Title:
Application of Mixture Rule to Finite Element Analysis for Forging of Cast
Mg-Zn-Y Alloys with Long Period Stacking Ordered Structure
Authors:
Ryo Matsumoto1,*, Masaaki Otsu2, Michiaki Yamasaki3, Tsuyoshi Mayama4, Hiroshi
Utsunomiya1 and Yoshihito Kawamura3
* Correspnding author (R. Matsumoto,
+81-6-6879-7500, Fax: +81-6-6879-7500)
E-mail:
[email protected],
Tel:
Affiliation:
1
Division of Materials and Manufacturing Science, Graduate School of Engineering, Osaka
University, 2-1 Yamadaoka, Suita 565-0871, Japan
2
Department of Mechanical Engineering, Faculty of Engineering, University of Fukui, 3-9-1
Bunkyo, Fukui 910-8507, Japan
3
Department of Materials Science and Engineering, Graduate School of Science and
Technology, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan
4
Priority Organization for Innovation and Excellence, Kumamoto University, 2-39-1
Kurokami, Kumamoto 860-8555, Japan
Abstract
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 measured 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 the flow stresses of Mg-Zn-Y alloys
with different volume fractions of LPSO phase. To examine the validity of the mixture rule,
the finite element analysis for tensile test and forging of as-cast Mg-Zn-Y alloy was carried
out using the estimated flow stresses of -Mg and LPSO single phase alloys on the basis of
mixture rule of the properties of Mg-Zn-Y alloy. The calculated load-stroke curves in tensile
test and forging agreed well with the experimental ones, and the deformation behaviour of
Mg-Zn-Y alloy was discussed.
Keywords: Forging, Magnesium alloy, Flow stress, Mixture rule, Finite element analysis
1. Introduction
Magnesium alloys are increasingly used in the automotive and electronics industries
for lightweight structural and functional parts due to the low density and high specific
strength. Mg-Zn-Y alloys which consist of a fine-grained -Mg matrix and a long period
stacking ordered (LPSO) structure exhibit excellent mechanical properties compared with
conventional Mg alloys, for example, high strength above 600 MPa in Mg97Zn1Y2 (at.%) RS
P/M (rapidly solidified powder metallurgy) [1-5]. Due to this, Mg-Zn-Y alloys are strongly
desired to apply to the automotive parts and other structural parts, however, amount of
investigations concerning the forming properties of these alloys, especially the forging
properties (forgeability, flow stress), is still small [6,7].
Some properties of Mg-Zn-Y two-phase (-Mg and LPSO) alloys such as yield stress
and hardness have been reported to satisfy with mixture rule [8,9]. If the flow stress of
Mg-Zn-Y alloys is satisfied with mixture rule, the flow stress of Mg-Zn-Y alloys with various
compositions can be predicted without any experiment, and is available in the computational
simulation such as finite element analysis for metal working processes because the flow stress
is one of inevitable input data for the finite element analysis. Furthermore, Mg-Zn-Y alloy
with optimum composition for forging process may be determined from the computational
simulation applying of mixture rule, and a new method for alloy design may be established.
To clarify deformation mechanism of metals in metal working processes,
inhomogeneity of metals has been considered as one of major solutions. Inhomogeneous
deformation behaviour of Mg-Zn-Y alloys was experimentally observed by high precision
markers [10]. In computational simulation technique, some methods for treatment of
heterogeneity of material have been proposed to realize high-accuracy calculation as well as
to clarify the deformation mechanism. To treat martensitic transformation induced by plastic
deformation of 18-8 stainless steel, the flow stresses of austenite and martensite phases were
considered in the finite element analysis of forging and deep drawing [11,12]. In the finite
element analysis of tensile deformation of aluminium alloy, anisotropy behaviour of the flow
stress was considered for high-accuracy analysis [13]. The free surface roughnening
behaviour was also analyzed by the finite element simulation considering material
inhomogeneity [14].
