J. Cent. South Univ. (2018) 25: 1560-1572

DOI: https://doi.org/10.1007/s11771-018-3848-y

Constitutive modeling for high temperature flow behavior of a high-strength manganese brass

WANG Meng-han(王梦寒), WEI Kang(危康), LI Xiao-juan(李小娟), TU Ao-zhe(涂奥哲)

School of Material Science and Engineering, Chongqing University, Chongqing 400044, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: The hot compressive deformation behaviors of ZHMn34-2-2-1 manganese brass are investigated on Thermecmastor-Z thermal simulator over wide processing domain of temperatures (923–1073 K) and strain rates (0.01–10 s^–1). The true stress–strain curves exhibit a single peak stress, after which the stress monotonously decreases until a steady state stress occurs, indicating a typical dynamic recrystallization. The analysis of deviation between strain-dependent Arrhenius type constitutive and experimental data revealed that the material parameters (n, A, and Q) for the ZHMn34-2-2-1 manganese brass are not constants but varies as functions of the deformation conditions. A revised strain-independent sine hyperbolic constitutive was proposed, which considered the coupled effects of strain rate temperature and strain on material parameters. The correlation coefficient and the average absolute relative error are used to evaluate the accuracy of the established constitutive model. The quantitative results indicate that the proposed constitutive model can precisely characterize the hot deformation behavior of ZHMn34-2-2-1 manganese brass.

Key words: constitutive modeling; manganese brass; activation energy; hot deformation

Cite this article as: WANG Meng-han, WEI Kang, LI Xiao-juan, TU Ao-zhe. Constitutive modeling for high temperature flow behavior of a high-strength manganese brass [J]. Journal of Central South University, 2018, 25(7): 1560–1572. DOI: https://doi.org/10.1007/s11771-018-3848-y.

1 Introduction

Hot deformation is a complicated plastic deformation processing method among a variety of metal processing technologies, and can be affected by various factors, such as temperature, strain rate, microstructure, plastic anisotropy. The extremely complex interaction and interdependency between these factors cause the quality discrepancy of finished product [1–3]. In the automobile industry, particularly in the manufacturing of components, where resistance to wear is the chief requirement, high-strength brass is regarded as an ideal choice due to the good balance between strength, toughness and a property of excellent corrosion resistance in an extreme working environment. High-strength brasses are α+β or β brasses which contain additions such as aluminum, silicon, iron, manganese, tin and nickel. These additions work as solute strengthening of α phase and β phase, precipitation strengthening and grain refinement [4, 5]. In the case of alloy containing additions of manganese and silicon, manganese silicide (Mn₅Si₃) intermetallic compounds are formed. The intermetallic compounds give the alloy a high wear resistance [6]. However, the manganese brass lacks sufficient workability when subjected to hot deformation. Due to the narrow forming temperature range, great deformation resistance and complex microstructures, comprehensive investigations on the high temperature deformation behavior of manganese brass are particularly important. Thus, a comprehensive study on hot deformation behavior of high-strength brasses is required for successfully deforming these materials to expected shape and size without introducing defect.

