ARTICLE

J. Cent. South Univ. (2019) 26: 1550-1562

DOI: https://doi.org/10.1007/s11771-019-4111-x

Flow behaviour constitutive model of CuCrZr alloy and 35CrMo steel based on dynamic recrystallization softening effect under elevated temperature

HUANG Yuan-chun(黄元春)^{1, 2}, LI Ming(李明)¹, MA Cun-qiang(马存强)³,XIAO Zheng-bing(肖政兵)^{1, 2}, LIU Yu(刘宇)¹

1. Light Alloy Research Institute, Central South University, Changsha 410012, China;

2. College of Mechanical and Electrical Engineering, Central South University, Changsha 410012, China;

3. Capital Aerospace Mechinery Corporation Limited, Beijing 100076, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: In order to study the effect of dynamic recrystallization on the metal flow behavior during thermal deformation, the elevated temperature compression experiments of CuCrZr alloy and 35CrMo steel are carried out using Gleeble-3810 thermal simulator. It is proved that the samples underwent obvious dynamic recrystallization behavior during thermal deformation by microstructure observation of deformed specimens. The size of recrystallized grains increases as the temperature improved and the strain rate decreased. Meanwhile, the net softening rate caused by dynamic recrystallization is determined based on the stress-dislocation relationship. It can be found that the value of net softening rate increases quadratically as the Z parameter decreases, and the dynamic recrystallization net softening rate of CuCrZr alloy and 35CrMo steel are calculated to be 21.9% and 29.8%, respectively. Based on the dynamic recrystallization softening effect proposed, the novel elevated temperature flow constitutive models of two different alloys are proposed, and the related parameters are well defined and solved in detail. The predicted values of the obtained models are agreed well with the experimental values.

Key words: CuCrZr alloy; 35CrMo steel; dynamic recrystallization; dynamic recrystallization softening effect; high temperature flow constitutive model

Cite this article as: HUANG Yuan-chun, LI Ming, MA Cun-qiang, XIAO Zheng-bing, LIU Yu. Flow behaviour constitutive model of CuCrZr alloy and 35CrMo steel based on dynamic recrystallization softening effect under elevated temperature [J]. Journal of Central South University, 2019, 26(6): 1550-1562. DOI: https://doi.org/ 10.1007/s11771-019-4111-x.

1 Introduction

In order to explore the relationship between the flow behavior and microstructure evolution of metal in hot plastic processing, numerical simulation technology has been widely used in the field of metal plastic processing. Nevertheless, obtaining accurate constitutive models that well describe the flow stress and strain is the key for accurate simulation calculations [1]. Therefore, it is very important to study the flow behavior of metal during elevated temperature deformation for designing and optimizing its processing technology, and improving the quality of metal products [2].

Thereby, research [3-12] on the prediction accuracy of elevated temperature flow constitutive model has been widely concerned. Some improved models were proposed based on several traditional constitutive models [13-18]. Among them, LIN et al [13] revised the Johnson-Cook (JC) model to predict the elevated temperature flow stress of high strength alloy steels. The improved Arrhenius model has been used by many researchers to predict the elevated temperature flow stresses of various alloys in recent years, such as 42CrMo steel [14], GCr15 steel [15], cast A356 aluminum alloy [16], duplex stainless steel [17], 17-4 PH stainless steel [18]. Since these models mainly consider macroscopic phenomena of materials during high temperature deformation, it is unable to explain the results of flow stress and microstructure evolution from a microscopic point of view. And it is not applicable to the elevated temperature flow behavior with strong nonlinearity. Although the artificial neural network model proposed by JI et al [19, 20] can accurately predict the nonlinear high- temperature flow stress. However, it can’t explain the microstructure evolution of the material during the deformation process from the microscopic point of view. The physically-based constitutive models consider the evolution of microstructures that cause changes in the flow stress of materials, such as dislocation migration, dynamic recovery, and dynamic recrystallization. Many physically-based constitutive models have been used to describe high temperature flow stresses in metals, such as dynamic recrystallization (DRX) models [8, 11, 21], Cellular-Automaton (CA) models [22], and Zerilli-Armstrong (ZA) models [23, 24]. However, the physically-based constitutive model is very complicated and requires a lot of experimental data to solve, so its practical application is also limited.