To establish forging process for Mg-Zn-Y alloys, the flow stresses of Mg-Zn-Y
alloys with different volume fractions of LPSO phase were measured by the upsettability test
in this study. The mixture rule for the flow stress of Mg-Zn-Y two-phase (-Mg and LPSO)
alloys was discussed and the flow stresses of -Mg and LPSO single phase alloys were
estimated from different composition alloys. The finite element analysis for forging of cast
Mg97Zn1Y2 (at.%) alloy having 26 vol.% LPSO phase was carried out using the estimated
flow stresses of -Mg and LPSO single phase alloys. The deformation behaviour of the Mg
alloy and the validity of the mixture rule on the finite element analysis were discussed.
2. Experimental procedures
2.1. Materials tested
The materials tested were as-cast Mg85Zn6Y9, Mg89Zn4Y7, Mg92Zn3Y5, Mg97Zn1Y2
and Mg99.2Zn0.2Y0.6 (at.%) alloys. The ingots were prepared by high-frequency induction
melting in an Ar atmosphere followed by homogenizing at 773 K for 10 h. Fig. 1 shows the
optical micrographs of as-cast Mg85Zn6Y9, Mg89Zn4Y7, Mg92Zn3Y5, Mg97Zn1Y2 and
Mg99.2Zn0.2Y0.6 alloys. The volume fractions () of the LPSO phase of Mg85Zn6Y9,
Mg89Zn4Y7, Mg92Zn3Y5, Mg97Zn1Y2 and Mg99.2Zn0.2Y0.6 alloys are estimated ~100, ~86, ~61,
~26 and ~1 vol.%, respectively. As shown in Fig. 1(e), small amount of the inescapable
intermetallic compounds was observed in the LPSO phase grain interior and grain boundary.
The density, specific heat and thermal conductivity of Mg-Zn-Y alloys are shown in Figs. 2, 3
and 4, respectively [15]. These material properties were used in the finite element analysis.
Figg. 1 Microstrructure of as-cast
a
Mg-Z
Zn-Y alloyss (: volumee fraction off LPSO phaase).
–3
Density [kg·m ]
2500
2000
1500
1000
500
0
0
20
40
60
80
100
Volume fraction
f
of LPSO phase
e  [%]
1200
–1
–1
Specific heat [J·kg ·K ]
Fig. 2 Denssity of Mg-Z
Zn-Y alloyss.
Mg
g 99.2Zn0.2Y0.6
1100
1000
Mg 97Zn
n1Y2
900
800
Mg 85Zn
n6Y9
700
250 300 3
350 400 450
4
500 55
50 600
Temperatu
ure [K]
Fig. 3 Specific heeat of as-casst Mg-Zn-Y alloys.
–1
–1
Thermal conductivity [W·m ·K ]
120
100
Mg 85Zn6Y9
80
60
Mg 97Zn1Y2
40
20
Mg 99.2Zn0.2Y0.6
0
250 300 350 400 450 500 550 600
Temperature [K]
Fig. 4 Thermal conductivity of as-cast Mg-Zn-Y alloys.
2.2. Upsettability test
The flow stresses of Mg-Zn-Y alloys were measured by the upsettability test [16]. In
the test, a cylindrical billet was compressed with concentrically grooved flat tools to restrict
the end surfaces of the billet, so that the influence of friction between the billet and the tool
during the test was removed. The average flow stress and average equivalent strain were
calculated by a finite element simulation from the measured load and reduction in height in
the experiment because the billet was deformed to a barrel shape and the equivalent strain in
the billet was not distributed uniformly [17]. Furthermore, to remove the influence of the
temperature change during the upsettability tests from the measured flow stress curves, a
calculation method proposed by Kada et al. [18] was applied. In this method, the isothermal
flow stress was calculated by combining experimental results from the upsettability test with
finite element analysis.
The initial shape of specimen for the upsettabiity test was cylinder with a diameter of
18 mm and a height of 27 mm. The specimen was heated in a furnace without protective gas
and was compressed in the temperature range of 473–773 K. To prevent the heated specimen
from rapidly cooling on the tool, the tools were heated to a temperature of 523 K when the
testing temperatures were higher than 523 K, while the tools were heated to a temperature of
473 K in case of the testing temperature of 473 K. The upsettability test was conducted on a
material testing machine (Shimadzu Autograph, AG-250kNISE). The compression speed was
8.3 mm/s; the initial strain 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 out to examine the
validity of the finite element analysis with applying the mixture rule. The sheet of as-cast
Mg97Zn1Y2 alloy with a gauge length of 10 mm, a width of 2 mm and a thickness of 1.6 mm
was deformed at an initial temperature of 573 K at a strain rate of 0.31 s-1 (see Fig. 10). The
atmosphere was kept to be as 573 K.
