![(a) Relationships between ln [sinh(ασ)] and lnε˙ at different temperatures; (b) Relationships between ln [sinh(ασ)] and 1000/T at different strain rates.](https://www.researchgate.net/publication/371658378/figure/fig4/AS:11431281407033728@1745748744881/a-Relationships-between-ln-sinhas-and-lne-at-different-temperatures-b_Q320.jpg)
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Citation: Xie, G.; Kuang, Z.; Li, J.;
Zhang, Y.; Han, S.; Li, C.; Zhu, D.; Liu,
Y. Thermal Deformation Behavior
and Dynamic Softening Mechanisms
of Zn-2.0Cu-0.15Ti Alloy: An
Investigation of Hot Processing
Conditions and Flow Stress Behavior.
Materials 2023,16, 4431. https://
doi.org/10.3390/ma16124431
Academic Editor: Frank Czerwinski
Received: 21 May 2023
Revised: 10 June 2023
Accepted: 11 June 2023
Published: 16 June 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
materials
Article
Thermal Deformation Behavior and Dynamic Softening
Mechanisms of Zn-2.0Cu-0.15Ti Alloy: An Investigation of Hot
Processing Conditions and Flow Stress Behavior
Guilan Xie 1, Zhihao Kuang 1, Jingxin Li 1, Yating Zhang 1, Shilei Han 1, Chengbo Li 1, Daibo Zhu 1,*
and Yang Liu 1,2,3,*
1School of Mechanical Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China;
xieguilan@xtu.edu.cn (G.X.); 17373241475@163.com (Z.K.); 202005501510@smail.xtu.edu.cn (J.L.);
202005501530@smail.xtu.edu.cn (Y.Z.); 16673234565@163.com (S.H.); csulicb@163.com (C.L.)
2
School of Materials Science and Engineering, Nanyang Technological University, Singapore 639798, Singapore
3Zhuzhou Smelter Group Co., Ltd., Zhuzhou 412005, China
*Correspondence: daibozhu@xtu.edu.cn (D.Z.); liuyang_225@126.com (Y.L.); Tel.: +86-731-5289-8553 (Y.L.)
Abstract:
Through isothermal hot compression experiments at various strain rates and temperatures,
the thermal deformation behavior of Zn-2.0Cu-0.15Ti alloy is investigated. The Arrhenius-type model
is utilized to forecast flow stress behavior. Results show that the Arrhenius-type model accurately
reflects the flow behavior in the entire processing region. The dynamic material model (DMM) reveals
that the optimal processing region for the hot processing of Zn-2.0Cu-0.15Ti alloy has a maximum
efficiency of about 35%, in the temperatures range (493–543 K) and a strain rate range (0.01–0.1 s
−1
).
Microstructure analysis demonstrates that the primary dynamic softening mechanism of Zn-2.0Cu-
0.15Ti alloy after hot compression is significantly influenced by temperature and strain rate. At
low temperature (423 K) and low strain rate (0.1 s
−1
), the interaction of dislocations is the primary
mechanism for the softening Zn-2.0Cu-0.15Ti alloys. At a strain rate of 1 s
−1
, the primary mechanism
changes to continuous dynamic recrystallization (CDRX). Discontinuous dynamic recrystallization
(DDRX) occurs when Zn-2.0Cu-0.15Ti alloy is deformed under the conditions of 523 K/0.1 s
−1
, while
twinning dynamic recrystallization (TDRX) and CDRX are observed when the strain rate is 10 s−1.
Keywords:
Zn-Cu-Ti alloy; hot compression; dynamic material model (DMM); flow stress behavior;
softening mechanism
1. Introduction
Zinc-based alloys are highly sought after due to their exceptional mix of qualities,
such as excellent ductility, weldability, outstanding corrosion resistance [
1
], and surface
aspect, making them suitable for various applications in the medical implantation, building
industry, and anti-corrosive fields [
2
–
4
]. Moreover, these alloys possess strong creep
resistance [
5
], which further increases their demand. The favorable hot workability of
these high-strength alloys also contributes to their popularity. Hot extrusion/rolling is the
main processing technique used for zinc-based alloys. However, the thermal deformation
behavior of the alloys can be affected by several factors [
6
,
7
], including temperature, strain
rates, and strain [
8
]. Thus, it is difficult to achieve the parameter specification of zinc-based
alloys [
9
–
11
]. Up to now, the thermal deformation behavior of Zn-Cu-Ti has not been
studied in depth. Little consideration has been given to the flow stress evolution and
constitutive relationship of the Zn-Cu-Ti alloy during hot deformation.
The constitutive equations and the processing maps are important methods to study the
hot workability characteristics and the deformation mechanisms of zinc-based
alloys [12,13]
.
Unfortunately, the traditional Arrhenius equations and 2D processing maps cannot reflect
the influence of strain, which has an obvious impact on flow stress [
14
]. In order to
Materials 2023,16, 4431. https://doi.org/10.3390/ma16124431 https://www.mdpi.com/journal/materials
Materials 2023,16, 4431 2 of 14
take the strain effect into account, modified Arrhenius equations are used to improve
the accuracy of the numerical simulation, overcoming the disadvantages of being time-
consuming and labor-intensive [
15
]. Meanwhile, it can also provide theoretical guidance
for the optimization and the predictions of numerical simulations. The 3D processing map
established by the dynamic material model (DMM) can reveal the reasonable processing
region to optimize the thermal working process [
16
,
17
] from which stable and unstable
regions can be found. Stable regions are accompanied by dynamic recrystallization [
18
]
(DRX) and dynamic recovery (DRV), which can uniform microstructures and improve
processability [
19
]. However, few systematic studies have revealed the evolution of the
microstructure of Zn-Cu-Ti alloys and optimized its forming parameters by 3D processing
maps. Therefore, it is of great significance to study flow stress behavior, constitutive
equation, processing maps (3D), and dynamic recrystallization (DRX) behavior of Zn-Cu-Ti
alloys during hot compression deformation.
