# Improving the Inner Surface State of Thick-Walled Tubes by Heat Treatments with Internal Quenching Considering a Simulation Based Optimization

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. State of the Art

#### 1.2. Internal Quenching

#### 1.3. This Work

## 2. Materials and Methods

#### 2.1. Samples

#### 2.2. Internal Quenching Heat Treatment

#### 2.3. Metallographic Characterization

#### 2.4. Residual Stress Measurement

#### 2.4.1. X-ray Diffraction Analysis

#### 2.4.2. Sachs Boring/EDM

## 3. FE-Simulation

#### 3.1. Model

#### 3.2. Material Parameters

## 4. Simulative Optimization Study

#### 4.1. Parameter Field for AISI 4140

#### 4.2. Parameter Field for AISI 1045

#### 4.3. AISI 4140, Discontinuous Cooling

#### 4.4. AISI 4140, Continuous Cooling

#### 4.5. AISI 1045, Continuous Cooling

## 5. Results and Discussion of Experiments and Validation

#### 5.1. Microstructure and Hardness

#### 5.2. Residual Stresses

#### 5.2.1. X-ray Diffraction Analysis

#### 5.2.2. Sachs EDM Measurements

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Input Parameters

Quantity | Unit | Phase | $\mathit{i}=0$ | $\mathit{i}=1$ | $\mathit{i}=2$ | $\mathit{i}=3$ | Source |
---|---|---|---|---|---|---|---|

$\lambda $ | $\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$ | $\gamma $ | $21.1$ | $-3.14\times {10}^{-2}$ | $2.87\times {10}^{-5}$ | 0 | [32] |

$\lambda $ | $\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$ | ${\alpha}_{F/P}$, ${\alpha}_{B}$, ${\alpha}_{M}$ | $44.23$ | $-5.97\times {10}^{-4}$ | $-1.79\times {10}^{-5}$ | 0 | [32] |

${c}_{p}$ | $\mathrm{J}/\left(\mathrm{k}\mathrm{g}\mathrm{K}\right)$ | $\gamma $ | 476 | $0.103$ | 0 | 0 | [32] |

${c}_{p}$ | $\mathrm{J}/\left(\mathrm{k}\mathrm{g}\mathrm{K}\right)$ | ${\alpha}_{F/P}$, ${\alpha}_{B}$, ${\alpha}_{M}$ | 233 | $0.586$ | 0 | 0 | [32] |

$\Delta {Q}_{\alpha \to \gamma}$ | $\mathrm{J}/\mathrm{k}\mathrm{g}$ | - | $8.3\times {10}^{4}$ | 0 | 0 | 0 | this work |

$\Delta {Q}_{\gamma \to \alpha}$ | $\mathrm{J}/\mathrm{k}\mathrm{g}$ | - | $-8.3\times {10}^{4}$ | 0 | 0 | 0 | this work |

E | $\mathrm{GPa}$ | $\gamma $ | $283.0$ | $-0.211$ | 0 | 0 | [33] |

E | $\mathrm{GPa}$ | ${\alpha}_{F/P}$, ${\alpha}_{B}$, ${\alpha}_{M}$ | $284.5$ | $-0.456$ | $7.29\times {10}^{-4}$ | $-4.43\times {10}^{-7}$ | [20] |

$\alpha $ | ${\mathrm{K}}^{-1}$ | $\gamma $ | $2.3\times {10}^{-5}$ | 0 | 0 | 0 | [32] |

$\alpha $ | ${\mathrm{K}}^{-1}$ | ${\alpha}_{B}$ | $7.9\times {10}^{-6}$ | $1.25\times {10}^{-8}$ | 0 | 0 | this work |

$\alpha $ | ${\mathrm{K}}^{-1}$ | ${\alpha}_{M}$ | $8.9\times {10}^{-6}$ | $1.09\times {10}^{-8}$ | 0 | 0 | this work |

${K}_{\gamma \to {\alpha}_{B}}$ | $\mathrm{MP}{\mathrm{a}}^{-1}$ | - | $7.4\times {10}^{-4}$ | 0 | 0 | 0 | this work |