2.4. Forging test
Forging test of as-cast Mg97Zn1Y2 alloy ( = 26%) was also carried out to examine
the validity of the finite element analysis with applying the mixture rule. The tool
arrangement for warm forging of as-cast Mg97Zn1Y2 alloy is shown in Fig. 5. Table 1 shows
the forging conditions. The initial shape of billet was cylinder with a diameter of 24 mm and a
height of 10 mm. The forging was carried out on a servo press (Komatsu Industrial Corp.,
H1F45) with an average forging speed of 80 mm/s at a temperature of 573 K under dry
condition.
16
Punch
Container
Mg
specimen
Knockout
punch
24
10
Stick
heaters
15º
Fig. 5 Schematic illustration of tool arrangement for warm forging of Mg-Zn-Y alloy.
Table 1 Forging conditions of Mg-Zn-Y alloy.
Billet material
As-cast Mg97Zn1Y2 alloy
Volume fraction of LPSO phase  [%]
26
Initial billet shape: diameter x height [mm] 24 x 10
Initial billet temperature [K]
573
Punch diameter [mm]
16
Punch temperature [K]
293
Container temperature [K]
573
Punch speed [mm/s]
80
Lubrication
Dry condition
3. Flow stress curve
Fig. 6 shows the isothermal flow stress curves of as-cast Mg-Zn-Y alloys having
different volume fractions of LPSO phase, prior to the occurrence of a crack in the billet at
various forging temperatures. The flow stress curves exhibited work hardening tendency at
average equivalent strain lower than 0.45 irrespective of forging temperature. The flow stress
mostly increased with increasing volume fraction of LPSO phase, however, the flow stresses
at a temperature of 773 K were almost same values irrespective of volume fraction of LPSO
phase. This may be affected that the billet temperature during upsetting was partly raised up to
around melting temperature due to heat generation by plastic deformation at an initial billet
temperature of 773 K. No phase transformation or formation occurred during forging, i.e. the
forged alloys consisted of two phases. In the comparatively coarse -Mg matrix grains of the
specimens after forging, profuse twins were 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) was satisfied 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 is discussed.
(1)
XMg-Zn-Y = (1- )X-Mg + XLPSO
where XMg-Zn-Y, X-Mg and XLPSO are the properties of Mg-Zn-Y two-phase alloy, -Mg single
phase alloy and LPSO single phase alloy, respectively, and  is the volume fraction of LPSO
phase. The flow stresses of Mg-Zn-Y alloys at average equivalent strains of 0.1 and 0.2 are
plotted in Fig. 7. The dashed lines are the fit lines of 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 were lower 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, the detailed
mechanism is not clear at present. A following assumption is considered. As shown in Fig.
1(e), the grains in as-cat Mg85Zn6Y9 alloy ( = 100%) show plate-like shapes with a flat
interface parallel to (0001) and as-cast Mg85Zn6Y9 alloy has longer mean free path [19] of
glide basal dislocations in comparison with Mg-Zn-Y two-phase alloys. Consequently, the
longer glide distance of basal slip resulted in lower flow stress in as-cast Mg85Zn6Y9 alloy
than that the expected one.
Except for the flow stresses of Mg99.2Zn0.2Y0.6 and Mg85Zn6Y9 alloys, the relation
between the flow stress and the volume fraction of LPSO phase shows the direct proportion
from the dashed lines in Fig. 7. Thus the mixture rule for the flow stress is assumable to
satisfy in Mg-Zn-Y two-phase alloys with the range of  = 26–81%. Fig. 8 shows the
isothermal flow stress curves of -Mg and LPSO single phase alloys estimated from those of
Mg89Zn4Y7, Mg92Zn3Y5 and Mg97Zn1Y2 alloys on the basis of the mixture rule shown as the
dashed line in Fig. 7. To examine the accuracy of the estimated flow stress curves, the
differences between the estimated and measured flow stresses of Mg89Zn4Y7, Mg92Zn3Y5 and
Mg97Zn1Y2 alloys are shown in Fig. 9. Relatively good agreement between the estimated and
measured flow stresses is found to be obtained.
