# Avrami Kinetic-Based Constitutive Relationship for Armco-Type Pure Iron in Hot Deformation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Experimental Details

^{−1}, 0.01 s

^{−1}, 0.1 s

^{−1}and 1 s

^{−1}at temperatures of 1273 k–1473 k, while warm compression tests relating to only DRV were also performed on the simulator with a strain rate of 1 s

^{−1}at temperatures of 973 k and 1023 k for Armco-type pure iron. Quartz plates were stuck on samples to reduce the friction during compression. The samples were heated to corresponding temperatures and kept for 3 minutes to ensure that the temperature uniformly distributed. The decrease in height was 60% at the end of the compression tests, after which these samples were quenched in water. The compressed samples were sectioned along the center axis by electro-spark wire-electrode cutting, then polished and chemically etched in a solution of 5% nitric acid and 95% alcohol to reveal the grain boundaries. The optical microstructures of these samples were observed by an optical microscope (Axio Observer A1m).

## 3. Analysis Methods

## 4. Flow Behavior and Microstructure Evolvement

## 5. Constitutive Relationship in Hot Deformation

#### 5.1. Constitutive Relationship of Part I

#### 5.1.1. Constitutive Models Only Relating to DRV Process

#### 5.1.2. Determination of Constant $k$

#### 5.2. Constitutive Relationship of Part II

#### 5.2.1. Avrami Kinetic Equation for DRX

#### 5.2.2. Determination of Critical Strain ${\epsilon}_{c}$

#### 5.2.3. Description of Fractional Softening

#### 5.2.4. Determination of Avrami Constant ${k}^{\ast}$ and Time Exponent ${n}^{\ast}$

^{−1}, 0.01 s

^{−1}, 0.1 s

^{−1}and 1 s

^{−1}are 3.595144, 3.224808, 2.939272 and 2.43113, respectively. The relation between the average values and $\dot{\epsilon}$ shown in Figure 9 is employed to determine the expression: ${n}^{\ast}={n}^{\ast}\left(\dot{\epsilon}\right)$. Through a linear fitting, ${n}^{\ast}\left(\dot{\epsilon}\right)$ is obtained as follows:

^{−1}, 0.01 s

^{−1}, 0.1 s

^{−1}and 1 s

^{−1}are −3.89581, 2.99362, 9.56383 and 14.89145, respectively. Therefore, except for the combined effects of $T$ and $\dot{\epsilon}$ on ${k}^{\ast}$ in the form of the $Z$ parameter, $\dot{\epsilon}$ also shows an individual effect, and ${k}^{\ast}={k}^{\ast}\left(\dot{\epsilon},T\right)$ can be expressed as follows:

#### 5.2.5. Constitutive Models Relating to the DRX and DRV Processes

#### 5.3. Constitutive Relationship of Part III

#### 5.3.1. Constitutive Model of Part III

#### 5.3.2. Determination of Constants $\alpha $, $n$ and ${Q}_{}$

^{−1}. The activation energy of Armco-type pure iron is lower than that of other steels, and this is mainly caused by the high Fe atom purity. Alloy elements in other steels usually decrease the mobility of Fe atoms and thus increase the value of ${Q}_{}$. According to the calculated value of ${Q}_{}$, the Z parameter for Armco-type pure iron can be expressed as follows:

^{12}. Taking the values of $\alpha $, $n$ and $A$ into Equation (27), the constitutive model for Armco-type pure iron in part III can be expressed as follows:

