Deformation Behavior and Constitutive Equation of 42CrMo Steel at High Temperature

: High-temperature reduction pretreatment (HTRP) is a process that can signiﬁcantly improve the core quality of a billet. The existing ﬂow stress data cannot meet the needs of simulation due to lack of high temperature data. To obtain the hot forming process parameters for the high-temperature reduction pretreatment process of 42CrMo steel, a hot compression experiment of 42CrMo steel was conducted on Gleeble-3500 thermal-mechanical at 1200–1350 ◦ C with the rates of deformation 0.001–10 s − 1 and the deformation of 60%, and its deformation behavior at elevated temperature was studied. In this study, the effects of ﬂow stress temperature and strain rate on austenite grain were investigated. Moreover, two typical constitutive models were employed to describe the ﬂow stress, namely the Arrhenius constitutive model of strain compensation and back propagation artiﬁcial neural network (BP ANN) model. The performance evaluation shows that BP ANN model has high accuracy and stability to predict the curve. The thermal processing maps under strains of 0.1, 0.2, 0.3, and 0.4 were established. Based on the analysis of the thermal processing map, the optimal high reduction process parameter range of 42CrMo is obtained: the temperature range is 1250–1350 ◦ C, and the strain rate range is 0.01–1 s − 1 .


Introduction
42CrMo steel, a representative low-alloy medium carbon steel, is widely used to fabricate mechanical parts such as axles and gears due to its good balance of high strength, toughness, hardenability, impact strength, and fatigue resistance [1].The construction of the high-temperature constitutive equation is necessary for the subsequent simulation to explore the internal defects healing evolution mechanism of 42CrMo steel in the HTRP process.The commonly used constitutive equation models are the Arrhenius model [2], the Johnson-Cook model [3], and the back propagation (BP) artificial neural network (ANN) constitutive model [4].In recent years, the development of ANN models has provided an effective method for predicting the thermal deformation behavior of materials, such as steel [5], titanium alloy [6], and aluminum alloy.
Defects deteriorate products' service performance and fatigue life [7].To control internal defects, researchers proposed many processes such as the soft reduction (SR) process [8,9] and heavy reduction (HR) process.Compared with the conventional process, the HR process applies a larger equivalent strain through a small reduction amount, pressuring the billet's internal shrinkage and porosity [10,11], breaking the dendrite [12], refining the grain, and improving density and segregation [13] based on the temperature gradient rolling.Liu et al. [11,13] found that large deformation, induced by temperature gradient

Experiment Procedure
The material was derived from the actual industrial production of the 42CrMo rolled bar.The chemical composition of the experimental steel was measured using a spark direct reading spectrometer (ARL4460, Beijing, China), and detailed results were presented in Table 1.The samples were heated to the deformation temperature at a rate of 10 • C/s.After holding for 10 s, the samples were isothermally compressed and subsequently quenched by argon, as shown in Figure 1a.The samples were subjected to hot compression deformation with a degree of deformation of 60% (9 mm).
A schematic illustration of the hot compression test is shown in Figure 1b.The size of experimental cylindrical compression samples was Φ10 mm × 15 mm.Both its surfaces and ends were mechanically polished to remove the surface oxidation layer.The hot compression test was conducted on a Gleeble-3500 (Data Science International, St. Paul, America), thermal-mechanical simulator system at five strain rates (0.001 s −1 , 0.01 s −1 , 0.1 s −1 ,1 s −1 , and 10 s −1 ) and four temperatures (1200 • C, 1250 • C, 1300 • C, and 1350 • C) in an argon atmosphere.W30Mo70 material with a size of Φ19.3 mm × 7 mm was used as the compression anvil.Tantalum sheet, graphite foil, and nickel-based lubricant were employed to eliminate friction.

The Flow Characteristics
The stress-strain curve is corrected by friction according to Appendix A, and the results are shown in Figure 2. The stress decreases with increasing temperature and decreasing strain rate.At the same strain rate, the peak-strain decreases with the increase in temperature.At the same temperature, the peak-strain increases with the increase in strain rate.There are two main characteristics of the curves: (1) The flow stress curves are divided into four stages according to the description of flow stress by Lin et al. [20].In the initial stage (work hardening stage), with the increase in strain, stress increases as the work hardening rate is bigger than the softening rate.In the second stage (transition stage), the softening effect is gradually strengthened, causing a decrease in the stress increase rate, and thus the flow stress reaches its peak.In the third stage (softening stage), the flow softening effect caused

The Flow Characteristics
The stress-strain curve is corrected by friction according to Appendix A, and the results are shown in Figure 2. The stress decreases with increasing temperature and decreasing strain rate.At the same strain rate, the peak-strain decreases with the increase in temperature.At the same temperature, the peak-strain increases with the increase in strain rate.

