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Available online at www.sciencedirect.com
Procedia Manufacturing 47 (2020) 237–244
2351-9789 © 2020 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the scientific committee of the 23rd International Conference on Material Forming.
10.1016/j.promfg.2020.04.205
10.1016/j.promfg.2020.04.205 2351-9789
© 2020 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the scientific committee of the 23rd International Conference on Material Forming.
Available online at www.sciencedirect.com
ScienceDirect
Procedia Manufacturing 00 (2019) 000–000
www.elsevier.com/locate/procedia
2351-9789 © 2020 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license https://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the scientific committee of the 23rd International Conference on Material Forming.
23rd International Conference on Material Forming (ESAFORM 2020)
Application of Friction Shear Test for Constitutive Modeling Evaluation
of Magnesium Alloy AZ31B at high Temperature
Vidal Sanabriaa,*, Felix Genschb, Soeren Muellera
aExtrusion Research and Development Center, Technische Universität Berlin, Berlin 13355, Germany
bINGWERK GmbH. Gustav-Meyer-Allee 25, Gebäude 17a, Treppe 5, Berlin 13355, Germany
* Corresponding author. Tel.: +49-30-314-72515; fax: +49-30-314-72503. E-mail address: vidal.sanabria@tu-berlin.de
Abstract
The experimental determination of the flow stress and its mathematical formulation are essential for the numerical simulation of metal forming
processes. The hot compression test is widely used to analyze the flow stress evolution as function of temperature, strain and strain rate. The
compression test is limited to a relative low strain (ε≤1) which is acceptable when the stress is minor influenced at higher strains. In the case of
magnesium alloys the flow stress is strongly influenced by the strain even at high strain (ε>1). In this work the thermo-mechanical behavior of
the magnesium alloy AZ31B was investigated to improve the constitutive modeling up to high strains. Experimental stress-strain curves obtained
from hot compression tests at different temperatures (450 °C-550 °C) and strain rates (0.01 1/s – 10 1/s) were applied to construct conventional
material models such as those proposed by Garofalo (Zener-Hollomon) and Hensel-Spittel. In addition, shear tests under sticking friction
conditions were carried out at high temperature (400 °C-500 °C) and different shear speeds (0.1 mm/s - 10 mm/s). During this test, the thin
contact subsurface of cylindrical specimens experiences a high plastic shear deformation, while the axial force and stroke are simultaneously
measured. Furthermore, a new constitutive modeling approach was proposed, which combine the Zener-Hollomon model and the experimental
result of the friction shear test to estimate the flow stress at low and high strain respectively. Numerical simulations of the friction shear test
applying the conventional models as well as the new constitutive formulation are presented in this study.
© 2020 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license https://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the scientific committee of the 23rd International Conference on Material Forming.
Keywords: Magnesium Alloy; Constitutive Model; Sticking Friction; Flow Stress; FEM
1. Introduction
Constitutive modeling of AZ31 magnesium alloy under hot
forming conditions has been the object of many studies in the
late years. The hot compression test has been commonly used
to obtain the stress-strain material behavior [1-5], while
investigations performed with the torsion test are uncommon
[6]. Contrary to aluminum alloys, magnesium alloys
experience a strong softening effect after the peak in the stress-
strain curve mainly due to dynamic recrystallization [7]. Due
to the limited strain (ε≤1) achieved in the hot compression test
a complete description of the stress reduction up to steady state
is not possible. Moreover, relative high strains (ε>1) have been
achieved with the torsion test but only at certain conditions [6]
due to the low ductility of AZ31. Thus, the stress reduction at
higher strains (ε>1) is commonly mathematically extrapolated
based on the available experimental results (0<ε≤1). According
to the literature, inaccurate measurements are obtained due to
significant inhomogeneous distribution of strain and strain rate
in the test specimen during compression test at 0.5 strain [5].
