application of mixture rule to finite element analysis for forging of cast mg–zn–y alloys with...
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Materials Science and Engineering A 548 (2012) 75– 82
Contents lists available at SciVerse ScienceDirect
Materials Science and Engineering A
jo ur n al hom epage: www.elsev ier .com/ locate /msea
pplication of mixture rule to finite element analysis for forging of cast Mg–Zn–Ylloys with long period stacking ordered structure
yo Matsumotoa,∗, Masaaki Otsub, Michiaki Yamasakic, Tsuyoshi Mayamad,iroshi Utsunomiyaa, Yoshihito Kawamurac
Division of Materials and Manufacturing Science, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita 565-0871, JapanDepartment of Mechanical Engineering, Faculty of Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui 910-8507, JapanDepartment of Materials Science and Engineering, Graduate School of Science and Technology, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, JapanPriority Organization for Innovation and Excellence, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan
r t i c l e i n f o
rticle history:eceived 14 December 2011eceived in revised form 10 March 2012ccepted 24 March 2012vailable online 8 April 2012
a b s t r a c t
To establish forging process for high strength Mg–Zn–Y alloys with a long period stacking ordered (LPSO)structure, the flow stresses of Mg–Zn–Y alloys with different volume fractions of LPSO phase were mea-sured by the upsettability test. Since mixture rule for the flow stress was satisfied in Mg–Zn–Y two-phase(�-Mg and LPSO) alloys, the flow stresses of �-Mg and LPSO single phase alloys were estimated from theflow stresses of Mg–Zn–Y alloys with different volume fractions of LPSO phase. To examine the valid-
eywords:orgingagnesium alloy
low stressixture rule
inite element analysis
ity of the mixture rule, the finite element analysis for tensile test and forging of as-cast Mg–Zn–Y alloywas carried out using the estimated flow stresses of �-Mg and LPSO single phase alloys on the basis ofmixture rule of the properties of Mg–Zn–Y alloy. The calculated load-stroke curves in tensile test andforging agreed well with the experimental ones, and the deformation behaviour of Mg–Zn–Y alloy wasdiscussed.
© 2012 Elsevier B.V. All rights reserved.
. Introduction
Magnesium alloys are increasingly used in the automotive andlectronics industries for lightweight structural and functionalarts due to the low density and high specific strength. Mg–Zn–Ylloys which consist of a fine-grained �-Mg matrix and a longeriod stacking ordered (LPSO) structure exhibit excellent mechan-
cal properties compared with conventional Mg alloys, for example,igh strength above 600 MPa in Mg97Zn1Y2 (at.%) RS P/M (rapidlyolidified powder metallurgy) [1–5]. Due to this, Mg–Zn–Y alloysre strongly desired to apply to the automotive parts and othertructural parts, however, amount of investigations concerning theorming properties of these alloys, especially the forging propertiesforgeability, flow stress), is still small [6,7].
Some properties of Mg–Zn–Y two-phase (�-Mg and LPSO) alloysuch as yield stress and hardness have been reported to satisfy withixture rule [8,9]. If the flow stress of Mg–Zn–Y alloys is satisfied
ith mixture rule, the flow stress of Mg–Zn–Y alloys with vari-us compositions can be predicted without any experiment, ands available in the computational simulation such as finite element
∗ Corresponding author. Tel.: +81 6 6879 7500; fax: +81 6 6879 7500.E-mail address: [email protected] (R. Matsumoto).
921-5093/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.msea.2012.03.088
analysis for metal working processes because the flow stress is oneof inevitable input data for the finite element analysis. Furthermore,Mg–Zn–Y alloy with optimum composition for forging process maybe determined from the computational simulation applying of mix-ture rule, and a new method for alloy design may be established.
To clarify deformation mechanism of metals in metal work-ing processes, inhomogeneity of metals has been considered asone of major solutions. Inhomogeneous deformation behaviourof Mg–Zn–Y alloys was experimentally observed by high preci-sion markers [10]. In computational simulation technique, somemethods for treatment of heterogeneity of material have beenproposed to realize high-accuracy calculation as well as to clarifythe deformation mechanism. To treat martensitic transformationinduced by plastic deformation of 18-8 stainless steel, the flowstresses of austenite and martensite phases were considered in thefinite element analysis of forging and deep drawing [11,12]. In thefinite element analysis of tensile deformation of aluminium alloy,anisotropy behaviour of the flow stress was considered for high-accuracy analysis [13]. The free surface roughening behaviour wasalso analyzed by the finite element simulation considering materialinhomogeneity [14].
