Hole Expansion Benchmark: DP980 Steel Sheet Analysis

BENCHMARK‐1 Hole Expansion of A High Strength Steel Sheet
Chair: Toshihiko Kuwabara (Tokyo University of Agriculture and Technology, Japan)
Co‐Chair: Tomoyuki Hakoyama (RIKEN, Japan)
Members: Taiki Maeda (Tokyo University of Agriculture and Technology, Japan)
Chiharu Sekiguchi (Tokyo University of Agriculture and Technology, Japan)
October 2, 2017
Revised on December 12, 2017
1. Overview
The objective of this benchmark (BM1) is to predict the deformation and fracture of a dual
phase steel sheet (1.2 mm thickness) with a tensile strength of 980 MPa (DP980) subjected
to hole expansion. A circular blank with a central hole (30 mm diameter) fabricated using a
wire‐electrical discharging machine is used to perform hole expansion forming to a set depth
with a circular punch (100 mm diameter), see Fig. 1. Formed parts will be sectioned to
measure thickness strain along the hole edge as well as along the three radial directions: 0
(RD), 45 (DD), and 90º (TD) from the rolling direction of the test sample. Additional parts will
be formed to “failure”, and the location and timing of the failure will be identified. These
measurements will be used to evaluate numerical predictions of the results submitted by
benchmark participants prior to the Numisheet 2018 Conference.
Fig. 1 A specimen after hole expansion forming. Material: cold rolled ultralow carbon steel
sheet. The punch and die geometry used is the same as that used in this benchmark,
see Fig. 2.
2. EXPERIMENTAL PROCEDURE
2.1 Blank material and material data
DP980 (1.2 mm thickness) is considered for this benchmark. Large blanks are trimmed to
circular blanks with a diameter of 215 mm. A detailed material characterization for the test
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sample is shown in Section 4 of this document, which includes the uniaxial and biaxial
tension test data. The data are also available in a separate Excel file (BM1_Material Data) to
allow analysts to use their own data fitting algorithms and/or employ alternate material
model options.
2.2 Hole Expansion Forming
Figure 2 shows the experimental apparatus used for the hole expansion forming. The punch
was 100 mm in diameter and the die and punch profile radii were 15 mm. The initial hole
diameter was d 0  30 mm, fabricated at the center of a circular blank using a wire‐electrical
discharging machine. The periphery of the blank was clamped using a triangular draw‐bead
with a diameter of 195 mm. The interface between the blank and punch head was lubricated
with Vaseline and 0.3 mm thick Teflon sheet, while no lubricant was applied to the interfaces
between the blank and the die/blank‐holder. The blank was clamped using a hydraulic
(a)
(b)
(c)
(d)
Fig. 2 Experimental apparatus for hole expansion experiment. (a) Schematic illustration of
the die geometry and specimen used for the hole expansion experiment. All
dimensions are in millimeters. (b) Geometry of the male and female beads. (c) Lower
die and punch. (d) Setup of a specimen; the center of the specimen is coaxial with that
of the punch as it is guided by a cylindrical block that stands at the punch center.
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cylinder, and the blank‐holding force was approximately 800 kN (revised on December 12,
2017). The punch speed was approximately 0.1 mm/s, and the forming was terminated when
the punch stroke reached 15 mm.
After the hole expansion forming, the sheet thickness along the radial lines at 0 (RD), 45 (DD),
and 90° (TD) from the RD, as well as along the hole edge, was measured using a micrometer
with a minimum readout of 0.001 mm.
3. BENCHMARK REPORT
All results must be reported using the benchmark report template, BM1_Report.xlsx, which
can be downloaded from the conference website. The file must be submitted by March 1,
2018, on the website under the “Benchmark Report Submission” tab on the right‐sided
menu.
3.1 General description
Please fill out the information:
A. Benchmark participant
Name, affiliation, address, e‐mail, phone‐number and fax number
B. Simulation Software
