Đường cong giới hạn gia công được sử
dụng trong phân tích gia công kim loại dạng tấm
nhằm xác định các giá trị ứng suất hoặc biến dạng tới
hạn mà tại các giá trị tới hạn này vật liệu sẽ bị hư
hòng khi chịu biến dạng dẻo, ví dụ như quá trình dập
kim loại. Bài báo này nhằm dự đoán giới hạn gia công
của tấm hợp kim nhôm AA6061-T6 dựa trên mô hình
nứt dẻo vi mô. Mô hình cơ sở được lập trình dưới
dạng chương trình vật liệu người dùng kết hợp với
mã phần tử hữu hạn trong phần mềm
ABAQUS/Explicit. Các thí nghiệm kéo đơn trục được
thực hiện để xác định ứng xử cơ tính của vật liệu. Các
tham số đầu vào của mô hình cơ sở được xác định
dựa trên phương pháp bán kinh nghiệm. Để đạt được
các trạng thái biến dạng khác nhau, các mẫu dập sâu
Nakajima được sử dụng để mô phỏng và kỹ thuật hồi
quy parapol ngược theo chuẩn ISO 124004-2:2008
được áp dụng để tính các giá trị biến dạng giới hạn.
Các kết quả đạt được thông qua mô phỏng số sẽ được
so sánh với các mô hình giải tích như M-K, Hill và
Swift.
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TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017
51
Abstract—The forming limit curve (FLC) is used
in sheet metal forming analysis to determine the
critical strain or stress values at which the sheet metal
is failing when it is under the plastic deformation
process, e.g. deep drawing process. In this paper, the
FLC of the AA6061-T6 aluminum alloy sheet is
predicted by using a micro-mechanistic constitutive
model. The proposed constitutive model is
implemented via a vectorized user-defined material
subroutine (VUMAT) and integrated with finite
element code in ABAQUS/Explicit software. The
mechanical behavior of AA6061-T6 sheet is
determined by the tensile tests. The material
parameters of damage model are identified based on
semi-experience method. To archive the various strain
states, the numerical simulation is conducted for the
Nakajima test and then the inverse parabolic fit
technique that based on ISO 124004-2:2008 standrad
is used to extracted the limit strain values. The
numerical results are compared with the those of M-
K, Hill and Swift analytical models.
Index Terms—forming limit curve, void growth,
Nakajima drawing, Dung model.
1 INTRODUCTION
ver many years, the aluminum alloy sheets
was widely applied in automotive and civil
industries because of their outstanding advantages
in high strength and light weight. Therefore, it is
necessary to accurately describe their forming
behaviors at large strains.
Manuscript Received on July 13th, 2016. Manuscript Revised
December 06th, 2016.
This research is funded by Vietnam National University Ho
Chi Minh City (VNU-HCM) under grant number C2017-20-05.
Nguyen Huu Hao author is with the Engineering Mechanics
Department, Ho Chi Minh City University of Technology,
VNU-HCM, Vietnam (e-mail: nguyenhuuhao@tdnu.edu.vn).
Nguyen Ngoc Trung author is with School of Mechanical
Engineering, 585 Purdue Mall, ME3011, Purdue University,
West Lafayette, IN 47907, USA. (e-mail:
trungnguyen@perdue.edu).
Vu Cong Hoa is with the Engineering Mechanics
Department, Ho Chi Minh City University of Technology,
VNU-HCM, Vietnam (e-mail: vuconghoa@hcmut.edu.vn).
The FLC curve is usually predicted by the
Marciniak-Kuczynski (M-K) theory model [1] that
based on an inconsistency in sheet. Beside the FLC
theory prediction, the Nakajima deep drawing
model is also applied widely in experiment and
numerical simulation to determine the forming
limit curve. Accordingly, the Nakajima test is
usually conducted for the several specimens to find
the various strain paths that presents forming
response of material from uniaxial to biaxial
stretched loading state. In this method, the limit
strains are determined by an inverse parabolic fit
[2, 3] or time-dependent technique [2, 4] at or after
the onset of necking.