${K}_{\gamma \to {\alpha}_{M}}$ | $\mathrm{MP}{\mathrm{a}}^{-1}$ | - | $6.04\times {10}^{-5}$ | 0 | 0 | 0 | this work |

$\begin{array}{c}\kappa \\ T\le 490{}^{\circ}\mathrm{C}\end{array}$ | - | - | $-0.01845$ | $3.29\times {10}^{-5}$ | - | - | this work |

$\begin{array}{c}\kappa \\ 490{}^{\circ}\mathrm{C}\le T\le 535{}^{\circ}\mathrm{C}\end{array}$ | - | - | $0.028$ | $-2.94\times {10}^{-5}$ | - | - | this work |

$\lambda $ | - | - | $12.44$ | 0 | 0 | 0 | this work |

$\Delta {\epsilon}_{\alpha \to \gamma}$ | - | - | $-0.092$ | 0 | 0 | 0 | this work |

$\Delta {\epsilon}_{\gamma \to {\alpha}_{B}}$ | - | - | $0.0076$ | 0 | 0 | 0 | this work |

$\Delta {\epsilon}_{\gamma \to {\alpha}_{M}}$ | - | - | $0.0093$ | 0 | 0 | 0 | this work |

${\sigma}_{y,\gamma}$ | $\mathrm{MPa}$ | $\gamma $ | $427.2$ | $-0.405$ | $7.60\times {10}^{-5}$ | 0 | [32] |

${R}_{0,\gamma}$ | $\mathrm{MPa}$ | $\gamma $ | $-351.1$ | $7.51$ | $-0.0062$ | 0 | [32] |

${R}_{1,\gamma}$ | $\mathrm{MPa}$ | $\gamma $ | $32.3$ | $-9.66\times {10}^{-4}$ | 0 | 0 | [32] |

${e}_{\gamma}$ | - | $\gamma $ | $410.5$ | $-0.277$ | 0 | 0 | [32] |

${\sigma}_{y,{\alpha}_{B}}$ | $\mathrm{MPa}$ | ${\alpha}_{B}$ | $1907.83$ | $-6.76$ | $0.014$ | $-9.82\times {10}^{-6}$ | [21] |

${R}_{0,{\alpha}_{B}}$ | $\mathrm{MPa}$ | ${\alpha}_{B}$ | $2000.0$ | 0 | 0 | 0 | [21] |

${R}_{1,{\alpha}_{B}}$ | $\mathrm{MPa}$ | ${\alpha}_{B}$ | 300 | 0 | 0 | 0 | [21] |

${e}_{{\alpha}_{B}}$ | - | ${\alpha}_{B}$ | 250 | 0 | 0 | 0 | [21] |

${\sigma}_{y,{\alpha}_{M}}$ | $\mathrm{MPa}$ | ${\alpha}_{M}$ | $1650.0$ | $-0.77$ | 0 | 0 | [21] |

${R}_{0,{\alpha}_{M}}$ | $\mathrm{MPa}$ | ${\alpha}_{M}$ | 0 | 0 | 0 | 0 | [21] |

${R}_{1,{\alpha}_{M}}$ | $\mathrm{MPa}$ | ${\alpha}_{M}$ | 0 | 0 | 0 | 0 | [21] |

${e}_{{\alpha}_{M}}$ | - | ${\alpha}_{M}$ | 0 | 0 | 0 | 0 | [21] |

Quantity | Unit | Phase | $\mathit{i}=0$ | $\mathit{i}=1$ | $\mathit{i}=2$ | $\mathit{i}=3$ | Source |
---|---|---|---|---|---|---|---|

$\lambda $ | $\mathrm{W}/\left(\mathrm{mK}\right)$ | $\gamma $ | $11.68$ | $0.01155$ | 0 | 0 | [22] |

$\lambda $ | $\mathrm{W}/\left(\mathrm{mK}\right)$ | ${\alpha}_{F/P}$, ${\alpha}_{B}$, ${\alpha}_{M}$ | $49.17$ | $-0.21$ | 0 | 0 | [22] |

