# Spatial Internal Material Load and Residual Stress Distribution Evolution in Synchrotron In Situ Investigations of Deep Rolling

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

^{2}around the contact point. Measurement positions were distributed in this area with a spatial resolution up to 50 µm. A “full-field” measurement consisted of complete, symmetrical scans to both sides of the contact point with up to 960 points, whereas for most measurements the symmetric properties of the strain fields were taken into account and only a “half-field” measurement was done on one-half of the strain field, consisting of 440 points. The experimental setup is shown in Figure 1b.

## 3. Theoretical Approach and Analysis

_{xx}= 0, ε

_{xz}= 0, and ε

_{xy}= 0. The approach is comparable to the orthogonal cutting setup and the strain state confirmed through FEM-simulation by Uhlmann et al. [5] for a similar transmission geometry. Thus, for further analysis the fundamental equation was reduced to Equation (1):

_{zz}(vertical z direction in the sample coordinate system), the longitudinal strain ε

_{yy}(horizontal y direction), and a shear strain component ε

_{yz}(yz direction). The strain values of the peaks were extracted with a pseudo-Voigt function. A mean error of ± 60 µm/m [µstrains] was achieved using the reduced equation at confidence interval of 95%.

## 4. Results and Discussion

#### 4.1. Stress Distribution during Deep Rolling

_{eq}(y,z) measured under load with 3000 N are shown in Figure 2. The result exhibits a clear mirror symmetry at the line below the contact point (y = 0). For a more in-depth discussion of the constituting stress fields in σ

_{yy}, σ

_{xx}, σ

_{yz}, and σ

_{zz}refer to the initial results from Meyer et al. [7].

_{zz}(y,z) stress component. The overall distribution is similar to the dynamic case with a moving contact point, as was measured in a previous study with this setup by Meyer et al. [7], but the residual stress generation and transition cannot be analyzed precisely without a second measurement after the contact.

_{eq}(y,z) during loading for the five different applied forces and with the constant contact geometry are displayed in Figure 3. This internal stress field is expected to be connected to the compressive residual stress state resulting after unloading. For increasing force, the intensity of the stress field and its distribution in depth increases, while the general shape and the local minimum under the contact point are more or less unchanged in the present range of tolerance for the determination of depth position. To compare the evolution of stress during loading and analyze the distribution characteristics, the depth profiles of σ

_{eq}(y = 0,z) have been extracted up to 4.15 mm.

_{max}was included in Equation (3). Since α, β, and λ are not connected to physical material or process parameters, z is to be considered dimensionless.

_{0}, as well as the width of the asymmetric function at a value of 500 MPa (W

_{500MPa}) were directly evaluated from the fit in order to determine field propagation in depth and analyze the influence of the force on these characteristic values. An example of the experimental stress distribution under load with 3000 N with the fitted function is shown in Figure 4. It can be observed that this model can achieve a very reliable fit of the stress distribution, while it should be noted that the distribution is tailored to the found stress distributions and not equally applicable to other line shapes without modification. The free fit parameters β, λ, and α are only slightly correlated to each other, which shows that the fitting function is not over parametrized, but it still allows fixing of the β-variable for one dataset during loading or after unloading, constraining the error for the extracted values.

#### 4.2. Stress State after Unloading

_{yy}, σ

_{xx}, and σ

_{zz}, but not the σ

_{yz}component, due to a geometrical shift of the center of gravity of the diffracting volume. These top layers, therefore, need to be neglected in the evaluation of the results. In Figure 6, a compressive stress of the σ

_{yy}component is found with a radial distribution around the contact point (about 2 mm around the center point), where maxima appear in the plastically displaced material to each side, whereas directly below the contact point, lower compressive stresses are generated and compensated with tensile stresses of around 30 MPa in depth.

_{yy}after unloading for the same five loading forces as given in Figure 4. The region with high compressive residual stress at the side of the contact point grows continuously with increasing applied force, as well as the level of compressive stresses.

_{yy}

_{,max}scales with the applied force, increasing from a maximum of −80(±25) MPa for 2000 N to −190(±30) MPa at 3900 N, an increase of −130(±30)% (Table 3). These values were calculated from the average of 5 points with the highest compressive stress in the σ

_{yy}(y,z) distribution.

#### 4.3. Comparison of Experimental Results with Theoretical Contact Mechanics Approach

_{yy}(z), σ

_{xx}(z), and σ

_{zz}(z) values were again combined into an equivalent stress. This evaluation leads to a comparable stress distribution as the measured internal material stresses and can also be described by a variation of the considered function, Equation (3). It can be observed that for every applied force, the calculated elastic stress distribution is much higher than the experimental data and the theoretical maximum equivalent stress value is higher by a factor ≈2.1 as compared with the experiment, whereas the depth position of maximum stress calculated from contact mechanics approach is closer to the surface, as shown in Figure 9.