The aim of this work is to investigate the thermal deformation behavior of Zn-2.0Cu-
0.15Ti alloy under various strains, strain rates, and deformation temperatures. The 3D
processing maps of Zn-2.0Cu-0.15Ti alloy are created to optimize the hot deformation pa-
rameters and find out the unstable deformation factors. The effects of thermal deformation
parameters on the DRX mechanism are analyzed by the electron backscatter diffraction
pattern (EBSD) and transmission electron microscopy (TEM), which provides scientific
guidance for the thermal deformation behavior and dynamic softening mechanism of
Zn-Cu-Ti alloy.
2. Experimental Procedure
Table 1lists the component (in wt.%) of Zn-Cu-Ti alloys. In our experiments, using
pure Zn (purity 99.995%), high-purity copper foil (99.99%), and high-purity titanium tablet
as starting materials for casting samples with a composition of Zn-2.0Cu-0.15Ti (wt.%
hereafter). The raw ingredients were melted at 973 K with the protection of argon (Ar) gas
in a graphite crucible. The alloy was poured (pouring temperature: 823 K) into the steel
mold preheated to 493 K. After natural cooling and solidification, an ingot was obtained.
The ingot underwent homogenization at 653 K for 10 h and was used to prepare all the
samples. The shape of each specimen for the compression testing is a cylinder (diameter:
8 mm, height: 12 mm). The samples were heated to the specified temperature (423, 473,
523, and 573 K) by using a Gleeble-3500 thermomechanical simulator (Dynamic Systems
Inc. America) with a heating rate of 10 K/s. The other experimental parameters were as
follows: the strain rates were 0.01 s
−1
, 0.1 s
−1
, 1 s
−1
, and 10 s
−1
, and the height of each
sample was compressed by 60%. All compression tests were carried out in a vacuum, and
in order to achieve a homogeneous temperature ahead of deformation, all samples were
maintained at the necessary temperature for 200 s. Additionally, after the compression test,
each sample was promptly water-quenched so that the microstructures could be preserved.
Figure 1depicts the hot compression experimentation process.
Table 1. Chemical composition of Zn-2.0Cu-0.15Ti alloy.
Component Cu Ti Impurity Zn
Content (wt.%) 2.03 0.152 0.045 Bal.
Materials 2023,16, 4431 3 of 14
Materials2023,16,xFORPEERREVIEW3of15
Tab le 1.ChemicalcompositionofZn-2.0Cu-0.15Tialloy.
ComponentCuTiImpurityZn
Content(wt.%)2.030.1520.045Bal.
Figure1.Schematicdiagramofhotcompressionexperimentprocess.
3.ResultsandDiscussion
3.1.AnalysisofFlowStressCurves
Figure2displaysthetruestress–straincurvesoftheZn-2.0Cu-0.15Tialloyobtained
fromthecompressiontestconductedatstrainrates0.01,0.1,1,and10s−1anddeformation
temperatures423,473,523,and573K.Theevolutionofflowstresscurvesisroughlydi-
videdintothreestages.Inthefirststage,flowstressincreaseswithincreasingstrainre-
sultingfromanincreaseindislocationdensity(workhardening).Inthesecondstage,the
increasingrateofflowstressslowsdownandreachesitspeakvalue,indicatingtheoccur-
renceofdynamicrecovery(DRV)andDRX.Inthethirdstage,flowstressgraduallydrops
untilreachingthefinalsteadyvalue.Thisphenomenoninhotworkingisusuallycaused
bydynamicrecrystallization(DRX)[20,21].
AsshowninFigure2,ontheonehand,itcanbeseenthatflowstressdecreaseswith
theincreaseindeformationtemperatureatagivenstrainrate.Thisisbecausehighertem-
peraturescanpromotetheoccurrenceofDRX,whichcausesadecreaseindislocationden-
sity.Ontheotherhand,flowstressincreaseswiththeincreaseinstrainratesatagiven
temperature.Thisisduetothefactthatwiththelimitedtimefordislocationannihilation
tomanifest,thedislocationdensitygraduallyrises.
Figure 1. Schematic diagram of hot compression experiment process.
Samples for electron backscatter diffraction pattern (EBSD) analysis were prepared
into round pieces with a thickness of 100
µ
m and a diameter of 3 mm. The round pieces
were mechanically and vibrationally polished for 0.5 h using a Buehler VibroMet2 Vibratory
Polisher (ITW, America), and then the Angle was reduced using a plasma thinning instru-
ment. SEM with a 200 Sirion field emission gun was used for the EBSD studies. OIM7.3
software was used to analyze the EBSD data. Transmission electron microscope (TEM)
images were collected using a Tecnai G220 transmission (FEI, America) electron microscope
at 200 kV. The shape of TEM samples was similar to EBSD samples, and twin-jet thinning
was applied in a mixed solution of HNO3:CH3OH = 1:3. The experimental parameters
were 30 V and −243 K, respectively.