${c}_{p}$ | $\mathrm{J}/\left(\mathrm{k}\mathrm{g}\mathrm{K}\right)$ | $\gamma $ | 491 | $0.115$ | 0 | 0 | [22] |

${c}_{p}$ | $\mathrm{J}/\left(\mathrm{k}\mathrm{g}\mathrm{K}\right)$ | ${\alpha}_{F/P}$, ${\alpha}_{B}$, ${\alpha}_{M}$ | $401.95$ | $0.285$ | 0 | 0 | [22] |

$\Delta {Q}_{\alpha \to \gamma}$ | $\mathrm{J}/\mathrm{k}\mathrm{g}$ | - | $8\times {10}^{4}$ | 0 | 0 | 0 | [22] |

$\Delta {Q}_{\gamma \to \alpha}$ | $\mathrm{J}/\mathrm{k}\mathrm{g}$ | - | $-8\times {10}^{4}$ | 0 | 0 | 0 | [22] |

E | $\mathrm{GPa}$ | $\gamma $ | $283.0$ | $-0.211$ | 0 | 0 | [33] |

E | $\mathrm{GPa}$ | ${\alpha}_{F/P}$, ${\alpha}_{B}$, ${\alpha}_{M}$ | $284.5$ | $-0.456$ | $7.29\times {10}^{-4}$ | $-4.43\times {10}^{-7}$ | [20] |

$\alpha $ | ${\mathrm{K}}^{-1}$ | $\gamma $ | $2.1\times {10}^{-5}$ | 0 | 0 | 0 | this work |

$\alpha $ | ${\mathrm{K}}^{-1}$ | ${\alpha}_{B}$ | $14.73\times {10}^{-6}$ | $1.25\times {10}^{-8}$ | 0 | 0 | this work |

$\alpha $ | ${\mathrm{K}}^{-1}$ | ${\alpha}_{M}$ | $6.2\times {10}^{-6}$ | 0 | 0 | 0 | this work |

${K}_{\gamma \to {\alpha}_{F/P}}$ | $\mathrm{MP}{\mathrm{a}}^{-1}$ | - | $5.2\times {10}^{-5}$ | 0 | 0 | 0 | this work |

${K}_{\gamma \to {\alpha}_{B}}$ | $\mathrm{MP}{\mathrm{a}}^{-1}$ | - | $5.2\times {10}^{-5}$ | 0 | 0 | 0 | this work |

${K}_{\gamma \to {\alpha}_{M}}$ | $\mathrm{MP}{\mathrm{a}}^{-1}$ | - | $5.2\times {10}^{-5}$ | 0 | 0 | 0 | this work |

$\begin{array}{c}{n}_{\gamma \to {\alpha}_{F/P}}\\ T\ge 650{}^{\circ}\mathrm{C}\end{array}$ | - | - | $3.6$ | $-0.004$ | 0 | 0 | this work |

$\begin{array}{c}{n}_{\gamma \to {\alpha}_{F/P}}\\ 580{}^{\circ}\mathrm{C}\le T\le 650{}^{\circ}\mathrm{C}\end{array}$ | - | - | $8445.5$ | $-42.409$ | $0.071\times {10}^{-5}$ | $-3.95\times {10}^{-5}$ | this work |

$\begin{array}{c}ln{b}_{\gamma \to {\alpha}_{F/P}}\\ T\ge 650{}^{\circ}\mathrm{C}\end{array}$ | - | - | $-5.76$ | $0.0058$ | 0 | 0 | this work |

$\begin{array}{c}ln{b}_{\gamma \to {\alpha}_{F/P}}\\ 580{}^{\circ}\mathrm{C}\le T\le 650{}^{\circ}\mathrm{C}\end{array}$ | - | - | $-20756.967$ | $102.60798$ | $-0.1689$ | $9.26\times {10}^{-5}$ | this work |

${n}_{\gamma \to {\alpha}_{B}}$ | - | - | $13.33$ | $-0.048$ | $4.91\times {10}^{-5}$ | 0 | this work |

$ln{b}_{\gamma \to {\alpha}_{B}}$ | - | - | $-56.34$ | $0.203$ | $-1.93\times {10}^{-4}$ | 0 | this work |

