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Full Length Article

Study on hot deformation behavior of homogenized
Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy using a combination of
strain-compensated Arrhenius constitutive model and finite element
simulation method
Li Hu, Mengwei Lang, Laixin Shi∗, Mingao Li, Tao Zhou∗, Chengli Bao, Mingbo Yang
College of Material Science and Engineering, Chongqing University of Technology, Chongqing 400054, China
Received 6 February 2021; received in revised form 29 April 2021; accepted 5 July 2021
Available online xxx

Abstract
Isothermal hot compression experiments were conducted on homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy to investigate hot deformation
behavior at the temperature range of 673–773 K and the strain rate range of 0.001–1 s − 1 by using a Gleeble-1500D thermo mechanical
simulator. Metallographic characterization on samples deformed to true strain of 0.70 illustrates the occurrence of flow localization and/or
microcrack at deformation conditions of 673 K/0.01 s − 1 , 673 K/1 s − 1 and 698 K/1 s − 1 , indicating that these three deformation conditions
should be excluded during hot working of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy. Based on the measured true stress-strain data, the
strain-compensated Arrhenius constitutive model was constructed and then incorporated into UHARD subroutine of ABAQUS software to
study hot deformation process of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy. By comparison with measured force-displacement curves,
the predicted results can describe well the rheological behavior of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy, verifying the validity
of finite element simulation of hot compression process with this complicated constitutive model. Numerical results demonstrate that the
distribution of values of material parameters (α, n, Q and ln A) within deformed sample is inhomogeneous. This issue is directly correlated
to the uneven distribution of equivalent plastic strain due to the friction effect. Moreover, at a given temperature the increase of strain rate
would result in the decrease of equivalent plastic strain within the central region of deformed sample, which hinders the occurrence of
dynamic recrystallization (DRX).
© 2021 Chongqing University. Publishing services provided by Elsevier B.V. on behalf of KeAi Communications Co. Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer review under responsibility of Chongqing University

Keywords: Mg-RE-Zn alloy; Hot deformation; Microstructure evolution; Constitutive model; Finite element simulation.

1. Introduction combination of low density and high strength [1,2]. Actu-

ally, the excellent mechanical properties of Mg-RE-Zn alloys
Unlike magnesium (Mg) and commercial Mg-Al-Zn al- mainly originate from the presence of the long period stack-
loys, which suffer from poor strength at room and elevated ing ordered (LPSO) phases, including Mg12 ZnRE (X) phase,
temperatures, Mg-RE-Zn (RE=rare earth elements) alloys Mg30 Zn60 RE10 (I) phase, Mg3 Zn3 RE2 (W) phase and so on
have attracted increasing attention in the fields of military [3]. The difference of crystal structure of these LPSO phases
and civilian applications because they possess a remarkable mainly comes from the stacking sequence along the c-axis
through an alternating stacking of face-centered cubic and
∗
hexagonal close-packed layers [4].
Corresponding authors.
E-mail addresses: shilaixin2016@cqut.edu.cn (L. Shi),
Nowadays, researchers have been trying to enhance the
zt19811118@cqut.edu.cn (T. Zhou). mechanical properties of Mg-RE-Zn alloys by composition

https://doi.org/10.1016/j.jma.2021.07.008
2213-9567/© 2021 Chongqing University. Publishing services provided by Elsevier B.V. on behalf of KeAi Communications Co. Ltd. This is an open access
article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer review under responsibility of Chongqing University