The constitutive equation, that is mathematically represented flow behavior of materials, is used as essential input to finite element code for simulating the response of material under prevailing loading conditions [7–9]. Enough accuracy for the establishment of constitutive model is of paramount importance, as it is the prerequisite for a feasible numerical simulation. As an ideal constitutive equation, a relatively reasonable number of material parameters should be contained. The equation should also enable precise modeling of processes within a wide range of stains, strain rates and temperatures [10–12]. Constitutive models are mainly divided into three categories, including phenomenological, physically based and artificial neural network (ANN) constitutive models [13]. Compared with physically based models, the phenomenological ones involve less material parameters and the required experiment can be conducted more easily. Although ANN model shows excellent performance, but it is merely a mathematical treatment and do not involve the physical nature of the hot deformation process, and the application of ANN model is inseparable from the availability of extensive, high-quality data, and characteristic variables. As a typical phenomenological model, the Arrhenius type equation has been extensively used in metals and alloys for it can describe the relationship between the temperature, flow stress and strain rate at evaluate temperature [14, 15]. However, the original Arrhenius equation is lack of suitability because the materials have obvious dynamic softening during hot deformation, since the effect of strain on the flow stress is not considered. SLOOFF et al [16] introduced a strain-dependent parameter into the hyperbolic sine constitutive equation to predict the flow stress in a wrought magnesium alloy. LIN et al [17] proposed a revised sine-hyperbolic constitutive equation which considering the activation energy (Q) and materials constant as functions of strain over a practical range of temperatures and strain rates of 42CrMo steel. Then the modified equation was successfully verified under different processing conditions by other researchers on various materials such as steels [2, 10, 11, 17, 18], magnesium alloys [19–21], aluminum alloys [3, 7, 22–24], titanium alloys [25, 26], copper alloys [27], intermetallic [28] and composites [29, 30].

The activation energy derived by constitutive analyses is usually used as an indicator to evaluate the difficulty of the hot deformation process. It also can furnish information on the microstructure and flow stress evolution in successive deformation processes to determine the deformation mechanism. Among the majority study [10–23], the activation energy usually calculated as constant for all applied hot deformation conditions. In fact, the activation energy represented mainly the free barrier to dislocation movement. As the dislocation movement is affected by external stress, temperature and microstructure during hot working process, the activation energy should not remain constant, but instead to be a function of the deformation parameters.

The object of this study is to characterize the flow behavior of ZHMn34-2-2-1 manganese brass and establish a constitutive relation between activation energy and deformation parameters. To this end, isothermal compression experiments were conducted at different temperatures and strain rates. The flow stress behavior was analyzed, and a revised constitutive model incorporating the effects of strain, strain rate and temperature was derived considering the compensation of strain, strain rate and temperature increment. The validity of the proposed constitutive model was also examined over the whole experimental range.

2 Material and experimental procedure

The material used in this investigation was ZHMn34-2-2-1 alloy and its nominal composition is listed in Table 1. As-cast cylindrical specimens with diameter of 5.6 mm and height of 8.4 mm were machined parallel to vertical axis for the hot compression tests. A lubricant composed of graphite and machine oil was employed to reduce friction between specimens and fixture. Hot compression tests were conducted on Thermecmastor-Z thermal simulator with selected strain rates of 0.01, 0.1, 1, 10 s^–1, where deformation temperatures range from 923 to 1073 K at an interval of 50 K. Each specimen was heated to the forming temperature at the rate of 10 K/s from ambient temperature to the deformation temperature. Prior to compression tests, the specimens were held for 3 min by thermocouple-feedback-controlled AC to eliminate the thermal gradients and decrease material anisotropy. The reduction of height was 60% at the end of compression test. The microstructures of deformed specimens were stabilized by cooling down in liquid nitrogen immediately after compression tests.

Table 1 Chemical compositions of ZHMn34-2-2-1(mass fraction, %)

3 Result and discussion

3.1 Friction correction

During the hot compression process, the stress state changes from one dimension to three dimension because of the interfacial friction between specimen and stroke. In practice, the middle parts of specimens are often formed with a drum. The negative friction effect results in a higher flow stress than its actual value in nature, even the lubricating measures are taken to minimize the friction. In order to accurately describe the flow behavior of the studied alloy, a correction of friction is necessary. Figure 1 shows the friction-corrected flow curves under different temperatures at strain rates of 0.01 s^–1. The corrected flow curves are lower than the measured ones at all the conditions, indicating a negative effect of interfacial friction on flow stress. Meanwhile, the decrease in flow stress is more remarkable with increasing strain. This can be ascribed to the increase in the size of interface between the specimen and stroke under hot compression with increasing deformation degree [23].