_{(Mises,Yield)}= 2200 N results as a threshold value to achieve a plastic deformation.

_{(exp,Yield)}= 1600 ± 400 N. At a lower applied load, no measurable residual stresses should be generated. This value is lower than the calculated theoretical limit. It can be assumed, that for the elastic-plastic deformation in the experimental case, the convex indenter shape can concentrate a higher amount of loading stress which leads to higher deformation than in the elastic calculation, where both contact surfaces flatten out in response, and therefore lead to local modifications of the contact geometry and create plastic deformation directly at the contact surface.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Internal full stress fields below the contact point at 3000 N external force in equivalent σ

_{eq}(y,z) stress.

**Figure 4.**Fit for experimental data of equivalent stress distribution σ

_{eq}

_{,3000N}(0,z) in depth during loading.

**Figure 5.**(

**a**) Equivalent stress maximum over applied force F[N] and (

**b**) width of fit function over applied force F[N] for internal equivalent stresses during loading.

**Figure 6.**Internal full residual stress fields below the contact point after unloading of 3000 N external force in equivalent σ

_{yy}(y,z).

**Figure 7.**Longitudinal residual stress maps (half fields) for σ

_{yy}(y,z) after unloading with 5 forces.

**Figure 8.**Longitudinal residual stress maximum σ

_{max}

_{,yy}as a function of the applied force F[N].

**Figure 9.**Comparison of experimental data of equivalent stress distribution σ

_{eq}(y,z) in depth at 3000 N load with theoretical elastic contact equivalent stress distribution.

**Figure 10.**Comparison of (

**a**) experimental maximum stresses and (

**b**) graphical representation of σ

_{eq}

_{,max}(0,z) and σ

_{yy}

_{,max}(y,z) over the maximum theoretical equivalent stress σ

_{eq}

_{,theo,max}(0,z).

Steel | AISI | C [%] | Si [%] | Mn [%] | P [%] | S [%] | Cr [%] | Mo [%] | |
---|---|---|---|---|---|---|---|---|---|

42CrMo4 | 4140H | 0.43 | 0.26 | 0.74 | 0.01 | < 0.001 | 1.09 | 0.25 |

**Table 2.**Fit values of the internal stress distribution σ

_{eq}(0,z) for Equation (3) during loading.

Force | β/α/λ | ${\mathit{\sigma}}_{\mathit{e}\mathit{q}}^{\mathit{m}\mathit{a}\mathit{x}}$ | W_{500MPa} | z_{0} | ${\overline{\mathit{R}}}^{2}$ |
---|---|---|---|---|---|

[MPA] | [µm] | [µm] | |||

2000 N | 1.52/0.15/0.23 | 610 ± 50 | 450 ± 60 | 440 ± 100 | 0.936 |

2500 N | 1.45/0.15/0.25 | 680 ± 40 | 600 ± 40 | 430 ± 90 | 0.965 |

3000 N | 1.44/0.15/0.22 | 810 ± 30 | 740 ± 30 | 390 ± 80 | 0.990 |

3400 N | 1.31/0.15/0.28 | 870 ± 20 | 930 ± 20 | 400 ± 70 | 0.995 |

3900 N | 1.31/0.15/0.30 | 940 ± 20 | 1110 ± 30 | 430 ± 70 | 0.995 |

Force | 2000 N | 2500 N | 3000 N | 3400 N | 3900 N |
---|---|---|---|---|---|

σ_{yy,max}[MPa] | −80(±25) | −120(±25) | −105(±35) | −175(±30) | −190(±30) |

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**MDPI and ACS Style**

Meyer, H.; Epp, J.
Spatial Internal Material Load and Residual Stress Distribution Evolution in Synchrotron In Situ Investigations of Deep Rolling. *Quantum Beam Sci.* **2020**, *4*, 3.
https://doi.org/10.3390/qubs4010003

**AMA Style**

Meyer H, Epp J.
Spatial Internal Material Load and Residual Stress Distribution Evolution in Synchrotron In Situ Investigations of Deep Rolling. *Quantum Beam Science*. 2020; 4(1):3.
https://doi.org/10.3390/qubs4010003

**Chicago/Turabian Style**

Meyer, Heiner, and Jérémy Epp.
2020. "Spatial Internal Material Load and Residual Stress Distribution Evolution in Synchrotron In Situ Investigations of Deep Rolling" *Quantum Beam Science* 4, no. 1: 3.
https://doi.org/10.3390/qubs4010003