Hot Deformation Behavior and Constitutive Analysis of As-Extruded Mg–6Zn–5Ca–3Ce Alloy Fabricated by Rapid Solidification

Abstract

The plasticity of Mg–6Zn–5Ca–3Ce alloy fabricated by rapid solidification (RS) at room temperature is poor due to its hexagonal-close-packed (HCP) structure. Therefore, hot deformation of RS Mg–6Zn–5Ca–3Ce alloy at elevated temperature would be a major benefit for manufacturing products with complex shapes. In the present study, hot deformation behavior of as-extruded Mg–6Zn–5Ca–3Ce alloy fabricated by RS was investigated by an isothermal compression test at a temperature (T) of 573–673 K and strain rate ($\dot{\epsilon }$) of 0.0001–0.01 s⁻¹. Results indicated that the flow stress increases along with the declining temperature and the rising strain rate. The flow stress behavior was then depicted by the hyperbolic sine constitutive equation where the value of activation energy (Q) was calculated to be 186.3 kJ/mol. This issue is mainly attributed to the existence of fine grain and numerous second phases, such as Mg₂Ca and Mg–Zn–Ce phase (T’ phase), acting as barriers to restrict dislocation motion effectively. Furthermore, strain compensation was introduced to incorporate the effect of plastic strain on material constants ($\alpha$,$Q$,$n$,$\ln A$) and the predicted flow stresses under various conditions were roughly consistent with the experimental results. Moreover, the processing maps based on the Murty criterion were constructed and visualized to find out the optimal deformation conditions during hot working. The preferential hot deformation windows were identified as follows: T = 590–640 K, $\dot{\epsilon }$ = 0.0001–0.0003 s⁻¹ and T = 650–670 K, $\dot{\epsilon }$ = 0.0003–0.004 s⁻¹ for the studied material.

1. Introduction

Magnesium (Mg) alloys, which exhibit low density, high specific stiffness, high specific strength as well as good damping properties, have great potential for applications in many aspects, such as aerospace, automobile, and 3C industries [1,2,3]. Among them, Mg–Zn (Zinc) alloys have been termed as one of the most promising alloys, as they possess superior strength, plasticity and corrosion resistance owing to the formation of metastable phase (MgZn₂ and Mg₂Zn₃) [4]. In recent years, alloying, known as a popular approach to improve the properties of magnesium alloys, has also been the focus of research on Mg–Zn alloys. Ping et al. [5] reported that Ca (calcium) element was beneficial to the refinement of grain size and its addition would significantly improve the mechanical properties and the bio-corrosion properties of Mg–Zn–Ca alloys. Moreover, the addition of Ce (Cerium) in Mg–Zn–Ca alloys resulted in the formation of a new Mg–Zn–Ce phase (T’ phase) in as-cast state, which contributes to high strength of Mg alloys [6,7]. Therefore, additions of Ca/Ce into Mg–Zn alloys surely possess great potential for practical applications of Mg alloy with enhanced mechanical properties.

