J. Cent. South Univ. Technol. (2010) 17: 7-12
DOI: 10.1007/s11771-010-0002-x 
Constitutive analysis of AZ31 magnesium alloy plate
YU Kun(余琨), CAI Zhi-yong(蔡志勇), WANG Xiao-yan(王晓艳),
SHI Ti(史褆), LI Wen-xian(黎文献)
School of Materials Science and Engineering, Central South University, Changsha 410083, China
? Central South University Press and Springer-Verlag Berlin Heidelberg 2010
Abstract:
nbsp; Abstract: The plastic deformation simulation of AZ31 magnesium alloy at different elevated temperatures (from 473 to 723 K) was performed on Gleeble-1500 thermal mechanical simulator at the strain rates of 0.01, 0.1, 1, 5 and 10 s-1 and the maximum deformation degree of 80%. The relationship between the flow stress and deformation temperature as well as strain rate was analyzed. The materials parameters and the apparent activation energy were calculated. The constitutive relationship was established with a Zener-Hollomon (Z) parameter. The results show that the flow stress increases with the increase of strain rate at a constant temperature, but it decreases with the increase of deformation temperature at a constant strain rate. The apparent activation energy is estimated to be 129-153 kJ/mol, which is close to that for self-diffusion of magnesium. The established constitutive relationship can reflect the change of flow stress during hot deformation.
Key words:
AZ31 magnesium alloy; hot deformation; flow stress; constitutive relationship；
1 Introduction
Magnesium alloy is lightweight material with high stiffness, excellent machinability and superior damping capacity. As such, it has a great potential to be used in automobiles to reduce vehicle mass and improve the application abilities [1-3]. However, the component forming of magnesium alloy is relatively difficult because of its limited number of slip systems with a hexagonal close-packed (HCP) lattice [4]. In magnesium, the 
basal slip takes place preferentially because the critical resolved shear stress (CRSS) for the basal slip is lower than that for non-basal slip. But the basal slip has only three independent slip systems that induce the poor ductility.
The understanding of metals and alloys behavior in hot deformation condition has a great effect for designers of metal-forming process (hot rolling, forging and extrusion) because of its effective role on metal flow pattern as well as the kinetics of metallurgical transformation [5]. It has been reported that high formability in magnesium alloys is obtained at elevated temperatures [6-7]. Generally, AZ31 alloy is a typical wrought magnesium alloy that is used as rolling and extrusion products. Therefore, the plastic deformation behavior of AZ31 magnesium alloy ingots is worthy of understanding [8]. CHENG et al [9] studied the flow stress of AZ31 magnesium at tensile tests and proposed a new mathematical model to calculate the flow stress curves, while stress-strain data were corrected for deformation heating in Ref.[10]. The effects of strain rate and deformation temperature on flow stress were discussed by SIVAPRAGASH et al [11]. Nevertheless, there exist some uncertainties about which deformation mode responsible for developing flow stress type, and which flow stress responsible for promoting a given deformation mode.
In this work, the plastic deformation simulation was performed on typical hot-rolled AZ31 magnesium alloy. The constitutive relationship was established with the experimental data. The material parameters during hot deformation were calculated to provide guidance to the reasonable development of high temperature plastic deformation processing technology of AZ31 magnesium alloy.
2 Experimental
The material used in this work was a commercial hot rolled AZ31 magnesium alloy plate with a thickness of 22 mm. The chemical composition of the alloy is shown in Table 1. The experimental specimens with 15 mm in length and 10 mm in diameter were machined with the compression axis parallel to the normal direction [12]. The specimens were deformed by uniaxial compression at constant strain rate on Gleeble-1500 thermal mechanical simulator. The temperatures were varied from 473 to 723 K. At every temperature, tests were performed at strain rates of 0.01, 0.1, 1, 5 and 10 s-1. To minimize the friction, solid lubricants were attached to the anvils [13]. The specimens were then hot- compressed at a constant strain rate to a true strain of 0.8. After the hot deformation, the specimens were water- quenched down to room temperature, to avoid microstructure modification during slow cooling from the testing temperature.