This is because the friction between punch and specimen
cannot be fully avoided producing thus the barrel effect. Well
known constitutive models such as power low [6], Garofalo or
Zener-Hollomon [1,2,4,5,8] and Hensel Spittel [8] have been
used to describe empirically the flow stress of AZ31
magnesium alloy. Even a more complex constitutive
formulation based on the combination of two models has been
proposed in order to fully describe the stress-strain
Available online at www.sciencedirect.com
ScienceDirect
Procedia Manufacturing 00 (2019) 000–000
www.elsevier.com/locate/procedia
2351-9789 © 2020 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license https://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the scientific committee of the 23rd International Conference on Material Forming.
23rd International Conference on Material Forming (ESAFORM 2020)
Application of Friction Shear Test for Constitutive Modeling Evaluation
of Magnesium Alloy AZ31B at high Temperature
Vidal Sanabriaa,*, Felix Genschb, Soeren Muellera
aExtrusion Research and Development Center, Technische Universität Berlin, Berlin 13355, Germany
bINGWERK GmbH. Gustav-Meyer-Allee 25, Gebäude 17a, Treppe 5, Berlin 13355, Germany
* Corresponding author. Tel.: +49-30-314-72515; fax: +49-30-314-72503. E-mail address: vidal.sanabria@tu-berlin.de
Abstract
The experimental determination of the flow stress and its mathematical formulation are essential for the numerical simulation of metal forming
processes. The hot compression test is widely used to analyze the flow stress evolution as function of temperature, strain and strain rate. The
compression test is limited to a relative low strain (ε≤1) which is acceptable when the stress is minor influenced at higher strains. In the case of
magnesium alloys the flow stress is strongly influenced by the strain even at high strain (ε>1). In this work the thermo-mechanical behavior of
the magnesium alloy AZ31B was investigated to improve the constitutive modeling up to high strains. Experimental stress-strain curves obtained
from hot compression tests at different temperatures (450 °C-550 °C) and strain rates (0.01 1/s – 10 1/s) were applied to construct conventional
material models such as those proposed by Garofalo (Zener-Hollomon) and Hensel-Spittel. In addition, shear tests under sticking friction
conditions were carried out at high temperature (400 °C-500 °C) and different shear speeds (0.1 mm/s - 10 mm/s). During this test, the thin
contact subsurface of cylindrical specimens experiences a high plastic shear deformation, while the axial force and stroke are simultaneously
measured. Furthermore, a new constitutive modeling approach was proposed, which combine the Zener-Hollomon model and the experimental
result of the friction shear test to estimate the flow stress at low and high strain respectively. Numerical simulations of the friction shear test
applying the conventional models as well as the new constitutive formulation are presented in this study.
© 2020 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license https://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the scientific committee of the 23rd International Conference on Material Forming.
Keywords: Magnesium Alloy; Constitutive Model; Sticking Friction; Flow Stress; FEM
1. Introduction
Constitutive modeling of AZ31 magnesium alloy under hot
forming conditions has been the object of many studies in the
late years. The hot compression test has been commonly used
to obtain the stress-strain material behavior [1-5], while
investigations performed with the torsion test are uncommon
[6]. Contrary to aluminum alloys, magnesium alloys
experience a strong softening effect after the peak in the stress-
strain curve mainly due to dynamic recrystallization [7]. Due
to the limited strain (ε≤1) achieved in the hot compression test
a complete description of the stress reduction up to steady state
is not possible. Moreover, relative high strains (ε>1) have been
achieved with the torsion test but only at certain conditions [6]
due to the low ductility of AZ31. Thus, the stress reduction at
higher strains (ε>1) is commonly mathematically extrapolated
based on the available experimental results (0<ε≤1). According
to the literature, inaccurate measurements are obtained due to
significant inhomogeneous distribution of strain and strain rate
in the test specimen during compression test at 0.5 strain [5].