To establish forging process for Mg–Zn–Y alloys, the flowstresses of Mg–Zn–Y alloys with different volume fractions of LPSOphase were measured by the upsettability test in this study. Themixture rule for the flow stress of Mg–Zn–Y two-phase (�-Mg
76 R. Matsumoto et al. / Materials Science and Engineering A 548 (2012) 75– 82
–Y alloys (�,volume fraction of LPSO phase).
aLt(edt
2
2
MwsoMta∼ipare
2
upstflfiiad
1008060402000
500
1000
1500
2000
2500D
ensi
ty [k
g·m
–3]
measured flow stress curves, a calculation method proposed byKada et al. [18] was applied. In this method, the isothermal flowstress was calculated by combining experimental results from theupsettability test with finite element analysis.
800
900
1000
1100
1200
Spe
cific
hea
t [J·
kg–1
·K–1
] Mg99.2Zn0.2Y0.6
Mg97Zn1Y2
Mg85Zn6Y9
Fig. 1. Microstructure of as-cast Mg–Zn
nd LPSO) alloys was discussed and the flow stresses of �-Mg andPSO single phase alloys were estimated from different composi-ion alloys. The finite element analysis for forging of cast Mg97Zn1Y2at.%) alloy having 26 vol.% LPSO phase was carried out using thestimated flow stresses of �-Mg and LPSO single phase alloys. Theeformation behaviour of the Mg alloy and the validity of the mix-ure rule on the finite element analysis were discussed.
. Experimental procedures
.1. Materials tested
The materials tested were as-cast Mg85Zn6Y9, Mg89Zn4Y7,g92Zn3Y5, Mg97Zn1Y2 and Mg99.2Zn0.2Y0.6 (at.%) alloys. The ingotsere prepared by high-frequency induction melting in an Ar atmo-
phere followed by homogenizing at 773 K for 10 h. Fig. 1 shows theptical micrographs of as-cast Mg85Zn6Y9, Mg89Zn4Y7, Mg92Zn3Y5,g97Zn1Y2 and Mg99.2Zn0.2Y0.6 alloys. The volume fractions (�) of
he LPSO phase of Mg85Zn6Y9, Mg89Zn4Y7, Mg92Zn3Y5, Mg97Zn1Y2nd Mg99.2Zn0.2Y0.6 alloys are estimated ∼100, ∼86, ∼61, ∼26 and1 vol.%, respectively. As shown in Fig. 1(e), small amount of the
nescapable intermetallic compounds was observed in the LPSOhase grain interior and grain boundary. The density, specific heatnd thermal conductivity of Mg–Zn–Y alloys are shown in Figs. 2–4,espectively [15]. These material properties were used in the finitelement analysis.
.2. Upsettability test
The flow stresses of Mg–Zn–Y alloys were measured by thepsettability test [16]. In the test, a cylindrical billet was com-ressed with concentrically grooved flat tools to restrict the endurfaces of the billet, so that the influence of friction betweenhe billet and the tool during the test was removed. The averageow stress and average equivalent strain were calculated by a
nite element simulation from the measured load and reductionn height in the experiment because the billet was deformed to barrel shape and the equivalent strain in the billet was notistributed uniformly [17]. Furthermore, to remove the influence
Volume f raction of LPSO phase φ [%]
Fig. 2. Density of Mg–Zn–Y alloys.
of the temperature change during the upsettability tests from the
600550500450400350300250700
Temperature [K]
Fig. 3. Specific heat of as-cast Mg–Zn–Y alloys.
R. Matsumoto et al. / Materials Science and Engineering A 548 (2012) 75– 82 77
6005505004504003503002500
20
40
60
80
100
120
Temperature [K]
Ther
mal
con
duct
ivity
[W·m
–1·K
–1]
Mg99.2Zn0.2Y0.6
Mg97Zn1Y2
Mg85Zn6Y9
dwiit
Fp
Fig. 4. Thermal conductivity of as-cast Mg–Zn–Y alloys.