Table 1 Chemical composition of AZ31 magnesium alloy (mass fraction, %)
3 Results and discussion
3.1 Flow stress behavior at elevated temperature
The flow stress-strain behaviors of the AZ31 magnesium alloy under various strain rates at different temperatures are shown in Fig.1. Typically, the flow stress of the alloy increases to a maximum stress value with increasing the strain. Then, it decreases and finally attains to stable value. This flow stress behavior is characteristic for hot work hardening accompanied by dynamic recrystallization (DRX) [14-15]. In more detail, specific differences in the shape of the curves are evident. In comparison with strain hardening and strain softening shows that the strain rate sensitivity is more pronounced. At low strain rate 
=0.01 s-1), the specimens have good ductility even at low temperature (T=473 K) with a large true strain value of about 1. Low strain rates are favorable for plastic deformation of AZ31 magnesium alloy. The ductility decreases evidently with the increase of strain rate = 0.1-1 s-1), the specimens crack at low temperature (T=473, 523 K). In the range of experimental temperature, all specimens crack with a true strain value of 0.5-0.6 when the strain rate is above 1 s-1. This relative insensitivity to deformation tempera- ture suggests that the high strain rate ＞5 s-1) flow stress in hot-rolled AZ31 magnesium alloy is not strongly affected by temperature.
=0.01 s-1；(b) =0.1 s-1；(c) =1 s-1；(d) =5 s-1；(e) =10 s-1It is demonstrated that the ductility of AZ31 magnesium alloy increased significantly with the decrease of strain rate, exhibiting obvious strain rate sensitivity. The characteristic failure of metals at low temperatures was by fracture through the crystals themselves, whereas failure at high temperature took place at crystal boundaries [16]. This is because more dislocation involves in motion at unit time with the increase of strain rate that increases the critical strain for the onset of DRX and results in more serious strain hardening [16]. The temperature dependences of peak stress and steady state stress are very similar. Owing to the similar behavior of peak stress and steady state stress, we confine our consideration to steady state stress.
During hot working process, the flow stress (σ) of the materials with different conditions of strain strongly depends on the deformation temperature (T) and the strain rate 
The investigation of different hot working data demonstrates that the mathematic relationship between σ and 
can be described as follows [17].
In the whole scale
(1)
where A, n and α are the material parameters, R is the gas constant, and Q is the apparent activation energy. This relationship also includes the dynamic equilibrium with work hardening and softening during hot working.
BARNETT [18] showed that Eq.(1) can be used to estimate the apparent activation energy of different metals or alloys. The linear regression gives an equation of the type y=A′x+B, in which A′ denotes β or n, 
(shown in Fig.2). The average correlation coefficient is above 0.97, and the mean value of α can be computed as 0.017 806 MPa-1.
Fig.2 Relationship between stress and strain rate for AZ31 magnesium alloy: (a) 
-σ; (b) -ln σ
The stress-strain curves and the flow properties from the compressive test are also strongly dependent on the experimental temperature. In general, the maximum stress value decreases with the increase of deformation temperature. This is because low strain rate and high temperature provide longer time for energy accumulation and higher mobilities at boundaries for the nucleation and growth of DRX grains and dislocation annihilation, thus reducing the flow stress level. So, effect of the deformation temperature and strain rate on flow stress traces can be explained in terms of DRX and dislocation mechanism. The deformation mechanism associated with the isothermal compression test is a thermally activated process. DRX phenomenon is sensitive to the deformation temperature and processing time [5].
Zener and Hollomon brought forward and proved that the relation between the strain rate and the deformation temperature can be described by a parameter Z, which can be defined as follows:
(2)
where Z is the Zener-Hollomon parameter combining the two control variables through an Arrhenius equation with apparent activation energy Q. In the constitutive analysis, the effects of deformation temperature and strain rate on flow stress are adequately expressed by the following expression [10,19]:
(3)
3.2 Constitutive analysis
In the plastic deformation processes, the deformation load and energy required are decided by the flow stress. While the most important use of a constitutive equation was to calculate the flow stresses during hot working [10]. From Eq.(3), the relationship among the flow stress, strain rate and deformation temperature at elevated temperature can be expressed by the hyperbolic sine relation:
(4)
In order to calculate Q, curves of 
vs ln[sinh(ασ)] are plotted as shown in Fig.3, and curves of ln[sinh(ασ)] vs 1/T are shown in Fig.4. It is obvious that the stresses obtained form the compressed tests can be approximated by a group of parallel and straight lines in the hot deformation conditions. The value of Q can evaluate by averaging the value of linear fitting.