The ductile fracture mechanism of metallic
materials and their alloys has been proved to be due
to the micro-void nucleation, growth and
coalescence in matrix material [5, 6]. A cylindrical
micro-void growth in rigid-plastic material based
ductile fracture criterion was proposed by
McClintock [5]. Dung [7] has modified the
McClintock model for the ellipsoidal and
cylindrical void growth in hardening matrix
material under the remoted stress field and has
proposed a constitutive model for porous ductile
material. Employing a ductile fracture model to
predict FLC is widely applied because it is
considered as an effective remedy for saving more
time than that of the experiment [3, 8].
In this study, we use a Dung’s porous ductile
material model [7], conjugated with the Hill’48
quadratic yield function to predict the FLC of
AA6061-T6 aluminum alloy. The ductile fracture
model is implemented by a vectorized user-defined
material subroutine (VUMAT) in
ABAQUS/Explicit software package. The seven
specimens with various waist width would be used
to numerical simulation and then the limit strains
were attained by the inverse parabolic fit in
accordance with ISO 12004-2:2008 standard. The
present results are compared with the those of
theory FLC models.
Forming limit curve determination of
AA6061-T6 aluminum alloy sheet
Nguyen Huu Hao, Nguyen Ngoc Trung, and Vu Cong Hoa
O
52 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017
2 CONSTITUTIVE MODEL
The sheet metal is usually showing an
anisotropy so that the von Mises equivalent stress
function in the original Dung’s model [7] is
replaced by the Hill48 quadratic criterion [9].
2 2
22 33 33 11
1/22 2 2 2
11 22 23 31 122 2 2
e F G
H L M N
= - -
-
(1)
Here ij , 1,2,3i j = are components of
Cauchy stress tensor, , , , , ,F G H L M N are
anisotropic coefficients.
0 0
90 0 0 0
0 90 45
90 0
1
1 1 1
1 2 3
2 1 2
R R
F , G , H ,
R R R R
R R R
N , L M
R R
= = =
= = =
(2)
The Lankford’s coefficients 0R , 45R and 90R
are determined by uniaxial tensile tests at 0o , 45o
and 90o in rolling direction.
Noting that for isotropic material, the
Lankford’s coefficients 0 45 90 1R R R= = = , stress
equivalent Hill’48 becomes stress equivalent von
Mises [10].
The hardening rule of matrix material,
pf f = (3)
Here p is equivalent plastic strain of matrix
material.
Gurson [11] has been introduced a yield
function based on mechanism of void nucleation,
growth and coalescences in matrix material. Based
on McClintock’s void growth model [5], Dung [7]
proposed not only a yield function that similar to
Gurson-Tvergaard-Needleman (GTN) model [12]
but also addition of a explicitly hardening
parameter n to consider hardening effects of matrix
material under deformation as follow:
2
2* *
1 22
3 1
2 cosh 1
3
ij ije
ff
n
f q q f
-
= - -
(4)
Where, the parameters 1q , 2q are proposed by
Tvergaard and Needleman [12], n is hardening
exponent of matrix material, e is Hill’48
equivalent stress, *f is function of void volume
fraction (VVF), ij is delta Kronecker.
*
if
if
c
F c
c c c F
u c
f f f
f f f
f f f f f f
f f
= -
- -
(5)
Here cf and Ff are critical and onset of fracture
void volume fraction, respectively, 1 2/uf q q= is
ultimate void volume fraction.
The evolution of void volume fraction is
computed as follow:
growth nucleationdf df df= (6)
Here, the void volume fraction growth of the
presence voids in matrix material:
*1 pgrowth ij ijdf f d = - (7)
Here pijd is plastic strain rate tensor.
The evolution of nucleated void volume fraction
during matrix material under deformation:
p
nucleationdf Ad= (8)
The number of nucleated voids A is a function
of equivalent plastic strain of matrix material.