Recently, rapid solidification (RS) has been treated as a superior method to refine the grain sizes and participations. Therefore, it benefits in modifying the comprehensive properties of Mg alloys. According to the previous study, the grain size of RS Mg–6Zn–5Ca alloy was refined significantly and the number of finely dispersed particles increased substantially, contributing to the compressive strength of the alloy up to 408 MPa at ambient temperature [8]. In addition, RS improved the mechanical properties of the Mg–Zn–Ca–Ce alloys at elevated temperatures due to the formation of meta-stable phases, such as Mg₂Ca and Mg₁₂Ce [4]. The aforementioned merits made RS Mg–6Zn–5Ca–3Ce alloy a potential Mg alloy. However, the subsequent processing capability of RS Mg–Zn–Ca–Ce alloy is still limited due to inadequate slip systems at ambient temperature [9,10,11] and plentiful hard-brittle phases [4]_. Fortunately, as the temperature increases, the workability of Mg alloys will be improved greatly owing to the activation of non-basal slip [12] and grain boundary slip [13]. Nowadays, many studies have been performed to investigate the hot deformation behavior of Mg–Zn–X alloy [14,15,16,17,18,19,20,21]. Liu et al. [17] calculated the activation energy of the casted Mg–3Zn–Ca–0.5Sr alloy (250.44 kJ/mol) and drew the processing map with a peak power dissipation efficiency of 42%, which was observed at 603–633 K and 0.005–0.03 s⁻¹. Gao et al. [18] researched the stress–strain curves of the Mg–2.3Zn–0.4Mn–0.2Ce alloy and concluded that the alloy exhibited a typical dynamic recrystallization (DRX) behavior with a single peak stress. Ma et al. [19] studied the thermal deformation mechanism of an extruded Mg–16Al alloy and identified the appropriate deformation window, where the temperature range was 633–673 K and the strain rate range was 0.001–0.1 s⁻¹. Wang et al. [20] reported that many fine grains were introduced by DRX around the coarse grains at 200 °C and 1 s⁻¹, increasing the engineering strain and promoting the work hardening.

However, few research works have been conducted previously with respect to flow behavior and workability of Mg–Zn–Ca–Ce alloys during hot deformation, especially for RS Mg–Zn–Ca–Ce alloys, which hinders their engineering application. Thus, understanding the hot deformation behavior and formulating some appropriate hot working conditions of RS Mg–Zn–Ca–Ce alloys are of great importance. In the present study, the true strain-stress curves of RS Mg–Zn–Ca–Ce alloy were firstly obtained by thermal compression experiments. Afterwards, constitutive equations and processing maps were calculated and constructed to investigate the deformation behavior and to identify the optimal processing window of the studied material.

2. Material and Methods

The Mg alloy with a composition of Mg–6Zn–5Ca–3Ce was produced by RS using pure (99.9%) magnesium, calcium, zinc, and cerium, and its chemical composition is shown in

Table 1. Nominal chemical composition of RS Mg–6Zn–5Ca–3Ce alloy.

Alloy	Zn (wt%)	Ca (wt%)	Ce (wt%)	Mg
Mg–6Zn–5Ca–3Ce	6	5	3	Balance

. The as-received RS flakes were hot pressed at 573 K in an Ar (Argon) atmosphere. Then, the hot-pressed billets were extruded in a YJ32-315A four-column hydraulic press (Jiangdong Machinery Co., Ltd., Chongqing, China) through a cylindrical die of 50 mm inner diameter at an extrusion speed of 0.1 mm/min, temperature of 653 K, and an extrusion ratio of 39:1.

The compressive specimens were machined with a diameter of 7 mm and a height of 10.5 mm along the extrusion direction. The process of thermal hot compression and the dimensions of specimen were schematically illustrated in Figure 1. Uniaxial compression tests were conducted at the temperature of 573 K, 623 K and 673 K as well as the strain rates of 0.0001 s⁻¹, 0.001 s⁻¹ and 0.01 s⁻¹ on the Gleeble-1500 thermo-mechanical simulator (Data Science International, INC, St. Paul, MN, USA). Graphite lubricant was applied on surfaces between the specimen and press parts to guarantee good friction condition. All specimens were then deformed to the total true strain of 0.7. Before compression, the specimens were heated to deformation temperatures at a heating rate of 5 K/s and then held for 300 s in order to obtain a stable and uniform temperature. In addition, for an easier understanding, the physical quantities and abbreviations encountered in this study are listed in

Table 2. Nomenclatures of the physical quantities and abbreviations.

$\dot{\epsilon }$	Strain rate(s−1)
$\epsilon$	Strain
Q	Activation energy (kJ/mol)
R	the universal gas constant (8.314 J/(mol*K))
σ	Flow stress (MPa)
A	Temperature-independent constant
α	Stress level parameter
β	Hyperbolic sine exponent
n	Stress exponent
Z	Zener-Holloman parameter
η	Power dissipation coefficient
P	Total power
G	Consumption of plastic deformation
J	Consumption of microstructural evolution
m	Strain rate sensitivity index
TEM	Transmission electron microscopy
DRX	Dynamic recrystallization
DRV	Dynamic recovery
DDRX	Discontinuous dynamic recrystallization

.