The Flow Characteristics
The stress-strain curve is corrected by friction according to Appendix A, and the results are shown in Figure 2. The stress decreases with increasing temperature and decreasing strain rate.At the same strain rate, the peak-strain decreases with the increase in temperature.At the same temperature, the peak-strain increases with the increase in strain rate.There are two main characteristics of the curves: (1) The flow stress curves are divided into four stages according to the description of flow stress by Lin et al. [20].In the initial stage (work hardening stage), with the increase in strain, stress increases as the work hardening rate is bigger than the softening rate.In the second stage (transition stage), the softening effect is gradually strengthened, causing a decrease in the stress increase rate, and thus the flow stress reaches its peak.In the third stage (softening stage), the flow softening effect caused There are two main characteristics of the curves: (1) The flow stress curves are divided into four stages according to the description of flow stress by Lin et al. [20].In the initial stage (work hardening stage), with the increase in strain, stress increases as the work hardening rate is bigger than the softening rate.In the second stage (transition stage), the softening effect is gradually strengthened, causing a decrease in the stress increase rate, and thus the flow stress reaches its peak.In the third stage (softening stage), the flow softening effect caused by dynamic recovery (DRV) and DRX is greater than that caused by work hardening (WH), resulting in a rapid decrease in flow stress.In the fourth stage (steady stage), when the deformation reaches a specific value, the competition between WH and dynamic softening reaches a dynamic equilibrium.It is worth noting that when the strain rate is greater than 0.1 s −1 , the stress has increased after reaching the steady state stage (ε > 0.5), which is caused by friction (2) Flow stress curve changed from a single peak to multiple peaks when .ε changed from 0.01 s −1 to 0.001 s −1 .
The curves show a single peak at high strain rates ( .ε ≥ 0.01 s −1 ) as the DRX rate is so slow that the first round of DRX is not completed when the second round of DRX has begun.
The curves show multiple peaks at a low strain rate ( .ε < 0.01 s −1 ).According to our previous simulation results of the recrystallization process through the multiphase field method [23], the reason is that the second round of DRX takes place after the first round of DRX is completed.
The curve reaches its first peak due to the first round of DRX.The curve reach to its first trough due to growth of each new DRX grain.Then, the deformation is not over, and the dislocation density in the newly formed grains has not reached the critical density of dynamic recrystallization.Therefore, the grains undergo secondary WH, and the second work hardening peak appears.With the increase in dislocation density, the second dynamic recrystallization begins, forming a second trough.Such complex WH and softening of DRX superimposed alternately, and the curve periodically appears multiple peaks until equilibrium [24,25].

Constitutive Modeling the Flow Stress of 42CrMo 3.2.1. Arrhenius Constitutive Model
The derivation of strain-compensated Arrhenius constitutive equation is divided into two steps.The first step is to solve the Arrhenius constitutive equation under a specific strain to obtain the material parameters.The second step is to conduct polynomial fitting of material parameters under all strains to obtain the final equation.
(1) Determination of material parameters According to the Zener and Hollomon [26] research, stress is related to strain, temperature, and strain rate.The Arrhenius model has been widely used to describe the stress-strain relationship.The method of deriving material parameters from Z parameter is shown in Figure 3, and the formula derivation is shown in Appendix B.
Metals 2021, 11, x 4 of 17 by dynamic recovery (DRV) and DRX is greater than that caused by work hardening (WH), resulting in a rapid decrease in flow stress.In the fourth stage (steady stage), when the deformation reaches a specific value, the competition between WH and dynamic softening reaches a dynamic equilibrium.It is worth noting that when the strain rate is greater than 0.1 s −1 , the stress has increased after reaching the steady state stage ( > 0.5), which is caused by friction.(2) Flow stress curve changed from a single peak to multiple peaks when  changed from 0.01 s −1 to 0.001 s −1 .
The curves show a single peak at high strain rates ( ≥ 0.01 s ) as the DRX rate is so slow that the first round of DRX is not completed when the second round of DRX has begun.
The curves show multiple peaks at a low strain rate ( < 0.01 s ).According to our previous simulation results of the recrystallization process through the multiphase field method [23], the reason is that the second round of DRX takes place after the first round of DRX is completed.
The curve reaches its first peak due to the first round of DRX.The curve reach to its first trough due to growth of each new DRX grain.Then, the deformation is not over, and the dislocation density in the newly formed grains has not reached the critical density of dynamic recrystallization.Therefore, the grains undergo secondary WH, and the second work hardening peak appears.With the increase in dislocation density, the second dynamic recrystallization begins, forming a second trough.Such complex WH and softening of DRX superimposed alternately, and the curve periodically appears multiple peaks until equilibrium [24,25].