2
1
exp
22
p
N N
NN
f
A
ss
-
= -
(9)
Where, Nf , Ns , N are the parameters relative
to the void nucleation during matrix material under
deformation.
3 NUMERICAL IMPLEMENTATION
Based on the numerical algorithm of Aravas
[13], the Dung’s porous ductile model is
implemented by a vectorized user-defined
subroutine (VUMAT) and conjugated with finite
element code of ABAQUS/Explicit software. The
implemented procedure for Dung’s model has been
completed by Hao et al. [14].
4 EXPERIMENTAL WORKS
The experimental works adopted in this section
to identify the mechanical behavior of AA6061-T6
aluminum alloy. The specimens to be designed and
tested according to the ASTM-E8 standard [15].
Tensile tests were accomplished with a thin
sheet that its nominal thickness of 2.0 mm. To
identify Lankford’s coefficients (R0, R45, R90),
having least three dog-bone specimens on each
direction of the rolling, transverse and 45 degrees
to rolling direction have used. The initial length of
the gage marks is 50 mm for all tests. The geometry
and dimension of dog-bone specimen are given in
Figure 1.
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017
53
Figure 1 Dog-bone specimen
The tensile tests help to obtain the mechanical
properties of AA6061-T6 aluminum alloy as shown
in TABLE 1. The difference of engineering stress-
strain behavior in three directions of 0o, 45o, 90o to
rolling direction is presented as Figure 2. The true
strain-stress curve that used to fit Swift hardening
rule is given in Figure 3.
Assuming that the isotropic hardening rule
obeys Swift model [16], fitting true strain-stress
curve, the hardening parameters ( K , 0 , n ) is
obtained as TABLE . The Lankford’s coefficients
are calculated by the eq. (10).
02
3 0 0
ln /
ln /
f
f f
w w
R
l w l w
= = (10)
Where 2 and 3 are the transverse and normal
strains, respectively. l0, lf, w0, wf (0 and f indexes
imply initial and final values) are the gage length
and width of dog-bone specimen, 0 ,45 ,90o o o = .
Figure 2. Experimental load behavior in various directions
Figure 3. True stress-strain curve
5 PARAMETER CALIBRATION
To apply the porous plastic material model to
prediction of ductile fracture, 8 parameters
1 2 0, , , , , , ,F C N N nq q f f f s f must be identified.
In general, any identification procedure that used to
identify all these parameters would be still
requirement of the computational time cost. In
addition, for each material type, may be have more
one set of material parameter (non-uniqueness of
the solution) [17-19]. A literature review of
material parameter identification for porous ductile
model is necessary in this work. On that basis, the
material parameters can be selected and calibrated
for Dung’s model.
Two parameters 1 1.5q = and
2
2 1 2.25q q= =
that proposed by Tvergaard [20] to correct result of
numerical calculation and original Gurson model.
The initial VVF parameter f0 is determined by
observation of micrograph of virgin material [21,
22] or calibration [23].For AA6061 aluminum
alloy, value of initial VVF is provided by several
researchers such as Agarwal et al. [22] (f0 =
0.0014), Xu et al. [21] (f0 = 0.0025), Shen et al.
[23] (f0 = 0.0005). Therefore, a suitable range for
the value of f0 VVF of AA6061-T6 can be lie in
(0.0005-0.0025).
The parameter sN can be explained through a
little metrology significance of nucleated strain
measurements. The distribution of nucleated strain
values εN is assumed to obey a normal distribution
with a standard deviation sN. Qualitatively, a low
standard deviation shows that the values of
nucleated strain εN tend to be close to the mean
(also called the expected value) of the data set,
while a high standard deviation indicates that the
values of nucleated strain εN are spread out over a
wider range of values. In this work, a good quality
of nucleated strain measurements is assumed to
obtain so that value of standard deviation sN of 0.05
is selected.
