The temperature dependency of the dynamic recrystallization (DRX) activation energy and strain rate exponent, which are major material properties in the Johnson–Mehl–Avrami–Kolmogorov (JMAK) model affecting the DRX phenomenon, are quantitatively presented in experimental and numerical ways. A finite element analysis-based optimization method was used to acquire the material properties of the JMAK model. The first and second stages of a three-stage hot forging process for a bearing outer race were used to obtain and verify all of the material properties, including the DRX activation energy and the strain rate exponent of the JMAK model, which were assumed to be constants or functions of temperature. The predicted grain size after the third stage obtained with the optimized material properties was compared with the experimental values to validate the acquired material properties and reveal the dependence of the two major material properties on temperature. The comparison showed that the difference between the measured and predicted grain sizes was significantly smaller for temperature-dependent material properties, indicating that the DRX activation energy and strain rate exponent are highly temperature-dependent.

Dynamic recrystallization (DRX) is a phenomenon typical of medium and low stacking-fault-energy metallic materials in hot metal forming. It can soften and restore material ductility and control the grain structure by replacing initially coarse grains by small DRX grains [1]. The characteristics of this microstructural evolution depend on the deformation conditions, which influence the grain size and the mechanical properties [2]. Thus, the ability to predict and control grain size and morphology during different metal forming processes allows the development of tailored mechanical properties and optimized products [3]. Because of the ambiguity of metallurgical effects on microscopic phenomena, there remain many difficulties in simulating hot metal forming processes in terms of hardness, strength or life-span with high accuracy.

In the case of hot metal forming processes, it is essential to determine an accurate microstructure evolution model, based on state variables such as strain, strain rate and temperature, which is appropriate for the forming conditions [4]. The kinetics of transformations have been described by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) phenomenological model since the 40s and many researchers have given a fundamental contribution to extend the range of applicability of the model [5]. JMAK model, with the associated material properties fixed as constants, has been widely used to predict the kinetics and grain size of DRX [6–8] because it gives acceptable results compared with semi-empirical and mesoscopic models [9–12]. The material constants of the equations in the JMAK model are usually obtained from experimental observations with regression techniques.

For example, Lin et al. [13] obtained the material constants of the JMAK model for DRX through investigating the influence of strain, strain rate and temperature on the compressive deformation characteristics of 42CrMo steel. Using a similar approach, Dehghan-Manshadi et al. [14] studied the hot deformation behavior of a 304 austenitic stainless steel to obtain the material constants using hot compression tests. Abbasi and Shokuhfar [15] and Momeni et al. [16] also modeled the hot deformation behavior of alloy steels using hot compression tests at different strain rates and temperatures. In their studies, the material properties of the JMAK model for DRX kinetics were assumed to be constant and the material constants were acquired by regression techniques.

It is established [17–20] that material properties are affected by state variables, especially temperature. For instance, Donati et al. [17] used different values of the strain rate exponent to model recrystallization of AA6060 aluminum alloy at different temperatures. Yanagida and Yanagimoto [20] considered the strain rate exponent as a linear function of temperature in the static recrystallization of steel. Yue et al. [18,20] also expressed the activation energy and strain rate exponent as temperature-dependent functions and predicted the DRX grain size. However, the dependence of the DRX activation energy and the strain rate exponent on temperature has never been revealed in a quantitative way.

In this study, experimental and numerical approaches used to identify the material properties in the JMAK model revealed a quantitative dependence of the DRX activation energy, denoted as Q, and the strain rate exponent, denoted as m, on temperature. A three-stage hot forging process of a bearing steel was used. The first and second stages were used to identify the material properties and the third stage was used to validate the accuracy of prediction of the grain size. All material properties, except Q and m, were assumed to be constant. On the contrary, Q and m were considered constants or functions of temperature defined by several material constants.