500
400
300
473K 573K
200
673K
100
773K
0
0.0
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
Flow stress [MPa]
Flow stress [MPa]
500
473K
300
200
773K
0
0.0
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
Flow stress [MPa]
500
400
473K
300
673K
200
100
773K
0
0.0
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
473K
400
573K
300
673K
200
773K
100
0
0.0
0.6
(c) Mg92Zn3Y5 alloy ( = 61%)
0.6
(b) Mg97Zn1Y2 alloy ( = 26%)
500
573K
573K
673K
100
0.6
(a) Mg99.2Zn0.2Y0.6 alloy ( = 1%)
Flow stress [MPa]
400
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
0.6
(d) Mg89Zn4Y7 alloy ( = 86%)
Flow stress [MPa]
500
473K
400
573K
300
673K
200
773K
100
0
0.0
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
0.6
(e) Mg85Zn6Y9 alloy ( = 100%)
400
400
350
350
300
250
473K
200
573K
150
673K
100
773K
Flow stress [MPa]
Flow stress [MPa]
Fig. 6 Isothermal flow stress curves of as-cast Mg-Zn-Y alloys having different volume
fractions of LPSO phase at various forging temperatures (: volume fraction of LPSO phase).
300
473K
573K
250
200
150
673K
100
773K
50
50
0
0
0
20
40
60
80 100
Volume fraction of LPSO phase  [%]
0
20
40
60
80 100
Volume fraction of LPSO phase  [%]
(a) Average equivalent strain: 0.1
(b) Average equivalent strain: 0.2
Fig. 7 Relation between isothermal flow stress and volume fraction of LPSO phase of as-cast
Mg-Zn-Y alloys at average equivalent strains of 0.1 and 0.2.
500
500
Flow stress [MPa]
Flow stress [MPa]
473K
400
300
200
100
0
0.0
473K 573K
673K
773K
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
400
673K
300
200
773K
100
0
0.0
0.6
(a) -Mg single phase ( = 0%)
573K
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
0.6
(b) LPSO single phase ( = 100%)
15
10
5
0
–5
–10
–15
–20
–25
0.0
Measured flow stress –
estimated flow stress [MPa]
Measured flow stress –
estimated flow stress [MPa]
Fig. 8 Isothermal flow stress curves of -Mg and LPSO single phase alloys estimated from
those of as-cast Mg89Zn4Y7 ( = 86%), Mg92Zn3Y5 ( = 61%) and Mg97Zn1Y2 ( = 26%)
alloys on the basis of the mixture rule.
573K
473K
773K
673K
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
0.6
Measured flow stress –
estimated flow stress [MPa]
(a) Mg97Zn1Y2 alloy ( = 26%)
15
10
5
0
–5
–10
–15
–20
–25
0.0
15
10
5
0
–5
–10
–15
–20
–25
0.0
673K
773K
473K
573K
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
0.6
(b) Mg92Zn3Y5 alloy ( = 61%)
573K
473K
773K
673K
0.1 0.2 0.3 0.4 0.5
Average equivalent strain
0.6
(c) Mg89Zn4Y7 alloy ( = 86%)
Fig. 9 Differences between measured and estimated isothermal flow stress curves of as-cast
(a) Mg97Zn1Y2 ( = 26%), (b) Mg92Zn3Y5 ( = 61%) and (c) Mg89Zn4Y7 ( = 86%) alloys.
4. Finite element analysis with consideration of mixture rule
4.1. Simulation method
To examine the validity of the mixture rule, the mixture rule was applied to the finite
element analysis for forming of as-cast Mg-Zn-Y two-phase alloy. Experimental results and
calculated ones employing different calculation methods were compared. One of the
calculation methods was the conventional one and another method was a newly proposed one
with consideration of the mixture rule.