Also, preliminary studies have shown that the metal softening effect caused by dynamic recrystallization during the thermal compression will undergo significant regular changes with changing of compression conditions (deformation temperature and strain rate). Thus, a novel elevated temperature flow constitutive model could be constructed with considering the regularity change caused by dynamic recrystallization. For that, the obtained model not only has a simple structural form, but also explains the microstructure change of the metal during the deformation process from a microscopic point of view. Obviously, for such an model to a given material, one must undergo dynamic recrystallization softening during thermal compression. Studies have shown that CuCrZr alloy [25] (as one of the main representatives of high- strength and high-conductivity copper alloy) undergoes dynamic recrystallization softening in the thermal compression experiments with deformation temperatures range from 600 to 750 °C and strain rates range from 0.001 to 1.0 s^-1, and 35CrMo steel [26] (as one of the main representatives of alloy structural steel) undergoes dynamic recrystallization softening in the thermal compression experiments with deformation temperatures range from 850 to 1150 °C and strain rates range from 0.01 to 10 s^-1.

Therefore, in this paper, CuCrZr alloy and 35CrMo steel were employed for the research, and samples were subjected to thermal compression. It was confirmed by metallographic microstructure observation that the samples underwent dynamic recrystallization during thermal compression. Based on the stress-dislocation relationship, the net softening rate caused by dynamic recrystallization alone was defined. The relationship between dynamic recrystallization softening effect and deformation temperature and strain rate was discussed. The elevated temperature flow constitutive model of two different alloys was constructed with the dynamic recrystallization softening effect considered, and the accuracy of the obtained models were evaluated by comparing the predicted values with experimental values.

2 Experiment

The materials used for the thermal compression were as-solution treated CuCrZr alloy (850 °C for 3.5 h) and electromagnetically as-cast 35CrMo steel. Their chemical compositions in mass fractions are shown in Table 1 (“—” represents not detected). Samples were machined into cylindrical rod specimens with a diameter of 10 mm and a height of 15 mm. A thermal compression experiment at an isothermal constant strain rate was performed on a Gleeble-3810 thermo-simulation machine. The samples of CuCrZr alloy were compressed at 650-800 °C with strain rate of 0.001-1.0 s^-1, and samples of 35CrMo steel were deformed at 950-1150 °C with strain rate of 0.01-10 s^-1. In order to reduce the effect of friction on the stress state and obtain accurate experimental results, a lubricant was applied to both ends of the sample. The sample was heated to a set deformation temperature at a rate of 5 °C/s and held for 2 min to ensure a uniform internal temperature of the sample. After the end of compression, it was rapidly quenched to retain the elevated temperature deformation structure. In addition, experiments at strain rates exceeding 0.1 s^-1 are adiabatic in nature. The effect of temperatures was ignored in this experiment. The deformed sample was thrown away along the line parallel to the compression direction, and then mechanically polished for corrosion. The microstructures under typical deformation conditions were observed with an optical microscope (OLYMPUS-DSX500).

Table 1 Chemical composition of CuCrZr alloy and 35CrMo steel (mass fraction, %)

3 Results

Figure 1 shows the true stress-strain curves of CuCrZr alloy and 35CrMo steel under different thermal deformation conditions. It can be seen that when the strain rate is constant, the flow stress and the peak strain decrease as the temperature increases, and the steady state region becomes longer. When the deformation temperature remains constant, the flow stress and peak strain increase with increasing strain rate, and the steady-state region becomes shorter. This is mainly due to the relatively high grain boundary migration, dislocation diffusion and atomic motion at higher temperatures, and increased energy accumulation and dislocation migration time at low strain rates, which are beneficial to the growth of DRX nucleation and recrystallized grains.

The original microstructure (Figures 2(a) and (e)) and the elevated temperature deformation structure are presented in Figure 2. It can be seen the two alloys underwent significant dynamic recrystallization and recrystallized grain growth under the indicated deformation conditions.

Figure 1 True stress-strain curves of CuCrZr alloy (a, b) and 35CrMo steel (c, d):

Figure 2 Typical microstructure of CuCrZr alloy (a, b, c, d) and 35CrMo steel (e, f, g, h) deformed at different conditions:

From Figure 2(b) (Figure 2(f)), Figure 2(c) (Figure 2(g)) and Figure 2(d) (Figure 2(h)), we can found that as the strain rate decreases and the temperature rises, the recrystallized grains show a tendency to grow. This indicates that a low strain rate and a high deformation temperature promote the occurrence of dynamic recrystallization.

