# Analysis on the Key Parameters to Predict Flow Stress during Ausforming in a High-Carbon Bainitic Steel

^{1}

^{2}

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## Abstract

**:**

_{2}(dimensionless parameters for dynamic recovery) as constants, the current models consider the quantitative relationship between key parameters and deformation temperature and strain. The results show that m is an exponential function related to temperature and strain, which decreases with the increase in strain. Meanwhile, k

_{2}is a temperature-dependent polynomial function that increases as the deformation temperature increases. Finally, a modified constitutive KME model was proposed to predict the austenitic plastic stress with strain. Using established m-ε and k

_{2}-T models, the predicted curves are in good agreement with the experimental measurements.

## 1. Introduction

_{2}and the deformation temperature, as well as the strain rate sensitivity exponent m of the KME model and strain, were clarified. Expressing the effects of k

_{2}and m on the flow stress as a function can improve the accuracy of the flow stress prediction model and provides guidance for the ausforming process of high-carbon bainitic steels.

## 2. Experiment

^{−1}. After deformation, all the cases were immediately cooled to room temperature at a rate of 5 °C/s. During the experiment, the change in the radial expansion of the cylinder sample was recorded.

## 3. Results and Discussions

#### 3.1. Flow Stress during Ausforming

#### 3.2. Modeling of Stress–Strain

_{0}depends on the geometric arrangement of dislocations [24]. In most cases, α is generally selected as 0.3~0.7 [4,25,26]. G, b, and ρ are the shear modulus, magnitude of the dislocation Burgers vector, and dislocation density, respectively. The classical KME model was used to evaluate the dislocation accumulation and annihilation during the deformation process, where the equation for the evolution of ρ reads [16]:

_{1}and k

_{2}are the dimensionless parameters used to characterize the generation of dislocations and dynamic recovery, respectively [27]. The production term, ${k}_{1}\sqrt{\rho}$, is considered to be related to the athermal hardening stage (stage Ⅱ) of the work hardening process [18]. At this stage, the material continues to deform with the increase in applied stress. When the strain reaches a certain extent, the moving dislocations are hindered, resulting in a pile-up of dislocation and generating forest dislocations. The forest dislocations and moving dislocations interact with each other, which can cause an increase in resistance to slipping dislocation, leading to the accumulation of dislocations again. The second term, ${k}_{2}\rho $, is associated with the dynamic recovery stage (stage Ⅲ) during the deformation process, which is due to the thermally activated process of recovery involving dislocation cross-slip (low temperature case) or dislocation climb (high temperature case) [18,28]. According to many previous works [15,18], k

_{1}can be calculated as:

_{0}is the shear modulus at 300 K of 81 GPa, T

_{M}is the melting temperature of 1810 K [29]. For face-centered cubic metals, the magnitude of the dislocation Burgers vector $b={a}_{\gamma}/\sqrt{2}$. The austenite lattice parameter [Å] ${a}_{\gamma}$ can be calculated as [30]:

_{c}is the carbon content (in atom fraction) and T is the deformation temperature (in °C).

#### 3.3. Validation of KME Model Parameters

_{2}and m, in the KME model and analyzes the quantitative functions of k

_{2}and m. According to the original KME model, the range of α for this experimental steel is 0.48~0.52. Referring to many studies [27,31,32], α was determined to be 0.5 during the following calculation and the intrinsic relationship between α and T was not explored further. k

_{1}is the parameter representing the dislocation storage term in the athermal state and can be calculated using Equations (5)–(8). As the deformation temperature increases, k

_{1}gradually decreases, denoting the recession of the dislocation superposition. On the contrary, k

_{2}is related to the thermal activation process and is a parameter representing dynamic recovery, which is strongly influenced by the deformation temperature and strain rate, and can be calculated using Equations (1)–(8). The value of k

_{2}in many different cases has been given in previous studies. Estrin et al. showed [18] that the annihilation coefficient k

_{2}ranged from 10 (room temperature) to 100 (high temperature) and increased with the increase in the deformation temperature. Hariharan et al. [12] determined the model parameters via least square fitting of the stress–strain curve and set k

_{2}as constant of 1.22. Although many studies have analyzed the contribution of temperature to the dislocation annihilation coefficient [12,18], few of them have described a quantitative relationship between k

_{2}and deformation temperature. This work attempted to illustrate that.

