FORMABILITY ANALYSIS OF MAGNESIUM ALLOY SHEETS AT ELEVATED TEMPERATURES WITH EXPERIMENTAL
AND NUMERICAL METHOD
1*
2
3
Lin WANG , L.C. Chan and T.C. Lee
Department of Industrial System Engineering, The Hong Kong Polytechnic University
1
2
3
[email protected] , [email protected] , [email protected]
Abstract
This paper describes a study in which thermo-mechanical properties of wrought magnesium were thoroughly investigated.
Formability of the magnesium based alloy sheet at the elevated temperatures was assessed with a forming testing machine
accompanied by a tailor-made temperature control system. The deformed laser printed circles on the magnesium sheet were
measured to plot a Forming Limiting Curve. The most used theoretical Forming Limiting Diagram (FLD) models were adopted
to predict the FLC and the suitability of the model was checked for the formability prediction of the hexagonal close-packed
(HCP) metals at the elevated temperatures.
Keywords: formability analysis, magnesium alloy, thermo-mechanical property
Magnesium and its alloys (such as ASTM-AZ31, ASTM-AZ60, ASTM-AZ61, ASTM-AZ91 etc.) exhibit low densities and
high specific strengths. These advantages are the driving force for the use of magnesium based materials in the automotive
industry, home appliances and portable digital products. The result is a significant weight saving and a great reduction of fuel
consumption. More recently, portable electronic appliance manufacturers have shown strong interest in implementing pressformed magnesium alloy parts into their products [1]. However, the number and usage of commercial wrought alloys is still
very limited due to their low ductility and large anisotropic coefficient.
The low ductility of Mg alloy results from the limited number of slip systems of hexagonal close-packed microstructure,
which suffocates their applications in the sheet metal forming operation. A large amount of earlier research confirmed that the
elevated temperature can increase the number of slip systems and change the slip direction, which can enhance the material
ductility and reduce the material anisotropy. The mechanical properties of the commercial sheet at elevated temperatures have
been investigated and the increased ductility under an elevated temperature can be explained by the dislocation glide in the
slip system and deformation twining [2]. Similar wok has been done to further investigate the material forming behaviour with
Acoustic Emission (AE), texture measurement, and Transmission Electron Microscopy (TEM) [3-6].
1. Experiment for tensile property at elevated temperatures
The material used in this study was a commercial magnesium based alloy AZ31B (Mg-3%, Al-1%, Zn) sheet with a
thickness of 0.8mm. The uniaxial tensile tests were conducted in an Instron thermal tensile machine at three temperatures
200ºC, 250ºC and 300ºC (Fig.1). The size of the specimen is shown in Fig. 2. The fixture is used in the tensile machine to
avoid the twist because the thin thickness and high working temperature.
Fig. 1 Instron high temperature tensile machine
Three tensile directions were tested at each temperature. These were direction parallel to the rolling direction, the one
transverse to the rolling direction and the one at a 45º angle to the rolling direction. The material anisotropic coefficient was
also measured when elongation was 15% instead of the conventional 20%, considering the low ductility of magnesium sheet at
room temperature.
36 mm
14 mm
3 mm
Fig. 2 Shape of specimen for tensile test
2. Experimental tensile tests results
Fig.3 shows the microstructure of the magnesium alloy sheet AZ31B with a thickness of 0.8mm. The grains are equiaxial
and the average grain size is 18µm. The grain size distribution is inhomogeneous at room temperature.
Fig. 3 Microstructure of Mg alloy AZ31B at room temperature
Fig. 4 shows the true stress strain curves at three temperatures 200ºC, 250ºC and 300ºC when the tensile direction was
parallel to the rolling direction and the strain rate kept constant. The tensile strength decreased and the total elongation of the
material increased when the temperature increased from room temperature 22ºC to 300ºC. Strain hardening phenomenon was
hard to observe when the temperature was raised to 250ºC and 300ºC. The total elongation at 300ºC was almost twice than
that at 200ºC.