Name of the FEM code, general aspects of the code, basic formulations (implicit or
explicit), element/mesh technology, type of elements, element adaptivity information,
number of elements (or initial and final number of elements if adaptivity is used), contact
property model and friction formulation
Simulation Hardware
CPU type, CPU clock speed, number of cores per CPU, number of CPUs, parallel or shared
processing, main memory, operating system and total CPU time
C. Material model
Yield function/Plastic potential, Hardening rule, Stress‐Strain Relation, and Failure Model
D. Remarks
3.2 Calculated results
Please fill out the information:
(1) Punch force (kN) vs. punch stroke (mm) up to and a bit beyond the point of maximum
load. The zero mm punch stroke is defined at the position when the specimen is just in
contact with the punch with zero interface force.
(2) Logarithmic thickness plastic strain  zp at a positon of S  2 mm distant from the hole
edge in the original blank vs. circumferential coordinate  (degree in the original
blank), see Fig. 3, at a punch stroke of 15 mm.
(3) Logarithmic thickness plastic strain  zp vs. S (mm), at a punch stroke of 15 mm, for
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the three radial directions: RD, DD and TD of the original blank, see Fig. 4.
(4) Punch stroke at the predicted onset of fracture (within 0.1 mm precision)
(5) Location of the predicted onset of fracture, (  (degree), S (mm)), in the original blank.
Fig. 3 Definition of the circumferential position,  , for the report of logarithmic thickness
plastic strain profile along the hole edge.  is defined to be zero at the RD.
Fig. 4 Definition of the positon S (mm) from the hole edge in the original blank for the
report of the logarithmic thickness strain profile along the three radial directions: RD,
DD and TD
4. UNIAXIAL AND BIAXIAL TENSION TESTS
4.1 UNIAXIAL TENSION DATA
Tensile tests are performed in every fifteen degrees, i.e., 0, 15, 30, 45, 60, 75, and 90°,
inclined from the rolling direction (RD) of the test sample. The strain rate is approximately
5 104 s1 . Stress–strain curves and r‐values are measured.
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Figure 5(a) and (b) shows the plastic‐work‐based proof stresses,  0.2(W) and  1.0(W) , and the
r‐values measured from the uniaxial tensile tests performed every 15° from the RD.  0.2(W)
and  1.0 (W) are defined as the uniaxial tensile stress at an instant when the same plastic
work per unit volume is consumed as that at a tensile strain of  0p  0.002 and 0.01,
900
1.2
 1.0(W)
850
1.0
800
0.8
r - value
Plastic-work-based proof stress
 0.2(W),  1.0(W) /MPa
respectively, in the RD, see Appendix. The r‐values were measured at a nominal tensile strain
of 0.1. These data are provided in a separate Excel sheet (BM1_Material Data).
750
700
600
0
30
60
0.4
Average
 0.2(W)
650
0.6
0.2
90
Angle from rolling direction  /°
0.0
0
30
60
90
Angle from rolling direction  /°
(a)
(b)
Fig. 5 Mechanical properties measured using uniaxial tension tests: (a)  0.2(W) and  1.0(W)
and (b) r‐values. The marks show an average of two specimens for each tensile direction.
Figure 6 shows the nominal stress  N vs. nominal strain  N curves for   0, 45, and 90°.
The number of specimens are two for each tensile direction. The data for the  N   N
curves are provided in a separate Excel sheet in file BM1_Material_Data.xlsx.
Figure 7 shows the true stress  vs. logarithmic plastic strain  p for   0, 45, and 90°,
compared with those approximated using Swift’s power law. The parameters of Swift’s power
law are determined for a strain range of 0.002   p   up ,  up is the logarithmic plastic strain
giving the maximum tensile force. The data for the    p curves are provided in a separate
Excel sheet in file BM1_Material_Data.xlsx.
frac
Logarithmic plastic strain components in the thickness (  Tfrac ), width (  W
), and longitudinal
frac
(  L ) directions in the localized necking area of fractured specimens are also measured for
  0, 45, and 90°, see Fig. 8.  Tfrac is determined by directly measuring the thickness at the
frac