This is because the friction between punch and specimen
cannot be fully avoided producing thus the barrel effect. Well
known constitutive models such as power low [6], Garofalo or
Zener-Hollomon [1,2,4,5,8] and Hensel Spittel [8] have been
used to describe empirically the flow stress of AZ31
magnesium alloy. Even a more complex constitutive
formulation based on the combination of two models has been
proposed in order to fully describe the stress-strain
238 Vidal Sanabria et al. / Procedia Manufacturing 47 (2020) 237–244
2 Vidal Sanabria et al. / Procedia Manufacturing 00 (2019) 000–000
experimental data [3]. Numerical simulations and subsequently
experimental validations are required to evaluate the accuracy
of the constitutive models. Bulk forming process such as
extrusion are too complex to evaluate the accuracy of the
constitutive model. Inaccurate modeling of thermal contact
resistance, convection heat transfer and friction during
extrusion may hinder the prediction of small changes in the
constitutive model [4]. For that reason a more simple
experiment should be applied for constitutive modeling
evaluation. During the axial friction test [9] a high shear
deformation is generated in a thin circumferential outer shell in
a cylindrical specimen while the axial displacement force and
stroke are simultaneously measured. Experimental results have
demonstrated that sticking friction is possible at high normal
pressure [10,11,12] and therefore it is referenced as friction
shear test in this work. Thus, the shear stress of the tested
material can be directly evaluated as also proposed in [13]. In
this study friction shear tests of AZ31B magnesium alloy were
carried out at high temperature and different sliding speeds.
Numerical simulations of the friction shear tests were
performed applying the well-known Garofalo (Zener-
Hollomon) and Hensel-Spittel constitutive models, which were
built based on hot compression tests. Additionally, a new
constitutive modeling approach is proposed to improve the
flow stress estimation at low and high strain values.
2. Experimental procedure
Axial friction tests under sticking conditions were carried
out at the Extrusion Research and Development Center ERDC
of TU Berlin. This method has been previously used to evaluate
the friction as well as the maximal shear stress of aluminum
and magnesium alloys [14]. For each test a cylindrical
specimen is first upset with a constant axial force (Fa) inside a
hollow cylinder (Fig. 1a). A graphite paper was placed in
between specimen and punch to reduce the friction. Sticking
condition between the specimen and the hollow cylinder can be
achieved at a high normal pressure (P). Subsequently, the
hollow cylinder moves with a constant speed while the
specimen remains fixed and compressed between the punches
(in this case with convex form). Since the sticking force
between specimen and hollow cylinder is higher or equals the
shear flow stress of the specimen a severe shear deformation
take place in the specimen under the contact surface [10,11].
a) b)
Fig. 1. Schema of axial friction test. (a) General assembly; (b) relative
position of specimen and hollow cylinder before and after stroke.
The shear stress is calculated dividing the measured friction
or shear force (Ff) by the nominal contact surface (A= πdL).
For this study, the specimens were extracted by means of wire
erosion process from homogenized cast billets of magnesium
alloy AZ31B (2,9% Al, 0,8% Zn, 0.25% Mn), provided by Otto
Fuchs KG Meinerzhagen. The initial diameter (di) and height
(Li) of the specimens were 7.8 mm and 10.5 mm respectively
(Fig 1b). Hollow cylinder and punches were manufactured
from hot working steel 1.2344 (46-48 HRC). The whole
assembly was heated up to a constant temperature by an electric
furnance while the specimen temperature was meassured with
a thermocopule placed at the top side of the specimen. Test
temperature of 400 °C and 500 °C as well as sliding speeds of
0.1 mm/s, 1 mm/s and 10 mm/s were selected. During each test
a nominal stroke length of 3.5 mm was set. Table 1 shows more
information about the axial force (Fa), yield stress (σy), axial
stress (σa) and normal pressure (σn=P) applied in the tests at
400 °C and 500 °C in order to achieve a normalized normal
pressure of P/σy=6 (enough pressure for sticking condition).
Assuming no friction during the specimen set up the
relationship between P, σa and σy can be estimated with the
equation 1. More details about the construction, testing
procedure and results of the axial friction test device can be
found in the literature [9].
Table 1. Experimental axial force and calculated axial and normal pressure.