Arrhenius Constitutive Model
The derivation of strain-compensated Arrhenius constitutive equation is divided into two steps.The first step is to solve the Arrhenius constitutive equation under a specific strain to obtain the material parameters.The second step is to conduct polynomial fitting of material parameters under all strains to obtain the final equation.
(1) Determination of material parameters According to the Zener and Hollomon [26] research, stress is related to strain, temperature, and strain rate.The Arrhenius model has been widely used to describe the stress-strain relationship.The method of deriving material parameters from Z parameter is shown in Figure 3, and the formula derivation is shown in Appendix B. (2) Strain compensated constitutive equation (

2) Strain compensated constitutive equation
The parameters obtained by the above calculation and fitting process can only predict the flow stress under specific strain.Thus, the influence of strain on the flow stress must be considered.The effect of strain can be indirectly reflected on the material constants.The material constants of constitutive equations under different deformation strains are calculated at intervals of 0.05 in the range of 0.05-0.6.The relationship between strain and material constants (α, n, Q, and A) can be well fitted by a polynomial (Equation ( 1), m = 8).Figure 4 shows the fitting results, and the fitting coefficients of polynomials are shown in Table 2.    Some research [27,28] shows that the Arrhenius-type constitutive equation has a large error between the predicted and actual values at high temperatures and low strain rates.Therefore, a strain rate correction of Z parameter is carried out, and the equation is as follows [29]: Metals 2021, 11, 1614 6 of 16 Strain compensated Arrhenius constitutive model can be expressed by Equation (3).

Back Propagation Artificial Neural Network Constitutive Model
The back propagation artificial neural network constitutive (BP ANN) has been an excellent solution for complex nonlinear problems [30].To better describe the flow stress of 42CrMo, the BP ANN model was employed in this research.A typical ANN consists of three units connected by weight values, namely, input layer, output layer, and hidden layer.
The ANN model with three input neurons, T, .
ε, and ε, and one output neuron, σ, is shown in Figure 5.The learning process of the BP ANN model is to continuously adjust the weights of neurons through error back propagation process.Neurons are equivalent to material parameters in conventional fitting methods, and weights are equivalent to parameter coefficients.The hidden layer contains 10 neurons.A logistic sigmoid function was employed as the activation function.The learning is based on the Levenberg-Marquardt algorithm.The learning rate is set as 0.001.The mean square error (MSE) was selected as the training objective function.MSE is expressed as Equation ( 4).Before training, all data need to be normalized in the [−1, 1] interval according to Equation (5).
where E i and P i are experimental and predicted values, respectively.
where X is the original data, X is the normalized data, and X max and X min are the maximum and minimum values of X, respectively.

Back Propagation Artificial Neural Network Constitutive Model
The back propagation artificial neural network constitutive (BP ANN) has been an excellent solution for complex nonlinear problems [30].To better describe the flow stress of 42CrMo, the BP ANN model was employed in this research.A typical ANN consists of three units connected by weight values, namely, input layer, output layer, and hidden layer.
The ANN model with three input neurons, T,  , and , and one output neuron, , is shown in Figure 5.The learning process of the BP ANN model is to continuously adjust the weights of neurons through error back propagation process.Neurons are equivalent to material parameters in conventional fitting methods, and weights are equivalent to parameter coefficients.The hidden layer contains 10 neurons.A logistic sigmoid function was employed as the activation function.The learning is based on the Levenberg-Marquardt algorithm.The learning rate is set as 0.001.The mean square error (MSE) was selected as the training objective function.MSE is expressed as Equation ( 4).Before training, all data need to be normalized in the [−1, 1] interval according to Equation (5).
where  and  are experimental and predicted values, respectively.
where  is the original data,  、 is the normalized data, and  and  are the maximum and minimum values of X, respectively.