Two parameters εN and fN are usually used as
TABLE 1
MECHANICAL PROPERTIES OF AA6061-T6 ALUMINUM
ALLOY
Young’s
modulus
( E )
Yield
stress
( 0 )
Poisson’s
ratio
( )
74.6 GPa 244 MPa 0.314
TABLE 2
MATERIAL PARAMETERS OF AA6061-T6 ALUMINUM
ALLOY
K (MPa) 0 n
Lankford’s
coefficients
R0 R45 R90
489.74 0.02 0.179 0.55 0.52 0.53
54 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017
the fitting parameters. In practice, it is difficulty to
recognize exactly the moment at void nucleation so
that the value of nucleated strain εN is relatively
selected based on onset of material damage [19].
Accordingly, a comparison of force vs.
displacement curve between experiment and finite
element method (FEM) result of pure Hill48
plasticity theory (no damage) is performed to
estimate the value of nucleated strain εN.
The mesh size of 0.5 mm x 0.5 mm at critical
zone is used to mesh for dog-bone specimen. The
displacement controlled load is applied to top edge
of specimen. The element type of 3D, reduced
integration, 8-nodes (C3D8R) used for dog-bone
specimen.
Figure 4. Graphics of determination of nucleated strain εN
Values of nucleated, critical and fractured VVF
(fN, fC, fF) are calibrated by matching load-
displacement curve of dog-bone specimen between
experiment and FEM.
The FEM simulations are performed for nine
values of fN = 0.01, 0.015, 0.02, 0.025, 0.03, 0.035,
0.04, 0.045, 0.05. A best matched result of load-
displacement curve between FEM and experiment
is selected to fit the values of fC and fF in next step.
The evolution stage of VVF from fC to fF
increased more rapid than that of previous period
due to the coalescence of micro-voids lead to quick
losing of loading carrying of matrix material. There
are 25 possible combinations of fC and fF from
TABLE . However, because of the constrain
C Ff f so that have only 24 runs in ABAQUS is
possible to obtain a best combination of (fC, fF) pair
that matches the experimental curve.
Finally, the best fit parameters for predicting of
ductile fracture are given in TABLE .
The displacement – load curve corresponding to
the best fitted material parameters are presented in
Figure 5.
Figure 5. The displacement – load curve after calibration
6 FORMING LIMIT CURVE
6.1 Nakajima test
The Nakajima’s type deep drawing is conducted
for the seven specimens with waist width w = 30,
55, 70, 90, 120, 145 and the circular shape as
Figure 6a. The setup of deep drawing is presented
in Figure 6b. The blank used mesh type of 3D, 8-
nodes, reduced integration (C3D8R) whereas the
punch, holder and die are assumed absolute hard
with 3D analytical rigid type. The initial mesh size
at analysis zone is 1.0 mm x 1.0 mm. Three
element layers through the thickness of blank are
used. The blank holding force 450holdF kN is used
to avoid any sliding phenomenon and early damage
at the blank holding region. The friction coefficient
between the blank and punch surfaces is 0.03
whereas the friction coefficient value of 0.1 on all
remain contact surfaces is adopted.
Figure 6 (a) Blank and (b) deep drawing setup (unit: mm)
TABLE 3
THE VALUES OF CRITICAL AND FRACTURE VVF FOR CALIBRATION
fC 0.015 0.035 0.06 0.08 0.15
fF 0.08 0.15 0.17 0.2 0.25
TABLE 4
BEST FIT VALUES OF MATERIAL PARAMETERS FOR DUNG MODEL
q1 q2 fF fC f0 εN sN fN
1.5 2.25 0.15 0.035 0.0018 0.09 0.05 0.03
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017
55
After the blank is clamped and the die is fixed,
the blank is stretched by moving the punch in
vertical direction until its fracture occurs.
(a)
(b)
Figure 7 The extracted path along cross section of W30 and
W120 specimens
The limit strains are then determined based on
cross-section method that its basic concept is the
analysis of the measured strain data along
predefined cross sections at onset necking time.