$\Delta {\epsilon}_{\gamma \to {\alpha}_{F/P}}$ | - | - | $0.0069$ | 0 | 0 | 0 | this work |

$\Delta {\epsilon}_{\gamma \to {\alpha}_{B}}$ | - | - | $0.00851$ | 0 | 0 | 0 | this work |

$\Delta {\epsilon}_{\gamma \to {\alpha}_{M}}$ | - | - | $0.0093$ | 0 | 0 | 0 | this work |

${\sigma}_{y,\gamma}$ | $\mathrm{MPa}$ | $\gamma $ | $174.91$ | $-0.152$ | 0 | 0 | [22] |

${R}_{0,\gamma}$ | $\mathrm{MPa}$ | $\gamma $ | $-351.1$ | $7.51$ | $-0.0062$ | 0 | [21] |

${R}_{1,\gamma}$ | $\mathrm{MPa}$ | $\gamma $ | $32.3$ | $-9.66\times {10}^{-4}$ | 0 | 0 | [21] |

${e}_{\gamma}$ | - | $\gamma $ | $410.5$ | $-0.277$ | 0 | 0 | [21] |

${\sigma}_{y,{\alpha}_{F/P}}$ | $\mathrm{MPa}$ | ${\alpha}_{F/P}$ | $392.18$ | $-0.33$ | 0 | 0 | [22] |

${R}_{0,{\alpha}_{F/P}}$ | $\mathrm{MPa}$ | ${\alpha}_{F/P}$ | 2000 | 0 | 0 | 0 | [21] |

${R}_{1,{\alpha}_{F/P}}$ | $\mathrm{MPa}$ | ${\alpha}_{F/P}$ | 300 | 0 | 0 | 0 | [21] |

${e}_{{\alpha}_{F/P}}$ | - | ${\alpha}_{F/P}$ | $410.5$ | 0 | 0 | 0 | [21] |

${\sigma}_{y,{\alpha}_{B}}$ | $\mathrm{MPa}$ | ${\alpha}_{B}$ | $645.27$ | $-0.676$ | 0 | 0 | [22] |

${R}_{0,{\alpha}_{B}}$ | $\mathrm{MPa}$ | ${\alpha}_{B}$ | 2000 | 0 | 0 | 0 | [21] |

${R}_{1,{\alpha}_{B}}$ | $\mathrm{MPa}$ | ${\alpha}_{B}$ | 300 | 0 | 0 | 0 | [21] |

${e}_{{\alpha}_{B}}$ | - | ${\alpha}_{B}$ | 250 | 0 | 0 | 0 | [21] |

${\sigma}_{y,{\alpha}_{M}}$ | $\mathrm{MPa}$ | ${\alpha}_{M}$ | 1598 | $-0.636$ | 0 | 0 | [22] |

${R}_{0,{\alpha}_{M}}$ | $\mathrm{MPa}$ | ${\alpha}_{M}$ | 0 | 0 | 0 | 0 | [21] |

${R}_{1,{\alpha}_{M}}$ | $\mathrm{MPa}$ | ${\alpha}_{M}$ | 0 | 0 | 0 | 0 | [21] |