The initial shape of specimen for the upsettabiity test was cylin-er with a diameter of 18 mm and a height of 27 mm. The specimen
as heated in a furnace without protective gas and was compressedn the temperature range of 473–773 K. To prevent the heated spec-men from rapidly cooling on the tool, the tools were heated to aemperature of 523 K when the testing temperatures were higher
0.60.50.40.30.20.10.00
100
200
300
400
500
Flow
stre
ss [M
Pa]
Average equivalent strain
473 K 573K 673 K
773K
0.60.50.40.30.20.10.00
100
200
300
400
500
Flow
stre
ss [M
Pa]
Average equivalent strain
473 K 573K
673K
773K
00.20.10.00
100
200
300
400
500
Flow
stre
ss [M
Pa]
Average eq
(a) Mg99.2Zn0.2Y0.6 alloy (φ = 1%)
(c) Mg92Zn3Y5 alloy (φ = 61%)
(e) Mg85Zn6Y9 allo
ig. 6. Isothermal flow stress curves of as-cast Mg–Zn–Y alloys having different volume fhase).
Fig. 5. Schematic illustration of tool arrangement for warm forging of Mg–Zn–Yalloy.
than 523 K, while the tools were heated to a temperature of 473 K incase of the testing temperature of 473 K. The upsettability test wasconducted on a material testing machine (Shimadzu Autograph,AG-250kNISE). The compression speed was 8.3 mm/s; the initialstrain rate at the beginning of compression was 0.31 s−1.
2.3. Tensile test
Tensile test of as-cast Mg97Zn1Y2 alloy (� = 26%) was carried outto examine the validity of the finite element analysis with apply-ing the mixture rule. The sheet of as-cast Mg97Zn1Y2 alloy with a
0.60.50.40.30.20.10.00
100
200
300
400
500
Flow
stre
ss [M
Pa]
Average equ ivalent strain
473 K 573K673 K
773K
0.60.50.40.30.20.10.00
100
200
300
400
500
Flow
stre
ss [M
Pa]
Average equ ivalent strain
473K 573K
673K773K
0.60.50.4.3uivalent strain
473K573K
673K773K
(b) Mg97Zn1Y2 alloy (φ = 26% )
(d) Mg89Zn4Y7 alloy (φ = 86% )
y (φ = 100% )
ractions of LPSO phase at various forging temperatures (�, volume fraction of LPSO
78 R. Matsumoto et al. / Materials Science and Engineering A 548 (2012) 75– 82
1008060402000
50
100
150
200
250
300
350
400
Volume fraction of LPSO ph aseφ [%]
Flow
stre
ss [M
Pa] 473K
573K
673K
773K
(a) Average equivalent strain: 0.1
1008060402000
50
100
150
200
250
300
350
400
Volume fraction of LPSO phas eφ [%]
Flow
stre
ss [M
Pa]
473 K573 K
673 K
773 K
Fig. 7. Relation between isothermal flow stress and volume fraction of LPSO p
Table 1Forging conditions of Mg–Zn–Y alloy.
Billet material As-cast Mg97Zn1Y2 alloy
Volume fraction of LPSO phase � [%] 26Initial billet shape: diameter × height [mm] 24 × 10Initial billet temperature [K] 573Punch diameter [mm] 16Punch temperature [K] 293Container temperature [K] 573
gw0
2
raofdrad
3
Mp
Fa
Punch speed [mm/s] 80Lubrication Dry condition
auge length of 10 mm, a width of 2 mm and a thickness of 1.6 mmas deformed at an initial temperature of 573 K at a strain rate of
.31 s−1 (see Fig. 10). The atmosphere was kept to be as 573 K.
.4. Forging test
Forging test of as-cast Mg97Zn1Y2 alloy (� = 26%) was also car-ied out to examine the validity of the finite element analysis withpplying the mixture rule. The tool arrangement for warm forgingf as-cast Mg97Zn1Y2 alloy is shown in Fig. 5. Table 1 shows theorging conditions. The initial shape of billet was cylinder with aiameter of 24 mm and a height of 10 mm. The forging was car-ied out on a servo press (Komatsu Industrial Corp., H1F45) with anverage forging speed of 80 mm/s at a temperature of 573 K underry condition.