Figure 1 Comparison between friction-corrected flow stress and measured flow stress at strain rate of 0.01 s^–1

3.2 Flow behavior

The corrected true stress–strain curves for ZHMn34-2-2-1 under deformation testing are depicted in Figure 2. It is clearly summarized that the effects of temperature and strain rate on the flow stress are significant for all tested conditions. The curves abruptly exhibit a peak stress once plastic deformation begins, followed by the flow monotonous decrease of flow stress until it ultimately reaches a plateaus. This shape of curves indicates the competing deformation mechanisms of work hardening, dynamic recovery (DRV) and dynamic recrystallization (DRX) [31]. In the first stage, the dislocation generation and multiplication are dominant, leading to the obviously work hardening.

Meanwhile, the dynamic recovery induced by dislocation climb, cross-slip and annihilation is weak due to the lower stacking fault energy [32]. The dynamic recovery mechanism is inadequate to balance work hardening. In consequence, the flow stress rapidly increases to a peak value. In the second stage, the dislocation density continuously rises, the driving force and rate of recovery also tend to increase. Meanwhile, the nucleation and growth of new grains occur during deformation. The dynamic recrystallization occurs when the accumulated dislocation density exceeds a critical strain. With the increasing strain, the dynamic softening becomes more and more predominant at this stage. At last, the steady state results from equilibrium between work hardening and dynamic softening (DRV and DRX).

It is noticeable that the flow behaviors are strongly sensitive to the strain rate and deformation temperature. From Figure 2, it can be found that the stress level decreases with an increase of deformation temperature at fixed strain rate. This can be mainly ascribed to the fact that lower strain rate provides sufficient time for energy accumulation, which can neutralize the work hardening effect [33]. However, under the relatively high strain rate, the reduction of deformation time restrains the growth of DRX grains and increases the work hardening effect [34]. From Figure 2, it also can be found that the flow stress increases with the decrease in deformation temperature. On one hand, the obstructions of dislocation motion and crystal slip become easy due to the increased averaged kinetic energy of atoms at relatively high deformation temperatures [35]. On the other hand, higher temperature promotes the mobility at grain boundaries which results in dislocation attenuation, the nucleation and growth of dynamically recrystallized grains [36]. So, the DRX is accelerated, and the flow stress decreases.

Figure 2 Comparison between experimental and predicted flow stress calculated from strain-dependent constitutive model at different strain rates:

3.3 Constitutive equation for flow behavior

According to the compression test, the material constants of constitutive equation can be derived from the obtained stress–strain data. Amongst the multitudinous phenomenological approaches, the Arrhenius equation is widely employed for its high accuracy to describe the relationship among strain rate, temperature and flow stress, especially at elevated temperatures [37]. The effects of temperature and strain rate on hot deformation behavior of metallic materials can be incorporated in a temperature-compensated strain rate factor, the Zener–Hollomon parameter (Z) [14] in the following exponent-type equation:

                           (1)

where denotes the strain rate (s^–1); R is universal gas constant (8.314 J/(mol·K)); T is absolute temperature in Kelvin (K); Q is the activation energy of hot deformation (kJ/mol).

In addition, the parameter Z can be expressed as a function of stress in power law, exponential law and hyperbolic sine law [15] as follows:

Z=AF(σ)                                (2)

               (3)

where σ is the characteristic stress (MPa), for instance peak stress, steady stress or a corresponding stress to a specific strain. Hereby we use the peak stress (σ_p) for calculation. A, α, β, n and n₁ are the material constants, α=β/n₁.

The value of α represents the stress reciprocal at which the material deformation changes from power to exponential stress dependence [29]. For the low stress level (ασ<0.8), the description of stress is appropriate for creep by power law. Contrarily, in the high stress level (ασ>1.2), the exponential law performs better for high strain rates and relatively temperatures. However, the hyperbolic sine law in Arrhenius type equation gives better approximations for the entire range of stress.