Fig.3 ln[sinh(ασ)] as a function of at different temperatures
Fig.4 Peak stress and steady-state flow stress as a function of reciprocal temperature
The apparent activation energy is measured to be 129-153 kJ/mol, which is in the range of the apparent activation energy of self diffusion (135 kJ/mol) [20-21]. The relationship of Q with the temperature and strain rate is shown in Fig.5. It is clear that Q changes little at high temperature, but the strain rates have a greater effect on Q. Q increases with the increase of T, especially in intermediate temperature (523-623 K). While the influence of is complex, Q increases with the increase of strain rate when ≤1 s-1, but decreases when ＞ 1 s-1.
Fig.5 Apparent activation energy as a function of temperature at different strain rates
The change of apparent activation energy with deformation temperature is similar to the effect of temperature on DRX. For the illustrative reflection the relations among deformation temperature, the strain rate and apparent activation energy, according to the experiment data, the three-dimensional graph is established, as shown in Fig.6. It is evident that there exists a strong consistent influence of temperature and stain rate on the apparent activation energy. This shows that a relationship of the apparent activation energy with the microstructure and deformation mechanism of the alloy is obtained. The increase of deformation temperature provides longer time for energy accumulation, higher mobility at boundaries, which release stress concentration and thus decrease the number of dislocation. This can result in the increase of activation energy, and the onset of dislocation climb accelerates this process.
From Eq.(3), the relationship between strain rate and flow stress can be expressed as follows:
(5)
Fig.6 Apparent activation energy in connection with deformation temperature and strain rate
where A is calculated by plotting the curve of nln[sinh(ασ)]-ln(Fig.3), A=4.97×1012 s-1.
From the definition of the double arch sine, the flow stress can be characterized by the function of strain rates and deformation temperature, and can also be described by the expression combining Zener-Hollomon parameter.
(6)
The steady state flow stress at elevated temperature is insensitive to the strain, so the influence of strain is negligible. Thus, the constitutive relationship of flow stress, strain rate and deformation temperature can be described by Z parameter.
(7)
(8)
where
(9)
The calculated flow stresses at a true strain of 0.6 are compared with the measured results (Fig.7). It can be seen that the calculated stresses coincided with the measured stresses in experimental condition at lower stress level, while they were higher than the measured stresses at higher stress level. This might be the result of materials parameters. The disadvantage of linear fitting lies in that the values of material parameters vary with different deformation temperatures and strain rates, which negatively influence the accuracy of stress-strain data.
Fig.7 Comparison between measured and calculated stresses at true strain of 0.6
4 Conclusions
(1) The flow stresses behavior is characterized by hot working accompanying with dynamic recrystallization. And the ductility of AZ31 magnesium alloy exhibits obvious strain rate sensitivity.
(2) The DRX occurs evidently during hot compressive deformation. The increase of temperature provides higher mobility at boundaries, which results in the increase of activation energy.
(3) In the analysis of the data obtained is true stress. The flow stresses usually vary with increasing the strain. The strain may have to be included in the formula for flow stress. However, the dependence of the flow stress on the strain is insignificant at elevated temperature. Therefore, the flow stress in hot working can evaluate approximately by the above formula for the true stress.
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Foundation item: Project supported by China-Canada-USA Collaborative Research and Development Project (Magnesium Front End Research and Development (MFERD))
Received date: 2009-02-25; Accepted date: 2009-05-09
Corresponding author: YU Kun, PhD; Tel: +86-731-88879341; E-mail: kunyugroup@163.com
(Edited by YANG You-ping)
Abstract: The plastic deformation simulation of AZ31 magnesium alloy at different elevated temperatures (from 473 to 723 K) was performed on Gleeble-1500 thermal mechanical simulator at the strain rates of 0.01, 0.1, 1, 5 and 10 s-1 and the maximum deformation degree of 80%. The relationship between the flow stress and deformation temperature as well as strain rate was analyzed. The materials parameters and the apparent activation energy were calculated. The constitutive relationship was established with a Zener-Hollomon (Z) parameter. The results show that the flow stress increases with the increase of strain rate at a constant temperature, but it decreases with the increase of deformation temperature at a constant strain rate. The apparent activation energy is estimated to be 129-153 kJ/mol, which is close to that for self-diffusion of magnesium. The established constitutive relationship can reflect the change of flow stress during hot deformation.