Please cite this article as: L. Hu, M. Lang, L. Shi et al., Study on hot deformation behavior of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy using a
combination of strain-compensated Arrhenius constitutive model and finite element simulation method, Journal of Magnesium and Alloys, https://doi.org/
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and/or by processing technologies. Xu et al. [5] fabricated a compensated Arrhenius model. Afterwards, the user subrou-
Mg-8.2Gd-3.8Y-1.0Zn-0.4Zr alloy sheet by virtue of vertical tine of UHARD in ABAQUS software was applied to code
direct chill casting, extrusion, hot rolling and peak-aging treat- the constructed strain-compensated Arrhenius model. Further-
ment. This sheet exhibited ultra-high strength (tensile yield more, the comparison of loading force between experimental
stress of 416 MPa and ultimate tensile strength of 505 MPa) and predicted results was performed to verify the reliability
and high ductility (elongation to failure of 12.8%). Further- of finite element simulation. Finally, simulated results, such
more, Xu et al. [6] tailored the strength (tensile yield stress as the distribution of equivalent plastic strain and material
of 466 MPa) and ductility (elongation to failure of 14.5%) parameters, were exhibited and analyzed to offer straightfor-
of Mg-8.2Gd-3.8Y-1.0Zn-0.4Zr alloy by means of hot extru- ward insights on hot deformation behavior of homogenized
sion, followed by forced-air cooling and an artificial aging Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy.
treatment. In addition, Yu et al. [7] successfully fabricated a
Mg-11Gd-4.5Y-1Nd-1.5Zn-0.5Zr alloy via hot extrusion, cold
rolling and aging treatment. The optimal mechanical proper- 2. Material and methods
ties obtained in their study are 502 MPa with regard to tensile
yield stress and 546.8 MPa as for ultimate tensile strength. The as-received plates of Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr al-
Yamada et al. [8] developed a Mg-11Gd-4Y-1.7Zn-0.5Zr alloy loy were fabricated via hot extrusion. The homogenized op-
by applying solution treatment, hot rolling and artificial aging eration (solution treatment) in the present study was con-
treatment. They reported that this material possesses a tensile ducted under the condition of heating temperature of 773 K
yield stress of ∼365 MPa and an ultimate tensile strength of and holding time of 12 h. Compressive samples with the
∼460 MPa. It is worth noting that hot deformation has been diameter of 8 mm and height of 12 mm were then cut
termed as an essential prerequisite for the fabrication of Mg- along the extrusion direction (ED) of plates by using electro-
RE-Zn alloys with excellent mechanical properties. There- discharge machining (EDM). Afterwards, isothermal hot com-
fore, for the purpose of alloy development and engineering pression experiments were performed on the Gleeble-1500D
applications, it is imperative to obtain a comprehensive un- thermo mechanical simulator at the temperature range of 673–
derstanding on the hot deformation behavior of Mg-RE-Zn 773 K and the strain rate range of 0.001–1 s − 1 . The
alloys. heating rate and hold time were chosen to be 2 K/s and
Actually, constitutive analysis of metallic materials has 300 s. All samples were compressed to the true strain of
been known as an effective approach to understand the hot de- 0.70 and then were quenched into water immediately to pre-
formation behavior of metals [9]. However, the flow behavior serve the hot-deformed microstructure. The correspondingly
of material is complicated during hot deformation as it would schematic illustration of the isothermal hot compression pro-
be significantly affected by deformation parameters including cess is demonstrated in Fig. 1(a). To conduct metallographic
deformation temperature, deformation degree and strain rate. observations on these undeformed and deformed samples, an
Therefore, an appropriate constitutive model should simul- optical microscopy (OM) was applied in the present study.
taneously consider the coupled effects of deformation tem- These samples were halved along their center axis and the
perature, strain and strain rate. As a modified form of hy- corresponding section planes were mechanical ground with
perbolic sine-typed Arrhenius model, strain-compensated Ar- 1200-grit SiC paper, followed by polishing in the chosen pol-
rhenius equation can meet the aforementioned requirements ishing solution, which contains 1 g oxalate, 1 ml nitric acid
and it has been widely applied in the studies of metallic and 98 mL H2 O. Figs. 1(b) and (c) depict the microstructure
materials during hot deformation [10,11]. For Mg alloys, Li characteristics of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr
et al. [12] established the strain-compensated Arrhenius equa- alloy before hot deformation. The correspondingly observed
tion for Mg-5Li-3Al-2 Zn alloy. The predicted mechanical surfaces, which are marked by green color in Fig. 1(a), are
responses at the temperature range of 573–623 K and the identified to be the ones parallel to ED and perpendicular
strain rate range of 0.001–1 s − 1 agree well with these mea- to ED. It is obvious that there mainly exist equiaxed grains
sured results. Fan et al. [13] compared the measured true within the homogenized material, demonstrating the occur-
stress-strain curves during hot compression and the predicted rence of complete recrystallization. In addition, a substantial
ones on the basis of strain-compensated Arrhenius equation. part of precipitate has been dissolved into the matrix of ho-
They found that the average absolute relative error (AARE) mogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy and only some
for the flow stress curve is as low as 0.030, showing a good chunky phases and cubic phases survive after solution treat-
accordance between predicted and experimental results. How- ment at 773 K/12 h. Based on the work of Li et al. [14] and
ever, literature search demonstrates that few investigations on Zhang et al. [15], the former ones belong to LPSO phase of
hot deformation behavior of Mg-RE-Zn alloys employ strain- 14H type, and the latter ones attribute to RE-rich phase. The
compensated Arrhenius model, let alone the corresponding correspondingly statistical analysis of grain size is conducted
finite element simulation. via quantitative metallography method and shown in Fig. 1(d).
Therefore, the present study aims to investigate the hot The average grain size on surface parallel to ED is identified
deformation behavior of homogenized Mg-8.5Gd-4.5Y-0.8Zn- to be 39.1 μm, which is quite close to the one (36.4 μm) on
0.4Zr alloy under specific deformation conditions (different surface perpendicular to ED, demonstrating the homogeneity
combinations of temperatures and strain rates) by using strain- of microstructure after solution treatment.