3. Results and Discussion
3.1. Analysis of Flow Stress Curves
Figure 2displays the true stress–strain curves of the Zn-2.0Cu-0.15Ti alloy obtained
from the compression test conducted at strain rates 0.01, 0.1, 1, and 10 s
−1
and deformation
temperatures 423, 473, 523, and 573 K. The evolution of flow stress curves is roughly
divided into three stages. In the first stage, flow stress increases with increasing strain
resulting from an increase in dislocation density (work hardening). In the second stage,
the increasing rate of flow stress slows down and reaches its peak value, indicating the
occurrence of dynamic recovery (DRV) and DRX. In the third stage, flow stress gradually
drops until reaching the final steady value. This phenomenon in hot working is usually
caused by dynamic recrystallization (DRX) [20,21].
As shown in Figure 2, on the one hand, it can be seen that flow stress decreases with
the increase in deformation temperature at a given strain rate. This is because higher
temperatures can promote the occurrence of DRX, which causes a decrease in dislocation
density. On the other hand, flow stress increases with the increase in strain rates at a given
temperature. This is due to the fact that with the limited time for dislocation annihilation
to manifest, the dislocation density gradually rises.
Materials 2023,16, 4431 4 of 14
Materials2023,16,xFORPEERREVIEW4of15
Figure2.Thetruestress–straincurvesinthehotcompressiontestsofZn-2.0Cu-0.15Tialloyat0.01–
10s−1withadeformationtemperatureof(a)423K,(b)473K,(c)523K,(d)573K.
3.2.DevelopmentofConstitutiveEquation
TheflowstresscausedbydeformationconditionscanbeevaluatedusingtheArrhe-
niusequations,whicharepresentedinEquations(1)and(2)andarecommonlyutilized
fordescribingthermaldeformationbehavior[22,23].
()exp
Q
AF
R
T
(1)
1,0.8
( ) exp( ), 1.2
sinh( ) for all
n
n
F
(2)
Foracertainstrain,thetruestress(MPa)representedby𝜎,wherethestrainrate(s−1)
isrepresentedby𝜀,iscalculatedbysomematerialconstants,includingA,n1,n,α,andβ,
whereα=β/n1,aswellasgasconstantR(8.314Jmol−1K−1),theactivationenergyQ(kJ
mol−1),andabsolutetemperatureT(K).
Meanwhile,combiningEquations(1)and(2)leadstotheZener–Hollomon(Z)pa-
rameter[17,24]:
[sinh( )] exp
nQ
A
R
T
(3)
exp [sinh( )]
n
Q
ZA
RT
(4)
Figure 2.
The true stress–strain curves in the hot compression tests of Zn-2.0Cu-0.15Ti alloy at
0.01–10 s−1with a deformation temperature of (a) 423 K, (b) 473 K, (c) 523 K, (d) 573 K.
3.2. Development of Constitutive Equation
The flow stress caused by deformation conditions can be evaluated using the Arrhenius
equations, which are presented in Equations (1) and (2) and are commonly utilized for
describing thermal deformation behavior [22,23].
.
ε=AF(σ)exp−Q
RT (1)
F(σ) =
σn1,ασ <0.8
exp(βσ),ασ >1.2
sinh(ασ)nfor all σ
(2)
For a certain strain, the true stress (MPa) represented by
σ
, where the strain rate (s
−1
)
is represented by
.
ε
, is calculated by some material constants, including A,n
1
,n,
α
, and
β
, where
α
=
β
/n
1
, as well as gas constant R(8.314 J mol
−1
K
−1
), the activation energy Q
(kJ mol−1), and absolute temperature T(K).
Meanwhile, combining Equations (1) and (2) leads to the Zener–Hollomon (Z) param-
eter [17,24]:
.
ε=A[sinh(ασ)]nexp−Q
RT (3)
Z=.
εexpQ
RT =A[sinh(ασ)]n(4)
By substituting Equation (2) into Equation (1) and taking the natural logarithm of both
sides, the following equation was obtained:
ln σ=1
n1
ln .
ε−1
n1
ln A1+Q
n1RT (5)
Materials 2023,16, 4431 5 of 14
σ=1
βln .
ε−1
βln A2+Q
βRT (6)
Assuming the material’s activation energy for the deformation is a fixed value un-
affected by temperature, according to the true stress–strain curve, the peak stress corre-
sponding to the alloy under various deformation circumstances is determined, and then the
ln
σ
-ln
.
ε
and
σ
-ln
.
ε
curves (Figure 3) can be plotted by linear regression processing according
to Equations (5) and (6). Here, the average slopes of the ln
σ
-ln
.
ε
and
σ
-ln
.
ε
curves are
shown by n
1
and
β
, respectively, n
1
= 10.601 and
β
= 0.086 MPa
−1
can be obtained, so that
α=β/n1= 0.007 MPa−1.
Materials2023,16,xFORPEERREVIEW5of15
BysubstitutingEquation(2)intoEquation(1)andtakingthenaturallogarithmof
bothsides,thefollowingequationwasobtained:
1
11 1
11
ln ln ln
Q
A
nn nRT
(5)
2
11
ln ln
Q
A
R
T
(6)
Assumingthematerial’sactivationenergyforthedeformationisafixedvalueunaf-
fectedbytemperature,accordingtothetruestress–straincurve,thepeakstresscorre-
spondingtothealloyundervariousdeformationcircumstancesisdetermined,andthen
thelnσ-ln𝜀andσ-ln𝜀curves(Figure3)canbeploedbylinearregressionprocessingac-
cordingtoEquations(5)and(6).Here,theaverageslopesofthelnσ-ln𝜀andσ-ln𝜀curves
areshownbyn1andβ,respectively,n1=10.601andβ=0.086MPa−1canbeobtained,so
thatα=β/n1=0.007MPa−1.