4 Discussion

4.1 Work hardening and dynamic recovery

During the thermal deformation of metals and alloys, the rate of increase and elimination of dislocations directly determines the variation of dislocation density in the matrix. Meanwhile, the change of the flow stress of metals and alloys is mainly affected by the density in the matrix. As reported, the variation of dislocation density of the metal during deformation can be expressed as follows without considering the influence of DRX [27, 28]:

                             (1)

Using the classical flow stress-dislocation density relationship, the relationship between the flow stress and dislocation density can be expressed as [29]:

                             (2)

where ρ is the dislocation density; b is the Berges vector; μ is the shear modulus; U is work hardening; α is the material property constant, which is usually 0.5; dρ/dε is the rate at which the dislocation density increases with strain in the matrix; Ω is the dynamic recovery coefficient and represents the softening behavior due to DRV. Integrating Eq. (1) and substituting it into Eq. (2), we can get the equation of flow stress considering only dynamic recovery:

          (3)

where σ₀ is the yield stress, and the yield stress value under different deformation conditions is generally the stress value added when the residual strain is 0.2% according to the true stress-strain curve; σ_sat is the saturation stress, and the saturation stress is the abscissa of the intersection of the tangent and the X-axis of the inflection point on the work hardening rate curve (θ=dσ/dε).

4.2 Dynamic recrystallization softening effect

The true stress-strain curve only considering work hardening and dynamic recovery can be calculated by Eq. (3), as shown by the σ_recov curve in Figure 3. The dynamic recovery curve (σ_recov) with only work hardening and dynamic recovery is generally considered to be the result from the increase of dislocation density during work hardening and the elimination of dislocation density in dynamic recovery. As the strain increases, the generation rate and the elimination rate of the dislocation density will eventually reach an equilibrium state. At this time, the macroscopic behavior is that the flow stress is stable at the saturation stress (σ_sat). In fact, before the flow stress reaches the equilibrium state of the saturation stress, dynamic recrystallization has begun to occur at the critical strain due to the large accumulation of dislocation density. The occurrence of DRX will further increase the speed of dislocation density elimination, and the softening effect of the flow stress curve will also increase. This ultimately leads to the difference between the σ_DRX curves and the σ_recov curves. Therefore, we can use the following formula to describe the softening effect of dynamic recrystallization under specific deformation conditions:

                       (4)

where η is the net softening rate caused by dynamic recrystallization; σ_sat is the saturation stress value of the dynamic recovery curve; and σ_ss is the steady state stress value of the dynamic recrystallization curve.

Figure 3 Schematic diagram of calculated dynamic recovery curve and experimental dynamic recrystallization curve

According to Eq. (4) the dynamic recrystallization net softening rate (η) of CuCrZr alloy and 35CrMo steel under different deformation conditions can be calculated. The three-dimensional graphs of the dynamic recrystallization net softening rate (η) as a function of deformation temperature and strain rate were made, as shown in Figure 4. The maximum and minimum values of the dynamic recrystallization net softening rate of CuCrZr alloy were 21.9% and 5.1%, respectively, and the maximum and minimum values of the net recrystallization softening rate of 35CrMo steel were 29.8% and 10.2%, respectively. As can be seen from Figure 4, the dynamic recrystallization net softening rate (η) decreases as the deformation temperature decreases or the strain rate increases. The dynamic recrystallization net softening rate (η) is relatively small at low temperatures and high strain rates. The main reason is that the atomic motion is relatively weak at lower temperatures and the time of energy accumulation at high strain rates is reduced, which makes certain obstacles to the occurrence of DRX. Eventually, the softening effect of flow stress is weakened. Higher temperatures and lower strain rates provide greater driving force for dislocation quenching and promote nucleation and growth of dynamically recrystallized grains. The appearance of DRX can significantly increase the elimination rate of dislocation density, resulting in a decrease in flow stress. Therefore, increasing the deformation temperature and decreasing the strain rate tend to promote dynamic recrystallization.