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Fig. 1. (a) Schematic illustration of isothermal hot compression process; (b) Microstructure on surface parallel to ED; (c) Microstructure on surface perpendicular
to ED; (d) Statistical analysis of grain size.

3. Results and discussion tribute to the flow softening effect. In particular, Mg alloys
with hexagonal close-packed (HCP) structure tend to occur
3.1. Flow stress behavior DRX during isothermal deformation because they usually pos-
sess relatively low stacking fault energy [18]. To validate this
Fig. 2 displays the measured true stress-strain curves of ho- issue, microstructures of all deformed samples were further
mogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy under different analyzed to clarify the occurrence of DRX.
deformation conditions. It is obvious that flow stress decreases
with increasing deformation temperature at a fixed strain rate, 3.2. Microstructure characterization
while flow stress increases with increasing strain rate at a
given temperature. In addition, it can be seen from Figs. 2(a)- Fig. 3 demonstrates the optical microstructures of hot-
(d) that, at the beginning of hot deformation, the work harden- deformed samples at low temperatures of 673 K and 698 K.
ing effect dominates the mechanical behavior and flow stress It is observed that numerous lamellar-shaped precipitates oc-
increases rapidly to the peak value. The investigations con- cur within deformed grains under these specific deformation
ducted by Wang et al. [16] and Sun et al. [17] have confirmed conditions. Similar phenomenon has also been reported by
that the work hardening effect is mainly attributed to the gen- Zhang et al. [15] and they confirmed that these phases are
eration, motion and multiplication of dislocation during hot identified to be 14H LPSO, whose formation is a diffusion-
deformation of Mg alloys. Soon after the arrival of peak value, dominated process. In addition, the pre-existing 14H LPSO
the flow softening effect dominates the mechanical behavior phases with chunky shape also survive after plastic defor-
and flow stress gradually decreases with increasing plastic mation, as marked by red arrows. It is interesting to note
strain in all true stress-strain curves. It has been generally that kink bands (KBs) occur within all deformed samples, as
accepted that the restoration mechanisms of dynamic recrys- shown in Fig. 3. Wang et al. [19] and Li et al. [20] have
tallization (DRX) and/or dynamic recovery (DRV) occurred both confirmed that KB can serve as an efficient mechanism
within hot-deformed sample of metallic materials mainly con- to accommodate plastic strain within individual grains of Mg-

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Fig. 2. True stress-strain curves of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy under different strain rates: (a) 0.001 s − 1; (b) 0.01 s − 1; (c) 0.1 s − 1;

(d) 1 s − 1 .