Performance Evaluation of Constitutive Models
In this work, two models, i.e., the Arrhenius model and the BP ANN model, were employed to predict the strain-stress of 42CrMo at elevated temperatures.The comparison of experimental and predicted data from different models is shown in Figures 6 and 7, respectively.
To quantitatively evaluate the performance of the above models, the R-value and average absolute relative error (AARE) value [4] are calculated by Equations ( 6) and ( 7), and the results are shown in Figure 8.The closer the R-value is to 1, the stronger the predictive ability is.The smaller the AARE value, the stronger the reliability of the prediction is [31].
where P i and E i are the experimental and predicted data, respectively, and E and P are the mean value.
It can be seen from Figures 6 and 7 that the Arrhenius model has poor fitting results at large strain rates, while the BP ANN model has a better predictive ability.The R-value and AARE value of the BP ANN model are calculated to be 0.9999 and 0.4165%, respectively (Figure 8b), which are both lower than those of the strain-compensated Arrhenius model (Figure 8a), indicating that the BP ANN model has higher prediction accuracy and stability.

Performance Evaluation of Constitutive Models
In this work, two models, i.e., the Arrhenius model and the BP ANN model, were employed to predict the strain-stress of 42CrMo at elevated temperatures.The comparison of experimental and predicted data from different models is shown in Figures 6 and  7, respectively.To quantitatively evaluate the performance of the above models, the R-value and average absolute relative error (AARE) value [4] are calculated by Equations ( 6) and (7), and the results are shown in Figure 8.The closer the R-value is to 1, the stronger the predictive ability is.The smaller the AARE value, the stronger the reliability of the prediction is [31].
where  and  are the experimental and predicted data, respectively, and E and P are the mean value.
It can be seen from Figures 6 and 7 that the Arrhenius model has poor fitting results at large strain rates, while the BP ANN model has a better predictive ability.The R-value and AARE value of the BP ANN model are calculated to be 0.9999 and 0.4165%, respectively (Figure 8b), which are both lower than those of the strain-compensated Arrhenius model (Figure 8a), indicating that the BP ANN model has higher prediction accuracy and stability.

Hot Processing Maps Principles
To further clarify the DRX, DRV, and flow instability behavior of 42CrMo, the hot processing map was drawn according to the dynamic materials model (DMM) [32,33].DMM defines that the total power of the input system, P, consists of two parts, namely the dissipation coordination J, and the dissipation G. Their relationship is Equation (8).

. Hot Processing Maps Principles
To further clarify the DRX, DRV, and flow instability behavior of 42CrMo, the hot processing map was drawn according to the dynamic materials model (DMM) [32,33].DMM defines that the total power of the input system, P, consists of two parts, namely the dissipation coordination J, and the dissipation G. Their relationship is Equation (8).
G and J represent the energy consumed by plastic deformation and by microstructure evolution, respectively.Strain rate sensitivity coefficient m decided their percent of P and can be expressed as.
J takes the maximum value J max when m = 1 under the ideal linear dissipation process.The energy consumption efficiency factor η is defined as the ratio of the dissipation coefficient J to the ideal dissipation coefficient J max [34].In general, the larger the η value is, the larger the energy proportion consumed by structural transformation is, and the better the working performance is.When drawing the thermal processing map, the instability region should also be avoided.Based on the maximum rate of entropy principle, the Prasad criterion was used to describe the flow instabilities (see Equation (11)).

Hot Processing Maps of 42CrMo
Figure 9 shows the processing maps of 42CrMo under different true strains.The contour lines represent the value of η, and the blue regions denote the flow instability.It is reported that a high value of η indicates that the microstructure consumes more energy, and DRV, DRX, or phase transition may occur [35].The regions of high η values, which facilitate hot working, are marked with red lines.These regions are mainly distributed in the range of strain rate 0.01-1 s −1 and temperature greater than 1250 • C. Liu et al. [23] studied the dynamic recrystallization of the HR process by the multi-phase field method and found that a low strain rate promoted dynamic recrystallization, which could explain the areas where high η values occur.