Microstructure characterization of the deformed specimens was conducted by transmission electron microscopy (TEM, Tecnai G220, FEI Company, Hillsboro, OR, USA) operated at 200 kV. The specimens for TEM observation were twin-jet electropolished using a solution of 11.6 g magnesium perchlorate, 5.3 g lithium chloride, 500 mL methanol and 100 mL 2-butoxyethanol, at −55 °C and 85 V, followed by ion milling. The TEM micrographs of as-extruded Mg–6Zn–5Ca–3Ce alloy fabricated by RS were shown in Figure 2. It is clear that after hot extrusion, the grain size was refined into the range of 350–1000 nm (compared with RS Mg–6Zn–5Ca–3Ce alloy [4]), which can be estimated directly from the TEM micrographs. Moreover, a host of particles with various sizes were distributed at the grain boundary and in the grain interior. Furthermore, the morphology of particles is quite different from each other, including spherical, elliptic and block particles. It was reported that these particles were Mg₂Ca and Mg–Zn–Ce phase (T’ phase) [22].

3. Results and Discussion

3.1. Flow Stress Behavior

The true stress–strain curves for as-extruded Mg–6Zn–5Ca–3Ce alloy fabricated by RS at different deformation temperatures ranging from 573 K to 673 K and different strain rates varying from 0.0001 s⁻¹ to 0.01 s⁻¹ are shown in Figure 3. Evidently, the flow stress exhibits an increasing tendency with increasing strain rate and decreasing temperature, especially at a low temperature (Figure 3d). Moreover, the peak stress appears under an extremely low strain (even less than 0.05), indicating that the significant work-hardening effect appears at the initial deformation stage. Furthermore, the peak stress at a low temperature is higher than that of high temperature at a certain strain rate since it has more severe strain hardening in a low temperature. At the initial stage of plastic deformation, flow stress rises rapidly but does not reach the peak stress due to working hardening introduced by dislocation multiplication, pileup and tangle [23,24]. Afterwards, it increases to the peak stress slightly indicating a steady state which is a dynamic competition process between work hardening and softening by the dynamic recovery (DRV) and DRX. After that, some serrations in the stress–strain curve are observed due to the discontinuous dynamic recrystallization (DDRX) [25]. It is worth noting that when the strain rate is 0.0001 s^–1, an evident decreasing trend occurs at a high temperature (623 K and 673 K), implying that dynamic softening dominates the mechanical response after reaching the peak value.

3.2. Constitutive Analysis Based on Arrhenius Equation

In order to predict the hot workability behavior of metallic materials, different constitutive models have been constructed in order to describe the relationship between the deformation parameters. Among them, the Arrhenius constitutive model has been termed as the prevailing one due to its straightforward format and high prediction accuracy [26]. The key equation for this model would be depicted by the hyperbolic sine equation:

$$\dot{\epsilon }=A{\left[\sinh (\alpha \sigma )\right]}^n\exp \left(\raisebox{1ex}{$-Q$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.\right)$$

(1)

For the low stress level (ασ < 0.8), the value of sinh(ασ) is similar to its argument ασ; While for the high stress level (ασ > 1.2), the value of sinh(ασ) is similar to the exp(βσ)/2. Thus Equation (1) can be converted into the following equations:

$$\dot{\epsilon }=A_1{\left(\alpha \sigma \right)}^{n_1}\exp \left(\raisebox{1ex}{$-Q$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.\right)\left(\alpha \sigma <0.8\right)$$

(2)

$$\dot{\epsilon }=A_2\exp \left(\beta \sigma \right)\exp \left(\raisebox{1ex}{$-Q$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.\right)\left(\alpha \sigma >1.2\right)$$

(3)

where $\dot{\epsilon }$ is strain rate (s⁻¹), R is the universal gas constant (8.314 J/(mol*K)), Q is the deformation activation energy, T is the absolute temperature (K), σ is the peak stress (MPa), and A, A₁, A₂, α (α = β/n₁), β and n are material constants [27,28].