In the conventional method, the properties of Mg-Zn-Y two-phase alloy such as the
flow stress, density, specific heat and thermal conductivity, obtained macroscopically from the
experiment, were uniformly given to all elements as a single phase material. On the other
hand, in the proposed method, each element was characterized by -Mg or LPSO phases. The
element numbers of their types were determined on the basis of their volume fractions. The
allocation of -Mg phase and LPSO phase type elements were periodical and almost uniform
in macroscopic view because the cast alloy with isotropic properties was treated. The
properties of -Mg and LPSO single phases obtained from the experiment, such as the density
(Fig. 2), specific heat (Fig. 3) and thermal conductivity (Fig. 4), were given to the
corresponding type elements. However, for the flow stress curves of each type element, the
estimated ones from the mixture rule shown in Fig. 8 were used because the flow stresses of
-Mg (Mg99.2Zn0.2Y0.6) and LPSO (Mg85Zn6Y9) single phases did not satisfy the mixture rule.
That is, the properties of -Mg and LPSO phases obtained from the mixture rule shown in
section 3 are separately applied to each element in accordance with volume fraction of each
phase in the proposed method whereas the properties of two-phase alloy obtained from
experiments are applied to all elements in the conventional method.
The rigid-plastic finite element analysis for plastic deformation [20] and heat
conduction finite element analysis for temperature change were carried out alternately to
calculate the stress, strain states and temperature distributions of the specimen at each
calculation step during forging. The changes of microstructure and texture were not taken into
account in the simulation. The initial size of each element was 200 m x 200 m. The
interface friction between the each phase is assumed to be sticking (no sliding). The
constitutive relation used in this study was a multilinear isotropic hardening determined from
the stress-strain curves shown as Fig. 8. The temperature dependence of stress-strain curve
was obtained by linear interpolation from the input curves.
A similar method has been proposed in the finite element analysis of forming of 18-8
stainless steel to realize high-accuracy calculation [11,12]. In the method, the flow stresses of
austenite and martensite phases were given to each element because martensitic
transformation was induced by plastic deformation during forging.
4.2. Simulation of tensile test of Mg-Zn-Y two-phase alloy
To examine the validity of the finite element analysis with applying the mixture rule,
tensile test of as-cast Mg97Zn1Y2 alloy ( = 26%) was analyzed. The test conditions were
described in section 2.3. In the simulation, the two-dimensional plane stress analysis was
conducted and the properties of -Mg and LPSO phases were given to each element based on
the volume fractions of -Mg and LPSO phases as shown in Fig. 10. The initial size of each
element was 200 m x 200 m.
Fig. 11 shows the calculated and experimentally obtained load-stroke results in
tensile test of as-cast Mg97Zn1Y2 alloy. The load-stroke curve calculated with the mixture rule
agreed well with the experimental one. Fig. 12 shows the changes of the calculated
temperatures and equivalent strains in tensile test of the specimen. The average temperature
and equivalent strain of the specimen were almost same with/without applying the mixture
rule during tensile test, while the differences of maximum and minimum values of
temperature and equivalent strain in the simulation with applying the mixture rule were larger
than the simulation results without applying the mixture rule. This means that the
inhomogeneous deformation induced by the difference of the properties of -Mg and LPSO
phases is promoted in the proposed simulation method with applying the mixture rule. The
proposed simulation method has a potential to express the inhomogeneous deformations of
-Mg and LPSO phases in Mg-Zn-Y two-phase alloy, and thus the method may be effective
to realize the finite element analysis with high accuracy in forming of Mg-Zn-Y two-phase
alloy.
In the casee of dual ph
hase alloys, deformatio
on generally
y starts withh the yieldin
ng of the
k hardeningg through stress
s
partitioning due to inhomo
ogeneous
soft phaase, followeed by work
plastic ddeformation
n between th
he soft and hard phasees. It is expeected that thhe interactio
on effect
betweenn alpha-Mg
g and LPSO
O phases conntributes to
o strengthen
ning of the dual phase alloy in
the begginning of the plastic region. H
However, ex
xperimentall results deealing with
h as-cast
Mg-Zn--Y alloys sh
howed thatt the mixtuure rule for the flow stress
s
of M
Mg-Zn-Y alloys was
suitablee in this casse, since caalculation b ased on continuum mechanics aggreed well with the
experim
mental resultts. On the contrary,
c
thiis agreemen
nt would sug
ggest that thhe interactio
on effect
betweenn alpha-Mg
g and LPSO
O phases in the as-cast state is inssignificant, because thee as-cast
alloys eexamined in
n this study have coarsee grains witth low dispeersion of thee hard LPSO
O phase.