#### 5.4. Integrated Constitutive Relationship for Hole Part

#### 5.4.1. Determination of Yield Stress ${\sigma}_{y}$ and Saturation Stress ${\sigma}_{rs}$

#### 5.4.2. Avrami Kinetic-Based Constitutive Model of Whole Part

## 6. Conclusions

- (1)
- Based on the KM model, the strain-hardening process relating only to DRV can be described by the equation: ${\sigma}_{r}{}^{2}={\sigma}_{y}^{2}\mathrm{exp}(-k\epsilon )+{\sigma}_{rs}^{2}\left[1-\mathrm{exp}(-k\epsilon )\right]$. ${\sigma}_{rs}^{}$ is determined by the part of the true stress-strain curve before the initiation of DRX. As for Armco-type pure iron, $\dot{\epsilon}$ and $T$ show combined effects on $k$, ${\sigma}_{y}^{}$ and ${\sigma}_{rs}^{}$ in the form of the Z parameter with expressions as: $k=\mathrm{exp}\left[0.01723{\left(\mathrm{ln}Z\right)}^{2}-1.18699\left(\mathrm{ln}Z\right)+22.28195\right]$, ${\sigma}_{y}=3.00921\mathrm{ln}Z-50.0975$ and ${\sigma}_{rs}=9.69716\mathrm{ln}Z-218.049$, respectively.
- (2)
- Considering the strain-hardening process, the model used to describe the fractional softening for alloys under DRX is modified as: $X=({\sigma}^{2}-{\sigma}_{r}^{2})/({\sigma}_{ds}^{2}-{\sigma}_{r}^{2})$. The volume fraction of DRX increases as long as the temperature increases and the strain rate decreases. The critical points for the onset of DRX are determined as the inflection point in the $\theta $ vs. $\sigma $ curve. As for Armco-type pure iron, the relation between ${\epsilon}_{c}$ and the Z parameter is expressed as: ${\epsilon}_{c}=2.74264\times {10}^{-4}{Z}^{0.18138}$.
- (3)
- The Avrami kinetic equation is used to describe the kinetic of DRX with the modified expression as: $X=1-\mathrm{exp}\left[-{k}^{\ast}{\left((\epsilon -{\epsilon}_{c})/\dot{\epsilon}\right)}^{{n}^{\ast}}\right]$, where ${k}^{\ast}={k}^{\ast}\left(\dot{\epsilon},T\right)$ and ${n}^{\ast}={n}^{\ast}\left(\dot{\epsilon},T\right)$. As for Armco-type pure iron, including an individual effect, the strain rate $\dot{\epsilon}$ also presents a combined effect with $T$ on ${k}^{\ast}$ in the form of the Z parameter, and the relation is expressed as: ${k}^{\ast}={Z}^{-0.3484}\mathrm{exp}(2.7331\mathrm{ln}\dot{\epsilon}+15.32807)$. ${n}^{\ast}$ is only the function of $\dot{\epsilon}$ with the expression: ${n}^{\ast}=-0.16406\mathrm{ln}\dot{\epsilon}+2.48095$.
- (4)
- Arrhenius-type equations are suitable for describing the flow behavior of the steady state. Regressing from the equations, the DRX activation energy for Armco-type pure iron is 383094 J mol
^{−1}, and the expressions for the $Z$ parameter and ${\sigma}_{ds}$ are determined as Equations (35) and (36), respectively. - (5)
- Based on strain hardening, fractional softening models and the modified Avrami kinetic equation, the constitutive model for alloys considering the effects of DRV and DRX is constructed as Equation (39). The constitutive model can be well used to describe the flow behavior of Armco-type pure iron. The DRV and DRX characters are clearly presented in these curves determined by this model.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\dot{\epsilon}$ | strain rate (s^{−1}) |

$T$ | temperature (K) |

$Q$ | activation energy (J mol^{−1}) |

$R$ | universal gas constant (8.31 J mol^{−1}K^{−1}) |

$Z$ | Zener–Hollomon parameter |

$\epsilon $ | strain |

${\epsilon}_{y}$ | yield strain |

${\epsilon}^{\ast}$ | strain at maximum softening rate |

${\epsilon}_{c}$, ${\sigma}_{c}$ | critical strain and critical stress (MPa) |

${\epsilon}_{p}$, ${\sigma}_{p}$ | peak strain and peak stress (MPa) |

$\sigma $, ${\sigma}_{ds}$ | flow stress (MPa) and steady value of it |

${\sigma}_{r}$, ${\sigma}_{rs}$ | recovery stress (MPa) and saturation value of it |

$\rho $ | average dislocation density corresponding to whole deforming process |

${\rho}_{r}$ | dislocation density corresponding to only DRV process |

${\rho}_{y}$ | dislocation density at yield point |

${\rho}_{ds}$ | dislocation density at the steady state of flow stress |

${\rho}_{rs}$ | dislocation density at the saturation state of recovery stress |

$G$ | shear modulus |

$b$ | burgers vector |

$X$ | recrystallized volume fraction |

${k}^{\ast}$ | Avrami constant |

$t$ | time (s) |

${t}_{50}$ | characteristic time |

${n}^{\ast}$ | time exponent |

$d$ | grain size |

$A$, ${A}_{1}$, ${A}_{2}$, $\alpha $, $\beta $, $n$, ${n}_{1}$, $k$, ${k}_{1}$, ${a}^{\ast}$, ${A}^{\prime}$, $q$, $v$ | constants |