The detail procedure of this method is given in ISO
12004-2:2008 – part 2 standard. Accordingly, the
principal strain average value of three extracted
paths along cross section of each specimen (Figure
7) are taken to fit an inverse parabola.
The best fit inverse parabola is limited by the fit
boundaries that presented through the inner (purple
dot square line) and outer (green solid line) fit
window limits as Figure 8.
The size of inner fit window (L0) is determined
by the highest peaks of the second derivative of the
second order parabola that regressed by three
consecutive points of principal strain data within a
range of 6 mm.
(a)
(b)
Figure 8 The curve fit of the principal strain data and the limit
strain determination . (a) W30 and (b) W120 specimens
The size of the outer limit window should have
at least 5 points and calculated as follows:
L R FW =W =W /2 (11)
Where WL left fit window width, WR right fit
window width
F 2 1W 10 1 / = (12)
With
2 2, 2,1 / 2 BL BR = (13)
1 1, 1,1 / 2 BL BR = (14)
The subscripts “BL” and “BR” are used for ε1
and ε2 of the left and right inner boundaries,
respectively.
After determination of fit boundaries, the
inverse best fit parabola is fitted by all data points
within fit window (WL and WR). The resulting
value in the crack position is the wanted limits for
principal strains ε1 and ε2.
56 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017
6.2 M-K model
The Marciniak-Kuczynski (M-K) model is
probably the most well-known and widely used to
predict analytical FLC curve [24]. Marciniak and
Kuczynski introduced imperfections into sheets to
describe necking condition. This theory based on
the material inhomogeneity assumption, i.e., there
is groove which is perpendicular to the axial of
maximum principal stress on the sheet surface (see
Figure 9). This initial inhomogeneity grows
continuously and eventually becoming a localized
necking. From the Fig.9, the zone (b) is groove
zone, it is assumed the zone (a) is homogeneous
zone and obey uniform proportional loading states.
The x, y, z axes correspond to rolling, transverse
and normal directions of the sheet, whereas 1 and 2
represent the principal stress and strain directions in
the homogeneous region. Meanwhile, the set of
axes aligned to the groove is represented by n, t, z
axes, where t is the longitudinal one. In the sheet
metal forming process, the material is firstly under
plastic deformation with constant incremental
stretching until maximum force happen. The M-K
model assumes the flow localization occurs in the
groove when a critical strain is reached in the
homogeneous region. Then, the values of strain
increment in two regions are compared with
specific criterion (e.g., dε1b 10dε1a) and finally the
material major and minor strain limits are obtained
on the forming limit diagrams.
Because of M-K model based on an assumption
of plane stress state so that Hill48 yield criterion
can be written as follow
1/2
0 902 2 0
1 2 1 2
90 0 0
1 2
1 1
i
R R R
R R R
= -
(15)
1/2
0 90 2 0
1 90 0 0
1 2
1
1 1
i
R R R
R R R
= = -
(16)
Figure 9 Marciniak-Kuczynski (M-K) model
Because of M-K model based on an assumption
of plane stress state so that Hill48 yield criterion
can be written as follow
1/2
0 902 2 0
1 2 1 2
90 0 0
1 2
1 1
i
R R R
R R R
= -
(17)
1/2
0 90 2 0
1 90 0 0
1 2
1
1 1
i
R R R
R R R
= = -
(18)
The behavior of material can be represented in
the form of power law
n m
i i iK = (19)
Where n hardening exponent, m strain rate
exponent.
The ratio of the principal stress and strain are
defined as follows:
2 2 2
1 1 1
,
d
d
= = = (20)
The associated flow rule is expressed by
1
1
id d
=
and 2
2
id d
=
(21)
The yield criterion can be rewritten as follows
1 2
90 1 0 90 1 2 0 2 0 90 1 2
3
90 1 0 2 90 01
i
i
d d
R R R R R R
d d
R R R R
=
- - -
- -
= =
(22)
Thus, the strain rate can be written as follows:
0 90 0 902
1 90 0 0 90
1
1
R R R Rd
d R R R R
-
= =
-
(23)
The ratio of strain rate can be calculated
3 /d dt t = (24)
90 0 90 0
3 1 2
90 0 90 0 0 901 1
R R R R
d d d
R R R R R R
= - = -
- - -
(25)
Introducing a new parameter β and using eq.