RE alloys during plastic deformation. Furthermore, except for Fig. 3(a), the corresponding grain size and volume faction of
samples deformed at strain rate of 1 s − 1 (Figs. 3(g) and DRX grains in Fig. 4(a) are relatively larger. This observation
(h)), DRX occurs within the remaining samples with “neck- is unsurprising because the increasing temperature contributes
lace structure” that fine DRX grains gather around coarse to the occurrence of DRX of metallic materials during plas-
un-DRX grains. The corresponding volume factions of DRX tic deformation. Besides, the microstructure characteristics of
grains remain relatively low, indicating that hot deformation deformed samples at high deformation temperature of 748 K,
at temperatures of 673 K and 698 K is unfavorable for the as shown in Figs. 5(a), (c), (e) and (g), are quite similar to the
occurrence of DRX. Lastly, it is obvious that under deforma- ones shown in Fig. 4. Moreover, under deformation condition
tion conditions of 673 K/0.01 s − 1 (Fig. 3(c)) and 698 K/1 of 773 K/0.001 s − 1 a complete DRX happens (Fig. 5(b)),
s − 1 (Fig. 3(h)), flow localization and microcrack, marked while DRX grains with necklace structure occupy a signif-
by black arrows, can be observed around grain boundaries icant volume fraction at deformation condition of 773 K/1
and within individual grains, respectively. Similarly, flow lo- s − 1 (Fig. 5(h)). It is worth noting that the precipitation of
calization also occurs at deformation condition of 673 K/1 14H LPSO phases with lamellar-shape does not occur within
s − 1 (Fig. 3(g)). These observed microstructure character- un-DRX and DRX grains at temperature of 773 K. This issue
istics demonstrate that homogenized Mg-8.5Gd-4.5Y-0.8Zn- should be attributed to the fact that deformation temperature
0.4Zr alloy is unsuitable for hot deformation under these three of 773 K is the solution temperature and therefore the precip-
deformation conditions. itation of 14H LPSO phases will not happen during plastic
The optical microstructures of samples deformed at inter- deformation.
mediate temperature of 723 K are shown in Fig. 4. It is ob-
vious that no flow localization and microcrack occur within 3.3. Strain-compensated Arrhenius constitutive model
all deformed samples and the pre-existing 14H LPSO phases
(marked by red arrows) also survive under this temperature. To efficiently predict the mechanical behavior of metal-
In addition, KBs and DRX grains can be observed within all lic materials during hot deformation, researchers have been
deformed samples. By comparison with DRX grains shown in trying to construct different constitutive models which could

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Fig. 3. Optical micrographs of deformed samples at low temperatures of 673 K and 698 K under different strain rates: (a) and (b) 0.001 s − 1; (c) and (d)
0.01 s − 1 ; (e) and (f) 0.1 s − 1 ; (g) and (h) 1 s − 1 .

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Fig. 4. Optical micrographs of deformed samples at intermediate temperature of 723 K under different strain rates: (a) 0.001 s − 1; (b) 0.01 s − 1; (c) 0.1
s − 1 ; (d) 1 s − 1 .

describe the relationship between flow stress and processing When the flow stress is at the high level, Eq. (1) can be
parameters. Among these reported constitutive models, Ar- further expressed as:
rhenius constitutive model has been termed as a popular one
ε˙ = A2 exp(βσ ) exp (−Q/RT ) (ασ > 1.6) (3)
because it possesses high simplicity and accuracy in describ-
ing the relationship among deformation temperature (T with where A2 =A/2n and β=nα.
unit K), strain rate (ε˙ with unit s − 1 ) and flow stress (σ with Adopting the operation of natural logarithm, Eq. (2) and
unit MPa) [21-23]. The key point about Arrhenius constitutive (3) are further formulated as:
model can be depicted by the hyperbolic sine-type equation
as follows: ln ε˙ = ln A1 + n ln σ − Q/RT (4)