Substituting all the expressions in Eq. (3) into Eq. (2) gives:

        (4)

where A₁ and A₂ are material constants related to temperature.After logarithm of both sides of Eq. (4), the following equations can be obtained:

                (5)

                  (6)

Relationship of stress and strain rate can be obtained based on the peak flow stress measured from true stress–strain curve, as plotted in Figure 3. The values of material constants n₁ and β can be evaluated from the mean slopes of lines in lnσ– plot and σ–plot by linear regression method, respectively. The values of n₁ and β are 3.8858 and 0.1507 MPa^–1, respectively. Subsequently, the value of α is evaluated using α=β/n₁ and equals to 0.0388 MPa^–1.

Taking the logarithm of both sides of Eq.(4) yields:

              (7)

Using previous estimated constant α value, the material constant n can be derived from average

slopes of linear plot betweenusing linear regression analysis under different temperatures as shown in Figure 3(c).Meanwhile, the mean value of n becomes 2.8038.

Figure 3 Relationship between stress and strain rate:

The activation energy Q can be expressed as the following equation by differentiating Eq.(7) at a given strain rate with respect to 1/T:

  (8)

where S is the mean slope of plots of ln[sinh(ασ)]–1/T at various strain rates.

Hence, the value of S can be derived by substituting the values of strain rates, temperatures and peak stresses into Eq. (8), as demonstrated in Figure 3(d). Based on average values of Q under different strain rates, the value of Q can be determined. Finally, the averaged value of Q is 182.5699 kJ/mol.

In addition, the value of A can be derived from the intercept of linear fitting of ln[sinh(ασ)]–T^–1 with previously estimated Q which is substituted in Eq. (7). Then A=8.559×10⁸ s^–1 can be easily calculated.

As illustrated in Figure 2, the experimental flow stress does vary with strain whereas the Arrhenius-type constitution relation neglects the effect of strain. Hence, compensation of strain should be taken into account for the sake of establishing a precise constitutive equation to predict the flow stress. It is suitable to incorporate the influence of strain in constitutive equation by assuming the material constants (α, n, Q and A) as polynomial functions of strain [16]. In this study, the material constants under different strains ranging from 0.1 to 0.8 with an interval of 0.05 were calculated, as plotted in Figure 4. These values were employed to fit polynomial functions.

The variations can be presented with good correlation and generalization by sixth order polynomial, as portrayed in Eq.(9).

 (9)

Figure 4 Variation of α(a), n (b), Q (c) and lnA (d) with true strain by polynomial fitting

Once the material constants are evaluated, the flow stress can be described by Zener–Hollomon parameter as:

   (10)

Applying the calculated material constants aforementioned into Eq. (10), the predicted flow stress values can be obtained. By comparing the predicted and experimental flow stress, the accuracy of the constitutive equation considering compensation of strain is verified, as illustrated in Figure 2. As observed from these figures, the predicted flow stresses are consistent with the experimental results under high temperatures (1073 K) as well as strain rate of 0.01 s^–1. Whereas, a considerable deviation between experimental and computed flow stress is observed at 1–10 s^–1 under lower temperatures (923–1023 K). At strain rates of 0.1 s^–1 and 1 s^–1, a more specific observation shows that the constitutive equation over calculated the flow stress. However, the contrary conclusion is obtained for the strain rate of 10 s^–1. Another noteworthiness is that the deviation values decrease when temperature increases.