350
300
US-12-200
US-12-250
True stress MPa
250
US-12-300
US-12-22
200
150
100
50
0
0
10
20
30
40
50
True strain %
Fig.4 True stress-strain curves for AZ31B alloys at
-2 -1
various temperatures when strain rate is 1.54×10 s
200
180
160
140
120
100
80
60
40
20
0
160
140
True stress MPa
True stress MPa
The uniaxial tensile tests were also carried out when the tensile direction was perpendicular to and at a 45º angle to the
rolling direction. The yield stress varied a lot when there were different angles between tensile direction and the rolling
direction. When the temperature increased to 300ºC, the differences among three tensile curves (0º, 45º and 90º to the tensile
direction) were reduced a lot compared with the curves at 200º and 250º. When the angle between the rolling direction and
tensile direction was 45º, the flow stress was found higher than the other two curves under the three temperatures. The flow
strength of the sample with 90º orientations was higher than the one with 0º orientation. This phenomenon was also
investigated by S.R. Agnew and O. Duygulu [4]. It was found to be related to the crystallographic texture.
US0-12-200
US45-12-200
10
15
True strain %
20
80
60
US0-12-250
US45-12-250
US90-12-250
40
0
0
25
10
90
80
70
60
50
40
30
20
10
0
True stress MPa
5
100
20
US90-12-200
0
120
20
True strain %
30
40
US0-12-300
US45-12-300
US90-12-300
0
5
10
15
20
True strain %
25
30
Fig. 5 Tensile curves when the angles between the rolling direction and tensile direction are 0º,
45º and 90º under three temperatures
The anisotropy coefficient of AZ31B was also measured when temperature varied. As the temperature increased, the
anisotropic coefficient dropped in all three orientations. The average R value, Lankford coefficient, showed a similar trend. The
trend to isotropy, when temperature increases, is caused by the involvement of more slip systems.
Anisotropic coefficient
2
1.8
R0
1.6
R45
1.4
R90
1.2
R_average
1
0.8
0.6
0.4
0.2
0
200
250
300
Temperature ºC
Fig. 6 Anisotropic coefficient of magnesium sheet under three temperatures
3. Experimental measure of Forming Limiting Diagram
Forming Limiting Diagram (FLD) is the most commonly used assessment method for the evaluation of sheet metal. To
achieve a more accurate experimental FLD, a stable isothermal condition should be provided. In this study an isothermal
condition was built using a tailor-made die set with inner heater. The heaters built in the die and punch were controlled with a
temperature controller Omron E5CN. There were four controllers available for the on-line control three heaters located in the
die, punch and blank holder. The controller was able to keep a stable temperature within ±5º through the temperature sensor
installed in the tool (Fig.7). A standard test method (ASTM E2218) was adopted to determine the FLC of Magnesium alloy
sheets AZ31 at the temperatures 200ºC, 250ºC and 300ºC. The various width specimens with laser marked grids were formed
to determine FLC curves through the measurement of deformed grids.
Temperature
controller
Die set with
inner heater
Forming testing
machine
Fig. 7 Hille forming testing machine with closed loop temperature controller
Marginal
Failed
Fig. 8 Specimens and types of grids for FLD measurement
Three types of grids have been plotted in the principal strain space to delineate the experimental Forming Limiting Curve
(FLC). If the strain status of the material lies under the curve, it is assumed that the formed product is necking free. The
experiment method to acquire FLC is time and cost consuming, although it is the most direct and practical way to be used until
now.
0.9
0.8
0.7
Major strain
0.6
0.5
0.4
0.3
Good
Marginal
0.2
Failed
Experimental FLC
0.1
0
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
M inor s train
Fig. 9 Experimental FLC of Mg alloy at 200ºC
4. Theoretical calculated FLD
On the other hand, numerical analysis is another effective tool that can be used to calculate the FLC. Various theoretical
models were proposed by the researchers. The current generally-used theoretical FLC models are listed as follows: i) diffuse
necking, proposed by Swift [7], ii) localized necking, introduced by Hill [8], iii) inhomogeneous model, developed by Marciniak
and Kuczynski [9], and iv) Vertex theory, proposed by Stören and Rice [10]. No matter which method is adopted, the choice of
a yield criterion is also the main factor affecting the calculated limit strain. The influence of the yield criteria on the FLC has
been predicted by many researchers based on the models mentioned above, such as Hill’s 93 [11] based on the M-K method
[12]; Hill’s 48 [13], Hill’s 79 [14], and the Hosford higher-order criterion [15] based on the M-K method; Hill’s 93 based on the
Swift and M-K method [16]; the Hosford higher-order criterion based on the modified Vertex theory [17], and so forth.
In this study, the Swift model and vertex theory with different yield criteria, e.g. Hill 48 [13], and Hill 90 [18], were developed
to predict the Forming Limiting Diagram (FLD) and compared with experimental results. Through the comparison, the model
which could calculate the closest theoretical FLC was chosen to predict the formability of magnesium alloy sheet at an
elevated temperature.