Temperature
[°C]
F
a
[kN]
σ
y
[MPa]
σ
a
[MPa]
σ
n
=P
[MPa]
400
23.95
66.4
467.5
401.0
500
10.80
30.4
214.8
184.4
𝑃𝑃=𝜎𝜎𝑎𝑎−𝜎𝜎𝑦𝑦 (1)
3. Experimental results
Selected experimental results obtained during the friction
shear tests at 400 °C and 500 °C are depicted in figures 2a and
2b respectively. This graphics show the evolution of the
sticking friction force (shear force) generated during the first
3 mm stroke of the test. In other words, the graphics quantify
the force needed to shear the contact subsurface of AZ31B at
400 °C and 500 °C at 0.1 mm/s, 1 mm/s and 10 mm/s of
deformation speeds.
a) b)
Fig. 2. Sticking friction force measured at (a) 400 °C; (b) 500 °C.
Vidal Sanabria et al. / Procedia Manufacturing 47 (2020) 237–244 239
Vidal Sanabria et al. / Procedia Manufacturing 00 (2019) 000–000 3
At the beginning the force increases proportional to the
stroke until a maximal value is reached. Before the shear test
takes place the specimen upsets and establishes sticking
friction contact to the hollow cylinder and therefore those
straight lines indicate the elastic deformation of the whole
assembly. Moreover, the peak of the curves shows the maximal
force needed to initiate the plastic deformation. It increases at
higher speeds due to the viscoplastic behavior of the
magnesium alloy AZ31B. Subsequently, the force decreases
drastically especially at 500 °C. The strong softening effect
observed after the peak is expected to occur due to the dynamic
recrystallization, which is accelerated at a higher temperature.
Finally, the force decreases slightly throughout the rest two-
thirds of the stroke until an equilibrium between work-
hardening and dynamic softening is achieved [7]. Magnesium
as a low stacking fault energy material exhibits dynamic
recrystallization during hot processing and this phenomenon
mainly determines the observed flow stress decline after the
peak stress.
Figure 3 shows the maximal shear stress of magnesium alloy
AZ31B generated at different deformation speeds (log. scale)
and temperatures. These values were calculated dividing the
maximal sticking friction force obtained in Fig. 2a and 2b by
the calculated contact area A. Similar procedure was performed
to calculate the shear stress at a stroke of 3, where steady state
is assumed (Fig. 4). The results show a logarithmic relationship
of the stress to the speed of deformation as expected for
viscoplastic materials. They also demonstrate the accuracy of
the data and the experimental procedure.
Fig. 3. Maximal shear stress at (a) 400 °C; (b) 500 °C.
Fig. 4. Shear stress at steady state at (a) 400 °C; (b) 500 °C.
4. Analytical calculation
Based on the experimental results the shear stress were
calculated. However, the local strain and strain rate (state
variables) at the subsurface deformation zone are still
unknown. Considering simple shear deformation the shear
strain γ and shear strain rate
can be estimated with equations
2 and 3 respectively. Where y is the relative displacement
between specimen and hollow cylinder, h is the sheared width
of the specimen and v the friction speed (Fig. 5).
𝛾𝛾=𝑦𝑦
ℎ (2)
𝛾𝛾=𝑣𝑣
ℎ (3)
The relative displacement y can be calculated removing the
elastic deformation from the measured stroke (Eq. 4). For that
a proportional factor k can be calculated (only for the straight
lines) for each temperature as showed in Fig. 6. According to
the linear fitting in Fig. 6 the elastic proportional factors k are
25.7 kN/mm at 400 °C and around 24 kN/mm at 500 °C.
𝑦𝑦=𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠− 𝐹𝐹
𝑘𝑘 (4)
Fig. 5. Definition of simple shear deformation.
Fig. 6. Determination of elastic proportional factor k of the system.
240 Vidal Sanabria et al. / Procedia Manufacturing 47 (2020) 237–244
4 Vidal Sanabria et al. / Procedia Manufacturing 00 (2019) 000–000
Table 2 resumes the calculated relative displacements for a
stroke 3 mm and a temperature of 500 °C (k=24 kN/mm).