To simplify the equations, natural logarithms are introduced into both sides of Equations (1)–(3), which conform to the following equations:

$$\ln \dot{\epsilon }=\ln A+n\ln \left[\sinh \left(\alpha \sigma \right)\right]-\raisebox{1ex}{$Q$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.$$

(4)

$$\ln \dot{\epsilon }=\ln A_1+n_1\ln \sigma -\raisebox{1ex}{$Q$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.$$

(5)

$$\ln \dot{\epsilon }=\ln A_2+\beta \sigma -\raisebox{1ex}{$Q$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.$$

(6)

In accordance with Equations (5) and (6), $\ln \dot{\epsilon }$ is linearly related to $\ln \sigma$ and σ, respectively. Thus, the average slope of $\ln \dot{\epsilon }-\ln \sigma$ and $\ln \dot{\epsilon }-\sigma$, n1 and β, can be achieved, where $n_1=5.26,\beta =0.1435$. Afterwards, α (α = β/n1), can be calculated as 0.0273. The relationships of $\ln \dot{\epsilon }-\ln \sigma$ and $\ln \dot{\epsilon }-\sigma$ are depicted in Figure 4a,b).

Because there are three variations in Equation (4), by calculating partial differential for temperature and strain rate, respectively, the linear relationship of $\ln \left[\sinh \left(\alpha \sigma \right)\right]-\frac{1000}{T}$ and $\ln \dot{\epsilon }-\ln \left[\sinh \left(\alpha \sigma \right)\right]$ are drawn in Figure 4c,d. Therefore, the activation energy Q can be obtained:

$$Q=R{\left[\frac{\partial \ln \dot{\epsilon }}{\partial \ln \left(\sinh \left(\alpha \sigma \right)\right)}\right]}_T{\left[\frac{\partial \ln \left(\sinh \left(\alpha \sigma \right)\right)}{\partial \left(\raisebox{1ex}{$1000$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\right)}\right]}_{\dot{\epsilon }}$$

(7)

In the above equation, ${\left[\frac{\partial \ln \dot{\epsilon }}{\partial \ln \left(\sinh \left(\alpha \sigma \right)\right)}\right]}_T$ is the slop of $\ln \dot{\epsilon }-\ln \left[\sinh \left(\alpha \sigma \right)\right]$ and ${\left[\frac{\partial \ln \left(\sinh \left(\alpha \sigma \right)\right)}{\partial \left(\raisebox{1ex}{$1000$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\right)}\right]}_{\dot{\epsilon }}$ is that of $\ln \left[\sinh \left(\alpha \sigma \right)\right]-\frac{1000}{T}$. It can be calculated that the mean slope of $\ln \left[\sinh \left(\alpha \sigma \right)\right]-\frac{1000}{T}$ is 6.09 and that of $\ln \dot{\epsilon }-\ln \left[\sinh \left(\alpha \sigma \right)\right]$ is 3.68. Accordingly, the current activation energy, Q, is estimated to be 186.3 kJ/mol, which is higher than that of the self-diffusion in pure Mg. Zheng et al. [29] researched the effect of the addition of MgO on the plastic deformation behavior of Mg alloy and indicated that the lower activation energy (150.3 kJ/mol) can be attribute to the fine grains and second phases. In this study, the moderate Q value can be attributed to the exceptional fine grain size and the occurrence of numerous second phases, such as Mg₂Ca and Mg–Zn–Ce phases (T’ phase). These precipitations with good thermal stability could act as barriers to restrict dislocation motion effectively [2].