4.1The theoretical FLD model based on Swift’s theory
Swift used the Cönsidere criterion to determine the limit strains in biaxial tension. He analyzed the case by loading a sheet
element along two perpendicular directions, and applied the Cönsidere criterion for each direction. The material was assumed
to obey the strain hardening power law, Swift then obtained the following expressions for the limit strain:
2
ε 1∗ =
⎛ ∂f ⎞
⎛ ∂f
⎟ +σ2⎜
⎟
⎜ ∂σ
⎝ ∂σ 1 ⎠
⎝ 2
2
⎞⎛ ∂f ⎞
⎛ ∂f ⎞⎛ ∂f ⎞
⎛ ∂f ⎞
⎟⎜
⎟
⎟⎟
⎟⎟⎜⎜
⎟⎟ + σ 1 ⎜⎜
σ 2 ⎜⎜
⎟⎜ ∂σ ⎟
⎠⎝ 1 ⎠ ⋅ n ; ε ∗ = ⎝ ∂σ 1 ⎠
⎝ ∂σ 1 ⎠⎝ ∂σ 2 ⎠ ⋅ n
2
2
2
2
2
⎛ ∂f ⎞
⎛ ∂f ⎞
⎛ ∂f ⎞
⎛ ∂f ⎞
⎜
⎟
⎜
⎟
⎟ +σ 2⎜
⎟
+
σ
σ
σ 1 ⎜⎜
1⎜
2⎜
⎟
⎟
⎟
⎜ ∂σ ⎟
⎝ ∂σ 1 ⎠
⎝ ∂σ 2 ⎠
⎝ ∂σ 1 ⎠
⎝ 2⎠
σ 1 ⎜⎜
(1)
Here, f is the yield criterion function.
By using Hill’s 48 yield criterion, the limit strain (2) was evaluated as a function of the loading ratio a and the material
parameters (hardening coefficient n, anisotropy coefficient r, etc.).
2r
r (1 + r90 )
2r
∂f
∂f
= (2 − 0 ⋅ α ) ⋅ σ 1 ,
= (2α ⋅ 0
− 0 ) ⋅σ1
∂σ 1
1 + r0
∂σ 2
r90 (1 + r0 ) 1 + r0
ε 1* =
ε 2* =
(1 + r0 − r0α ) ⋅ [1 + r0 + α 2 ⋅ ( r0 r90 )(1 + r90 ) − 2αr0 ]
*n
r (1 + r90 )
(1 + r0 − r0α ) 2 + α ⋅ (α 0
− r0 ) 2
r90
(1 + r0 − r0α ) ⋅ [α + αr0 − α 2 r0 + α ⋅ (r0 r90 )(1 + r90 ) − r0 ]
*n
r (1 + r90 )
− r0 ) 2
(1 + r0 − r0α ) 2 + α ⋅ (α 0
r90
(2)
If Hill’s 90 yield criterion was used in the Swift model, the limit strain on the right side of the FLC would be delineated with
equation (3) ignoring the thickness strain. In this paper, only the biaxial principal stress states are considered, which means
σ1>0, σ2>0.
Here assuming σ1> σ2>0,
m
−2
∂f
= {m(1 + α ) m−1 + (1 + 2r45 ) ⋅ m(1 − α ) m−1 + cos 2γ ⋅ (1 + α 2 ) 2 ⋅
∂σ 1
[−2b cos 2γ ⋅ α 3 + [(−4a + 2b cos 2γ ) + ( m − 2)(2a + b cos γ )]α 2 +
( −2b cos 2γ ( m − 1))α + m(b cos 2γ − 2a )]}σ 1m−1
m
−2
∂f
= {m(1 + α ) m−1 + (1 + 2r45 ) ⋅ m(1 − α ) m−1 + cos 2γ ⋅ (1 + α 2 ) 2 ⋅
∂σ 2
[(m( 2a + b cos 2γ )α 3 + (−2b( m − 1) cos 2γ )α 2 +
(3)
((b cos γ − 2a) ⋅ ( m − 2) + ( 4a + 2b cos 2γ ))α − 2b cos 2γ ]}σ 1m−1
m
⎛ 2σ ⎞
Here, m can be calculated from equation ⎜⎜ b ⎟⎟ = 2(1 + r45 ) , among which σ b is the yield stress in equibiaxial tension. And
⎝ σ 45 ⎠
γ is the direction between σ 1 and the rolling direction.