Additionally, the average shear strain and shear strain rate for
different speeds are also presented. Experimental results have
shown that the shear width depends on the deformation
temperature, speed and stroke [11]. Since, the microstructure
of specimens tested in this study has not been analyzed, an
average shear width of h=400 µm observed in previous studies
was assumed as reference.
Table 2. Average displacement, shear strain and shear strain rate at 500 °C.
Speed [mm/s]
0.1
1
10
F [kN]
3.0
4.29
5.83
y [mm]
2.87
2.82
2.76
7.1
7.0
6.9
[1/s]
0.25
2.5
25
5. Numerical analysis
Selected sticking friction tests carried out at 500 °C were
numerically simulated applying the FEM-based software
DEFORM 2D. The geometrical model used for all simulations
is depicted in Fig. 7. The axisymmetric model of the already
upset specimen (workpiece) was meshed applying different
element sizes in three zones as shown in Fig. 7. Small element
size (40 µm) was set at the friction subsurface zone, where a
severe shear deformation was expected. Heat transfer between
workpiece and tools was set at 11 kW/m2K. Moreover, the
friction was modeled according to tresca model with a constant
friction factor of m=0.3 between punch and specimen, while
m=2 and sticking condition between workpiece and hollow
cylinder. In order to reproduce the experimental load
conditions, a constant axial force of 10.8 kN was applied
between punch and specimen (Table 1). Additionally, the
sliding speed of the hollow cylinder was set at 0.1 mm/s,
1 mm/s and 10 mm/s with a step increment of 1x10-4 mm/step.
Tools were modeled as rigid objects without mesh, while the
workpiece as plastic body. The Garofalo or Zener-Hollomon
(Z-H) as well as the Hensel-Spittel (H-S) constitutive models
were evaluated during the simulations. Both formulations were
based on experimental data collected from hot compression
tests [8]. Constant parameters of Z-H formulation (Eq. 5)
considering stress values at a constant strain (ε=1) and a range
temperature from 450 °C to 550 °C are presented in table 3 [8].
Fig. 7. 2D axisymmetric geometrical model applied for FE-simulations.
Where A and n are constants, while Q, α, R,
𝜀𝜀
and T are the
deformation activation energy, stress multiplier, gas constant,
strain rate and temperature (°K) respectively. Garofalo model
is one of the default formulations used by DEFORM and
therefore parameters of table 3 can be directly introduced for
the constitutive modeling.
Table 3. Modeling parameters of AZ31B according Garofalo or Z-H (ε=1).
Temperature [°C]
Q [J/mol]
A [1/s]
α [MPa]
n [-]
450 - 550
218921
9.1486x1012
0.048
4.331
𝜎𝜎=1
𝛼𝛼sinh−1[[1
𝐴𝐴𝜀𝜀𝑒𝑒(𝑄𝑄
𝑅𝑅𝑅𝑅)]1
𝑛𝑛] (5)
In addition, table 4 shows the parameters needed to
construct the H-S approach (Eq. 6) under the same range of
temperature (450°C - 550°C) [8]. Where A and m1 to m9 are
regression coefficients. In order to model the H-S equation in
DEFORM the stress was first analytically calculated at
different strains (ε≤5), strain rates (<100 1/s) and temperatures
(450 °C to 550 °C). Then the stress values were introduced in
the software in form of tables, where the values can be
automatically interpolated or extrapolated (logarithmic). For
strains higher than 5 the stress was assumed constant.
Table 4. Modeling parameters of AZ31B according to Hensel-Spittel.
T [°C]
A
m
1
m
2
m
3
m
4
m
5
m
7
m
8
m
9
450 - 550
2450
-56E-4
-0.52
0.12
-0.1
45E-5
0.04
35E-6
8E-3
𝜎𝜎=𝐴𝐴𝑒𝑒𝑚𝑚1𝑇𝑇𝑇𝑇𝑚𝑚9𝜀𝜀𝑚𝑚2𝑒𝑒(𝑚𝑚4𝜀𝜀
⁄ )(1+𝜀𝜀)𝑚𝑚5𝑇𝑇𝜀𝜀𝑚𝑚7𝜀𝜀𝜀𝜀𝑚𝑚3𝜀𝜀𝑚𝑚8𝑇𝑇 (6)
Experimental and H-S modeled stress-strain curves for the
strain rates 0.1 1/s, 1 1/s and 10 1/s at 500 °C at low and high
strain range are depicted in Fig. 8a and 8b respectively.