_{2}with temperature was obtained as shown in Figure 3. The value of k

_{2}increases slightly at a deformation temperature of 400 °C~650 °C, and then increases significantly at a deformation temperature of 700 °C~900 °C. It can be observed that k

_{2}does not increase monotonically with the increase in temperature before 650 °C, due to the decrease in dynamic recovery in the dynamic strain aging temperature range [13]. Previous studies have shown that dynamic strain aging results in an increase in dislocation and a delay in the recovery of dislocation structures due to a decrease in the mobility of the dislocations [33,34,35]. For the sake of simplicity, the dislocation annihilation coefficient k

_{2}can be fitted using a polynomial model k

_{2}-T, listed as follows:

#### 3.4. Verification and Discussion

_{2}combined with the KME model. As shown in Figure 2, when the deformation temperature is higher than 700 °C, a plateau appears in the stress–strain curve, indicating the dynamic recovery is dominant at a high deformation temperature. In this time, the positive coefficient k

_{1}of dislocation density tends to be small, while the negative coefficient k

_{2}extensively increases (Figure 3). There is no plateau in the stress–strain curve at a relatively low deformation temperature, which is due to the domination of the work hardening, corresponding to larger k

_{1}and smaller k

_{2}. Speculatively, it is essential to establish quantitative models of k

_{2}and m correlated to strain and deformation temperature. Other verifications were also performed in steels with different compositions and deformation temperatures.

_{2}-T and m-ε equations combined with KME model gives a good but insufficient prediction result for bainitic steels with different chemical compositions. It is not easy to develop a constitutive equation which is suitable for both a large deformation temperature range and a large chemical composition range. Most of the existing constitutive models of metallic materials cannot fulfill this requirement. The KME model and the k

_{2}-T, m-ε equations established in this work should be applicable to austenite in a relatively wide range of chemical compositions and temperatures. In addition, considering the influence of the SRS exponent on the accuracy of flow stress, the quantitative relationship between m and both strain and temperature deserves further investigation. Since we only use an empirical fitted model to correlate m to strain, there are still some small errors in the prediction results. In a future step, the temperature-dependent parameters A, B, and C in the m-ε model may be replaced by the form of function m (ε, T). Regardless, the revised key parameters of m and k

_{2}in the KME model provide a promising indication with which to predict austenite flow stress in high-strength bainitic steels, especially in similar cases to the one presented here.

## 4. Conclusions

- (1)
- The SRS exponent, m, plays a significant role in predicting flow stress during ausforming, and its value decreases with the increase in strain. Based on the experimental measurements, an exponential model of m correlated with strain was established, which contains three fitted parameters that are dependent on the deformation temperature.
- (2)
- The dislocation annihilation coefficient increases with the increase in deformation temperature. A simplified polynomial model could be used to describe the quantitative relationship between k
_{2}and temperature. For the present material, which can be expressed as:

- (3)
- The combination of current k
_{2}-T, m-ε equations in this work with classical KME model can provide a comparative evaluation for the austenite flow stress in the test material and other steels with different compositions and deformation temperatures.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Comparison between predicted and experimental values (m=5.97 and m=15.73 are constant values taken from existing studies; adapted from: [15]).

**Figure 6.**Experimental values vs. predictions by the modified model, where the dotted line represents the standard error of the experiment.

**Figure 7.**Verification of the current model using experimental values from references [36] vs. predictions by the modified model, where the dotted line represents the standard error of the experiment.

T/°C | A | B | C |
---|---|---|---|

900 | 0.8342 | 0.04345 | 2.92506 |

850 | 0.96437 | 0.04842 | 2.85075 |

800 | 1.00118 | 0.05824 | 2.83002 |

750 | 0.9567 | 0.08106 | 2.79579 |

700 | 1.15806 | 0.10067 | 2.60082 |

650 | 1.07383 | 0.07838 | 2.66202 |

600 | 1.22245 | 0.09011 | 2.55322 |

550 | 1.47495 | 0.11025 | 2.39155 |

500 | 1.63259 | 0.12457 | 2.28242 |

450 | 1.70311 | 0.11914 | 2.28477 |

400 | 1.76159 | 0.11823 | 2.30097 |

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**MDPI and ACS Style**

Wang, L.; Hu, H.; Wang, W.; He, P.; Li, Z.; Xu, G.
Analysis on the Key Parameters to Predict Flow Stress during Ausforming in a High-Carbon Bainitic Steel. *Metals* **2023**, *13*, 1526.
https://doi.org/10.3390/met13091526

**AMA Style**

Wang L, Hu H, Wang W, He P, Li Z, Xu G.
Analysis on the Key Parameters to Predict Flow Stress during Ausforming in a High-Carbon Bainitic Steel. *Metals*. 2023; 13(9):1526.
https://doi.org/10.3390/met13091526

**Chicago/Turabian Style**

Wang, Lifan, Haijiang Hu, Wei Wang, Ping He, Zhongbo Li, and Guang Xu.
2023. "Analysis on the Key Parameters to Predict Flow Stress during Ausforming in a High-Carbon Bainitic Steel" *Metals* 13, no. 9: 1526.
https://doi.org/10.3390/met13091526