2Characterization of material propertiesIn the JMAK model, the grain size after DRX, dDRX, is calculated as follows [16,21]:

where ε,ε˙ and T are the state variables of effective strain, effective strain rate and temperature, respectively, d0 and R are the input variables of initial grain size and gas constant, respectively, and a, h, n, m, c and Q are material properties to be determined experimentally and numerically. Note that n is the strain hardening exponent. Among all the material properties, Q and m, the most influencing (or temperature-sensitive) material properties on DRX [16,22], were assumed to be functions of temperature as follows [19,21]:where Q1, Q2 and Q3 and m1, m2 and m3 are the material constants to be determined.Fig. 1 shows a conceptual procedure of the methodology for identifying the material properties, including the coefficients, to describe the temperature dependence of m and Q, as well as the other constants such as a, h and n, which are all called material constants in this study. The target values of the DRX grain size necessary for optimal material identification were obtained experimentally. The measured grain size at the ith sample point, denoted as dDRX,exp(i), was applied as the associated target value in the root-mean-square error formula [23] to define the following objective function:

where dDRX,pre(i) is the predicted grain size at the ith sample point.The objective function, φ, was iteratively reduced to minimize the difference in error between the predicted and associated measured grain sizes through optimizing the material constants as the design variables. Additional details concerning the approach can be found in [24–26]. To reveal the temperature dependency of Q and m, five different optimization cases were studied (Table 1). In Case 1, constant values for m and Q were applied to the model (Q1=Q2=0,m2=m3=0). In Case 2, m was assumed to be a function of temperature with constant Q, while Q was assumed to be a function of temperature with constant m in Case 3. In Cases 2–4, a, h and n obtained in Case 1 were used while all material constants, including a, h, n, Qi's and mi's, were calculated simultaneously in Case 5, giving more degrees of freedom to the objective function to be minimized.

Optimization cases.

Case | T-dependency | Material constants to be determined | Remarks |
---|---|---|---|

1 | Constant Q and m | a, h, n, Q3, m1 | Q1=Q2=0m2=m3=0 |

2 | Constant Q and T-dependent m | m1,m2,m3 | Q1=Q2=0a, h, n and Q3 are fixed |

3 | Constant m and T-dependent Q | Q1, Q2, Q3 | m2=m3=0a, h, n and m1 are fixed |

4 | T-dependent Q and T-dependent m | m1,m2,m3,Q1,Q2,Q3 | a, h and n are fixed |

5 | T-dependent Q and T-dependent m | a, h, n, m1,m2,m3,Q1,Q2,Q3 | – |

The aforementioned procedure was applied to identify the material constants of 100CrMnSi6 steel [27]. The first and second stages of the three-stage hot forging process used in this study are shown in Fig. 2, which can be assumed to be axisymmetric. Hot-forged specimens were prepared using a mechanical press with a capacity of 1000 tons. The chemical composition of the 100CrMnSi6 material is given in Table 2. The initial workpieces were heated at 1150°C for 5min, resulting in a homogenous initial grain size ranging from 230 to 240μm. Thus, the initial grain size for the first stage was assumed to be 235μm. Different values of the initial grain size are required to obtain the exponent of the initial grain size, denoted as h in Eq. (1). Therefore, the first and second stages were used together to identify the materials constants. After each stage of the hot forging process (Fig. 2), the workpieces were quenched rapidly to halt microstructural evolution. The grain sizes of a plane section were then measured at the specific sample points depicted in Fig. 3. They were used as the control points for both finite element predictions and optimal material identification. All numerical simulations were done using a thermoviscoelastic finite element method [22,28–30]. As the size of elements has a significant effect on the optimization time, it is of great importance to determine the minimum number of element that not only ensures calculation accuracy but also reduces optimization time. Thus, analysis for a 6000 quadrilateral finite element model was carried out using a Coulomb friction coefficient of 0.2 [22]. The flow stress information of the forged steel was that used previously [27]. The gas constant and c value used in Eq. (1) were 8.314472J/kmol and zero, respectively.

The grains of prior austenite before the first stage are illustrated in Fig. 4. Saturated aqueous picric acid with sodium tridecylbenzenesulfonate as the wetting agent was used to reveal the prior austenite grain boundaries [31]. The measured grain sizes were averaged over a small measuring circle [32] having a diameter of 1000μm. Note that the average values of the predicted grain sizes, dDRX,pre, were calculated over the same circle (Fig. 3(c)). The target grain size for the first and second stages are given in Table 3 and the microstructures of some selected sample points are shown in Fig. 5.

Prior austenite grains at different sample locations indicated in Fig. 3 of the initial workpiece.

Measured grain sizes at the sample locations indicated in Fig. 3.