Temperature Effect
The samples were eroded in a supersaturated picric acid solution for an optical microscope (OM) observation to study the prior austenite grain boundaries.Deformation temperature affects recrystallized grains size.Figure 10 shows the prior austenite microstructure results at different temperatures under strain rate 0.01 s −1 .The statistical results of Figure 10a-d are ~72 µm, ~112 µm, ~224 µm, and ~295 µm, respectively.High deformation temperature promotes grain boundary migration and increases grain size.The original grain was compressed along the deformation direction and elongated along the radial direction.Moreover, the deformed large grains are surrounded by newly generated DRX small grains, and the microstructure presents the characteristics of coexistence of original grains and recrystallization grains.
reported that a high value of  indicates that the microstructure consumes more energy, and DRV, DRX, or phase transition may occur [35].The regions of high  values, which facilitate hot working, are marked with red lines.These regions are mainly distributed in the range of strain rate 0.01-1 s −1 and temperature greater than 1250 °C.Liu et al. [23] studied the dynamic recrystallization of the HR process by the multi-phase field method and found that a low strain rate promoted dynamic recrystallization, which could explain the areas where high  values occur.

Temperature Effect
The samples were eroded in a supersaturated picric acid solution for an optical microscope (OM) observation to study the prior austenite grain boundaries.Deformation temperature affects recrystallized grains size.Figure 10 shows the prior austenite microstructure results at different temperatures under strain rate 0.01 s −1 .The statistical results of Figure 10a-d are ~72 μm, ~112 μm, ~224 μm, and ~295 μm, respectively.High deformation temperature promotes grain boundary migration and increases grain size.The original grain was compressed along the deformation direction and elongated along the radial direction.Moreover, the deformed large grains are surrounded by newly generated DRX small grains, and the microstructure presents the characteristics of coexistence of original grains and recrystallization grains.

Strain Rate Effect
The strain rate also affects the recrystallized grains.Figure 11 shows the prior austenite microstructure results at different strain rates under 1350 °C.The statistical results of Figure 11a-d

Strain Rate Effect
The strain rate also affects the recrystallized grains.Figure 11 shows the prior austenite microstructure results at different strain rates under 1350 • C. The statistical results of Figure 11a-d are ~60 µm, ~83 µm, ~183 µm, and ~295 µm, respectively.A large strain rate means large dislocation density and high deformation energy, which drives recrystallization.At the same time, a large strain rate means short recrystallization time, which hinders the subsequent recrystallization grain growth.

Grain Size Prediction Model
The average grain sizes of austenite at different strain rates were counted and shown in Figure 12a.The grain size decreases with the increase in strain rate at a specific temperature.Additionally, the grain size decreases with the decrease in temperature at a specific strain rate.The empirical equation can fit the relationship between austenite grain size and Z parameter d 0 = C 1 Z C 2 .The fitting equation is shown in Equation ( 12), and the fitting curve is shown in Figure 12b.

Grain Size Prediction Model
The average grain sizes of austenite at different strain rates were counted and shown in Figure 12a.The grain size decreases with the increase in strain rate at a specific temperature.Additionally, the grain size decreases with the decrease in temperature at a specific strain rate.The empirical equation can fit the relationship between austenite grain size and Z parameter  =   .The fitting equation is shown in Equation (12), and the fitting curve is shown in Figure 12b.

Figure 1 .
Figure 1.(a) Hot compression experiment procedure diagram and (b) schematic illustration of the hot compression test.

Figure 1 .
Figure 1.(a) Hot compression experiment procedure diagram and (b) schematic illustration of the hot compression test.

Metals 2021, 11 , x 3 of 17 Figure 1 .
Figure 1.(a) Hot compression experiment procedure diagram and (b) schematic illustration of the hot compression test.

Figure 3 .
Figure 3. Flow chart of Arrhenius constitutive model parameters solving.

Figure 3 .
Figure 3. Flow chart of Arrhenius constitutive model parameters solving.

Figure 5 .
Figure 5. Schematic of the BP ANN with a single hidden layer.Figure 5. Schematic of the BP ANN with a single hidden layer.

Figure 5 .
Figure 5. Schematic of the BP ANN with a single hidden layer.Figure 5. Schematic of the BP ANN with a single hidden layer.

Figure 8 .
Figure 8.Comparison of predicted and experimental flow stress by (a) strain-compensated Arrhenius model and (b) BP ANN model.

Table 2 .
Coefficients of the polynomial for material constants α, n, Q, and A.

Table 2 .
Coefficients of the polynomial for material constants α, n, Q, and A.