tip of the fracture surface using a tool microscope in an accuracy of 0.001 mm.  W
is
frac
determined using a DIC system just outside of the localized neck.  L is calculated as
 Tfrac   Wfrac . The data for  Tfrac ,  Wfrac , and  Lfrac are provided in a separate Excel sheet in file
BM1_Material_Data.xlsx.
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1200
0°
Nominal stress  N /MPa
Nominal stress  N /MPa
1200
1000
800
600
400
Experimental #1
Experimental #2
Maximum stress
200
0
0.00
0.04
0.08
0.12
0.16
45°
1000
800
600
400
Experimental #1
Experimental #2
Maximum stress
200
0
0.00
0.20
0.04
Nominal strain  N
0.08
0.12
0.16
0.20
Nominal strain  N
(a) 0°
(b) 45°
Nominal stress  N /MPa
1200
90°
1000
800
600
400
Experimental #1
Experimental #2
Maximum stress
200
0
0.00
0.04
0.08
0.12
0.16
0.20
Nominal strain  N
(c) 90°
Fig. 6 Nominal stress vs. nominal strain curves for   0, 45, and 90°
1400
0°
1200
True stress  /MPa
True stress  /MPa
1400
1000
800
600
Experimantal #1
Experimantal #2
Swift's power low
p 0.12
 =1525(-0.0009+ ) /MPa
p
(for  =0.002 ~0.083)
400
200
0
0.00
0.02
0.04
0.06
0.08
Logarithmic plastic strain 
True stress  /MPa
1000
800
600
Experimental #1
Experimental #2
Swift's power low
p 0.12
 =1481(-0.0012+ )
/MPa
p
(for  =0.002 ~0.081)
400
200
0.10
p
0
0.00
0.02
0.04
0.06
0.08
Logarithmic plastic strain 
(a) 0°
1400
45°
1200
0.10
p
(b) 45°
90°
1200
1000
800
600
Experimental #1
Experimental #2
Swift's power low
p 0.11
 =1522(-0.0014+ ) /MPa
p
(for  =0.002 ~0.084)
400
200
0
0.00
0.02
0.04
0.06
0.08
Logarithmic plastic strain 
0.10
p
(c) 90°
Fig. 7 True stress vs. logarithmic plastic strain curves, compared with those approximated using
Swift’s power low
6
frac
. The pictures were taken
Fig. 8 Schematic diagram for the measurement of  Tfrac and  W
for a tensile specimen in the RD (   0°).
4.2 BIAXIAL TENSION DATA
The specimen geometry used in the biaxial tension tests and a method of determining a
contour of plastic work are given in Appendix.
Figure 9 shows the stress points that form the contours of plastic work associated with the
reference plastic strain of  0p  (a) 0.002 and (b) 0.01; the stress values are normalized by
the 0 associated with each work contour. Each data point represents an average of the
data measured from two specimens; the difference between these two measured data was
less than 1% of the flow stress. The maximum value of  0p , for which the work contour has a
full set of stress points measured using cruciform specimens, was  0p  0.012. The values of
the stress points forming the work contours are provided in a separate Excel sheet
(BM1_Material Data).
Figure 9 also includes the theoretical yield loci based on the von Mises (Von Mises, 1913), Hill’s
quadratic (Hill, 1948), and the Yld2000‐2d yield function (Barlat et al., 2003; Yoon et al., 2004)
with an exponent of a  6, the value typically suggested for BCC metals. The material
1.50
DP980
DP980
1.25
1.25
1.00
1.00
0.75
0.50
0.25
y / 0
y / 0
1.50
0p = 0.002
Experimental
von Mises
r-Hill '48
Yld2000-2d
(a = 6)
0.75
0.50
0.25
0.00
0.00 0.25 0.50 0.75 1.00 1.25 1.50
0p = 0.01
Experimental
von Mises
r-Hill '48
Yld2000-2d
(a = 6)
0.00
0.00 0.25 0.50 0.75 1.00 1.25 1.50
 x / 0
x / 0
(a)
(b)
Fig. 9 Measured stress points forming the contour of plastic work at (a)  0p  0.002 and (b)
 0p  0.01, compared with those theoretical yield loci calculated using the von Mises,
Hill ‘48, and the Yld2000‐2d yield functions.
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parameters i ( i  1~8) of the Yld2000‐2d yield function were determined using r0 , r45 , r90 ,
and rb , and  0 /  0 ,  45 /  0 ,  90 /  0 , and  b /  0 , where r and   are the r‐value
and uniaxial tensile flow stress measured at an angle  from the RD, respectively, and rb
and  b are the ratio of the plastic strain rates,  yp / xp , and the flow stress at a balanced
biaxial tension, respectively, associated with a particular value of  0p . The values of i ( i 
1~8) are provided in a separate Excel sheet (BM1_Material Data).
The material parameters of the Hill '48 yield function were determined using the r0 , r45 , and
r90 .
135
DP980
y
90
45