ε˙ = A[sinh (ασ )]n exp (−Q/RT ) (1)

ln ε˙ = ln A2 + βσ − Q/RT (5)
where R refers to the universal gas constant (8.31 J
mol−1 K−1 ) and Q represents the activation energy of spe- Accordingly, as shown in Figs. 6(a) and (b), the relation-
cific material. In addition, A, α and n are material con- ships of ln ε˙−σ and ln ε˙− ln σ can be obtained and depicted
stants. In fact, the function sinh (x) can be explicitly expressed by fitting the true stress values at different temperatures in
assinh (x)=(exp(x) − exp(−x))/2. The value of sinh (x) is homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy. Afterwards,
similar to its argument x when x < 0.8, while its value of the material constant α can be determined by the equation
sinh (x) is close to the one of exp (x)/2 when x > 1.6. Conse- α=β/n.
quently, when the flow stress is at the low level, Eq. (1) can Taking the operation of natural logarithm on both sides of
be further expanded as: Eq. (1) yields the following equation:
ε˙ = A1 σ n exp (−Q/RT ) (ασ < 0.8) (2) ln [sinh (ασ )] = ln ε˙/n + Q/nRT − ln A/n (6)
where A1 is still a material constant and it can be expressed Moreover, by applying the operation of differentiation on both
as A1 =Aα n . sides of Eq. (6), the value of activation energy Q can be

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Fig. 5. Optical micrographs of deformed samples at high temperatures of 748 K and 773 K under different strain rates: (a) and (b) 0.001 s − 1; (c) and (d)
0.01 s − 1 ; (e) and (f) 0.1 s − 1 ; (g) and (h) 1 s − 1 .

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Fig. 6. Relationships between: (a) ln ε˙−σ ; (b) ln ε˙− ln σ ; (c) ln ε˙−ln[sinh(ασ )]; (d) ln[sinh(ασ )]-1000T-1 ; (e) ln Z − ln [sinh (ασ )]; (f) Correlation between
the experimental and predicted flow stresses of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy.

evaluated by the following equation: perature and strain rate on the mechanical behavior during hot
Q=R[∂ ln ε˙/∂ ln [sinh (ασ )]]T [∂ ln [sinh (ασ )/∂ (1/T )]ε˙ (7) deformation:

where the first term on the right side of Eq. (7) can be deter- Z =ε˙ exp(Q/RT ) (8)
mined on the base of the relationship of ln ε˙−ln[sinh(ασ )], Introducing Eq. (8) into Eq. (1) and then conducting the op-
as shown in Fig. 6(c), and the second term on the right eration of natural logarithm leads to Eq. (9) as:
side of Eq. (7) can be obtained by means of the relationship
ln Z = ln A + n ln [sinh (ασ )] (9)
ofln[sinh(ασ )]-1000T-1 , as shown in Fig. 6(d).
In the present study, Zener-Holloman parameter (Z) is in- It is clear that the value of ln A and n can be obtained by
troduced in order to reflect the influences of deformation tem- separately calculating the intercept and slope of fitting curves,

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Fig. 7. Relationships between material constants ((a) n; (b) α; (c) Q and (d) ln A) and true strain by using fifth-order polynomial fitting method.