Further investigatiion of the interactionn between alpha-Mg and
a LPSO phases is now in
progresss, and somee results from
m viewpoinnt of microsstructure willl be reporteed in future work.
o Mg-Zn-Y
Y specimen for tensile test
t and finiite element analysis mo
odel.
Fig.. 10 Shape of
600
FEM (Mg 97
oy)
9 Zn1Y2 allo
500
EM
FE
(–M
Mg
+LPS
SO)
Load [N]
400
300
Experime
ent
200
100
0
0
0.0
0.5 1.0
0 1.5 2.0 2.5 3.0 3.5 4.0
Stroke [m
mm]
mentally obbtained load
d-stroke currves in tensiile test of Mg-Zn-Y
M
Fig. 11 Calculated and experim
alloy.
Temperature [K]
584
582
–Mg+LPSO
Mg 97Zn1Y2
Maximum
Average
580
Minimum
578
576
574
572
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Stroke [mm]
(a) Temperature change
Equivalent strain
0.30
0.25
0.20
–Mg+LPSO
Mg 97Zn1Y2
Maximum
Average
0.15
0.10
Minimum
0.05
0.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Stroke [mm]
(b) Equivalent strain
Fig. 12 Changes of calculated temperatures and equivalent strains in tensile test of Mg-Zn-Y
alloy.
5. Application to finite element analysis of forging
The finite element analysis with applying the mixture rule was applied to warm
forging of as-cast Mg97Zn1Y2 alloy ( = 26%). The forging conditions were described in
section 2.4. Fig.13 shows the finite element analysis model in forging. The two-dimensional
axisymmetric analysis was conducted and the properties of -Mg and LPSO phases were
given to each element based on the volume fractions of -Mg and LPSO phases. The initial
size of each element was 200 m x 200 m. Heat transfer coefficients at the specimen-tool
contact interfaces and free surfaces were determined respectively as 10000 W·m-2·K-1 and 16
W·m-2·K-1 from the heating and cooling tests of the billet. The frictional condition of the
specimen-tool interface was  = 0.20.
Fig. 14 shows the calculated and experimentally obtained load-stroke results in
forging of as-cast Mg97Zn1Y2 alloy. The load-stroke curve calculated with the mixture rule
agreed well with the experimental one as well as the simulation result of tensile test. The
calculated temperature distributions of the billet with/without applying the mixture rule are
shown in Fig. 15. The temperature of the proposed method around the punch corner was
higher than that of the conventional method. Fig. 16 shows the calculated temperature
changes in forging of the billet. Although the average temperature of the billet was almost
same during forging with/without applying the mixture rule, the maximum and minimum
temperatures with applying the mixture rule were higher than the conventionally calculated
ones. If the same strain is given to the element in the finite element analysis, the heat
generatiion of plasstic deformation in LP
PSO phase is larger than
t
that inn Mg97Zn1Y2 alloy
becausee the flow stress of th
he LPSO pphase is hig
gher than that
t
of Mgg97Zn1Y2 allloy. The
temperaature with applying th
he mixturee rule tends to be higher than the conven
ntionally
calculatted ones. Thus
T
the load
l
calcullated with the mixturre rule waas lower th
han that
conventtionally calcculated oness, and agreeed well with
h the experim
mental one..
To discusss the influen
nce of the aallocation of -Mg and
d LPSO phaases and/or analysis
of forging with a complicated
c
d shape, threee-dimensio
onal finite element
e
anaalysis with applying
a
the mixxture rule is
i needed. However, the three-d
dimensional finite elem
ment analy
ysis with
applyingg the mixtu
ure rule is difficult to cconduct on the
t finite element simuulation codee used in
this studdy. Further investigatio
ons on the validity of the applicaation of the mixture ru
ule to the
finite ellement analy
ysis are a fu
uture work.
