# A Continuum Model for the Effect of Dynamic Recrystallization on the Stress–Strain Response

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## Abstract

**:**

## 1. Introduction

^{−1}the steady state is reached after approximately 2.5 s and at a strain rate of 0.02 s

^{−1}it is reached after approximately 100 s. Clearly the recrystallization rate at high strain rate is much larger than the rate at low strain rate, i.e., on average 1% versus 50% recrystallized volume per second. As of yet there is no clear consensus on the physical mechanisms causing the observed increase.

## 2. Continuum Mechanical Model for Hot Forming

#### 2.1. Yield Stress Description

#### 2.1.1. Work Hardening and Dynamic Recovery

#### 2.1.2. Evolution of Mean Free Path on Hardening Behavior

#### 2.2. Effect of Dynamic Recrystallization on the Yield Stress

- Nucleation Depending on the prevailing mechanism, i.e., grain boundary bulging, twinning, or nucleation from subgrains, a new dislocation-poor grain nucleates into the dislocated microstructure.
- Growth Driven by the difference in dislocation density within the new grain and its surroundings the grain boundary migrates increasing the size of the new grain at the expense of dislocation-rich surroundings
- Growth stagnation Concurrent hardening, due to ongoing deformation during the growth-phase, increases the dislocation density within the grain to be almost equal to the dislocation density of the surroundings effectively halting growth
- Hardening Continued deformation increases the dislocation density within the grain such that itself becomes a site for “new” nucleation.

#### 2.2.1. Recrystallization Rate

#### 2.2.2. The Amount of Recrystallizing Grains

#### 2.2.3. The Average Size of the Recrystallizing Grains

#### 2.2.4. Average Grain Boundary Velocity and Driving Pressure

#### 2.2.5. Coupling of DRX to the Bergström Equation

## 3. Model Results and Discussion

#### Grain Size Prediction

## 4. Conclusions

- The model is capable of accurately describing the stress–strain behavior of AISI 316LN over a wide range of temperatures and strain rates.
- The high strain rate DRX-induced softening seen during hot torsion of HSLA-steel is appropriately captured.
- Grain boundary migration velocity at higher strain rates can be predicted by adding the elastic energy, imparted by the applied dynamic stress, to the driving pressure thereby accurately predicting the extreme differences in recrystallization rate at high strain rate.
- The predicted average steady state grain size is in good agreement with the expected power-law relation with the steady state stress.

## 5. Future Work

## Author Contributions

## Conflicts of Interest

## Abbreviations

AISI | American Iron and Steel Institute |

HSLA | High-Strength Low-Alloy |

DRX | Dynamic Recrystallization |

DDRX | Discrete Dynamic Recrystallization |

CDRX | Continuous Dynamic Recrystallization |

PDRX | Post Dynamic Recrystallization |

FEM | Finite Element Method |

DACE | Design and Analysis of Computer Experiments |

TEM | Transmission Electron Microscopy |

BMIS | Boundary Migration Induced Softening |

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**Figure 2.**(

**a**) Change of the distribution of recrystallizing grains from the onset of DRX to the steady state. (

**b**) Detail of the effect of several mechanisms on the distribution at the steady state showing that the net effect is zero, leading to a constant distribution.

**Figure 3.**(

**a**–

**c**) Model validation for AISI 316LN for strain rates ranging from ${10}^{-3}$ s${}^{-1}$–${10}^{1}$ s${}^{-1}$ and temperatures ranging from 900 ${}^{\circ}$C–1100 ${}^{\circ}$C, model results shown in solid line. (

**d**) Model validation for HSLA-steel for strain rates ranging from 0.02 s${}^{-1}$–2 s${}^{-1}$ and a temperature of 900 ${}^{\circ}$C.

**Figure 4.**Predicted recrystallization rate HSLA-steel for strain rates 0.02 s${}^{-1}$ and 2 s${}^{-1}$, the maximum recrystallization rate at the higher-strain rate is approximately 74 times that of the lower strain rate.