The deviation is contributed to the simplifications, both n and S are considered to be materials constants. The activation energy Q value became a constant independent of the deformation conditions because the calculation is only based on the mean values of n and S. Based on the experimental data, the values of n under different temperatures at a given strain are plotted in Figure 5(a), which shows discrepant tendencies with strain increasing. Also can be noticed that the values of n increase with temperature increasing, while the deviations between the values and average value at different temperatures are nonlinear relations. Therefore, the value of n can be considered as a variable related to deformation temperature and strain. The values of S at each strain rate are calculated as shown in Figure 5(b), where they increase with the strain rate increasing. At the strain rate of 0.01–1 s^–1, the values show approximately tendency with the average value while the strain increasing. Especially at the strain rate of 1 s^–1, the values almost track the average value. The values increase obviously at 1–10 s^–1. At the strain rate of 10 s^–1, the values are significantly deviating from the average value, especially at large strain conditions. It is evident that n and S are not material constants, but vary with temperature and strain, strain rate and strain respectively, which indicate strong dependencies on deformation conditions. The variations of n with deformation temperature and S with strain rate have also been observed [24]. The values of Q under different strain rates at a given strain are calculated as shown in Figure 5(c), which has the same deviation tendency according to the values of S. The activation energies at different temperatures and strain rates are calculated as plotted in Figure 6. At the strain rates of 1–10 s^–1, the activation energy increases rapidly, while the deviation between calculated activation energy and average activation energy are dramatically larger than that of 0.01–0.1 s^–1. The activation energy increase with the strain rate is contributed to the dislocation multiplication. The high dislocation density increases the possibility of dislocation tangling and thus reduces the mobility of dislocation, leading to enhancement of the flow stress [38].

The variation of n and S resulting the deviation of Q, the activation energy is closely related with strain rate, temperature and strain. Thus, it can be indicated that the activation energy in hot deformation is not adequately to be considered solely by a constant, but treated as a function of strain rate and deformation temperature. Hence, the constitutive equation should be modified by taking into account the coupled effects of thermo- mechanical factors on the evolution of the activation energy to ameliorate the predictability through the applied deformation conditions.

3.4 Modification of constitutive equation

As discussed in Section 3.3, the material parameters (n, A, and Q) in Arrhenius-type constitutive model are more reasonable to be regarded as variables. To overcome the shortcomings of the original equation, a revised equation which considers the coupled effects of the strain rate, deformation temperature and strain on the materials parameters (n, A, and Q) is proposed as follows:

   (11)

where σ_p is the peak stress, the material variable n is a function of deformation temperature and strain, the material variables A, and Q are functions of the strain rate, temperature and strain.

Figure 5 Relationship between n and (a), S and(b), Q and(c), lnA and(d) under different strains

Figure 6 Evolution of activation energy value (kJ/mol) under different temperatures and strain rates:

Taking natural logarithm on both sides of Eq. (11) yields

 (12)

Taking partial differential of Eq. (12) at given strain rate and strain,

     (13)

                       (14)

The term P can be assumed to zero approximately according to previous study [39]. Then Eq. (13) can be simplified as follows:

                  (15)

Based on the multivariate polynomial fitting, an excellent relationship between n, temperature and strain is shown in Figure 7(a). Likewise, the good relationship between S, strain rate and strain, as shown in Figure 7(b). The material variables n and S can be expressed by the following correlation formulas:

                      (16)

(17)

In an attempt to gain more insight into the effect of deformation conditions on activation energy, the variations in activation energy value as a function of deformation temperature, strain rate and strain are shown in Figure 6(b) and Figure 7(d). The mean error for the activation energy under different temperature and strain rate is determined to be 6.303%, while the mean error for the activation energy under different strain and strain rate is calculated to be 0.87%, which demonstrate an excellent agreement with the experimental data.

Likewise, the good relationship between lnA, strain and strain rate can be derived from Eq. (18), as shown in Figure 7(c).

              (18)

Figure 7 Variation of n (a), S (b), lnA (c) and Q (d) by polynomial fitting

The comparison of experimental data and predicted data after modified is shown in Figure 8. An accordance between experimental and computed values is satisfactory for most of experimental conditions. The revised constitutive equation could track the deformation behavior more accurately than the former strain-dependent model.

3.5 Verification of modified constitutive equation

In order to quantify the predictive validity of modified constitutive equation incorporated the compensation of strain, strain rate and temperature increment, correlation coefficient (R), root mean square error (RMSE) and average absolute relative error (AARE) are introduced. The corresponding calculations are as follows:

              (19)

                 (20)

               (21)

where E is the experimental flow stress and P is the predicted flow stress;andare the mean values of E and P, respectively; N is the total number of data employed in this work.