. Flow stress curve
Fig. 6 shows the isothermal flow stress curves of as-castg–Zn–Y alloys having different volume fractions of LPSO phase,
rior to the occurrence of a crack in the billet at various forging
0.60.50.40.30.20.10.00
100
200
300
400
500
Flow
stre
ss [M
Pa]
Average equivalent strain
473K 573 K673K
773K
(a) α-Mg single phase (φ = 0%)
ig. 8. Isothermal flow stress curves of �-Mg and LPSO single phase alloys estimated fromlloys on the basis of the mixture rule.
(b) Average equivalent strain: 0.2
hase of cast Mg–Zn–Y alloys at average equivalent strains of 0.1 and 0.2.
temperatures. The flow stress curves exhibited work hardeningtendency at average equivalent strain lower than 0.45 irrespec-tive of forging temperature. The flow stress mostly increasedwith increasing volume fraction of LPSO phase, however, the flowstresses at a temperature of 773 K were almost same values irre-spective of volume fraction of LPSO phase. This may be affectedthat the billet temperature during upsetting was partly raised upto around melting temperature due to heat generation by plas-tic deformation at an initial billet temperature of 773 K. No phasetransformation or formation occurred during forging, i.e. the forgedalloys consisted of two phases. In the comparatively coarse �-Mg matrix grains of the specimens after forging, profuse twinswere observed. However, twinning was not found and some kink-deformation bands were observed in the LPSO phase region [4].
Since it was reported that mixture rule shown as Eq. (1) wassatisfied with the yield stress and hardness in Mg–Zn–Y two-phase(�-Mg and LPSO) alloys [8,9], the mixture rule for the flow stress isdiscussed.
XMg−Zn−Y = (1 − �)X�-Mg + �XLPSO (1)
where XMg–Zn–Y, X�-Mg and XLPSO are the properties of Mg–Zn–Ytwo-phase alloy, �-Mg single phase alloy and LPSO single phasealloy, respectively, and � is the volume fraction of LPSO phase. Theflow stresses of Mg–Zn–Y alloys at average equivalent strains of0.1 and 0.2 are plotted in Fig. 7. The dashed lines are the fit linesof the plotted marks of � = 26, 61 and 81%. The flow stresses of� = 1% at 573 K and 673 K were slightly higher than the dashed lines,while the flow stresses of � = 100% at 473 K, 573 K and 673 K werelower than the dashed lines. The interaction in the boundary of
�-Mg and LPSO phases makes the flow stress in Mg–Zn–Y two-phase alloys to be higher such as composite materials, however, thedetailed mechanism is not clear at present. A following assumptionis considered. As shown in Fig. 1(e), the grains in as-cat Mg85Zn6Y90.60.50.40.30.20.10.00
100
200
300
400
500
Flow
stre
ss [M
Pa]
Average equivalent strain
473K573K
673K
773K
(b) LPSO single phase (φ = 100%)
those of cast Mg89Zn4Y7 (� = 86%), Mg92Zn3Y5 (� = 61%) and Mg97Zn1Y2 (� = 26%)
R. Matsumoto et al. / Materials Science and Engineering A 548 (2012) 75– 82 79
0.60.50.40.30.20.10.0–25–20–15–10
–505
1015
Mea
sure
d flo
w s
tress
–es
timat
ed fl
ow s
tress
/MP
a
Average equivalent strai n
473K
573K
673K773K
0.60.50.40.30.20.10.0–25–20–15–10
–505
1015
Mea
sure
d flo
w s
tress
–es
timat
ed fl
ow s
tress
/MP
a
Average equ ivalent strain
473K
573K
673K
773K
0.60.50.40.30.20.10.0–25–20–15–10
–505
1015
Mea
sure
d flo
w s
tress
–es
timat
ed fl
ow s
tress
/MP
a
Average equ ivalent strain
473K573K
673K
773K
(a) Mg97Zn1Y2 alloy (φ (b) Mg= 26%) 92Zn3Y5 allo y (φ = 61%)
(c) Mg89Zn4Y7 alloy (φ = 86%)
F of as-a
at[prt
aoisFsadsflsa
ig. 9. Differences between measured and estimated isothermal flow stress curveslloys.
lloy (� = 100%) show plate-like shapes with a flat interface parallelo (0 0 0 1) and as-cast Mg85Zn6Y9 alloy has longer mean free path19] of glide basal dislocations in comparison with Mg–Zn–Y two-hase alloys. Consequently, the longer glide distance of basal slipesulted in lower flow stress in as-cast Mg85Zn6Y9 alloy than thathe expected one.