which demonstrating the relationship of ln Z − ln [sinh (ασ )] Table 1

as shown in Fig. 6(e). Material parameters obtained by polynomial fitting method.
Based on these determined material constants (α, Q, ln A α 0-5 n0-5 Q0-5 A0-5
and n), the predicted true stresses at various deformation con- 0.01034 6.21059 3.3903 × 105 52.85177
ditions can be obtained, and their relationship to the corre- 0.00838 −25.13826 −5.0151 × 105 −96.67385
spondingly measured ones can be depicted in Fig. 6(f). Obvi- −0.03711 106.93182 2.1491 × 106 460.10313
ously, Arrhenius constitutive model exhibits good capability 0.18186 −275.19208 −7.7822 × 106 −1636.14731
−0.32017 374.30845 1.3345 × 107 2732.47292
on predicting the mechanical behavior of homogenized Mg- 0.19306 −198.91101 −8.0543 × 106 −1623.04676
8.5Gd-4.5Y-0.8Zn-0.4Zr alloy.
However, it is worth noting that extensive researches have
confirmed that material constants (α, Q, ln A and n) are son between measured true stress values and the correspond-
not unchanged during hot deformation of metallic materials ingly predicted ones at various plastic strains are depicted
and they are actually sensitive to the applied plastic strain in Fig. 8. The good consistency between experimental and
[11,16,24,25]. To compensate the strain effect into the evolu- predicted true stress-strain curves firmly confirms the neces-
tion of aforementioned material parameters, the values of α, sity of applying the strain-compensated Arrhenius constitu-
Q, ln A and n are repeatedly calculated under selected plastic tive model during isothermal hot deformation of homogenized
strain values, which ranges from 0.05 to 0.65 with an inter- Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy.
val of 0.05. In the present study, fifth-order polynomials are
subsequently applied to represent the relationships between α = α0 + α1 ε + α2 ε 2 + α3 ε 3 + α4 ε 4 + α5 ε 5 (10)
material constants and plastic strain, as shown in Eq. (10)-
(13). The corresponding values of these polynomial constants n = n0 + n1 ε + n2 ε 2 + n3 ε 3 + n4 ε 4 + n5 ε 5 (11)
are listed in Table 1. Besides, Fig. 7 demonstrates the vari-
ants in these material parameters under different plastic strains
Q = Q0 + Q1 ε + Q2 ε 2 + Q3 ε 3 + Q4 ε 4 + Q5 ε 5 (12)
of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy. Evidently,
plastic strain indeed influences the evolution of material con-
stants in Arrhenius constitutive model. Lastly, the compari- ln A = A0 + A1 ε + A2 ε 2 + A3 ε 3 + A4 ε 4 + A5 ε 5 (13)

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Fig. 8. Comparison between experimental and predicted true stress-strain curves at different strain rates: (a) 0.001 s − 1; (b) 0.01 s − 1; (c) 0.1 s − 1; (d) 1
s − 1.

Fig. 10. Comparison of experimental and predicted force-displacement

Fig. 9. Finite element model of isothermal hot compression. curves at 673 K/0.001 s − 1 and 673 K/1 s − 1 .

3.4. Finite element model and simulation results ther simulated by using the commercial ABAQUS software.
Fig. 9 exhibits the constructed finite element model and the
In the present study, isothermal hot compression process corresponding mesh grid, where the specimen was discretized
of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy was fur- by 8-node thermally coupled brick, trilinear displacement and

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Fig. 11. Distribution of material parameters of the strain-compensated Arrhenius constitutive model at true strain of 0.70: (a) and (b) α; (c) and (d) n; (e) and
(f) Q; (g) and (h) ln A.

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Fig. 12. Equivalent plastic strain distribution of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy during isothermal hot compression process under different
deformation conditions: (a)-(c) 673 K/0.001 s − 1 ; (d)-(f) 673 K/1 s − 1 .