Equations(7)and(8)arepresentedusingthepartialderivativemethodappliedtothe
logarithmofEquation(3)provided:
ln
[sinh( )]
ε
nασ (7)
Q ln[sinh(ασ)]
Rn (1/ T)
(8)
Thevalueofbothparameters(𝜎,𝜀)isusedfortheplotsofln[sinh(ασ)]-ln𝜀andln
[sinh(ασ)]-1000/T(Figure4)bylinearregressionprocessingaccordingtoEquations(7)and
(8).Byusingthelinearfiingmethod,theaveragevaluesofnandbarecalculatedtobe
7.70001and2.534,respectively.Qiscalculatedtobe130.69kJmol−1.Aiscalculatedtobe
3.33×1012s−1bysubstitutingthecorrespondingdataintoEquation(5).
Then,therelationshipofσ-ZcanbedefinedbyEquation(9),andtheformulacanbe
wrienas:
1
12
2
nn
1Z Z
ln 1
AA
(9)
Figure3.Linearrelationshipof(a)lnσ-lnεand(b)σ-lnεatdifferenttemperatures.
Figure 3. Linear relationship of (a) lnσ-ln .
εand (b)σ-ln .
εat different temperatures.
Equations (7) and (8) are presented using the partial derivative method applied to the
logarithm of Equation (3) provided:
n=∂ln .
ε
∂[sinh(ασ)] (7)
Q
Rn =∂ln[sinh(ασ)]
∂(1/T)(8)
The value of both parameters (
σ
,
.
ε
) is used for the plots of ln [sinh(
ασ
)]-ln
.
ε
and
ln [sinh(
ασ
)]-1000/T (Figure 4) by linear regression processing according to
Equations (7) and (8)
.
By using the linear fitting method, the average values of n and b are calculated to be 7.70001
and 2.534, respectively. Q is calculated to be 130.69 kJ mol
−1
. A is calculated to be
3.33 ×1012 s−1by substituting the corresponding data into Equation (5).
Materials2023,16,xFORPEERREVIEW6of15
Figure4.(a)Relationshipsbetweenln[sinh(ασ)]andlnεatdifferenttemperatures;(b)Relation-
shipsbetweenln[sinh(ασ)]and1000/Tatdifferentstrainrates.
Theeffectofstrainonthermaldeformationbehaviorisdisregardedbasedonthe
aforementionedconstitutiveequations.Nevertheless,itcanbefoundfromFigure2that
thestrainplayedaconsiderableroleinthetruestress.Thismeansthattakingtheimpact
ofstrainintoaccountontheconstitutiveequationsisessential.
Thematerialconstants(A,n,Q,anda)areknowntobefunctionsofthestrainand
canbedescribedusingpolynomialfunctions[25,26].Toobtainthevaluesofthesecon-
stants,thestrainisvariedfrom0.05to0.7inincrementsof0.05,andthematerialconstants
arecomputedateachstrain.Therelationshipbetweenthematerialconstantsandstrainis
thenestablishedthroughpolynomialfiingtechniques,asshowninFigure5.Itwasfound
thatasix-orderpolynomialfunctioncanaccuratelydepicttheimpactofstrainonmaterial
constants,asshowninEquation(10).Figure5summarizestherelationshipbetweenthe
materialconstantsandstrain.Thevalidityoftheseventh-orderpolynomialmodeliscon-
firmedbytheresultsofpolynomialfiing,whicharepresentedinTab le2.
23456
0123456
23456
0123456
23456
0123456
23456
0123456
ln
ah h h h h h h
ni i i i i i i
Qj j j j j j j
A
kk k k k k k
(10)
Tab le 2.Coefficientsofthepolynomialequations.
anQln
A
00.0025h09.0261i025.071j014.882k
10.1749h185.08i13023j1454.91k
21.6323h 21147.2i 227214j 24230.1k
36.8839h35542.6i3113308j317877k
414.666h 412569i 4238415j 437778k
515.409h513604i5246903j539081k
66.3395h 65665i 699939j 615752k
Figure 4.
(
a
) Relationships between ln [sinh(
ασ
)] and ln
.
ε
at different temperatures; (
b
) Relationships
between ln [sinh(ασ)] and 1000/T at different strain rates.
Materials 2023,16, 4431 6 of 14
Then, the relationship of
σ
-Z can be defined by Equation (9), and the formula can be
written as:
σ=1
αln
Z
A1
n+"Z
A2
n+1#1
2
(9)
The effect of strain on thermal deformation behavior is disregarded based on the
aforementioned constitutive equations. Nevertheless, it can be found from Figure 2that
the strain played a considerable role in the true stress. This means that taking the impact of
strain into account on the constitutive equations is essential.
The material constants (A,n,Q, and a) are known to be functions of the strain and can
be described using polynomial functions [
25
,
26
]. To obtain the values of these constants,
the strain is varied from 0.05 to 0.7 in increments of 0.05, and the material constants are
computed at each strain. The relationship between the material constants and strain is
then established through polynomial fitting techniques, as shown in Figure 5. It was found
that a six-order polynomial function can accurately depict the impact of strain on material
constants, as shown in Equation (10). Figure 5summarizes the relationship between the
material constants and strain. The validity of the seventh-order polynomial model is
confirmed by the results of polynomial fitting, which are presented in Table 2.
a=h0+h1ε+h2ε2+h3ε3+h4ε4+h5ε5+h6ε6
n=i0+i1ε+i2ε2+i3ε3+i4ε4+i5ε5+i6ε6
Q=j0+j1ε+j2ε2+j3ε3+j4ε4+j5ε5+j6ε6
ln A=k0+k1ε+k2ε2+k3ε3+k4ε4+k5ε5+k6ε6
(10)
Materials2023,16,xFORPEERREVIEW7of15
Figure5.Relationshipbetweenthefiedparametersandthestrain:(a)a;(b)n;(c)Q;(d)lnA.