(22)
90 0
1 90 0
1
1 1
i
R Rd
d R R
= =
-
(26)
The ratio of initial thickness between (b) and (a)
zones
0
0
0
b
mk
a
t
f
t
= (27)
Because of thickness strain 3 0ln /t t = so that
the current thickness of sheet can be calculated as
0 3expt t = . The present thickness ration is
determined as follow
0 3 3
0
expb b b a
a a
t t
t t
= - (28)
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017
57
or
3 30 expmk b amkf f = - (29)
The equilibrium condition requires that the applied
load remains constant between (a) and (b) zone,
therefore
1 1a bF F= (30)
If the sheet width is a unity then
1 1a a b bt t = (31)
or
1 1a mk bf = (32)
From eq. (18)
a ia mk b ibf = (33)
From eq. (19)
n nm m
a ia ia ia mk b ib ib ibd f d = (34)
From eq. (29)
3 30 exp
n nm m
a ia ia ia b a b ib ib ibmk
d f d = -
(35)
From eq. (23), (30) and (31), the strain relation
between the (a) and (b) zones is given as follow
3 30 exp
m
n a
a ia ia
a
m
n b
b a b ib ibmk
b
d
f d
= -
(36)
In general, the equilibrium equation (36) can be
solved numerically by using the supplementary
equations (18), (23) and (26). Given a stress ratio in
(a) zone (αa) and a finite increment of strain is also
imposed in (a) zone ( εa = 0.001). The values of
hardening exponent n = 0.179 and of strain rate
exponent m = 0 are chosen. Ratio of initial
thickness between (b) and (a) zones 0 0.996mkf = .
Then, the numerical computation is performed by
using a computational program, e.g. MatLab
language, to determine the limit strain of each
strain path in the FLC. The limit strains in (a) zone
(ε1a, ε2a) are determined once condition (dε1b/dε1a >
10) is satisfied.
6.3 Hill model
Hill [25] proposed a model to describe the curve
on the left side of the FLC (ε2 < 0) based on the
local necking condition. Principal strains are
calculated as follows,
1
1
n
=
and 2
1
n
=
(37)
Where 2 1/d d = denotes strain ratio.
According to eq. (37), the FLC calculated based on
Hill’s model is only dependent on the hardening
coefficient of n = 0.179 and strain ratio that lie
in range from -0.5 to zero.
6.4 Swift model
Swift [16] introduced a criterion for predicting
FLC based on the onset of diffuse necking
criterion.
2
1 2
2 1
1 2 2
n
=
-
(38)
2
1 2
2 1
1 2 2
n
=
-
(39)
It is important to remark that, for plane strain (β
= 0) and equibiaxial tension (β = 1). Similar to Hill
model, given hardening coefficient of n = 0.179
and strain ratio that lie in range from zero to 1.0,
the right side of FLC curve is plotted in Fig. 10.
Figure 10. FLC curve of AA6061-T6 aluminum alloy sheet
Finally, the FLC curve of AA6061-T6 sheet is
obtained by the Nakajima deep drawing simulation
using a porous ductile model and the analytical
model as shown in Figure 0. The results show that
the FLC of three models are consistent with each
other at plane strain state and the FLC curve shape
of Dung-Hill48 model is agree with that of Hill and
Swift models. While that the M-K model displays a
big shift large compare to two remaining models.
7 CONCLUSION
In this study, we present a FLC determination
of AA6061-T6 aluminum alloy sheet. Material
properties and anisotropy coefficients were
obtained from tensile test. Applying Dung’s porous
ductile model to determined FLC through
numerical simulation of Nakajima deep drawing.