Except for the flow stresses of Mg99.2Zn0.2Y0.6 and Mg85Zn6Y9lloys, the relation between the flow stress and the volume fractionf LPSO phase shows the direct proportion from the dashed linesn Fig. 7. Thus the mixture rule for the flow stress is assumable toatisfy in Mg–Zn–Y two-phase alloys with the range of � = 26–81%.ig. 8 shows the isothermal flow stress curves of �-Mg and LPSOingle phase alloys estimated from those of Mg89Zn4Y7, Mg92Zn3Y5nd Mg97Zn1Y2 alloys on the basis of the mixture rule shown as theashed line in Fig. 7. To examine the accuracy of the estimated flow
tress curves, the differences between the estimated and measuredow stresses of Mg89Zn4Y7, Mg92Zn3Y5 and Mg97Zn1Y2 alloys arehown in Fig. 9. Relatively good agreement between the estimatednd measured flow stresses is found to be obtained.Fig. 10. Shape of Mg–Zn–Y specimen for tensil
cast (a) Mg97Zn1Y2 (� = 26%), (b) Mg92Zn3Y5 (� = 61%) and (c) Mg89Zn4Y7 (� = 86%)
4. Finite element analysis with consideration of mixturerule
4.1. Simulation method
To examine the validity of the mixture rule, the mixture rulewas applied to the finite element analysis for forming of as-castMg–Zn–Y two-phase alloy. Experimental results and calculatedones employing different calculation methods were compared. Oneof the calculation methods was the conventional one and anothermethod was a newly proposed one with consideration of the mix-ture rule.
In the conventional method, the properties of Mg–Zn–Ytwo-phase alloy such as the flow stress, density, specific heatand thermal conductivity, obtained macroscopically from the
experiment, were uniformly given to all elements as a single phasematerial. On the other hand, in the proposed method, each elementwas characterized by �-Mg or LPSO phases. The element numbers oftheir types were determined on the basis of their volume fractions.e test and finite element analysis model.
80 R. Matsumoto et al. / Materials Science and Engineering A 548 (2012) 75– 82
4.03.53.02.52.01.51.00.50.00
100
200
300
400
500
600Lo
ad [N
]
Stroke [mm]
FEM(α–Mg
+LPSO)
Experiment
FEM (Mg97Zn1Y2 alloy)
Fo
Tpc�a(ffsp�svta
[csctsbchFo
ycmt
4
a(Sapoe
s
(a) Temperature change
(b) Equivalent strain
4.03.53.02.52.01.51.00.50.0572
574
576
578
580
582
584
Stroke [mm]
Tem
pera
ture
[K]
Maximum
Average
Minimum
α–Mg+LPS OMg97Zn1Y2
4.03.53.02.52.01.51.00.50.00.00
0.05
0.10
0.15
0.20
0.25
0.30
Stroke [mm]
Equ
ival
ent s
train
Maximum
Average
Minimum
α–Mg+LPSOMg97Zn1Y2
ig. 11. Calculated and experimentally obtained load–stroke curves in tensile testf Mg–Zn–Y alloy.
he allocation of �-Mg phase and LPSO phase type elements wereeriodical and almost uniform in macroscopic view because theast alloy with isotropic properties was treated. The properties of-Mg and LPSO single phases obtained from the experiment, suchs the density (Fig. 2), specific heat (Fig. 3) and thermal conductivityFig. 4), were given to the corresponding type elements. However,or the flow stress curves of each type element, the estimated onesrom the mixture rule shown in Fig. 8 were used because the flowtresses of �-Mg (Mg99.2Zn0.2Y0.6) and LPSO (Mg85Zn6Y9) singlehases did not satisfy the mixture rule. That is, the properties of-Mg and LPSO phases obtained from the mixture rule shown inection 3 are separately applied to each element in accordance witholume fraction of each phase in the proposed method whereashe properties of two-phase alloy obtained from experiments arepplied to all elements in the conventional method.
The rigid-plastic finite element analysis for plastic deformation20] and heat conduction finite element analysis for temperaturehange were carried out alternately to calculate the stress, straintates and temperature distributions of the specimen at each cal-ulation step during forging. The changes of microstructure andexture were not taken into account in the simulation. The initialize of each element was 200 �m × 200 �m. The interface frictionetween the each phase is assumed to be sticking (no sliding). Theonstitutive relation used in this study was a multilinear isotropicardening determined from the stress–strain curves shown inig. 8. The temperature dependence of stress–strain curve wasbtained by linear interpolation from the input curves.