3.3.PerformanceEvaluations
Figure6displaysthattheArrhenius-typemodelhasgoodpredictabilityoftheflow
behaviorofZn-2.0Cu-0.15Tialloyundervariousdeformationconditions.However,toen-
surethereliabilityofthepredictionsoftheArrhenius-typemodelinourwork,themean
absolutepercentageerror(MAPE),correlationcoefficient(R),andrelativeerrorparame-
tersareusedtoevaluatethereliability.Theformulacanbewrienas:
1
22
1=1
=
n
ii
i=
nn
ii
i= i
AABB
R
A
ABB
(11)
n
i1
1
MAPE(%) = 100
n
ii
i
AB
A (12)
wherenisallthedata;AiandBiindicatethemeasureddataandthepredicteddata,re-
spectively.
A
and
B
representtheaveragevaluesofAiandBi,respectively.
Generally,Risusedtoreflecttheclosenessofthecorrelationbetweenvariables[27].
Inourstudy,theclosertheRistoone,themoreaccuratethepredictionofthemodel.
Furthermore,theMAPEistypicallyusedtoreflectthepracticalpredictionerrors.The
smallerthevalueofMAPE,thebeerperformanceforthepredictedmodel.Itcanbeob-
servedfromFigure7athatRwascalculatedat0.986.Moreover,theMAPEwas6.48%,and
therelativeerrorpercentagewasusedtoinvestigatethepredictabilityperformanceofthe
model.Onecanthereforeobtainthefollowing:
Relative error = 100%
ii
i
AB
A (13)
Figure 5. Relationship between the fitted parameters and the strain: (a)a; (b)n; (c)Q; (d) lnA.
Materials 2023,16, 4431 7 of 14
Table 2. Coefficients of the polynomial equations.
a n Q lnA
h0=0.0025 i0=9.0261 j0=25.071 k0=14.882
h1=0.1749 i1=85.08 j1=3023 k1=454.91
h2=−1.6323 i2=−1147.2 j2=−27, 214 k2=−4230.1
h3=6.8839 i3=5542.6 j3=113, 308 k3=17, 877
h4=−14.666 i4=−12, 569 j4=−238, 415 k4=−37, 778
h5=15.409 i5=13, 604 j5=246, 903 k5=39, 081
h6=−6.3395 i6=−5665 j6=−99, 939 k6=−15, 752
3.3. Performance Evaluations
Figure 6displays that the Arrhenius-type model has good predictability of the flow
behavior of Zn-2.0Cu-0.15Ti alloy under various deformation conditions. However, to
ensure the reliability of the predictions of the Arrhenius-type model in our work, the mean
absolute percentage error (MAPE), correlation coefficient (R), and relative error parameters
are used to evaluate the reliability. The formula can be written as:
R=∑n
i=1Ai−ABi−B
q∑n
i=1Ai−A2∑n
i=1Bi−B2(11)
MAPE(%) = 1
n
n
∑
i=1
Ai−Bi
Ai
×100 (12)
where n is all the data; A
i
and B
i
indicate the measured data and the predicted data,
respectively. Aand Brepresent the average values of Aiand Bi, respectively.
Materials2023,16,xFORPEERREVIEW8of15
FromFigure7b,thevalueoftherelativeerrorparameterrangesfrom30.04to30.15%.
Combiningtheabovecalculationparameters,theresultshowsthatpredictedandmeas-
uredvaluesoftheArrhenius-typemodelhavegoodagreementinourstudy.
Figure6.Comparisonsbetweenthemeasuredandthepredictedtruestressatdifferenttempera-
tures:(a)423K;(b)473K;(c)523K;and(d)573K.
Figure7.(a)Correlationbetweentheexperimentalandpredictedflowstressvaluesobtainedfrom
Arrhenius-Typeconstitutivemodel;(b)StatisticalanalysisoftherelativeerrorbyArrhenius-Type
constitutivemodel.
3.4.ProcessingMaps
Atpresent,PrasedproposedhotprocessingmapsbystudyingthetheoryofDMM
[28,29]andcontinuummechanics.Ingeneral,thehotprocessingmapscanbeobtainedby
combiningthepowerdissipationmapandtheinstabilitymap.Additionally,inaccord-
ancewiththetheoryofDMM,thetotalpower(P)oftheinputsystemismadeupoftwo
components:dissipatedco-contentJanddissipatedcontentG[20],whichareproduced
by,respectively,structuralchangeandplasticdeformation.Theequationappearstobe
thefollowing:
Figure 6.
Comparisons between the measured and the predicted true stress at different temperatures:
(a) 423 K; (b) 473 K; (c) 523 K; and (d) 573 K.
Generally, Ris used to reflect the closeness of the correlation between variables [
27
].
In our study, the closer the Ris to one, the more accurate the prediction of the model.
Furthermore, the MAPE is typically used to reflect the practical prediction errors. The
Materials 2023,16, 4431 8 of 14
smaller the value of MAPE, the better performance for the predicted model. It can be
observed from Figure 7a that Rwas calculated at 0.986. Moreover, the MAPE was 6.48%,
and the relative error percentage was used to investigate the predictability performance of
the model. One can therefore obtain the following:
Relative error =Ai−Bi
Ai×100% (13)
Materials2023,16,xFORPEERREVIEW8of15
FromFigure7b,thevalueoftherelativeerrorparameterrangesfrom30.04to30.15%.