a=
1
4
⎛ 2σ b
⎜
⎜σ
⎝ 90
m
⎛ 2σ
⎞
⎟ −⎜ b
⎜ σ
⎟
⎝ 0
⎠
m
⎞
⎟ ;
⎟
⎠
b=
1
2
⎛ 2σ b
⎜
⎜ σ
⎝ 0
m
⎛ 2σ
⎞
⎟ +⎜ b
⎜σ
⎟
⎝ 90
⎠
m
⎛ 2σ
⎞
⎟ −⎜ b
⎜σ
⎟
⎝ 45
⎠
⎞
⎟
⎟
⎠
m
The limit strain calculated by Swift model based on Hill’s 90 yield criterion is shown as follows:
ε 1* =
( p + qc) 2 + α ( p + qc)( p + qd )
α ( p + qd ) 2 + ( p + qc)( p + qd )
*
;
∗
n
=
∗n
ε
2
( p + qc) 2 + α ( p + qd ) 2
( p + qc) 2 + α ( p + qd ) 2
(4)
Here,
p = m(1 + α ) m−1 + (1 + 2r45 ) ⋅ m(1 − α ) m−1 ;
m
q = cos 2γ ⋅ (1 + α 2 ) 2
−2
c = [ −2b cos 2γ ⋅ α 3 + ( m − 4)(2a − b cos 2γ )α 2 + ( −2b cos 2γ (m − 1))α + m(b cos 2γ − 2a)]
d = [(m( 2a + b cos 2γ )α 3 − 2b( m − 1) cos 2γα 2 + (m(b cos 2γ − 2a ) + 8a )α − 2b cos 2γ ]
With the adoption of Hill’s 93 yield criterion, the theoretical prediction of the limit strains using Swift’s theory (5) was derived
by Banabic [16] as follows:
∂f
2
= r = 2σ 90
+ ασ 0σ 90 [c + p + q − ( pt + qαt )]
∂σ 1
∂f
= s = 2ασ 02 + σ 0σ 90 [c + p + q − ( pt + qαt )]
∂σ 2
(5)
Here, t = σ 1 σ b is introduced for the calculation of the limit strain while Hill’s 93 yield criterion is used. The limit strain is
calculated as follows:
ε 1∗ =
αr 2 + r ⋅ s
r2 +α ⋅r ⋅s
, ε 2∗ = 2
2
2
r +α ⋅s
r + α ⋅ s2
(6)
The pairs ε 1* and ε 2* for various loading ratios α can be calculated to represent the associated FLC. It is obvious that the
FLC, constituted by the limit strain values, varies with the used yield criterion. The limit strain predicted with the Swift model
emphasizes the right side of the FLD. The Hill model was used to calculate the limit strain located on the left side of the FLD,
which assumed that the necking direction was coincident with the direction of zero-elongation.
4.2 The theoretical FLD model based on Hill’s theory
The limit strain obtained with the Hill model (7) was used here to describe the curve on the left side of the FLD, while the
power law σ = kε n was used.
ε 1*
=
∂f
∂σ 1
∂f
∂f
+
∂σ 1 ∂σ 2
n,
ε 2*
=
∂f
∂σ 2
∂f
∂f
+
∂σ 1 ∂σ 2
n
(7)
It can be seen that
ε 1* + ε 2* = n
(8)
This is the equation of a line parallel with the second bisectrix of the principal strain coordinate system ε1, ε2 and
intersecting the vertical axis at the point (0, n). According to Equations 7 and 8, the FLD calculated on the basis of Hill’s model
is not dependent on the yield criterion, but only on the hardening coefficient.
Besides the Swift and Hill models, the bifurcation theory with consideration of the vertex theory, which can cover the whole
range of the FLD, has aroused the interest of researchers in this field more and more. The vertex theory with several yield
criteria were also studied as follows.
4.3 The theoretical FLD model based on the Stören and Rice’s theory
The Vertex theory, first proposed by Stören and Rice [10], had been further discussed and developed by Zhu, Weinmann
and Chandra [17] with consideration of the moment equilibrium in addition to the force equilibrium condition. Later, the
analytical expression of the theoretical FLC was developed by Chow, Jie and Hu [19] with the adoption of Hosford’s high-order
yield criterion, Hill’s quadratic yield criterion and the von Mises yield criterion. The calculated strain varied greatly with the yield
criteria.