Fig. 8. Experimental and modeled (H-S) stress-strain curves at different strain
rates (0.1 1/s, 1 1/s and 10 1/s) at 500 °C. (a) Low strain; (b) high strain.
5.1. Numerical results applying standard constitutive models
Fig. 9a and 9b compare the experimental and simulated
shear friction forces performed at 500 °C applying the Z-H and
H-S models respectively. The stroke measured experimentally
was corrected with Eq. 4 in order to plot only the plastic relative
movement (y). The hyperbolic sine law (Eq. 5) is a strain
independent formulation and therefore allows the stress
Vidal Sanabria et al. / Procedia Manufacturing 47 (2020) 237–244 241
Vidal Sanabria et al. / Procedia Manufacturing 00 (2019) 000–000 5
calculation based only on the strain rate and temperature
distribution. Since each friction shear test was performed at a
constant speed no significant changes of the strain rate
distribution during the whole stroke is expected. For this
reason, almost constant forces were obtained during each
simulation (Fig. 9a). In the case of the simulation performed at
10 mm/s, the force decreased 4 % at the end of the stroke
(3 mm) due to the significant heat generation (10_Z-H in Fig.
9a).
On the other hand, the results obtained with the H-S model
(strain dependent) show curve paths more similar to the
experimental measurements. Thus, curves with a peak and a
subsequently force reduction could be observed (Fig. 9b). Figs.
10a and 10b depict the effective strain and strain rate
distribution simulated with the Z-H and H-S models at 500 °C
and sliding speed 10 mm/s. The results were taken at a relative
movement of y=0.3 mm, which was the position where the
maximal force was estimated (Fig. 9b). Moreover, the mapping
observed in Fig. 10 was performed at the mid height of the
specimen and from the center (radius=0) to the friction surface
(radius=4 mm). Simulated results show different estimations
applying the Z-H and H-S models. Using the Z-H model, higher
strain and strain rate values were concentrated at the shear
deformation zone (Fig. 10a,b). However, a lower and more
distributed strain and strain rate were simulated with the H-S
formulation.
a) b)
Fig. 9. Experimental and simulated sticking friction force carried out at
500 °C and different sliding speeds (0.1 mm/s, 1 mm/s and 10 mm/s)
applying (a) Z-H model ε=1; (b) H-S model.
a) b)
Fig. 10. Numerical simulation of the (a) effective strain and (b) effective
strain rate applying Z-H and H-S constitutive models at the mid height of the
specimen and at y=0.3 mm.
Fig. 11. Effective stress simulated applying Z-H and H-S models (500 °C,
y=0.3, v=10 mm/s)
Although higher strain rates were obtained with the Z-H
model, the maximal stress was lower than the calculated with
the H-S (Fig 11). It is possible because the Z-H modeling was
performed considering the flow stress at the constant strain ε=1
and neglecting maximal stress at ε≈0.2 (Fig. 8). These results
suggest that a more realistic strain and strain rate distribution
were obtained with the H-S model. Nonetheless, significant
differences between experimental and simulated sticking
friction forces were obtained applying the H-S model (Fig. 9b).
5.2. Alternative constitutive modeling
Instead of using a unique equation to describe the particular
stress-strain plastic behavior of AZ31B, the new suggested
approach applies three functions corresponding to different
strain sections (Fig. 12). The three sections are divided by two
points (P1 and P2) with fixed strains (εσ1=0.012 and εσ2=0.23)
observed in all experimental curves. Therefore this empirical
formulation can be called strain-dependent multifunction
model (SDM). Additionally, σ1 and σ2 are the stress σ(T,𝜀𝜀)
calculated by means of the Z-H model at the mentioned strains.