${e}_{{\alpha}_{M}}$ | - | ${\alpha}_{M}$ | 0 | 0 | 0 | 0 | [21] |

## References

- Vormwald, M.; Schlitzer, T.; Panic, D.; Beier, H.T. Fatigue strength of autofrettaged Diesel injection system components under elevated temperature. Int. J. Fatigue
**2018**, 113, 428–437. [Google Scholar] [CrossRef] - Dewangan, M.K.; Panigrahi, S.K. Residual stress analysis of swage autofrettaged gun barrel via finite element method. J. Mech. Sci. Technol.
**2015**, 29, 2933–2938. [Google Scholar] [CrossRef] - Perl, M.; Saley, T. Swage and hydraulic autofrettage impact on fracture endurance and fatigue life of an internally cracked smooth gun barrel Part I – The effect of overstraining. Eng. Fract. Mech.
**2017**, 182, 372–385. [Google Scholar] [CrossRef] - Krug, T.; Lang, K.H.; Fett, T.; Löhe, D. Influence of residual stresses and mean load on the fatigue strength of case-hardened notched specimens. Mater. Sci. Eng. A
**2007**, 468–470, 158–163. [Google Scholar] [CrossRef] - Liu, H.; Liu, H.; Bocher, P.; Zhu, C.; Sun, Z. Effects of case hardening properties on the contact fatigue of a wind turbine gear pair. Int. J. Mech. Sci.
**2018**, 141, 520–527. [Google Scholar] [CrossRef] - Bepari, M. 2.3 Carburizing: A Method of Case Hardening of Steel. In Comprehensive Materials Finishing; Hashmi, S., Ed.; Elsevier: Oxford, UK; Waltham, MA, USA, 2017; pp. 71–106. [Google Scholar] [CrossRef]
- Kiefer, D.; Schüssler, P.; Mühl, F.; Gibmeier, J. Experimental and Simulative Studies on Residual Stress Formation for Laser-Beam Surface Hardening. HTM J. Heat Treat. Mater.
**2019**, 74, 23–35. [Google Scholar] [CrossRef][Green Version] - Areitioaurtena, M.; Segurajauregi, U.; Urresti, I.; Fisk, M.; Ukar, E. Predicting the induction hardened case in 42CrMo4 cylinders. Procedia CIRP
**2020**, 87, 545–550. [Google Scholar] [CrossRef] - Kobasko, N.I.; Aronov, M.A. 12.07—Intensive Quenching. In Comprehensive Materials Processing; Hashmi, S., Ed.; Elsevier: Oxford, UK, 2014; pp. 253–269. [Google Scholar] [CrossRef]
- Kobasko, N.I. Intensive Quenching Systems: Engineering and Design; ASTM MNL; ASTM International: West Conshohocken, PA, USA, 2010; Volume 64. [Google Scholar]
- Rath, J. Maximierung der Randnahen Druckeigenspannung von Stählen mit Hilfe einer Hochgeschwindigkeits-Abschreckanlage. Ph.D. Thesis, Universität Bremen, Bremen, Germany, 2012. [Google Scholar]
- Kobasko, N. Intensive Steel Quenching Methods. In Theory and Technology of Quenching—A Handbook; Springer: New York, NY, USA, 1992; pp. 367–389. [Google Scholar]
- Habschied, M.; de Graaff, B.; Klumpp, A.; Schulze, V. Fertigung und Eigenspannungen. HTM J. Heat Treat. Mater.
**2015**, 70, 111–121. [Google Scholar] [CrossRef] - Muehl, F.; Dietrich, S.; Schulze, V. Internal Quenching: Ideal Heat Treatment for Difficult to Access Component Sections. HTM J. Heat Treat. Mater.
**2019**, 74, 191–201. [Google Scholar] [CrossRef] - Eigenmann, B.; Macherauch, E. Röntgenographische Untersuchung von Spannungszuständen in Werkstoffen. Tl. 3. Mater. Wiss. u. Werkstofftech.
**1996**, 27, 426–437. [Google Scholar] [CrossRef] - Peiter, A. Eigenspannungen 1. Art: Ermittlung und Bewertung; Triltsch: Düsseldorf, Germany, 1966. [Google Scholar]
- Kaiser, D.; Damon, J.; Mühl, F.; de Graaff, B.; Dietrich, S.; Schulze, V. Experimental investigation and finite-element modeling of the short-time inductive quench-and-temper process. J. Mater. Process. Technol.