Fig. 13 Finite
F
elemeent analysis model of warm
w
forgin
ng of Mg-Znn-Y alloy.
Forging load [kN]
200
FEM (–Mg+LPS
SO)
FEM (Mg
g 97Zn1Y2 allo
oy)
150
100
50
0
Experiment
curs at punc
ch
(crack occ
stroke of about 3.0m
mm)
0
0
0.0
0.5 1.0
0 1.5 2.0 2.5 3.0 3.5 4.0
P
Punch
stroke [mm]
ging load-sttroke curvees in forging
g of cast
Fig. 14 Calculated and experiimentally obbtained forg
Mg-Zn--Y alloy.
Fig. 15 Callculated tem
mperature diistributions in forging of
o cast Mg--Zn-Y alloy..
750
Temperature [K]
700
–
–Mg+LPSO
Mg
g 97Zn1Y2
650
Maximum
Avera
age
600
550
Minim
mum
500
0
0.0
0.5 1.0
0 1.5 2.0 2.5 3.0 3.5 4.0
P
Punch
stroke [mm]
Fig. 16 Calculated
C
temperature
t
e changes in
n forging of cast Mg-Znn-Y alloy.
6. Concclusions
The flow stresses
s
of as-cast
a
Mg- Zn-Y alloyss with differrent volumee fractions of
o LPSO
phase w
were measu
ured by thee upsettabiility test. The
T mixturee rule for the flow stress
s
of
Mg-Zn--Y two-phasse (-Mg and
a LPSO) aalloys was discussed and
a the finitte element analyses
of tensiile test and
d forging of
o Mg-Zn-Y
Y two-phasse alloys was
w carried out with applying
a
mixturee rule. The following
f
co
onclusions w
were obtain
ned.
(1) Mixxture rule fo
or the flow stresses off Mg-Zn-Y two-phase (-Mg andd LPSO) allloys was
apprroved in th
he range off volume fr
fraction of LPSO phasse of 26–8 1 vol.%. The
T flow
stressses of-M
Mg and LPS
SO single phhase alloys were estim
mated from tthe flow strresses of
Mg--Zn-Y two-p
phase alloys with differrent volumee fractions of
o LPSO phhase.
(2) The mixture ru
ule for the properties off Mg-Zn-Y two-phase alloys was aapplied to the
t finite
elem
ment analysis for formiing by givinng the prop
perties to each element in accordan
nce with
voluume fraction
n of LPSO phase.
p
In thhe proposed
d method, in
nhomogeneoous deformaations of
-M
Mg and LPSO phases in
n Mg97Zn1Y 2 alloy hav
ving 26 vol.% LPSO phhase were analyzed.
a
Goood agreemen
nts of the lo
oad-stroke rresults in teensile test an
nd forging oof Mg97Zn1Y2 alloy
was obtained between the calculated oones and ex
xperimental ones.
Acknow
wledgemen
nt
This work
k was supported by thhe Kumamo
oto Prefectu
ure Collabooration of Regional
R
Entities for the Ad
dvancementt of Technoological Ex
xcellence, Japan Scienc
nce and Tecchnology
Agencyy (JST).
References
[1] Y. Kawamura, K. Hayashi, A. Inoue and T. Masumoto: Rapidly solidified powder
metallurgy Mg97Zn1Y2 alloys with excellent tensile yield strength above 600 MPa,
Materials Transactions, 42-7 (2001), pp. 1172-1176.
[2] E. Abe, Y. Kawamura, K. Hayashi and A. Inoue: Long-period ordered structure in a
high-strength nanocrystalline Mg-1at%Zn-2at%Y alloy studied by atomic-resolution
Z-contrast STEM, Acta Materialia, 50-15 (2002), pp. 3845-3857.
[3] S. Yoshimoto, M. Yamasaki and Y. Kawamura: Microstructure and mechanical
properties of extruded Mg-Zn-Y alloys with 14H long period ordered structure, Materials
Transactions, 47-4 (2006), pp. 959-965.
[4] M. Yamasaki, K. Hashimoto, K. Hagihara and Y. Kawamura: Effect of multimodal
microstructure evolution on mechanical properties of Mg-Zn-Y extruded alloy, Acta
Materialia, 59-9 (2011), pp. 3646-3658.