Similar to the experimental results the first part of the curve (I)
correspond to a straight line modeled with eq. 7. Where, the
point 1 (P1) is located at the end of the section I. Thus, the
equation 7 is valid for the strain range 0≤ε≤0.012. The section
II is modeled with the ellipse equation (8) with center (f,j) and
mayor axis 𝑎𝑎 parallel to the σ-axis (Fig. 12). The minor axis 𝑏𝑏
is parallel to ε-axis and can be calculated with equation 9.
Fig. 12. Three different sections (I, II, III) for constitute modeling of AZ31B.
242 Vidal Sanabria et al. / Procedia Manufacturing 47 (2020) 237–244
6 Vidal Sanabria et al. / Procedia Manufacturing 00 (2019) 000–000
Finally, the Box-Lucas function (10) was selected to model
the segment III, where c and β (c=2.3, β=2) as well as
∆
𝜎𝜎
(0.27•𝜎𝜎2) are used to control the softening effect after P2 (Fig.
12).
𝜎𝜎𝐼𝐼=(𝜎𝜎1
𝜀𝜀𝜎𝜎1)𝜀𝜀 (7)
(𝜀𝜀−𝑓𝑓)2
𝑏𝑏2+(𝜎𝜎𝐼𝐼𝐼𝐼−𝑗𝑗)2
𝑎𝑎2=1 (8)
𝑏𝑏=√(𝜀𝜀𝜎𝜎1−𝜀𝜀𝜎𝜎2)2
1−(𝜎𝜎1
𝜎𝜎2)2 (9)
𝜎𝜎𝐼𝐼𝐼𝐼𝐼𝐼=𝜎𝜎2−∆𝜎𝜎𝑐𝑐[𝑒𝑒(−𝛽𝛽𝜀𝜀𝜎𝜎2)−𝑒𝑒(−𝛽𝛽𝜀𝜀)] (10)
Table 4 resumes the Z-H parameters calculated from the
experimental stress-strain curves [8] at low strains (εσ1=0.012
and εσ2=0.23) applying standard procedure [8].
Table 5. Modeling parameters of AZ31B according Z-H at low strain.
Temperature
[°C]
Strain
[-]
Q
[J/mol]
A
[1/s]
α
[MPa]
n
[-]
450 - 550
0.012
286293
1.4x1019
0.034
9.381
450 - 550
0.023
207359
3.77x1013
0.022
4.857
Fig. 13a depicts the stress-strain curves calculated with the
SDM model at low stress (0≤ε≤0.012). Good agreement
between experimental and modeled results can be observed.
During compression test inhomogeneous strain and strain rate
distribution inside the specimen introduce significant
inaccuracy to this experimental test at higher strains. Duo to the
friction the barrel effect can not be fully avoided and therefore
the strain and strain rates are much higher at the center of the
specimen. Numerical simulations of hot compression test of as
cast AZ31 magnesium alloy show that at ε=0.5 the strain and
strain rate in the center of specimen is at least 80 % higher that
the assumed during the experimental test (𝜀𝜀=1) [5]. For that
reason the experimental results obtained from the hot
compression test (ε>0.23) were not taken as reference to fit the
Box-Lucas equation (10).
a) b)
Fig. 13. Experimental and modeled (SDM) stress-strain curves at different
strain rates (0.1 1/s, 1 1/s and 10 1/s) at 500 °C. (a) low strain; (b) high strain.
The parameters c and β (Eq. 10) were changed to obtain a
stronger softening effect (faster stress decreasing) after peak
(Fig. 13b). Moreover, the new model was modified with a
particular combination of parameters c and β and subsequently
used to simulate the friction shear test. This iterative process
was performed until a good agreement between experimental
and simulated sticking friction force was obtained.
5.3. FE-Simulation applying the alternative constitutive model
The new strain-dependent multifunction model (SDM) (Eq.
7-10) was integrated in the FE-based software DEFORM in
table formatting as was also implemented for the H-S model.