temperature, reduced integration, hourglass control element

(C3D8RT) and the punches were meshed by 4-node 3-D bi-  
∂σ −Q(ε) sinh [(α(ε)σ )]
linear rigid quadrilateral element (R3D4). During hot com- HARD(3) = = 
∂T ε˙ α(ε)n(ε)RT 2 1 + sinh2 [(α(ε)σ )]
pression, the bottom punch would keep fixed, while the upper
punch moved down along the negative direction of Y-axis, as (17)
shown by the purple arrow in Fig. 9. Besides, reference point Fig. 10 demonstrates the experimental and predicted force-
named as “RP” on the bottom punch was used to apply the displacement curves of homogenized Mg-8.5Gd-4.5Y-0.8Zn-
boundary condition, while reference point named as “RP-1 0.4Zr alloy under deformation conditions of 673 K/0.001 s − 1
on the upper punch was selected and used to export the data and 673 K/1 s − 1 . It is obvious that the numerical force-
of force and displacement at each simulation increments. displacement curves exhibit high consistency with regard to
As the strain-compensated Arrhenius constitutive model is the experimental results. This observation verifies the validity
not yet included in the commercial ABAQUS software, user of incorporating the strain-compensated Arrhenius constitutive
subroutine of UHARD is applied in the present study in order model into the ABAQUS UHARD subroutine. Furthermore,
to code this constructed constitutive model. Within UHARD Fig. 11 demonstrates the distribution of material parameters
subroutine, there are four variables (SYIELD, HARD(1), of the constructed strain-compensated Arrhenius constitutive
HARD(2) and HARD (3)), which need to be mandatorily model for homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy. It
defined [26]. Actually, SYIELD represents the yield stress, can be found that a large value of α occurs within the central
HARD(1), HARD(2) and HARD (3) stand for the values of region of deformed sample at true strain of 0.70, while the
derivatives of the yield stress with respect to the strain, strain remaining material parameters (n, Q and ln A) are relatively
rate and temperature, separately. The specific expressions of small at the central region of deformed sample. Eqs. (10)-
these variables are as follows: (13) have indicated that the evolution of these material pa-
  1 
1 Z (ε) n(ε) rameters will be influenced by the value of equivalent plastic
−1
SYIELD =σ = sinh (14) strain. Therefore, the numerical results about the distribution
α(ε) A(ε)
of equivalent plastic strain at true strain of 0.70, as shown
in Fig. 12, would be further analyzed. Obviously, equivalent
HARD(1) = 0 (15)
plastic strain distributes unevenly within all deformed speci-
mens. This issue is due to the friction effect occurred on the
 
∂σ contact surfaces between the specimen and punches. Besides,
HARD(2) = even though the same true stain was employed, sample de-
∂ ε˙ T
  formed at 673 K/0.001 s − 1 sustains evidently larger equiva-
sinh1−n(ε) [(α(ε)σ )] Q(ε)
=  exp lent plastic strain in the central region than the one deformed
α(ε)A(ε)n(ε) 1 + sinh2 [(α(ε)σ )] RT at 673 K/1 s − 1 . It has been generally accepted that greater
(16) equivalent plastic strain directly correlates to larger stored en-

12
JID: JMAA
ARTICLE IN PRESS [m5+;August 4, 2021;19:31]
L. Hu, M. Lang, L. Shi et al. Journal of Magnesium and Alloys xxx (xxxx) xxx

ergy during hot deformation of metallic materials [27]. There- 51701034), the Scientific and Technological Research Pro-
fore, the existence of DRX in Fig. 3(a) and the non-existence gram of Chongqing Municipal Education Commission
of DRX in Fig. 3(g) can be explained that sufficient stored (Grant Nos. KJQN201801137, KJ1600922), the Basic and
energy is an essential prerequisite of the occurrence of DRX. Advanced Research Project of Chongqing Science and
Technology Commission (Grant Nos. cstc2017jcyjAX0062,
Conclusions
cstc2018jcyjAX0035), the Chongqing University Key Labo-
ratory of Micro/Nano Materials Engineering and Technology
In the present study, hot deformation behavior of homog-
(Grant Nos. KFJJ2003).
enized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy was thoroughly in-
vestigated via isothermal hot compression experiments at the References
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Acknowledgments (2019) 108019.

This work was supported by the National Natu-

ral Science Foundation of China (Grant Nos. 51805064,

13