Sample point | x (mm) | y (mm) | Grain size (μm) |
---|---|---|---|

S2-1 | 38.0 | 1.0 | 50.9 |

S2-2 | 37.5 | 2.0 | 50.4 |

S2-3 | 32.6 | 7.0 | 54.5 |

S2-4 | 17.3 | 2.0 | 49.8 |

S2-5 | 17.4 | 2.0 | 50.9 |

S2-6 | 17.2 | 3.0 | 60.2 |

S2-7 | 17.2 | 5.0 | 64.7 |

S2-8 | 17.2 | 8.0 | 64.6 |

S2-9 | 41.3 | 8.0 | 54.5 |

S2-10 | 36.2 | 8.0 | 53.5 |

S2-11 | 31.4 | 8.0 | 61.5 |

S3-1 | 23.0 | 8.0 | 26.2 |

S3-2 | 33.0 | 5.0 | 22.0 |

S3-3 | 34.0 | 10.0 | 34.6 |

S3-4 | 39.0 | 16.0 | 36.9 |

S3-5 | 40.0 | 20.0 | 29.5 |

S3-6 | 41.0 | 24.0 | 27.9 |

S3-7 | 42.0 | 25.0 | 25.3 |

S3-8 | 43.0 | 16.0 | 25.5 |

S3-9 | 43.0 | 9.0 | 23.0 |

S3-10 | 44.0 | 26.0 | 27.2 |

S3-11 | 44.0 | 25.0 | 25.4 |

S3-12 | 44.0 | 8.0 | 22.5 |

Prior austenite grains at the sample points indicated in Fig. 3 for the forging process.

The optimization was iteratively conducted for all of the cases and continued until no additional decrease in the objective function was achieved (Fig. 6). The minimum values of the objective functions for Cases 1–5 were 0.928, 0.847, 0.839, 0.822 and 0.770, respectively. The optimized material constants are listed in Table 4, together with the constraints and initial guesses.

Constraints, initial guesses and optimized values of constants.

Case | Material constants to be determined | Constraints | Initial guesses | Optimized values | Material constants fixed |
---|---|---|---|---|---|

1 | a | 1.0×103<a<1.0×104 | 5.0×103 | 9.3×103 | Q1=Q2=0m2=m3=0 |

h | 0.0<h<1.0 | 5.0 | 0.261 | ||

n | −1.0<n<0.0 | −0.5 | −0.085 | ||

m1 | −1.0<m1<0.0 | −0.5 | −0.065 | ||

Q3 | 50.0<Q3<100.0 | 60.0 | −76.49 | ||

2 | m1 | −2.0<m1<2.0 | −0.5 | −0.85 | a=9.3×103h=0.261n=−0.085Q3=−76.49Q1=Q2=0 |

m2 | 0.0<m2<1.0×10−3 | 5.0×10−4 | 0.18×10−4 | ||

m3 | 5.0×102<m3<1.5×103 | 7.5×102 | 1.096×103 | ||

3 | Q1 | −2.0×10−6<Q1<2.0×10−6 | 0.0 | −0.44×10−6 | a=9.3×103h=0.261n=−0.085m1=−0.065m2=m3=0 |

Q2 | −2.0×10−2<Q2<2.0×10−2 | −0.5×10−2 | −0.90×10−3 | ||

Q3 | −76.0<Q3<−73.0 | −75.0 | −74.82 | ||

4 | Q1 | −2.0×10−6<Q1<2.0×10−6 | −0.44×10−6 | −0.16×10−6 | a=9.3×103h=0.261n=−0.085 |

Q2 | −2.0×10−2<Q2<2.0×10−2 | −0.90×10−2 | −0.77×10−3 | ||

Q3 | −76.0<Q3<−73.0 | −74.82 | −74.83 | ||

m1 | −2.0<m1<2.0 | −0.85 | −0.66 | ||

m2 | 0.0<m2<1.0×10−3 | 0.18×10-4 | 0.28×10−4 | ||

m3 | 5.0×102<m3<1.5×103 | 1.096×103 | 0.79×103 | ||

5 | a | 8.0×103<a<1.0×104 | 9.3×103 | 8.1×103 | – |

h | 0.2<h<0.3 | 0.261 | 0.261 | ||

n | −0.1<n<0.0 | −0.085 | −0.035 | ||

Q1 | −0.5×10−6<Q1<0.5×10−6 | −0.44×10−6 | 0.48×10−6 | ||

Q2 | −1.1×10−2<Q2<−0.9×10−2 | −0.90×10−3 | −0.91×10−3 | ||

Q3 | −75.0<Q3<−74.0 | −74.82 | −74.93 | ||

m1 | −0.9<m1<−0.8 | −0.85 | −0.84 | ||

m2 | 0.1×10−3<m2<0.2×10−3 | 0.18×10−4 | 0.15×10−4 | ||

m3 | 1.0×103<m3<1.2×103 | 1.096×103 | 1.08×103 |

Notably, the constraints in Cases 1–3 were applied to achieve feasible optimized solutions. For Cases 4 and 5, the initial values and constraints were generated according to the optimized values of Cases 2 and 3. The ranges of material constants for Case 5 were narrow to reduce the required iterations and computational time because of the increased number of design variables to be optimized.