The numerical results applying the SDM and H-S models as
well as the experimental results are compared in Fig. 14. For
the three different sliding speeds, the simulated sticking friction
forces are very similar to the experimental measurements. It
was also observed that accurate sticking friction forces were
simulated both at the peak (Fig. 15a) and at high strain (Fig.
15b) with the new SDM model, improving the results obtained
with the Z-H and H-S formulations.
Lower peak forces were simulated with the SDM model
because the stress calculated at low strain (ε≈0.2) and based on
the Z-H model (σ2 in Fig. 12) is lower than the calculated with
the H-S equation. This difference is particularly noticeable at
high strain rates as shown in Fig. 16.
Fig. 14. Experimental and simulated sticking friction force carried out at
500 °C and different sliding speeds (0.1 mm/s, 1 mm/s and 10 mm/s)
applying the SDM and H-S models.
a) b)
Fig. 15. Difference between experimental and simulated sticking friction
force (a) at the peak; (b) at steady state.
Vidal Sanabria et al. / Procedia Manufacturing 47 (2020) 237–244 243
Vidal Sanabria et al. / Procedia Manufacturing 00 (2019) 000–000 7
Fig. 16. Modeling of maximal stress applying H-S and Z-H models.
Accurate result obtained with the SDM model at high strains
(steady state) are explained because the softening effect of the
Box-Lukas function was adapted to the experimental friction
results. Thus, it was assumed that the force reduction path
measured with the friction shear test was more representative
than that obtained with the hot compression test (ε>0.23) for
this alloy.
Since the specimen remains under a permanent hydrostatic
condition a severe deformation (high strain) at the friction
subsurface can be achieved without fracture. Unfortunately, an
exact correlation between the relative movement (y) and the
strain, as well as the sliding speed and the strain rate have not
been developed. However, this challenge is getting more
attention in the last years [13]. Simulations results allow the
estimation of the effective strain and strain rate inside the tested
specimens. Fig. 17a depicts the variation of the strain
distribution that take place in the mid height of the specimen
during the friction shear test at different relative movements
(y). According to the results maximal strains of 0.2 and 7.6
were estimated at y=0.2 mm and y=3 mm respectively.
Additionally, an increment of the sheared width was noticed at
higher strains. Moreover, the strain rate distribution simulated
at the mid height of the specimen is presented in Fig. 17b.
Maximal values of strain rate of 0.16 1/s and 19 1/s were
simulated at 0.1 mm/s and 10 mm/s respectively.
a) b)
Fig. 17. Numerical simulation of the a) effective strain and b) effective strain
rate at the mid height of the specimen applying the new constitutive approach.
6. Conclusions
In this study experimental axial friction tests under sticking
conditions were applied on cylindrical specimens to evaluate
the accuracy of constitutive models of the AZ31B magnesium
alloy. During each test a high shear deformation takes place at
the contact subsurface of the specimens. Thus, the shear force
and the corresponding stroke can be directly measured.
Experiments were carried out at 400 °C and 500 °C and at
sliding speeds of 0.1 mm/s, 1 mm/s and 10 mm/s. Based on hot
compression tests performed from the same batch of AZ31B
magnesium alloy, as used for the friction tests, conventional
Zener-Hollomon (Z-H) and Hensel-Spittel (H-S) material
models were constructed. Additionally, a new constitutive
modeling approach was proposed and applied in order to allow
a more accurate stress estimation at low and high strains. The
proposed strain-dependent multifunction model (SDM) applies
the Z-H equation to calculate the stress at constant low strains.
Experimental results showed an inflexion of the stress-strain
curves at ε=0.012, while a maximal stress value was observed
at ε=0.23 approximately. Moreover, the experimental friction
shear tests were applied to estimate the softening behavior
taking place after the maximal stress. Numerical simulations
performed with the new modeling approach gave more
accurate results than the conventional models, both at, and
behind the peak value, of the curve.
Acknowledgements
The authors are grateful for the financial support of the
German Research Foundation (DFG) under the contract no.
MU 2968/18-1.
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