For the purpose of reflecting the influence of deformation temperature and strain rate, the Zener–Holloman parameter (Z) will be introduced. Combined with the hyperbolic sine function (Equation (1)), the Z parameter is shown in Equation (8).

$$Z=\dot{\epsilon }\exp \left(\frac{Q}{RT}\right)=A{\left[\sinh \left(\alpha \sigma \right)\right]}^n$$

(8)

The relationship of $\ln \left[\sinh \left(\alpha \sigma \right)\right]$ and $\ln Z$ can be illustrated in Figure 5. Based on linear fitting, the intercept ($\ln A$) can be obtained and subsequently the $A$ can be calculated as $8.8\times {10}^{11}$. Ultimately, the σ can be calculated by the following equation:

$$\sigma =\frac{1}{\alpha }\ln \left\{{\left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}+{\left[{\left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}+1\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}$$

(9)

Substitute the calculated value, $\alpha$,$Q$,$n$ and $A$ into Equation (1), and the constitutive equation for hot compression of as-extruded Mg–6Zn–5Ca–3Ce fabricated by the RS alloy can be calculated as:

$$\dot{\epsilon }=8.8\times {10}^{11}{\left[\sinh \left(0.0273\sigma \right)\right]}^{3.13}\exp \left(-\frac{186300}{8.314T}\right)$$

(10)

However, intensive research has demonstrated that plastic strain significantly influences material constants within the entire strain range [30,31]. In order to predict the hot deformation behavior and flow stress more accurately, the material constants ($\alpha$,$Q$,$n$ and $\ln A$) in various strains at an interval of 0.05 are calculated via the coded Matlab program. The corresponding calculated results are shown in

Table 3. Calculated material constants ($\alpha$,$Q$,$n$ and $\ln A$) of as-extruded Mg–6Zn–5Ca–3Ce alloy fabricated by RS.

Material Constant	$\mathit{\alpha }$	$\mathit{Q}\left(\mathit{J}/\mathit{m}\mathit{o}\mathit{l}\right)$	$\mathit{n}$	$\mathit{l}\mathit{n}\mathit{A}$
$\epsilon$	$\dot{\epsilon }=A{\left[\sinh \left(\alpha \sigma \right)\right]}^n\exp \left(\raisebox{1ex}{$-Q$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.\right)$
0.05	0.0277	188,910	3.9056	27.9870
0.1	0.0278	187,360	3.5316	27.7654
0.15	0.0278	190,060	3.5626	28.2665
0.2	0.0281	180,540	3.3224	26.5097
0.25	0.0278	180,420	3.3674	26.5022
0.3	0.0281	177,710	3.2620	26.0034
0.35	0.0279	171,520	3.1663	24.8966
0.4	0.0278	165,520	3.1060	23.7974
0.45	0.0279	164,540	3.0390	23.6428
0.5	0.0280	168,580	3.0782	24.3824
0.55	0.0278	163,070	3.0099	23.3730
0.6	0.0276	160,340	2.9864	22.8882
0.65	0.0275	157,600	2.9442	22.4021
0.7	0.0272	159,140	2.9786	22.7239

. Clearly, there indeed exists an obvious difference in strain varying from 0.05 to 0.7. Thus, it is necessary to take the strain effect into account.

According to the principle of regression analysis, the relationship between the material constants and ε could be depicted by polynomial fitting. In the present study, a sixth order polynomial is applied and the containing terms are shown in

Table 4. The relationship between material constants and ε using sixth degree polynomial.

Material	Mg–6Zn–5Ca–3Ce
constant	$Z=\dot{\epsilon }\exp \left(\raisebox{1ex}{$Q$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.\right)$
	$\sigma =\frac{1}{\alpha }\ln \left\{{\left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}+{\left[{\left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}+1\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}$
$\alpha$	$0.02797-0.0111\epsilon +0.15334{\epsilon }^2-0.79727{\epsilon }^3+1.95385{\epsilon }^4-2.27157{\epsilon }^5+1.0031{\epsilon }^6$
$Q$	$197.3-315.3\epsilon +3979.9{\epsilon }^2-23050.1{\epsilon }^3+60675.7{\epsilon }^4-73992.7{\epsilon }^5+34052.1{\epsilon }^6$
$n$	$4.75-25.59\epsilon +209.6{\epsilon }^2-887.1{\epsilon }^3+1939.8{\epsilon }^4-2097.3{\epsilon }^5+887.6{\epsilon }^6$
$\ln A$	$29.3-51.9\epsilon +679.6{\epsilon }^2-4024.6{\epsilon }^3+10717.1{\epsilon }^4-13164.3{\epsilon }^5+6090.1{\epsilon }^6$