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**Figure 1.**Schematic illustration of the microstructure evolvement and the division of the true stress-strain curve in the hot deformation process. According to the difference of the microstructure evolvement, the true stress-strain curve ($\sigma $) is divided into three parts (part I, part II and part III). Part I undergoes the dynamic recovery (DRV) process. Part II is the initiation of the dynamic recrystallization (DRX) process while these un-recrystallized regions still undergo the DRV process. Part III is the steady state; all the original microstructure has been recrystallized and reaches a dynamic equilibrium in this part. The dashed curve (${\sigma}_{r}$ ) represents the work-hardening behavior of part I and the un-recrystallized regions in part II.

**Figure 2.**Microstructure of Armco-type pure iron deformed at different conditions: (

**a**) $\dot{\epsilon}$ = 1s

^{−1}, $T$ = 1423K; (

**b**) $\dot{\epsilon}$ = 1s

^{−1}, $T$ = 1373K; (

**c**) $\dot{\epsilon}$ = 1s

^{−1}, $T$ = 1023K; and (

**d**) $\dot{\epsilon}$ = 1s

^{−1}, $T$ = 973K. (

**a**,

**b**) relate to the completed DRX process in the hot deformation, while (

**c**,

**d**) relate to the DRV process with the deformation temperature lower than that of the hot deformation. The inserted figures are the corresponding true stress- strain cures.

**Figure 3.**$d{\sigma}_{}{}^{2}/d\epsilon $ vs. ${\sigma}^{2}$ plots derived from the true stress-strain curves of Armco-type pure iron: (

**a**) $\dot{\epsilon}$ = 0.001 s

^{−1}; (

**b**) $\dot{\epsilon}$ = 0.01 s

^{−1}; (

**c**) $\dot{\epsilon}$ = 0.1 s

^{−1}; and (

**d**) $\dot{\epsilon}$ = 1 s

^{−1}. The part before the DRX critical point of each curve refers to the DRV process and is fitted by the linear curve of the dot line. These linear curves are employed to determine the values of $k$ and ${\sigma}_{rs}$. The method to determine the DRX critical points will be discussed later.

**Figure 4.**Original true stress-strain curves and calculated flow stress curves considering only the DRV process of Armco-type pure iron: (

**a**) $\dot{\epsilon}$ = 0.001 s

^{−1}; (

**b**) $\dot{\epsilon}$ = 0.01 s

^{−1}; (

**c**) $\dot{\epsilon}$ = 0.1 s

^{−1}; and (

**d**) $\dot{\epsilon}$ = 1 s

^{−1}. The red solid curves are the original true stress-strain curves while the dot lines are the curves calculated through Equation (8). The dot lines describe the work-hardening behavior of part I and the un-recrystallized regions in part II.

**Figure 5.**The relationship between the Z parameter and $k$. The curve is obtained from polynomial fitting in the order of two. Here, the $Z$ parameter is in the form of $Z=\dot{\epsilon}\mathrm{exp}\left(383094/(8.31T)\right)$ which is calculated in following chapter.

**Figure 6.**$-\partial \theta /\partial \sigma $ vs. $\sigma $ curves of Armco-type pure iron: (

**a**) $\dot{\epsilon}$ = 0.001 s

^{−1}; (

**b**) $\dot{\epsilon}$ = 0.01 s

^{−1}; (

**c**) $\dot{\epsilon}$ = 0.1 s

^{−1}; and (

**d**) $\dot{\epsilon}$ = 1 s

^{−1}. According to the Poliak method, the DRX critical point is supposed to be the peak point of each curve [30]. The dashed lines correspond to the peak stresses of each curve.

**Figure 7.**The relationship between the Z parameter and ${\epsilon}_{c}$. The linear curve is obtained from linear fitting. Here, the $Z$ parameter is in the form of $Z=\dot{\epsilon}\mathrm{exp}\left(383094/(8.31T)\right)$.

**Figure 8.**$\mathrm{ln}\left(\mathrm{ln}\left(1/\left(1-X\right)\right)\right)$ vs. $\mathrm{ln}\left(\left(\epsilon -{\epsilon}_{c}\right)/\dot{\epsilon}\right)$ curves of Armco-type pure iron: (

**a**) $\dot{\epsilon}$ = 0.001 s

^{−1}; (

**b**) $\dot{\epsilon}$ = 0.01 s

^{−1}; (

**c**) $\dot{\epsilon}$ = 0.1 s

^{−1}; and (

**d**) $\dot{\epsilon}$ = 1 s

^{−1}. The values of $X$ are calculated out through Equation (16). The dot lines are the linear fitting results of each curve. After the linear fitting, ${n}_{1}$ and $\mathrm{ln}{k}^{\ast}$ are obtained as the slope and intercept, respectively.