[5] K. Hagihara, A. Kinoshita, Y. Sugino, M. Yamasaki, Y. Kawamura, H.Y. Yasuda and Y.
Umakoshi: Effect of long-period stacking ordered phase on mechanical properties of
Mg97Zn1Y2 extruded alloy, Acta Materialia, 58-19 (2010), pp. 6282-6293.
[6] R. Matsumoto, M. Yamasaki, M. Otsu and Y. Kawamura: Forgeability and flow stress of
Mg-Zn-Y alloys with long period stacking ordered structure at elevated temperatures,
Materials Transactions, 50-4 (2009), pp. 841-846.
[7] M. Noda, T. Mayama and Y. Kawamura: Evolution of mechanical properties and
microstructure in extruded Mg96Zn2Y2 alloys by annealing, Materials Transactions, 50-11
(2009), pp. 2526-2531.
[8] K. Hagihara, A. Kinoshita, Y. Sugino, M. Yamasaki, Y. Kawamura, H.Y. Yasuda and Y.
Umakoshi: Plastic deformation behavior of Mg97Zn1Y2 extruded alloys, Transactions of
Nonferrous Metals Society of China, 20-7 (2010), pp. 1259-1268.
[9] T. Sakamoto, K. Takashima and M. Otsu: Micro-mechanical testing of Mg-Zn-Y alloys,
Proceedings of the 2nd Pan Yellow Sea Rim International Symposium on Magnesium
Alloys, (2007), 47.
[10] T. Morikawa, Y. Mitani and K. Higashida: Inhomogeneous deformation observed using
high-precision markers drawn by electron beam lithography in a magnesium alloy with
LPSO phase, Materials Science Forum, 638-642 (2010), pp. 1574-1578.
[11] J. Cortes, Y. Mitani, M. Tokuoka, M. Shiraishi and K. Osakada: Simulation of cold
forging process of 18-8 stainless steel considering metallurgical factors, Advanced
Techonology of Plastisity 1990, Vol.1 (1990), pp. 65-70.
[12] K. Shinagawa, K. Mori and K. Osakada: Finite element simulation of deep drawing of
stainless steel sheet with deformation-induced transformation, Journal of Materials
Processing Technology, 27-(1-3) (1991), pp. 301-310.
[13] H. Utsunomiya, K. Masui, S. Kaneko and T. Sakai: Consideration of spatial
inhomogeneity in flow stress in finite element analysis, Advanced Techonology of
Plastisity 2008, (2008), pp. 1860-1865.
[14] T. Furushima, K. Nakata, K. Manabe and S. Alexandrov: Constitutive modeling of free
surface roughening in sheet metal considering microscopic inhomogeneity based on
hardness distribution, Journal of Solid Mechanics and Materials Engineering, 3-12 (2009),
pp. 1285-1296.
[15] M. Yamasaki and Y. Kawamura: Thermal diffusivity and thermal conductivity of
Mg-Zn-are earth element alloys with long-period stacking ordered phase, Scripta
Materialia, 60-4 (2009), pp. 64-267.
[16] H. Kudo, K. Sato and K. Aoi: A cold upsettability test, CIRP Annals – Manufacturing
Technology, 16-4 (1968), pp. 309-318.
[17] K. Osakada, T. Kawasaki and K. Mori: A method of determining flow stress under
forming conditions, CIRP Annals – Manufacturing Technology, 30-1 (1981), pp.
135-138.
[18] O. Kada, T. Miki, M. Toda and K. Osakada: Calculation of isothermal flow stress by
combination of FEM and simple compression test, CIRP Annals – Manufacturing
Technology, 47-1 (1998), pp. 185-188.
[19] K. Hagihara, A. Kinoshita, Y. Fukusumi, M. Yamasaki and Y. Kawamura: Materials
Science Forum, 706-709 (2012), pp. 1158-1163.
[20] K. Osakada, J. Nakano and K. Mori: Finite element method for rigid-plastic analysis of
metal forming – formulation for finite deformation, International Journal of Mechanical
Sciences, 24-8 (1982), pp. 459-468.