3Results and discussion3.1Validation of the material constantsA schematic diagram of the third stage of the forging process used to validate the optimized material constants is illustrated in Fig. 7. Using the optimized material constants from Case 1, the predicted and experimental grain sizes at the third stage of the forging process were compared (Fig. 8). The prior austenite grains after the third stage are shown in Fig. 9. The predicted grain sizes and their corresponding experimental values had an average error of 11.7% when the constants m and Q for Case 1 were used.

Prior austenite grains at three selected sample points (indicated in Fig. 8) after the third stage.

To study the temperature sensitivity of m and Q, the minimum objective function values, denoted as φmin, for the five cases were compared (Fig. 10). The dependence of the strain rate exponent on temperature (Case 2) reduced the value of φmin remarkably. However, the T-dependent DRX activation energy (Case 3) provided almost the same reduction in the objective function. In other words, m and Q showed similar sensitivity to temperature. In Case 4, i.e., where both m and Q were temperature-dependent, the φmin value was only slightly reduced compared with Cases 2 and 3. Note that the number of design variables in Case 4 is six, which is more than that of Cases 2 and 3. The greater the number of design variables, the greater the freedom of the objective function to be minimized. Therefore, the φmin value in Case 5, for which the number of design variables is nine, decreased noticeably.

Fig. 11(a) compares the measured grain sizes at 11 sample points after the third stage of the forging process with the predicted grain sizes for all of the test cases. The graphs show a smaller difference in grain size between the predictions and experiments in the cases when both m and Q were assumed to be functions of temperature, especially for Case 5, in which the number of design variables was higher.

To quantitatively investigate the accuracy of the predicted grain size, the error index between the experimental and predicted values for the ith sample point was defined as follows:

where dDRX,exp and dDRX,pre are the experimental and predicted grain sizes, respectively. As seen in Fig. 11(b), with the temperature-dependent Q and m in the JMAK model, the average error of the predicted grain sizes with respect to the experimental values reduced from 11.7 to 5.8%. The results indicate that the optimized material constants in the JMAK model for predicting the DRX grain size, including all of the coefficients needed to express the strain rate exponent and activation energy as functions of temperature, can be obtained through the optimization procedure. Thus, the most accurate JMAK model to predict the grain size after recrystallization of the 100CrMnSi6 bearing steel studied in this paper is as follows:where m(T) and Q(T) are calculated as follows:Consequently, it is recommended that the strain rate exponent and DRX activation energy in the JMAK model should be expressed as functions of temperature for more accurate predictions of microstructural evolution.

4ConclusionThe temperature sensitivity of the DRX activation energy, Q, and the strain rate exponent, m, in the JMAK model for predicting the grain size after DRX was investigated using a finite element analysis-based optimization technique. The first and second stages of a three-stage hot forging process were used to obtain the microstructural material constants of the 100CrMnSi6 bearing steel. The material constants were acquired by minimizing the objective error function between the measured and predicted grain sizes at specific sample points.

To study the T-sensitivity of Q and m, the optimal material constants were identified for five cases; these included all combinations of constants and T-dependent Q and m in the JMAK model. The results revealed that Q and m exhibited almost the same sensitivity to temperature. Moreover, the objective function had the minimum value when the material constants for Case 5 were used, for which both Q and m were assumed to be functions of temperature and all of the other material constants were considered simultaneously as design variables during the optimization. Notably, considerably advanced solutions could be achieved when at least one of Q and m was expressed in terms of temperature. This implied that these material properties may offset each other during the optimization.

The grain sizes after the third stage were predicted using the material constants obtained from the different optimization cases and compared with the experimental values. The minimum error between the predicted and measured grain sizes was observed in Case 5, implying that the JMAK model with T-dependent Q and T-dependent m can provide more accurate predictions of grain size.

Data availabilityThe raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.

Conflicts of interestThe authors declare no conflicts of interest.

This work was supported by the BK 21 Plus project, WC300, and the industrial technology innovation project of Korea. The authors acknowledge the Se-il Forging Company for providing the experimental facilities.

*et al*.