# Model-Based Residual Stress Design in Multiphase Seamless Steel Tubes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- external cooling only, that is, the coolant is applied only to the tube’s outer surface;
- (ii)
- both, external and internal cooling for mainly martensitic microstructure, that is, an additional cooling is applied through a cooling device from inside the tube;
- (iii)
- and both external and internal cooling, mainly to adjust a bainitic microstructure.

## 2. Materials and Methods

#### 2.1. Model Description

^{2}K. When internal cooling is applied, ${\alpha}_{is}$ is calculated as described above using the work of [15], and is not applied by eight baskets, but by one inner cooling device. The setup is implemented in the FE software ABAQUS with the user subroutine FILM for the heat transfer coefficient, and a more detailed description of the manufacturing setup can be found in the work of [16].

_{S}; see the work of [25]. The measured values for the Greenwood–Johnson parameter are ${K}_{M}=8.9\times {10}^{-5}$ MPa

^{−1}for martensite phase transformation and ${K}_{B}=8.83\times {10}^{-5}$ MPa

^{−1}for bainite.

#### 2.2. Design Strategy

## 3. Results

#### 3.1. Strategy 1

_{t}and axial σ

_{z}direction of ~900 MPa decaying towards the inner surface are characteristic for external cooling. Comparing this stress distribution for the dimension of 200 × 22.65 mm with a smaller dimension of 177.8 × 12.65 mm investigated in the work of [16] reveals that this trend is independent of the tube’s dimension for the given steel class and cooling intensity.

^{γ}in Equation (12) the work of [19]. The variable $\gamma $ is a constant growth parameter and $z$ is the already transformed product phase fraction.

^{γ}was modified in this work to account for the multiphase transformation to ${\left(1-{\displaystyle \sum}_{i=1}^{N}{z}_{i}\right)}^{\gamma}$ for I = 1, …, N product phases. This reduces the remaining austenite fraction also by the previously formed martensite fraction, which is no longer available for bainite formation.

#### 3.2. Strategy 2

#### 3.3. Strategy 3

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Dependency of the transformation start temperature of martensite ${M}_{s}$ (blue, λ = 0.1) and of bainite ${B}_{s}$ (red, λ = 1.1) for different applied stresses.

**Figure 3.**Comparison of high-energy X-ray diffraction residual stress measurements with simulation results for an industrial case.

**Figure 4.**Strategy 1. (

**a**) Temperature evolution using external cooling only. (

**b**) Resulting residual stresses and plastic strain in tangential direction after quenching as a function of distance from the tube’s outer surface.

**Figure 5.**Strategy 1. (

**a**) Snapshots at different times of phase distributions over distance from outer surface (green: martensite; blue: bainite). (

**b**) Front view of Figure 5a depicting phase distributions as a function of distance from the tube’s outer surface.

**Figure 6.**Strategy 2. (

**a**) Temperature evolution using external and internal cooling. (

**b**) Resulting residual stresses and plastic strain in tangential direction after quenching as a function of distance from the tube’s outer surface.

**Figure 7.**Strategy 2. (

**a**) Snapshots at different times of phase distributions over distance from outer surface (green: martensite; blue: bainite). (

**b**) Front view of Figure 7a depicting phase distributions as a function of distance from the tube’s outer surface.

**Figure 8.**Strategy 3. (

**a**) Temperature evolution using internal and external cooling. (

**b**) Resulting residual stresses and plastic strain in tangential direction after quenching over distance from the outer tube’s surface.

**Figure 9.**Strategy 3. (

**a**) Snapshots at different times of phase distributions over distance from outer surface (green: martensite; blue: bainite). (

**b**) Front view of Figure 9a depicting phase distributions as a function of distance from the tube’s outer surface.

Title | Internal Cooling | External Cooling | Cooling Speed |
---|---|---|---|

(kgm^{−2}s^{−1}) | (kgm^{−2}s^{−1}) | (ms^{−1}) | |

Strategy 1 | - | 80 | 0.2 |

Strategy 2 | 100 | 10 | 0.2 |

Strategy 3 | 100 | 10 | 0.1 |

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**MDPI and ACS Style**

Leitner, S.; Winter, G.; Klarner, J.; Antretter, T.; Ecker, W.
Model-Based Residual Stress Design in Multiphase Seamless Steel Tubes. *Materials* **2020**, *13*, 439.
https://doi.org/10.3390/ma13020439

**AMA Style**

Leitner S, Winter G, Klarner J, Antretter T, Ecker W.
Model-Based Residual Stress Design in Multiphase Seamless Steel Tubes. *Materials*. 2020; 13(2):439.
https://doi.org/10.3390/ma13020439

**Chicago/Turabian Style**

Leitner, Silvia, Gerald Winter, Jürgen Klarner, Thomas Antretter, and Werner Ecker.
2020. "Model-Based Residual Stress Design in Multiphase Seamless Steel Tubes" *Materials* 13, no. 2: 439.
https://doi.org/10.3390/ma13020439