The correlation coefficient reflects the strength of linear relation between the experimental and predicted values. However, sometimes higher value of R may not necessarily indicate a better performance due to the tendency of the equation could be biased. Root mean square error is the standard error and the average absolute relative error is an unbiased estimator for evaluating the predictability of the constitutive equation.

Hence, if the values of RMSE and AARE are in low level, the performance of the proposed constitutive equation will be better. The values of R, RMSE, and AARE between experimental and predicted data from the modified constitutive equation were found to be 0.997, 0.853 MPa and 2.363%, respectively, as shown in Figure 9(b). For comparison, the statistical values obtained from the former constitutive equation are also shown in Figure 9(a). It can be seen that the value of R from the modified constitutive equation is relatively higher, meanwhile, the values of RMSE and AARE are in low level. A more concrete observation shows that the AARE values of modified model are lower through the whole strain range and temperatures in Figure 10. Compared with the strain-dependent model, the AARE values of the revised model are significantly lower under most of the deformation conditions. The result of comparison indicated that the predictability of the constitutive equation has been ameliorated after modification.

Figure 8 Comparison between experimental and predicted flow stress derived from revised model at different strain rates:

Figure 9 Correlation between experimental and original predicted flow stress (a) and correlation between experimental and modified predicted flow stress (b)

4 Conclusions

1) The constitutive behavior of ZHMn34-2-2-1 alloy has been analyzed through isothermal compression tests at temperature ranges of 923–1073 K and strain rates of 0.01–10 s^–1. The deformation characteristics of ZHMn34-2-2-1 alloy indicate that the flow stress increases when strain rate increases or deformation temperature decreases. Typical work hardening and dynamic recrystallization features can be observed from the flow curves.

Figure 10 Comparison between AARE values of strain-dependent model and modified model under different conditions:

2) The deformation conditions have a great effect on material parameters of ZHMn34-2-2-1 alloy for hot deformation. The material parameters (n, A, and Q) in Arrhenius type constitutive model are more reasonable to be regarded as variables. A revised constitutive equation which considered the effect of strain, strain rate and temperature on material parameters provides great predictions of the deformation behavior of ZHMn34-2-2-1 alloy at elevate temperature.

3) The reliability of the proposed constitutive equation was quantified in terms of the standard statistical method. The statistical values of R, RMSE and AARE derived from modified constitutive equation were 0.997, 0.853 MPa and 2.363%, respectively, confirming a better predictability in describing the constitutive relationships compared to strain-compensated constitutive equation.

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(Edited by HE Yun-bin)

中文导读

高强锰黄铜高温热流变行为及本构方程的建立

摘要：通过Thermecmastor-Z热模拟试验机研究了ZHMn34-2-2-1锰黄铜在温度为923~1073 K和应变速率为0.01~10 s^–1的加工范围内的热压缩变形行为。真应力–应变曲线显示随着应变增加应力达到峰值，随后单调递减达到稳态，表现出明显的动态再结晶行为。将建立的基于应变补偿的Arrhenius本构模型预测结果和实验数据进行偏差分析，结果表明在该本构模型中，ZHMn34-2-2-1锰黄铜材料参数(n, A and Q)随着变形条件的波动，不能被简单地看作一组常数。随后提出了考虑变形条件对材料参数耦合效应补偿的改进型本构模型。利用相关系数和平均绝对相对误差对已建立本构模型的精度进行评价，结果表明所提出的本构模型可以准确地描述ZHMn34-2-2-1锰黄铜的热变形特性。

关键词：本构模型；锰黄铜；热激活能；热变形

Foundation item: Project(2012ZX04010-081) supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China

Received date: 2017-01-16; Accepted date: 2017-09-27

Corresponding author: WANG Meng-han, PhD, Associate Professor; Tel: +86–13637957075; E-mail: cquwmh@163.com