The limit strain on the left hand side of the FLD or the negative strain ratio region is expressed as follows:
ε 1* =
(1 + R0 )[rσa −2 + R90 (1 + rε )(1 − rσ )a−2 ] f a (rσ )
(a − 1)n − 1
+
(a − 1)(1 + rε )(1 + rε rσ )[rσa−2 + (R90 + R0 rσa−2 )(1 − rσ )a−2 ] (a − 1)(1 + rε )
(9)
On the right hand side of the FLC or the positive strain ratio region, it is:
ε 1* =
(1 + R0 )[rσa−2 + R90 (1 − rσ )a−2 ] f a (rσ )
(a − 1)n − 1
+
(a − 1)(1 + rε rσ )[rσa−2 + (R90 + R0 rσa−2 )(1 − rσ )a−2 ] (a − 1)(1 + rε rσ )
(10)
R0- anisotropic coefficient in the rolling direction
R90- anisotropic coefficient in the transverse rolling direction
rε- principal strain ratio
2r + 1
rσ- principal stress ratio rσ = ε
2 + rε
a- constant, a=2, 6 and 8 represent different yield criteria
f (rσ)- deformation theory of plasticity
σ eq
sgn (σ 1 )
a
a 1/ a
f (rσ ) =
=
R90 + R0 rσ + R0 R90 1 − rσ
(11)
1/ a
σ 1 [R90 (1 + R0 )]
While a = 2 and R0, R90 ≠ 1, the Equation 10 is simplified to the Hill quadratic yield criterion. Hosford’s high order yield
criterion was also represented in the equation when a=6 or 8.
[
]
5. Comparison of the experimental FLC to the theoretical FLC
0.7
0.8
0.6
0.7
0.6
0.4
0.3
-0.4
0.5
Major strain
Major strain
0.5
Sw if t model w ith Hill 48
0.2
Sw if t model w ith Hill 90
Sw if t model w ith Hill 93
0.1
Hill model
Experimental FLC
0
-0.3
-0.2
-0.1
0
0.1
M inor s train
Vertex theory w ith Hill
48
Vertex theory w ith
Hosf ord a=6
Vertex theory w ith
Hosf ord a=8
Experimental FLC
0.4
0.3
0.2
0.1
0
0.2
0.3
Fig. 10 Comparison of experimental FLC and
Swift model predicted ones
0.4
0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
M inor s train
Fig. 11 Comparison of experimental FLC and
Vertex model predicted ones
The experimental FLC at 200ºC was compared with the theoretical ones. In Fig. 10 when the strain ratio is negative, the
large difference between the Hill model predicted curve and the experimental one indicates that the Hill model is not suitable to
describe Mg alloy FLC. The curves calculated by the Swift model with Hill 48, Hill 90 and Hill 93 yield criteria cannot match the
experimental one well probably because the necking criterion of the hexagonal close packed metal is not accordance with
Considère’s, the one Swift model based on. The curves predicted by the Vertex theory are displayed in Fig. 11. The
experimental FLC has a better coincident with the curve predicted by the Vertex model and the Hosford yield criterion, in which
the exponent a was 6. There is a slight discrepancy between the curves predicted by the Vertex model when the exponent a in
Hosford equals to different values, 6 or 8. Hosford thought that the yield criterion could give a better approximation by a=6 for
FCC material and a=8 to BCC material [20]. The non-basal slip systems were activated at elevated temperatures to the
hexagonal crystals such as Mg alloy and there were only a few basal slip systems at the room temperature. The increase of
slip systems did improve the material ductility and formability.
In this study, a tailor-made instrument was built for theFLC measurement of magnesium alloy at elevated temperatures.
This facility (Fig.7) can provide an almost isothermal environment for the testing machine, which is very useful to such FLC
testing. The tensile properties at different temperatures reveal the great influence of temperature on this material. The
isothermal forming process should be developed indispensably due to the great improvement of the material ductility at a
higher temperature, for instance 250~300ºC.
The numerical calculated FLCs vary with the use of yield criterion. The Swift model is limited to the right side of FLD, while
Vertex theory can cover both sides. Vertex theory accompanying with Hosford’s yield criterion is more consistent with the
experimental result. The theoretical model using Vertex theory and Hosford’s yield criterion, when exponent a=6, is more
accessible to calculate the FLC of FCC material conventionally. It can give a great approximation to the HCP metal probably
because the activity of non-basal slip system increases with increasing temperatures.
Acknowledgement
The work described in this paper was supported by Industrial Technology Fund of the Hong Kong Special Administrative
Region, China under the project no. UIM/150 / ZM-17.
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