**Figure 6.**(

**a**) Grain size evolution for three temperatures at strain rate ${10}^{-3}$ s${}^{-1}$ in which the solid lines represent ${D}_{a}$ and the dotted lines represent $\overline{D}$. (

**b**) Predicted relation between normalized steady state stress and grain size.

**Table 1.**Model parameter values. Fitted by least squares optimization to the hot compression stress–strain data of Zhang et al. [16].

$\mathit{\alpha}$ | 0.5 | ${\mathit{h}}_{0}$ | 9.05 × 10^{5} mm^{−1} | ${\mathit{K}}_{0}$ | 3.05 × 10^{5} mm^{−1} | ${\mathit{m}}_{0}$ | 5.69 × 10^{9} mm^{4}K/Js |

b | 2.8 × 10^{−7} mm | ${h}_{s}$ | 1.90 × 10^{5} mm^{−1} | ${c}_{k}$ | 2.59 × 10^{12} mm^{−1} | ${Q}_{m}$ | 219 kJ/mol |

M | 3 | Q | 411 kJ/mol | ${n}_{k}$ | −0.51 | ${c}_{n}$ | 3.90 × 10^{9} s^{−1}mm^{−3} |

${\mu}_{ref}$ | 8.7 × 10^{4} N/mm^{2} | ${f}_{0}$ | 11. 22 | ${c}_{ds}$ | 1.5 × 10^{−3} MPa | ${Q}_{n}$ | 95.7 kJ/mol |

${c}_{\mu}$ | 26.3 MPa/K | ${c}_{f}$ | 3.1 × 10^{4} | ${n}_{ds}$ | 0.81 | ${a}_{d}$ | 5.16 × 10^{2} |

${\rho}_{0}$ | 2.55 × 10^{6} mm^{−2} | ${n}_{f}$ | −0.2 | ${Q}_{ds}$ | 74.0 kJ/mol | ${D}_{0}$ | 1× 10^{−3} mm |

**Table 2.**Model parameter values. Fitted by least squares optimization to the hot torsion test data of Roucoules et al. [29].

$\mathit{\alpha}$ | 0.5 | ${\mathit{\rho}}_{0}$ | 6.27 × 10^{5} mm^{−2} | ${\mathit{n}}_{{f}}$ | −0.08 | ${\mathit{c}}_{\mathit{n}}$ | 2.86 × 10^{5} s^{−1}mm^{−3} |

b | 2.8 × 10^{−7} mm | ${h}_{s}$ | 1.74 × 10^{5} mm^{−1} | ${c}_{ds}$ | 1.69 MPa | ${a}_{d}$ | 1.69 × 10^{2} |

M | 3 | ${f}_{0}$ | 1.5 | ${n}_{d}$ | 1.92 | ||

$\mu $ | 7.97 × 10^{4} N/mm^{2} | ${c}_{f}$ | 12.53 | ${m}_{0}$ | 2.53 mm^{4}/Js |

**Table 3.**Comparison between the driving pressure contributions from dislocation density and dynamic stress.

$\dot{\mathit{\epsilon}}$ [s^{−1}] | P($\mathit{\rho}$) [MPa] | P(${\mathit{\sigma}}^{*}$) [MPa] | ${\mathit{\sigma}}^{*}$ [MPa] |
---|---|---|---|

0.02 | 0.19 | 0 | 0 |

0.2 | 0.25 | 0.67 | 8.34 |

2 | 0.30 | 9.47 | 31.48 |

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**MDPI and ACS Style**

Kooiker, H.; Perdahcıoğlu, E.S.; Van den Boogaard, A.H.
A Continuum Model for the Effect of Dynamic Recrystallization on the Stress–Strain Response. *Materials* **2018**, *11*, 867.
https://doi.org/10.3390/ma11050867

**AMA Style**

Kooiker H, Perdahcıoğlu ES, Van den Boogaard AH.
A Continuum Model for the Effect of Dynamic Recrystallization on the Stress–Strain Response. *Materials*. 2018; 11(5):867.
https://doi.org/10.3390/ma11050867

**Chicago/Turabian Style**

Kooiker, H., E. S. Perdahcıoğlu, and A. H. Van den Boogaard.
2018. "A Continuum Model for the Effect of Dynamic Recrystallization on the Stress–Strain Response" *Materials* 11, no. 5: 867.
https://doi.org/10.3390/ma11050867