as well as the specific constitutive equations of as-extruded Mg–6Zn–5Ca–3Ce alloy fabricated by RS. Moreover, the regression curves of material constants are presented in Figure 6.

Based on material constants under different strains in Table 3 and Table 4, the flow stress at a specific deformation condition can be calculated using Equation (10) and the calculated stress–strain curves of as-extruded Mg–6Zn–5Ca–3Ce alloy fabricated by RS can be obtained, as shown in Figure 7. These calculated flow stresses under various deformation conditions globally agree with the correspondingly experimental results.

3.3. Processing Maps

The power dissipation map is usually applied to describe the microstructure evolution during the hot deformation. The thermal deformation material is considered as a non-linear energy dissipation unit in the dynamic material model (DMM) [12,16,32]. During hot deformation, the total power P is converted into the G content and J co-content, where G is dissipation energy consumed by plastic deformation and J is consumed by microstructure evolution Thus, the total power P can be expressed as:

$$P=\sigma \dot{\epsilon }=G+J={\displaystyle {\int }_0^{\dot{\epsilon }}\sigma }d\dot{\epsilon }+{\displaystyle {\int }_0^{\sigma }\dot{\epsilon }d\sigma }$$

(11)

The relationship among flow stress σ, the strain rate stress $\dot{\epsilon }$ and the dynamic constitutive equation under a given temperature can be depicted as:

$$\sigma =K{\dot{\epsilon }}^m$$

(12)

where m represents the strain rate sensitivity index and K is the material constant. Moreover, m is the proportion of dissipative power between G and J, which is defined as:

$$m=\frac{dJ}{dG}={\left[\frac{\partial \left(\ln \sigma \right)}{\partial \left(\ln \dot{\epsilon }\right)}\right]}_{\dot{\epsilon },T}$$

(13)

Supposing that the material is an ideal linear dissipation body, J obtains to the maximum as m = 1 (that is ideally plastic flow):

$$J_{\max }=\frac{1}{2}\sigma \dot{\epsilon }$$

(14)

As for a complicated alloy system, the relationship between flow stress and strain rate will be more consistent with the Murty criterion which is widely applicable to various $\sigma -\dot{\epsilon }$ versus. Thus, the G content can be written as:

$$G={\displaystyle {\int }_0^{\dot{\epsilon }}\sigma d\dot{\epsilon }=}{\displaystyle {\int }_0^{{\dot{\epsilon }}_{\min }}\sigma d\dot{\epsilon }+{\displaystyle {\int }_{{\dot{\epsilon }}_{\min }}^{\dot{\epsilon }}\sigma d\dot{\epsilon }=}}{\left(\frac{\sigma \dot{\epsilon }}{m+1}\right)}_{\dot{\epsilon }={\dot{\epsilon }}_{\min }}+{\displaystyle {\int }_{{\dot{\epsilon }}_{\min }}^{\dot{\epsilon }}\sigma d\dot{\epsilon }}$$

(15)

where ${\dot{\epsilon }}_{\min }$ means the minimum strain rate during hot deformation.