Combiningtheabovecalculationparameters,theresultshowsthatpredictedandmeas-
uredvaluesoftheArrhenius-typemodelhavegoodagreementinourstudy.
Figure6.Comparisonsbetweenthemeasuredandthepredictedtruestressatdifferenttempera-
tures:(a)423K;(b)473K;(c)523K;and(d)573K.
Figure7.(a)Correlationbetweentheexperimentalandpredictedflowstressvaluesobtainedfrom
Arrhenius-Typeconstitutivemodel;(b)StatisticalanalysisoftherelativeerrorbyArrhenius-Type
constitutivemodel.
3.4.ProcessingMaps
Atpresent,PrasedproposedhotprocessingmapsbystudyingthetheoryofDMM
[28,29]andcontinuummechanics.Ingeneral,thehotprocessingmapscanbeobtainedby
combiningthepowerdissipationmapandtheinstabilitymap.Additionally,inaccord-
ancewiththetheoryofDMM,thetotalpower(P)oftheinputsystemismadeupoftwo
components:dissipatedco-contentJanddissipatedcontentG[20],whichareproduced
by,respectively,structuralchangeandplasticdeformation.Theequationappearstobe
thefollowing:
Figure 7.
(
a
) Correlation between the experimental and predicted flow stress values obtained from
Arrhenius-Type constitutive model; (
b
) Statistical analysis of the relative error by Arrhenius-Type
constitutive model.
From Figure 7b, the value of the relative error parameter ranges from 30.04 to 30.15%.
Combining the above calculation parameters, the result shows that predicted and measured
values of the Arrhenius-type model have good agreement in our study.
3.4. Processing Maps
At present, Prased proposed hot processing maps by studying the theory of
DMM [28,29]
and continuum mechanics. In general, the hot processing maps can be obtained by combin-
ing the power dissipation map and the instability map. Additionally, in accordance with
the theory of DMM, the total power (P) of the input system is made up of two components:
dissipated co-content Jand dissipated content G[
20
], which are produced by, respectively,
structural change and plastic deformation. The equation appears to be the following:
P=σ.
ε=G+J=Z.
ε
0σd.
ε+Z.
ε
0
.
εdσ(14)
where
σ
and
.
ε
represent the flow stress and the strain rate, the sensitivity of strain rate (m)
can be expressed using the following equation:
m=dJ
dG =∂ln σ
∂ln .
ε(15)
Typically, the power dissipation efficiency (
η
) is used to assess an alloy’s capacity for
power dissipation, which is as follows [30]:
η=J
Jmax
=2m
m+1(16)
When m
≤
0, there is no power dissipation in the entire system. When 0 < m < 1, it
is considered to be in a steady-state flow regime. When m = 1, the value of J is equal to
its peak value (J
max
). A 3D contour map of power dissipation can express the changes in
η
under various deformation temperatures, strain, and strain rates. Generally, the higher
the power consumption efficiency is, the better the performance of alloy processing can
be [
31
], which demonstrates that DRX may occur. However, processes such as adiabatic
Materials 2023,16, 4431 9 of 14
shear banding and crack growth usually result in structural instability. According to the
extreme principle of irreversible thermodynamics proposed by Ziegler [32], the following
is an expression for the unstable criterion:
ξ(.
ε) = ∂ln(m
m+1)
∂ln .
ε+m<0 (17)
The flow behavior of the material can become unstable when the values of the param-
eters (
ξ
) mentioned above fall below zero. Figures 8and 9illustrate, respectively, the 3D
maps of power dissipation and flow instability for the strain range (0.1–0.7), temperature
range (423–573 K), and strain rate range (0.01–10 s
−1
). The colored grids of the power dissi-
pation map represent
η
, and the shaded regions of the flow instability map can be used to
identify the unstable area. The results suggest that the power dissipation efficiency (
η
) and
the flow instability zone change significantly under different deformation conditions. With
the increase in strain (from 0.1 to 0.7), the value of
η
declines (Figure 8). Moreover, it can be
found that the peak values of the efficiency region are the strain rate range (
0.01–0.1 s−1
)
and the temperature range (493–543 K) (Figure 8b,c).
From Figure 9a, the shaded regions increase rapidly at lower strain levels (0.1–0.3).
Meanwhile, it can be seen that the shaded regions are mostly presented at low temperatures
(423–473 K) and strain rates (0.1–10 s
−1
) (Figure 9b,c). To ensure excellent processability,
the shaded regions should be avoided. According to the analysis based on the power
dissipation map and the unstable map, the optimal processing region for the hot processing
of Zn-2.0Cu-0.15Ti alloy is the temperature range (493–543 K) and the strain rate range
(0.01–0.1 s−1).
Materials2023,16,xFORPEERREVIEW10of15
Figure8.The3DpowerdissipationmapsofZn-2.0Cu-0.15Tialloy:(a)strainsection,(b)strainrate
section,and(c)temperaturesection.
Figure9.The3Dflow-instabilitymapsofZn-2.0Cu-0.15Tialloy:(a)strainsection,(b)strainrate
section,and(c)temperaturesection.
3.5.MicrostructureEvolution
Figure10(a1,a4)displaystheEBSDmapsofthespecimensdeformedat1s−1andthe
deformationtemperatureof423–573K.InFigure10(a1,a4),thelow-anglegrainbounda-
ries(LAGBs,2–15°)aredescribedasfineblacklines,whilethehigh-anglegrainbounda-
ries(HAGBs,≥15°)aredescribedasthickblacklines.Itisdominatedbyelongatedgrains
whenthespecimendeformedat423k(Figure10(a1)).Withincreasingdeformationtem-
perature(from423to473K),elongatedgrainsgraduallydisappearandarereplacedby
Figure 8.