A similar method has been proposed in the finite element anal-sis of forming of 18-8 stainless steel to realize high-accuracyalculation [11,12]. In the method, the flow stresses of austenite andartensite phases were given to each element because martensitic
ransformation was induced by plastic deformation during forging.
.2. Simulation of tensile test of Mg–Zn–Y two-phase alloy
To examine the validity of the finite element analysis withpplying the mixture rule, tensile test of as-cast Mg97Zn1Y2 alloy� = 26%) was analyzed. The test conditions were described inection 2.3. In the simulation, the two-dimensional plane stressnalysis was conducted and the properties of �-Mg and LPSOhases were given to each element based on the volume fractions
f �-Mg and LPSO phases as shown in Fig. 10. The initial size of eachlement was 200 �m × 200 �m.Fig. 11 shows the calculated and experimentally obtained load-troke results in tensile test of as-cast Mg97Zn1Y2 alloy. The
Fig. 12. Changes of calculated temperatures and equivalent strains in tensile test ofMg–Zn–Y alloy.
load–stroke curve calculated with the mixture rule agreed wellwith the experimental one. Fig. 12 shows the changes of the cal-culated temperatures and equivalent strains in tensile test of thespecimen. The average temperature and equivalent strain of thespecimen were almost same with/without applying the mixturerule during tensile test, while the differences of maximum and min-imum values of temperature and equivalent strain in the simulationwith applying the mixture rule were larger than the simulationresults without applying the mixture rule. This means that the inho-mogeneous deformation induced by the difference of the propertiesof �-Mg and LPSO phases is promoted in the proposed simulationmethod with applying the mixture rule. The proposed simulationmethod has a potential to express the inhomogeneous deforma-tions of �-Mg and LPSO phases in Mg–Zn–Y two-phase alloy, andthus the method may be effective to realize the finite element anal-ysis with high accuracy in forming of Mg–Zn–Y two-phase alloy.
In the case of dual phase alloys, deformation generally startswith the yielding of the soft phase, followed by work hardeningthrough stress partitioning due to inhomogeneous plastic defor-mation between the soft and hard phases. It is expected that theinteraction effect between alpha-Mg and LPSO phases contributesto strengthening of the dual phase alloy in the beginning of theplastic region. However, experimental results dealing with as-castMg–Zn–Y alloys showed that the mixture rule for the flow stress ofMg–Zn–Y alloys was suitable in this case, since calculation based oncontinuum mechanics agreed well with the experimental results.
On the contrary, this agreement would suggest that the interactioneffect between �-Mg and LPSO phases in the as-cast state is insignif-icant, because the as-cast alloys examined in this study have coarse
R. Matsumoto et al. / Materials Science and Engineering A 548 (2012) 75– 82 81
α-Mg LPSO
200μ
m
200μm
Mg billet (plastic body)
Punch(rigid body)
Container(rigid body)
Mg97Zn1Y2
(a) α-Mg and LPSO with applying mixture rule
method)
(b) Mg97Zn1Y2 withoutapplying mixture rule (conventional method)
odel of warm forging of Mg–Zn–Y alloy.
ggpb
5
aftaauesmto
llwtwthcawittpa
4.03.53.02.52.01.51.00.50.00
50
100
150
200
Punch stroke [mm]
Forg
ing
load
[kN
]Experiment
(crack occurs at punch stroke of about 3.0mm)
FEM (α–Mg+LPSO)
FEM (Mg97Zn1Y2 alloy)
(proposed
Fig. 13. Finite element analysis m
rains with low dispersion of the hard LPSO phase. Further investi-ation of the interaction between �-Mg and LPSO phases is now inrogress, and some results from viewpoint of microstructure wille reported in future work.
. Application to finite element analysis of forging
The finite element analysis with applying the mixture rule waspplied to warm forging of as-cast Mg97Zn1Y2 alloy (� = 26%). Theorging conditions were described in Section 2.4. Fig. 13 showshe finite element analysis model in forging. The two-dimensionalxisymmetric analysis was conducted and the properties of �-Mgnd LPSO phases were given to each element based on the vol-me fractions of �-Mg and LPSO phases. The initial size of eachlement was 200 �m × 200 �m. Heat transfer coefficients at thepecimen-tool contact interfaces and free surfaces were deter-ined respectively as 10000 W m−2 K−1 and 16 W m−2 K−1 from
he heating and cooling tests of the billet. The frictional conditionf the specimen-tool interface was � = 0.20.