**2019**, 279, 116485. [Google Scholar] - Liščić, B. (Ed.) Quenching Theory and Technology, 2nd ed.; CRC Press: Boca Raton, FL, USA; IFHTSE: Zurich, Switzerland, 2010. [Google Scholar]
- Desalos, Y. Comportement Dilatométrique et Mécanique de L’austénite Métastable d’un Acier A 533; Rapport Technique N • 95349401; IRSID: Saint-Germainen-Lay, France, 1981. [Google Scholar]
- Kaiser, D. Experimentelle Untersuchung und Simulation des Kurzzeitanlassens unter Berücksichtigung Thermisch randschichtgehärteter Zustände am Beispiel von 42CrMo4. Ph.D. Thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany, 2019. [Google Scholar]
- Mioković, T. Analyse des Umwandlungsverhaltens bei ein- und mehrfacher Kurzzeithärtung bzw. Laserstrahlhärtung des Stahls 42CrMo4. In Schriftenreihe Werkstoffwissenschaft und Werkstofftechnik; Vol. Band 25/2005; Shaker Verlag: Aachen, Germany, 2005. [Google Scholar]
- Graja, P. Rechnerische und Experimentelle Untersuchungen zum Einfluß Kontinuierlicher und Diskontinuierlicher Wärmebehandlungsverfahren auf die Wärme- und Umwandlungseigenspannungen und Verzüge von un- und Niedriglegierten Stählen. Ph.D. Thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany, 1987. [Google Scholar]
- Majorek, A. Der Einfluärmeübergangs auf die Eigenspannungs- und Verzugsausbildung beim Abschrecken von Stahlzylindern in Verdampfenden Flüssigkeiten. Ph.D. Thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany, 1996. [Google Scholar]
- Van Bohemen, S.M.C.; Sietsma, J. Modeling of isothermal bainite formation based on the nucleation kinetics. Int. J. Mater. Res.
**2008**, 99, 739–747. [Google Scholar] [CrossRef] - Avrami, M. Kinetics of Phase Change. I General Theory. J. Chem. Phys.
**1939**, 7, 1103–1112. [Google Scholar] [CrossRef] - De Graaff, B.; Autenrieth, H.; Hoffmeister, J.; Schulze, V. Investigation on Short Time Tempering by Induction Heating of the low alloyed AISI 4140 steel. In Proceedings of the European Conference on Heat Treatment, Munich, Germany, 12–15 May 2014. [Google Scholar]
- Wever, F.; Rose, A. (Eds.) Atlas zur Wärmebehandlung der Stähle; Verl. Stahleisen: Düsseldorf, Germany, 1961. [Google Scholar]
- Denis, S.; Gautier, E.; Simon, A.; Beck, G. Stress–phase-transformation interactions—basic principles, modelling and calculation of internal stresses. Mater. Sci. Technol.
**1985**, 1, 805–814. [Google Scholar] [CrossRef] - Said Schicchi, D.; Hunkel, M. Effect of Pre-strain and High Stresses on the Bainitic Transformation of Manganese-boron Steel 22MnB5. Metall. Mater. Trans. A
**2018**, 49, 2011–2025. [Google Scholar] [CrossRef] - Van Bohemen, S.M.C.; Hanlon, D.N. A physically based approach to model the incomplete bainitic transformation in high-Si steels. Int. J. Mater. Res.
**2012**, 103, 987–991. [Google Scholar] [CrossRef] - Ravi, A.M.; Sietsma, J.; Santofimia, M.J. Exploring bainite formation kinetics distinguishing grain-boundary and autocatalytic nucleation in high and low-Si steels. Acta Mater.
**2016**, 105, 155–164. [Google Scholar] [CrossRef][Green Version] - Schwenk, M. Numerische Modellierung der Induktiven Ein- und Zweifrequenzrandschichthärtung. Ph.D. Thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany, 2012. [Google Scholar]
- Ahrens, U. Beanspruchungsabhängiges Umwandlungsverhalten und Umwandlungsplastizität Niedrig Legierter Stähle mit Unterschiedlich Hohen Kohlenstoffgehalten. Ph.D. Thesis, Universität Paderborn, Paderborn, Germany, 2003. [Google Scholar]