The 3D power dissipation maps of Zn-2.0Cu-0.15Ti alloy: (
a
) strain section, (
b
) strain rate
section, and (c) temperature section.
Materials 2023,16, 4431 10 of 14
Materials2023,16,xFORPEERREVIEW10of15
Figure8.The3DpowerdissipationmapsofZn-2.0Cu-0.15Tialloy:(a)strainsection,(b)strainrate
section,and(c)temperaturesection.
Figure9.The3Dflow-instabilitymapsofZn-2.0Cu-0.15Tialloy:(a)strainsection,(b)strainrate
section,and(c)temperaturesection.
3.5.MicrostructureEvolution
Figure10(a1,a4)displaystheEBSDmapsofthespecimensdeformedat1s−1andthe
deformationtemperatureof423–573K.InFigure10(a1,a4),thelow-anglegrainbounda-
ries(LAGBs,2–15°)aredescribedasfineblacklines,whilethehigh-anglegrainbounda-
ries(HAGBs,≥15°)aredescribedasthickblacklines.Itisdominatedbyelongatedgrains
whenthespecimendeformedat423k(Figure10(a1)).Withincreasingdeformationtem-
perature(from423to473K),elongatedgrainsgraduallydisappearandarereplacedby
Figure 9.
The 3D flow-instability maps of Zn-2.0Cu-0.15Ti alloy: (
a
) strain section, (
b
) strain rate
section, and (c) temperature section.
3.5. Microstructure Evolution
Figure 10(a1,a4) displays the EBSD maps of the specimens deformed at 1 s
−1
and the
deformation temperature of 423–573 K. In Figure 10(a1,a4), the low-angle grain boundaries
(LAGBs, 2–15
◦
) are described as fine black lines, while the high-angle grain boundaries
(HAGBs,
≥
15
◦
) are described as thick black lines. It is dominated by elongated grains when
the specimen deformed at 423 k (Figure 10(a1)). With increasing deformation temperature
(from 423 to 473 K), elongated grains gradually disappear and are replaced by smaller
rounded grains or equiaxed grains (Figure 10(a2)). As the deformation temperature rises, it
shows that the DRX begins to occur. When T = 523 K, an increase in the amount of recrys-
tallization fraction was observed, and some sub-grain boundaries from the original grains
can be observed (Figure 10(a3)). Meanwhile, some fine grains develop along the original
grain boundaries. The phenomenon is in line with the typical feature of discontinuous-type
DRX (DDRX) induced by strain-induced boundary migration [
33
]. During deformation,
the increase in HAGBs will hinder the continuity of dislocation slip [
34
] and cause stress
concentration. In order to reduce stress concentration, the grain boundary bulges and
migrates locally to form a distortion-free recrystallized nucleation. When T = 573 K, the
recrystallized grains further grow, tending to equiaxed grain formation. The complete
processing of the DRX is depicted in Figure 10(a4).
In Figure 10(b1,b4), blue denotes the fully recrystallized grains, yellow symbolizes the
subgrains, and red represents the deformed microstructures. Figure 10(b1) shows that the
microstructure underwent deformation with only a few subgrains and recrystallized grains.
Low deformation temperatures cause inadequate recrystallization time. When T = 523 K
(Figure 10(b3)), the deformed structure is replaced by the sub-grains and disappears. When
the temperature exceeds 523 K (Figure 10(b4)), full recrystallized grains formed along the
original grain boundaries can be detected. Therefore, the DRX volume percent is enhanced
with increasing deformation temperature [34,35].
From Figure 10(c1,c4), it can be found that the proportion of LAGBs declines, and the
proportion of HAGBs increases gradually as the deformation temperature increases [
36
,
37
].
The proportion of HAGBs increases from 48.5% to 68.1% when the deformation temperature
rises from 423 to 573 K. This is because higher deformation temperature favors DRX, which
causes subgrain boundaries to absorb dislocations and merge into large subgrains [
38
]. In
Materials 2023,16, 4431 11 of 14
addition, when T = 423 K, the maximum pole intensity of the basal texture in Figure 10(c1)
is 27.068. However, when T = 573 K, the maximum pole intensity suddenly decreases to
5.947. The result indicates that the deformation temperature plays a significant role in the
dynamic softening of the alloy [39].
Materials2023,16,xFORPEERREVIEW12of15
Figure10.EBSDmaps(a1–a4),recrystallizationgraindistributionmaps(b1–b4)andmisorientation
angle(c1–c4)ofthespecimensdeformedwithstrainratesof1s−1at(a1–c1)423K,(a2–c2)473K,(a3–
c3)523K,and(a4–c4)573K.
Figure11.TEMmicrographsofZn-2.0Cu-0.15Tialloydeformedat(a)423K/0.1s−1;(b)423K/1s−1;
(c)523K/0.1s−1;(d)523K/10s−1.
Figure 10.
EBSD maps (
a1
–
a4
), recrystallization grain distribution maps (
b1
–
b4
) and misorientation
angle (
c1
–
c4
) of the specimens deformed with strain rates of 1 s
−1
at (
a1
–
c1
) 423 K, (
a2
–
c2
) 473 K,
(a3–c3) 523 K, and (a4–c4) 573 K.
