAT-FE which is a tool that considers the effects of microstructure on macroscopic material properties as shown in Figure 6. It should be noted that this study does not consider the heterogeneity of cylindrical pad material.

_{2}CuMg [60] and Al

_{2}O

_{3}[61] in this paper are given in Table 1. This article assumes that the cylindrical pad is a homogeneous aluminum alloy 2024-T3.

_{2}O

_{3}inclusions, 65 μm diameter, and 6% volume ratio has been studied first.

## 4. Numerical Results and Discussion

#### 4.1. Completely Randomly Distributed Inclusions

#### 4.1.1. Stress Peak and Its Location

#### 4.1.2. Stress Peak Location Characteristic

#### 4.2. Randomly Distributed and Manually Placed Inclusions

#### 4.2.1. Effect of Inclusions Type

_{2}CuMg has a more obvious effect than Al

_{2}O

_{3}. This means that in the heavy load condition, the presence of inclusions may accelerate the distortion of the contact edge of the fretting fatigue contact member. The effect of inclusion on the surface tensile stress is not obvious. However, the inclusion will affect the normal and shear stresses.

#### 4.2.2. Effect of Distance from Surface

#### 4.2.3. Effect of Inclusion Size

#### 4.2.4. Effect of Inclusion Shape

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgements

## Conflicts of Interest

## References

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**Figure 10.**(

**a**) Mesh sensitivity diagram (effect of mesh refinement on shear stress distribution on contact interface); (

**b**) shear stress distribution near contact interface for homogenous case with 2 μm contact element size.

**Figure 12.**Stress distribution of Case 3 below the contact surface, (

**a**) Mises stress, (

**b**) tensile stress, and (

**c**) shear stress.

**Figure 13.**(

**a**) Comparison of stress peaks between heterogeneous materials and homogenous materials, and (

**b**) their location of all cases.

**Figure 15.**Stress distribution of Case 5 below the contact surface, (

**a**) Mises stress, (

**b**) tensile stress, and (

**c**) shear stress.

**Figure 16.**(

**a**) Mises stress, (

**b**) tensile stress, (

**c**) normal stress, (

**d**) shear stress distribution on the contact surface for different inclusion materials cases.

**Figure 17.**(

**a**) Mises stress, (

**b**) tensile stress, (

**c**) normal stress, (

**d**) shear stress distribution on the contact surface for different distance cases.

**Figure 18.**(

**a**) Mises stress, (

**b**) tensile stress, (

**c**) normal stress, (

**d**) shear stress distribution on the contact surface for different inclusion size cases.

**Figure 19.**(

**a**) Mises stress, (

**b**) tensile stress, (

**c**) normal stress, (

**d**) shear stress distribution on the contact surface for different inclusion shape cases.

Material | Modulus (GPa) | Poisson’s Ratio |
---|---|---|

Aluminum alloy 2024-T3 | 72.1 | 0.33 |

${\mathsf{A}\mathrm{l}}_{2}\mathrm{CuMg}$ | 120.5 | 0.2 |

${\mathsf{A}\mathrm{l}}_{2}{\mathsf{O}}_{3}$ | 380 | 0.2 |

Number | Volume Ratio | Type | Size (µm) | Aspect Ratio | ${\mathit{E}}^{\ast}\left(\mathbf{GPa}\right)$ | ${\mathit{\mu}}^{\ast}$ |
---|---|---|---|---|---|---|

Case 1 | 4% | Al_{2}CuMg | 44 | 1 | 73.7005 | 0.32584 |

Case 2 | 2% | Al_{2}O_{3} | 44 | 1 | 74.269 | 0.32791 |

Case 3 | 4% | 76.615 | 0.32579 | |||

Case 4 | 6% | 78.2946 | 0.3243 | |||

Case 5 | 4% | Al_{2}O_{3} | 23 | 1 | 76.5524 | 0.32503 |

Case 6 | 65 | 76.5046 | 0.32537 | |||

Case 7 | 4% | Al_{2}O_{3} | 53.889 | 1.5 | 76.318 | 0.3256 |

Case 8 | 62.225 | 2 | 76.163 | 0.32255 | ||

Case 9 | 4% | Al_{2}O_{3} | 23 to 65 | 1 | 76.2028 | 0.32532 |

Case 10 | 4% | Void | 44 | 1 | 69.1 | 0.327 |

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**MDPI and ACS Style**

Deng, Q.; Bhatti, N.; Yin, X.; Abdel Wahab, M.
Numerical Modeling of the Effect of Randomly Distributed Inclusions on Fretting Fatigue-Induced Stress in Metals. *Metals* **2018**, *8*, 836.
https://doi.org/10.3390/met8100836

**AMA Style**

Deng Q, Bhatti N, Yin X, Abdel Wahab M.
Numerical Modeling of the Effect of Randomly Distributed Inclusions on Fretting Fatigue-Induced Stress in Metals. *Metals*. 2018; 8(10):836.
https://doi.org/10.3390/met8100836

**Chicago/Turabian Style**

Deng, Qingming, Nadeem Bhatti, Xiaochun Yin, and Magd Abdel Wahab.
2018. "Numerical Modeling of the Effect of Randomly Distributed Inclusions on Fretting Fatigue-Induced Stress in Metals" *Metals* 8, no. 10: 836.
https://doi.org/10.3390/met8100836