Figure 4 Relationship between deformation temperature, strain rate and softening ratio (η):

It can be seen from the above analysis that the dynamic recrystallization softening effect of CuCrZr alloy and 35CrMo steel is mainly related to the deformation temperature and the strain rate. Thus, the combined effects of deformation temperature and strain rate can be expressed by the Zener-Hollomon parameter [30-32]:

                         (5)

where is the strain rate (s^-1); T is the thermodynamic temperature (K); Q is the thermal deformation activation energy, and R is the gas constant (8.31 J/(mol·K)). By linear regression analysis [30-34], it can be calculated that the thermal deformation activation energy Q of CuCrZr alloy is 244.94 kJ/mol, and the thermal deformation activation energy Q of 35CrMo steel is 402 kJ/mol. Therefore, the graph of the dynamic recrystallization net softening rate (η) as a function of the lnZ as shown in Figure 5 can be plotted with the lnZ as the X axis and the η value as the Y axis. It can be obtained by quadratic fitting:

 (6)

               (7)

4.3 Establishment of constitutive model

As can be seen from Figure 3, since there is only work hardening and dynamic recovery in the stage before the dynamic recrystallization critical strain, the σ_recov curve and the σ_DRX curve are coincident with each other. At the stage after the critical strain, the difference between the σ_recov curve and the σ_DRX curve is gradually increased from 0 to η due to the occurrence of dynamic recrystallization. Therefore, we can establish a new constitutive model to predict the elevated temperature flow stress of different alloys based on the dynamic softening effect described above. The new constitutive model established is as follows:

                 (8)

where ε_c is the critical strain; η is the net softening rate of dynamic recrystallization; ε_ss is the strain value of the flow stress curve just entering the steady state, which can be obtained by linear regression analysis of the true stress-strain curve; σ_recov is the flow stress considering only work hardening and dynamic recovery. According to the formula, the physical meaning of n is the speed of the dynamic recrystallization softening rate, so n can be called the dynamic recrystallization softening coefficient.

Figure 5 Relationship between Zener-Hollomon parameter and softening ratio (η):

In order to accurately calculate the predicted value of the flow stress, one must have the values of n and ε_c determined accurately.

4.3.1 Determination of softening coefficient n

It can be seen from the analysis in Section 4.1 that the dynamic recrystallization has different softening ability to the flow stress curve under different deformation conditions, thus resulting in different softening rates. Taking the logarithm of both sides of Eq. (8) gives:

               (9)

According to the true stress-strain curve, the relationship diagrams of of CuCrZr alloy and 35CrMo steel were obtained as shown in Figure 6. Linear regression analysis can be used to obtain n values under different conditions. From the calculated data, it is known that the n value increases as the temperature increases and the strain rate decreases.

Furthermore, the relationship curves between the softening coefficient n and the Z parameter were pictured as presented in Figure 7. After linear fitting the corresponding curves, we can get:

               (10)

             (11)

4.3.2 Dynamic recrystallization critical condition

Generally, the critical conditions of DRX softening behavior can not be gained from the experimental elevated temperature flow stress curve directly, but it can be determined from the studied of work hardening rate [35, 36]. POLIAK et al [37] and MIRZADEH et al [38] proposed a method of fitting θ-σ curve and lnθ-ε curve using a third- order polynomial to determine the critical condition of DRX initiation, based on which the critical conditions of dynamic recrystallization were calculated by fitting a third-order polynomial (θ=Aσ³+Bσ²+Cσ+D) to the θ-σ curve and the lnθ-ε curve, and then using the formula σ_c=-B/3A or the formula ε_c=-B/3A. In fact, there is a deviation between the fitted curve and the original curve during the three-fitting process, which leads to a corresponding deviation of the critical point of dynamic recrystallization (△σ or △ε), as shown in Figure 8. And this deviation phenomenon changes with the deformation condition, which has a certain influence on the dynamic recrystallization critical condition.

Figure 6 Relationship between softening coefficient and flow stress during dynamic recrystallization stage:

Figure 7 Relationship between softening coefficient and Zener-Hollomon parameter:

Figure 8 Difference in curves before and after fitting by a third order polynomial:

In order to improve the accuracy of dynamic recrystallization critical conditions, the Poliak- Jonas criterion [39] was employed in the present paper. The critical condition of dynamic recrystallization was determined as the minimum value in the -(dθ/dσ)-σ curve as shown in Figure 9.

As presented above, many methods can be applied to determine the critical conditions for the starting of DRX, but the Poliak-Jonas criterion has the highest accuracy. By comparing the correlation factors (R²) and the inflection point deviation values (△σ or △ε) of the other two fitting methods, it can be found that the θ-σ curve method (compared to the lnθ-ε curve method) determines the critical condition of DRX initiation more accurately.

