# On the Transient Effects at the Beginning of 3D Elastic-Plastic Rolling Contacts for a Circular Point Contact Considering Isotropic Hardening

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## Abstract

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## 1. Introduction

## 2. Method and Numerical Modeling

#### 2.1. Rolling Contact Simulation Using a Semi-Analytical Method

#### 2.2. Model Setup

## 3. Results and Discussion

#### 3.1. Change of the Pressure Distribution Due to a Change in Conformity

#### 3.2. Plastic Strains and Associated Plastic Deformations

#### 3.3. Development of the Strain Components

## 4. Conclusions

- The strain state at the very beginning of the rolling path is characterized by the vertical initial indentation. In contrast, during rolling, plastification occurs significantly at the leading edge due to the isotropic hardening behavior. The result is a different strain state in the steady-state regime. The transition between the two strain states takes place due to the decaying influence of the initial indentation as the distance from the start of rolling increases.
- The profile of the plastic deformation is only influenced to a minor extend by transient effects. The deep indentation at the beginning, as well as the shoulders at the beginning and end of the rolling path, are rather determined by the spacial distribution of the plastic strains, especially the shear strain ${\u03f5}_{xz}^{{\scriptscriptstyle p}}$ with a change of sign at the beginning of the rolling path.
- The history of the pressure distribution is mainly a result of the previously described shape of the plastic deformation of the surface, and therefore the conformity of the contact. Certainly, an increase in pressure is coupled with a change in stresses and strains, and thus in plastic deformation, but for the model considered here, these influences on the transient behavior seem to be very small.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a | contact radius given by Hertzian theory |

B, C, n | swift isotropic hardening law parameters |

E | Youngs’s modulus |

F | applied load |

h, ${h}_{0}$ | surface separation, initial gap |

i, j | tensor indices |

k, l | indices of the surface grid |

p | contact pressure |

${p}_{H}$ | maximum contact pressure given by Hertzian theory |

${p}_{max}$ | maximum contact pressure |

R | radius of the sphere |

s | deviatoric stress tensor |

u | total surface deformation |

${u}^{r}$ | plastic surface deformation |

x, y, z | space coordinates |

$\Gamma $, ${\Gamma}_{c}$ | computational domain, contact area |

$\Delta $ | mesh size |

$\delta $ | rigid body displacement |

${\u03f5}_{}^{{\scriptscriptstyle p}}$ | plastic strain tensor |

${\u03f5}_{\mathit{eff}}^{{\scriptscriptstyle p}}$ | effective plastic strain |

$\nu $ | Poisson’s ratio |

${\sigma}_{VM}$ | yield stress |

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**Figure 1.**Sectional view of the plastic surface deformation of an elastic–plastic plane that was rolled over by a rigid sphere. The red arrows indicate the loading, rolling and unloading directions and positions. The contact radius given by Hertzian theory is denoted by a.

**Figure 3.**Hardening curve of AISI 52100 bearing steel for the isotropic swift law, normalized by the maximum contact pressure ${p}_{H}$ = 4.39 GPa given by Hertzian theory.

**Figure 4.**Longitudinal profile of the plastic surface deformation ${u}^{r}$ in the x–z plane for the vertical indentation (dashed red line) and for a rolled length of 18a (solid black line).

**Figure 5.**(

**a**) History of the maximum pressure ${p}_{max}$ during rolling in the transient regime; (

**b**) pressure distributions for the initial indentation and for a rolled length of $0.6a$ and $1.3a$, centered at $x/a=0$.

**Figure 6.**Pressure distribution p (solid black line) and plastic surface deformation ${u}^{r}$ (dashed red line) for the moments of (

**a**) the initial indentation, (

**b**) a rolled length of $1.3a$ and (

**c**) a rolled length of $4a$.

**Figure 7.**Plastic strains ${\u03f5}_{ij}^{{\scriptscriptstyle p}}$ in the x–z plane at depth $z=0.48a$ for the initial indentation (dotted lines), a rolled length of $18a$ (solid lines) and a synthesized strain state (dashed lines, see end of Section 3.2). The plastic strains ${\u03f5}_{xy}^{{\scriptscriptstyle p}}$ and ${\u03f5}_{yz}^{{\scriptscriptstyle p}}$ are zero in the x–z plane and are not shown for a clear view.

**Figure 8.**Longditudinal profile of the plastic deformations of the surface ${u}_{ij}^{r}$ in the x–z plane for each plastic strain ${\u03f5}_{ij}^{{\scriptscriptstyle p}}$ for a rolled length of $18a$ (solid lines) and for the synthesized stress state (dashed line, see end of Section 3.2).

**Figure 9.**Schematic representation of the relationship between plastic strain ${\u03f5}_{xz}^{{\scriptscriptstyle p}}$ and the resulting plastic deformation of the surface ${u}_{xz}^{r}$.

**Figure 10.**Longitudinal profile of the plastic strain ${\u03f5}_{xz}^{{\scriptscriptstyle p}}$ and the corresponding plastic deformation of the surface ${u}_{xz}^{r}$ in the x–z plane for a rolled length of $18a$ (solid lines) and the synthesized strain state (dashed line, see end of Section 3.2). The gray arrow pairs schematically show the direction of plastic surface deformation for the locally present plastic strain.

**Figure 11.**Deviatoric stresses ${s}_{ij}$ in the x–z plane for a depth of $0.48a$ for the moment of the initial indentation (solid lines) and a rolled length of $4a$ (dashed lines). The contact centers are shifted to $x/a=0$.

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**MDPI and ACS Style**

Juettner, M.; Bartz, M.; Tremmel, S.; Wartzack, S. On the Transient Effects at the Beginning of 3D Elastic-Plastic Rolling Contacts for a Circular Point Contact Considering Isotropic Hardening. *Lubricants* **2022**, *10*, 47.
https://doi.org/10.3390/lubricants10030047

**AMA Style**

Juettner M, Bartz M, Tremmel S, Wartzack S. On the Transient Effects at the Beginning of 3D Elastic-Plastic Rolling Contacts for a Circular Point Contact Considering Isotropic Hardening. *Lubricants*. 2022; 10(3):47.
https://doi.org/10.3390/lubricants10030047

**Chicago/Turabian Style**

Juettner, Michael, Marcel Bartz, Stephan Tremmel, and Sandro Wartzack. 2022. "On the Transient Effects at the Beginning of 3D Elastic-Plastic Rolling Contacts for a Circular Point Contact Considering Isotropic Hardening" *Lubricants* 10, no. 3: 47.
https://doi.org/10.3390/lubricants10030047