Figure 11 displays the TEM micrographs of Zn-2.0Cu-0.15Ti alloy under various
deformation conditions. According to the results, the deformation temperature and strain
rate are closely related to DRX [
40
,
41
]. Dislocation tangling and walls can be observed
in the sample, along with original subgrains forming inside without obvious dynamic
recrystallization grains in Figure 11a in the case of 423 K and 0.1 s
−1
. The softening of the
alloy was mainly achieved through dislocation slip, cross-slip, climb, and recombination
inside the grain [
42
]. When T = 423 K and
.
ε
= 1 s
−1
, a polygonal structure appeared clearly
inside the grain due to the segmentation effect of dislocation grids or walls (Figure 11b),
indicating the occurrence of continuous dynamic recrystallization (CDRX) inside the alloy.
Within the strain rate range of 0.01–10 s
−1
, the elevated strain rate can enhance the effect of
CDRX, due to the decline in the CRSSnon-basal/CRSSbasal ratio [
12
,
43
], which leads to
CDRX occurring easily, weakening DDRX. At 523 K/0.1 s
−1
(Figure 11c), an obvious curved
grain boundary was observed on the original grain boundary. These bow-outs were the
nucleation sites of discontinuous dynamic recrystallization (DDRX) [
34
,
36
]. Additionally,
the elevated temperature increases dislocation mobility, which makes more dislocations
tend to accumulate near grain boundaries (GBs). The sliding and moving rates of the GB
are correspondingly increased due to higher driving forces at high temperatures. As a
result, flow stress decreases (Figure 2c). However, as the strain rate increases to 10 s
−1
,
twinning dynamic recrystallization (TDRX) and CDRX are observed (Figure 11d). This
Materials 2023,16, 4431 12 of 14
indicates that twinning is generated during the thermal deformation process, and twins
can form much easier than slip at a high strain rate (10 s
−1
), due to the highly effective
interface velocity [
13
] of the twin. A twin boundary (TB) will hinder the dislocation motion,
thus increasing the strain energy around the twinning and providing the driving force
for the nucleation of TDRX on TB. Meanwhile, the high strain rate increases deformation
inhomogeneity, making dislocation slip more difficult, which leads to a significant increase
in flow stress (Figure 2c) and the weakening of DDRX [13,44].
Materials2023,16,xFORPEERREVIEW12of15
Figure10.EBSDmaps(a1–a4),recrystallizationgraindistributionmaps(b1–b4)andmisorientation
angle(c1–c4)ofthespecimensdeformedwithstrainratesof1s−1at(a1–c1)423K,(a2–c2)473K,(a3–
c3)523K,and(a4–c4)573K.
Figure11.TEMmicrographsofZn-2.0Cu-0.15Tialloydeformedat(a)423K/0.1s−1;(b)423K/1s−1;
(c)523K/0.1s−1;(d)523K/10s−1.
Figure 11.
TEM micrographs of Zn-2.0Cu-0.15Ti alloy deformed at (
a
) 423 K/0.1 s
−1
; (
b
) 423 K/1 s
−1
;
(c) 523 K/0.1s−1; (d) 523 K/10 s−1.
4. Conclusions
The hot deformation behavior of Zn-2.0Cu-0.15Ti alloy was investigated at strain rates
0.01, 0.1, 1, and 10 s
−1
and deformation temperatures 423, 473, 523, and 573 K, using the
Arrhenius model and 3D processing maps. The following are the primary conclusions:
1.
The Arrhenius-type model is utilized to forecast flow stress behavior. The results
show the Arrhenius model can accurately predict the flow stress behavior of Zn-2.0Cu-
0.15Ti alloy.
2.
Three-dimensional processing maps are generated at different strains based on DDM
theory. The ideal processing domain for Zn-2.0Cu-0.15Ti alloy is the temperature
range from 493 to 543 K and strain rate range from 0.01 to 0.1 s−1.
3.
The softening mechanism of Zn-2.0Cu-0.15Ti alloy has diversification, including
CDRX, DDRX, and TDRX, and is activated at T = 423–573 K and
.
ε
= 0.01–10 s
−1
.
CDRX is activated at low deformation temperature (423 K) and high strain rate (1 s
−1
)
and is inhibited with increasing deformation temperature and decreasing strain rate.
However, when the deformation temperature increases (523 K) and the strain rate
declines (0.1 s
−1
), DDRX becomes the primary softening mechanism, and is weakened
with increasing strain rate. CDRX and TDRX become the main softening mechanisms
when the strain rate is 10 s−1.
Author Contributions:
Conceptualization, G.X.; Methodology, G.X.; Software, J.L. and Y.Z.; Formal
analysis, C.L.; Investigation, C.L.; Data curation, S.H.; Writing—original draft, Z.K.; Project adminis-
tration, D.Z. and Y.L.; Funding acquisition, D.Z. and Y.L. All authors have read and agreed to the
published version of the manuscript.
Materials 2023,16, 4431 13 of 14
Funding:
This research was funded by Natural Science Foundation of Hunan Province, China
(Grant Nos. 2021JJ30673 and 2022JJ30570), the Research Foundation of Zhuzhou Smelter Group Co.
(grant number ZYKJ202309), and the Project of the Education Department of Hunan Province (Grant
No. 20B569).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
The data are available from the corresponding author upon reason-
able request.
Acknowledgments:
This work was supported by the Natural Science Foundation of Hunan Province,
China (Grant Nos. 2021JJ30673 and 2022JJ30570), the Research Foundation of Zhuzhou Smelter Group
Co (grant number ZYKJ202309), and the Project of the Education Department of Hunan Province
(Grant No. 20B569).
Conflicts of Interest: The authors declare no conflict of interest.
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