The relationship between the dynamic recrystallization critical strain and the Z parameter were shown in Figure 10, and linear fitting of the curves we can get:

               (12)

               (13)

4.4 Model prediction and error analysis

In order to verify the correctness of the established constitutive model, the predicted stress values of the two alloys were compared with the experimental stress values. As shown in Figure 11, we can see that the predicted values agree well with the experimental values, indicating the correctness of the model.

Figure 9 Curves of θ vs σ (a) and –(dθ/dσ) vs σ (b) for CuCrZr alloy at a strain rate of 0.001 s^-1

In order to further estimate the accuracy of the model, the accuracy of the model is evaluated by the correlation coefficient, R, and the mean absolute error,.

Figure 10 Relationship between dynamic recrystallization critical strain and Zener-Hollomon parameter:

             (14)

                 (15)

where σ is the experimental value; σ_p is the calculated value of the model; N is the sample size; and are their mean values. As shown in Figure 12, the correlation coefficients R between the experimental high temperature flow stress values and predicted values of CuCrZr alloy and 35CrMo steel are 0.9939 and 0.9981, respectively, and the average absolute errors are 2.83% and 2.15%, respectively. The results show that the new constitutive model can accurately reflect the elevated temperature flow behavior of different alloys, and has certain reference value for the formulation of temperature, deformation speed and strain parameters in thermoforming processes such as forging and extrusion.

Figure 11 Comparison of measured and predicted values of CuCrZr alloy (a, b) and 35CrMo steel (c, d):

Figure 12 Correlation coefficient between measured and predicted values of CuCrZr alloy and 35CrMo steel:

5 Conclusions

1) The true stress-strain curves and elevated temperature deformation microstructures of CuCrZr alloy and 35CrMo steel were obtained. It was found that both alloys showed obvious dynamic recrystallization under the compression conditions; the size of the recrystallized grains are getting bigger as the temperature increases and the strain rate decreases.

2) The dynamic recrystallization net softening rate was defined. The maximum values of dynamic recrystallization net softening rate of CuCrZr alloy and 35CrMo steel were 21.9% and 29.8%, respectively, and the minimum values were 5.1% and 10.2%, respectively. And dynamic recrystallization net softening rate increases as the deformation temperature increases or the strain rate decreases. The net softening rate caused by dynamic recrystallization can be expressed as a quadratic function of the Zener-Hollomon parameter.

3) Three methods were systematically compared for calculating the critical conditions of dynamic recrystallization, and the Poliak-Jonas criterion turned out to have the highest accuracy, followed by the θ-σ curve method, and the worst is the lnθ-ε curve method.

4) The elevated temperature flow constitutive model based on the dynamic recrystallization softening effect can predict the flow stress of CuCrZr alloy and 35CrMo steel accurately. The correlation coefficients between the predicted value and the experimental value were 0.9939 and 0.9981, respectively, and the average absolute errors were 2.83% and 2.15%, respectively.

Conflicts of interest: The authors declare no conflict of interest.

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(Edited by YANG Hua)

中文导读

基于动态再结晶软化效应的CuCrZr和35CrMo高温流变行为本构模型

摘要：为了研究热变形过程中动态再结晶对金属流变行为的影响，采用Gleeble-3800热模拟机对CuCrZr合金和35CrMo钢进行了高温压缩实验。通过观察热变形后样品的金相组织，证明了样品在热变形过程中经历了明显的动态再结晶，且动态再结晶晶粒尺寸随着变形温度的升高和应变速率的降低而增大。以应力-位错密度的关系为理论依据，定义了由动态再结晶所引起的净软化作用η。η与Z参数之间呈三次递减关系。计算得到的CuCrZr合金和35CrMo钢的最大η值分别为21.9%和29.8%。基于所提出的动态再结晶软化效应，构建了两种合金的高温流动行为本构模型。并对模型参数进行了定义和求解。所建模型的预测值与实验所得的数据高度吻合。

关键词：CuCrZr合金；35CrMo钢；动态再结晶；动态再结晶软化作用；高温流动本构模型

Foundation item: Project(2019zzts525) supported by the Fundamental Research Funds for the Central Universities, China; Projects(U1837207, U1637601) supported by the National Natural Science Foundation of China

Received date: 2018-08-29; Accepted date: 2019-05-31

Corresponding author: LIU Yu, PhD; Tel: +86-731-88876315; E-mail: csuliuyu@csu.edu.cn; ORCID: 0000-0002-1806-6011