The inverse parabolic fit technique that based on
58 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017
ISO 12004-2:2008-part standard is used to achieve
the limit strain values in forming process. Using the
famous theory models of the FLC calculation by
M-K, Swift and Hill, the analytical FLC curve is
proposed. The analytical FLC curve shape of Hill
and Swift models agrees with that of the numerical
data whereas the predicted FLC curve in biaxial
loading states using M-K model is fairly large
deviation from that of remaining models.
REFERENCES
[1] Z. Marziniak and K. Kuczynski, "Limit strain in the
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TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017
59
Nguyen Huu Hao received
the B.E. (2004), Mechanical
Engineering, Military
Technical Academy (MTA),
Vietnam, M.E. (2012) degree
in engineering mechanics from
Ho Chi Minh City University
of Technology - VNU-HCM.
He is a Ph.D. Candidate at Department of
Engineering Mechanics, Ho Chi Minh City
University of Technology - VNU-HCM. His
current interests include the modelling and
predicting of ductile fracture of metallic materials.
Nguyen Ngoc Trung B.E.
(2001, Aeronautical
Engineering, Ho Chi Minh
City University of Technology
- VNU-HCM, Vietnam), M.Sc.
(2003, Mechanics of
Construction, Liège
University, Belgium), Ph.D.
(2008, Aerospace Engineering, Konkuk University,
South Korea)
He is current working as a postdoctoral research
associate at Purdue University, IN, USA. His
research interests include constitutive modeling of
materials, materials design, and mechanics of
advanced manufacturing processes.
Vu Cong Hoa received the
B.E. (1995), M.E. (2000,), and
Dr.Eng. (2006) in Mechanical
Design from Chonbuk
National University, South
Korea.
He is an Associate Professor,
at Department of Engineering
Mechanics, Ho Chi Minh City University of
Technology - VNU-HCM. His current interests
include estimating of microscopic ductile fracture
of metallic materials and micro-mechanical
behavior of composite materials.
60 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017
Tóm tắt - Đường cong giới hạn gia công được sử
dụng trong phân tích gia công kim loại dạng tấm
nhằm xác định các giá trị ứng suất hoặc biến dạng tới
hạn mà tại các giá trị tới hạn này vật liệu sẽ bị hư
hòng khi chịu biến dạng dẻo, ví dụ như quá trình dập
kim loại. Bài báo này nhằm dự đoán giới hạn gia công
của tấm hợp kim nhôm AA6061-T6 dựa trên mô hình
nứt dẻo vi mô. Mô hình cơ sở được lập trình dưới
dạng chương trình vật liệu người dùng kết hợp với
mã phần tử hữu hạn trong phần mềm
ABAQUS/Explicit. Các thí nghiệm kéo đơn trục được
thực hiện để xác định ứng xử cơ tính của vật liệu. Các
tham số đầu vào của mô hình cơ sở được xác định
dựa trên phương pháp bán kinh nghiệm. Để đạt được
các trạng thái biến dạng khác nhau, các mẫu dập sâu
Nakajima được sử dụng để mô phỏng và kỹ thuật hồi
quy parapol ngược theo chuẩn ISO 124004-2:2008
được áp dụng để tính các giá trị biến dạng giới hạn.
Các kết quả đạt được thông qua mô phỏng số sẽ được
so sánh với các mô hình giải tích như M-K, Hill và
Swift.
Từ khóa - đường cong giới hạn gia công, tăng
trưởng lỗ hổng vi mô, dập Nakajima, mô hình N. L.
Dung
Xác định đường cong giới hạn gia công
tấm hợp kim nhôm AA6061-T6
Nguyễn Hữu Hào, Nguyễn Ngọc Trung, Vũ Công Hoà
Các file đính kèm theo tài liệu này:
- forming_limit_curve_determination_of_aa6061_t6_aluminum_allo.pdf