x
0p = 0.002
Experimental
von Mises
r-Hill '48
Yld2000-2d
(a = 6)
0
-45
0
15
30
45
60
Loading direction  / °
75
90
Direction of plastic strain rate  / °
Direction of plastic strain rate  / °
Figure 10 compares the directions of the plastic strain rates, measured at  0p  0.002 and
 0p  0.01, with those calculated using selected yield functions. The Yld2000‐2d yield
function with a  6 is consistent with the measured data, while those predicted by the von
Mises and Hill ’48 yield functions deviate slightly from the measured data. The value of 
for each stress path is provided in a separate Excel sheet (BM1_Material Data).
(a)
135
DP980
y
90
45


x
0p = 0.01
Experimental
von Mises
r-Hill '48
Yld2000-2d
(a = 6)
0
-45
0
15
30
45
60
Loading direction  / °
75
90
(b)
Fig. 10 Directions of the plastic strain rates measured at (a)  0p  0.002 and (b)  0p  0.01,
compared with those calculated using the von Mises, Hill ’48, and the Yld2000‐2d yield
functions.
APPENDIX: METHOD OF BIAXIAL TENSION TEST
A1 Specimen geometry
Figure A1 shows the geometry of the cruciform specimen used in the biaxial tension tests.
The stress measurement error is estimated to be less than 2% when using the cruciform
specimen with the geometry and strain measurement positions as shown in Fig. A1
(Hanabusa et al., 2010 and 2013).
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Fig. A1 Geometry of the cruciform specimen and the strain measurement positions used in
the biaxial tensile tests
A2 Measurement of contours of plastic work
Seven linear stress paths,  x :  y  4:1, 2:1, 4:3, 1:1, 3:4, 1:2, and 1:4, are applied to the
specimens using a servo‐controlled biaxial tensile testing machine developed by Kuwabara et
al. (1998), where  x and  y are the true stress components in the RD and TD of the test
sample, respectively. The von Mises’ strain rate is approximately 5 104 s1 . Details of the
testing procedures are given in ISO 16842 (2014).
Contours of plastic work in the stress space (Hill and Hutchinson, 1992; Hill et al., 1994) are
measured to identify a proper material model for the test sample, see Fig. A2. First, the true
stress vs. logarithmic plastic strain curve ( 0   0p curve) is measured from the uniaxial
tension test for the RD, and is selected as the reference data for work hardening; the plastic
work per unit volume W 0 associated with selected values of  0p are determined. Next,

y

x
90
Wx
Wy
p
Wy
y
x


Wy
p
p
y
WX
x
y
x
W0 =Wx+Wy
0

Wx
Wy
0
y
p

W0

(a)
p
0
p

x
(b)
Fig. A2 Schematic diagrams for the data analysis of the biaxial tension test. (a) A method for
determining the stress points forming a contour of plastic work. (b) The definitions of
  arctan( y /  x ) and   arctan( yp / xp ) .
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the stress points that give the same plastic work as W 0 are determined from the biaxial
tension test data and the uniaxial tension test data in the TD. Consequently, these stress
points form a contour of plastic work associated with  0p in the first quadrant of the
 x   y stress space.
Moreover, the direction of the plastic strain rate,   arctan( yp / xp ) , is measured for each
linear stress path with a loading angle of   arctan( y /  x ) , to check whether the
normality flow rule is established for the work contour.
References
Barlat, F., Brem, J.C., Yoon, J.W., Chung, K, Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.H.,
Chu, E., 2003. Plane stress yield function for aluminum alloy sheets—part 1: theory. Int. J.
Plasticity 19, 1297‐1319.
Hanabusa, Y., Takizawa, H., Kuwabara, T., 2010. Evaluation of accuracy of stress
measurements determined in biaxial stress tests with cruciform specimen using numerical
method. Steel Research Int. 81 (9), 1376‐1379.
Hanabusa, Y., Takizawa, H., Kuwabara, T., 2013. Numerical verification of a biaxial tensile test
method using a cruciform specimen. J. Mater. Processing Technol. 213, 961‐970.
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the Royal Society of London A193(1033), 281‐297.
Hill, R., Hecker, S.S., Stout, M.G., 1994. An investigation of plastic flow and differential work
hardening in orthotropic brass tubes under fluid pressure and axial load. Int. J. Solids
Struct. 31, 2999‐3021.
Hill, R., Hutchinson, J.W., 1992. Differential hardening in sheet metal under biaxial loading: a
theoretical framework. J. Appl. Mech. 59, S1‐S9.
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a cruciform test piece.
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work‐hardening in cold‐rolled steel sheet under biaxial tension. J. Mater. Process Technol.
80‐81, 517‐523.
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Nachrichten, math.‐phys. Klasse, 582‐592.
Yoon, J.W., Barlat, F., Dick, R.E., Chung, K., Kang, T.J., 2004. Plane stress yield function for
aluminum alloy sheets—part II: FE formulation and its implementation. Int. J. Plasticity 20,
495‐522.
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