**Figure 9.**The relationship between ${n}^{\ast}$ and $\dot{\epsilon}$. The linear curve is obtained from linear fitting.

**Figure 10.**The relationship between the Z parameter and ${k}^{\ast}$. These linear curves are obtained from the linear fitting of points with the same strain rates. Here, the $Z$ parameter is in the form of $Z=\dot{\epsilon}\mathrm{exp}\left(383094/(8.31T)\right)$. Insert: the relationship between $\dot{\epsilon}$ and the intercepts of these linear fitting curves.

**Figure 11.**S-curves of Armco-type pure iron calculated by Equation (22): (

**a**) $\dot{\epsilon}$ = 0.001 s

^{−1}; (

**b**) $\dot{\epsilon}$ = 0.01 s

^{−1}; (

**c**) $\dot{\epsilon}$ = 0.1 s

^{−1}; and (

**d**) $\dot{\epsilon}$ = 1 s

^{−1}.

**Figure 12.**Original true stress-strain curves and calculated flow stress curves considering eDRV and DRX processes of Armco-type pure iron: (

**a**) $\dot{\epsilon}$ = 0.001 s

^{−1}; (

**b**) $\dot{\epsilon}$ = 0.01 s

^{−1}; (

**c**) $\dot{\epsilon}$ = 0.1 s

^{−1}; and (

**d**) $\dot{\epsilon}$ = 1 s

^{−1}. The red solid curves are the original true stress-strain curves while the dot lines are the flow stress curves calculated through Equation (23).

**Figure 13.**The relationships between $\dot{\epsilon}$, ${\sigma}_{ds}$ and $T$: (

**a**) $\mathrm{ln}\dot{\epsilon}$ and $\mathrm{ln}{\sigma}_{ds}$; (

**b**) $\mathrm{ln}\dot{\epsilon}$ and ${\sigma}_{ds}$; (

**c**) $\mathrm{ln}\dot{\epsilon}$ and $\mathrm{ln}[\mathrm{sinh}(\alpha {\sigma}_{ds})]$; and (

**d**) $\mathrm{ln}[\mathrm{sinh}(\alpha {\sigma}_{ds})]$ and $1/T$. The linear curves are obtained from linear fitting.

**Figure 14.**Relationship between Z parameter and ${\sigma}_{ds}$. The linear curve is got from linear fitting.

**Figure 15.**The relationships between ${\sigma}_{y}$, ${\sigma}_{rs}$ and the $Z$ parameter. The linear curves are obtained from linear fitting.

**Figure 16.**Calculated flow stress curves using the Avrami kinetic-based constitutive model of Armco-type pure iron: (

**a**) $\dot{\epsilon}$ = 0.05 s

^{−1}; (

**b**) $\dot{\epsilon}$ = 0. 1 s

^{−1}; (

**c**) $\dot{\epsilon}$ = 0.5 s

^{−1}; and (

**d**) $\dot{\epsilon}$ = 1 s

^{−1}.

C | Si | Mn | P | S |
---|---|---|---|---|

0.0019 | 0.0294 | 0.1298 | 0.0056 | 0.0038 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Fan, Q.; Zhang, X.; Zhou, Z.; Xia, Z.; Qian, Z.
Avrami Kinetic-Based Constitutive Relationship for Armco-Type Pure Iron in Hot Deformation. *Metals* **2019**, *9*, 365.
https://doi.org/10.3390/met9030365

**AMA Style**

Zhang Y, Fan Q, Zhang X, Zhou Z, Xia Z, Qian Z.
Avrami Kinetic-Based Constitutive Relationship for Armco-Type Pure Iron in Hot Deformation. *Metals*. 2019; 9(3):365.
https://doi.org/10.3390/met9030365

**Chicago/Turabian Style**

Zhang, Yan, Qichao Fan, Xiaofeng Zhang, Zhaohui Zhou, Zhihui Xia, and Zhiqiang Qian.
2019. "Avrami Kinetic-Based Constitutive Relationship for Armco-Type Pure Iron in Hot Deformation" *Metals* 9, no. 3: 365.
https://doi.org/10.3390/met9030365