The power dissipation coefficient, η, is defined as $\raisebox{1ex}{$J$}\!\left/ \!\raisebox{-1ex}{$J\max $}\right.$ which is a dimensionless parameter to access the power dissipation. Thus, η can be expressed as:

$$\eta =\frac{J}{J_{\max }}=\frac{P-G}{J_{\max }}=2\left(1-\frac{1}{\sigma \dot{\epsilon }}{\displaystyle {\int }_0^{\dot{\epsilon }}\sigma d\dot{\epsilon }}\right)=2\left[1-\left({\left[\frac{\sigma \dot{\epsilon }}{m+1}\right]}_{\dot{\epsilon }={\dot{\epsilon }}_{\min }}+{\displaystyle \underset{{\dot{\epsilon }}_{\min }}{\overset{\dot{\epsilon }}{\int }}\sigma \mathrm{d}\dot{\epsilon }}\right)/\sigma \dot{\epsilon }\right]$$

(16)

On account of the maximum principles of irreversible thermodynamics, an instability criterion was constructed by Murty et al. to predict the sprout of flow instability as follows [32,33,34]:

$$2m<\eta$$

(17)

The unstable flow caused by the adiabatic shear will occur when all power is converted into viscoplastic heat (J = 0), which means η = 0. Thus, only when the domains locate in $0<\eta <2m$ and $0<m\le 1$, will the stable material flow with DRX or DRV appear.

Finally, the power dissipation maps were established and the processing maps could be constructed by superimposing the power dissipation maps and instable maps using Matlab 2020 and Origin 8.0, as shown in Figure 8 and Figure 9. The values on counter lines symbolize the power dissipation efficiency (η) which can describe the microstructure evolution during hot deformation [35]. The shaded areas highlighted by red color mean the unstable regions while the areas in blue represent stable areas. It is clear that lower power dissipation efficiency lies in regions with lower temperature and higher strain rate, which exhibits the typically unstable flow. In the case of a relatively low temperature, DRX can hardly happen, and thus extensive dislocations are inevitably accumulated with increasing plastic deformation, causing severe working hardening [36]. Therefore, this condition trends to induce unstable flow of metallic materials. Moreover, it is too fast for a sufficient DRX to occur at a high strain rate, which contributes to dislocation multiplication, pile-ups and tangle. Consequently, this condition is also unfavorable for the hot working of metallic materials. Thus, choosing appropriate hot deformation conditions is a perquisite for ensuring that sufficient DRX occurs in the material and it possesses enough power dissipation efficiency. For the studied material, there exist two optimal deformation domains, which are the domain Ⅰ (T = 590–640 K, $\dot{\epsilon }$ = 0.0001–0.0003 s⁻¹) and domain Ⅱ (T = 650–670 K, $\dot{\epsilon }$ = 0.0003–0.004 s⁻¹), as shown in Figure 9. The power dissipation efficiency of these two domains is relatively large, where DRX would take place more completely.

4. Conclusions

The present study focused on the hot deformation behavior of as-extruded Mg–6Zn–5Ca–3Ce alloy fabricated by RS via isothermal compression tests at different temperatures and strain rates. Based on true stress–strain curves, the constitutive equations, power dissipation maps and processing maps using Murty criteria were established and the optimal deformation parameters were identified. Finally, the preferable process conditions were identified. The conclusions are as follows:

• During isothermal compression, the flow stress of the alloy increases significantly with the increasing strain rate and the declining temperature.
• The Arrhenius constitutive equation of the studied material can be described as $\dot{\epsilon }=8.8\times {10}^{11}{\left[\sinh \left(0.0273\sigma \right)\right]}^{3.13}\exp \left(-\frac{186300}{8.314T}\right)$. Strain compensation is considered to introduce the effect of plastic strain on material parameters. The calculated flow stresses at various deformation conditions are generally consistent with the experimental results.
• The dissipation maps and processing maps of as-extruded Mg–6Zn–5Ca–3Ce alloy fabricated by RS in the temperature range of 573–673 K and the strain rate range of 0.0001–0.01 s⁻¹ were constructed. The preferential processing domains can be identified, which lie in the domain Ⅰ (T = 590–640 K, $\dot{\epsilon }$ = 0.0001–0.0003 s⁻¹) and domain Ⅱ (T = 650–670 K, $\dot{\epsilon }$ = 0.0003–0.004 s⁻¹).