Fig. 14 shows the calculated and experimentally obtainedoad–stroke results in forging of as-cast Mg97Zn1Y2 alloy. Theoad–stroke curve calculated with the mixture rule agreed well
ith the experimental one as well as the simulation result ofensile test. The calculated temperature distributions of the billetith/without applying the mixture rule are shown in Fig. 15. The
emperature of the proposed method around the punch corner wasigher than that of the conventional method. Fig. 16 shows the cal-ulated temperature changes in forging of the billet. Although theverage temperature of the billet was almost same during forgingith/without applying the mixture rule, the maximum and min-
mum temperatures with applying the mixture rule were higher
han the conventionally calculated ones. If the same strain is giveno the element in the finite element analysis, the heat generation oflastic deformation in LPSO phase is larger than that in Mg97Zn1Y2lloy because the flow stress of the LPSO phase is higher than that ofFig. 15. Calculated temperature distribu
Fig. 14. Calculated and experimentally obtained forging load-stroke curves in forg-ing of Mg–Zn–Y alloy.
Mg97Zn1Y2 alloy. The temperature with applying the mixture ruletends to be higher than the conventionally calculated ones. Thusthe load calculated with the mixture rule was lower than that con-ventionally calculated ones, and agreed well with the experimentalone.
To discuss the influence of the allocation of �-Mg and LPSOphases and/or analysis of forging with a complicated shape, three-dimensional finite element analysis with applying the mixture ruleis needed. However, the three-dimensional finite element analysiswith applying the mixture rule is difficult to conduct on the finite
element simulation code used in this study. Further investigationson the validity of the application of the mixture rule to the finiteelement analysis are a future work.tions in forging of Mg–Zn–Y alloy.
82 R. Matsumoto et al. / Materials Science an
4.03.53.02.52.01.51.00.50.0500
550
600
650
700
750Te
mpe
ratu
re [K
]
Maximum
Minimum
Average
Mg97Zn1Y2
α–Mg+LPSO
6
ut(ycw
(
(
A
oE
[
[
[
[
[
[
[
[
[
[19] K. Hagihara, A. Kinoshita, Y. Fukusumi, M. Yamasaki, Y. Kawamura, Mater. Sci.Forum 706–709 (2012) 1158–1163.
[20] K. Osakada, J. Nakano, K. Mori, Finite element method for rigid-plastic analysis
Punch stroke [mm]
Fig. 16. Calculated temperature changes in forging of Mg–Zn–Y alloy.
. Conclusions
The flow stresses of as-cast Mg–Zn–Y alloys with different vol-me fractions of LPSO phase were measured by the upsettabilityest. The mixture rule for the flow stress of Mg–Zn–Y two-phase�-Mg and LPSO) alloys was discussed and the finite element anal-ses of tensile test and forging of Mg–Zn–Y two-phase alloys wasarried out with applying mixture rule. The following conclusionsere obtained.
1) Mixture rule for the flow stresses of Mg–Zn–Y two-phase (�-Mgand LPSO) alloys was approved in the range of volume fractionof LPSO phase of 26–81 vol.%. The flow stresses of �-Mg andLPSO single phase alloys were estimated from the flow stressesof Mg–Zn–Y two-phase alloys with different volume fractionsof LPSO phase.
2) The mixture rule for the properties of Mg–Zn–Y two-phasealloys was applied to the finite element analysis for form-ing by giving the properties to each element in accordancewith volume fraction of LPSO phase. In the proposed method,inhomogeneous deformations of �-Mg and LPSO phases inMg97Zn1Y2 alloy having 26 vol.% LPSO phase were analyzed.Good agreements of the load–stroke results in tensile test andforging of Mg97Zn1Y2 alloy was obtained between the calcu-lated ones and experimental ones.
cknowledgment
This work was supported by the Kumamoto Prefecture Collab-ration of Regional Entities for the Advancement of Technologicalxcellence, Japan Science and Technology Agency (JST).
d Engineering A 548 (2012) 75– 82
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