**Figure 1.**Internal Quenching device and schematical temperature evolution during the process overlaid on a schematic TTT-diagram.

**Figure 3.**FE-model for the Internal Quenching heat treatment, including the mesh and boundary conditions (

**a**) and 3D model for the simulation of the cutting process and electrolytic removal (

**b**).

**Figure 4.**Experimental and modeled bainite volume fractions at different isothermal temperatures for AISI 4140.

**Figure 5.**Experimental and modeled ferrite/perlite (

**a**) and bainite (

**b**) volume fractions at different isothermal temperatures for AISI 1045.

**Figure 6.**Experimentally measured hardness as a function of the isothermal temperatures for AISI 4140 (

**a**) and AISI 1045 (

**b**).

**Figure 7.**Varied process parameters (

**left**) and the investigated temperature fields of the internal and the external surface in the case of a continuous (

**bottom right**) and a discontinuous (

**top right**) cooling of the external surface for AISI 4140.

**Figure 8.**Simulated axial (

**a**) and tangential (

**b**) residual stresses at the inner surface in dependence of the internal and external quenching temperature.

**Figure 9.**Simulated stress development during the Internal Quenching heat treatment in tangential direction (

**a**) and the resulting bainite content in the sample (

**b**) at different external quenching temperatures (

**b**).

**Figure 10.**Simulated axial (

**a**) and tangential (

**b**) residual stresses at the inner surface in dependence of the internal quenching temperature and internal cooling rate during a continuous cooling of the external surface with $3\mathrm{K}/\mathrm{s}$.

**Figure 11.**Influence of two different outer cooling rates on the resulting bainite content and residual stresses as a function of distance from surface using the same cooling conditions at the inner surface.

**Figure 12.**Simulated axial (

**a**) and tangential (

**b**) residual stresses at the inner surface in dependence of the heat transfer coefficient $\alpha $ and the outer cooling rate.

**Figure 13.**Simulated martensite distribution and the corresponding development of the tangential stresses in dependence of the heat transfer coefficient $\alpha $ using an outer cooling rate of $3\mathrm{K}/\mathrm{s}$.

**Figure 14.**Experimental measured and simulated temperatures at the inner and outer surface of the optimal heat treatment strategies for both investigated steels in the corresponding TTT-diagrams ((

**a**): AISI 4140; (

**b**): AISI 1045) [27].

**Figure 15.**Simulated phase, hardness and residual stress distribution after the described heat treatments for both AISI 4140 (

**a**) and AISI 1045 (

**b**).

**Figure 16.**Comparison between measured and simulated hardness as a function of the surface distance after the described optimal heat treatments for AISI 4140 (

**a**) and AISI 1045 (

**b**).

**Figure 17.**AISI 4140: Comparison between measured and simulated residual stresses at the inner surface as a function of the surface distance (

**a**) and the outer surface along the height (

**b**) of the sample after the described heat treatment.

**Figure 18.**AISI 1045: Comparison between measured and simulated residual stresses at the inner surface as a function of the surface distance (

**a**) and the outer surface along the height (

**b**) of the sample after the described heat treatment.

**Figure 19.**Comparison of the Sachs EDM measurements and the simulated residual stress development for AISI 4140 (

**a**) and AISI 1045 (

**b**).

C | Si | Mn | Cr | Mo | Fe | |
---|---|---|---|---|---|---|

AISI 4140 | 0.43 ± 0.01 | 0.21 ± 0.003 | 0.82 ± 0.014 | 1.13 ± 0.015 | 0.18 ± 0.002 | bal. |

AISI 1045 | 0.48 ± 0.02 | 0.25 ± 0.003 | 0.612 ± 0.011 | 0.162 ± 0.018 | 0.01 ± 0.001 | bal. |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mühl, F.; Klug, M.; Dietrich, S.; Schulze, V. Improving the Inner Surface State of Thick-Walled Tubes by Heat Treatments with Internal Quenching Considering a Simulation Based Optimization. *Processes* **2020**, *8*, 1303.
https://doi.org/10.3390/pr8101303

**AMA Style**

Mühl F, Klug M, Dietrich S, Schulze V. Improving the Inner Surface State of Thick-Walled Tubes by Heat Treatments with Internal Quenching Considering a Simulation Based Optimization. *Processes*. 2020; 8(10):1303.
https://doi.org/10.3390/pr8101303

**Chicago/Turabian Style**

Mühl, Fabian, Moritz Klug, Stefan Dietrich, and Volker Schulze. 2020. "Improving the Inner Surface State of Thick-Walled Tubes by Heat Treatments with Internal Quenching Considering a Simulation Based Optimization" *Processes* 8, no. 10: 1303.
https://doi.org/10.3390/pr8101303